Food Physics: Physical Properties - Measurement and Applications [2 ed.] 3031273974, 9783031273971

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Ludger O. Figura Arthur A. Teixeira

Food Physics

Physical Properties - Measurement and Applications Second Edition

Food Physics

Ludger O. Figura • Arthur A. Teixeira

Food Physics Physical Properties - Measurement and Applications Second Edition

Ludger O. Figura Food Engineering Hochschule Osnabrück, University of Applied Sciences Osnabrück, Germany

Arthur A. Teixeira University of Florida Gainesville, FL, USA

ISBN 978-3-031-27397-1 ISBN 978-3-031-27398-8 https://doi.org/10.1007/978-3-031-27398-8

(eBook)

# The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2007, 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, 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. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The demands on food production are increasing constantly. Food shall be fresh and healthy, contain only few additives, be stable and safe. Foods shall be produced in a way that is sustainable and ensures the needs of a growing world population. In order to meet these challenges, cooperation between science and craftsmanship is necessary to develop efficient, energy-saving, and low-waste processes that fit into individual geographical situations. In addition, there is a need for efficient forms of learning that enable us to extract, understand, and scientifically evaluate the required content from the wealth of digitally available information so that we can contribute to that task throughout our working life. The new edition of Food Physics wants to make a contribution to this. The core statements of more than 1000 current publications have been included, each chapter begins with simple physics and, with the help of examples, introduces practical measurement technology and food characterization. At the end of each chapter, there is a list of technical applications that can be used for in-depth study or for finding ideas for your own investigations in projects, bachelor and master theses. The practitioner will probably start a chapter with these applications and then scroll forward if necessary. The chapter on-line sensors and the sections on electromagnetic and optical properties have been expanded, and a separate chapter has been devoted to the texture of food. For efficient learning, didactic elements were integrated into the book, in the E-book you can continue reading directly in the cited literature with the help of links. However, you still have to read and to study by yourself. With this in mind, we wish all readers the beneficial experience of developing new understanding and using this together with own ideas for a sustainable global food production. Quakenbrück, Germany Spring 2023

Ludger O. Figura Arthur A. Teixeira

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Contents

1

Water Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Water Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Solid Boundary Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Adsorption Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Porous Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Sorption Isotherms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Moisture Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Hygroscopicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 BET Equation for Foods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 GAB Equation for Food . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Sorption Enthalpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 Other Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12 Shelf Life of Food Related to Water Activity . . . . . . . . . . . . . . 1.13 Laboratory Determination of Sorption Isotherms . . . . . . . . . . . . 1.14 Standard for Sorption Isotherms . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 7 11 14 18 23 27 29 32 34 39 41 48 51 53

2

Mass Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Weighing and Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Laboratory Methods for Determining Density . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 59 77 99

3

Disperse Systems: Particle Characterization . . . . . . . . . . . . . . . . . 3.1 Particle Size Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Measurement of Particle Size Distributions . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

101 103 131 142

4

Rheological Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Elastic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Stress, Pressure, Uniaxial Tension, Young’s Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Three-Dimensional Stress, Bulk Compression . . . . . . . 4.1.3 Shear, Shear Modulus . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Transverse Strain, Poisson’s Ratio . . . . . . . . . . . . . . . .

145 145 146 151 152 154 vii

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4.2 4.3

Rheological Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Viscous Behavior, Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Shear Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Newtonian Flow Behavior . . . . . . . . . . . . . . . . . . . . . 4.3.3 Non-Newtonian Flow Behavior . . . . . . . . . . . . . . . . . . 4.3.4 Comparison: Newtonian and Non-Newtonian Fluids . . . 4.3.5 Pseudoplastic Flow Behavior . . . . . . . . . . . . . . . . . . . 4.3.6 Thixotropic Flow Behavior . . . . . . . . . . . . . . . . . . . . . 4.3.7 Dilatant Flow Behavior . . . . . . . . . . . . . . . . . . . . . . . . 4.3.8 Rheopectic Flow Behavior . . . . . . . . . . . . . . . . . . . . . 4.3.9 Plastic Flow Behavior . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.10 Overview: Non-Newtonian Flow Behavior . . . . . . . . . . 4.3.11 Model Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.12 Ostwald–de Waele Law . . . . . . . . . . . . . . . . . . . . . . . 4.3.13 Model Functions for Plastic Fluids . . . . . . . . . . . . . . . 4.4 Temperature Dependency of Viscosity . . . . . . . . . . . . . . . . . . . 4.5 Viscosity of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Rheological Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 Rotational Rheometers . . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 Oscillation Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.3 Further Measurement Systems . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155 157 159 164 166 167 168 169 169 170 170 173 175 177 180 183 186 187 193 193 208 213 219

5

Texture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Test Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Stress Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Creep Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Deborah’s Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Oscillating Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Fracture Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Texture Profile Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

223 224 230 234 236 239 240 242 244 245

6

Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Interfacial Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Curved (Convex/Concave) Interfaces . . . . . . . . . . . . . . 6.1.2 Temperature Dependence of Interfacial Tension . . . . . . 6.1.3 Concentration Dependence of Interfacial Tension . . . . . 6.1.4 Emulsions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.5 Liquid–Liquid–Gas Interface . . . . . . . . . . . . . . . . . . . . 6.1.6 Solid–Liquid–Gas Interface . . . . . . . . . . . . . . . . . . . . . 6.2 Kinetic Phenomena at Interfaces . . . . . . . . . . . . . . . . . . . . . . . 6.3 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

247 248 250 256 260 263 265 269 271 272 281

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7

Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Steady State Diffusion in Solids . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Conductivity, Conductance, Resistance . . . . . . . . . . . . . . . . . . . 7.3 Steady State Transport Through Solid Multi-layers . . . . . . . . . . 7.4 Food Packaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Composite Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Permeation as a Molecular Process . . . . . . . . . . . . . . . . . . . . . . 7.6 Temperature Dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Measurement of Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Analogous Transport Phenomena: Heat and Electricity . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

285 285 289 290 292 295 298 299 300 304 307

8

Thermal Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Heat and Enthalpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Thermodynamics: Basis Principles . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Laws of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . 8.4 Heat capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Ideal Gases and Ideal Solids . . . . . . . . . . . . . . . . . . . . 8.4.2 Heat Capacity of Real Solids . . . . . . . . . . . . . . . . . . . . 8.5 Classification of Phase Transitions . . . . . . . . . . . . . . . . . . . . . . 8.6 Heat Transfer in Food . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.1 Heat Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.2 Heat Conduction Transfer . . . . . . . . . . . . . . . . . . . . . . 8.6.3 Convection Heat Transfer . . . . . . . . . . . . . . . . . . . . . . 8.6.4 Heat Transfer by Phase Transition . . . . . . . . . . . . . . . . 8.6.5 Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.6 Thermal Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.7 Measurement of Thermal Conductivity and Thermal Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Caloric Value of Foods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.1 Caloric (Energy) Requirement of the Human Body . . . . 8.7.2 Caloric Value of Food . . . . . . . . . . . . . . . . . . . . . . . . 8.7.3 Measurement of Caloric (Combustion) Values . . . . . . . 8.8 Thermal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.1 Thermogravimetry (TG) . . . . . . . . . . . . . . . . . . . . . . . 8.8.2 Heat Flow Calorimetry . . . . . . . . . . . . . . . . . . . . . . . . 8.8.3 Combustion Calorimetry . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

311 312 314 318 318 323 325 327 328 332 333 335 344 349 351 360

Electrical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Electrical Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Temperature Dependence of Electrical Conductivity . . . 9.1.2 Electrolyte Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Conductivity of Solid Foods . . . . . . . . . . . . . . . . . . . . 9.2 Capacitance and Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . .

403 403 407 408 414 416

9

362 364 364 369 373 374 374 379 392 395

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9.3

Impedance and Admittance . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Frequency Dependance of Impedance . . . . . . . . . . . . . 9.4 Measurement of Electrical Conductivity and Impedance . . . . . . 9.5 Zeta Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

418 420 420 423 425 426 427

10

Magnetic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Magnetic Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.2 Ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.3 Diamagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Hysteresis in Magnetization . . . . . . . . . . . . . . . . . . . . 10.2.2 Metal Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.3 Lorentz Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.4 Induction Cooker . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Magnetic Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Electron Spin Resonance . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

431 432 432 433 433 435 437 439 440 440 442 450 451

11

Electromagnetic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Electric Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Temperature Dependency . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Frequency Dependency . . . . . . . . . . . . . . . . . . . . . . . . 11.2.3 Complex Permittivity . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Microwaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Conversion of Microwaves into Heat . . . . . . . . . . . . . . 11.3.2 Penetration Depth of Microwaves . . . . . . . . . . . . . . . . 11.3.3 Microwave Heating . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Terahertz Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 NIR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

455 455 457 461 463 465 467 468 470 473 474 475 479

12

Optical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.1 Measurement of Refractive Index . . . . . . . . . . . . . . . . 12.2 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Reflection, Absorption, and Transmission . . . . . . . . . . . . . . . . . 12.4 Scattering and Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Colorimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.1 Color as a Result of Selective Absorption . . . . . . . . . . 12.5.2 Physiology of Color Vision . . . . . . . . . . . . . . . . . . . . . 12.5.3 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

483 483 488 490 494 496 497 497 499 500 506

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13

UV and X-rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 UV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 X-ray . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 X-ray Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.2 Imaging Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

513 513 515 516 517 519

14

Radioactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Types of Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.1 Activity and Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.2 Measurement of Ionizing Radiation (α-, β-, γ-) . . . . . . . 14.1.3 Natural Radioactivity . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

523 524 525 528 530 537

15

Acoustic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.1 Speed of Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.2 Loudness and Volume . . . . . . . . . . . . . . . . . . . . . . . . 15.1.3 Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Acoustic Quality Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Ultrasonic Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

539 539 540 543 544 545 545 547

16

On-line Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Control–Directing–Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Working Principles of On-line Sensors . . . . . . . . . . . . . . . . . . . 16.3.1 Piezoresistive Sensors . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.2 Mechanical Oscillation . . . . . . . . . . . . . . . . . . . . . . . . 16.3.3 Induction (Flow Measurement, Metal Detection) . . . . . 16.3.4 Ultrasound Transit Time . . . . . . . . . . . . . . . . . . . . . . . 16.3.5 Capacity and Permittivity . . . . . . . . . . . . . . . . . . . . . . 16.3.6 Refraction and Absorption . . . . . . . . . . . . . . . . . . . . . 16.4 Chemo- and Bio-sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

551 552 555 559 559 560 564 566 567 571 572 578 578

Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625

1

Water Activity

Water is an important component of nearly all food materials and plays a decisive role in dictating the physical properties, as well as quality, microbial, chemical, and biochemical degradation of the food material [1–4]. For most food materials, unless the moisture content is reduced below 50% (wet basis), much of the water content is freely available to behave physically as pure water with properties such as vapor pressure equal to pure water. As moisture content is lowered further, a point will be reached at which the water becomes less active in that it cannot act physically or chemically as pure water. For example, it cannot freeze or act as a solvent or reactant. In this state, it is considered to be bound water. The way in which water is bound to the internal structure of the food, the degree to which it is freely available to act as a solvent, to vaporize or freeze, or the degree to which it is chemically bound and unavailable can all be reflected by an ability to specify the water activity of a food material. The matrix presented in Table 1.5 attempts to illustrate the range of conditions under which water may be bound and the role it can play under each condition. The water activity of a food can be thought of as the equilibrium relative humidity of the food material. When a food sample comes into equilibrium with the atmosphere surrounding it, the water activity in the food sample becomes equal to the relative humidity of the atmosphere surrounding it. Once this equilibrium is reached, the food sample neither gains nor loses moisture over time. A more comprehensive definition of water activity will be given in this chapter.

1.1

Water Activity

To understand the concept of water activity, let’s first consider an aqueous solution at a given temperature. The vapor pressure p above the solution is in the range between zero and the saturation vapor pressure ps of water at this temperature. If we have the solution in a closed vessel and track the vapor pressure over time in the gas space of the vessel, we observe an increase in vapor pressure up to an equilibrium # The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. O. Figura, A. A. Teixeira, Food Physics, https://doi.org/10.1007/978-3-031-27398-8_1

1

2

1

Water Activity

Fig. 1.1 Vessel with aqueous solution of concentration c. The water vapor pressure p in the gas chamber above the liquid is detected with a sensor

value p. How high this value p is depends on how “free” the water molecules of the solution are. If the water molecules are restricted in their “freedom” by interaction with dissolved substances in the solution, the equilibrium vapor pressure is lower. This equilibrium vapor pressure can thus be used as an indicator of the freedom (also: reactivity, activity) of the water molecules of a solution. One refers to the relative vapor pressure, i.e. the quotient of vapor pressure p and saturation vapor pressure ps above the solution as the water activity aW of the solution (Fig. 1.1) aW =

p : ps

ð1:1Þ

In an aqueous solution in which the freedom of the water molecules is not restricted, the water activity has its maximum value aW = 1. This is the case when there are no interactions or when the concentration of the solution is zero, i.e. the solution consists of 100% pure water. In thermodynamics, the reactivity of a substance is expressed by means of the chemical potential of the substance i, generally μi =

∂G ∂ni

= μ0i þ RT  ln ai :

ð1:2Þ

p,T,nj

In the case of water, the chemical potential is therefore a function of water activity aW [5]: μW =

μ μ0 G n R T

∂G ∂nW

= μ0W þ RT  ln aW :

ð1:3Þ

p,T,nS

chemical potential in Jmol-1 chemical potential under standard conditions in Jmol-1 Gibbs energy in J amount of substance in mol general gas constant (8.314 JK-1mol-1) temperature in K (continued)

1.1 Water Activity

p aW

3

pressure in Pa water activity

In our thought experiment with the aqueous solution in a closed vessel, we had tacitly assumed that no other gases were present. Let us now mentally allow the vessel to contain air in addition to water vapor: Now there may be interactions between the different molecules of the gas phase, i.e. between water molecules and the molecules of air such as N2, O2, CO2, and other air components. Through such interactions, the above-called water vapor partial pressure can be changed, e.B. lowered. For this reason a distinction is made between the fugacity f and the partial pressure p of a gas. f = γ  p:

ð1:4Þ

The fugacity is the actual measured pressure of a gas, which can be lower than the partial pressure of the gas as a result of interactions between the gas molecules. The quotient of fugacity and partial pressure is called the fugacity coefficient of the gas concerned [6]. γ=

f : p

ð1:5Þ

With the assumption that air behaves like an ideal gas at normal pressure and the assumption that the water vapor molecules in atmospheric air also behave like molecules of an ideal gas—i.e., there are no interactions between the molecules of the gas phase—i.e., because of γ = 1 and therefore f = p we can dispend the distinction between fugacity and partial pressure. In the following, we always speak of partial pressure of the water molecules in the gas phase, i.e. of the water vapor partial pressure. Let’s look again at the closed vessel in which our aqueous solution and the gas space with this water vapor partial pressure are located: Let us now replace in our thoughts a part of the liquid water with a few moles of a solid, soluble substance. The ratio of the molar concentrations of the solution is now nW V ðnW þnS Þ V

=

nW : nW þ nS

ð1:6Þ

Assuming that the solution behaves like an ideal solution, i.e. there are no interactions between the molecules, then the relative water vapor pressure of the solution is p nW = : ps n W þ n S i.e., the water activity of an ideal solution is calculated

ð1:7Þ

4

1

Water Activity

nW : nW þ nS

aW =

ð1:8Þ

This relation is also known as the law of Raoult. It states that the water activity (or also: the relative water vapor pressure) depends on the ratio of the amount of water to the total amount of material of the ideal solution. This ratio is also called mole fraction or relative mass fraction. Because of n=

N : NA

ð1:9Þ

the water activity of the ideal solution depends on the number of molecules N aW =

nW V ðnW þnS Þ V

=

NW V ðN W þN S Þ V

=

NW NW þ NS

ð1:10Þ

with n=

m : M

ð1:11Þ

Raoult’s law can be written aW = aW p p n N NA M m V W S

nW mS nW þ M S

ð1:12Þ water activity vapor pressure in Pa saturation vapor pressure in Pa amount of substance in mol number of particles Avogadro constant molar mass in kgmol-1 mass in kg volume in m3 subscript for water subscript for solute

i.e., for the water activity it plays a role which molecular weight M the solute has. Substances with a small molecular weight, e.g. simple sugars, lower the water activity more than the same mass of a higher molecular weight substance, e.g. soluble starch. Because of the influence of the number of molecules, salts such as NaCl, which dissociate in two ions in aqueous solutions, have twice the effect as the same amount of substance of a substance that goes undissociated into solution. The vapor pressure reduction by dissolved materials is also referred to as a colligative property because of the number of molecules dependence. We will recall

1.1 Water Activity

5

this when we talk about boiling point increase and freezing point reduction of solutions. These simple considerations made here help us to understand the concept of water activity and its causes. Raoult Law is based on ideal solutions. However, the experimental measurement of the water activity of aqueous solutions shows that only very diluted solutions of, e.g., sucrose or NaCl behave like ideal solutions. With increasing concentration, there are numerous interaction effects between dissolved molecules and water molecules, which lead to non-ideal behavior and make it very difficult to calculate the water activity of foods from recipe information [7]. Bottom Line The relative water vapor pressure of an aqueous solution is a measure of the thermodynamic activity of the water phase of the solution. This property of liquid and solid food is referred to as water activity. Their value is between 0 and 1. In the case of non-ideal solutions, Raoult’s law has the form aW = γ W 

nW nW þ nS

ð1:13Þ

with γW aW n

activity coefficient of water water activity amount of substance in mol

In the following example, the water activity of a sugar solution according to Raoult is calculated and compared with the experimentally determined value.

Example Calculation of the water activity of a sugar solution of 9 g of sucrose and 100 g of water according to Raoult: mS 9g = = 0:0263 mol M S 342:3 g  mol - 1 m 100 g nW = W = = 5:55 mol M w 18:016 g  mol - 1 nW atheor = W nW þ nS 5:55 atheor = = 0:995 W 5:55 þ 0:0263

nS =

By experiment we measure aexp W = 0:995, i.e. the activity coefficient is (continued)

6

1

γW =

Water Activity

aexp 0:995 W = = 1: 0:995 atheor W

Example Calculation of the water activity of a sugar solution of 200 g of sucrose and 100 g of water according to Raoult: mS 200 g = = 0:584 mol M S 342:3 g  mol - 1 m 100 g nW = W = = 5:55 mol M w 18:016 g  mol - 1 nW atheor = W nW þ nS 5:55 atheor = = 0:904 W 5:55 þ 0:584 nS =

By experiment we measure aexp W = 0:860, i.e. the activity coefficient is γW =

aexp 0:86 W = = 0:955: theor 0:90 aW

These examples of aqueous sucrose solutions show the range of water activity in which Raoult’s law is valid. Prediction of the water activity of an 8.3% (m/m) sucrose solution by means of Raoult’s law succeeds well. Apparently, the solution behaves similarly to an ideal solution (γ W = 1). In the case of the 66.7% (m/m) sucrose solution, the deviations from the behavior of an ideal solution are clearly visible, the activity coefficient is γ W = 0.995. Raoult’s law was derived for ideal solutions. A diluted sugar solution can apparently be regarded as an ideal solution up to a certain limit. For higher accuracies or for higher concentrations, there are other models for calculating the water activity of solutions [2, 5, 8–11]. A selection of models is listed here (Tables 1.1, 1.2, and 1.3). There are numerous other approaches to mathematically model the water activity of special systems, e.g. salt mixtures or aqueous carbohydrate mixtures, refer [2, 8– 12].

1.2 Solid Boundary Surfaces

7

Table 1.1 Water activity calculation formula for non-electrolyte solutions Raoult (1882)

W aW = γ W  nWnþn S Suitable for low concentrations

n—amount of substance in mol

Norrish (1966)

log axWw = - k i ð1 - zW Þ2 Suitable for high concentrations

zw—mole fraction water Ki—Norrish constant

Grover (1947)

aW = 1:04 - 0:1 

ðsi  ci Þ2

ðsi  ci Þ þ 0:0045  i

i

Empirical model for sweets, confectionery Money and Born (1951)

1:0 aw = ð1:0þ0:27n Þ Empirical model

c—concentration s—sucrose equivalent s (lactose) = 1.0 s (invert sugar) = 1.3 n—amount of substance sugar per 100 g water

Table 1.2 Water activity calculation formula for electrolyte solutions Pitzer

aW = exp - 0:01802  ϕ 

ϕ—osmotic coefficient M—molality of solution

Mi i

aW = exp (-0.01802  ϕ  z  M )

Bromley

ϕ—osmotic coefficient z—ion number M—molality of solution

Table 1.3 Water activity calculation formula for multicomponent solutions Ross (1975)

aW =

ðaW,i Þ i

Ferro-Fontan-Benmergui-Chirife (1980)

Ii

aW =

ðaW,i Þ I i

aW, i—Water activity of component i aW, i—Water activity of component i I—Ionic strength of component i

Attention In the case of real gases, instead of pressure, we speak of fugacity. The quotient of fugacity and partial pressure is called the fugacity coefficient. With real solutions we use activity of the solute instead of concentration. The quotient of activity and concentration is called activity coefficient of the solute. Water activity, on the other hand, does not describe the solute, but the solvent of the aqueous solution.

1.2

Solid Boundary Surfaces

Let us now consider a solid surface which is contact to gaseous phase containing water molecules. When gaseous water molecules hit the solid surface and begin to adhere to it, the process is called adsorption with respect to the solid. When the same molecules are already present on the solid surface and are escaping from the surface

8 Fig. 1.2 Solid surface in adsorption equilibrium with the surrounding atmosphere, schematic. 1: gaseous phase, 2: sorbate, 3: sorbent

1

Water Activity

1 2 3

by being attracted to the fluid phase, the process is called desorption with respect to the solid (Fig. 1.2). For example, when a fresh moist food sample is being dried to produce a dehydrated product, it is placed into an environment of very low relative humidity. Under that condition, the free water molecules on the surface begin to escape into the relatively dry fluid atmosphere surrounding it in attempt to reach equilibrium relative humidity, and the food sample is undergoing desorption. Alternatively, when a previously dehydrated food sample is being rehydrated by being placed into a relatively humid environment or surrounded by water, the water molecules from the surrounding fluid will be attracted to the dry surface and begin to adhere to it in attempt to reach equilibrium relative humidity, and the food sample is undergoing adsorption. The rate at which these adsorption and desorption processes take place is governed largely by the physical and chemical characteristics of the surface boundary. For example, how readily the fluid molecules can either adhere or escape from the surface can depend upon special features of the surface physical structure, such as roughness, smoothness, and porosity (having a porous structure affecting absorbance, etc.). Besides physical adsorption, there can also be chemical binding that can enhance or interfere with the surface adsorption/desorption process (chemisorption), as shown in Fig. 1.3. These phenomena of adsorption and absorption will be explored more fully in the following section. If there is a chemical bond between the sorptive—here our water molecule—and the adsorbate—here our solid food—one speaks chemisorption, while the formation of physical and secondary bonds (e.g., capillary forces, electrostatic forces, dipole– dipole interactions, hydrogen bonds) is summarized under the term physisorption (Fig. 1.3). Table 1.4 is listing terms of sorption [13]. Water that is bound in solid foods can be divided into different groups based on the type of binding. This can be done by the binding heat, which is the energy necessary for the desorption of the adsorbed water. A high heat of binding is a sign of a strong bond. If you want to remove such bound water from a body by evaporation, you have to apply the bond heat in addition to the heat of evaporation. Table 1.5 shows some examples of water bound to different degrees to the solid surface. With increasing binding strength, the binding heat increases and the mobility of the water molecules decreases. Water molecules with a very small bond behave similarly to “free” water molecules. In order to evaporate free water molecules from solids, only the heat of evaporation need to be used, since the binding heat is zero.

1.2 Solid Boundary Surfaces

9

physisorption

H O H

O O H H H H Cl

-

Cl Cl-

-

chemisorption

H

+HOH

O

O

R

Si O

O

Cl-

capillary dipole forces bonding

hydrogen bonding

HO O

Si

OH O

chemical bonding

gaseous water molecules

adsorption

absorption

adsorption

desorption

adsorbate

Fig. 1.3 Terms used in sorption, by binding (top) by place of residence (down left), by direction of process (down right) Table 1.4 Terms of sorption

rf. Fig. 1.2 Gaseous phase Solid material Bound molecules

Term Sorptive Sorbents Sorbate

These considerations apply not only to flat interfaces where water molecules adsorb, but also to modified or structured surfaces with cracks or pores. In particular, pores with a small diameter, which are open to the gas phase, can change the adsorption behavior of solids. More about this in Sect. 1.4. Definition Interfaces exist between non-miscible phases, which can be solid, liquid, or gaseous. Interfaces bordering a gas phase are called surfaces. The theory of adsorption of fluid phases at solid-state interfaces includes adsorption from the gas phase and also from the liquid phase. Separation processes based on the adsorption of substances from liquids (example: drinking water treatment) can therefore be treated with the same laws. In the following, however, we want to focus on the balance between solid food and the gas phase.

Binding Binding type Water mobility Heat of binding in J/g H2O Examples

Description

Wet surfaces of solids

Adhering water droplets Water dropping from bodies None Nonstoichiometric Free 0 Starch gel, Gelatin gel

Decreased 0–300 Wet filter tissue

Nearly free 0

Wet sand

0–1000

Water in fine capillaries Physical

Solid surfaces in atmospheric air

100–3300

Water as a solute Adsorbed water Water between Water sitting on molecules surfaces Physicochemical

Capillary water Water in coarse capillaries Mechanical

Table 1.5 Types of water binding and related heats of binding [14]

Crystalline glucosemono- hydrate

Crystal water Water belonging to a crystal lattice Chemical Stoichiometric Nonmobile 300–2200

Ca(OH)2

1000–6000

Water of constitution Water in a compound

10 1 Water Activity

1.3 Adsorption Equilibrium

11

Definition Fluids are: liquids and gases.

1.3

Adsorption Equilibrium

The term adsorption equilibrium refers to the steady state condition that is ultimately reached when adsorption and desorption are going on at the same time. This is a dynamic state, in which molecules are leaving the surface in a desorption process, while other molecules are attaching themselves to the surface in an adsorption process. Eventually, a dynamic equilibrium state is reached when the number of molecules leaving the surface and those attaching themselves to the surface is the same, and the number of molecules resting on the surface remains constant on average. This is adsorption equilibrium. The various factors affecting the rate at which this equilibrium can come about are discussed in the following subsections. Another way to describe the behavior of water in foods is by the mechanisms of molecular adsorption, corresponding to “bound water” and capillary adsorption corresponding to “free water” described earlier. Molecular adsorption occurs under very low water activity when water molecules adhere to specific points in the molecular structure of the cell walls within the solid food material. When the distance between the water molecule and the cell wall becomes small enough, the force of attraction is large enough to draw the water molecule into the structure of the cell wall. The force of attraction at such low moisture content is so high that an “adsorption compression” results in a net decrease in volume of the solid-water aggregate. As the moisture content increases, the molecular attraction lessens and there is a volume increase, which is roughly equal to the volume of water added. Because of the initial adsorption compression, however, the total volume of the aggregate remains smaller than that of the sum of the constituents. Normally, it is undesirable to bring water activity of food materials to such low levels that irreversible damage from adsorption compression will occur. The extent and nature of the surface on which adsorption compression can take place are likely the primary factors governing molecular adsorption. Molecular attraction can be due to electronic and van der Waals attraction, but it is mostly due to hydrogen bonding in the case of water in foods. Thus, the greater the number of ionic or polar type molecules, the more water is held in the food material in this form. Molecular adsorption is the primary cause of swelling in hygroscopic food materials, such as starches. At still higher moisture contents, where the vapor pressure has not yet reached the saturation point, most of the available attraction sites have been filled with water molecules, and further holding of water molecules is possible only through the formation of “water bridges,” chains of water molecules extending between those molecules which have been directly adsorbed.

12

1

Water Activity

In the case of solid food materials with non-porous surfaces when placed in contact with a gaseous phase at a different relative humidity, adsorption or desorption will take place freely at the solid–fluid boundary surface, as described earlier. During adsorption, gaseous molecules will be attracted to the solid surface and begin diffusion to the interior once the surface becomes saturated with gas-phase molecules. In the case of desorption, water molecules from within the solid phase will be attracted to the surface, and freely escape into the gaseous phase once they reach the surface. The degree to which the molecules disperse themselves about the surface before escaping to the gaseous phase will depend upon the strength of bonding at the surface. In the case of a non-porous surface with no interference from porosity, this bonding strength is a function of the energy or enthalpy of adsorption or heat of vaporization. Both adsorption and desorption processes take place at the same time, but very different rates, depending on the initial difference in relative humidity of the two phases. At a constant temperature and partial pressure of the gaseous phase, equilibrium will be reached when the results from both processes compensate for themselves, and conditions at the boundary surface remain constant. If partial pressure of the gaseous phase is increased, the equilibrium is disrupted, and adsorption will begin once again until the surface becomes saturated with gaseous molecules forming a complete monolayer of molecules. This is known as monomolecular adhesion of a complete monolayer. If partial pressure is increased even further, then further adsorption from the gaseous phase causes formation of multiple molecular layers. These multiple layers are held by much weaker bonds and begin to give rise to term “free water” referred to drying technology that can be most easily evaporated. When only lower layers are present approaching only the monolayer, then the binding forces are very strong giving rise to “bound water” that is difficult to remove by evaporation. Table 1.5 shows how these levels of binding depend upon the way the water is held within the material. For each partial pressure p of a component, the corresponding interface assignment m is set. The adsorption equilibrium is characterized by a fixed relationship between interface assignment and partial pressure. m = fkt ðpÞ m p

mass of sorbate in kg, mg, μg partial pressure of sorptive in Pa

Definition A substance coming from the gas phase that sorbs and condenses at an interface is called an sorptive.

1.3 Adsorption Equilibrium

13

If the partial pressure is increased, the occupancy increases. Finally, the entire solid surface is covered with a complete layer of the adsorpt. In this state, one speaks of a monomolecular occupation of the interface. The solid interface is covered with a monolayer of the gas-phase molecules. Definition A continuous layer of adsorpt molecules on an interface is called a monolayer. This chapter discusses monolayer formed by water molecules. If the partial pressure is increased further, further adsorption of molecules from the gas phase takes place, whereby further layers of molecules settle on the monolayer, the so-called multiple layers. The adsorptive binding of molecules in the multiple layers is significantly reduced compared to the monolayer. Molecules at higher altitudes with a bonding heat close to zero behave similarly to free molecules in the gas phase. If the adsorpt is water, this water is also referred to as “free water.“This is to express that, in contrast to bound water (lower layers), these layers have practically no binding to the solid. For drying technology and for the shelf life of food, it is essential whether there is free or bound water and how strong this water binding is. We come back to this in Sect. 1.6. If the sorptive load at the interface is drawn graphically above the partial pressure for a given temperature, a so-called sorption isotherm is obtained. In the sorption isotherm, all equilibrium states are summarized at different partial pressures or interface assignments. Sorption isotherms provide important information in the field of drying and preservation of food. Some examples of simple sorption isotherms are available in [15]. The influence of temperature on the sorption equilibrium is understandable from kinetic gas theory: A higher temperature is synonymous with a higher speed or average kinetic energy of the gas molecules. The relationship between temperature and the velocity distribution of the molecules of an ideal gas is shown in the appendix using Maxwell-Boltzmann’s equation. A higher energy leads to the fact that a larger number of molecules can leave the solid surface contrary to the sorptive bond and a new equilibrium state with a lower interface load is achieved. The appearance of the sorption isotherm of a food thus changes when the temperature is increased or decreased.

14

1

Water Activity

Definition Sorption isotherm: Graphical representation of equilibrium sorbate load over the partial pressure of the sorptive. Following are examples of adsorption equilibria in processing technology: • • • • •

Activated carbon filter for the removal of pollutants from air Adsorption of oxygen to zeolite for air separation Drying of air by water vapor adsorption to silica gel Unpackaged icing sugar adsorbs water vapor from the ambient air. Nitrogen oxides are adsorbed on the catalyst surface of the combustion engine in order to be degraded there. Definition A steady state process is a process in which the state variables are constant over time.

1.4

Porous Surfaces

Solids with pores are called porous bodies. The porosity of a body is the ratio of the pore volume to the total volume of the solid body. ε= ε VP VK

VP VK

ð1:14Þ porosity volume of pores in m3 total volume of solid body in m3

Attention Information such as porosity, water content, alcohol content, relative humidity are often in %. Avoid mistakes by providing complete information such as % (V/V ), Vol.%, wt.%, % RH, % (m/m). Surfaces with porous structure contain voids that promote transport of water by capillary absorption. Depending on the size and number of pores, the specific area of a porous surface is significantly larger than that of the same mass with a smooth surface. The quotient of area and mass of the sample is called specific surface area Am.

1.4 Porous Surfaces Table 1.6 Pore size classes according to IUPAC

15

Term Micropores Mesopores Macropores

Pore radius in nm 25

A distinction is made between closed and open pores. While closed pores lie inside the solid and have no opening to the gas phase, open pores have an opening to the gas phase and thus provide additional space for adsorption. In contrast to the outer surface of a solid, this additional interface created by open pores is called the inner surface. For highly porous substances such as activated carbon, the outer specific surface area Am = 3–5 m2∙g-1, but the inner specific surface area may be over 1000 m2∙g-1 [13]. Capillary absorption occurs when voids in the cellular structure are of the size to hold water in liquid form by forces of surface tension. Pore sizes can be classified like in would Table 1.6 [16]. There are proposals for other classifications, e.g. to distinguish between nanopores and micropores [17]. In the case of natural products and also in food, the non-porous case is rare. The far more common case is the adsorption of molecules from the gas phase on porous solid surfaces [18, 19]. Examples are the adsorption of water molecules from the ambient air on the surfaces of baked goods or at the interfaces of powder particles. The pore radii of a material are not uniform but are subject (as well as, e.g., particle sizes) to a size distribution [20]. Likewise, the pore shape is subject to distribution. In micropores, there is a very high adsorption potential due to the small distance between opposite pore walls. Micropores are almost always covered with traces of adsorpt, only in high vacuum can adsorbed molecules also be removed from micropores. In mesopores, capillary condensation occurs. The adsorpt (e.g., water) deposited in these pores has a lower vapor pressure than the free adsorptive (e.g., free water). Therefore, a gaseous component condenses in these pores already at a partial pressure that is well below the saturation vapor pressure. This pressure, the vapor pressure of the adsorpt in a capillary, can be calculated using Kelvin’s equation. For a cylindrical pore with the pore radius rP is: ln

p 2  σ  Vm = r P  RT p0

ð1:15Þ

with V V Vm 1 = = = R n  Rs M m  Rs ρ  Rs

ð1:16Þ

16

1

Water Activity

it is ln

p 2σ = p0 r P  ρ  R s  T

p p0 R Rs M r T rP n V Vm

ð1:17Þ

vapor pressure above curved interface saturation vapor pressure universal gas constant in JK-1mol-1 specific gas constant in JK-1kg-1 molar mass of liquid in kg × mol-1 density of the liquid in kg × m-3 temperature in K pore radius in m amount of liquid substance in mol volume in m3 molar volume in m3∙mol-1

This is the Kelvin’s equation for calculating the relative vapor pressure of droplets. Droplets have a convex-shaped surface whereas liquid surfaces in capillaries have a concave-shaped surface. In mathematics the difference is the sign of the radius only. That is why we have to put into Kelvin’s equation here negative values of the cylindrical pores, e.g. ln

p 1:0682 nm p = = 0:899: = - 0:10682 → p0 - 10 nm p0

For water vapor as an adsorptive at 20 °C, it is ln

1 1 2σ 2  72:25  10 - 3 N  m - 1 p =  =  rP 999 kg  m 3  461:9 J  K - 1  kg - 1  293:15 K p0 r P ρ  Rs  T

so ln

p 1 =  1:0682 nm p0 r P

Example Relative water vapor pressure in cylindrical pores: For a pore with a diameter of 20 nm, a radius of r P = - 10 nm (negative sign due to the concave curvature of the interface) is put into Kelvin’s equation (continued)

1.4 Porous Surfaces

17

ln

p 1 =  1:0682 nm = - 0:10682 p0 - 10 nm p = e - 0:10682 = 0:89868 p0

The vapor pressure of water in a pore of 20 nm diameter is lowered to 89.9% of the vapor pressure of free water. For other pore radii at 20 °C it is: rP in nm 1 5 10 50 100 1000

p/p0 0.344 0.808 0.899 0.979 0.989 0.999

It becomes clear that vapor pressure reduction, and thus capillary condensation plays a role in mesopores, i.e. in nanostructured surfaces. For this reason, a mesoporous substance has a higher hygroscopicity than the corresponding non-porous solid. Nanostructured surfaces can also be used to increase food safety [21]. In the case of macropores, the effect of reducing vapor pressure is negligible. Pore size and pore size distribution are determined using the so-called porosimetry [20, 22]. Bottle or Flask-Shaped Pores Reference to cylindrical-shaped pores in the previous section is only an ideal case to help explain the mechanism of capillary adsorption. A way to explain hysteresis of the sorption isotherm is to assume surfaces having pores with the shape of a bottle or flask, whose opening has a considerably smaller radius than the bottom of the pore (see Fig. 1.4) [23]. Recall that the relative vapour pressure required for capillary adsorption will depend upon the pore radius. In the case of such bottle-shaped pores, the smaller radius at the top of the pore will govern the vapour pressure needed for the process of desorption (drying), in which water molecules must be drawn out from Fig. 1.4 The capillary radius on adsorption (I) is different from that on desorption (II) when a pore has the shape of a bottle or flask, schematic

I

II

18

1

Water Activity

Table 1.7 Four main types of sorption isotherms I

II

III

IV

m

m

m

m

p

p

p

Freundlich

Langmuir

BET/GAB

p

BET/GAB with pores

the bottom of the pore. However, in the case of the adsorption process when the material starts out in the dry form and the pores are initially empty of free water, the water molecules adsorb at the larger radius at the bottom of the pore. This will require a greater vapour pressure to reach the same level of moisture content during desorption than was required during adsorption. This difference in vapour pressure that is needed to reach the same level of moisture content depending on the direction of the process (adsorption or desorption) is often given as a possible explanation for the hysteresis observed in most sorption isotherms of food materials (graph IV in Table 1.7). Sorption isotherms will be discussed at some length in the following sections.

1.5

Sorption Isotherms

Sorption isotherms are graphical plots of the equilibrium between surface adhesion forces and the partial pressure of the gaseous adsorbent at the boundary surface over a range of partial pressures at a constant temperature. Four classic types of sorption isotherms encountered in scientific studies are shown in Table 1.7 along with names of the mathematical models that are used to characterize each type [15, 24]. Freundlich Model The Freundlich model is given by Eq. (1.4) and is intended to characterize the isotherm when it shows nearly no saturation behavior (Type I) when adhesion at the boundary surface takes place. Since the Freundlich model is a simple power law equation, taking logarithms of both sides will produce a linear Eq. (1.4) from which the Freundlich constants (aF and bF) in the model can be obtained by linear regression of a log-log plot of Eq. (1.5). The constant bF is taken from the slope of the straight-line log-log plot, and the constant aF can then be found by substitution.

1.5 Sorption Isotherms

19

The Freundlich model is the model of choice when sorption isotherms are to be analyzed in regions of very low partial pressure. However, when regions of higher partial pressure are important, other models like the Langmuir and BET/GAB models described in the following discussion are better suited. m = aF  p b F

ð1:18Þ

lgm = lgaF þ bF  lgp

ð1:19Þ

or

m aF, bF p

mass of adsorbent in kg Freundlich constants (0 < bF < 1) partial pressure in Pa

Langmuir Model The Langmuir model focuses on characterizing the saturation behavior of the sorption process. This is the region in the sorption isotherm where the curve tends to flatten out (Type II). This region of the isotherm is often explained by realizing that it normally covers the range of partial pressures over which molecular adhesion at the boundary surface is a saturated monolayer. Under this condition a relatively wide shift in partial pressure may produce relatively little change in molecular adhesion at the surface. The model is based upon the assumption that in adsorption equilibrium the rates of adsorption k and desorption k´ must produce an equal end result, as reflected in Eqs. (1.6) and (1.7): k  p  ð1- mÞ = k 0  m: m=

ð1:20Þ

kp : k  p þ k0

ð1:21Þ

p , pþb

ð1:22Þ

Over the range of the curve it is m = mmax where M mmax k, k′ b p

mass of adsorbent in kg maximum mass of adsorbent in kg rate constants Langmuir constant in Pa partial pressure in Pa

The Langmuir model starts out with the assumption of a homogeneous monomolecular layer in which the adsorbent is held with maximum adhesion at the surface

20 Table 1.8 Adsorption models and its range of application

1 General: mmmax =

Water Activity

p pþb

For small p

For middle p

m mmax

=

m = mmax 1þ1 b

For higher p p m mmax = pþb

m mmax

≈

m = mmax  k 0  pbF

m mmax

m = aF  p Freundlich model

m = mmax Langmuir model

1 1þbp p b

m=kp Henry’s Law

p

bF

≈

p p

= const:

mmax. The model parameters (b, mmax) can be determined by writing the equation in the form of 1 1 b 1 þ  : = m mmax mmax p

ð1:23Þ

b and This is the equation of a straight line in a graph of m1 versus 1p. The slope mmax 1 intercept mmax will give the constants. The saturation behavior described by the Langmuir model applies to gas and liquid phase adsorption, particularly in the case of chemisorption (refer Fig. 1.3) when the monomolecular layer covering the boundary surface cannot be exceeded. If the monolayer were to become covered with additional molecular layers, this would give rise to multilayer adsorption, as in the case of physisorption (refer Fig. 1.3). In this case the Langmuir model would fail, and the Brunauer, Emmett, and Teller (BET) model [25], described next, should be used instead. Table 1.8 gives an overview of the applicability of these various models at increasing partial pressure. Note that the Langmuir model changes into the Freundlich model as partial pressures decrease from the intermediate range. At very low partial pressures the Freundlich model becomes practically identical with Henry’s Law.

BET Model Once multilayer molecular adsorption is reached, further increase in partial pressures will cause the isotherm to depart from the relatively flat region characterized earlier by the Langmuir model, and it will begin to increase dramatically reflecting the weakening bonds of multilayer adhesion. When the isotherm is examined over the full range of partial pressure, it will take on a sigmoid shape (Types III and IV in Table 1.7). The best known and most widely used mathematical representation of the complete adsorption phenomenon in biological materials is given by the BET Eq. (1.10), after Brunauer, Emmett, and Teller (5), because it mathematically characterizes the entire isotherm over all three regions: 1 p C-1 p = þ  V  ð p S - p Þ V a  C V a  C pS

ð1:24Þ

1.5 Sorption Isotherms

21

with V

1 =m ρ

ð1:25Þ

and the abbreviation p =φ pS

ð1:26Þ

it is p pS

m  1-

p pS

=

1 C-1 p þ  ma  C ma  C pS

ð1:27Þ

respective φ 1 C-1 = þ φ m  ð1 - φÞ ma  C ma  C

ð1:28Þ

where p r j Va ma C m V ps

partial pressure in Pa density of the adsorbent in kg m-3 relative partial pressure volume of the monolayer in m3 mass of the monolayer in kg BET constant mass of adsorbent in kg volume of adsorbent in m3 saturation vapor pressure in Pa

The BET equation can undergo a linear transformation by creating a graphical plot of m1  1 -φ φ against φ, as shown in Fig. 1.5. This produces a straight line having an equation in the form y = a + b  φ. Fig. 1.5 BET plot of adsorbed mass versus relative vapor pressure

22

1

Water Activity

The BET constant C can be obtained from the slope and intercept of this straight line using the expression, C = ba þ 1, the monolayer mass ma from ma =

a

b a

1 1 = : aþb þ1

ð1:29Þ

The BET model [25] has a strong advantage over the Freundlich or Langmuir models because of its validity over the full range of partial pressures. It has become a very useful tool for estimating the volume or mass of the monolayer on the surface boundary of powders or materials with porous surfaces with mesopore size. For example, if we adsorb nitrogen onto a powder surface, the monolayer area can be calculated from the known area of a nitrogen molecule AN 2 = 1.62 × 10-19 m2. If water vapor is used as the adsorbent, then the relative partial pressure pp = φ takes on S the meaning of equilibrium relative humidity (or water activity, which will be explained further. When isotherm data points taken from an adsorption process are plotted on the same graph as those taken from a desorption process on the same material at the same temperature, the two isotherms will follow different pathways revealing the hysteresis behavior shown in Type IV isotherms (Table 1.7). Sorption isotherm hysteresis is far from being understood [5]. As explained earlier, the presence of bottle-shaped pores with openings that are much narrower than their interior is a possible cause of this type of hysteresis. Another possible cause is that irreversible changes may occur in the food material during the adsorption/desorption experiment. Both desorption and adsorption isotherms can be characterized well by the BET equation. The desorption isotherm relates to a drying process, in which a fresh food sample with initial high moisture content is exposed to a gaseous surrounding of lower and lower partial pressures. The desorption isotherm relates to a rehydration process, in which an initially dry food sample is exposed to a surrounding fluid of higher and higher partial pressure. Understanding the distinction between both cases is a necessary prerequisite for undertaking the experimental procedures used to obtain the sorption isotherm for a given material. With the help of the BET theory, the surface area of solids can be determined. For this purpose, nitrogen molecules are adsorbed on the surface, e.g., of a powder sample, and the mass (of the adsorbed molecules) in the state of monolayer covering is determined. From the determined mass, the number of adsorbed nitrogen molecules can be calculated and from the known area requirement of a nitrogen molecule (AN 2 = 1.6210-19 m2) the entire monolayer area. This gas adsorption method for determining the specific surface area of porous or powdery solids is also known as the BET method [26].

1.6 Moisture Content

23

GAB Model The GAB model (after Guggenheim, Anderson, de Boer) is a semi-theoretical multimolecular adsorption model intended for use over a wide range of water activity and can be written as a 3-parameter model: aW 1 C-1 = þ  aW m  C  k m m  ð1 - k  a W Þ a aC aW m k C ma

ð1:30Þ

water activity mass of adsorbate in kg GAB correction factor (0.7–1.0) Guggenheim constant mass of monolayer in kg

The GAB isotherm equation is an extension of the 2-parameter BET model which takes into account the modified properties of the adsorbate in the multilayer region and bulk liquid (“free” water) properties through the introduction of a third parameter k. If k is less than 1, a lower sorption is predicted than that by the BET model and allows the GAB isotherm to be successful up to water activity of 0.9. The GAB equation reduces to the BET equation when k = 1. The constants in the GAB equation (k and C) are temperature dependent and are the means by which we can extract information when we construct and refer to sorption isotherms at different temperatures.

1.6

Moisture Content

For general purposes, the moisture content of a food is normally expressed simply as the percent moisture in the food substance. Mathematically, this is the ratio of the mass of water contained in the food sample (adsorbent) over the total mass of food sample containing the moisture (adsorbate), expressed as a percent. However, moisture content used as the variable plotted on the vertical axis of vapor sorption isotherms often is expressed as the ratio of mass of water mW (adsorbate) divided by mass of dry matter mdm, only, (absorbent).

24

1

Table 1.9 Conversion of moisture content on dry basis and wet basis

Water Activity

Dry basis (db) W xW:db = mmdm

Wet basis (wb) W xW,wb = mmtotal

xW,db =

xW:wb =

mW mtotal - mW

mW mdm þmW

Division by mW yields xW,db = 1 1 - 1

xW,wb =

Possibilities to present the result xW (db) xW on dry basis xW in g H2O/g dry matter

xW (wb) xW on wet basis xW in g H2O/g total

xW,wb

1

1 xW,db þ1

Attention Water content of a foodstuff can be expressed based on dry matter or based on wet matter. Based on dry matter mass water xW = mass of dry matter

Based on wet matter water xW = massmass of total food

xW =

xW =

mW mdm

mW mtotal

These two different methods for expressing moisture content in a food sample are known as “wet basis” (wb) and “dry basis” (db), respectively. The distinction between both methods of expression, as well as the ability to quickly calculate one from the other must be well understood. The example below shows a vertical bar representing a 100 g food sample composed of 20 g water and 80 g dry matter and shows how these quantities are used to correctly express moisture content on either basis. For example, in this case the moisture content on a wet basis is 20/100 = 20%. But, on a dry basis it is 20/80 = 25%. Refer also to Table 1.9 for further elaboration on the difference between the two expressions and how to convert the quantities from one to another.

1.6 Moisture Content

25

Example A food consists of 20 g of water and 80 g of dry matter. water

dry matter

20g

80g

The water content of the food based on dry matter (db) is 20 g = 0:25 = 25% 80 g The water content of the food based on total weight (wb) is 20 g = 0:20 = 20% 100 g The water content of food is one of the most important quality parameters and is of crucial importance for the production processes and the shelf life or safety of food. In order to avoid misunderstandings when specifying the water content, fixed designations have proven their worth. Table 1.9 shows some names and the conversion of the information. Attention Water content and water activity are not the same thing. The relationship between the two is given by the sorption isotherm. The content of ingredients is often given in %. Examples are the water content or the alcohol concentration of a food or beverage. Expressing these simply as a percentage without specifying the basis upon which the percentage is calculated will lead to confusion. To say that alcohol concentration of beer is 5.5%, or water content of crispbread is 15%, it is unclear whether these are about volume or weight percentages. In order to avoid misunderstandings, the water content of a food is

26

1

Water Activity

therefore often expressed with the addition of wt.%, mass percentage or (m/m). This makes it clear that the reference value is a mass. In order to specify which mass is the reference value (dry matter, db, or total weight, which includes the water content, wb), the abbreviations, db and wb are added to the percentage number (see above) (Table 1.10). The expression for the water content of our crispbread is therefore: xW = 15% (m/m) db or xW = 15 wt.% based on the dry matter. When determining the water content or the dry matter content by drying methods in the laboratory [27] the results are often reported on a wet basis (wb). The conversion to dry basis (db) or vice versa is shown in Table 1.9 and is shown here using two examples: Example Moisture content of corn flakes (dry basis). A sample of corn flakes has a moisture content of 7.5% (wet basis). Express this moisture content on a dry basis. xW,db =

mW 7:5 kg 7:5 kg = = = 0:081 100 kg - 7:5 kg 92:5 kg mtotal - mW

so xW,db = 8:1 g H2 O=100 g dm = 8:1%ðdbÞ or more simply xW,db =

1 = 0:081 -1

1 0:075

Table 1.10 Percentages with different bases and examples of their use Relative mass fraction xi = mmi

Relative volume fraction

15% (m/m) 15% by weight

5% (V/V) 5% by volume

xi =

Vi V

Mole fraction xi = nni 5% (n/n) 40% by mole

Relative number xi =

Ni N

45% by number

Relative water vapor pressure φ = pp S

50% r.h. 50% relative humidity

1.7 Hygroscopicity

27

Example Moisture of corn flakes (wet basis). A sample of corn flakes has a moisture content of 21.2% (dry basis). Express the moisture content on a wet basis. xW,wb =

mW 21:2 kg = = 0:175 100 kg þ 21:2 kg mdm þ mW

xW,wb = 17:5 g H2 O=100 g = 17:5%ðwbÞ When we refer to moisture content in food sorption phenomena in this book, we always mean moisture content on a dry basis.

1.7

Hygroscopicity

Hygroscopicity is a term used to describe how readily a material will take up moisture when subjected to a given shift (change) in relative humidity. The term is often used in industrial practice to indicate materials that quickly become problematic in the presence of the least increase in surrounding relative humidity. For example, certain powdered ingredients are said to very hygroscopic if they become sticky and cause caking problems that prevent free flowing from the storage hopper in a food process. In this regard the term is used in a relative qualitative sense, and no quantitative scale has ever been assigned to hygroscopicity. We have no means to quantify what is the difference between strong and weak hygroscopic behavior. It may be possible to consider hygroscopicity as a material property that could be quantified if we understand that this type of moisture uptake behavior in response to change in relative humidity can be observed from the shape of the sorption isotherm for this material. For example, a highly hygroscopic powder will show a much higher uptake of water when the relative humidity surrounding the powder is shifted from 45% to 75% than would a powder with lower hygroscopicity. To help better understand this, consider the four different sorption isotherms shown in Fig. 1.6. Each isotherm belongs to a different material, and all four Fig. 1.6 Exposing four materials exhibiting different degrees of hygroscopicity to a shift in water activity from aW1 to aW2 . The same shift causes different amounts of water uptake Δx

28

1

Water Activity

materials exhibit different degrees of hygroscopicity. Let us assume that we shift the equilibrium relative humidity (water activity) from water activity aW1 to water activity aW2 on all four materials. Then, we can observe the different quantities of water uptake experienced by each material as an indicator of different degrees of ΔxW . hygroscopicity. The mean slope of the sorption isotherm in this range is Δa W It clearly can be seen that material 2 will have greater water uptake than material 1. The slope of the secant in the segment of the curve between the starting point and final point of the shift in water activity can be taken as a water uptake potential (wPvalue) of the given material. The wP-value serves only as an indicator of the moisture difference Δx, which occurs on a given change ΔaW. It does not take into account the level of moisture already absorbed into the material earlier. So, when we examine Fig. 1.6 once again, we would have to note that although material 1 is a very good sorbent, when we make our shift in water activity aW1 → aW2 experiment, material 2 will show the higher water uptake. Therefore, we can say that material 2 has a greater water wP than material 1 in the range of water activities of our experiment. When we use wP as an indicator of hygroscopicity, the points aW1 and aW2 have to be clearly specified. Also, a constant temperature was assumed throughout this example. For infinitesimally small changes in water content or water activity, the secant slope (mean slope) of the sorption isotherm turns into the tangent slope: W P = lim

ΔxW

aw → 0 ΔaW

=

dxW : daW

ð1:31Þ

The hygroscopicity wP of a material can thus be specified by the tangent slope of the sorption isotherm in a specified point (aW/xw). Bottom Line Hygroscopicity of a material is the slope of the water vapor adsorption isotherm for that material. For a comparison of materials, it is necessary to specify in which state (aW, xw, T, p) the comparison should be made.

Example When water activity is increased from 0.45 to 0.75, a substance shows an increase in water content from 20% (m/m) db to 40% (m/m) db. The hygroscopicity is WP =

ΔxW 0:4 - 0:2 = = 0:66 ðdbÞ 0:75 - 0:45 ΔaW

1.8 BET Equation for Foods

29

This example provides the hygroscopicity in a range aW = 0.45 to 0.75 or water content between 20 and 40% (m/m) db, i.e. an average value for this part of the sorption isotherm. It also shows the need for conventions and indications such as “water content related to dry matter” or “water content related to weighting.”

1.8

BET Equation for Foods

In the case of water vapor sorption in foods, when the isotherms are obtained as plots of moisture content (db) versus water activity, the BET equation gets the form 1 aW 1 C-1  = þ a x W 1 - aW xW,a  C xW,a  C W aW xW C xW, a

ð1:32Þ

water activity moisture content (db) in kg/kg dry matter BET constant monolayer moisture content (db) in kg/kg dry matter

If the quantity x1W  1 -aWaW were plotted against water activity aW, a straight line is obtained (see Fig. 1.7). The monolayer moisture content xW,a as well as the BET constant can then be obtained from the slope and intercept of the line. The value of xW,a is usually the moisture content at which the water is tightly bound with water molecules in a single monolayer, and it cannot participate as a solvent. Thus, it is the moisture content that should be reached for maximum stability of dehydrated foods (see also Sect. 1.12). The evaluation of the plot is performed again by obtaining the intercept and slope of the straight-line curve in the diagram. intercept : a =

slope : b =

1 xW,a C

C-1 xW,a C

Because of ba = C - 1 or C = ba þ 1 the moisture content of the food at the point where a complete monolayer exists is

Fig. 1.7 BET plot of adsorbed moisture content versus water activity 1 aw ⋅ x 1 − aw

aw

30

1

xW,a =

a

Water Activity

1 1 = : aþb þ1

ð1:33Þ

b a

For the water activity at reaching the monolayer is [28] because of xW ðaW,a Þ = xW,a : aW,a 1 C-1 1  = þ a : xW,a 1 - aW,a xW,a  C xW,a  C W,a p C-1 aW,a = : C-1

ð1:34Þ ð1:35Þ

The monolayer moisture content xW,a and BET constant C are the only two parameters needed to describe the sorption isotherm for a given food product. Published tables or on-line databases listing values of these parameters for various food stuffs can be found in references [29–33]. Also, sorption enthalpies needed for estimating energy requirements in the engineering design of drying or dehydration processes can be obtained from the Arrhenius temperature dependency of these parameters. Example BET—isotherm for a food product: The moisture content (dry basis) of the product in equilibrium with different relative humidity is listed below: Construct the BET plot from these data, and calculate the values for the monolayer moisture content xW,a and BET constant C. φ/% r.h. 0 10 20 30 50 75

xW/% (db) 0.0 2.5 4.3 5.2 8.3 18.8

First we prepare a table of values; then we draw the BET diagram (Fig. 1.8): φ/% r.h. 0 10 20 30 50 75

xW/% (db) 0.0 2.5 4.3 5.2 8.3 18.8

xW (db) 0 0.025 0.043 0.052 0.083 0.188

aW 0 0.1 0.2 0.3 0.5 0.75

 1 -aWaW

aW 1 - aW

1 xW

0 0.11 0.25 0.43 1.0 3.0

– 4.4 5.8 8.3 12.0 16.0

1.8 BET Equation for Foods

31

For the slope, we get b = 19.0 and for the intercept a = 2.4. So the monolayer moisture content is xW,a =

1 1 = = 0:0467 a þ b 2:4 þ 19:0

And the BET constant C is C=

b 19:0 þ 1= þ 1 = 8:92: a 2:4

The water activity with monolayer reached is aW,a =

p C-1 C-1

p aW,a =

8:9 - 1 = 0:25: 8:9 - 1

BET: One-Point Method For the purpose of rough estimation, the straight-line BET plot can be approximated by constructing a straight line from the origin of the coordinate axes through a single data point. The single data point should be taken at value of water activity at which the monolayer is fully developed (saturated) but no multiple layer formation exists. As a rule of thumb for most foods, this value is normally chosen in the range of water activity between 0.3–0.4. The BET parameters for the monolayer moisture content xW,a and BET constant C can be estimated from the slope of this straight line and will normally serve sufficiently well for most purposes of approximating the profile of the sorption isotherm for the given food substance, as shown by Eq. (1.32). Fig. 1.8 Evaluation of the BET diagram: The slope of the straight line provides the value of the constant b, and the intercept provides the value for the constant a

20

10

1 aw ⋅ x 1 −aw

0.1

0.5

aw

1

32

1

xW,a =

Water Activity

1 : b

ð1:36Þ

Because of a = 0 a value for C is not needed in the BET—one-point method Example Estimation of the monolayer water content of a food using the BET—onepoint method: A food product has a moisture content of 5.2% (db) in equilibrium with air at a relative humidity of 30% r.h. We construct the one-point BET straight line, and estimate the monolayer moisture content xw,a for this food. With that single point we get the following: 1 aW 1 0:3 0:3  =  = = 0:3  27:5 = 8:25 x w 1 - aW 0:052 1 - 0:3 0:036 φ/% r.h. 30

xW/% (db) 5.2

xW (db) 0.052

aW 0.3

 1 -aWaW

aW 1 - aW

1 xW

0.43

8.2

Therefore, without diagram we see the slope will be b = 8:25 0:3 = 27:5. 1 1 The monolayer water content is xW,a = b = 27:5 = 0:036 = 3:6%ðm=mÞ db. The same result we get by xW  (1 - aW). It is 0.052  0.7 = 0.036. The BET model has its strengths in the range of monomolecular adsorption (around aW = 0⋯0.45). For higher water activities we have adsorption of multiple layers and here the GAB model is more accurate.

1.9

GAB Equation for Food

The GAB model is most appropriate when we deal with water activities above 0.40.5. A small disadvantage of this model is the increased mathematical complexity in dealing with a 3-parameter equation compared to the BET model. It is 1 k  aW 1 C-1  = þ  k  aW xW,a  C xW,a  C x W 1 - k  aW

ð1:37Þ

C-1 1 aW 1  = þ a : xW 1 - k  a W xW,a  C  k xW,a  C W

ð1:38Þ

or

aW xW

water activity moisture content (db) in kg∙kg-1 dry matter (continued)

1.9 GAB Equation for Food

33

C k xW, a

Guggenheim constant GAB correction factor (0.7–1.0) monolayer moisture content (db) in kg∙kg-1 dry matter

By knowing the GAB parameters k,C and xW,a, complete sorption isotherms of a material can be drawn or aW – xW – values can be calculated. To calculate the water content of a food at a given water activity, we use the GAB equation in the form xW,a  C  k  aW : ð1 - k  aW Þð1 þ ðC - 1Þ  k  aW Þ

ð1:39Þ

xW,a  C  k  aW : ð1 - k  aW Þð1 - k  aW þ C  k  aW Þ

ð1:40Þ

xW = or xW =

By taking experimental data of xW versus aW and bringing them into a function like Eq. (1.37) we can determine the coefficients a, b, g of the polynomial, and from it we can calculate the GAB parameters [2]. aW = α  a2W þ β  aW þ γ xw

ð1:41Þ

with α=

k 1 -1 : xW,a C

ð1:42Þ

β=

2 1 1: C xW,a

ð1:43Þ

1 : xW,a  C  k

ð1:44Þ

γ= So we get the GAB parameters xW,a =

k= C=

1 β2 - 4α  γ

1 2

:

ð1:45Þ

2  α  xW,a : β  xW,a þ 1

ð1:46Þ

1 : xW,a  γ  k

ð1:47Þ

With the aid of appropriate computer software, these parameters can be calculated within a few seconds for daily routine use of the BET or GAB models. With the use

34

1

Water Activity

of such software and appropriate data input, food engineers and scientists can quickly determine the information needed to design and specify optimum processing, packaging, storage, and handling conditions with respect to water activity requirements for shelf-life stability of foods.

1.10

Sorption Enthalpy

Up to this point, we had only looked at a single temperature or kept the temperature of our samples constant. Let’s now go one step further and take sorption isotherms of a food at different temperatures. The evaluation of the BET parameters shows that they depend on the temperature. If the BET constant C is drawn logarithmically above the reciprocal value of the absolute temperature (the so-called Arrhenius diagram), a straight line is obtained, from whose slope the excess enthalpy of adsorption, or monolayer-bonding enthalpy ΔhC, can be determined. The slope (m) of the straight line on the graph of log C over 1/T is: m= -

ΔhC 2:3  Rs

ð1:48Þ

with ΔhC = Δhs - Δhvap

ð1:49Þ

Δhs = ΔhC þ Δhvap

ð1:50Þ

respective

ΔhC Δhvap Δhs T Rs

specific bonding enthalpy of H2O monolayer in J kg-1 (desorption) specific vaporization enthalpy H2O in J kg-1 (desorption) specific sorption enthalpy H2O monolayer in J kg-1 (desorption) absolute temperature in K specific gas constant of H2O kJK - 1 mol - 1 = 461:4 J  g - 1 Rs = MR = 8:314 18:02 gmol - 1

The specific sorption enthalpy Δhs, mono will be found at the drying condition which leaves only the monolayer, but fully intact. At this point the monolayer is a complete layer of water molecules tightly bound to the boundary surface of the food material, and the enthalpy at this point marks the distinction between “free” and “bound” water in the food. As explained earlier, the BET theory covers a monolayer only and there is no regard for multilayers (like in the GAB model, see below). That means water in the second layer is treated like “free” water. So in the BET theory the “excess” monolayer-bonding enthalpy ΔhC disappears and plays no further role. In this case, water is freely available to behave as normal water (vaporize, condense, freeze, sublime and thaw), and only the enthalpy of vaporization (in the case of desorption) or condensation (in the case of adsorption) applies. In the opposite

1.10

Sorption Enthalpy

35

direction when the “excess” monolayer-bonding enthalpy ΔhC takes on high values, the bonding strength at the monolayer surface becomes very strong. Further water removal beyond this point through normal drying processes becomes very costly and time consuming. Therefore, it is of critical importance to identify this point of distinction with respect to the “excess” monolayer-bonding enthalpy ΔhC in order to design and specify optimum drying, storage, and packaging conditions to assure long-term stability of dehydrated foods. A fast method for quickly estimating this “excess” monolayer-bonding enthalpy ΔhC is possible by comparing water activity of a food at different temperatures and determining the temperature dependency of water activity for the given food. This can be done with the Clausius-Clapeyron equation: ∂ ln pS ΔH Δh = : = ∂T RT 2 RS T 2

ð1:51Þ

∂ ln pS Δhvap : = ∂T Rs T 2

ð1:52Þ

∂ ln pS  aW ΔhS : = ∂T Rs T 2

ð1:53Þ

Δhvap ΔhC ∂ ln pS ∂ ln aW þ : þ = ∂T ∂T Rs T 2 Rs T 2

ð1:54Þ

∂ ln aW ΔhC : = ∂T Rs T 2

ð1:55Þ

here is

d ln aW =

ΔhC  dT: Rs T 2

aW1

T1

ΔhC  ∂T: Rs T 2

ð1:57Þ

ΔhC 1 1 :  T2 T1 Rs

ð1:58Þ

∂ ln aW = aW2

ln aW2 - ln aW1 = -

T2

ΔhC lgaW2 - lgaW1 =: 1 1 2:3  Rs T2 T1 or

ð1:56Þ

ð1:59Þ

36

1

ΔðlgaW Þ ΔhC =: 2:3  Rs ΔðT1 Þ

Water Activity

ð1:60Þ

i.e., the slope of the lg aW over 1/T diagram is m= -

ΔhC : 2, 3  Rs

ð1:61Þ

The net isosteric sorption enthalpy is ΔhC = - 2, 3  m  Rs

ð1:62Þ

ΔhS = ΔhC þ Δhvap :

ð1:63Þ

because of

we distinguish between net isosteric sorption enthalpy ΔhC and total sorption enthalpy ΔhS. The isosteric enthalpy of sorption ΔhC determined in this way (net isosteric heat of sorption) applies to a defined water content xW of the food. If one determines for different water contents xW, it shows that ΔhC with decreasing water content increases strongly. A closer look at the isosteres ln aW above 1/T shows that it is only approximately a linear function. In fact, however, the slope is slightly dependent on the temperature. Example Starch powder monolayer sorption enthalpy. From two sorption isotherms taken at different temperatures we read the water activities at moisture content xW = 10% (db): ϑ/°C 30 80

aW 0.25 0.45

We transform these data to. i 1 2

ϑ/°C 30 80

T/K 303.15 353.15

-1 1 T =K 3.29910-3 2.83210-3

aW 0.25 0.45

log aW -0.602 -0.347

From equation (continued)

1.10

Sorption Enthalpy

37

ΔðlgaW Þ ΔhC =2, 3  Rs Δ T1 we get ΔhC = - 2:3  Rs 

Δ log aW Δ T1

or ΔhC = 2:3  Rs 

lgaW2 - lgaW1 1 1 T2 - T1

so ΔhC = - 2:3  0:461 kJ  K - 1  kg - 1 

- 0:34678 - ð - 0:6020Þ 2:832  10 - 3 K - 1 - 3:299  10 - 3 K - 1

that is ΔhC = 579 kJ  kg - 1 : The result is a positive enthalpy. It means that ΔhC = 579 kJ  kg-1 has to be introduced into the material to the sorption binding strength of the monolayer molecules. To bring the water molecules form the monolayer phase into the gaseous phase according to Eq. (1.13) we need the total enthalpy of ΔhS = ΔhC þ Δhvap = 579 kJ  kg - 1 þ 2200 kJ  kg - 1 = 2779 kJ  kg - 1 : When we calculate a monolayer sorption enthalpy based on two points only (here 30° and 80 °C) we should not overestimate the precision of the result. Although in this example we got the information that the heat needed for desorption of the monolayer molecules is about 20 to 30% higher than for free water: ΔhC 579 = = 0:26: Δhvap 2200 If the enthalpy is replaced by the specific free enthalpy Δg = Δh - T  Δs (specific Gibbs enthalpy) in the Arrhenius plot, the temperature dependence can be recognized. ΔðlgaW Þ Δh - T  Δs : =2:3  Rs Δ T1 Then from the slope of the Arrhenius plot (lg aW over 1/T ), then we get:

38

1

m= -

Water Activity

ΔgC 2:3  Rs

and from that specific Gibbs enthalpy ΔgC = ΔhC - T  ΔsC This makes it possible to determine the change in the specific entropy of sorption ΔsC in addition to the specific enthalpy of sorption ΔhC. Negative values for ΔgC indicate that there is an exergetic process, i.e. that the sorption takes place spontaneously. Similarly, the temperature dependence of the BET constant C and the Guggenheim correction factor k can be determined to obtain information about the sorption process [34]. Unlike the BET model, the GAB model takes into account the fact that when multi-molecular layers begin to develop, adsorbed water molecules do not instantly behave as free water. But instead, bonding forces gradually weaken with increasing multilayers, such that vapor pressure is slightly reduced from that of pure water. This is taken into account by the parameter k, which is used as a multiplier coefficient that serves as a correction factor (with values between 0.7 and 1.0) in order to take this behavior into account. The BET model neglects this effect. From the temperature dependency of the GAB parameter k, we can determine the excess enthalpy from bonding of the multilayers: ΔhC

C = C0  eRs T

ð1:64Þ

respectively Δh

k = k 0  eRs T

ð1:65Þ

Bottom Line The drying of a food is the reduction of the water content and thus also the water activity. The sorption isotherm provides information on what the water activity must be at a specified water content. The shelf life is usually higher the lower the water activity of the dried food. For deviations from this rule, see text. Therefore, the specific sorption enthalpy of the monolayer can be determined from the temperature dependency of the constants (C, C′) in the BET equation. The “excess” monolayer-bonding enthalpy ΔhC disappears and plays no further role in the case of multilayer adsorption. In this case, other values of ΔhC apply depending on the adhesive bonding of the second, third, and additional multiple layers. With each additional layer, less and less bonding energy (enthalpy) is needed until the adsorbed water becomes freely available to behave as normal water (vaporize,

1.11

Other Models

39

condense, freeze, sublime, and thaw), and only the enthalpy of vaporization (or condensation) applies. Knowledge of sorption enthalpy gives indications of the binding strength, as well as the type of bond between water and the solid food. It is therefore helpful in estimating the shelf life of food or the design of necessary packaging. In addition, the sorption enthalpy provides information for specifying or designing the most appropriate drying processes. Since the focus when drying solid foods is often on the removal of multilayers, the GAB model is preferable here. Bottom Line During desorption, i.e. drying, the isosteric enthalpy of water increases: at the beginning of drying, we have to apply the isosteric enthalpy of water in different multilayers, then of water in the monolayer.

1.11

Other Models

Several other models to mathematically describe the profile of sorption isotherms in foods have been proposed and reported in the literature [15, 17]. They differ in regard to their physical approach, method for mathematical derivation, and number of parameters. Tables 1.11, 1.12, and 1.13 contain a listing of these other models organized by the number of parameters required of each model. Models with only two parameters are listed in Table 1.11, 3-parameter models are listed in Table 1.12, and 4-parameter models in Table 1.13. If water-rich foods are treated approximately Table 1.11 Two-parameter models for sorption isotherms Freundlich (1906) Langmuir (1916)

xW = A  aBW xW = xmax

aW aW þB

Smith (1947) Oswin (1946)

xW = A + B  (ln(1 - aW))

Henderson (1952)

xW =

Brunauer, Emmett, Teller (1938) (BET)

xW =

xW = A

1 xW



B

aW 1 - aW

lnð1 - aW Þ -A

1 B

xW,a CaW ð1 - aW Þð1þðC - 1ÞaW Þ aW 1 C-1 1 - aW = xW,a C þ xW,a C

 aW

1 B

Halsey (1948)

xW = -

A ln aW

Chung, Pfost (1967)

xW = -

ln

Iglesias, Chirife (1978)

ln xW þ ðxW 2 þ x0:5W Þ2 = A  aW þ B

Lewicki (2000)

xW = A 

1 B

ln aW -A 1

1 aW

-1

B-1

-1

40

1

Water Activity

Table 1.12 Three-parameter models for sorption isotherms Cubic model

xW = p1 þ p2  aW þ p3  a2W þ p4  a3W

Guggenheim (1966), Andersen (1946), De Boer (1953) (GAB) GAB formulated by Bizot: Chung, Pfost (1967)—modified [36]

xW =

Henderson (1952)—modified [36]

xW =

Halsey (1948)—modified [36]

xW = -

Oswin (1946)—modified [36]

x W = ðA þ B  T Þ

aW xW

xW,a CkaW ð1 - kaW Þð1 - kaW þCkaW Þ 2 = α  aW þ β  aW þ γ  C

xW = -

1 B

ln aW ðTþC Þ -A

ln

lnð1 - aW Þ - AðTþCÞ A ln aW

1 B

1 B

aW 1 - aW

C

Table 1.13 Four-parameter models for sorption isotherms Peleg (1993)

xW = A  aCW þ B  aD W

DLP (double log polynomial) Isse (1993)

xW = A  ln - ln a3W ln xW = A  ln

aW 1 - aW

þ B  ln - ln a2W

þ C  lnð- lnðaW ÞÞ þ D

þ B, q > 0, q < 0

like aqueous solutions, the sorption isotherm can also be calculated on the basis of Raoult‘s law [35]: aW =

nW : nW þ nS

ð1:66Þ

1 : 1 þ nnWS

ð1:67Þ

aW =

xW aW xm x0.5, m A, B, C, D

moisture content (db) in kg∙kg-1 dry matter water activity monolayer moisture content (db) in kg∙kg-1 dry matter moisture content in kg∙kg-1 (db) at aW = 0.5 constants

Definition A model is a representation of reality. It is a simplified reproduction of an effect, behavior, or object in which the properties considered essential are emphasized and the aspects considered incidental are disregarded in order to simplify understanding. A mathematical model is a mathematical expression (continued)

1.12

Shelf Life of Food Related to Water Activity

41

of the relationship between variables that is capable of predicting the outcome for given values of the variables. Because of the simplifications, there are always limitations to the applicability of a model.

1.12

Shelf Life of Food Related to Water Activity

Water plays an important role in the shelf life of food. For spoilage caused by microbial activity and biochemical reactions, water activity—not the water content of the food—plays the decisive role. The reason for this is that water activity characterizes the availability of water molecules, while water content is a sum parameter for the water it contains, both bound and free water. For this reason, knowledge of the water activity of a food is important and for this reason sorption isotherms are needed, from which the relationship between water activity and water content of a food can be seen. In addition to the kinetics of microbial inactivation, kinetics of quality-degradation reactions such as enzymatic tanning or non-enzymatic browning are also dependent on water activity. In addition to numerous other factors, water activity also influences the kinetics of oxidation processes. Figure 1.9 schematically shows the dependence of the speed of some spoilage reactions on water activity. If you move on the abscissa of Fig. 1.9 from right to left, it can be seen that with decreasing water activity, the speed of the spoilage reactions decreases and these can be gradually brought to a standstill. An exception is oxidation, where the speed of the reaction increases again as soon as the monolayer cover is undercut (Fig. 1.9). This phenomenon is explained by the fact that the monolayer performs a protective

Fig. 1.9 Relative rate (Vrel) of different spoilage reactions as a function of water activity aW in food. 1 lipid oxidation, 2 non-enzymatic browning reactions, 3 hydrolytic reactions, 4 enzymatic reactions, 5 molds, 6 yeasts, 7 bacteria. The dashed line indicates the sorption isotherm of the sample material. From [43]

42

1

Water Activity

function against atmospheric oxygen. As soon as the monolayer is “damaged,” this protective function is no longer given. For this reason, the knowledge of the monolayer water content, as a lower limit during drying, is also important for fatty foods. In addition to water activity, the glass transition temperature of a food is a critical parameter for the kinetics of conversion processes in food. The glass transition is a so-called phase transition of the second order, it is characterized in that the substance shows a solid–liquid transition without absorbing melting enthalpy. Such a phase transition occurs in substances that are non-crystalline solidified. This solid, non-crystalline phase is also called amorphous phase or glass. A glass can also be understood as a solidified liquid, i.e. as a substance with an extremely high viscosity that drops extremely when passing through the glass transition temperature. Numerous solid foods show glass transitions. Typical examples are confectionery such as sweets, which are created by solidifying highly concentrated solutions. Many powders created by rapid drying processes (spray drying, roller drying, freeze drying) and frozen foods created by rapid cooling processes show glass transitions. But pasta and baked goods can also contain non-crystalline-solid areas and are then prone to physicochemical changes when the glass transition temperature is exceeded. Undesirable changes of this kind are the loss of flow properties in powders, clumping in bulk solids, the stickiness of carbohydrate-rich solids, or the delayed crystallization of ingredients. When passing through the glass transition temperature, the fluidity of the material and the mobility of the molecules in the substance increase sharply, and the water activity also increases by leaps and bounds. For this reason, the speed of qualityreducing reactions can increase by leaps and bounds at this temperature. In order to exclude such transformations in affected foods, the temperature of the food should remain well below its glass transition temperature. For this reason, knowledge of the glass transition temperature is important for solid foods. The glass transition temperature depends on the composition of the solid, i.e. on the recipe. Here, the water content of the food again plays a major role. An initial estimation of the glass transition temperature of a simple system can be made after with the help of the composition and the glass transition temperatures of the pure components [5]: 1 = Tg x Tg k

i

xi T g,i

ð1:68Þ mass fraction glass transition temperature in K constant of Gordon Taylor equation

1.12

Shelf Life of Food Related to Water Activity

43

Example Estimation of the glass transition temperature of a sugar product from 90% (m/ m) sucrose and 10% (m/m) water: 1 = Tg

i

x x xi = W þ S T g,i T g,W T g,S

1 0:1 0:9 1 = þ = 0:0033697 = T g 138:15 K 340:15 K 296:8 K ϑg = 23:6 ° C A higher water content thus leads to a reduction in the glass transition temperature and can thus limit the stability of a food or its shelf life. On the other hand, the transfer of a system to a “safe” glass condition can be used to maintain quality. Details on the glass condition, on phase diagrams of food [37–40] and their experimental measurement are discussed in Chap. 8. Bottom Line In the case of solid substances, a distinction is made between the crystalline state and the glassy state. Crystalline substances pass into the liquid phase at their melting point, glass-like substances soften as soon as the glass transition temperature is reached. The stability and shelf life of food thus depend on the glass transition temperature, among other factors. The glass transition temperature, in turn, depends on the water content of the food, i.e. the adsorption of water influences the glass transition temperature. In practice, the shelf life of food is the time between harvest or production and intended consumption. In European law, the minimum shelf life is understood to mean the period of time within which “the food” retains its specific properties if stored correctly [41]. According to the European Food Information Regulation, the minimum shelf life is shown in the form of a date that is affixed to the packaging. This best-before date thus characterizes the period of time guaranteed by the manufacturer after packaging, within which the food retains its specific properties, i.e. does not show excessively high quality losses.

44

1

Water Activity

Bottom Line The minimum shelf life of a food is the length of time it can remain “on the shelf” before any quality degradation can occur that would compromise consumer acceptance. Manufacturers of a food product will stamp a “best by” date on the package serving as a guarantee, that with proper storage, the desirable properties of the food remain preserved until that date. With perishable food products in which microbial spoilage is of greatest concern, shelf life is the length of time within which such a food can be safely consumed when held under proper storage conditions. Dried or intermediate moisture foods have sufficiently low water activity that microbial growth is not possible allowing them to have much longer shelf life. The length of a specified shelf life depends on the type and speed of the microbial and quality-degradation reactions taking place in the food. A distinction is made between microbiological, chemical, biochemical and physical processes that lead to a loss of quality. Often these reactions are also summarized as spoilage-causing. If you want to roughly divide food into shelf life classes, you can differentiate between shelf life in days, weeks, and years. While, e.g., untreated beverages such as milk or fruit juices (aW = 0.99) have a shelf life of only a few days, jams (aW = 0.85) can be stored for weeks or months and low-fat biscuits or rusks (aW = 0.2) can be stored for years. However, shelf life depends not only on water activity, but also on the type and number of microorganisms present and the storage temperature. As we know, the shelf life of milk or fruit juices can be increased to several weeks by thermal inactivation of the microorganisms (pasteurization, sterilization), while the water activity remains at its value of 0.99. On the other hand, the shelf life of food can be reduced by storing it under improper conditions, such as being left unpackaged or held at too high a temperature. If biscuits or rusks are stored unpacked in ambient air, they absorb water from the atmosphere and their water activity increases making them stale. Depending on the water activity that occurs, spoilage reactions can occur, which reduce the shelf life of the baked goods from a few months or years to only a few weeks or days. From these examples, we can see that shelf life is not determined solely by water activity. However, the water activity of a food is a prerequisite for a number of spoilage reactions. The value of the water activity of a non-sterilized food is thus an indication of the shelf life, i.e. the time in which the food loses its specific properties due to spoilage, unless technical measures are taken against it, such as cooling or pasteurization. Table 1.14 shows such indicative values for the shelf life of foods of different water activity. Microbial spoilage reactions are the multiplication of microorganisms (e.g., bacteria, fungi) and the associated formation of microbial metabolic products or toxins. Chemical and biochemical spoilage reactions include fat oxidation,

1.12

Shelf Life of Food Related to Water Activity

Table 1.14 Rule of thumb values for the shelf life of foods of different water activity, explanations in the text

Water activity aW > 0.95 aW ≈ 0.85 aW < 0.75 aW < 0.65 aW < 0.6

45

Shelf life of food amounts to few... Days Weeks Months Years Decades

Table 1.15 Rule of thumb values of water activity for propagation of microorganisms [4] aW 0.97 0.95 0.94 0.93 0.92 0.91 0.90 0.86 0.80 0.77 0.61 1.4

Now a cylindrical extension overring is slipped onto the 1000 cm3 cylinder and more sample material is filled in. The cylinder is mounted on the tapping device and moved for a fixed number of tappings. In German testing standards [39] a number of 2500 with a frequency of 250 s-1 is specified. After this the sample material is adjusted to 1000 cm3 again and weighed. The tapped bulk density should be recorded with the parameters of its measurement. The difference between bulk density and maximum tapped bulk density provides information about the ability of the bulk material to be compressed by gravity or pressure. Powders can be characterized for this property by the Hausner ratio, which is the quotient of tapped bulk density over untapped bulk density (see Table 2.11). Porosity Also, the volume of the hollow void space (pores) can be calculated. The ratio of the volume of the void space (pores) and the total volume of the bulk is called porosity ε. VH : VB

ð2:77Þ

V VB - VS =1- S : VB VB

ð2:78Þ

ρ= so ε= because mB ≈ mS = m ε=1-

VS
m m
VB

ð2:79Þ

ρB : ρS

ð2:80Þ

so ε=1-

With ε as a relative volume of the hollow pore space, and α as a relative volume of the solid particle space it is evident that: α þ ε = 1: VB VH VS

ð2:81Þ total volume bulk in m3 volume of hollow space in bulk in m3 volume of solid material in bulk in m3 (continued)

96

2

Mass Density

Fig. 2.15 Particle of a powder, schematic, with open (II) and closed (I) pores

density of bulk in kg
m-3 density of solid in kg
m-3 mass of bulk in kg mass of solid in kg porosity of bulk relative volume of solids in bulk

ρB ρS mB mS ε α

When describing porosity, we took into account the volume of pores in a powder. On closer inspection, we have to distinguish between open pores and closed pores (Fig. 2.15). Open pores are detected when measuring cavity volume, while closed pores are not detected. If closed pores exist in the solid particles of a fill, they will change the solids density but not the bulk density. Conversely, the open pores have an influence on the bulk density but not on the solids density. A distinction is made here between external porosity and internal porosity of particles. The porosity of an unconsolidated mass of bulk agricultural materials such as silage, straw, and hay is a very important physical property that is needed in air flow and heat flow processes, as well as other applications. Measuring the solid volume of the particles in these types of materials can be very difficult. They are not suitable for being in contact with liquids used with pycnometer methods. For these types of materials, their natural porosity can be determined directly with the use of a simple apparatus called porosity tanks. In this apparatus, two tanks of equal volume that can be closed air-tight are connected by a manifold of tubing with shut-off valves and a manometer as shown in Fig. 2.16. Tank 2 is filled with sample material and sealed. Valves 2 and 3 are closed, and compressed air is brought into tank 1. When a suitable manometer displacement is achieved, valve 1 is also closed, and the pressure in tank 1 p1 is read and recorded from the manometer. Then, valve 2 is quickly opened, and the new lower equilibrium pressure in the system p3 is measured and recorded from the manometer. From the perfect gas law at constant temperature, the mass of air added to the system when tank 1 was initially pressurized m can be expressed as: m=

p1
V 1 : RS
T 1

ð2:82Þ

2.2 Laboratory Methods for Determining Density

97

Fig. 2.16 Differential pressure gas pycnometer (porosity tanks) [17]

to manometer valve 1

valve 2

air in

valve 3 air out

tank 1

tank 2

After valve 2 is opened, this same mass becomes distributed between the empty volume of tank 1 m1 and the available hollow pore spaces in tank 2 m2 containing the sample. Then, by conservation of mass: m = m1 þ m2 :

ð2:83Þ

p
V p
V p1
V 1 = 3 1þ 3 2: RS
T 1 RS
T 1 RS
T 1

ð2:84Þ

and ε= ε p V m T RS

V 2 p1 - p 3 = V1 p1

ð2:85Þ porosity of bulk pressure in Pa volume in m3 mass in kg temperature in K specific gas constant in J
K-1
kg-1

This differential pressure gas pycnometer can be operated with dry compressed air in the simplest case [17]. When measuring, make sure that the temperature of the gas pycnometer is constant. For approximation of air as ideal gas, see example in Sect. 2.1.

98

2

Mass Density

Definition The term density is used in numerous contexts, e.g. energy density, population density, particle density, electron density, defect density, star density, charge density, momentum density, flux density, storage density, stock density, area density, distribution density, and much more. This chapter was about mass density.

Further Reading Bakery products: measuring volume after Fornet Alcohol content of beverages by pycnometry Milk: density by means of a hydrometer Fruit and vegetable juices: relative density Beer and beer wort: relative density Density determination with X-ray scanner Powder: moisture and density determination by means of microwaves Density and viscosity determination by mechanical resonance Porosity: measurement by pycnometer Density measurement by means of ultrasound Potato starch content by underwater weighing Piezoelectric MEMS resonator for density and viscosity Lysimeter: online weighing of soil and crops Mango: ripeness detection by means of relative density Density determination of food by means of X-ray image processing methods

[40] [41] method 37.001 [41] method L01.00-28 [41] method L31.00-1 [41] method L36.00-3 [42] [43, 44] [45] [46] [47] [30] [48] [49] [50] [42]

Summary From the density of food, first conclusions can be drawn about the composition of a food, e.g. the concentration of an ingredient or the dry matter content. The density is used for quality characterization and process control. Classical density determinations require a volume determination and a mass determination, therefore the knowledge of basic rules of weighing is necessary. There (continued)

References

99

are numerous food-specific and industry-specific measurement methods for gaseous, liquid, semi-solid, and solid materials, some of which are explained in this chapter using examples. In addition to the methods based on a mechanical, static force equilibrium, there are resonance techniques that use the influence of mass on the frequency of an oscillator. At the end of the chapter, application examples are listed, which can be used for further studies and as suggestions for your own scientific work.

References 1. Becker P, Bettin H (2016) Atome zählen für Masse und Stoffmenge. PTB-Mitteilungen 126 2. Meschede D (2006) Gerthsen Physik. Springer, Berlin 3. OIML International Recommendation R111 (2004) International Organization of Legal Metrology, Paris 4. Kohlrausch F (1996) Praktische Physik, vol 3. Teubner, Stuttgart 5. VDI-Wärmeatlas (2013) Springer 6. Kohlrausch F (1996) Praktische Physik, vol 1. Teubner, Stuttgart 7. Kessler H-G (1996) Lebensmittel- und Bioverfahrenstechnik: Molkereitechnologie. Verlag A. Kessler, Munich 8. Hui YH (1991) Data sourcebook for food scientists and technologists. VCH, New York 9. Singh RP, Heldman DR (2001) Introduction to food engineering. Academic Press, San Diego 10. Barbosa-Canovas GV (ed) (2005) Food powders - physical properties, processing, and functionality. Kluwer, New York 11. Hayes (1987) Food engineering data handbook. Wiley, New York 12. Wächter (2012) Tabellenbuch zur Chemie. Wiley VCH, Weinheim 13. Tschubik IA, Maslow AM (1973) Wärmephysikalische Konstanten von Lebensmitteln und Halbfabrikaten. VEB Fachbuchverlag, Leipzig 14. Lewis MJ (1996) Physical properties of foods and food processing systems. Woodhead Publishing, Cambridge, UK 15. Gruda Z, Postolski J (1980) Gefrieren von Lebensmitteln. VEB Fachbuchverlag, Leipzig 16. Grigull U (1989) Zustandsgrößen von Wasser und Wasserdampf in SI-Einheiten. 0-800 Grad Celsius+ 0-1000 bar, Berlin 17. Mohsenin NN (1986) Physical properties of plant and animal materials. Gordon and Breach Science Publishers, New York. https://doi.org/10.1124/mol.105.016220.marker 18. Peleg M, Bagley EB (1983) Physical properties of foods. AVI Publishing, Westport, CT 19. Töpel A (2004) Chemie und Physik der Milch. Behr’s Verlag, Hamburg 20. von Rymon Lipinski GW, Schiweck H (2007) Handbuch Süßungsmittel. Behr’s Verlag, Hamburg 21. Choi Y, Okos MR (1986) Effects of temperature and composition on the thermal properties of foods. J Food Process Appl 1 22. Fricke BA, Becker BR (2001) Evaluation of thermophysical property models for foods. HVAC&R Res 7(4):311. https://doi.org/10.1080/10789669.2001.10391278 23. Souci SW, Fachmann W, Kraut H (2016) Die Zusammensetzung der Lebensmittel, NährwertTabellen. Wissenschaftliche Verlagsgesellschaft (WVG), Stuttgart 24. Kell GS (1975) Density, thermal expansivity, and compressibility of liquid water from 0 °C to 150 °C: correlations and tables for atmospheric pressure and saturation reviewed and expressed on 1968 temperature scale. J Chem Eng Data 20

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2

Mass Density

25. Bettin HF, Spieweck F (1990) Die Dichte des Wassers als Funktion der Temperatur nach Einführung der Internationalen Temperaturskala von 1990. PTB Mitteilungen 100 26. Li L, Bo M, Li J, Zhang X, Huang Y, Sun CQ (2020) Water ice compression: principles and applications. J Mol Liq 315:113750. https://doi.org/10.1016/j.molliq.2020.113750 27. D'ans J, Lax E (1949) Taschenbuch für Chemiker und Physiker. Springer, Heidelberg 28. Standards E (2011) DIN 51757 Prüfung von Mineralölen und verwandten Stoffen Bestimmung der Dichte. Beuth, Berlin 29. Nielsen S (2014) Food analysis. Springer, New York 30. Unterwassergewicht (1995) EG-Verordnung Nr 97/95 31. DIN 12790 - Laborgeräte aus Glas - Aräometer (2018) Beuth, Berlin 32. Lauth JG, Kowalczyk J (2016) Einführung in die Physik und Chemie der Grenzflächen und Kolloide. Springer Spektrum, Berlin 33. Lide DR (ed) (2008) Handbook of chemistry and physics, 99th edn. CRC Press, Boca Raton, FL 34. DIN EN ISO 1183 Kunststoffe - Verfahren zur Bestimmung der Dichte von nicht verschäumten Kunststoffen - Teil 2: Verfahren mit Dichtegradientensäule (2004) Beuth, Berlin 35. Moshenin NN (1986) Physical properties of plant and animal material. Gordon and Breach Science Publishers, New York 36. Fukushima H, Katsube K, Hata Y, Kishi R, Fujiwara S (2007) Rapid separation and concentration of food-borne pathogens in food samples prior to quantification by viable-cell counting and real-time PCR. Appl Environ Microbiol 73(1):92 37. Kessler HG (2002) Food and bio process engineering: dairy technology, 5th edn. A. Kessler, München 38. ISO 8130-3 Coating powders — Part 3: Determination of density by liquid displacement pycnometer (2021) Beuth, Berlin 39. DIN EN 1237 fertilizers - determination of bulk density (tapped) (ISO 5311) (1992) Beuth, Berlin 40. AACC-Method 10-05. Guidelines for measurement of volume by rapeseed displacement (2000) St. Paul, Minnesota 41. Amtliche Sammlung von Untersuchungsmethoden nach § 64 LFGB Beuth, Berlin 42. Kelkar S, Boushey CJ, Okos M (2015) A method to determine the density of foods using X-ray imaging. J Food Eng 159:36. https://doi.org/10.1016/j.jfoodeng.2015.03.012 43. Kent M, Kress-Rogers E (1986) Microwave moisture and density measurements in particulate solids. Trans Inst Meas Control 8(3):161 44. Austin J, Rodriguez S, Sung PF, Harris M (2013) Utilizing microwaves for the determination of moisture content independent of density. Powder Technol 236:17. https://doi.org/10.1016/j. powtec.2012.06.039 45. Abdallah A, Reichel EK, Heinisch M, Clara S, Jakoby B (2014) Symmetric plate resonators for viscosity and density measurement. Procedia Eng 87:36. https://doi.org/10.1016/j.proeng.2014. 11.260 46. Sereno AM (2007) Determination of particle density and porosity in foods and porous materials with high moisture content. Int J Food Prop 10(3):455 47. Hoppe N, Schonfelder G, Hauptmann P (2002) Ultrasonic density sensor for liquids - its potentials and limits. Tech Mess 69(3):131. https://doi.org/10.1524/teme.2002.69.3.131 48. Manzaneque T, Ruiz-Díez V, Hernando-García J, Wistrela E, Kucera M, Schmid U, SánchezRojas JL (2014) Piezoelectric MEMS resonator-based oscillator for density and viscosity sensing. Sens Actuators A 220:305. https://doi.org/10.1016/j.sna.2014.10.002 49. Liu X, Xu C, Zhong X, Li Y, Yuan X, Cao J (2017) Comparison of 16 models for reference crop evapotranspiration against weighing lysimeter measurement. Agric Water Manag 184:145. https://doi.org/10.1016/j.agwat.2017.01.017 50. Wanitchang P, Terdwongworakul A, Wanitchang J, Nakawajana N (2011) Non-destructive maturity classification of mango based on physical, mechanical and optical properties. J Food Eng 105(3):477. https://doi.org/10.1016/j.jfoodeng.2011.03.006

3

Disperse Systems: Particle Characterization

Systems that are made up of particles in an ambient medium are called disperse systems. These include, e.g. for powders with which we have already worked in the first chapters (water activity of powders, porosity of powders, etc.). Powders consist of small solid particles surrounded by a gaseous phase. In the disperse system “powder,” the solid particles are the disperse phase while the gaseous phase represents the continuous phase (Fig. 3.1). There are numerous examples of disperse systems with which we come into contact on a daily basis. Mists and sprays, e.g., consist of liquid droplets in a gaseous phase, while emulsions consist of liquid droplets in a continuous liquid phase. Since both disperse phase and continuous phase can occur solid, liquid, or gaseous state, there are numerous possible combinations. Table 3.1 shows examples of disperse systems. Many foods, pharmaceuticals, and cosmetics are disperse systems, e.g. emulsions, suspensions, or foams. Definition Disperse means “finely distributed,” a dispersion is thus a fine distribution of one phase in another phase. Another meaning of dispersion in physics is the wavelength dependence on optical quantities.

Example Fog and mist are disperse systems. By increasing the temperature, the disperse system passes into a single-phase, non-disperse system: air, transparent. Disperse systems are always multiphase systems. The disperse phase is sometimes called the inner phase. The continuous phase is then called the outer phase. If the continuous phase has a solubility for the disperse phase, it may happen that the # The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. O. Figura, A. A. Teixeira, Food Physics, https://doi.org/10.1007/978-3-031-27398-8_3

101

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3 Disperse Systems: Particle Characterization

Fig. 3.1 Disperse system, schematic, gray: disperse phase, white: continuous phase

Table 3.1 Examples of disperse systems State of disperse phase Solid Solid

State of continuous phase Gaseous Liquid

Generic term for disperse system Powder, bulk solids Suspension

Liquid

Liquid

Emulsion

Liquid Gaseous Gaseous

Gaseous Liquid Solid

Aerosol Foam Solid foam

Examples Cornstarch, sugar, grain Vegetable puree, cocoa drink, rapeseed honey Milk, mayonnaise, salad dressing, butter, margarine Spray, fog Whipped cream, ice cream Marshmallows, toast, meringue

disperse phase dissolves. Then the disperse system passes into a non-disperse, single-phase system. Conversely, disperse phases can form due to precipitation, insolubility, or crystallization. Example Add a little water to ouzo or pastis, the clear liquid becomes a solution with a milky appearance (Louche effect). The change in concentration has produced an emulsion. It is a disperse system of droplets of the essential oils of the anise seed in an aqueous solution. For the identification of disperse systems, information on the size and shape of the particles is required. The particle size of an emulsion, suspension, or foam strongly influences the physical properties of such systems, e.g. the creaming tendency of cow’s milk is lowered when the fat droplets are brought to smaller diameters. The process of homogenization therefore increases the stability of the emulsion against creaming. The particles of a disperse system do not necessarily have to be spherical. While liquid droplets and small gas bubbles are usually spherical due to the capillary pressure (see Chap. 6), this is hardly true for solid particles. Solid particles can have a fibrous shape or be, e.g., edgy or crystalline. Unground starch grains often have an ellipsoidal shape, particles in a powder created by crushing processes have edgy shapes, which depend on the crushing process used.

3.1 Particle Size Distributions

103

In addition, the particle size and shape of a real product are not uniform. We cannot simply tell the particle size of a disperse system, but must specify the particle size distribution with the help of suitable distribution parameters. The same applies to the particle shape. For these reasons, we will start this chapter with an introduction to distribution functions.

3.1

Particle Size Distributions

Distributions play an important role for biological and technical quantities. Variables such as the age of the population, the spatially different harvest quality, or the data streams of the Internet can be described by distribution functions and associated statistical parameters. First, let us imagine a bowl full of glass balls (marbles) of different diameters. The total number of balls is the population (No). We determine the diameter of each of the spheres and assign the balls to different size classes designated by having diameters above diameter x. If we count the number of balls N in a size class above diameter x, we get a number distribution of our sample. Often, the relative number is plotted, which is the quotient of the number in the category over the total number of spheres in the population, N/No. This gives the percentage of the population in each size category (Fig. 3.2). By summing up the relative number in the individual size classes, we get the distribution sum function Q. It runs from 0 to 1 (which is equivalent to 0% to 100%) and represents the distribution of the sphere diameter in a sigmoid curve. In Fig. 3.3, the distribution sum function Q (left) and the distribution density function q (right) of a distributed quantity x are shown. The distribution sum has a sigmoid curve, while the distribution density shows a bell-shaped curve. The figure illustrates also that the distribution density function q is the derivative of the distribution sum function Q. A distribution can be characterized by its central value and its spread. Figure 3.4 shows two distributions that differ in central value and spread for populations of two different particulate materials. The combination of the distribution density function, q, and the distribution sum function, Q, characterizes the differences between these two distributions.

Fig. 3.2 Distribution function: Plotted is the number N of particles occurring in a size class x

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3 Disperse Systems: Particle Characterization

Fig. 3.3 Distribution sum function Q(x) (left) and the distribution density function q(x) (right) of a property x, e.g. the sphere diameter x

Fig. 3.4 Median and statistical spread of two different distributions

Fig. 3.5 Mode xh, median x50, and arithmetic mean x of a distribution in comparison

Various expressions can be used as central values. Figure 3.5 shows the mode, median, and arithmetic mean as common central values. Median The median of a set of numbers is the value that divides the ordered series into two equal halves. For odd numbers n of measured values, the median value is found in the middle, for even numbers n of measured values is calculated as the average value of the two inner measured values.

3.1 Particle Size Distributions n odd x50,r = xnþ1 2

105 n even x50,r =

xn þxnþ1 2

2

2

Example The median of the series of numbers 48 51 55 60 66 75 87 is 60 (
xarith = 63:1). The median of the series of numbers 48 51 55 60 66 75 76 87 is 63 (
xarith = 64:8). The median of the series of numbers 48 51 55 60 66 75 76 187 is 63 (
xarith = 77:3). The example shows that the median is different from the arithmetic mean xarith . While the arithmetic mean takes all values into account equally, the median is insensitive to extreme values. This is shown in the example above by the fact that the median value of the last two series of numbers is the same, although the values of the number series differ. The median value is therefore referred to as a characteristic value with a greater robustness than the arithmetic mean. In distribution functions, the median is read by reading the value at the abscissa (x) coinciding with the midpoint value on the ordinate axis (Q = 0.5). In the case of a particle size distribution, the value of x at this point would be the median of the distribution. Example From the sum distribution of our glass sphere example, we read the median: x50 = 12 mm. This means that 50% of the sample has a particle size below 12 mm and 50% of the sample has a particle size above 12 mm.

Mode The maximum of the distribution density function q(x) is called mode of distribution. The corresponding value on the abscissa is the mode value xh,r . The mode of a distribution indicates the value that is most represented in terms of quantity. In our example distribution of the number of glass spheres, the mode would be the glass ball diameter that has occurred most often. It is therefore also the most likely diameter found when sampling the glass balls. In addition to monomodal (one maximum) distribution density functions, bimodal (two maxima) or multi-modal distributions can occur. Disperse systems are then called mono-disperse, bi-disperse, tri-disperse systems, etc. When a variety of modes occur, it is called a polydisperse system (Fig. 3.6).

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3 Disperse Systems: Particle Characterization

Fig. 3.6 Monomodal distribution (left) and bimodal distribution (right)

Quantiles In Fig. 3.4, we have already read the x50 value (the median) from the sum distribution: It was the particle size for Q = 0.50. Similarly, we can also read the particle sizes for other values of Q, e.g. for 25% or for 75%: x25 is the particle size at Q = 0.25. x75 is the particle size at Q = 0.75. The parameters x25, x50, x75 are also referred to as quartiles (first, second, third quartile) of a distribution. Similarly, other parameters are in use, such as Terciles: x33, x66, x99. Quartiles: x25, x50, x75. Quintiles: x20, x40, x60, x80. Sextiles: x17, x34, x50, x67, x84. ... Deciles: x10, x20, x30, x40, x50, x60, x70, x80, x90. Percentiles: x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, etc. These parameters are summarized under the generic term quantiles. The quantiles are robust characteristic values. Common synonyms for commonly used quantiles are [1]: x25 x75 x50 x10 x90

Lower quartile, first quartile Upper quartile, third quartile Second quartile Lower decile Upper decile

With the help of quantiles, parameters for the width of distributions can be formed, e.g.

3.1 Particle Size Distributions

107

Calculation

Name Quartile mean

x75 þx25 2

x75 - x25 x75 - x25 2 x90 þx10 2 x90 þx10 x50 1 x84 - x16  100% 2  x50 x5,0 x5,3 resp: x90,0 x90,3 x60,0 x60,3 resp: x10,0 x10,3

Interquartile range, middle fifty Quartile distance Decile mean Span Coefficient of variation Uniformity index Non-uniformity index

Up to this point, we have learned some parameters for the central value and the spread of distributions. For symmetric distribution functions, mode, median, and arithmetic mean coincide. The normal distribution according to Gauss is an example of such a function. However, most real distribution functions are not such symmetric distributions and therefore require the differentiation of these central values. There are also other parameters for characterizing the form of distribution functions or for deviations from the normal distribution like the so-called skewness and the kurtosis of a distribution, which can also be formed with the help of quantiles [1]. Types of Quantities We started the chapter with the example of glass balls. The spheres had been put into categories based on their diameter and then we had used the number of spheres per category to create a number distribution. These categories often are called classes. If we are dealing with a large number of particles, it saves time to weigh the balls instead of counting them. The difference is that as quantity we now use not the number but the mass (or volume). The different types of quantities which can be used are listed in Table 3.2, index r is used to indicate the type of quantity used. Distribution functions of a sample, which are based on different types of quantities, differ in appearance and thus by central and spread parameters. The following example helps to understand that very different curves can be created even when the sample is the same. For this reason, it is essential to indicate in each distribution on which type of quantity it is based.

Table 3.2 Types of quantities

Type of quantity Number Length Area Volume Mass a

Index r r=0 r=1 r=2 r=3 r = 3a

Use with distribution functions Very common Rare Frequently Frequently Very common

Mass and volume have the same dimension, they can be converted into each other with the help of density and therefore have the same index

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3 Disperse Systems: Particle Characterization

Example A simple particle sample consists of 9 spheres: 3 balls with 1 mm diameter, 3 balls with 2 mm diameter, and 3 balls with 3 mm diameter. We calculate the number distribution and the volume distribution. For this purpose, the balls are first put into size classes. Then the quantities in the classes are determined. From the number the relative number N/Ntotal and the relative volume V/Vtotal are calculated. Cumulation of the data gives the sum distribution Q. Q0 is the sum distribution to the quantity type number (r = 0), and Q3 is the sum distribution to the quantity type volume (r = 3) (Fig. 3.7). Index i 1 2 3

Index i 1 2 3

Class (mm) 0.5–1.5 1.5–2.5 2.5–3.5

Class (mm) 0.5–1.5 1.5–2.5 2.5–3.5

Number N 3 3 3 Ntotal = 9

Volume V = N  πd6

3

N N total

Q0 =

3/9 = 0.33 3/9 = 0.33 3/9 = 0.33

0.33 0.66 1.0

i

Q3 =

V V total

i

3

1:6 56:6

= 0:03

0.03

3

12:6 56:6

= 0:22

0.25

3

42:4 56:6

= 0:75

1.0

V = 3  π ð1 6mmÞ = 1:6 mm3 V = 3  π ð2 6mmÞ = 12:6 mm3 V = 3  π ð3 6mmÞ = 42:4 mm3 Vtotal = 56.6 mm3

N N total

V V total

In the field of food engineering particle size distributions often are determined by dynamic laser diffraction (refer Sect. 1.2 and Chap. 12). By this measurement method the particle volume is recorded, i.e. volume distributions (r = 3) are obtained. In classical sieve analysis, the particle fractions are weighed, i.e. mass distributions (r = 3) are obtained. However, there are also electrical counting methods for particle size analysis, so that number distributions (r = 0) are obtained. Fig. 3.7 Number distribution and volume distribution (dashed) of the same sample, here 9 spherical particles with 3 different diameters, have a completely different appearance

3.1 Particle Size Distributions

109

Image analysis methods in which the (shadow) area of particles is determined can be used for area distributions. In addition, it is possible to convert different distributions into each other (see below). Example A sieve analysis of a sum distribution shows that the median x50,3 = 75 μm. This means that 50% of the sample has a particle size below 75 μm and 50% of the sample has a particle size above 75 μm. The index 3 in x50,3 stands for r = 3 and reminds us that the sieve analysis provides a mass distribution. So 50% of the sample here means 50% of the sample mass. Another laboratory has obtained the median value x50,0 = 75 μm from a particle size measurement. The index r = 0 indicates that it is a number distribution, i.e. 50% of the particles have a particle size below 75 μm and 50% of the particles have a particle size above 75 μm. With 50% of the particles here is meant “50% of the total number of particles.” Have you noticed the difference?

Bottom Line A graphical presentation of a particle size distribution can be represented as a number distribution or volume distribution. The particle size is on the abscissa, the quantities such as number or volume are plotted on the ordinate. The type of quantity selected determines the measuring method and vice versa. The particle size distribution can be represented graphically as a distribution sum or as a distribution density.

Terminology of Particle Size Distributions To get particle size distributions, the particles are classified according to their size. Then the quantities in the individual size classes are determined. The amount of particles in a class is called fraction. Each fraction contains particles which are larger than the lower interval limit xi - 1 and less than or equal to the upper interval limit xi . Fraction i is often characterized by the arithmetic mean xi of these interval limits. The fraction i therefore includes all particles with the diameter x, for which applies: xi ≥ x > xi-1 (Fig. 3.8). Δxi = xi - xi-1 xi = xi þx2i - 1

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3 Disperse Systems: Particle Characterization

Fig. 3.8 Illustration of interval limits, interval (class), and interval width (class width)

xi-1 xi Δxi xi i

lower interval limit upper interval limit interval width (class width) arithmetic mean of interval interval index

The particle size sum distribution curve is now created by a diagram in which the particle size x is plotted on the abscissa and the quantity fraction is plotted on the ordinate, which is less than (or equal to) this particle size Qr, i(xi) xi. This fraction Q(xi) is obtained by subsuming the quantities of the individual fractions for which is x ≤ xi. This is the reason for the function being called distribution sum function. General Qr,i ðxi Þ =

particle fraction ðxmin ⋯xi Þ all particles ðxmin ⋯xmax Þ

ð3:1Þ

in the case of laser diffraction (quantity type: volume r = 3) Q3,i ðxi Þ =

particle volume fraction ðxmin ⋯xi Þ particle total volume ðxmin ⋯xmax Þ

ð3:2Þ

with ΔVi as the volume of the individual classes Q3,i =

1 V

i

ΔV i

ð3:3Þ

i=1

in the case of or sieve analysis (quantity type: mass, r = 3) Q3,i ðxi Þ =

particle mass fraction ðxmin ⋯xi Þ particle total mass ðxmin ⋯xmax Þ

with Δmi as mass of the individual classes

ð3:4Þ

3.1 Particle Size Distributions

111

Q3,i =

1 m

i

Δmi

ð3:5Þ

i=1

r i x m V Q q

index for type of quantity index for class particle size in m mass of particles in kg volume of particles in m3 distribution sum distribution density in m-1

r The difference quotient ΔQΔxr ðixi Þ or the differential dQ dx is referred to as distribution density q(xi). The application of q(xi) above the particle size x is called distribution density function [2].

qr ðxi Þ =

dQr ðxi Þ : dx

ð3:6Þ

qr ðxi Þ ffi

ΔQr ðxi Þ : Δxi

ð3:7Þ

Attention In the real world of particle size analysis, the class width is not infinitesimal small (dx) but has finite values specified by the measurement method (e.g., laser diffraction or sieve analysis). So we work here with ΔQr ðxi Þ Δxi

ð3:8Þ

Particle Dimensions When working with particle size distributions, the question soon arises what is particle size. To understand we first consider a single particle as in Fig. 3.9. For

Fig. 3.9 Platon’s bodies: I tetrahedron, II hexahedron, III octahedron, IV pentagon dodecahedron, and V icosahedron

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3 Disperse Systems: Particle Characterization

Fig. 3.10 Examples of bodies with different main dimensions: cylinder, rotational ellipsoid, and pyramid

Fig. 3.11 Projection of a cuboid particle to determine the particle size. Depending on the measuring direction, different dimensions are obtained

b

b a

c

c

a

b

a

c a

b b

a

c

c this purpose, clear geometric dimensions such as the main dimensions of a particle are suitable. These are the lengths of a particle in the x, y, and z directions. Highly symmetrical bodies such as spheres, in which the length is the same in all three spatial directions, cause fewer problems here than those bodies that are of different lengths in x, y, and z directions. These include, e.g., cuboids and cylinders, see Fig. 3.10. If we look at a cuboid body under a microscope from different viewing angles, we find that the observed dimensions depend on the direction of measurement, ref. Fig. 3.11. Depending on the measuring direction, we get a different value for the projection surface, the circumference, the longest or shortest tendon, the maximum diameter, etc. This is because the cuboid is an anisometric body, in contrast to the sphere, which as an isometric body having the same dimensions in all spatial

3.1 Particle Size Distributions Table 3.3 Geometric equivalent diameters

113

Equivalent diameter Volume equivalent sphere diameter

Calculation

Surface equivalent sphere diameter

dA =

A π

Projection area equivalent diameter

dP =

4S π

Projection perimeter equivalent diameter

d Pe =

dV =

3

6V π

U π

Sieve mesh size

directions. The sphere represents the body with the highest symmetry, the symmetry is even higher than that of Platonic bodies (Fig. 3.9). The mathematical treatment of spherical particles therefore is easy. For this reason, the spherical shape is often used as an approximation for real particle shapes. The sphere is also used as a reference body to indicate the particle size by a so-called equivalent diameter. Equivalent Diameter An equivalent diameter is the diameter of a sphere that has the same volume as the actual particle under consideration. With the help of an equivalent diameter, we can quantify the particle size e.g. of an ellipsoid particle (Fig. 3.10) as the diameter of a spherical particle that has the same volume as our ellipsoid particle. This equivalent diameter is called the diameter of the sphere with the same volume. Analogously we can define particle surface or circumferential equivalent diameters (see Table 3.3). The advantage of using equivalent diameters is the mathematic simplification. Now you have one clearly defined diameter for a particle, although the particle has different lengths in all spatial directions. The use of equivalent diameters is a mathematic approximation and, as with all approximations, we have to be aware that this approximation will cause a certain error. Attention Using equivalent diameters like “diameter of a sphere with the same volume” is an approximation which is well or less well fulfilled depending on the application. At, e.g., ellipsoid starch grains this approximation is better fulfilled than with needle-shaped crystals. The sieve mesh size is also an equivalent diameter. In particle size analysis by sieving (cf. Sect. 1.2) it is assumed that a particle which has passed through a certain sieve opening has a diameter that is less than or equal to this sieve mesh size. Most sieve openings are the square-shaped spaces on a screen mesh, but some can be round or slit-like. Therefore, it becomes clear that we have an approximation here, too. Its validity has to be checked product by product.

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3 Disperse Systems: Particle Characterization

Table 3.4 Physical equivalent diameters Equivalent diameter Diameter of a sphere with the same descent rate in the Stokes range Diameter of a sphere with the same descent rate in the Newton range Diameter of a sphere with the same light scattering Sieve mesh size

Calculation d ST =

18η ðρS - ρF Þg

c

d N = 0:33 ðρ -ρFρ Þg S F

 c2

Chap. 12

To be determined experimentally: Rate of descent c Rate of descent c Scattered light intensity Particle fraction on sieve

Table 3.5 Examples of equivalent diameters Measurement Sieving Sedimentation Laser diffraction Flow resistance Weighing

Equivalent diameter used Sieve mesh size Diameter of stokes equivalent sphere Diameter of light scattering equivalent sphere Diameter of surface equivalent sphere Diameter of mass equivalent sphere

Symbol in Fig. 3.12 dsieve dStokes dV dA dW

Fig. 3.12 Symbols of equivalent diameters used in Table 3.5

In addition to geometric equivalent diameters, there are physical equivalent diameters. Here, we compare the particle shape with a spherical particle that behaves the same as the real particle with regard to a certain physical property. Thus, we can specify the particle size of, e.g., our ellipsoidal particle as the diameter of a sphere that shows the same rate of descent to a specified experiment. Some physical equivalent diameters are listed in Table 3.4. V A U d

volume in m3 area in m2 circumference in m diameter in m (continued)

3.1 Particle Size Distributions

115

η g c ρF ρS

dynamic viscosity of the fluid phase in Pa∙s gravitational acceleration m∙s-2 rate of descent in m∙s-1 density of fluid phase in kg∙m-3 density of particles in kg∙m-3

If we want to use the volume equivalent sphere to represent size of a particle, we have to determine the volume of the particle and use the corresponding determination equation in Table 3.3 to get the diameter of the equivalent sphere. If we want to calculate the surface equivalent diameter, we have to determine the surface area of the particle, and if we want to determine the Stokes equivalent diameter, we need to determine the rate of descent c of the particle, etc. Conversely, the preferred equivalent diameter should be selected according to which measurement method is available. If we have a way to measure Stokes descent rates of particles, we will opt for Stokes equivalent diameter (refer Table 3.4). Example The equivalent diameter of a sphere with the same volume shall be specified: (a) for an ideal cube with edge length a = 0.7 μm (b) for an ideal tetrahedron with edge length a = 0.7 μm (c) for an ideal octahedron with edge length a = 0.7 μm Solution: V=

π 3 d 6 V

dV =

3

6V π

(a) The volume is V = a3

dV =

3

6V = π

dV = 0:7 μm 

3

3

6  a3 =a  π

3

6 π

6 = 0:87 μm π

(continued)

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3 Disperse Systems: Particle Characterization

p a3  2 12

(b) The volume is V =

3

dV =

6V = π

d V = 0:7 μm  (c) The volume is V =

dV =

p 6  a3  2 =a  12π p 2 = 0:43 μm 2π

3

3

3

p

2 2π

p a3  2 3

3

6V = π

dV = 0:7 μm 

3

p p 3 2  6  a3  2 2 =a  3π π p 2 2 = 0:68 μm π

3

Example The equivalent diameter of a surface equivalent sphere shall be specified: 1. For an ideal cube with edge length a = 0.7 μm 2. For an ideal tetrahedron with edge length a = 0.7 μm 3. For an ideal octahedron with edge length a = 0.7 μm: Solution: A = π  d2A dA =

A π

(continued)

3.1 Particle Size Distributions

117

(a) The area is A = 6  a2 6  a2 =a  π

A = π

dA =

6 π

6 = 0:97 μm π

d A = 0:7 μm  p (b) The area is A = 3  a2

dA =

p 3  a2 3 =a  π π p 3 = 0:52 μm π

p

A = π

dA = 0:7 μm  (c) The area is A = 2 

p

dA =

3  a2

A = π

d A = 0:7 μm 

p 2 3 3  a2 =a  π π p 2 3 = 0:74 μm π

2

p

Particles are often analyzed using imaging techniques. One common method is to take numerous snapshots of the particle surface in different orientations to obtain images of the particle surface from different directions of projection. The projection area can be imagined as the area of shadow caused by the particle. Image analysis software derives the particle size from these projected areas. The objective is to determine how closely the particle shape can approach that of a sphere because spherical shapes are the simplest to use for mathematical calculations. This can be done by finding the diameter of the smallest circle that completely encloses the projected area of the particle and comparing the particle projected area with the total area enclosed by the circle (projected area of a sphere). As would be expected, the difference between these areas will be relatively small for rounded-body objects compared to particles with projected areas that deviate widely from a circular shape (see example in Fig. 3.13). Here image analysis can use a number of equivalent diameters too [3] (Table 3.6).

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3 Disperse Systems: Particle Characterization

Fig. 3.13 Projection of a single particle: The Martin diameter xMa divides the projection surface into two equal halves, Feret diameter: xF, longest chord: xC,max

Table 3.6 Lengths from projection surfaces [4] Feret diameter Martin diameter Longest chord

Definition Vertical distance between two tangents Length of the segment that halves the projection area Largest length of projection area

Symbol xFe xMa xC,max

Particle Form So far, we have dealt with regularly shaped particles. However, we often must deal with particles having irregular shapes. Powders produced by spray drying often consist of ellipsoidal particles. Powders produced by crushing processes often have very irregularly shaped particles. Regularly shaped particles similar to Platon bodies (Fig. 3.9) are obtained by crystallization of solids. In the case of crystals, a distinction is made between shape and habitus. The shape depends on the structure of the solid-state lattice (e.g., octahedral, tetrahedral, prismatic, cubic), while the habitus is dominated by the manufacturing process. A cubic crystal (the shape is cubic) can have a cuboid, platelet-shaped or needle-like habitus. To characterize particle shape, indicators such as aspect ratio (ratio of length to width or ratio of largest to smallest dimension) or form factors are used as a basis upon which to compare the particle shape with the shape of a sphere. These are taken from projected areas of the body in all three dimensions (Fig. 3.14). The ratio of specific surface area AV of a particle and the specific surface area of the surface equivalent sphere is the form factor used in DIN 66141 [5]. It is for spheres and cubes fr = 1, for elongated shapes fr > 1 and runs toward infinity for long thin particles such as needles (Table 3.7).

3.1 Particle Size Distributions

119

Table 3.7 Sphericity and form factor examples

Isometric particles

Sphericity φWA [6] 1

Form factor fr [5] 1

Oblong particles

1

Needle-shaped particles

→0

→1

fr =

AV : π  d 2V

Examples Sphere, cube, octahedron, icosahedron Ellipsoids, short cuboids, short cylinders Elongated cuboids or cylinders

ð3:9Þ

The reciprocal of fr is called sphericity φWA after Wadell [6]. φWA has a value range from 0 to 1. Spherical particles have φWA = 1, other particle form has φWA < 1, the minimum value of φWA = 0 is reached for needle-shaped particle with infinite length, refer Table 3.7. φWA =

π  d 2V : AV

ð3:10Þ

With the help of equivalent diameters, further form characteristics can be derived to describe the particle shape, e.g. circularity, concavity, or convexity [7] (Table 3.8). Since particle shape and size are not uniform for all particles of a collective, form factors are derived from particle form factor distributions (Table 3.7). For a comparison of sphericities for different shapes and forms see Table 3.8. Statistical Moments The so-called statistical moments are suitable for characterizing distribution functions. They are formally structured analogous to moments of mechanics. Examples from mechanics are torque, mass moment of inertia, geometric moment of inertia, etc. In order to understand the concept of statistical moments, we first want to familiarize ourselves with integral mean values. A weighted mean is formed by weighting the incoming values according to their occurrence. For particle sizing, we use intervals with the width Δxi and the middle of the interval xi . The relative amount of particles in an interval now determines the weighting of that interval. In the case of sieve analysis, this is easy to imagine because the relative mass ΔQi = mmi is actually determined by weighing. Now xi is multiplied by its “relative weight” ΔQi: xi  ΔQi

ð3:11Þ

By subsuming these products for all intervals used, we get the so-called weighted mean value x.

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3 Disperse Systems: Particle Characterization

Fig. 3.14 Projection areas of a soybean. Quantities such as sphericity and particle size depend on the direction of measurement

Table 3.8 Sphericity of particles, examples [8, 9]

Particle form Sphere Cylinder (h = d ) Octahedron Cylinder (h = 2d ) Cube Tetrahedron Needle Maize kernel Sucrose crystal Wheat kernel Oat kernel Soy bean Potatoes

i

Sphericity after Wadell 1.000 0.874 0.846 0.832 0.806 0.670 →0 0.655 0.848 0.833 0.555 0.860 0.780

i

x=

xi  ΔQi = i=1

xi  q  Δxi

ð3:12Þ

i=1

In the case of infinitesimal small intervals, this is the integral mean xmax

x  qðxÞ  dx

x= xmin

With an index for the type of quantity r = 3 or r = 0, we get

ð3:13Þ

3.1 Particle Size Distributions

121

Type of quantity r=3

Integral mean x3 =

r=0

x0 =

r=r

xr =

xmax xmin xmax xmin xmax

x  q3 ðxÞ  dx x  q0 ðxÞ  dx x  qr ðxÞ  dx

xmin

Let us first consider the case of the number distribution (r = 0): x0 =

xmax

x  q0 ðxÞ  dx is the integral mean of length x from a number distribution

xmin

q0. It is the same what we know as arithmetic mean of x. This expression is called the statistical moment M1, 0 of a number distribution. Such integral means can be formed not only for lengths (k = 1), but also for the squares (k = 2), cubes (k = 3), etc. Table 3.9 lists the equations for the calculation of such statistical moments Mk,r. Parameter r indicates the type of quantity in the distribution function (Table 3.2) and parameter k indicates the dimension of property x. The table does not list the statistical moments for k = 0 because of x0 = 1 and they all have the same value, i.e. Mk, r = M0, r = M0, 1 = M0, 2 = M0, 3 = 1 A general formulation of the kth moment of a qr-distribution is xmax

M k,r =

xk  qr ðxÞ  dx

ð3:14Þ

xmin

and provides the integral mean value xk of a given distribution. Example Known statistic moments from a number distribution are: The statistical moment M1,0 is identical to the arithmetic mean of x1 The statistical moment M2,0 is identical to the arithmetic mean of x2 The statistical moment M3,0 is identical to the arithmetic mean of x3 The variance (= square of the standard deviation σ) of a number distribution can also be represented with the help of statistical moments: xmax

ðx - xÞ2 q0 ðxÞ  dx = M 2,0 - M 21,0

σ = 2

xmin

for distributions with r > 0 the variance is [7]:

ð3:15Þ

122

3 Disperse Systems: Particle Characterization

Table 3.9 Nomenclature of integral mean values and statistical moments

Integral mean xmax

x0 = x20

=

x30 = x40 = x1 = x21 = x31

=

x2 =

x  q0 ðxÞ  dx

xmin xmax xmin xmax xmin xmax xmin xmax xmin xmax xmin xmax xmin xmax

M2, 0

r=0 k=2

x3  q0 ðxÞ  dx

M3, 0

r=0 k=3

x4  q0 ðxÞ  dx

M4, 0

r=0 k=4

x  q1 ðxÞ  dx

M1, 1

r=1 k=1

x2  q1 ðxÞ  dx

M2, 1

r=1 k=2

x3  q1 ðxÞ  dx

M3, 1

r=1 k=3

x  q2 ðxÞ  dx

M1, 2

r=2 k=1

x2  q2 ðxÞ  dx

M2, 2

r=2 k=2

x3  q2 ðxÞ  dx

M3, 2

r=2 k=3

x  q3 ðxÞ  dx

M1, 3

r=3 k=1

x2  q3 ðxÞ  dx

M2, 3

r=3 k=2

x3  q3 ðxÞ  dx

M3, 3

r=3 k=3

x2 2 =

x3 = x23

=

x33 =

xmin xmax xmin xmax

xmin xmax xmin xmax xmin

General: xkr =

xmax xmin

r=0 k=1

x2  q0 ðxÞ  dx

xmin xmax

x2 3 =

Statistical moment M1, 0

General: x  qr ðxÞ  dx k

M k,r =

xmax

xk  qr ðxÞ  dx

xmin

xmax

σ 2r

ðx - xÞ2 qr ðxÞ  dx = M 2,r - M 21,r

=

ð3:16Þ

xmin

Statistical moments are useful during converting a distribution of a given quantity type into another quantity type, or when converting characteristic values from such distributions. If we want to convert an existing distribution qr into the distribution qs, we use:

3.1 Particle Size Distributions

123

x s - r  qr ð xÞ : M s - r,r

qs ð x Þ =

ð3:17Þ

The required statistic moment M can be obtained from the moment relationship M k,s =

M kþs - r,r : M s - r,r

ð3:18Þ

with k r s

dimension of x type of quantity of the given distribution type of quantity of the wanted distribution

The moment relationship allows the calculation of the kth moment of the desired distribution with quantity type s from two moments of a given distribution with quantity type r. So with two known moments from one distribution, each integral mean of a different distribution can be calculated. For r = 0 (number distribution), the moment relationship is M k,s =

M kþs,0 : M s,0

ð3:19Þ

Example A number distribution q0 shall be converted into a mass distribution q3. with r=0 s=3 from qs ð x Þ =

xs - r  qr ð xÞ M s - r,r

qs ð x Þ =

x3 - 0  q0 ðxÞ M 3 - 0,0

we get

and the desired distribution is (continued)

124

3 Disperse Systems: Particle Characterization

q3 ð x Þ =

x3  q0 ð xÞ M 3,0

Example A mass distribution q3 shall be converted into a number distribution q0. with r=3 s=0 from qs ð x Þ =

xs - r  qr ð xÞ M s - r,r

qs ð x Þ =

x0 - 3  q3 ðxÞ M 0 - 3,3

q0 ð x Þ =

x - 3  q3 ð xÞ M - 3,3

we get

and

using the moments relation M k,s =

M kþs - r,r M s - r,r

with k= -3 s=3 we get M - 3,3 =

M - 3þ3 - 0,0 M 0,0 1 = = M 3 - 0,0 M 3,0 M 3,0

so the desired function is (continued)

3.1 Particle Size Distributions

125

q0 ð x Þ =

M 3,0  q3 ðxÞ : x3

There are a number of parameters for particle distributions that can be expressed by statistical moments. Table 3.10 lists some examples. From the graphical representation of the distribution in Fig. 3.15, the mode and the median of a particle size distribution can be read: The mode is the particle diameter belonging to the maximum of the bell-shaped distribution density function q(x). The median is the particle diameter at Q(x) = 0.5. The first moment M1,r of the distribution is the “balance point” of the distribution density function qr. Figure 3.15 illustrates the coexistence of these different parameters of a distribution. Specific Surface, Sauter Diameter, De Brouckere Diameter In addition to particle size, the specific surface area is another important parameter for characterizing particle fineness. A specific surface is the quotient of surface area Table 3.10 Moment notation for particle characterization

p xk,r = k M k,r p xk,0 = k M k,0 x1,0 = M 1,0 p x2,0 = 2 M 2,0 p x3,0 = 3 M 3,0 M rþ1,0 M r,0 2,0 x1,1 = M 1,,1 = M M 1,0 M 3,0 x1,2 = M 1,2 = M 2,0 4,0 x1,3 = M 1,3 = M M 3,0

x1,r = M 1,r =

Fig. 3.15 Median (x50,r), mode (xmod), and 1.moment ( ) of a distribution density function [4]

( ) ( )

Mean particle size Arithmetic mean particle size

Weighed mean particle size

126

3 Disperse Systems: Particle Characterization

to mass of a particle. Occasionally, the quotient of surface area to volume of a particle is also called a specific surface. The following table compares both definitions. With the density of the particles, ρS both properties can be converted into each other: AV = ρS  Am

ð3:20Þ

Specific surface area of bodies Designation Specific surface area based on volume

AV =

Definition

Specific surface area based on mass

Am =

A V A m

SI unit m-1 m2∙kg-1

For a spherical particle with the diameter d it is AV =

π  d2 3 π 6d

ð3:21Þ

6 d

ð3:22Þ

so AV =

It can be seen that the specific surface area grows inversely proportional to the particle diameter and can be a suitable property for characterizing the particle size. Example Assuming sugar icing (particle density) (ρS = 1500 kg  m-3) consists of uniform, spherical particles with diameter d = 10 μm the specific surface area can be estimated: AV = Am =

6 6 = = 6  105 m - 1 d 10 μm

AV 6  105 m - 1 = = 400 m - 2  kg - 1 ρS 1500 kg  m - 3

If the specific surface area is related to the bulk density of the sugar icing ρB = 700 kg  m-3 the result is Am =

AV 6  105 m - 1 = = 857 m - 2  kg - 1 ρB 700 kg  m - 3

For non-spherical shapes, the Heywood factor f is introduced

3.1 Particle Size Distributions

127

AV =

A 6 = f V x

f A V x

Heywood factor particle surface particle volume particle size

Thus, the specific surface area for a particle collective is xmax

x2  q0 ðxÞ  dx AV =

xmin xmax

:

ð3:23Þ

x  q0 ðxÞ  dx 3

xmin

In moment notation it is AV =

M 2,0 M 3,0

ð3:24Þ

Sauter Diameter The Sauter diameter d32 is again an equivalent diameter. It is the diameter of a spherical particle that has the same specific surface area AV based on volume as the particle collective under consideration. 6 AV

ð3:25Þ

6 6 = 2,0 AV 6M M 3,0

ð3:26Þ

M 3,0 M 2,0

ð3:27Þ

d32 = in moment notation d3,2 = so d3,2 =

Sometimes the Sauter diameter is called the surface weighted mean.

128

3 Disperse Systems: Particle Characterization

De Brouckere Diameter The De Brouckere diameter d4,3 is the equivalent diameter of a sphere with the same volume as the particle under consideration. Sometimes it is called volume weighted mean. In moment notation it is d 4,3 =

M 4,0 : M 3,0

ð3:28Þ

Example A particle collective consists of 3 spheres with a diameter of 1 μm, 3 spheres with a diameter of 2 μm, and 3 spheres with a diameter of 3 μm. The diameters d4, 3 and d3, 2 and the specific surface AV can be calclated with the help of statistic moments. p The arithmetic mean values xk,0 = k M k,0 are: 1 1 1  ð1 μmÞ1 þ  ð2 μmÞ1 þ  ð3 μmÞ1 = 2:00 μm 3 3 3

x1,0 = M 1,0 = x2,0 =

2

M 2,0 =

1 1 1  ð1 μmÞ2 þ  ð2 μmÞ2 þ  ð3 μmÞ2 = 2:16 μm 3 3 3

x3,0 =

3

M 3,0 =

1 1 1  ð1 μmÞ3 þ  ð2 μmÞ3 þ  ð3 μmÞ3 = 2:29 μm 3 3 3

x4,0 =

4

M 4,0 =

1 1 1  ð1 μmÞ4 þ  ð2 μmÞ4 þ  ð3 μmÞ4 = 2:39 μm 3 3 3

The moments with r = 0 are M 1,0 = 2:00 μm M 2,0 = ð2:16 μmÞ2 M 3,0 = ð2:29 μmÞ3 M 4,0 = ð2:39 μmÞ4 with xk,r = M k,r =

M rþk,0 M r,0

we get

d 4,3 =

M 4,0 ð2:39Þ4 = μm = 2:72 μm M 3,0 ð2:29Þ3

(continued)

3.1 Particle Size Distributions

d 3,2 = AV = 6 

129

M 3,0 ð2:29Þ3 = μm = 2:57 μm M 2,0 ð2:16Þ2

ð2:16 μmÞ2 M 2,0 =6  = 2:33  106 m - 1 M 3,0 ð2:29 μmÞ3

Attention For diameters calculated from moments different nomenclatures are used: The Sauter diameters are abbreviated x1,2 and also d3, 2. Both are identic: 3,0 x1,2 = M 1,2 = M M 2,0 = d 3,2 . Analog for the De Brouckere diameter: x1,3 = M 1,3 =

M 4,0 M 3,0

= d4,3 .

Empirical Distribution Functions Particle size distributions of disperse systems have to be determined experimentally. The resulting distribution curves Qr(x) over x can be modeled with empirically found mathematical relationships. For many process engineering applications, it is important to know “what is the mean” and “how broad” is the distribution. Options used to statistically quantify these characteristics about distributions are statistical parameters, such as the “norm” of a distribution and the standard deviation from the norm. For every day work in engineering it is useful to have linear functions instead of non-linear curves that require more complicated mathematical equations. Particle size distributions can be transformed into linear functions by using appropriate transformation model functions. In the following subsections, we will learn about some options to represent complete distribution functions by two parameters only. GGS Distribution The function after Gates, Gaudin, and Schuhmann is a power-law function. By plotting values onto logarithmic paper, the distribution function appears as a straight line which can be easily quantified by a slope and an intercept: Q r ð xÞ =

x

m

xmax

lgQr ðxÞ = m  lgx - m  lgxmax x ≤ xmax

ð3:29Þ ð3:30Þ

130

3 Disperse Systems: Particle Characterization

When a particle size distribution function can be fitted with a GGS distribution only two parameters are sufficient to represent the total curve. These are the slope m, representing the width of the original distribution and xmax which locates the point of the curve where Qr (x) = 1. Log-Normal-Distribution qr ð x Þ =

σ

1 p

2π

 exp -

1 x-x 2 σ

2

ð3:31Þ

with the median x50, r taken for x, σ as standard deviation of x and c=

x-x σ

it is c2 1 qr ð cÞ = p  e - 2 2π

ð3:32Þ

Using special grid paper a log-normal-distribution appears as a straight line in the diagram. Also, a particle size distribution can be plotted as a straight line which is characterized by its median x50, r and the standard deviation σ of the distribution. RRSB Distribution The RRSB distribution (after Rosin, Rammler, Sperling, and Benett) is a two-parameter exponential function: 1 - Qr ðxÞ = exp -

x x0

n

ð3:33Þ

with R r ð xÞ = 1 - Q r ð xÞ

ð3:34Þ

it is lg lg

1 = n  lgx - n  lgx0 þ lg ðlgeÞ = n  lgx þ d R

ð3:35Þ

Using special grid paper (RRSB net) our particle size distribution appears as a straight line in the diagram which can be characterized by x′ and the slope n only. There is no guarantee that an experimental particle size distribution can be represented with one of the models mentioned. Whether a model can be used or choosing what model will best fit the data depends on the food sample properties that result from the process of its preparation, e.g. milling process or crystallization process.

3.2 Measurement of Particle Size Distributions

3.2

131

Measurement of Particle Size Distributions

To determine the particle size, there are a number of different physical measurement methods, e.g. electrical impulse methods, optical methods such as microscopy or laser scattering or gravimetric size classification. In order to obtain reproducible size distributions, sampling technology and sample division are of great importance. Disperse systems such as emulsions, foams, and suspensions can undergo changes due to contact with diluents (coalescence, flocculation, dissolution, etc.). Powders can change due to contact with humidity or solvents. The method of sampling shall ensure that the sample is representative of the disperse system under consideration. There are special methods for dividing the sample into similar analytical samples [4]. It cannot be emphasized enough that despite all the progress in computing power and automatic evaluation of particle size distributions, the manual errors in the handling of the samples must not be ignored. In order to create more safety here, standard samples and comparison with different measuring methods are advisable. Optical Methods Optical methods of particle size measurement include microscopic methods and camera methods, which are operated with automatic image processing system. Microscopic methods, both light microscopic and electron microscopic, have the advantage that the particles can be visually assessed in terms of their shape and size. Their main disadvantage is that comparatively few particles of a sample can be considered at one time. When we realize that a powder sample can consist of billions of particles and we look at a few 10 or 100 of them only, it becomes clear that the risk of misinterpretation is high here. Example A spherical particle with a diameter of 10 μm has a volume of -6 π  d3 π  10  10 m = VK = 6 6

3

= 5:2  10 - 16 m3

1 g of a powder with a solids density of 1300 kgm-3 has a volume of 10 - 3 kg -7 3 V = mρ = 1300 m kgm‐3 = 7:7  10 This corresponds to 7.7  10-7m3/5.2  10-16m3 = 1.5  109 particles with a diameter of 10 μm. Scattered light methods have become widely used in which monochromatic laser light is scattered on the particles to be determined and the scattered light distribution is analyzed [10, 11]. Figure 3.16 illustrates that scattering of light on a particle can be caused by reflection, refraction, or diffraction. In laser diffraction (low angle laser light scattering, LALLS) the effect is used that the diffraction angle increases with decreasing particle diameter. This diffraction angle is measured and the particle size

132 Fig. 3.16 Scattering of light waves (1) on a single particle (2): diffraction (3), absorption and refraction (4), and reflection (5)

3 Disperse Systems: Particle Characterization

1

2

3 4

5 is calculated with the help of the theory of Fraunhofer. The light intensity of a certain deflection angle is a measure of the number of particles of this diameter. Fraunhofer’s theory only accounts for the contour of the particle, but not to the material of which the particle is made. This results in good validity in cases where the wavelength of the light used is much greater than the particle size. When using visible light, e.g., a He-Ne laser with a wavelength of 680 nm, Fraunhofer’s theory provides good results for particle sizes above about 50 μm. For smaller particle sizes, Mie´s theory is used, and the refractive properties of the particle, i.e. h. the ability to refract and absorb, are required as input variable. This can be achieved by using the complex refractive index. The real part of the complex refractive index (see Chap. 12) characterizes the refraction of light in the particle while the imaginary part of the complex refractive index describes the light absorption of the particle. Mie’s theory assumes the scattering particles to be spherical, smooth, and homogeneous in terms of refractive index. Particle size analysis using laser diffraction based on the Mie theory provides good results for small particles up to a size of 0.1 μm. Smaller particles can be examined, e.g., by photon correlation spectroscopy (PCS) or dynamic light scattering (DLS). Flow cells in the beam allow the characterization of particles while they flow through the device by a carrier liquid. Liquid dispersants are used as carrier fluids for emulsions and suspensions, while powders, dusts, and aerosols are moved into the laser beam by an air jet. In this way, particle size distributions based on very large particle numbers can be created in less than a minute. Figure 3.17 shows the principle of a laser diffractometer schematically. By installing several detectors (forward and backward scattering) and the use of several wavelengths, the performance of such devices can be further increased. The particle size distributions output by laser diffractometers are volume-based (r = 3) size distributions. Conversion into distribution curves of other types of quantities r is also possible. A limitation of laser diffraction is that, like all optical particle measurement methods, the concentration of the disperse phase must not be too high. In order for the diffraction angle to be assigned to the particle diameter, multiple diffraction must be avoided. This is best achieved by sufficient dilution of the sample. If irregularly shaped particles are examined by laser diffraction, it should be noted that the particle

3.2 Measurement of Particle Size Distributions

133

Fig. 3.17 Principle of laser scattering, schematic: I light source, II sample, III diffraction light, IV lens, V focal plane, VI detector V, and VII light intensity distribution at the detector

diameter output is an equivalent diameter, namely the diameter of a spherical particle with the same scattering light behavior. Dynamic Light Scattering (DLS): Photon Correlation Spectroscopy (PCS) When observing dormant samples with very small particles, it is found that the particles perform slight movements that counteract sedimentation. This Brownian molecular movement depends on the diffusion coefficient of the particle. Assuming spherical particles subjected to a Stokes law flow, the diffusion coefficient is inversely proportional to the particle radius. Stokes–Einstein relationship is given by: D= D kB T r η

kB  T 6π  r  η

ð3:36Þ diffusion coefficient in m2 s-1 Boltzmann’s constant temperature in K particle radius in m dynamic viscosity in Pa s

By evaluating the laser light scattering caused by Brown´s motion of the particles, particle diameters up to the order of 1 nm can be determined from the Stokes– Einstein relationship. This technique is called dynamic light scattering (DLS),photon correlation spectroscopy (PCS), or quasi-elastic light scattering (QELS) and is used for suspended colloids, micelles, emulsions, i.e. proteins, fats, and carbohydrates. The particle diameter determined here for a particle collective again is an equivalent

134

3 Disperse Systems: Particle Characterization

Fig. 3.18 Electrozone counting method, schematic

diameter: it is the diameter of a spherical particle with the same diffusion behavior as the particles examined [12]. Electro-Resistive Techniques If particles suspended in an electrolyte solution are flowing through a narrow opening, the electrical resistance changes during particle passage. The change in resistance correlates with the volume of the particle in the counting opening. If we count the change in resistance and classify it by size, particle size number distributions can be created very quickly. In Fig. 3.18, the measuring principle of such electrozone counters is shown. The counting orifice is realized by a narrow glass capillary with a defined diameter. In order to cover a larger measuring range, several capillaries with different diameters are used. Electrozone counters are widely used, originally developed as blood cell counting devices, they can now also be used for nanoparticles [13]. A technical limitation is that the particles must be dispersed in an electrical conductive liquid. Size Classification and Weighing: Sieve Analysis For powders and bulk solids, sieving is the simplest method for size classification. The quantity of particles in each size class is generally determined by weighing. It is therefore not a measurement of the particle size, but a gravimetric determination of the quantities resulting from size classification. The equivalent diameter used here is the sieve opening width. When specifying the screen opening width, it must be indicated whether it is a wire mesh sieve (DIN ISO 3310 [14]) or perforated sheet screen with square holes, round holes, or slotted holes. Analytical sieving is carried out by placing the sample on a set of sieves (see Fig. 3.19). The sieve set consists of different screens, the mesh size of which decreases from top to bottom. By a suitable type of sieve movement, the particles fall through those sieves whose sieve opening width is greater than the particle size and remain on the upper sieve of those sieves whose sieve opening width is below the particle size. Among the various types of screen movement, a distinction is made between plan sieving (horizontal movement), throwing sieving (vertical movement), and combined movements (tumbling sieving). The sieve movement serves to give each particle opportunities to pass through a free sieve opening and it counteracts blockage of the sieve openings. Vibration sieving is often used, in which a set of

3.2 Measurement of Particle Size Distributions Fig. 3.19 Sieving machine. 1 engine housing, 2 vibrating plate, 3 set of sieves (sieve tower), 4 straps for mounting of sieve tower, 5 pan, 6 sieve and 7 cover

135

7 4

6

3

5 2

1

screens performs vertical movements with selectable frequency and amplitude. In addition to the sieve movement, spherical or cube-shaped solids can be used as screening aids. They are intended to support the screening of difficult-to-flow powders or agglomerating particles. However, the screening aids must not change the particle size sought, i.e. they must not lead to the crushing of the primary particles. In the so-called air jet screen, compressed air serves as a screening aid. Those particles that are on a sieve with mesh size xi-1 after sieving are greater than xi–1 and less than or equal to xi . The quantity of particles that remains on a sieve is called residue. The amount of particle that has passed through a sieve is called passage. Particles in the size category xi - 1 ≤ x < xi have a size between xi-1 and xi. To characterize the size category by a diameter the arithmetic mean of the interval xi = xi þx2i - 1 is used. Although it would be desirable to use as many narrow intervals as possible, i.e. numerous screens, the effort is kept within limits. A set of sieves usually consists of only 5–10 screens. It can be seen that this is a comparatively simple method with limited particle size resolution. In addition to the dry sieving described here, there is wet screening. It is preferred at very small particle sizes using suitable liquid dispersants, if necessary with ultrasonic support.

136

3 Disperse Systems: Particle Characterization

Attention A dispersant is used for dispersing, while a solvent is used to dissolve a substance. In particle size analysis, particles must not be dissolved, but dispersed. For this reason a liquid is needed in which the sample is not soluble.

Sieve Analysis Sieving of particles is a process for classifying the particles into size classes. This process is used in the production of powders and bulk granular solids as well as on a laboratory scale to analyze an existing powder (analytical sieving, sieve analysis, sizing by sieving). For this purpose, a weighed sample quantity is successively applied to sieving with decreasing mesh size. Let us consider one of these sieves: some of the particles cannot pass through the sieve openings due to their size and form a residue on the sieve, which is easy to determine by weighing. The part of the particles that have passed through the sieve due to their size (the passage) falls on a subsequent sieve, on which a separation into residue and passage takes place again. A sieve analysis is usually carried out in such a way that the sample is placed on a set of sieves and, after sieving each sieve is weighed individually. From the difference between the weight and the weight of the empty sieve, the fraction of the sample is obtained, containing particles whose size lies between the upper and lower sieve mesh size. In this method of particle size analysis, the sieve mesh size is used as the equivalent diameter. The amount of particle fractions is determined by weighing, i.e. we create a distribution of the quantity type “mass,” i.e. r = 3. With Eq. (3.5), we get the distribution sum Q3,i and from that we calculate distribution density for each by q3 ð x i Þ =

ΔQ3 ðxi Þ : Δxi

ð3:37Þ

Table 3.11 shows an example of how to get from the values of a sieve analysis to the values of Q3 and q3. In Table 3.12 this is shown in tabular form according to DIN ISO 9276 [5]. In Table 3.12, Q3 is also referred to as D (passage). The mathematical complement to the passage is the residue R. R=D-1

ð3:38Þ

D þ R=1

ð3:39Þ

because of

3.2 Measurement of Particle Size Distributions

137

Table 3.11 Sieve analysis, example. In the left column, the sieve set, the index i indicates the mesh size xi and the associated particle fraction with a particle size of xi ≥ x > xi - 1 i 8 7 6 5 4 3 2 1

xi/μm 500 400 315 200 125 100 63 50 0

Δxi/μm 100 85 115 75 25 37 13 50

mi/g 0.6 2.2 12.2 34.8 17.0 25.2 4.9 3.1

mi m

0.006 0.022 0.122 0.348 0.170 0.252 0.049 0.031

Q3,i 1.000 0.994 0.972 0.850 0.502 0.333 0.080 0.031 0,000

q3,i/μm-1 0.6∙10-4 2.6∙10-4 10.6∙10-4 46.4∙10-4 68.0∙10-4 68.1∙10-4 37.7∙10-4 6.2∙10-4

Table 3.12 Sieve analysis, table display according to DIN ISO 9276 [15] i 1 2 3 4 5 6 7 8

Δxi/μm 50 13 37 25 75 115 85 100

xi/μm 0 50 63 100 125 200 315 400 500

mi

m=

i i=1 ges i=1

Δmi

i i=1

Ri = 1 - Di q3,i = i

mi m

Δxi

Δmi/m 0.031 0.049 0.252 0.170 0.348 0.122 0.022 0.006

Q3,i = D 0 0.031 0.080 0.333 0.502 0.850 0.972 0.994 1.000

R 1.0 0.969 0.920 0.667 0.498 0.150 0.028 0.006 0.000

q3,i/μm-1 6.2∙10-4 37.7∙10-4 68.1∙10-4 68.0∙10-4 46.4∙10-4 10.6∙10-4 2.6∙10-4 0.6∙10-4

total mass (sample mass)

Δmi

Q3,i ðxi Þ =

Δmi/g 3.1 4.9 25.2 17.0 34.8 12.2 2.2 0.6 0

mass fraction in class i in kg relative mass fraction in class i sum of mass fractions with x ≤ xi

mi m

m=

xi /μm 25 57 82 113 163 258 358 450

Δmi m

= Di

passage (distribution sum) residue sum distribution density class index

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3 Disperse Systems: Particle Characterization

Example Let us look at the passage and residue on sieve 6 in Table 3.12. It has a mesh size of 200 μm (x6 = 200 μm). Through this sieve, 85% of the sample has passed. The reason is that 85% of the sample has a particle size less than or equal to 200 μm (Q3,6 = 0.85). So 15% of the sample with a size larger than 200 μm lies on the sieve 6, i.e. the residue is R = 0.15. In Fig. 3.20, the corresponding graphical representation of the distribution sum is shown schematically. The diagram in Fig. 3.20 shows the representation Q above the particle size, the distribution sum function Q. The function value is dimensionless and runs from Q = 0⋯1, i.e. Q = 0 % ⋯100%. Reason for this is the standardization xmax

Qr ðxmax Þ =

qr  dx = 1:

ð3:40Þ

xmin

The value of Q = 0.5 on the abscissa, divides the distribution into two equal halves, 50% above and 50% is below. The corresponding particle size on the abscissa is called the median (cf. Fig. 3.20). The distribution density function q(x) is the first derivative of the function Q(x). The maximum of the q(x)-function corresponds to the inflection point of the Q(x)-function. The abscissa value associated with the maximum of the q(x)-function is called the mode. The figure shows that the mode value and median value are not identical (there is a special case when both have the same value). It should be noted that q is created by division with Δx (thus the unit is m-1 or μm-1). Therefore, the value of q depends on the selected class width Δx of the sieve analysis. The ordinate value of the q(x)-distribution can thus vary from laboratory to laboratory and must not be misinterpreted. In some particle size software instead of the distribution density q over x the representation ΔV V over x is preferred, then the ordinate is scaled in % (V/V) [16].

Fig. 3.20 Distribution sum function with central values

3.2 Measurement of Particle Size Distributions

139

Sedimentation Sedimentation is a particle size distribution method in which the particles are allowed to free fall in a fluid of lesser density than that of the particles, much like falling through air in a parachute. During free fall, the particles are exposed to competing forces. These forces are the acceleration force of gravity (particle weight), acting downward in direction of free fall and the buoyancy force and drag force caused by friction acting upward in the opposite direction in which the sample falls. According to Stokes’ law, drag force increases with relative velocity. At the beginning of free fall, the particles will accelerate until they reach their terminal velocity when the drag force becomes equal to particle weight force. At that point, there is no longer any acceleration, and the particles will continue free fall at their constant terminal velocity. Eventually, the particles reach the bottom as a sediment. At low speeds (laminar case), the frictional force according to Stokes is: F R = 6π  r  η  v:

ð3:41Þ

F G = ðρK - ρF Þ  V K  g:

ð3:42Þ

ð ρK - ρ F Þ  V K  g : 6π  r  η

ð3:43Þ

4 π  r3 3

ð3:44Þ

2  g  r 2 ð ρK - ρF Þ : 9η

ð3:45Þ

In balance with weight force

the rate of descent is v= with the volume of a sphere V= the rate of descent is v=

with radius replaces by the particle diameter d=2  r

ð3:46Þ

the rate of descent is v= FG FR r

g  d 2 ð ρK - ρF Þ : 18  η

ð3:47Þ weight force in N friction force in N radius of spherical particle in m (continued)

140

3 Disperse Systems: Particle Characterization

Fig. 3.21 Sedimentation of spherical particles in a liquid

d V g v ρK ρF η

diameter of spherical particle in m volume of spherical particle in m3 gravitational acceleration in ms-2 velocity of particle in ms-1 density of particle in kgm-3 density of fluid in kgm-3 dynamic viscosity of fluid in Pas

Since friction force is affected by particle diameter, particles with different diameters will fall (sink) at different velocities and reach the bottom at different times. A classification according to particle sizes takes place with increasing sedimentation time [4]. From the rate of descent, the particle size can be concluded (Fig. 3.21). Using the sedimentation method particle size measurements can be carried out by placing the analytical sample in a cylinder filled with liquid to allow the particles to sink and form the sediment. After suitable times, the sedimentation pattern in the stationary cylinder is determined. The determination can be carried out optically (turbidity measurement, X-ray absorption) or by gravimetrically determining the solids or the density of samples taken. The method can also be applied to non-spherical particles. The determined equivalent diameter then is the diameter of a sphere with the same Stokes behavior. Air Classification Separation by air classification is commonly used in the process industries to separate particles having different terminal velocities (hence, particle size diameters) in air. The scientific principle is the same as sedimentation because it is also based on Stokes´ law. However, it works backward from sedimentation. Instead of allowing particles to free fall in still air, the mixture of particles is placed to rest upon a perforated screen surface. Then, air is blown upward from beneath the perforated screen at a controlled velocity. As soon as the velocity exceeds the terminal velocity of the smallest size particles, the drag force acting upward will exceed the weight of the particles, and they will become airborne, rising upward to be captured by an appropriate particle collection system. Samples of these particles can then be taken for analysis to measure particle size. When higher air flow velocities are used, then

3.2 Measurement of Particle Size Distributions

141

larger size particles will become airborne, and so on. In the oil seed crushing and grain processing industries, air classification is widely used to automatically separate the lightweight (low density) hulls, husks, and shells from the relatively higher density seeds, grains, and kernels. In this way, the seeds continue down-stream to the oil extraction or other process operations free of contamination from the unwanted shells or hulls. Other Techniques Particle size determination is possible by measuring the specific surface area of the particles by nitrogen adsorption. Due to the known size of N2 molecules, the increase in mass during adsorption (see Chap. 1, Sorption) can be used to calculate the BET surface [17] of a solid sample. Also, the specific surface area of a compressed powder like pharmaceutical tablets can be measured that way. Another way to determine the specific surface area is to determine the gas flow resistance of a powder bed according to Blaine [18]. Particle sizes can also be determined by acoustic methods since sound waves are attenuated by the particles of a suspension [19]. Particle sizing by ultrasonic attenuation spectroscopy (UAS) is possible [20– 23]. An advantage of ultrasonic methods is the possibility to measure emulsions in an undiluted state. For online particle sizing sensors, optical sensors and MEMS (micro electronic mechanical systems) can be used (see Chap. 16). Looking back at the techniques of particle size determination presented, we can divide these into three groups. Group Measurement of collectives Counting methods Classification with subsequent measurement

Examples Laser diffraction, dynamic light scattering, ultrasound spectroscopic Electrozone and optical counters, microscopy with image analysis, time-of-flight counter Air classification, sieving, sedimentation, centrifugation with subsequent measurement of mass, turbidity, etc.

These different methods lead to different equivalent diameters for the particle size under consideration. Even when directly observing particles with a camera or microscope, it is necessary to determine in advance what is meant by particle size. In addition, when particle sizes and shapes are calculated using statistical methods, it is necessary to define which type of quantity is used, e.g. whether there is a number distribution or a volume distribution. Main parameters for the characterization of distributions are center of the distribution and its width. These information can be displayed graphically, by quantiles from the distribution sum or by statistical moments. While the statistical moments offer advantages from a mathematical point of view, different nomenclatures in use can be confusing for beginners. For this purpose, study of the presented examples here is recommended.

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3 Disperse Systems: Particle Characterization

Further Reading Functional nanoparticles in food Natural nanoparticles Particle sizing by acoustic spectroscopy Chocolate: Particle size and sensory quality Coconut milk emulsion: Particle size distribution modeling Structural investigation of Pickering emulsions Brownies: Influence the sugar particle size on sensory properties Characterization of double emulsions Inline particle size measurement during drying Particle size flow properties of pharmaceutical powders Lactose: Influence of particle size on dehydration and caking Edible packaging with nanotechnology

[24, 25] [26] [27] [28–30] [31] [32, 33] [34] [35] [36] [37] [38, 39] [40]

Summary In bulk solids, powders, emulsions, suspensions, and other disperse systems, the geometric dimensions of the particles play a decisive role in the product properties. In this chapter, the basic concepts of particle size distributions are explained step by step and explained with examples. The associated measurement methods are presented comparatively and the classic analysis sieving is calculated using an example. At the end of the chapter, application examples are listed, which can be used for further studies and as suggestions for your own investigations

References 1. Ferschl F (1980) Deskriptive Statistik. Springer, Heidelberg 2. Leschonski K (1984) Representation and evaluation of particle size analysis data. Part Part Syst Charact 1:89. https://doi.org/10.1002/ppsc.19840010115 3. Bhandari B, Bansal N, Zhang M, Schuck P (2013) Handbook of food powders: processes and properties. Woodhead Publishing, Cambridge, UK. https://doi.org/10.1533/9780857098672 4. Stieß M (1995) Mechanische Verfahrentechnik 1. Springer, Berlin. https://doi.org/10.1007/9783-662-08600-1 5. DIN ISO 9276 (2004) Representation of results of particle size analysis - Part 1: Graphical representation. Beuth, Berlin. https://doi.org/10.31030/9560648 6. Wadell H (1933) Volume, shape and roundness of rock particles. J Geol 41:310 7. Rumpf H (1975) Particle technology. Springer, Berlin 8. Kurzhals H-A (2003) Lexikon Lebensmitteltechnik. Behr’s Verlag, Hamburg 9. Moshenin NN (1986) Physical properties of plant and animal material. Gordon and Breach Science Publishers, New York

References

143

10. Xu R (2015) Light scattering: a review of particle characterization applications. Particuology 18:11. https://doi.org/10.1016/j.partic.2014.05.002 11. Rhodes M (2008) Introduction to particle technology. Wiley, Hoboken, NJ. https://doi.org/10. 1002/9780470727102 12. ISO 22412 particle size analysis – dynamic light scattering (DLS) (2017) Beuth, Berlin 13. Figueiredo M, Ferreira P, Campos E (2015) Electrozone sensing goes Nano. Wiley, Hoboken, NJ, pp 1–19. https://doi.org/10.1002/9780470027318.a9521 14. DIN ISO 3310 (2017) Test sieves - Technical requirements and testing. Beuth, Berlin. https:// doi.org/10.31030/2743363 15. DIN ISO 9276 (2004) Darstellung der Ergebnisse von Partikelgrößenanalysen - Teil 1. Beuth, Berlin. https://doi.org/10.31030/9560648 16. Sommer K (2000) 40 Jahre Darstellung von Partikelgrößenverteilungen—und immer noch falsch? Chem Ing Technol 72(8):809. https://doi.org/10.1002/1522-2640(200008)72:83.0.CO;2-K 17. DIN ISO 9277 (2014) Bestimmung der spezifischen Oberfläche von Festkörpern mittels Gasadsorption - BET-Verfahren. Beuth, Berlin. https://doi.org/10.31030/2066286 18. DIN 66126 (2015) Bestimmung der spezifischen Oberfläche disperser Feststoffe mittels Gasdurchströmung - Blaineverfahren. Beuth, Berlin. https://doi.org/10.31030/2313103 19. Babick F, Ripperger S (2004) Schallspektroskopische Charakterisierung konzentrierter Emulsionen. Chem Ing Technol 76(1-2):30. https://doi.org/10.1002/cite.200406145 20. Povey MJW (2017) Applications of ultrasonics in food science - novel control of fat crystallization and structuring. Curr Opin Colloid Interface Sci 28:1. https://doi.org/10.1016/j.cocis. 2016.12.001 21. Yang H, Su M, Wang X, Gu J, Cai X (2016) Particle sizing with improved genetic algorithm by ultrasound attenuation spectroscopy. Powder Technol 304:20. https://doi.org/10.1016/j.powtec. 2016.08.027 22. McClements DJ (2009) Ultrasonic characterization of foods and drinks: principles, methods, and applications. Crit Rev Food Sci Nutr 37:1. https://doi.org/10.1080/10408399709527766 23. Holmes MJ, Povey MJW (2017) Ultrasonic particle sizing in emulsions. Wiley, Hoboken, NJ. https://doi.org/10.1002/9781118964156.ch2 24. Szakal C, Roberts SM, Westerhoff P, Bartholomaeus A, Buck N, Illuminato I, Canady R, Rogers M (2014) Measurement of nanomaterials in foods: integrative consideration of challenges and future prospects. ACS Nano 8(4):3128. https://doi.org/10.1021/nn501108g 25. Krishna VD, Wu K, Su D, Cheeran MCJ, Wang J-P, Perez A (2018) Nanotechnology: review of concepts and potential application of sensing platforms in food safety. Food Microbiol 75:47. https://doi.org/10.1016/j.fm.2018.01.025 26. Rogers MA (2016) Naturally occurring nanoparticles in food. Curr Opin Food Sci 7:14. https:// doi.org/10.1016/j.cofs.2015.08.005 27. Bonacucina G, Perinelli DR, Cespi M, Casettari L, Cossi R, Blasi P, Palmieri GF (2016) Acoustic spectroscopy: a powerful analytical method for the pharmaceutical field? Int J Pharm 503(1–2):174. https://doi.org/10.1016/j.ijpharm.2016.03.009 28. Dahlenborg H, Millqvist-Fureby A, Bergenstahl B (2015) Effect of particle size in chocolate shell on oil migration and fat bloom development. J Food Eng 146(0):172. https://doi.org/10. 1016/j.jfoodeng.2014.09.008 29. Renshaw RC, Dimitrakis GA, Robinson JP, Kingman SW (2019) The relationship of dielectric response and water activity in food. J Food Eng 244:80. https://doi.org/10.1016/j.jfoodeng. 2018.08.037 30. Rohm H, Böhme B, Skorka J (2018) The impact of grinding intensity on particle properties and rheology of dark chocolate. LWT 92:564. https://doi.org/10.1016/j.lwt.2018.03.006 31. Jena S, Das H (2006) Modeling of particle size distribution of sonicated coconut milk emulsion: effect of emulsifiers and sonication time. Food Res Int 39(5):606. https://doi.org/10.1016/j. foodres.2005.12.005

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32. Wei ZH, Huang QR (2019) Edible Pickering emulsions stabilized by ovotransferrin-gum Arabic particles. Food Hydrocoll 89:590. https://doi.org/10.1016/j.foodhyd.2018.11.037 33. Berton-Carabin CC, Schroen K (2015) Pickering emulsions for food applications: background, trends, and challenges. Annu Rev Food Sci Technol 6:263. https://doi.org/10.1146/annurevfood-081114-110822 34. Richardson AM, Tyuftin AA, Kilcawley KN, Gallagher E, O’Sullivan MG, Kerry JP (2018) The impact of sugar particle size manipulation on the physical and sensory properties of chocolate brownies. LWT 95:51. https://doi.org/10.1016/j.lwt.2018.04.038 35. Pulatsu ET, Sahin S, Sumnu G (2018) Characterization of different double-emulsion formulations based on food-grade emulsifiers and stabilizers. J Dispers Sci Technol 39(7): 996. https://doi.org/10.1080/01932691.2017.1379021 36. Reimers T, Thies J, Dietrich S, Quodbach J, Pein-Hackelbusch M (2019) Evaluation of in-line particle measurement with an SFT-probe as monitoring tool for process automation using a new time-based buffer approach. Eur J Pharm Sci 128:162. https://doi.org/10.1016/j.ejps.2018. 11.026 37. Tay JYS, Liew CV, Heng PWS (2017) Powder flow testing: judicious choice of test methods. AAPS PharmSciTech 18(5):1843. https://doi.org/10.1208/s12249-016-0655-3 38. Crisp JL, Dann SE, Edgar M, Blatchford CG (2010) The effect of particle size on the dehydration/rehydration behaviour of lactose. Int J Pharm 391(1-2):38. https://doi.org/10.1016/j. ijpharm.2010.02.012 39. Carpin M, Bertelsen H, Dalberg A, Bech JK, Risbo J, Schuck P, Jeantet R (2017) How does particle size influence caking in lactose powder? J Food Eng 209:61. https://doi.org/10.1016/j. jfoodeng.2017.04.006 40. Katiyar V, Ghosh T (2021) Nanotechnology in edible food packaging. Springer, Singapore. https://doi.org/10.1007/978-981-33-6169-0

4

Rheological Properties

Rheology is the branch of physics in which we study the way in which materials deform or flow in response to applied forces or stresses. The material properties that govern the specific way in which these deformation or flow behaviors occur are called rheological properties. The Greek philosopher and scholar, Heraclitus (550–480 BC) once said “πάντα ῥεῖ ” (“everything flows”). In the context of physics, “flow” can be defined as continuous deformation over time, and it can be said that all materials can flow. Therefore, the ability to flow is not only possessed by gases and liquids, but also by solids to a varying degree. Indeed, we all know examples of solids which are capable of continuous deformation over time (flow), like asphalt on a road surface after long-term usage. It is also evident that temperature can have a strong influence on the ability of materials to flow. For example, the asphalt road surface will deform at a faster rate when carrying traffic during time periods of elevated temperatures. In this chapter, the rheology of both solids and liquids will be studied, but in the context of being at opposite ends of a continuum (continuous spectrum) of rheological behavior exhibited by matter in different forms. Because liquids and gases are the forms in which matter can flow most easily, the flow of liquids and gases will be covered in greater depth. After an introduction of ideal rheological behavior in solids and liquids, non-ideal behavior is treated, and concepts like viscoelasticity will be discussed. At the end of the chapter, texture measurement and quantification in food will be discussed as an important practical application of rheology of solids to food technology.

4.1

Elastic Properties

Solid bodies can deform elastically as a result of stress. With a uniaxial load, the body shows a change in length, when load is applied from all directions, such as under pressure, a body will show a change in volume. In the case of a shear load, a change in shape occurs. Figure 4.1 shows the types of load that can be applied to a # The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. O. Figura, A. A. Teixeira, Food Physics, https://doi.org/10.1007/978-3-031-27398-8_4

145

146

4 Rheological Properties

Fig. 4.1 Load types of technical mechanics: I tension, II compression, III shear, IV bending, and V torsion

Fn Fz

Ft Fy

Fy

Fx

Ft Fz

Fn I

J

II

III

Fig. 4.2 Deformations in response to load types: I uniaxial, II pressure, and III shear

body. If the deformation is reversible, i.e. the body returns to its original dimensions and to its thermodynamic initial state after the load has been removed, this is referred to as elastic behavior. In uniaxial loading, the force acts perpendicular to the loaded surface and is therefore called normal force or tensile force when being stretched or compressive force when being compressed (I and II in Fig. 4.2). In shearing, the force acts parallel to the surface, it is also called tangential force (III in Fig. 4.2). Figure 4.1 in addition to these types of stress, bending and torsion are known in technical mechanics. Bending can be understood as a combination of uniaxial tensile and compressive loads, torsion as a combination of load types I, II, and III.

4.1.1

Stress, Pressure, Uniaxial Tension, Young’s Modulus

A one-dimensional tensile stress is caused by a one-dimensional force on a surface directed away from the body. The quotient of force and the size of the surface area on which it acts is called mechanical stress σ. The unit of stress is 1 N∙m-2 = 1 Pa. The reaction of a body to a tensile or compressive stress is a change in length. The ratio of observed change in length Δl to the original length l is called strain ε. With an ideal elastic body, the strain increases linearly with the tension. The stress–strain curve of

4.1 Elastic Properties

147

Fig. 4.3 The stress F/A causes a decrease in length Δl of the sample. The materials modulus of elasticity E rules the extent of deformation

F

A

F

A

Δl l

Δl l

an ideally elastic body is thus a straight line that runs through the coordinate origin. When the tension is reduced, the ideally elastic body releases its elastic energy without delay and completely. The stress–strain curve of an ideally elastic body thus applies to both increasing and decreasing load or stretching. The linear relationship between stress and strain is called Hooke‘s law (Fig. 4.2). σ = E
ε:

ð4:1Þ

σ=

dF : dA

ð4:2Þ

ε=

Δl : l

ð4:3Þ

F A α ε l Δl E

force in N area in m2 stress in N
m-2 strain length in m change in length in m Young’s modulus in N
m-2

The proportionality constant in Hooke’s (Eq. 4.1) law is called Young s modulus, also known as modulus of elasticity. The modulus of elasticity corresponds mathematically to the slope of the linear curve on the stress–strain graph (ref. Fig. 4.4) E=

dσ = tan α: dε

ð4:4Þ

When a stress–strain diagram is constructed to illustrate shear loading, the slope of the straight line or modulus of elasticity is called Shear modulus, G. In the case of bulk loading from all directions (pressure), the modulus of elasticity is called bulk modulus, K. If the stress is directed away from the body, it is called tension, and when directed toward the body, it is called compression (unit N∙m-2 or Pa) or often simply pressure. The change in length Δl that the body shows in response to a compressive stress now has a negative value (decrease in length) (Fig. 4.3).

148

4 Rheological Properties

Fig. 4.4 Stress–strain diagram of a material with a clear yield strength Re

Re H Re L

V/

N mm 2

E

D

dV dH

H /% A Hookean body is an ideal model body because its stress–strain behavior follows Hooke’s law. However, most solid bodies show deviations from ideal elastic behavior, e.g. non-linear behavior or a non-completely reversible behavior. Figure 4.4 shows the stress–strain diagram of a material that behaves like a Hookean body at low tensile stresses, but shows non-linear and non-reversible behavior above a certain load ReL. A tensile stress above ReL leads to a non-reversible, i.e. permanent deformation of the body. Such a deformation is called a plastic deformation. As the tensile load continues to rise, a point is reached beyond which the body can no longer respond to any further stress and is ruptured. This is called the rupture point. The curve section between ReL and the rupture point is the area of plastic deformation. ReH is known as yield strength or yield point. It is the point at which the stress does not rise or begins to fall with increasing strain. The point of maximum tension is also called tensile strength of the body. Occasionally, ReH and ReL are also referred to as the upper and lower yield strength. Definition An ideally elastic body, a so-called Hooke body, shows a linear stress–strain curve passing through the coordinate origin. The strain is completely reversible, i.e. it decreases along the linear curve without delay and completely when the load is removed.

Figure 4.5 shows the stress–strain diagram of a material with a less clear transition from elastic to plastic behavior. Such a smooth curve makes it difficult to specify a precise yield stress. A rule of convention has been adopted for this purpose which specifies the yield stress as the stress at which a straight line originating from a designated offset of strain and drawn parallel to the original straight-line portion of the stress–strain diagram, intersects the actual stress–strain curve. This is illustrated in Fig. 4.5 for a designated offset strain of 0.2%. Analogously, the stress at 0.1% offset can be taken as the technical limit of elasticity. In this material example, the rupture stress, B, is lower than the measured maximum tensile stress. Such small strain response is typical of metals such as aluminum, which are stiff and exhibit very small strain in response to applied stresses. In

4.1 Elastic Properties

149

Fig. 4.5 Stress–strain diagram of a solid with hardto-detect yield strength

Rm Rp0.2

B

V/

N mm 2 E

D

dV dH

H /%

0.2%

Table 4.1 Terms for interpretation of σ - ε-diagrams Elastic range

Criterion σ - ε-function is a line through origin Strain and energy storage reversible

Non-elastic range

Strain and energy storage not or not completely reversible

Yield strength

Maximum stress in σ - ε-function

0.2% yield strength 0.1%-yield strength Rupture point

Permanent strain reaches 0.2%

Young’s modulus

Slope of the σ - ε-function in the linear range

dσ dε dσ dε dσ dε dσ dε dσ dε

= constant ≠0 ≠ constant ≠0 =0

Permanent strain reaches 0.1%

Technical elastic limit

Stress or strain at which rupture occurs

Rupture strain, rupture stress Modulus of elasticity, E-modulus

contrast, food materials are relatively soft, and a much larger value of strain would need to be chosen as the designated offset strain. Table 4.1 compiles the terms. The stress–strain behavior of a material can also be represented as a function of “force over change in length (deformation)” and shown on a force-deformation diagram. σ = E
ε:

ð4:5Þ

F Δl =E
: A l

ð4:6Þ

F=E


A
Δl: l

F = D
Δl:

ð4:7Þ ð4:8Þ

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4 Rheological Properties

Table 4.2 Magnitude of mechanical moduli Material Silicon elastomer Gummy bears Carrot Almond Wood (beech, parallel) Ice (-4 °C) Quartz glass Stainless steel X5CrNi18-10 Diamond Cherry pit

G-modulus in MPa 0.003

– 3.7 33 77 500

E-modulus in MPa 0.01 0.07 7 20 15 9.9 76 200 1100 190

Poisson’s ratio 0.5

– 0.33 0.17 0.3 0.1

References [5] [6] [6] [6] [7] [7] [7] [5] [5] [6]

stress in N
m-2 Young’s modulus in Pa strain force in N area in m2 change in length in m length in m Hooke’s constant in N
m-1

α E ε F A Δl l D

Hooke´s constant is proportional to Young´s modulus but depends on the geometric dimensions (l and A) of the sample: D=E


A dσ A =
l dε l

ð4:9Þ

In the case of severe deformations, it must be taken into account that the surface A of the sample is no longer constant. This means that the quotient of force and area (the stress) is also not constant during a load. The mechanical examination of solid foods or “soft solids” often leads to such strong deformations, which lead to a change in the geometry of the sample [1]. In Chap. 5 (texture), we will come back to this. The elasticity of materials is routinely determined in materials science by means of compression tests or tensile tests. Table 4.1 shows some terms used to describe stress–strain curves. The magnitude of some E-modulus can be taken from Table 4.2. Remark In 1675, Robert Hooke announced some inventions in the postscript of his work A description of helioscopes, and some other instruments, Hooke’s Law was initially published in this encrypted form: ceiiinosssttuv [2].

4.1 Elastic Properties

151

In the case of natural products and foodstuffs, it should be noted that the mechanical properties may depend on the direction of the load. Such material properties are called anisotropic properties. Material properties that are the same in all spatial directions are called isotropic properties. Example The modulus of elasticity of wood is anisotropic. Parallel to the fiber, the modulus of elasticity is much higher than perpendicular to the fiber. Similarly, fibrous foods (plant stems or leaves, meat and fish) show anisotropic material properties.

4.1.2

Three-Dimensional Stress, Bulk Compression

By applying a compressive pressure from three directions (x,y,z), the volume of a solid, liquid, or gaseous material is reduced (Fig. 4.6). Hereby the system absorbs energy, the so-called expansion work. An ideal elastic material shows proportionality between load and deformation, i.e. between volume change and pressure and returns to its initial state after the load has been removed. The volume reduction of a body is called bulk compression. The proportionality variable pressure dp and relative deformation is called compressibility κ. The reciprocal compressibility is the bulk modulus of elasticity K: -

1 dV = κ
dp =
dp V K

ð4:10Þ

κ= -

1 dV V dp

ð4:11Þ

K= -

V
dp dV

ð4:12Þ

Fig. 4.6 Compression of a cube by three-dimensional compressive stress

Fz

Fy

Fy

Fx Fz

152

4 Rheological Properties

Fig. 4.7 Compression of a solid by isotropic pressure. Left: isotropic compressibility. Right: Example of a change in shape due to anisotropic compressibility

volume in m3 pressure in Pa bulk modulus in Pa compressibility in Pa-1

V p0 K κ

Similarly, a body shows a volume increase (volume dilation) when applying all-sided tensile stress. This occurs, e.g., during vacuum applications or during relaxation (decompression) after a pressure treatment. At the molecular level, the elastic volume change consists of a change in the molecular distances or lattice spacings. Real materials show deviations from the ideal elastic compression or dilatation behavior. These deviations can consist of non-linear behavior or non-reversible behavior. For example, materials can show plastic deformation or fracture as a result of three-dimensional stress. Solid materials with anisotropic mechanical properties have K-values which are dependent on the direction of stress. To express this mathematically, the bulk module K is written as a tensor. The tensor has 21 linear-independent components. At higher symmetry, as in the case of orthotropic or transverse isotropic material properties, it may have less, e.g. nine linear-independent components Kij [3]. K=

K xx

K xy

K xz

K yx K zx

K yy K zy

K yz K zz

ð4:13Þ

In the special case of isotropic material properties, all components Kij have the same value and the tensor notation can be omitted. In the case of anisotropic compressibility, the dimensions of a body change differently in different spatial directions under the same pressure on all sides. The result is a change in shape of the body (Fig. 4.7). In the case of isotropic compressibility, the shape of the solid does not change as a result of a pressure application. During high-pressure treatment of food [4] significant volume changes can occur.

4.1.3

Shear, Shear Modulus

If a tangential force is applied to the surface of a body, a deformation of the body occurs, which is called shear strain. In Fig. 4.8, this type of deformation is illustrated

4.1 Elastic Properties

153

Fig. 4.8 Shear deformation of a body by angle γ under influence of the tangential force Ft

A

J

x

Ft

y

on a cube-shaped body. The extent of the deformation is expressed by the shear angle γ. The quotient of tangential force and area is called shear stress (τ). It has like a perpendicular stress the physical unit N∙m-2 or Pa. τ=

Ft : A

ð4:14Þ

In the case of an ideally elastic body, there is a linear relationship between the shear stress τ and the tangent of the shear angle. In the ideal elastic body, the shear is completely reversible, i.e. when the shear stress is removed, the body returns to its geometric and thermodynamic initial state. The proportionality quantity of this law of elasticity (Eq. 4.15) is called shear modulus G. τ = G
tan γ:

ð4:15Þ

For small angles, the approximation applies tan γ = γ

ð4:16Þ

τ=G
γ

ð4:17Þ

so

Ft A τ G γ

tangential force in N area in m2 shear stress in N
m-2 shear modulus in N
m-2 angle of deformation in rad

For materials with anisotropic properties, the G-modulus can be written as a tensor analogous to the E-modulus and K-modulus. For isotropic materials, the shear deformation is independent of the spatial direction. Real materials show deviations from the ideal elastic shear behavior. These deviations can consist of non-linear behavior or non-reversible behavior. For example, materials can show plastic deformation or breakage in response to an applied shear stress. The shear modulus is also known as modulus of rigidity.

154

4.1.4

4 Rheological Properties

Transverse Strain, Poisson’s Ratio

When a material shows an elastic change in length Δl simultaneously a material specific change in thickness Δd can be observed. The relative change in thickness Δd d is called transverse strain. Transverse strain is proportional to longitudinal strain. The proportionality factor is called Poisson´s ratio μ. Poisson’s ratio is a dimensionless quantity that indicates the ratio of relative transverse strain to relative longitudinal elongation. μ= -

μ= -

εq : ε

εq =ε

ð4:18Þ Δd d Δl l

d Δd l Δl ε εq μ

:

ð4:19Þ initial thickness in m change in thickness in m-1 initial length in m change in length in m strain, longitudinal strain transverse strain Poisson’s ratio

The elastic and compressible properties of a material are related to each other via that Poisson’s ratio μ. For the relation between μ and the elastic modulus (G-, E-, and K-modulus) it is [5]. E 2ð 1 þ μ Þ

ð4:20Þ

E : 3ð1 - 2μÞ

ð4:21Þ

G= and K=

For positive stress Poisson’s ratio is between 0 ≤ μ ≤ 0.5. Let us look at the two extreme values: Solids in which the elastic properties dominate over compressibility (case A) have a high Poisson’s ratio up to μ = 0.5. Solids, on the other hand, in which the elastic properties are subordinate to compressibility (case B), have a very small value for Poisson’s ratio (see table below). Thus, materials can be characterized on the basis of Poisson’s ratio.

Case A Case B

K High Small

E Small High

G Small High

μ μ = 0.5 μ=0

4.2 Rheological Models

155

In addition, the Poisson number of a material can be used to calculate missing moduli [8]. 3K ð1 - 2μÞ E 3E
K = : = 2ð1 þ μÞ 9 K - E 2ð1 þ μÞ

ð4:22Þ

2ð1 þ μÞ E E
G : = =G 3ð1 - 2μÞ 9G - 3E 3ð1 þ 2μÞ

ð4:23Þ

9G
K = 2Gð1 þ μÞ = 3K ð1- 2μÞ: 3K þ G

ð4:24Þ

E 3K - 2G E - 2G 1 - 3K : = = 2 2G 2ð3K þ GÞ

ð4:25Þ

G=

K=

E=

μ=

For cases A and B, this results in μ=0 E 3 G= = K 2 2 E E K= = 3 3ð1 - 2μÞ

μ = 0.5

E=3
K

E=3
G

4.2

E E = 3 2ð 1 þ μÞ E K= =1 3ð1 - 2μÞ

G=

Rheological Models

For a simpler mathematical description of the rheological behavior of materials, idealized model materials are often used. We have already used the ideally elastic material (Hooke´s body) to get to know the terms stress, stain, and shear. There is also a model for the ideal rigid, totally incompressible body, a model body for the ideal viscous material, a model body for the ideal plastic material, and a model body for the ideal, frictionless flowing fluid. Table 4.3 lists those five ideal model materials. Table 4.3 Model materials with ideal mechanical properties

Model body Pascal Newton St. Venant Hooke Euclid

Rheological properties Ideal frictionless flowing Ideal viscous flowing Ideal plastic Ideal elastic Ideal rigid (incompressible)

156

4 Rheological Properties

W

W

II

W

W

W

W

I

W

W

III

IV

Fig. 4.9 Element symbols for the basic model materials: I Hooke element, II fraction element, III Newton element, and IV St. Venant element

Less simple and non-ideal behavior of materials can be modeled by interconnecting these ideal materials. Similar to electrical engineering, elements with switching symbols are used for this purpose, which are connected in series or in parallel to each other. Figure 4.9 shows the symbols of basic elements. Figure 4.9 shows examples of some model materials composed by ideal elements. For example, a Maxwell body consists of a Hooke element and a Newton element connected in series. The Kelvin body consists of a Hooke element and a Newton element connected in parallel. A Kelvin body can be used to model the behavior of a viscoelastic material (Table 4.4). Table 4.4 Examples of composed models Model Hook

Symbol

With fracture element added

Newton

(continued)

4.3 Viscous Behavior, Flow

157

Table 4.4 (continued) Model St. Venant

Symbol

With fracture element added

Maxwell

Prandtl

Kelvin

4.3

Viscous Behavior, Flow

When a sample of material is subjected to a shear stress there is a shear deformation which we can measure by the shear angle γ. When under a constant shear stress the shear angle grows continuously—instead of taking a fixed value—this is called viscous behavior or flow. While ideally elastic materials show a temporally constant shear angle under a given shear stress, ideally viscous bodies react with a constantly increasing shear

158

4 Rheological Properties

Fig. 4.10 Two-platen model: Flow of a material between a stationary and a moving plate

A

J

x

Ft

y

angle, i.e. with a constant shear rate γ_ . Figure 4.10 shows the flow of a substance between a stationary plate and a plate moved at a constant speed. While the elastic behavior is described by Hooke’s law τ = G
γ, the ideal viscous behavior is a proportionality between shear stress and shear rate τ = η
γ_ . The proportionality constant is called dynamic viscosity η. Later we will get to know further terms such as Newtonian viscosity, Bingham viscosity, relative viscosity, apparent viscosity, etc. (Sects. 4.3.9–4.3.12). With shear stress τ=

Ft A

ð4:26Þ

it is ideal elastic behavior: τ=G
γ

ð4:27Þ

τ = η
γ_

ð4:28Þ

and ideal viscous behavior:

Ft A τ G γ γ_ η

tangential force in N area in m2 shear stress in N
m-2 shear modulus in N
m-2 angle of deformation in rad shear rate in s-1 viscosity in N
m-2
s

Solids generally exhibit elastic behavior due to the strong interaction forces between their atoms or molecules. In contrast, fluids such as liquids and gases in which the intermolecular interactions are much weaker than in solids, show viscous behavior. However, some materials exhibit a transitional behavior in which they may gradually change from solid to liquid behavior. In this way, solids can also be made to flow if the shear stress exceeds the elastic limit (see Sect. 4.1.1). Conversely, flowable systems can also show elastic properties (see Sect. 4.4). In the following, simplest case, ideal viscous flow is treated first. Initially, we will treat only the vortex-free so-called laminar flow.

4.3 Viscous Behavior, Flow

4.3.1

159

Shear Rate

When a shear stress τ is applied to a viscous body, the body shows a constantly increasing shear angle γ over time, i.e. a constant shear rate γ_ . The shear rate is defined as: γ_ =

dγ dt

and thus has the SI unit s-1. In this case, the shear rate is calculated as the quotient of velocity of the moving plate in Fig. 4.10 over the distance between the moving and resting plates. Shear rate can also be calculated with reference to angular velocities and rotational movement as shown in Fig. 4.11. If we look at the angle γ in Fig. 4.11, we can write tan γ =

ds r

for small angles, the approximation applies tan γ = γ i.e., γ=

ds r

ð4:29Þ

so dγ ds = dt dt
r

ð4:30Þ

Fig. 4.11 Angular velocity and shear angle

ds r J

160

4 Rheological Properties

with the circumferential velocity v in Fig. 4.11 ds dt

ð4:31Þ

v γ_ = : r

ð4:32Þ

v= can be written for the shear rate:

The shear rate γ_ therefore is the quotient of the circumferential velocity and the distance between the circumference and the center of the rotational movement. At the same time, one can consider it as the angular velocity of a rotational movement. With a look on Fig. 4.10, we can also write tan γ =

dx y

ð4:33Þ

then with approximation tanγ = γ it is γ_ =

dx v = y
dt y

ð4:34Þ

That is, here, too, the shear rate as the quotient results from the velocity of the moving plate in Fig. 4.10 and the distance between the moving and resting plates. For this reason, the shear rate is sometimes referred to as the velocity gradient in the two-platen model. The lateral shear velocity gradient is the quotient of the change in velocity vx and the path perpendicular to it, in Fig. 4.10 that means γ_ =

dvx Δvx v2 - v1 v2 - 0 v = ≈ = = dy Δy y2 - y1 y2 - 0 y

ð4:35Þ

If we consider the material between the plates in Fig. 4.10 as a point on the circumference located a large distance from the center of the circle in the rotational movement illustrated in Fig. 4.11, both approaches come together. Because of the different ways of looking at movement, there are a number of synonymous terms for the shear rate. Here is a summary of these terms. Definition Shear rate: synonymous terms used shear rate rate of shear shear velocity, shear speed shear velocity gradient, velocity gradient lateral angular velocity gradient, angular velocity

4.3 Viscous Behavior, Flow

161

Because the shear rate has the SI unit s-1 terms ending with velocity or speed are confusing and should be avoided. However, in many fields of engineering, the term “cycles per second or minutes, etc.” often is used. Different names are also used for the shear angle and the shear stress or force. Here are some of them. Definition Shear angle (in rad): Angle of deformation that a material shows under shear stress. Synonym: angular deformation.

Definition Shear force (in N): The tangential force acting on a surface. Synonymous terms: tangential force, transverse force.

Definition Shear stress τ in N
m‐2. The quotient of shear force and area. Synonyms used: tangential stress. Shear rates occur in nature and technology in a wide range of orders of magnitude. Examples of typical shear rate orders of magnitude are compiled in Table 4.5. Some examples for calculating approximate values for shear rates in the field of food technology are listed below. Example Shear rate when spreading or painting: With value from Table 4.5, we get Spreading speed v = 0.5 m
s-1 v=1m
s

-1

v = 0.1 m
s-1

Thickness y = 2 mm

Shear rate γ_ =

y = 0.2 mm

γ_ =

y = 1 mm

γ_ =

dv dy dv dy dv dy

= 250 s - 1 = 5000 s - 1 = 100 s - 1

162

4 Rheological Properties

Table 4.5 Magnitudes and examples of shear rates [9, 10] Process Sedimentation of small solid particles Sedimentation of larger solid particles Flow caused by surface tension Flow caused by gravity Extrusion processing

Shear rate γ_ =s - 1 10-6–10-4 10-4–10-1

Example Fruit juice, medical suspensions

Roller drum processing Pouring Dipping

101–102 101–102 101–102

Chewing/swallowing Flow-through pipe Stirring/mixing Painting, brushing Surface smearing Spraying (aerosols) Wet milling Roller (pressing) Homogenization (highpressure nozzle) Lubrication Calendering

101–102 100–103 101–103 102–104 102–104 103–106 103–105 104–106 105–106

Spice particles in dressings, suspensions, paint pigments Glazes, paintings, coatings, print colors Dripping off coated materials Snacks, cereals, tooth paste, noodles, pet food, polymers Rolling of dough Food, cosmetics pouring out of containers Dipping of food in brine, of confectionary in chocolate Food, feed, pharmaceutical products Pumping of liquid food Stirring of liquid food Paint, lip gloss, nail polish, spreads Skin creams, salves, bath gels, hand lotion Spray drying, spray painting, fuel injection Food suspensions, mustard Printing of newspapers Milk homogenization

103–107 103–107

Machine mount Polymer film coating

10-2–10-1 10-1–101 100–103

Example Shear rate when rollering: With value from Table 4.5, we get Roller Radius 50 cm Rotational frequency 20 RPM Radius 20 cm Rotational frequency 100 RPM

Thickness y = 100 μm

y = 25 μm

Circumferential velocity v = ω
r 20
2π v=
0:5 m 60 s v = 1 m
s-1 100
2π
0:25 m 60 s -1 v = 2:6 m
s v=

Shear rate γ_ =

dv dy

=

1 m
s - 1 10 - 4

γ_ =

dv dy

=

2:6 m
s - 1 25
10 - 6

= 10000 s - 1

≈ 105 s - 1

4.3 Viscous Behavior, Flow

163

Example Shear rate for liquid flow in a pipe: During laminar flow of an incompressible, ideal viscous fluid in pipe, the shear stress is given by τW = r
Δp 2l : and the shear rate γ_ W =

4
V_ : π
r3

In tubular flow in a tube with a diameter of 10 cm and a flow rate of 10 l/ min, the shear rate at the wall of the tube is γ_ W =

4
V_ 4
10
10 - 3 m3 4
10 - 3 4
10 - 3 = = = 3 3 3 6 π
r 6 s
π
5
10 6 s
π
53 60 s
π
ð0:05 mÞ

= 1:7 s - 1 :

Example Shear rate during sedimentation: According to Stokes law, for Reynolds number < 1 and without interactions between particles and fluid, the terminal velocity of a particle falling through a fluid can be given by v = For sand particles with

d2
g 18
η

ρK - ρfl .

d = 0:5 μm: in water, there is a difference between fluid and particle density of ρK - ρfl = 1500 kg
m - 3 - 1000 kg
m - 3 = 500 kg
m - 3 : With the dynamic viscosity of water of about η = 1 mPa
s there will be a terminal velocity of 2

v=

5
10 - 7 m
9:81 m
s - 2
500 kg
m - 3 = 6:8
10 - 8 m
s - 1 18
10 - 3 N
m - 2
s

so the shear rate is (continued)

164

4 Rheological Properties

γ_ =

4.3.2

dv 6:8
10 - 8 m ffi 0:1 s - 1 ffi d_γ 5
10 - 7 s
m

Newtonian Flow Behavior

When there is a linear dependence between the shear stress and the shear rate of a material, we are observing ideal viscous behavior or Newtonian flow behavior. Fluids that behave in this way are called Newtonian fluids. If the shear stress is applied above the shear rate in a diagram, the so-called flow curve is obtained (Fig. 4.12 top diagram). In Newtonian fluids, the flow curve is a straight line passing through the origin. The Newtonian flow law is τ = η
γ_

ð4:36Þ

The slope of the flow curve is called dynamic viscosity η η=

Fig. 4.12 Flow curve (top) and viscosity curve (bottom) of a Newtonian fluid. With a Newtonian fluid, the slope, i.e. the viscosity, is the same at all shear rates

dτ Δτ bzw:η ≈ d_γ Δ_γ

W 'W

'J•

J•

K

J•

4.3 Viscous Behavior, Flow Table 4.6 Newtonian fluids, examples

165 η20°C/mPa
s 0.0148 0.0177 1.002 1.20 2 84 1490

Material Carbon dioxide Nitrogen Water Ethanol Milk Olive oil Glycerol

The SI unit of dynamic viscosity η is N
m-2
s-1 or Pa
s. In order to determine the dynamic viscosity of a fluid, the ratio of shear stress and dτ shear rate must be measured and quotient η = d_ γ formed. For Newtonian fluids, this quotient is the same for all shear rates, i.e. it is sufficient to determine the quotient at a single shear rate. With the help of this one point you draw a line from the origin in a dτ Δτ τ versus γ_ diagram. The application of the quotient η = d_ γ ≈ Δ_γ over the shear rate is called the viscosity function of the fluid. Table 4.6 lists some fluids, which in good approximation behave like Newtonian fluids. A distinction must be made between dynamic viscosity η and kinematic viscosity ν, which can be calculated from dynamic viscosity using density (see Chap. 2). ν=

η : ρ

ð4:37Þ dynamic viscosity in Pa∙s-1 kinematic viscosity in m2
s-1 density of fluid in kg∙m-3 fluidity in Pa-1∙s-1

η ν ρ ϕ

The reciprocal quantity of dynamic viscosity is called fluidity. ϕ=

1 η

ð4:38Þ

Example The dynamic viscosity η of water at room temperature is approximately 1 mPa
s. The density ρ of water at room temperature is about 1000 kg∙m-3. Thus, the kinematic viscosity of water at room temperature is (continued)

166

4 Rheological Properties

ν=

N
m-2
s η 1 mPa
s 10 - 3 Pa
s = = 10 - 6 = ‐3 ‐3 ρ 1000 kg
m kg
m‐3 1000 kg
m

= 10 - 6

kg
m
s - 2
m - 2
s - 1 = 10 - 6 m2
s - 1 : kg
m‐3

The conversion of the formerly used unit Poise (P) for the dynamic viscosity is 10 - 5 N
s = 10 - 1 Pa
s 10 - 4 m2 Pa
s = 1 mPa
s

1 P = 1 dyn
s
cm - 2 = 1 cP = 10 - 2 P = 10 - 3 1 cP = 1 mPa
s

Attention The unit Poise (P) should not be confused with the unit Poiseuille (Pl). 1 Pa
s = 1 Pl. The conversions of the formerly used unit Stokes (St) for the kinematic viscosity is 1 St = 1cm2
s - 1 = 10 - 4 m2
s - 1 1 cSt = 10 - 6 m2
s - 1 In Newtonian fluids, the flow curve is a linear function starting in the origin, i.e. the slope which is the dynamic viscosity is the same for all shear rates. This simple case does not apply to most biological materials and foods. Here, the flow curve is not a linear function starting in the origin, i.e. the viscosity depends on the shear rate or the applied shear stress. This is called non-Newtonian behavior. Definition Laminar flow: In the two-platen model, it is assumed that all liquid layers move laterally and glide past each other like plates. This flow without transverse movement, without flow vortices is called laminar flow. A fluid flow that involves transverse movements, i.e. vortices, is called turbulent flow.

4.3.3

Non-Newtonian Flow Behavior

A typical characteristic of a non-Newtonian fluid is that the flow curve is not a straight line through the origin but depends upon shear rate. With such fluids, the

4.3 Viscous Behavior, Flow

167

Fig. 4.13 Flow functions: Newtonian (2), pseudoplastic (1), dilatant (3), Bingham Plastic (5), and mixed behavior (4)

Fig. 4.14 Viscosity functions relating to figure above, schematic: Newtonian (2), pseudoplastic (1), dilatant (3), Bingham plastic (5), and mixed behavior (pseudoplastic with yield stress) (4)

viscosity depends on the shear rate. Typical deviations from the linear course (curve 2) are shown in Fig. 4.13. A pseudoplastic fluid exhibits a downward or convex shape of the flow curve (curve 1) in which the slope decreases with increasing shear rate. A dilatant fluid is reversed and exhibits an upward or concave shape of the flow curve (curve 3). Here, the slope increases with increasing shear rate. Curve 5 is linear, but not through the coordinate origin. In order for a measurable shear rate to occur here, a minimum initial shear stress must be applied. This minimum shear stress is called the yield stress of the fluid. Fluids exhibiting this type of behavior are known as Bingham plastic fluids. In addition, there are numerous more complicated curves that can be thought of as superimpositions or combinations of these basic types. For example, curve 4 can be interpreted as a mixed flow behavior fluid, exhibiting pseudoplastic behavior, but only after an initial yield stress has been applied (combination of pseudoplastic and Bingham plastic) (Fig. 4.14).

4.3.4

Comparison: Newtonian and Non-Newtonian Fluids

While Newtonian fluids have the same viscosity at different loads (shear rates), the viscosity of non-Newtonian fluids depends on the strength of the load or the load time. If we compare three different fluids at the low shear rate γ_ 1 in Fig. 4.15 so fluid 3 has the highest viscosity. If we compare them at shear rate γ_ 2 , fluid 1 has a higher viscosity than fluid 3, and at the highest shear rate, fluid 1 has greater viscosity than both fluids 2 and 3. The viscosity of fluid 1, on the other hand, is the same at all shear rates. Fluid 1 is a Newtonian fluid, while fluids 2 and 3 are different pseudoplastic

168

4 Rheological Properties

Fig. 4.15 Newtonian (1) fluid and non-Newtonian fluid (2,3) in comparison

K

1 2 3

J&1

J&2

J&

J&3

fluids. This example shows that in the case of non-Newtonian fluids, the specification of a single viscosity alone is not sufficient to describe the flow behavior. The viscosity must be specified at a certain shear rate. Advantageous is the indication of the entire viscosity function instead of the single value for η. Attention Simple viscosity meters which do not allow to control the shear rate are suitable for Newtonian fluids only not for non-Newtonian fluids.

4.3.5

Pseudoplastic Flow Behavior

Pseudoplastic flow behavior occurs when we observe shear stress increasing at a diminishing rate with increasing shear rate. On a shear stress–shear rate diagram, the flow behavior curve has a convex profile in which the tangential slope is decreasing with increasing shear rate. This means the viscosity is decreasing with increasing shear rate. Sometimes this observation is referred to as “shear-thinning” behavior. Figures 4.13 and 4.14 show the flow function and the viscosity function of a pseudoplastic fluid. It is a characteristic of pseudoplastic fluids that the increasing flow resistance (shear stress) seems to decrease when the fluid is subjected to higher shear rates. This behavior is assumed to be caused by decreasing molecular interactions within the molecular structure of the fluid during flow. In the case of flowing macromolecules, the additional effects of unfolding and reorientation of molecules may be occurring. When these effects are completely reversible, we call this “true” pseudoplasticity. This means that if we first increase shear rate to a given value, and then decrease it back again, the viscosity will be fully restored to the same value as before. An aqueous xanthan gum solution shows a reversible structural degradation during shear, i.e. true pseudoplasticity while, e.g., the protein gel of a set yogurt is irreversibly degraded by shear.

4.3 Viscous Behavior, Flow

169

Definition True pseudoplasticity is when the viscosity decreases reversibly as a result of structural degradation due to shear, i.e. after the load has been removed, the material returns to its initial viscosity. Non-true pseudoplasticity is when the viscosity decreases irreversibly as a result of permanent structural degradation due to shear.

4.3.6

Thixotropic Flow Behavior

Thixotropic flow behavior occurs when the viscosity of a fluid decreases over time while subjected to a constant shear rate. The viscosity therefore does not decrease with increasing shear rate—as with pseudoplasticity—but with increasing time during which the shear load takes place. A distinction is also made here between true thixotropy and non-true thixotropy. If, after removing the shear load, the viscosity gradually rises back to the initial value, it is true thixotropy. If the viscosity remains permanently low, it is an irreversible structural change of the material and one speaks of non-true thixotropy.

4.3.7

Dilatant Flow Behavior

Dilatant flow behavior occurs when we observe shear stress increasing at an increasing shear rate. On a shear stress–shear rate diagram, the flow behavior curve has a concave profile in which the tangential slope is increasing with increasing shear rate. This means the viscosity is also increasing with increasing shear rate. Sometimes this observation is referred to as “shear-thickening” behavior. This flow behavior is seen in highly concentrated suspensions. Figures 4.13 and 4.14 show examples of the flow behavior curve and viscosity curve for a dilatant fluid. It is assumed that under increasing shear rate the liquid between the solid particles is squeezed out and the friction between the particles increases. This increasing friction accounts for the increasing shear stress and increasing viscosity experienced as shear rate increases. Also, the squeezing causes a dilatation (an increase in volume) of the suspension, and therefore the phenomenon is called dilatant flow behavior. When these effects are completely reversible, we call this “true” dilatancy. This means that if we first increase shear rate to a given value, and then decrease it back again, the viscosity will be fully restored to the same value as before. Sometimes the effect of shearthickening is not completely reversible. This can be explained by a permanent damage to the molecular structure in the material under investigation. In that case we have no “true” dilatancy, but an apparent dilatancy (Fig. 4.16).

170

4 Rheological Properties

Fig. 4.16 Time dependency of viscosity, schematic: Thixotropy (1), Rheopexy (3). Newtonian Fluids (2) have a constant viscosity

4.3.8

Rheopectic Flow Behavior

Rheopectic flow behavior is observed when shear stress or viscosity increases over time at a constant shear rate. The reason for the increasing viscosity (or shear stress) is also assumed to stem from the intermolecular interactions causing friction to increase with time at constant shear rate within the molecular structure of the material. When shear rate stops, the original structure is restored, as well as the initial viscosity. This is called “true” rheopexy. If the viscosity is not restored because of irreversible structure damage, the behavior is called apparent rheopexy. When observing what appears to be rheopectic behavior, it is important that we interpret correctly what we observe. For example, when whipping liquid heavy cream into whipped cream, we observe the need for more and more effort to maintain the same whipping speed (viscosity increasing over time at constant shear rate). This would appear to be rheopectic behavior. But, that is not the case here. In whipping cream, we start out with a liquid of relatively low viscosity that becomes transformed into a soft-solid foam as a result of incorporating air into the cream producing a gas-liquid emulsion. We now have a totally new material with different rheological properties, and not a liquid with rheopectic behavior.

4.3.9

Plastic Flow Behavior

If a material only begins to flow when a yield stress is exceeded, it is called plastic flow behavior. When the material is stressed below the yield point, the substance behaves like a solid, i.e. it can be deformed a little elastically. If the shear stress exceeds the yield point, the viscous deformation begins, i.e. the material begins to flow. Think of ketchup, which requires an initial impulse of shear stress induced by a shake of the bottle in order to flow. If the shear load is reduced again and the yield point falls below the yield point, the flow process stops and the body retains its current shape and shape. The shape of the material has been permanently deformed by this process, which is why it is called plastic deformation. The yield stress (yield point) is therefore a minimum shear stress for the flow of the material in question. In the flow curve, the yield point can be read as the yield stress on the ordinate axis beyond which the flow occurs (see Fig. 4.13), in the viscosity curve (see Fig. 4.14) the material shows a transition from initially infinitely

4.3 Viscous Behavior, Flow

171

high viscosity to a finite viscosity. Examples of plastic materials are short pastry, butter, clay, chocolate melt. But also e.g. metals can be plastically deformed. They are materials with a very high yield point. Below the yield point, they behave like elastic solids, above them like highly viscous fluids. The production of aluminum beverage cans is based on the plastic deformation of a flat aluminum sheet. The deformation (the so-called deep drawing) can take place at ambient temperature (cold forming) or at elevated temperature. Like viscosity, the yield point decreases with increasing temperature. Steel forging processes are therefore carried out at temperatures of, e.g., 800 °C, at higher temperatures steel can also be cast. The plasticity or the yield point is caused by structure-forming interactions between the molecular components of the material. If these interactions are strong, the system has a high yield point and high shear forces or high temperatures are required to get the material flowing. With low structure-forming forces, lower shear stresses are sufficient to exceed the yield point. Whipped cream is an example of a material with a low but clearly recognizable yield point: The whipped cream changes its shape only by external action (shear load), after which the shape is preserved. The yield stress is low means it can be exceeded by a relatively low shear stress. Other materials have such a low yield point that they already begin to flow under the influence of gravity, apparently “by themselves.” Examples are unwhipped cream, oil, water. It looks like that materials do not have a yield point. To be precise, we must have to say that the yield point is very low, or imperceptible without sensitive instruments. Definition The yield point is the minimum shear stress that must be exceeded for flowing to take place. Thus, a portion of mayonnaise on a plate retains its shape, i.e. it does not flow. If the yield point is exceeded, e.g. by spreading the mayonnaise with a spoon, then it changes its shape, i.e. it flows. A drop of vegetable oil on a flat surface, on the other hand, seems to spread “by itself.” It does not seem to have a yield point. With a sensitive rheometer, however, the value of the yield point can also be determined for such materials. It is so low that a flow can be caused by gravity alone.

Example Thickness of a chocolate coating. What is the thickness of a coating layer on a food made with liquid milk chocolate with a yield stress of 30 Pa? The weight force FG of a layer with the layer thickness d is equivalent to the tangential force acting at the surface A (shear stress). As long as this shear stress is greater than the yield point of the liquid chocolate, the chocolate flows (continued)

172

4 Rheological Properties

off and the layer thickness decreases, along with the weight of the layer. As soon as the layer thickness decreases to the point where the weight of the remaining coating leaves a shear stress equal to the value of the yield point τ0, the flow stops, i.e. a layer of thickness d0 remains on surface A. FG = m
g = ρ
A
d
g FG = τ
A ρ
A
d 0
g = τ0
A ρ
d 0
g = τ0

τ0 A d0 ρ g FG m

yield point in Pa area in m2 thickness in m density in kg∙m-3 gravitational acceleration in m
s-2 weight force in N mass in kg

From this balance of forces, the thickness of the remaining layer can be roughly calculated: The shear stress when the chocolate layer flows down with the layer thickness d0 is: d0 =

τ0 : ρ
g

At a yield point of τ0 = 30 Pa this results in a layer thickness of d0 =

30 Pa 30 N
m - 2 = 3 2 1520 kg
m
9:81 m
s 1520 kg
m - 3
9:81 m
s - 2

30 kg
m
s - 2
m - 2 1520 kg
m - 3
9:81 m
s - 2 30 d0 =
10 - 3 m = 2
10 - 3 m = 2 mm 1:52
9:81 d0 = 2 mm =

With this simple approach and the assumption of a constant temperature, the layer thickness is proportional to the yield point. Figure 4.17 shows the estimated magnitude of achievable layer thicknesses during brushing, dripping, coating, etc. In practice, a constant temperature is usually not given. Especially in industrial processes, the temperature dependence of the material properties is used to shorten the time of the processes (Table 4.7).

4.3 Viscous Behavior, Flow

173

Fig. 4.17 Thickness d0 of a chocolate coating

d0

A

Table 4.7 Yield points and layer thicknesses, examples of simple estimates τ0/Pa Thickness d0

3 200 μm

30 2 mm

150 1 cm

4.3.10 Overview: Non-Newtonian Flow Behavior Non-Newtonian behavior is when the flow curve of a material is not a straight line through the origin (see Fig. 4.13) or is dependent on time. That is, these are all those cases in which the flow curve is curved or does not pass through the coordinate origin due to the presence of a yield point or if the appearance of the flow curve changes with the duration of the shear load. Figure 4.18 shows the subdivision of the different cases of non-Newtonian behavior and the related terminology. In cases where the viscosity changes with the duration of the stress (i.e., the viscosity is timedependent), a distinction is made between thixotropy and rheopexy. If the viscosity is not dependent on the duration of the stress but on the shear rate, a distinction is made between pseudoplasticity and dilatancy. Materials which show plastic flow are also non-Newtonian materials, even when the viscosity is not dependent on shear stress and time. Flow behavior like this is called Bingham plastic flow, and the material is called a Bingham plastic fluid. Flowable materials with elastic properties, the so-called viscoelastic substances, are not ideal fluids, i.e. non-Newtonian fluids.

non-NEWTONian non time dependant

time dependant

pseudoplastic

dilatant

plastic

visco-elastic

thixotropic

rheopectic

A

B

C

D

E

F

Fig. 4.18 Non-Newtonian flow behavior, schematic of behavior categories and their terminology. Examples are: (a) paint, tomato paste, (b) crystallized honey, (c) whipped egg white, solid butter, lipstick, (d) wheat dough, and (e) ketchup

174

4 Rheological Properties

Examples Pseudoplastic materials: ketchup, tomato paste, starch paste Dilatant materials: starch suspension, crystallized honey Plastic materials: cream cheese, butter, short paste Viscoelastic materials: wheat flour yeast dough, camembert Thixotropic materials: pudding, ketchup Rheopectic materials: gel-forming materials

Bottom Line Terms for the characterization of flow behavior. Newtonian Non-Newtonian Pseudoplastic Dilatant Plastic Thixotropic Rheopectic

the flow curve is a straight line through the origin the flow curve is not a straight line through the origin viscosity decreases with increasing shear rate (shearthinning) viscosity increases with increasing shear rate (shearthickening) the fluid has a yield point viscosity decreases over time at a constant shear rate viscosity increases over time at a constant shear rate

Examples for Misinterpretations Apparent dilatancy or apparent rheopexy: Vortex formation (increase in flow resistance) in the rotational rheometer Air input during the examination in the rotational rheometer Solvent evaporation Chemical reaction Gel formation during the examination Apparent pseudoplasticity or apparent thixotropy: By particle size change in a dispersion during the examination in the rotational rheometer Real materials normally have a more complicated flow behavior which can be explained with various mixtures of the different ideal behaviors described up to here. Sometimes approximations can help, like treating the Bingham plastic fluid as a Newtonian fluid once the initial yield stress has been discounted. For many food science and engineering applications, it is convenient to transform complicated flow behavior curves and viscosity curves into mathematical equations with constant parameters. Then only a few numbers (parameters) are sufficient to represent and to specify the complete curves. For this purpose, model functions have to be derived, and tested and challenged to be sure of their validity.

4.3 Viscous Behavior, Flow

175

4.3.11 Model Functions Rheological model functions are nothing more than the mathematical equations derived to describe the various flow behavior curves on shear stress–shear rate diagrams. First we will focus on model functions for fluids without yield stress. Then, we will address fluids with yield stress. Pseudoplastic fluids are assumed to have an initial viscosity at the origin η ðγ_ = 0Þ = η0 which then decreases with increasing shear rate. The decrease is assumed to be the consequence of (reversible) loss of structure/network in the material (in the German language pseudoplastic is called “strukturviskos” because of this interpretation). When the molecular structure/network in the fluid has reached a steady state (where the intermolecular forces acting to build the structure/network and those acting to break it down by shear are in equilibrium) no further decrease of viscosity is observed. This viscosity is called equilibrium viscosity η ðγ_ = 1Þ = η1 : Figure 4.19 shows a typical profile for a flow curve of this type. The viscosity curve (which is the derivative of the flow behavior curve) can also be constructed and is given in Fig. 4.20. In this type of pseudoplastic material, the viscosity is changing Fig. 4.19 Flow function during structure loss in a pseudoplastic fluid, schematic

Kf

K0

W J& Fig. 4.20 Viscosity function during structure loss in a pseudoplastic fluid, schematic: the viscosity decreases from η0 to η1

K0

K Kf

J&

176

4 Rheological Properties

Table 4.8 Model functions for fluids without yield stress, examples Viscosity function model η = γτ_

Name Newton

Flow function model τ = η
γ_

Ferry

τ=

η0 1þC
τ

Steiner–Steiger– Ory

τ=

De haven

τ=

Ostwald–de Waele (power law)

τ = K ow
γ_ n

ηs = K ow
γ_ n - 1

Sisko

τ = η1
γ_ þ b
γ_ n

ηs = η1 þ b
γ_ n - 1

Ellis I

τ = η0 þ K
γ_ n - 1 γ_

ηs = η0 þ K
γ_ n - 1

Ellis II

τ=


γ_

ηs =

η0 1þC
τ

1 C þA
τ2

γ_

ηs =

1 C þA
τ2

η0 1þC
τn

γ_

ηs =

η0 1þC
τn

η0

A-1


γ_

η0

ηs =

1þτ τ

1þ

1=2

A-1

τ τ1= 2

τ: Shear stress in Pa γ_ : Viscosity in Pa
s γ_ : Shear rate in s-1 C: Constant in Pa-1 η0: Initial viscosity in Pa
s (for C = 0) Newton) C = η1 : Constant in 0 (Pa
s)-1 A: Constant in Pa-1 η0: Initial viscosity in Pa
s C: Constant in Pa-n n: Flow behavior index (for n = 1) Ferry) Kow: Consistency coefficient in Pa
sn n: Flow behavior index (for n = 1) Newton) b: Constant in Pa
sn η1: Equilibrium viscosity in Pa
s η0: Initial viscosity in Pa
s K: Constant in Pa
sn η0: Initial viscosity in Pa
s A: Constant τðη1 Þ - τðη0 Þ 2

τ1=2 =

η1: Equilibrium viscosity in Pa
s - η1 γ_ τ = η1 þ η01þ τ

Peek–McLean– Williamson Reiner– Philippoff

τ = η1 þ

Reiner

τ=

ηs =

τm

1-

η0 - η 1

1þðττm Þ

η1 η0 - η1 - τ 2 κ η0
e

γ_

2

γ_

ηs =

η0 - η 1 τ 1þτm

þ η1

η0 - η 1

1þðττm Þ

2

τm = τ

η0 þη1 2

þ η1 κ=

φ1 - φ0 dφ d ðτ2 Þ

: Coefficient of

pseudoplasticity ϕ = 1η: Fluidity

from η0 to η1 with increasing shear rate. For a mathematical description of these sigmoid shaped curves, a number of model functions have been developed. In general, most of these functions require two or three parameters. These parameters are constants in the model function, whose numerical values make the function specific for a given material. Therefore, in order to use these model functions, these parameters have to be known. Normally, model parameters are measured experimentally, or they be obtained from other investigators working on the same material, who report their findings in the scientific literature. Table 4.8 lists

4.3 Viscous Behavior, Flow

177

some model functions like these. There are cases where a model function takes on the same form as that of a simpler more easily recognized model function, like Newton’s law. In other words, these model functions can be thought as more or less sophisticated modifications of basic functions like those which describes Newtonian flow behavior. A classic question posed by investigators working on rheology of fluids is, “Which model is the right one for my material and/ or application?” The answer will depend on factors like: • For what purpose is the viscosity needed? • In what range of shear rates will the work be done? • How precise must the data be? These models differ in their ability for preciseness or “goodness of fit” in approximating experimental data. For example, the model functions after Ferry, Steiner–Steiger–Ory, De Haven, Ostwald–de Waele, Ellis I, and Sisko are valid only in the region η < η1 where the fluid structure is not completely broken down. For synthetic polymers (but not those in the form of dispersions or gels), the models after Carreau, Cross, Ellis-Sisko, Phillips-Deutsch, Reiner-Philippoff, Krieger–Dougherty are recommended [10].

4.3.12 Ostwald–de Waele Law A very useful and not too complicated model function is the power-law equation after Ostwald–de Waele. It is a power law with the exponential parameter n, which is called flow exponent. It reads τ = K ow
γ_ n Kow γ_ τ n

ð4:39Þ consistency coefficient in Pa
sn shear rate in s-1 shear stress in pa flow exponent

The flow behavior index n indicates the deviation of the flow curve from a straight line, i.e. from Newtonian behavior. When the flow behavior index is less than unity (n < 1) the function describes a pseudoplastic flow behavior curve (convex profile). When the flow behavior index is greater than unity (n > 1) the function describes a dilatant flow behavior curve (concave profile). For Newtonian fluids, the flow behavior index becomes 1, and the consistency coefficient becomes the viscosity. In other words, for Newtonian fluids is n = 1 and Kow = η.

178

4 Rheological Properties

Definition The flow exponent of a material characterizes the curvature of the flow curve.

Definition The consistency coefficient Kow also is called Ostwald–Faktor Kow in the Ostwald–de Waele law. The higher the consistency factor, the tougher the fluid. For non-Newtonian fluids, the consistency coefficient Kow has the meaning of a mathematical parameter with the physical unit Pa
sn. From the unit it can be seen that Kow it is not same as viscosity, which has the unit Pa
s. Its value is determined experimentally. In case of a Newtonian fluid we have Kow = η. The Ostwald–de Waele law allows the calculation of ηs of a material at a given shear rate or a given shear stress. For this purpose, the Ostwald–de Waele law is used in the logarithmic form: From τ = K ow
γ_ n we get lgτ = lgK ow þ n
lg_γ . In the double-logarithmic (log-log) graphical representation, a straight line results. The slope of the straight line is the flow behavior index (exponent) n. The value of Kow we get from lgK ow = lgτ - n
lg_γ :

ð4:40Þ

At a shear rate of γ_ = 1 s - 1 this simplifies to lgK ow = lgτ

ð4:41Þ

i.e. by reading the ordinate value for γ_ = 1 s - 1 we get Kow (refer Fig. 4.21). This results in the dynamic viscosity Fig. 4.21 Evaluation of an Ostwald–de Waele flow curve

.

4.3 Viscous Behavior, Flow

179

Table 4.9 Examples of flow exponents and consistency coefficients for selected materials Material Tomato juice 12.8% (m/m) dm Tomato sauce 25.0% (m/m) dm Tomato paste 30.0% (m/m) dm Condensed milk, 10% (m/m) fat UF concentrate (protein concentrate from cheese whey or whole milk) 1–99% (m/m) Xanthan solution 1% (m/m) in water Xanthan solution 0.5% (m/m) in water Xanthan solution 0.25% (m/m) in water Xanthan solution 0.125% (m/m) in water Pure water

ηs =

ϑ/ ° C 32 32 32 10 5–50

Kow/Pa
sn 2.0 12.9 18.7 0.123 20–16,000

n 0.43 0.4 0.4 0.861 1.13–0.17

Source [11] [11] [11] [12] [13]

– – – – 20

10 3 0.4 0.14 0.001

0.18 0.24 0.35 0.5 1

[13] [14] [14] [14] [15]

dτ K ow
γ_ n = γ_ d_γ

ð4:42Þ

so ηs = K ow
γ_ n - 1

ð4:43Þ

With knowledge of Kow and n, the dynamic viscosity of a material can be calculated for any shear rate. Table 4.9 shows some examples of Ostwald–de Waele values reported in the literature. Attention In the case of non-Newtonian fluids, the viscosity depends on the load, i.e. the shear rate. A recommended notation for the viscosity of non-Newtonian fluids is the co-indication of the shear rate, e.g. in one of these forms: η γ_ = 50 s - 1 = 24 Pa
s η50 s - 1 = 24 Pa
s η50 = 24 Pa
s

Definition Apparent viscosity is a term used to distinguish between the constant viscosity of Newtonian fluids (slope of a straight line), and some corresponding value that can be obtained when the flow behavior curve is not straight, and there is no constant slope. Apparent viscosity is taken to be the slope of a straight line through the origin intersecting the curve at a specified constant shear rate. Such (continued)

180

4 Rheological Properties

a value, when taken at a specified shear rate, is often referred to as “apparent viscosity,” ηs. In other words, apparent viscosity ηs is the slope of the τ - γ_ dτ graph at γ_ , i.e. ηs = d_ γ ðγ_ Þ.

4.3.13 Model Functions for Plastic Fluids The models best suited to describe plastic flow behavior (those with a yield stress) are the models after Bingham, Casson, Heinz, Herschel–Bulkley, Schulmann– Haroske–Reher, Tscheuschner and Windhab (Table 4.10). In the use of models like these, the model parameters are often given special names. For example, the parameter τ0 in the Bingham model is given the name Bingham yield stress, and the parameter η of this model is named the Bingham viscosity. By use of this naming convention, we know to which model the parameter belongs. Otherwise, the parameters per se would not give any indication as to what model they belonged. In many of these models that describe flow behavior with a yield stress, the viscosity term η sometimes is called plastic viscosity, and the parameter K often is Table 4.10 Model functions for plastic fluids, examples Bingham

τ = τ0 þ η
γ_

Casson

τ2 = τ0 2 þ ðη
γ_ Þ2

Heinz

τ3 = τ0 3 þ ðη
γ_ Þ3

Casson (general)

τn = τ0 n þ ðη
γ_ Þn

1

1

1

2

2

2

1

1

1

τ0: Bingham yield stress in Pa η: Bingham viscosity in Pa
s (plastic viscosity) τ0: Casson yield stress in Pa η: Casson viscosity in Pa
s τ0: Heinz yield stress in Pa η: Heinz viscosity in Pa
s τ0: Casson yield stress in Pa η: Casson viscosity in Pa
s

Herschel–Bulkley

τ = τ0 þ K
γ_ n

Schulmann– Haroske–Reher

τn = τ0 n þ ðK
γ_ Þm

1

1

1

τ0: Yield stress in Pa K: Plastic viscosity in Pa
s (consistency coefficient) n: Flow exponent τ0: Yield stress in Pa K: Plastic viscosity in Pa
s (consistency coefficient) m, n: Constants

with τ0 = 0: τ = η
γ_ Newton τ = η
γ_ Newton τ = η
γ_ Newton τ = η
γ_ Newton For

n = 1): Bingham n = 2): Casson n = 3): Heinz τ = K
γ_ n Ostwald–de Waele τ = K 0
γ_ k Ostwald–de Waele

4.3 Viscous Behavior, Flow

181

Table 4.11 Herschel–Bulkley parameters for dairy products, examples ϑ/ ° C 10 10 10 10 10

Material Buttermilk 0.4% (m/m) fat Yogurt, stirred 1.5% (m/m) fat Yogurt, stirred 13.5% (m/m) fat Sour cream % (m/m) fat Cream cheese 20% (m/m) fat in dry matter

τ0/Pa 1.299 4.34 11.03 25.17 67.47

Kow/Pa
sn 1.607 11.05 6.91 21.53 14.65

n 0.410 0.285 0.408 0.532 0.688

Quelle [16] [16] [16] [16] [16]

called consistency coefficient, refer Table 4.10. For a very small yield stress τ0 the approximation τ0 = 0 leads to simpler model function. So, for τ0 = 0 the models after Bingham, Casson, and Heinz transform into the basic Newtonian model. In the same way, the models after Herschel–Bulkley and Schulmann–Haroske–Reher, for the case where τ0 = 0 would transform into the Ostwald–de Waele model, refer Table 4.10. Casson´s model (the general Casson’s model) represents stronger dependency on the flow behavior index n than other models like Bingham, Casson, or Heinz. In the case of no yield stress, where τ0 = 0, Casson’s general model will transform into the basic Newtonian model. The models after Bingham, Casson, and Heinz can be seen as special cases of the Casson’s general model. In the same way, the Ostwald–de Waele model obviously is a special case of the Herschel–Bulkley model. The model function after Schulmann–Haroske–Reher is a power-law function like Ostwald–de Waele, but with 4 parameters instead of two. Table 4.10 gives an overview of these model functions. If we compare Bingham, Casson, and Heinz model functions, we can see that these can be understood as special cases of the general Casson model function. The general Casson model function is thus very versatile and can even take the form of the Newton model function with n = 1 as flow exponent and yield point τ0 = 0. Likewise, the Ostwald–de Waele model function can be understood as a special case of the Herschel–Bulkley model function. Table 4.11 gives numerical values of the specific flow behavior constants for several viscous foods. The model functions of Tscheuschner and Windhab, specially developed for melted chocolate, use further parameters (ηstr1 and τ1) to increase the quality of approximation between experimental and mathematical curves (Table 4.12).

Table 4.12 Models of Tscheuschner and Windhab Tscheuschner

τ0: Yield stress in Pa η1: Final viscosity in Pa
s γ_ Str1 = 1 s - 1

τ = τ0 þ η1
γ_ þ ηStr1
γ_γ_n r

γ_ r = Windhab

τ = τ0 þ η1
γ_ þ ðτ1 - τ0 Þ
1- e

- γ_γ_

γ_ γ_ Str1

ηStr1 = ηðγ_ Str1 Þ γ_  = γ_ ðτ Þ τ = τ0 þ ðτ1 - τ0 Þ
1- 1e τ1: extrapolated yield stress in Pa

182

4 Rheological Properties

For molten chocolate the International Office of Cocoa, Chocolate and Sugar Confectionery (IOCCC) recommended in 1973 Casson’s model for shear rates from 5 to 60 s-1. However, since the year 2000, the Windhab model has been recommended for shear rates in the range γ_ = 2 . . . 5 s - 1 at ϑ = 40 ° C [17]: γ_

τ = τ0 þ η1
γ_ þ ðτ1 - τ0 Þ
1- e - γ_ 

ð4:44Þ

shear rate in s-1 shear stress in Pa equilibrium viscosity in Pa∙s constant in s-1 constant in Pa

γ_ τ η1 γ_  τ1

According to Windhab, a shear-induced restructuring takes place in the shearing of chocolate with increasing shear rate. This restructuring manifests itself in a transition from an initial viscosity to a point of equilibrium viscosity η1, refer Fig. 4.22. At this point with the shear stress τ = τðγ_  Þ the flow curve transits into a linear flow function, the extrapolation of which to the ordinate provides the so-called extrapolated yield point τ1 (refer Fig. 4.22). With the shear rate γ_  at this point we get γ_

τ = τ0 þ η1
γ_ þ ðτ1 - τ0 Þ
1- e - γ_  :

ð4:45Þ

τ = τ0 þ η1
γ_ þ ðτ1 - τ0 Þ
1- e - 1 :

ð4:46Þ

τ = τ0 þ η1
γ_ þ ðτ1 - τ0 Þ
0:632:

ð4:47Þ

so

Above this shear stress or shear rate, chocolate melts show a linear flow curve, i.e. they behave like a Bingham fluid from here on [18]. Fig. 4.22 Determining the extrapolated yield stress τ1 for the Windhab model

Kf

W W1 W0

.

J

4.4 Temperature Dependency of Viscosity

183

Bottom Line The characteristics of ideal materials, which we got to know at the beginning of the chapter can be expressed very shortly on the basis of their material properties: η=0 η = const. τ0 > 0 E = const. G→1

Pascal material Newton material St. Venant material Hook material Euklid material

4.4

Temperature Dependency of Viscosity

Since the viscosity of fluids is based on intermolecular interactions, it is a temperature-dependent quantity. As the temperature rises, the thermal motion of the molecules increases, which reduces the temporal mean of the bond strength to the neighboring molecule. For this reason, the viscosity of liquids decreases with increasing temperature and increases with decreasing temperature. For low-viscosity substances, this behavior can be represented with an Arrheniusanalogous approximation, the so-called Andrade equation B

Eo

η = A
eT = A
eRT

ð4:48Þ

1 η E 1 = a R T Tr ηr

ð4:49Þ

or ln η ηr T Tr Ea R A, B, C

dynamic viscosity in Pa∙s at T dynamic viscosity in Pa∙s at Tr temperature in K reference temperature in K activation energy in J∙mol-1 universal gas constant in J
K-1
mol-1 constants

If we draw viscosities η measured logarithmically above the reciprocal absolute temperature T1 , a straight line with the slope b = ERa is obtained. The quantity EA is called activation energy or shear activation energy in analogy to the Arrhenius approach. In systems with unknown molecular weight, EA is not determined, instead we work with slope b as a material parameter for the

184

4 Rheological Properties

temperature-related change in viscosity. If the viscosity ηr is known at any reference temperature Tr, the viscosity η can be calculated at a different temperature T in this way. ln

1 1 η : =b T Tr ηr

ð4:50Þ

η = ηr
eb
ðT - T r Þ : 1

1

ð4:51Þ

Since this is an approximation and the temperature dependence of the viscosity of real substances deviates from the linear behavior of Arrhenius behavior, the smaller the temperature difference between T and Tr the better the result. Example The dynamic viscosity of water at 33.4 °C is required. With the reference value ηr = 1:0 mPas ϑr = 20 ° C T r = 293:15 K and at

ϑ = 33:4 ° C T = 306:55 K with the help of Andrade’s equation we get ln 1 306:55 K

-

1 293:15 Kr

η ηr

= 2036:8 K
- 1:49
10

=b
-4

K

1 T

-1

-

1 Tr

= 2036:8 K


= - 0:037

1 1 η = eb
ðT - T r Þ = e - 0:3037 = 0:738 ηr

η = ηr
0:738 = 1:0 mPas
0:738 = 0:738 mPas In Vogel’s equation [19] an additional parameter the so-called Vogel temperature C is used. B

η = A
eTþC

ð4:52Þ

η 1 1 =B ηr T þ C Tr þ C

ð4:53Þ

or ln

4.4 Temperature Dependency of Viscosity

185

η ηr T Tr Ea R A, B, C

dynamic viscosity in Pa∙s at T dynamic viscosity in Pa∙s at Tr temperature in K reference temperature in K activation energy in J∙mol-1 universal gas constant in J
K-1
mol-1 constants

In addition, there are over 100 other, mostly empirically equations for calculating viscosity as a function of temperature [20–22]. Grigull [23] uses the following empirical equation (Vogel type) for water at ϑ in ° C b

ηðϑÞ = a
eϑþc

ð4:54Þ

with a = 0:0318 mPa
s b = 484:3726 ° C c = 120:2202 ° C

Example The dynamic viscosity of water at 33.4 °C is required. With b

ηðϑÞ = a
eϑþc and the above constants for water are 484:3726 ° C

ηð33:4 ° CÞ = 0:0318 mPa
s
e33:4 ° Cþ120:2202 ° C = 0:0318 mPa
s
e3:153 = 0:0318 mPa
s
23:407 ηð33:4 ° CÞ = 0:744 mPa
s For many polymers and highly viscous substances, the Williams–Landel–Ferry equation (WLF equation) is used instead of the Arrhenius relationship or Vogel’s equation [24]: log

C
ðT - T r Þ η =- 1 : ηr C 2 þ ðT - T r Þ

ð4:55Þ

The material constants C1, C2 depend on the selected reference temperature, often the glass transition temperature Tg of a material is chosen as the reference temperature.

186

4 Rheological Properties

log

C0
T - T g η = - 01 ηg C2 þ T - T g

η ηr T Tr Tg C 1 , C 01 C 2 , C 02

ð4:56Þ dynamic viscosity in Pa∙s at T dynamic viscosity in Pa∙s at Tr temperature in K reference temperature in K glass transition temperature in K constants material constants in K

If a fluid is cooled, its viscosity increases. When the glass transition temperature Tg is reached, the material has such a high viscosity that it behaves as a solid. This non-crystalline state of a solid is called the glassy state. If the temperature is increased again, a sharp decline in viscosity takes place at the glass transition temperature: the glass softens into a viscous flowing material. Glass transitions are examined with the help of thermal analysis. Therefore, in Chap. 8 on thermal properties we will hear again about the glass transition temperature. Since numerous physical, chemical, and biological transformations in a food are related to molecular mobility, many temperature correlated phenomena can be described more precisely with WLF kinetics than with activation kinetics according to Arrhenius. These include the viscosity of carbohydrate melts, crystallization rates, browning reactions, oxidations [25]. Even the stability of food in the sense of temperature-dependent shelf life can be predicted with the WLF equation [26]. On the other hand, the WLF concept has limits, e.g. it must not be overlooked that the WLF material constants are not “universal,” as in the field of synthetic polymers, but depend on the composition of the food, e.g. on water content and water activity [27, 28].

4.5

Viscosity of Solutions

If macromolecular substances are dissolved in a solvent, e.g. water, the viscosity of the solution increases with increasing concentration of the solution. The ratio of viscosity of the solution and viscosity of the pure solvent is called relative viscosity or viscosity ratio. ηrel =

η : ηsolvent

ð4:57Þ

It is a ratio without a unit. If the viscosity of the solution is reduced by the value of the pure solvent, we get

4.6 Viscoelasticity

187

η - ηsolvent : ηsolvent

ð4:58Þ

η -1 ηsolvent

ð4:59Þ

or

which is called viscosity relative increment. It is a measure for the viscosity effect of the solute. The former term “specific viscosity“for this size should be avoided. If the viscosity relative increment is applied above the concentration of the dissolved material and the function extrapolated to the hypothetical value c = 0, the limit value for an infinite dilution, the so-called Staudinger index of the dissolved material, is obtained. Staudinger index = lim

c→0

1 η -1 : c ηsolvent

ð4:60Þ

The former term “intrinsic viscosity“for the Staudinger index shall be avoided because it is not a viscosity in the unit Pa
s. The Staudinger index is related to the hydrodynamic volume of the dissolved substance. Therefore, the Staudinger index can be used to give information about the molar mass of the dissolved substance, for this purpose the Kuhn–Mark–Houwink relationship is used, cf. [29]. If the solute is not a pure substance, average molecular weights of the dissolved substance are obtained in this way. Definition Staudinger Index: Extrapolated viscosity of a solution at infinite dilution. Although this is unusual for an index, the Staudinger index has a unit, namely the unit of an inverse concentration, e.g. (g/100 cm3 solvent)-1.

4.6

Viscoelasticity

To understand viscoelastic materials, we first consider a soft, elastic solid, e.g. camembert, marshmallows, yeast dough. The material reacts to a shear stress with deformation. Typically the deformation does not occur instantly, but gradually. Likewise, when the stress is removed, the material gradually returns to its initial form. Figure 4.23 shows such behavior schematically. Here, a shear stress τ is applied abruptly, kept constant for some time and then suddenly set back to zero (rectangle shear signal). The deformation γ is the response signal of the body. An ideal elastic material would react to the shear stress with an instantaneous deformation and return to the initial state just as instantaneously when the stress is removed. A material that as in Fig. 4.23 follows a shear stress with a delayed

188

4 Rheological Properties

Fig. 4.23 Viscoelasticity: A material loaded with a rectangular shear stress signal (τ - t-curve) shows a timedelayed deformation γ

Fig. 4.24 Maxwell element: shear stress curve in response to a rectangular deformation signal

deformation, but returns to its initial state, is called viscoelastic [30]. At the beginning of the chapter, we got to know simple mechanical models with which the rheological behavior of materials can be mapped. For viscoelastic behavior, the Maxwell element and the Kelvin element are suitable in the simplest case each consisting of a spring and a damper (Fig. 4.9). The spring models the elastic behavior of the body, the damper the viscous flow resistance, which slows down the elastic deformation [9]. If a Maxwell element is loaded with a rectangular deformation signal, it shows a typical, time-delayed course of the shear stress in response (see Fig. 4.24), which can be calculated as follows: The applied stress is the same for the Hook element and the Newton element, so τ = τHooke = τNewton the deformation of the Maxwell element we get by addition

4.6 Viscoelasticity

189

γ = γ Hooke þ γ Newton dγ d γ_ = = ðγ Hooke þ γ Newton Þ dt dt with G = const: 1 dτ τ d 1 γ_ =
τ þ γ Newton =
þ G dt η dt G or η dτ τ þ
= η
γ_ G dt η with λrel = G dτ τ þ λrel
dt with γ = const:or γ_ = 0 we get dτ =0 dt τ 1 dt =dτ λrel τ þ λrel


τ

τ0

t

τ =dτ

1 dt λrel 0

τ t ln = τ0 λrel τ = τ0
e

-λt

rel

This means that the shear stress drops exponentially over time, the stress relaxation is a first-order decay function. From the relaxation curve we can determine the relaxation time λrel of the material. It is the period of time after which the stress has fallen to 1/e of the initial value. Materials that relax slowly have a greater relaxation time. Materials that relax immediately have a shorter relaxation time. Assuming γ = const., that means that the shear modulus of the viscoelastic material changes in time, we can therefore write: G=

τ0

γ const


e

-λt

rel

= G0
e

-λt

rel

:

This formulation shows that the shear modulus of a Maxwell material is not constant (as with a Hookean material), but changes over time. Note the indices: here τ0 is the maximum stress, it must not be confused with the yield point of the material. In physics, the symbol τ for the relaxation time is often used. To avoid confusion with the shear stress τ, in rheology the symbol λrel for the relaxation time is used. Viscoelastic materials that look solid at first glance can flow slowly, this is called creep. Let us consider a Kelvin element (Fig. 4.25) that is loaded with a constant strain. The applied strain is the same for the Hook element and the Newton element

190

4 Rheological Properties

Fig. 4.25 Kelvin element: reaction to a rectangle stress signal

γ = γ Hooke = γ Newton the stress of results additively: τ = τHooke þ τNewton i.e., τ = G
γ þ η
γ_ d_γ 1 dτ
= γ_ þ λret
dt G dt with γ = const:, or γ_ = 0 is 1 dτ
= γ_ G dt integration results to 0

τ0

t

1
dτ = G

γ_
dt 0

1 γ - τ0 = - t G λ e ret - 1 t τ0 γ =
1 - e - λret G The deformation of the Kelvin element thus proceeds in the form of a (1 - e-time)function asymptotic to a maximum value. Figure 4.25 illustrates the behavior. This behavior is often described with a reciprocal quantity to the shear modulus, the so-called compliance J. With

4.6 Viscoelasticity

191

1 G

ð4:61Þ

1 - t
1- e λred : G

ð4:62Þ

J= we have J=

Here λred is the retardation time. This is the time it takes to reach (1 - e-1)-fold of the final value, i.e. 63.2% of the final value. Attention Relaxation time λrel and retardation time λred are not the same. To improve the approximation between experimental data and model, the Kelvin element can be supplemented by a Maxwell element in series (see Fig. 4.26) which is called a Burgers model. The Burgers model can be further optimized by using several Kelvin elements in parallel. Finally, a fracture element and a St. Venant element (for the yield point) can be added to model the behavior of real materials. Definition The differential equations for the Kelvin model were formulated by O.E. Meyer in 1874 and extended by W. Vogt in 1892. W. Thomson (Lord Kelvin) described the behavior of viscoelastic bodies as early as 1865. Therefore the Kelvin/Vogt model (German standard DIN 1342) is the same as the Kelvin model [10]. In addition, there are viscoelastic materials that do not completely return to their initial state after deformation, but show a permanent deformation. The behavior of such plasto-viscoelastic materials can be modeled by adding a St. Venant element for Fig. 4.26 Burgers model for viscoelastic materials

192

4 Rheological Properties

Table 4.13 Systematics of Rheology of Solids and Fluids Material state Designation Ideal body Material properties Real materials

Solid Liquid Gaseous Solid Fluid Euklid, Hooke Newton, Pascal Elasticity, plasticity Viscosity Non-ideal Newtonian, non-ideal elastic, viscoelastic, elastoviscous, viscoplastoelastic, etc.

Fig. 4.27 Real behavior as mixed cases of ideal behavior

the yield point to the Maxwell model, the Kelvin model, or the Burger model. In the chapter about texture, we will come back to these mechanical models (Table 4.13). Rheology as a theory of flow generally describes the deformation behavior of bodies under the influence of forces or stresses. In order to be able to describe the often complicated deformation behavior of real bodies analytically, one tries to present the real behavior as mixed cases of ideal bodies [9]. In Table 4.3, we have seen the basic bodies for ideal elastic behavior, ideal viscous behavior, and ideal plastic behavior. For illustration, Fig. 4.27 shows the ideal cases as vertices of a triangle. The mixed cases, i.e. real materials, lie within this triangle, depending on the degree of their elasticity, viscosity, and plasticity. The experimental study of the deformation behavior of viscoelastic foods is dealt in the following Sect. 4.7.

4.7 Rheological Measurement

193

Fig. 4.28 Rheological systematics of ideal and non-ideal materials

Figure 4.28 shows a different view of the rheological classification of materials. Here, mixed cases are also distinguished, which lie between “ideal-elastic” and “ideal-viscous.” With the plasticity on its own axis, we get from this tabular representation to the triangle representation in Fig. 4.27.

4.7

Rheological Measurement

There is any number of different types of laboratory instrumentation by which to measure rheological properties of materials. Perhaps the most commonly used type of instruments for studying the rheology of liquids are rotational viscometers also called rheometers [31]. They will be described first in this section. Other types of instruments, based on different principles like capillary types and falling sphere type, will be described later.

4.7.1

Rotational Rheometers

All rotational devices consist of a rotating body (rotor) and a stationary body (stator). The liquid sample is always placed between the rotor and stator, which causes the sample to experience shear during operation of the instrument. Within the broad category of rotational instruments, the coaxial cylinder system is perhaps most widely used. It consists of a smaller cylinder (bob) within a slightly larger cylinder (cup), such that an annular gap is left between the two cylinders, into which the liquid sample is placed. This system is called coaxial cylinder measuring system [32].

194 Fig. 4.29 Coaxial cylinder systems after Searle (left) and Couette (right). The sample S is located in the annular gap

4 Rheological Properties

M S

S

M

Fig. 4.30 Rotational rheometer, Searle-type. 1 motor, 2 torque meter, 3 temperature-controlled cylindrical beaker, 4 rotating cylinder (bob), 5 annular space for liquid sample

1 2

3 4 5

Fig. 4.31 Mooney–Ewart system: Rotating bob with conical frontal surface in a double-walled, temperaturecontrolled cylindrical cup

When the cup is fixed and the bob is rotated, this is called a Searle-type coaxial cylinder system. When the cup is rotated around a fixed bob, this is called a Couettetype coaxial cylinder system, refer Fig. 4.29. The Searle-type is technically easy to implement and widely used, see Fig. 4.30. If we want to achieve that the same shear takes place on the lower frontal surface of the rotating cylinder as on the cylinder jacket, we can make the lower cylindrical frontal surface conical, see Fig. 4.31. If we then omit the cylinder jacket surface, we

4.7 Rheological Measurement

195

get a cone–plate measuring system, see Fig. 4.35. Here, the flat cone rotates over a stationary plate. If a circular plate is used instead of the rotating cone, this is referred to as the plate–plate measuring system. Rotational rheometers allow to vary the speed of the rotor, and by this the shear rate applied to the sample. By electronic measurement of the torque needed to achieve a given angular velocity, the shear stress can be measured. From the rotational speed and the geometry, the shear rate can be calculated. From the torque and the geometry the shear stress can also be calculated. A plot of shear stress over shear rate gives the flow curve, as described previously (Fig. 4.12). Modern instruments allow a choice between controlled shear rate mode (CSR) and controlled shear stress mode (CSS). In the CSR mode, the shear rate is programmed and presented to the sample, and the resulting shear stress is measured. In the CSS mode, the shear stress is programmed and presented to the sample, and the resulting shear rate is measured. Many instruments also allow a means of operation to perform measurements in an oscillating mode. Under an oscillating mode of operation, the rotor reverses direction periodically at a programmed frequency, meaning it rotates back and forth, first clockwise, then counter-clockwise, performing a swinging movement with the sample. This mode of operation is recommended for materials with viscous, as well as elastic properties, see Sect. 4.7.2. Cylindric System Geometry When using coaxial cylindrical rheometers, it is necessary to know the general relationships between the parameters of the rotating cylinder and the rheological quantities of shear stress and shear rate. First let us calculate the shear stress when a cylinder is rotated as shown like in Fig. 4.32. With the measured torque M M=r
F and the area A, A=2
π
r
h

Fig. 4.32 Left: torque on the rotary cylinder, right: jacket surface of the cylinder

M r

F

2π r

h

196

4 Rheological Properties

the following can be computed: τ=

F M = A 2
π
r2
h

ð4:63Þ shear rate in s‐1 shear stress in Pa area of cylinder surface in m2 tangential force in N torque in Nm radius of cylinder in m height of cylinder in m tangential velocity in m∙s-1 angular velocity in s-1 viscosity in Pa∙s

γ_ τ A F M r h v ω η

The shear rate within the annular sample space is: γ_ = -

dv dω =
r dr dr

dv d dω
r = ðω
r Þ = dr dr dr

ð4:64Þ ð4:65Þ

That means, at a constant speed of rotation ω, the shear rate γ_ depends on the distance r. From Eq. (4.64) we know τ=

M 2
π
r2
h

ð4:66Þ

so 1= 2

M r= 2
π
h


τ-

1= 2

ð4:67Þ

and dr M = dτ 2
π
h with (4.66) this results to

1= 2




-

1 3=
τ- 2 2

ð4:68Þ

4.7 Rheological Measurement

197

dr τ
2
π
r2
h = dτ 2
π
h

1= 2




-

1 r 3=
τ- 2 = 2 2τ

ð4:69Þ

i.e., dr dτ =r 2τ

ð4:70Þ

with (4.64) written as dω
r dr

γ_ = f ðτÞ = follows for the ω(r) function: dω = -

dτ dr 1
f ðτ Þ = f ðτ Þ
τ r 2

ð4:71Þ

Integration over the thickness of the annular shear space in Fig. 4.33 gives: ωa = 0

τa

1 dω = 2 ωi = Ω

f ðτ Þ
τi

dτ τ

ð4:72Þ

this is τa

1 Ω= 2

f ðτ Þ
τi

dτ τ

ð4:73Þ

ω

Fig. 4.33 Radii, shear stress, and angular velocity of the rotating cylinder

M

Ri Ra

ωaτa

ωτ i i

198

4 Rheological Properties

τ R ω Ω Index i Index a

shear stress in Pa radius of cylinder in m angular velocity in s-1 angular velocity of inner cylinder in s-1 inner cylinder (bob) outer cylinder (cup)

This is the general function between angular velocity Ω of the inner cylinder (bob) and the resulting shear rate in a Searle-type rheometer. The equation can be solved only by substituting the behavior of the fluid f(τ). To do this we first choose a Newtonian fluid. After that it will be shown how the equation also works with an Ostwald–de Waele fluid. For a Newtonian Fluid τ η

γ_ = f ðτÞ =

ð4:74Þ

So Eq. (4.73) becomes: τa

1 Ω= 2

τi

τa

dτ 1 f ðτÞ = τ 2

τi

τ dτ 1
=η τ 2η

τa

dτ

ð4:75Þ

τi

that means Ω=

1 ðτ - τa Þ 2η i

ð4:76Þ

with Eq. (4.66) we get a term called Margules’ equation Ω=

1 1 4π
h
η R2i - R2a

ð4:77Þ

Margules´ equation provides the relationship between angular velocity Ω of the rotor and the resulting torque M (angular moment) when a Newtonian fluid with viscosity η is in the annular space between Ri and Ra, refer Fig. 4.33. For an Ostwald-de-Waele Fluid Here we have instead of Eq. (4.74) γ_ = f ðτÞ = so

τ K OW

1 n

ð4:78Þ

4.7 Rheological Measurement

199 τa

1 Ω= 2

τi

τa

dτ 1 f ðτ Þ = τ 2

τi

1 n

τ K OW

dτ τ

ð4:79Þ

i.e., Ω=

1 1 n τni - τna 1 n 2
K OW

ð4:80Þ

with Eq. (4.66) we get Ω=

η 1 2
K OW n

M 2
π
h
R2i

1 n

-

M 2
π
h
R2a

1 n

ð4:81Þ

or Ω=

η M 1 2 2
K OW n 2
π
h
Ri

1 n




1-

Ri Ra

2

ð4:82Þ

This is a general version of the relationship between angular velocity Ω of the rotating body and torque M in the concentric cylinder system. It applies to Newtonian and non-Newtonian fluids. For n = 1 it converts into Margules equation. Of course, we can also use more complicated model laws instead of the Ostwald– de Waele law (refer Tables 4.8 and 4.10) if this is required by the material behavior (the flow curve). If the annular gap of the concentric cylinder system is very small, i.e. (Ra - Ri) trel t < trel t ≈ trel

De De < 1 De > 1 De ≈ 1

Texture

Material is Viscous Elastic Viscoelastic

thousand years, the same body can look like a fluid (low De number). This means that if we consider large (historical) periods in addition to laboratory tests (small periods), solids such as window glass, glaciers, and even mountains can be understood not only as solids but also as flowing. Heraclitus’ saying pantha rhei (“everything flows”) seems to already mean this. Bottom Line Solid materials have a large De number; liquid materials have a small De number. Surprisingly, the duration of observation plays a role.

Remark Deborah’s number is named after the clairvoyant, Deborah, mentioned in Judges 5,5 (The Bible) where she said “The mountains flow before the Lord.”

5.6

Oscillating Load

Just as with shear, we can put an oscillating load on a body. Let us apply an oscillating strain on a material in uniaxial direction and measure the resulting stress. With the strain ε=E
σ

ð5:41Þ

which is oscillating, e.g., in the form of a sinus function ε = ε0
sin ωt

ð5:42Þ

the strain rate is ε_ = i.e.

dε d = ðε0
sin ωt Þ dt dt

ð5:43Þ

5.6 Oscillating Load

241

ε_ = ε0
ω
cos ωt

ð5:44Þ

σ = kelastic
ε þ kviscous
ε_

ð5:45Þ

σ = E
ε þ ηE
ε_

ð5:46Þ

σ = E0
ε þ E 00
ε_

ð5:47Þ

the stress is

or

or

with E0 =

σ cos δ ε

ð5:48Þ

E 00 =

σ sin δ ε

ð5:49Þ

and

we can write E =

σ = ε

E0 2 þ E00 2

ð5:50Þ

This introduces the complex elasticity module E, which consists of the real part E and the imaginary part E″. ′

ε ω t σ σ0 kelastic, kviscous E E E′ E″ δ

Strain Angular frequency in s-1 Time in s Stress in Pa Stress amplitude in Pa Constants Modulus of elasticity in Pa Complex modulus of elasticity in Pa Storage modulus of elasticity in Pa Loss modulus of elasticity in Pa Phase shift

So-called DMA instruments (dynamic mechanical analysis) close the gap between measuring instruments that bring uniaxial compression stresses to the sample and devices that exert oscillating shear stresses. Instruments used for

242

5

Texture

conducting DMA put uniaxial oscillating load on a sample, and frequency and amplitude can be varied. These DMA measurements are fast and only need small material samples [12, 13].

5.7

Fracture Tests

Higher stress or strain on solid and semi-solid materials often leads to fracture (rupture) of the material. Before breakage, the behavior of the materials is usually far from the behavior of an ideal elastic material. First there are deviations from the elastic behavior leading to plastic deformation (flow) and finally to the irreversible separation of bonds between the molecules (or atoms) of the solid material. This applies analogously to uniaxial compression and shear. In addition, the rate of the loading plays a role in the investigation of fracture behavior. There are a number of methods for characterizing fracture behavior [9]; most simple are uniaxial compressive stress or uniaxial tensile stress. Again, we have to choose between: • Loading by stress (tensile or compressive) up to breakage, recording of strain or • Loading by strain (positive or negative) up to breakage, recording of stress The measured fracture strain (rupture strain), fracture stress, or fracture force is specified. In addition, the fracture work Wf and the volume-related fracture work can be determined, i.e., the mathematical product of stress and strain (shaded area in Fig. 5.12). σ f
εf =

F f sf Wf =
A L0 V0

F f
sf = W f σf εf Ff L0 Wf sf A V0

ð5:51Þ ð5:52Þ Fracture stress in Pa Fracture strain in s Fracture force in N Initial length in m Fracture work in J Fracture deformation in m Area in m2 Initial volume in m3

5.7 Fracture Tests Fig. 5.12 Compression fracture test, schematic. Top: Stress–strain diagram, bottom: Force–displacement diagram. The shaded area symbolizes the fracture work

243

Vf

V

Hf

H

Ff

F

sf

s

The choice of testing tools is of particular importance here. For example, the sharpness of a blade, the angle of a cone, and the diameter of a ball, which are used for the fracture test, influence the measurement result. In addition, the attachment of the sample plays a role. If we want to measure the fracture strain by an uniaxial tensile test, different sample fastenings for, e.g., a foil, a noodle, or a sausage are required. The sample attachments must be selected in such a way that they do not affect the measurement result as far as possible, e.g., by damaging the sample. If the sample breaks first at the clamping point, this is an indication of damage to the sample due to the attachment. In the bending fracture test, the sample is loaded at 3 points (Fig. 5.13). Here, not only the shape of the test tool but also the geometry of the entire arrangement must be specified when reporting the measurement result. By the shape of the support points (round, sharp, etc.), the test can be adapted to the question of investigation. The 3-point bending test is particularly suitable for samples that are long compared to their thickness. In the bending stress, the lower part of the sample is strained while a compression takes place in the upper area (see Fig. 5.13).

244

5

Texture

Fig. 5.13 Three-point bending test. N neutral axis, C compressed axis, E strained axis

C N E

5.8

Texture Profile Analysis

In instrumental texture profile analysis (TPA), loads are repeatedly applied to a sample and the sample behavior is recorded. These can be all combinations of load and measurements presented in the sections above. Often these tests attempt to imitate the real mechanical stress of a food, e.g., during chewing or cutting. To imitate the scenario in the oral cavity, a fixed number of chewing movements are put to the sample and the measurement signal is recorded over time. The evaluation can be used to describe the time-dependent behavior of the food between the tongue and the palate or between the teeth from the first to the last bite, and to determine cutting force, breaking force, or stickiness of the product. Further Reading Theme Cake: texture profile analysis and stress relaxation characteristics of quince sponge cake Cheese: rheological properties Stickiness in foods—test methods Food texture affected by ohmic heating Gels: relation between mechanical properties and texture Meat texture: MRI-aided texture analysis Banana: texture kinetics during ripening

References [14] [15–18] [19] [20] [21] [22] [23]

Summary This chapter deals with texture as a subfield of solid-state rheology. With the help of axial loads, typical solid-state properties such as elasticity and fracture can be measured, but also relaxation and retardation phenomena, which are (continued)

References

245

used to describe the viscoelasticity of materials. At the end of the chapter, application examples are listed which can be used for further studies and as suggestions for ongoing investigations.

References 1. ISO 5492 sensory analysis - vocabulary (2009) Beuth, Berlin 2. Bourne MC (2002) Food texture and viscosity - concept and measurement. Academic Press, San Diego 3. Joyner HS (2018) Explaining food texture through rheology. Curr Opin Food Sci 21:7–14. https://doi.org/10.1016/j.cofs.2018.04.003 4. Jowitt REF, Hallstrom B, Meffert HFT, Spiess WEL, Vos G (eds) (1983) Physical properties of foods. Elsevier Applied Science, London 5. Bourne MC (2002) Physics and texture. In: Bourne MC (ed) Food texture and viscosity concept and measurement. Academic Press, San Diego. https://doi.org/10.1016/B978012119062-0/50003-6 6. Chen J, Rosenthal A (2015) Food texture and structure. Woodhead Publishing, Sawston, pp 3–24. https://doi.org/10.1016/b978-1-78242-333-1.00001-2 7. Moshenin NN (1986) Physical properties of plant and animal material. Gordon & Breach Science Publishers, New York 8. Lillford PJ (2016) Oral perception of food texture. Elsevier, New York. https://doi.org/10.1016/ b978-0-08-100596-5.03431-4 9. Day L, Golding M (2016) Food structure, rheology, and texture. Elsevier, New York. https:// doi.org/10.1016/B978-0-08-100596-5.03412-0 10. Steffe J (1996) Rheological methods in food process engineering. Freeman Press, East Lansing, MI 11. Reiner M (1964) The Deborah number. Phys Today 17(1):62 12. Zhong Q, Daubert CR (2013) Food rheology. In: Kutz M (ed) Handbook of farm, dairy and food machinery engineering. Academic Press, San Diego, pp 403–426. https://doi.org/10.1016/ B978-0-12-385881-8.00015-X 13. Meyvis TKL, Stubbe BG, Van Steenbergen MJ, Hennink WE, De Smedt SC, Demeester J (2002) A comparison between the use of dynamic mechanical analysis and oscillatory shear rheometry for the characterisation of hydrogels. Int J Pharm 244(1):163–168. https://doi.org/10. 1016/S0378-5173(02)00328-9 14. Salehi F, Kashaninejad M (2018) Texture profile analysis and stress relaxation characteristics of quince sponge cake. J Food Meas Charac 12(2):1203–1210. https://doi.org/10.1007/s11694018-9734-3 15. Fox PF, Guinee TP, Cogan TM, McSweeney PLH (2017) Fundamentals of cheese. Science. https://doi.org/10.1017/CBO9781107415324.004 16. Gunasekaran S, Ak MM (2003) Cheese rheology and texture. CRC Press, Boca Raton, FL 17. Tariq S, Giacomin AJ, Gunasekaran S (1998) Nonlinear viscoelasticity of cheese. Biorheology 35(3):171–191. https://doi.org/10.1016/S0006-355X(99)80006-7 18. Ray CA, Gholamhosseinpour A, Ipsen R, Hougaard AB (2016) The effect of age on Cheddar cheese melting, rheology and structure, and on the stability of feed for cheese powder manufacture. Int Dairy J 55:38–43. https://doi.org/10.1016/j.idairyj.2015.11.009 19. Adhikari B, Howes T, Bhandari BR, Truong V (2001) Stickiness in foods: a review of mechanisms and test methods. Int J Food Prop. https://doi.org/10.1081/JFP-100002186

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5

Texture

20. Gavahian M, Tiwari BK, Chu Y-H, Ting Y, Farahnaky A (2019) Food texture as affected by ohmic heating: mechanisms involved, recent findings, benefits, and limitations. Trends Food Sci Technol 86:328–339. https://doi.org/10.1016/j.tifs.2019.02.022 21. Santagiuliana M, Piqueras-Fiszman B, van der Linden E, Stieger M, Scholten E (2018) Mechanical properties affect detectability of perceived texture contrast in heterogeneous food gels. Food Hydrocoll 80:254–263. https://doi.org/10.1016/j.foodhyd.2018.02.022 22. Bajd F, Skrlep M, Candek-Potokar M, Sersa I (2017) MRI-aided texture analyses of compressed meat products. J Food Eng 207:108–118. https://doi.org/10.1016/j.jfoodeng.2017.03.026 23. Nannyonga S, Bakalis S, Andrews J, Mugampoza E, Gkatzionis K (2016) Mathematical modelling of color, texture kinetics and sensory attributes characterisation of ripening bananas for waste critical point determination. J Food Eng 190:205–210. https://doi.org/10.1016/j. jfoodeng.2016.06.006

6

Interfaces

The contact surface between two phases is called an interface. Phase is understood here to be a homogeneous state form of a substance, which can be distinguished from another phase by this recognizable separation surface. A prerequisite for the formation of an interface is the incomplete miscibility of the phases involved. If two substances are completely miscible into each other (such as two gases or a pinch of salt in water) no permanent interfaces are formed. Assuming three states of aggregation (solid, liquid, and gaseous), the following interfaces are possible (Table 6.1): Interfaces at solid materials are also referred to as solid interfaces due to their lack of mobility. In Chap. 1, the sorption at solid interfaces is treated. Fluid–fluid interfaces are referred to as fluid interfaces, sometimes also referred to as liquid interfaces, because of their mobility. Those interfaces in which one of the phases involved is gaseous are called surfaces. Then, we speak of surface tension instead of interfacial tension. Definition An interface is the area that is located at the boundary of two phases. It separates these phases (solid, liquid, and gaseous) from each other. Surfaces belong to the interfaces. Since the physical properties of the interfaces differ from the physical properties within the substance of the same material, the surface layer of a phase is sometimes referred to as an independent phase, the so-called interfacial phase [1]. This is especially true in the case of molecular layers that are adsorbed at an interface. Examples of this are water molecules on solid surfaces (see Chap. 1) as well as adsorbed surfactants or emulsifiers at liquid interfaces. While a molecule inside a substance is surrounded by the same molecules, this is not the case at interfaces. At the interface to another phase, interactions with non-identical molecules occur. As a result, the molecules at the interface differ in their energetic state and their physical # The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. O. Figura, A. A. Teixeira, Food Physics, https://doi.org/10.1007/978-3-031-27398-8_6

247

248

6 Interfaces

Table 6.1 Classification of interfaces Fluid–fluid interfaces Liquid–gaseous Liquid–liquid

Solid interfaces Solid–gaseous Solid–liquid Solid–solid

e.g., beer–CO2 e.g., water–oil

e.g., NaCl–air e.g., sugar–oil e.g., glass–plastic

behavior from molecules within the substance. The difference in energetic state is the cause of interfacial phenomena such interfacial tension. For phases that are not composed of molecules but of individual atoms, e.g., noble gases or metals, this applies analogously. In this chapter, we will first look at two-phase systems and look at three-phase systems.

6.1

Interfacial Tension

A molecule M in the substance of a liquid is surrounded on all sides by molecules of the same substance. The intermolecular interaction forces on the molecule M are the same in all spatial directions, so that the resulting force on M is zero (Fig. 6.1). At the interface, on the other hand, the interaction forces do not compensate each other in all spatial directions, so that a resulting force is created. If the intermolecular interaction forces in the liquid phase are greater than the intermolecular interaction forces between the molecules of the liquid phase and the adjacent gas phase (in the case of water-air, as in Fig. 6.1), then a force F results for a liquid molecule M, which is directed into the liquid phase. If we want to increase the interface area, we have to bring liquid molecules from the volume to the interface against this force F. Enlargement or creation of an interface is therefore associated with a work or energy expenditure (work = force
distance), while the reduction of the interfacial area is associated with an energy gain. Bottom Line Molecules in interfaces have a higher energy than molecules within the same substance. The surface-specific energy of interfaces is given in J
m-2 and also referred to as interfacial tension.

M

F Fig. 6.1 Intermolecular forces within the substance and at the interface. A molecule M experiences a net force F at the interface, which is directed into the substance

6.1 Interfacial Tension

249

Fig. 6.2 Creation an element of interface dA

db

s dA

F

Since interfaces are high in energy, free liquid droplets tend to take on a spherical shape. The spherical surface is a minimal surface. At a given volume, all other threedimensional shapes have a larger interface, i.e., a greater energy than the sphere (refer to Platon’s bodies in Chap. 3). However, most liquid droplets do not have an ideal spherical shape, because of other influences such as gravity, adhesive forces, or frictional forces in a flow (e.g., air friction). Let us look at an interface represented by the thin wire of length, s, shown in Fig. 6.2. The area of this interface can be enlarged by moving the wire a distance, db, represented by the cross-hatched area in Fig. 6.2. We can see that the creation of this additional interface size, dA = s
db, is related to an energy expenditure of dE = F
db. This area-related energy is called specific interfacial energy σ which can also be called area-specific interfacial energy or specific interfacial energy. σ = lim

ΔE

ΔA → 0 ΔA

σ=

=

dE dA

F
db F = s
db s

F A σ E b s

ð6:1Þ ð6:2Þ force in N area in m2 interfacial tension in N∙m-1 specific interfacial energy in J∙m-2 length in m width in m

The surface-specific interfacial energy is identical to the interfacial tension. The units of surface-related energy and interfacial tension are identical: 1

Nm N J =1 2 =1 m m m2

. The energy for the generation of an interface dA or ΔA can thus be calculated according to dE = σ
dA

ð6:3Þ

250

6 Interfaces

or ΔE = σ
ΔA

ð6:4Þ

The energy used to generate an interface ΔA is thus directly proportional to the value of the interfacial tension between the two phases. In processes where interface creation occurs, knowledge of interfacial tension is helpful, as well as knowledge how to reduce it. A lower interfacial tension means that the energy for the interfacial formation process is lower. Figure 6.1 shows the forces on a molecule in a flat interface. The forces tangential to the interface cancel each other out to zero. The forces perpendicular to the interface do not compensate for each other, so there is a resulting net force directed into the liquid volume. In curved interfaces, the tangential forces similarly do not fully compensate for each other. This is the reason why curved interfaces have additional effects.

6.1.1

Curved (Convex/Concave) Interfaces

When a beaker is overfilled with water, a curved interface between the water and air is formed at the top surface, and the water reaches a slightly higher level at the center than at the rim of the beaker. It looks as though an invisible thin skin would retain the water at the top of the beaker. This invisible “thin skin” is caused by what we call “surface tension” and is a result of the interfacial energy discussed previously. This phenomenon is also the reason why the molecules in the inner volume are exposed to a slight pressure which would not be there without that “skin.” This pressure is called capillary pressure. The curvature effect is caused by the fact that at a curved interface, the tangential forces acting on a molecule in the interface do not compensate to zero. This produces a concave interface resulting from an increase in the forces that are directed inward into the volume of liquid (see Fig. 6.3). Fig. 6.3 Forces on a molecule in a liquid interface. With a concave curved interface, the resulting force directed into the liquid volume is greater than with a flat interface

6.1 Interfacial Tension

251

Fig. 6.4 Cross-section of a spherical droplet of fluid whose radius is increased by dr

dr

r

Virtual displacement: When we want to increase the volume of a liquid drop by adding a given amount of liquid to it (see Fig. 6.4), we have to do work against the capillary pressure pσ . This form of work is called volume work dWP dW P = - pσ
dV WP Eσ pσ V

ð6:5Þ volume work in N∙m specific interfacial energy in N∙m pressure in N∙m-2 volume in m3

For a sphere with V = 43 π
r 3 the dependence of the volume on the radius is dV d 4 =
π
r 3 = 4π
r 2 dr dr 3

ð6:6Þ

dW P = - pσ
4π
r 2
dr:

ð6:7Þ

used in Eq. (6.5) gives

The droplet, whose radius was increased by dr, gets a larger interface. The interfacial energy required for this is dE σ = σ
dA:

ð6:3Þ

2 Writing the differential dA dr for a sphere with A = 4π
r we get

dA d =
4π
r 2 = 8π
r dr dr

ð6:8Þ

dA = 8π
r
dr

ð6:9Þ

so

252

6 Interfaces

That means dEσ = σ
8π
r
dr

ð6:10Þ

According to energy conservation, the volume work performed is equal to the interfacial energy Eσ , i.e., dEσ þ dW P = 0

ð6:11Þ

dE σ = - dW P

ð6:11Þ

σ
8π
r
dr = - - pσ
4π
r 2
dr

ð6:12Þ

or

used in Eq. (6.10) gives

so pσ =

2σ : r

ð6:13Þ

This is called Laplace’s equation. The pressure in an interfacial sphere is called capillary pressure according to Laplace. The radius r in Laplace’s equation is the radius of curvature of the observed interface. Convex curved interfaces have positive curvature radii, while concave curved interfaces have negative radii of curvature. Example Mist consists of fine water droplets distributed in air. In a spherical drop of water with a radius of 70 nm and σ = 70 mN
m - 1 the pressure according to Laplace is pσ =

2σ 2
70mN
m - 1 2
10 - 3 N
m - 1 = = 20
105 Pa ≈ 20bar: = r 70
10 - 9 m 10 - 9 m

The capillary pressure according to Laplace is the overpressure of the phase with the concave-shaped interface compared to the adjacent phase. If phases 1 and 2 have absolute pressures p1 and p2, then the difference in absolute pressures provides the overpressure in the concave phase. Concave-shaped particles are typically droplets or gas bubbles.

6.1 Interfacial Tension

253

p1 - p2 =

2σ 12 = pσ r1

ð6:14Þ

interfacial tension between phase 1 and 2 in N m-1 radius of curvature on phase 1 in m absolute pressure in phase 1 (droplet) in Pa absolute pressure in phase 2 (surrounding) in Pa capillary pressure (overpressure of phase 1) in Pa

σ 12 r1 p1 p2 pσ

Bottom Line The pressure inside a drop of liquid or in a gas bubble is increased compared to the environment. The smaller the radius of curvature, the greater the pressure difference to the environment. Laplace’s equation is also applicable to concave-shaped interfaces. Because negative values for the radius of curvature are used, the capillary pressure is negative; i.e., there is a condition of underpressure compared to the other phase. At a flat interface, the radius of curvature is infinitely large and the Laplace’s equation returns the value zero; i.e., the pressure is the same on both sides of the interface. Table 6.2 summarizes these cases. Attention It is easy to get confused with the terms concave and convex: A concaveshaped interface looks convex to a viewer localized behind it. Think of a cereal bowl turned upside down on a table. If you were inside the bowl, the curved surface above would appear concave, but outside the bowl, it will appear convex. To avoid this confusion, we must be clear “on which side we are,” i.e., for which of the two phases we perform the calculation.

Table 6.2 Capillary pressure of differently curved interfaces Interface Appearance of the interface, schematic Interface of phase 1 is... Radius of curvature r1 p1 - p2 =

2σ 12 r1

In phase 1 there is compared to phase 2 Example:

Curved Phase 2

Curved Phase 2

Not curved Phase 2

Phase 1 concave r1 < 0 0 >0

phase 1 flat r1 = 1 0

Underpressure

Overpressure

Same pressure

Gas bubble (phase 2) in water (phase 1)

Droplet (phase 1) in air (phase 2)

Flat water surface

254

6 Interfaces

Example We have an air bubble with a radius of 1.4 mm in water. We imagine to sit inside the air bubble: We are in phase 1 and see an concave interface with radius r1 of curvature. Laplace’s equation p1 - p2 = 2σ r1 = pσ gives a positive -3

2
70 10 value: pσ = 2σ r1 = 1:4
10 - 3 Pa = 100 Pa, i.e., the pressure in phase 1 is greater than in phase 2, i.e., we are sitting in overpressure. Now we change sides and enter the water phase. We are now in phase 2 and see the air bubble interface in front of our eyes in the form of a convex interface, i.e., r2 = - 1.4 mm. Laplace’s equation p1 - p2 = 2σ r2 = pσ gives

pσ =

2σ 2
70 10 - 3 = Pa = - 100 Pa: r2 - 1:4
10 - 3

So we are sitting in underpressure. The pressure in phase 2 is lower than in phase 1. If a drop does does not have an ideal spherical shape, its shape can be described, for approximately, e.g., as an ellipsoid with 2 main radii of curvature r+ and r-. Then Laplace’s equation is pσ = σ


1 1 : þ rþ r -

ð6:15Þ

For spherical drops r+ = r- = r resulting again in Eq. (6.13): pσ = 2σr . Capillary pressure causes the vapor pressure of particles to be higher than the vapor pressure of the flat phase [2]. This effect is higher the smaller the particle diameter is. Kelvin’s equation used in Chap. 1 allows the calculation of the vapor pressure of a liquid drop as a function of its radius. Example Vapor pressure of a spherical water droplet with a diameter of 20 nm at 20 °C according to Kelvin: ln

1 p 1 2
σ 2
72:25
10 - 3 N
m - 1 =
=
: rp 999 kg
m - 3
461:9 J
K
kg - 1
293:15 K p0 r p ρ
R s
T

(continued)

6.1 Interfacial Tension

255

ln

p 1 =
1:0682 nm = 0:10682 p0 10 nm

p = e0:10682 = 1:1127 p0 The vapor pressure of a water droplet with a diameter of 20 nm is 11.1% higher than the vapor pressure of water, which is free with a flat surface. Because of this increased vapor pressure, small droplets evaporate more quickly in a gas phase than larger droplets. Small droplets in a liquid, e.g., in an emulsion, have a higher solubility than in the flat phase because of this increased vapor pressure. Since this effect is independent of the physical state and is based exclusively on the curvature of the interface, it also occurs with solid substances. This means that small solid particles have a higher vapor pressure and thus a higher solubility or a higher sublimation pressure than the same substance in a flat phase. Example Two drops of liquid of different sizes are brought to the touch, so that a pressure equalization of their capillary pressures can take place. What will happen? (a) the larger drop becomes smaller and the smaller one larger (b) the larger drop becomes larger; the smaller one becomes smaller and disappears completely (c) both drops take on the same size. Answer: The smaller drop has a higher pressure. Therefore, the smaller drop fills the larger one until the small one is completely gone. Answer b is correct. A disperse system with different particle diameters such as a spray—is not energetically stable. The different pressures of the particles with different diameters lead to a balancing process in which small particles disappear in favor of larger particles. This effect, known as Ostwald ripening, occurs in liquid, solid, and gaseous particles. Therefore, small particles often have a short life time because the energy of the particles depends on their radii. The disappearance of the small particles, i.e., the coarsening of the disperse system, leads to a reduction in the total interfacial energy of the system [3].

256

6.1.2

6 Interfaces

Temperature Dependence of Interfacial Tension

The interfacial tension between two phases decreases with increasing temperature. The surface tension of liquids goes to zero when the critical temperature is reached. Above the critical point, there is no longer an interface between the liquid and gaseous phases. If we assume a liquid approximately composed of cube-shaped molecules of edge length l, the volume of one mole of these molecules, the so-called molar volume, is V m = N A
l3

ð6:16Þ

a molecule in the surface requires the area l2 =

Vm NA

2 3

ð6:17Þ

One mole of this substance therefore requires the area Am 1

2

Am = N A
l2 = N A 3
V m 3

ð6:18Þ

The surface energy of a mole of these molecules is 1

2

σ m = σ
Am = σ
N A 3
V m 3

ð6:19Þ

with the molar volume M ρ

Vm =

ð6:20Þ

The surface energy of a mole is σm = σ
N A

1 3

M
ρ

2 3

ð6:21Þ

According to Eötvös, the temperature dependence of the molar interfacial energy is σ m = kE
ðT C- T θ - T Þ

ð6:22Þ

Equalizing (6.21) and (6.22) provides the equation for the temperature dependence of the interfacial tension according to Ramsay and Shields [4] σ
N A 3
M 3
σ - 3 = k E
ðT C- T θ - T Þ 1

2

2

ð6:23Þ

6.1 Interfacial Tension

257

Fig. 6.5 Temperature dependence of interfacial tension

i.e., σ = N A - 3
M - 3
ρþ3
kE
ðT C- T θ - T Þ 1

2

2

ð6:24Þ

For substances with a strong association of liquid molecules (as is the case with water), the correction factor χ is introduced: σ = N A - 3
ðM
χ Þ - 3
ρþ3
kE
ðT C- T θ - T Þ 1

2

2

kE T TC Tθ NA Vm ρ σ M χ

ð6:25Þ

Eötvös coefficient temperature in K critical temperature in K material constant in K Avogadro’s number molar volume in m3
mol-1 density in kg
m-3 interfacial tension in N
m-1 molar mass in kg
mol-1 correction factor

Figure 6.5 schematically represents the temperature dependence of the interfacial tension of a pure substance. Example Calculation of the surface tension of water at 20 °C. with data for water kE TC Tθ

Eötvös coefficient for H2O: 7.5 J
K-1 mol-1 critical temperature for H2O: 647.1 K material constant for H2O: 6

(continued)

258

6 Interfaces

Avogadro’s constant 6.023∙1023 mol-1 density of H2O: 998.2 kg∙m-3 molar mass of H2O: 18.015∙10-3 kg∙mol-1 correction factor for molecular association of water at 20 °C: 0.47)

NA ρ M χ

we get 2

σ=

ð998:2 kg
m - 3 Þ3
7:5 N
m
K - 1
mol - 1 6:023
1023
mol - 1

1 3


0:47
18:015
10 - 3 kg
mol - 1

2 3


ð647:15 K - 6 K - T=KÞ this numerical equation can be summarized to σ = 0:2135

mN
ð368:0 - ϑ= ° CÞ m

for 20 °C we get σ = 0:2135mN
m - 1
ð368:0 - 20:0Þ σ = 74:3mN
m - 1 This numerical equation is based on fundamental quantities, such as Avogadro’s number and molecular weight, and easily provides values for the surface tension of water at different temperatures. However, not taking into account the temperature dependence of the density and the association factor diminishes the utility of that equation. A better equation can be produced by fitting the experimental data to a mathematical function without attaching importance to any physical meaning of the mathematical parameters (“curve fitting”). Table 6.4 shows which improvements of fitting quality can be achieved. Three numerical value equations were used as examples (refer Table 6.3) and the deviations from the experimentally determined values of the surface tension listed. A comparison of the calculated values with the experimentally obtained values shows that C provides the smallest deviations. Also, the simple linear regression according to B provides a better fit than A. While B and C are purely empirical adaptations, A is derived from fundamental variables. However, it still contains the empirically determined correction factor χ, which depends on the temperature. The quality of A can be improved if we take into account the temperature dependence of the correction factor χ and the density. However, by this, the equation becomes more complicated. Why this comparison of the models? For process engineering calculations, material data are always required. The interfacial tension here serves as an example of such a material property. If we

6.1 Interfacial Tension

259

Table 6.3 Equations for calculating the surface tension of water (examples) A

Ramsay, Shields [4]

B C

Kohlrausch [5] International steam tables [6]

σ = 0:2135 mN m
ð368:0 - ϑ= ° CÞ -1 σ/mN
m = 76.056 - 0.1675
ϑ/ ° C σ=mN
m - 1 = B


TC - T TC

μ


1þb


TC - T TC

with B = 235.8
10-3 Nm-1 b = -0.625 μ = 1.256 TC = 647.096 K Table 6.4 Comparison of functions for calculating the surface tension of water Experimental value [5] ϑ/° C 0.01 10 20 30 40 50 60 70 80 90

σ exp/mN∙m-1 75.65 74.22 72.74 71.20 69.60 67.95 66.24 64.49 62.68 60.82

Calculated with A σ cal/ mN∙m1 Deviation 78.57 3.9% 76.43 3.0% 74.30 2.1% 72.16 1.3% 70.03 0.6% 67.89 -0.1% 65.75 -0.7% 63.62 -1.3% 61.49 -1.9% 59.35 -2.4%

Calculated with B σ cal/ mN∙m1 Deviation 76.05 0.5% 74.38 0.2% 72.71 0.0% 71.03 -0.2% 69.36 -0.4% 67.68 -0.4% 66.01 -0.4% 64.33 -0.2% 62.66 0.0% 60.98 0.3%

Calculated with C σ cal/ mN∙m1 Deviation 75.65 -0.001% 74.22 0.006% 72.74 0.000% 71.20 -0.003% 69.60 0.001% 67.95 -0.002% 66.24 0.005% 64.49 -0.006% 62.68 -0.003% 60.82 0.003%

find what we are looking for in food material data collections, we often have to be satisfied with the value of a material of similar composition, and we have to accept the value as an approximation. Another possibility is measurement of the required property in compliance with the rules of food physics measurement technology. This results in a true value for the material actually present. However, further measurements will be necessary as soon as the temperature changes or, e.g., the water content is to be changed slightly. An alternative is the use of mathematical functions which take into account the influence of, e.g., temperature and composition. Such functions are referred to as mathematical models and are used for automatic calculation of the desired material data and ensure that the process calculation is not based on incorrect data. If there are several functions from which to choose, a decision is necessary as to which model should be used. Deviation between the calculated and experimentally determined value is often used as a decision criterion for the quality of the function to be chosen. Such a comparison is shown in Table 6.4. In addition to a comprehensible decision for one of the models, a quantitative measure about the quality of the adaptation can be obtained by giving the average or maximum deviation of the data used. An indication of the

260

6 Interfaces

reliability of the process calculation then can be made by means of error propagation calculation. In process calculations in which dozens of parameters and material data are included, it is highly recommended to quantify the uncertainty of the calculation in this way. Bottom Line When physical substance data of foods and their components are missing, approximate values must be used for process calculations. In order to be able to quantitatively specify the resulting uncertainties, it is recommended that substance data of the individually available material be experimentally recorded and expressed in the form of mathematical functions of their parameters, such as temperature, pressure, or composition. The quality of the function used can then be expressed by comparing the calculated and measured values, thus providing a measure of the quality of the approximation. The values of the surface tensions of some foods are listed in the appendix.

6.1.3

Concentration Dependence of Interfacial Tension

Amphiphilic molecules—surfactants—emulsifiers Substances that arrange themselves at the interface between two phases and thus cause a change in interfacial tension are referred to as interfacial active substances. This effect is particularly evident in amphiphilic molecules. Due to their molecular structure, even small amounts of these dissolved substances lead to a significant reduction in interfacial tension. Such substances are, therefore, also called surfactants. Amphiphilic molecules have both lyophilic (“solvent-friendly”) and lyophobic (“solvent-hostile”) groups. If the solvent is water, we speak of hydrophilic (“water-friendly”) and hydrophobic (“water-hostile”) groups. While the lyophilic molecular parts enter into strong interactions with the solvent molecules, the energetically most favorable state for the lyophobic molecular parts is the avoidance of interactions with the molecules of the solvent. For this reason, the amphiphilic molecules accumulate in the boundary layer, known as boundary layer adsorption [7]. In the boundary layer, the molecules orient themselves in such a way that the lyophilic molecular parts interact strongly with the solvent and the lyophobic molecular parts avoid interaction with the solvent. An interface occupied by interface-active molecules in that way is in an energetically favorable situation; therefore, its specific interfacial energy (see Sect. 6.1) is lower than in the absence of these molecules. With increasing concentration of the surface-active substance, the occupancy of the interface increases and the interfacial tension continues to decrease (see Fig. 6.6). When complete monomolecular occupation of the interface is achieved, there is no more space in the interface for further molecules; the maximum interface occupancy is reached. On further increase in the surfactant concentration, the

6.1 Interfacial Tension

261

Fig. 6.6 Concentration dependence of interfacial tension by surface-active substances

V

cmc

c

surfactant molecules therefore start to form associates in the volume of the solvent, so-called micelles. The concentration where micelle formation begins is called critical micelle formation concentration, cmc. Micelles can be spherical, rod shape, tubular, or lamellar. Micelles can interact with each other and can form structures. In addition, there are phase transitions in which the structure of the micellar aggregates changes [8]. Definition Micelles are colloidal aggregates formed by association of amphiphilic molecules in solutions. Since the addition of surface-active substance beyond the cmc no longer changes the occupancy at the interface, the interfacial tension remains unchanged above this concentration. The interfacial tension achieved at the cmc is therefore a characteristic quantity for the interfacial active substance used. In Fig. 6.7, the achievement of the maximum interface occupancy is shown in a glass beaker, schematically. The concentration dependence of the interfacial tension for surface-active substances can be described mathematically with the Szyszkowski equation [3]: σ 0 - σ = a
ln 1 þ σ σ0 c b a

c b

ð6:26Þ

interfacial tension in N
m-1 interfacial tension bei c = 0 concentration of the surface-active substance in kg
m-3 material constant in m3
kg-1 material constant in N
m-1

Analogous to the cmc, the temperature at which micelles form is called the critical micelle temperature (cmt). If the temperature of a surfactant solution is lowered, micelles form when passing through the cmt. Conversely, on increasing the

262

6 Interfaces

Fig. 6.7 Once the maximum interface occupation is reached, the interfacial tension remains constant

Fig. 6.8 Surfactants in an aqueous solution: Monomers in solution (I). Micelles in solution (II). Liquid crystal suspension (III). Krafft point (K ). after [9]

temperature, cmt can be recognized by the beginning clearness of a cloudy, colloidal micelle dispersion. The intersection of the cmt curve and the cmc curve in the surfactant phase diagram is called Krafft point; the critical micelle formation temperature is also called Krafft temperature (Fig. 6.8) [10]. The structure of the micelles is ruled by the most energetically favorable state of the amphiphilic molecules in the solvent at given concentration and temperature. Since during micelle formation both the entropy and the energy of the system change, the free enthalpy (Gibbs energy; see Chap. 8) indicates the stability of the

6.1 Interfacial Tension

263

system. The minimum of free enthalpy determines the stability of the system [7]. Association and orientation of the molecules take place in such a way that lyophilic interactions are in favor before interactions of lyophobic molecular parts and solvent molecules. For example, hydrophilic groups are interacting with the water molecules, while hydrophobic groups are facing each other. If lamellar bilayers form aggregates with a cavity, this is referred to as vesicles. Attention Vesicles and micelles are different forms of association aggregates. Vesicles consist of concentrically arranged lamellar molecular bilayers and have a solvent-filled cavity. Vesicles are also called liposomes. Micelles and vesicles are used as capsules and vehicle systems for poorly soluble active ingredients. For example, casein micelles can be used for microencapsulation [11] or water-insoluble substances are introduced into body cells with the help of vesicles [8]. There are substances which, in contrast to surfactants, increase the interfacial tension of a liquid when dissolved. In the case of aqueous systems, these are mainly strong electrolytes (inorganic salts) and hydroxyl-rich compounds such as carbohydrates. The reason for this is the strong interaction of these substances with water (hydration) and thus an accumulation of the substances in the liquid volume, i.e., a depletion of the substances at the interface.

6.1.4

Emulsions

An emulsion is a disperse system of two immiscible liquid phases. Disperse means that one phase is distributed in the form of small particles - here liquid droplets—in the other liquid phase. A prerequisite for the formation of an emulsion is the non-miscibility of the two phases. Otherwise, a solution would be formed and not a disperse system. If the disperse phase consists of oil and the continuous phase of water, it is called an O/W emulsion (oil-in-water emulsion). During forming of an emulsion, interface is created, as the following example shows. Example In a beaker, 100 ml of oil and 100 ml of water are layered on top of each other. Assume the beaker has diameter of 10 cm. The size of the circular oil–water interface is A0 = π
r 2 = π
ð5 cmÞ2 = 7:85
10 - 3 m2 (continued)

264

6 Interfaces

An O/W emulsion is prepared from this. The resulting oil droplets are spherical with a uniform diameter of 20 μm. What is the area of oil–water interface in the emulsion? Solution: First we calculate the volume of a single oil droplet: Vd =

4 4 π
r 3d = π
10 - 5 m 3 3

3

=

4 π
10 - 15 m3 3

Then the surface of this oil droplet: Ad = 4π
r 2 = 4π
10 - 5 m

2

= 4π
10 - 10 m2

Now we calculate the number of oil droplets by dividing the existing volume of oil by the volume of a droplet: N=

V oil 100 ml 100
10 - 6 m3 3 = =4 =
1011 = 2:387
1010 - 15 3 Vd Vd 4π π
10 m 3

The area of the interface in the emulsion results from the sum of all oil droplet interfaces: Atotal = N
Ad = 2:387
1010
4π
10 - 10 m2 = 30 m2 If we compare this size of the interface with the initial interface in the beaker, we get Atotal 30 m2 = = 3821 A0 7:85
10 - 3 m2 i.e., during the production of the emulsion, the area of the oil–water interface is increased by some thousands. The interfacial energy of the emulsion is calculated according to ΔE = σ
ΔA

ð6:4Þ

By lowering the interfacial tension σ, the interfacial energy of the emulsion can be reduced. For this purpose, suitable surface-active substances (see Sect. 6.1.3) are used. Analogous to the O/W emulsions, such as milk, cream, or salad dressing, there are water/oil (W/O) emulsions, such as butter, in which water droplets are dispersed in an oil phase. By adding of suitable co-surfactants, we get optically transparent emulsions. Such clear or translucent preparations are called microemulsions. That is because it was assumed that the emulsified particles are smaller than the wavelength of visible light,

6.1 Interfacial Tension

265

i.e., below 400 nm. In contrast to emulsions, microemulsions are considered thermodynamically stable. Since no droplets can be detected even in the electron microscope, it is questionable whether it is an emulsion at all or rather a singlephase, ternary colloidal system of water, oil, and surfactant [12]. Cosmetic products with encapsulated oil in capsules with diameters in the range of 1–100 nm are sometimes referred to as nanoemulsions [13, 14]. The capsules consist, e.g., of a phosphatidylcholine single membrane that encloses a liquid lipid core. In order to distinguish between capsules and conventional emulsion droplets, preparations with nanoscale capsules should be called nano-dispersions. Pickering emulsions are emulsions in which solid particles are used as surfaceactive substances. With suitable solids the amount of classic emulsifiers can be reduced [15]. In addition, it is possible to disperse an emulsion in another phase. In this way, e.g., O/W/O emulsions or W/O/W emulsions, so-called multiple emulsions can be formed [16–18]. Definition Multiple emulsions are emulsions in which the dispersed phase itself is an emulsion. There are numerous options to make such emulsions usable. One option is to have the inner phase carry an active ingredient and the outer phases used to estabish the desired flow behavior of the emulsion. By use of co-surfactants, viscosity-increasing thickeners, and gelling agents, the number of technological options can be further increased. While three or more different substances are used in such systems, we still have a two-phase system (liquid–liquid). In the following section, we want to learn about interfacial phenomena in which three phases are involved.

6.1.5

Liquid–Liquid–Gas Interface

The shape of a liquid drop floating on the surface of another liquid is determined by the interfacial tensions between all three phases. The phases involved are gas phase, liquid A, and liquid B. In Fig. 6.9, such a drop shape is shown schematically. In the

Fig. 6.9 Interfacial tensions at the contact point of three phases: 1 gas phase. 2 Liquid A. 3 Liquid B

V 13 V 12 M1 M3 M2

V 23

1 3 2

266

6 Interfaces

Fig. 6.10 Vectors of interfacial tensions, shifted in parallel to apply the cosine theorem

V13

J

D E

V12

V23 E

equilibrium of forces, the tension σ 12 is just as great as the result of the vector addition of forces σ 13 and σ 23. The contact angle φ3 takes the value at which this equilibrium is realized. With the help of the Cosine theorem for a isosceles triangle c2 = a2 þ b2 - 2a
b
cos γ

ð6:27Þ

can be formulated (Fig. 6.10): σ 12 2 = σ 13 2 þ σ 23 2 - 2σ 13
σ 23
cos γ

ð6:28Þ

α þ β = φ3 and γ þ φ3 = 180∘ i:e: γ = 180∘ - φ3

ð6:29Þ

cos γ = - cos φ3

ð6:30Þ

σ 12 2 = σ 13 2 þ σ 23 2 þ 2σ 13
σ 23
cos φ3

ð6:31Þ

because of

is

therefore

this results in the contact angle cos φ3 =

σ 12 2 - ðσ 13 2 þ σ 23 2 Þ 2
σ 13
σ 23

ð6:32Þ

Three cases can be distinguished in which the contact angle φ3 is above 90°, below 90°, or near 0°. A contact angle above 90° leads to ellipsoid drops of phase 3 floating on phase 2. In an extreme case, the contact angle is 180°; then the floating drops have spherical shape. In the other extreme case, the contact angle is 0°; then phase 3 forms a thin film on phase 2. This is called spreading. By the use of spreading, very thin films can be created down to the thickness of a single monolayer. In the other cases with contact angles between 0 and 90°, phase 3 forms “lenses” that float on phase 3. Whether these lenses are more flat or more curved is determined by the contact angle φ3, i.e., from the vectorial addition of the interfacial tensions involved. Table 6.5 compiles these cases with some examples.

6.1 Interfacial Tension

267

Table 6.5 Cases with different contact angles Case I

Cos φ3 1

φ3 0

Phase 3 forms... Film on phase 2

II

Positive

0°–90°

Lens on phase 2

III

Negative

90°–180°

Ellipsoid/sphere on phase 2

Example Gasoline on water Benzene on water Olive oil on water Decane on water Water on silicone oil

Definition Wetting is the formation of a small contact angle between a liquid and a solid interface. Complete wetting is given at a contact angle of 0°. Then the liquid forms a film on the solid surface. Complete non-wetting is given at a contact angle of 180°. The concept of wetting can also be understood from an energetic point of view. The creation of an interface is associated with an energy expenditure. With three phases involved, there are three types of interfaces that can be formed: solid–liquid, solid–gaseous, and liquid–gaseous. Depending on the molecular structure of the phases involved, the energy expenditure for each type is different. By varying the areas of the three interfaces, the total interfacial energy of the system can be reduced to a minimum value. This state of equilibrium can be characterized by the contact angle; refer Fig. 6.13. Definition Complete wetting, i.e., a contact angle of 0° leads to the spreading of a liquid on the subphase. The spreading pressure is the difference between the adhesion energy and the cohesion energy. Substances with positive spreading pressure are called film formers. Liquids whose adhesion on another phase is associated with a greater energy gain than the cohesion of the liquid will spread and form a film on the other phase. The tendency toward spreading is characterized by the spreading pressure. With φ3 = 0 the vector σ 12 acts for adhesion and the vector sum (σ 13 + σ 23) for cohesion. The difference between the two is the spreading pressure π: π = σ 12 - ðσ 13 þ σ 23 Þ

ð6:33Þ

If the spreading pressure π is positive, the liquid spreads on the subphase.

268

6 Interfaces

Example A drop of a liquid hydrocarbon shall be placed on a water surface. Decane and benzene are available. What will happen? Will the drop form a film on the surface of the water or swim like a lens on the water? Solution: The material data for the interfacial tensions at room temperature are used to calculate the spreading pressure: π = σ 12 - ðσ 13 þ σ 23 Þ σ 12 σ 13 σ 23

air–water σ 12 = 72.4 mN
m-1 Air–benzene σ 13 = 28.7 mN
m-1 Water–benzene σ 23 = 33.7 mN
m-1

air–water σ 12 = 72.4 mN
m-1 Air–decane σ 13 = 23.9 mN
m-1 Water–decane σ 23 = 51.2 mN
m-1

Spreading pressure of decane on water: π=mN
m - 1 = 72:4 - ð23:9 þ 51:2Þ = - 2:7 Spreading pressure of benzene on water: π=mN
m - 1 = 72:4 - ð28:7 þ 33:7Þ = 10:0 It turns out that the spreading pressure of benzene on water is positive; i.e., the benzene droplet will spread on the water surface. The drop decane, on the other hand, will form a lens-like drop on the water surface. The spreading pressure π is also called surface pressure. By analogy to the equation of state for ideal gases p
V =n
R
T

ð6:34Þ

can be derived for surface pressure π π
A=n
R
T

ð6:35Þ

With a constant amount of substance n and a constant temperature T, there are analogues to the Boyle–Mariotte law p

1 V

ð6:36Þ

6.1 Interfacial Tension

269

i.e., π

1 A

ð6:37Þ

This means that the surface pressure of a film increases due to compression of the film (reduction of the area A) and vice versa. With a so-called Langmuir trough, the π - A isotherm of a film can be recorded with the help of a Langmuir balance. In this way, aggregate states and polymorphic states of films can be characterized [19]. If we dissolve benzene from the above example in a high-molecular-weight substance and give a drop on a water surface, then this substance can be distributed with the help of the spreading of benzene to a monomolecular layer on the water surface. After evaporation of the spreading liquid, a monolayer of this substance remains on the water surface. This monolayer can be transferred to a solid substrate, e.g., to a polymer or a glass. Such a film on a solid surface is called Langmuir– Blodgett film. Film formation techniques like this make it possible to modify the wetting properties or the biofunctionality of materials [20, 21].

6.1.6

Solid–Liquid–Gas Interface

We now replace the underlying liquid phase with a solid material. When a small amount of a liquid is applied to this solid surface, the shape of the liquid is again determined by the interfacial stresses involved (refer Fig. 6.11). In equilibrium, Young‘s equation applies: σ 12 = σ 23 þ σ 13
cos φ

ð6:38Þ

σ 12 - σ 23 = σ 13
cos φ

ð6:39Þ

i.e.,

This is called wetting tension σ B. σ B = σ 12 - σ 23 = σ 13
cos φ

ð6:40Þ

The wetting tension has a positive value if the adhesion of the liquid on the solid surface is energetically more advantageous than the cohesion of the liquid. With a negative wetting tension, it is reversed. In this case, the liquid lying on the solid tends

Fig. 6.11 Interfacial tensions at the point of contact of three phases. 1 gas, 2 solid, 3 liquid

270

6 Interfaces

to contract and forms drops. The cosine of the contact angle is positive in this case; i.e., the contact angle is between 0° and 90°. cos φ =

σ 12 - σ 23 σ 13

ð6:41Þ

If we look again on Fig. 6.11 and take the arrows as vectors, we see that the → wetting tension is the horizontal component of the vector σ 13 , i.e., σ B = σ 13
cos φ:

ð6:42Þ

In Table 6.6 are compared three cases, in which the contact angle φ is above 90°, below 90°, or at 0°. The wetting of a solid surface according to the listed cases I, II, and III is shown in Fig. 6.12 schematically. The difference between a well-wetting liquid and a low wetting liquid is illustrated in Fig. 6.13. Table 6.6 Different contact angles on a solid surface Case I II III

σ 12 σ 23 ≥ σ 13 0 > σ 12 σ 23 < σ 13 0 < σ 12 σ 23 < σ 13

I

cos φ cos φ ≥ 1

φ =0

0 > cos φ > -1 0 < cos φ φ > 180° 90° > φ > 0°

II

Consequence Complete wetting, creep of the liquid, film forming Low or no wetting, liquid forms drops Good or partial wetting, liquid forms flat drops

III

Fig. 6.12 Wetting of a solid surface: I film formation, II partial wetting, III roll off of a liquid

Fig. 6.13 Contact angle φ of a well wettable (I) and a poor wettable (II) solid surface. The contact angle is below 90°. With a non-wettable solid surface, the contact angle is above 90° (III)

6.2 Kinetic Phenomena at Interfaces

6.2

271

Kinetic Phenomena at Interfaces

The extent to which an interface is occupied with surface-active substances can depend on the speed of the processes involved. Examples of such processes are • • • •

Formation or degradation processes of surface-active substances Molecular orientation of surface-active substances Formation or degradation of associates such as micelles or vesicles Adsorption or desorption at interfaces

In these cases, the interfacial occupancy depends on the time. Interfacial tension, contact angle, and spreading pressure are functions of time. To illustrate we can imagine the adsorption of proteins at an interface, e.g., the adsorption of a milk protein at the interface to a fat droplet. For the speed of adsorption, first the speed of mass transport to the interface must be determined. This means that the kinetics of the interface stabilization depend on the diffusion rate of the molecule. This in turn depends on molecular weight, viscosity of the solution, and, thus, the temperature. In addition, the molecular alignment, i.e., the orientation of the surface-active molecule, can play a role in adsorption. If the protein forms micelles, its availability for interfacial adsorption depends on the balance between speed of micelle formation and degradation. This equilibrium, as well as the speed of equilibrium adjustment, can depend on factors such as pH or electrolyte content. The interaction of all those processes leads to the kinetics of adsorption and thus to kinetics of interface stabilization. As soon as there is an equilibrium between the processes, the interface occupancy is constant. Such a state of equilibrium exists at a given temperature for each concentration of the adsorbent. These equilibrium states can be represented with adsorption isotherms, analogous to the isotherms presented in Chap. 1. If a strong bond, such as chemisorption, is found between adsorbent and adsorbate, these interfacial adsorption isotherms have the form of the Langmuir type (see Chap. 1 on Water activity) [22]. Kinetic phenomena in interfacial stabilization play a role in the production of emulsions, in which newly formed interfaces must be stabilized in such a short time that reversing processes such as droplet coalescence are minimized. Here, it is important to avoid delays in the kinetics of the interface occupation. The reverse effect, the depletion of surface-active substances, is also called negative adsorption. Here, an increase in interfacial tension is observed when the concentration of surface-active substance is increased. Definition Dynamic interfacial tension: In contrast to an interfacial tension which is constant over time (static interfacial tension), a time-varying interfacial tension is referred to as dynamic interfacial tension.

272

6.3

6 Interfaces

Measurement

Measurement of Interfacial Tension One of the methods used for measurement of interfacial (surface) tension is known as the bow wire method. Figure 6.14 shows a schematic of how this method is applied. A piece of wire of length l is lifted for a distance dh out of a liquid interface. This causes the interfacial surface area to increase by dA = 2
l
dh.The force F necessary for lifting the wire is measured and used to calculate the surface tension. When using a flat plate instead of the wire, one speaks of the plate measuring method according to Wilhelmy (Fig. 6.15). In the ring method according to Du Noüy, a circular wire ring is used instead of a wire bracket. When the ring is pulled out of the interface, the interface is increased by the amount dA = 2
l
dh, in the case of the ring-shaped wire it is l = π
d. By determining the necessary force F, the interfacial tension can be determined by σ=

dE F
dh F = = : dA cos φ
2
l
dh cos φ
2
π
d

ð6:43Þ

The contact angle φ indicates the wetting of the ring by the measuring liquid. With complete wetting of the ring, φ = 0, it is σ=

Fig. 6.14 Schematic for measuring surface tension with the bow wire. 1 Forcemeter, 2 bow wire, 3 horizontal piece of wire, 4 liquid sample

dE F
dh F = = dA 2
l
dh 2
π
d

ð6:44Þ

1 F

4

2 3

l

h

F

Fig. 6.15 Wilhelmy plate for measuring interfacial tension, schematic

I

z

M

6.3 Measurement

273 interfacial tension in N
m-1 interface energy in Nm
m-2 area in m2 force in N wetting angle height in m length in m

σ E A F φ h l

The value thus obtained is to be multiplied by a tabulated correction factor that eliminates the effect of the volume of liquid hanging on the ring. The correction factor according to Harkins and Jordan in turn depends on the density of the liquid and the ring diameter [23]. The ring method is often used to determine the surface tension of aqueous liquids. For a ring made of platinum-iridium, it is assumed that the liquid ideally wets the wire or the plate. That is, we use the approximation that the contact angle between liquid and wetted solid is φ = 0. For most aqueous solutions, this approximation is well fulfilled. If we want to measure the interfacial tension between two liquids with the ring method, we must not neglect buoyancy effects and wetting angles of the fluids. With the plate method the maximum force F is measured with a wetted plate that is used to “raise” the underlying liquid. The interfacial tension is σ=

dE F
dh : = dA cosφ
2
ðz þ lÞ
dh

ð6:45Þ

dE F : = dA 2
ðz þ lÞ

ð6:46Þ

With φ = 0 it is σ=

Conversely, if we want to determine the contact angle between liquid and plate, a liquid of known surface tension is used. In this way, the contact angle φ of a material or a surface coating can be determined. φ = arc cos

F σ
2
ðz þ lÞ

ð6:47Þ

Definition The contact angle is the angle between liquid interface and solid surface. It is also known as the wetting angle. When a solid material is completely wetted by the liquid, the contact angle is 0°.

274

6 Interfaces

Fig. 6.16 Capillary method for determining the surface tension

rK M

h FG

In the capillary method, the height to which a liquid rises within a capillary tube is measured and used to calculate the surface tension. The wetting of a capillary creates a curved liquid surface (Fig. 6.16) with a pressure difference between the two sides of that interface (see Table 6.2). The liquid column in the capillary rises until the hydrostatic pressure of the column corresponds to the pressure above the curved surface. The surface tension can be calculated from the pressure equilibrium. The capillary pressure above the concave fluid meniscus is according to Laplace (refer Sect. 6.1.1). pK =

2
σ
cos φ : rK

ð6:48Þ

Here, φ is the contact angle, i.e., the angle between the liquid surface and the wall of the capillary (see Fig. 6.16) and is compensated by the hydrostatic pressure of the fluid column below the meniscus p = Δρ
h
g,

ð6:49Þ

where Δρ is the difference in density of liquid phase and gas phase. Equalizing the pressures provides 2
σ
cos φ = Δρ
h
g rK

ð6:50Þ

and thus for interfacial tension σ=

Δρ
h
g
r K : 2
cos φ

ð6:51Þ

With the approximations Δρ ≈ ρFl and φ = 0 which is well fulfilled in aqueous systems in glass capillaries, the surface tension therefore is σ=

ρFl
h
g
r K 2

ð6:52Þ

6.3 Measurement

275 interfacial tension in N
m-1 density of the liquid in kg
m-3 gravitational acceleration in m
s-2 capillary radius in m wetting angle height in m length in m

σ ρFl g rK φ h l

Example In a 1 mm capillary, a liquid rising height of 28 mm is determined for an aqueous solution at room temperature. The surface tension of the liquid thus is σ=

998kg
m - 3
28
10 - 3 m
9:81m
s - 2
0:5
10 - 3 m 2

σ = 68:5mN
m - 1

Example Estimation: How high will an aqueous solution with a surface tension of 70 mN∙m-1. rise in a capillary with D = 0.75 mm inner diameter at room temperature? With h=

2
σ rK
ρ
g

it is h=

4
70
10 - 3 N
m - 1 = 38:1 mm 0:75
10 - 3 m
998 kg
m - 3
9:81 m
s - 2

For estimations like this we can use: h≈

4
70
10 - 3 N
m - 1 280
10 - 4 m 28
10 - 3 m = = 3 D=m D=m
10 kg
m - 3
10 m
s - 2 D=10 - 3 mm (continued)

276

6 Interfaces

i.e., h≈

28 mm D=mm

i.e., for further capillary diameters: D/mm 0.5 1 2 5 10

h/mm 56 28 14 5.6 2.8

Drop Shape Analysis The measurement of the shape of droplets offers further possibilities for the determination of interfacial tension and contact angles. Pendant Drop Method Under the influence of gravity, a hanging drop does not have a spherical shape but an ellipsoidal shape. The height-dependent hydrostatic pressure in the drop causes the deviation from the spherical shape. With a microscopic drop contour analysis, the surface tension of a liquid can be calculated in this way at known droplet volume and known densities of the liquid and gaseous phase. In balance of forces, the Laplace pressure in the hanging drop is as high as the hydrostatic pressure of the droplet liquid pσ = σ


1 1 =ρ
g
h þ r1 r2

ð6:53Þ

i.e., σ=

ρ
g
h 1 r1

þ r12

ð6:54Þ

Sessile Drop Method If the drop is located on a solid material, microscopic techniques can be used to determine the contact angle between the liquid and the material. From the contact angle, conclusions can be drawn about the interfacial tensions of the substances involved (cf. Table 6.6).

6.3 Measurement

277

Fig. 6.17 Spinning drop method, schematic. 1 temperature control jacket,2 rotating drop, 3 liquid in rotating capillary

1

2

3

2rz

Y

Fig. 6.18 Droplet formation at the end of a capillary

Spinning Drop Method If a liquid-filled capillary, in which a drop of a liquid that cannot be miscible with the ambient phase, is brought into rotation, the shape of the drop changes depending on the rotation frequency (see Fig. 6.17). With known density difference, the interfacial tension between the two liquids can be determined from the video drop contour analysis. Drop Volume Techniques The volume of liquid droplets escaping from a capillary depends on the interfacial tension of the liquid (Fig. 6.18). A drop leaves a capillary orifice when the weight force of the drop can overcome the tension force [24], i.e., m
g = 2π
r
σ

ð6:55Þ

At a known density of the drop liquid, the interfacial tension can be determined from the drop radius: 2π
r
σ = m
g

ð6:56Þ

2π
r
σ = V
ρ
g

ð6:57Þ

σ=

V
ρ
g 2π
r

ð6:58Þ

In order to improve the accuracy, a predetermined correction factor C is used. It is to be determined by calibration with a substance of known interfacial tension. σ=

V
ρ
g 2π
r
C

ð6:59Þ

278

6 Interfaces

Fig. 6.19 Stalagmometer for the determination of the surface tension

Specially shaped glass capillaries for falling droplets, so-called stalagmometers (Fig. 6.19), are used for the rapid determination of surface tension. A fixed volume of the liquid to be examined is slowly dropped through the stalagmometer. By counting the drops, the volume of the individual drop can be determined.

σ ρ g r V V_

interfacial tension in N
m-1 density of the droplet-forming liquid in kg
m-3 gravitational acceleration in m
s-2 drop radius in m drop volume in m3 flow in m3
s-1

ΔN Δt C

number of drops duration in s correction factor

Drop Volume Tensiometer In a continuously operating drop volume tensiometer, a precision pump is used to deliver a defined volume flow rate which leads to a continuous generation of falling drops. ΔV ΔN
V V_ = = : Δt Δt

ð6:60Þ

By counting the drops per unit of time Δt, e.g., by means of a photodiode, the volume V of a drop can be determined:

6.3 Measurement

279

V_
Δt ΔN

ð6:61Þ

V_
Δt
ρ
g N
2π
r
C

ð6:62Þ

V= then we can calculate σ=

Instead of droplets falling into air (measuring surface tension), droplets falling into liquid (e.g., water droplets in oil) or rising drops (e.g., oil droplets in water) can also be examined. With a continuously operating drop volumeter, the so-called dynamic interfacial tension can be measured. Increasing the flow of the pump leads to faster drop generation. The time span between two escaping drops is the time that a drop has spent hanging on the capillary. This is called life time of the interface or age of the interface. By increasing the flow of the pump, it is possible to create droplets with very short life time up to the time the interfacial tension is measured. In this way, it is possible to determine whether the interfacial tension depends on the life time of the droplets. If this is the case, the continuous droplet volume tensiometer can be used to study the interfacial kinetics of the present system. Bubble Point Tensiometer With a bubble point tensiometer, the air pressure is measured that is necessary to produce gas bubbles of defined size in a liquid. According to the Laplace’s equation, the surface tension between liquid and gas can be calculated from the measured internal bubble pressure. Since the interfacial tension is temperature-dependent (Table 6.4) such measurements must always be carried out under temperature-controlled conditions. Dynamic Measurement of Interfacial Tension The continuous droplet volume tensiometer for determining the time-resolved interfacial tension on droplets has been described in the previous section. Another way to examine surfaces dynamically is an overflowing cylinder technique [25, 26]. With the help of a pump, the liquid to be examined is allowed to climb up a cylinder until it overflows. In the overflowing surface, the surface tension is measured, e.g., using Wilhelmy technique. By increasing the flow of the pump, the overflow speed and thus the surface pressure increase and the surface lifetime is reduced. A Langmuir–Pockels trough (commonly: Langmuir trough) consists of a shallow tub that is filled with the liquid to be examined. With the help of a movable barrier, a film of surface-active substances can be compressed or expanded. During compression, the interfacial tension is continuously measured, which gives information about occupancy densities and film states. The speed of the moving barrier can be changed, which allows the study of kinetic effects [27–29]. Contact angle measurements can also be carried out dynamically. For this purpose, the contact angle is measured during movement of a drop at an interface.

280

6 Interfaces

The direction of movement can be wetting or dewetting. Often a hysteresis is observed between wetting and dewetting which can be used to characterize a solid interface, e.g., its roughness. Dynamic investigations of interfacial effects are also possible by the drop contour analysis of lying or hanging droplets whose volume is not constant but pulsating. With the pulsating drop method the interface experiences a continuously alternating dilation and contraction. By varying the pulsation frequency, time-dependent effects of surface-active substance on can be investigated. The dynamic behavior of interfacial films can be described with variables such as interfacial elasticity and viscosity. As shown in Chap. 4 both quantities can be described together by using the complex interfacial elasticity. The complex modulus of interfacial elasticity consists of a real part for interfacial elasticity and an imaginary part for interfacial viscosity. In interfacial rheology, often the interfacial viscoelasticity as a function of the pulsation frequency is investigated [30, 31]. Further Reading Theme Espresso coffee foam: interfacial rheology Alginate capsules: interfacial elasticity Curcuma encapsulation by coacervation Double emulsion in gel droplets from Ca-alginate Egg white: dynamic interface measurement by pendant drop technique Emulsions with droplets

dQ T

ð8:13Þ

Equation 8.13 is an expression of the second law of thermodynamics, stating that if a system is not in equilibrium the entropy S tends to increase and to reach a maximum. It can remain constant in the best case (for reversible operations). In all spontaneous processes, i.e., in natural processes, the entropy increases: dS > 0. With the help of entropy, state functions can be formulated. For example, free energy F, as the difference between the internal energy and the product of temperature and entropy is also known as Helmholtz energy. F=U -T
S

ð8:14Þ

Also, free enthalpy, G, s the difference between enthalpy and the product of temperature and entropy, and is known as Gibbs energy. G=H -T
S

ð8:15Þ

Both F and G indicate the maximum amount of work a system can deliver. Maximum means that the conversion would be completely reversible. In this understanding F and G have the property of a potential. We already know the concept of potential from the section on transport variables in Chap. 7. A potential curve as in Fig. 8.2 has curve sections with positive and negative slope as well as extreme points. From the derivative, i.e., the slope of the potential curve, it is possible to read how a system will behave when the variable x changes. In mechanics, the concept of potential is used, for example, in connection with the position energy (potential energy) dependent on the height of a body. As long as the

320

8

Thermal Properties

Fig. 8.2 Potential of a system—schematically—as a function of a variable φx. The point O indicates the position of a stable equilibrium

body is not at the minimum point of its potential energy, a force acts on it that drives it towards this minimum. The direction and magnitude of this force can be calculated from the derivation of the potential curve. At the minimum of the curve (point O in Fig. 8.2), the slope of the potential curve and thus the force is zero. Definition The concept of potential can be illustrated by the position energy of a body in the Earth’s gravitational field: The position energy is calculated from the scalar →

→

product E = m g
h . If the height is increased by dh, the position energy changes by dE = mg
dh. The term dE dh = mg is the potential of the body in the gravitational field. The product of potential and dh provides the energy change dE = dE dh
dh of the body. Analogously, in thermodynamics we have dG = dG dx
dx , this means that the change in the free enthalpy G is obtained from the product of thermodynamic potential dG dx and the change dx. In order to familiarize ourselves with the use of thermodynamic potential, we first consider the dependence of enthalpy H on pressure and temperature.with dH = T
dS þ Vdp

ð8:16Þ

or dH =

∂H ∂p


dp þ T

∂H ∂T


dT

ð8:17Þ

p

in an isothermal case with dS = 0, the partial differential dH becomes

∂H ∂p T

and

with Eq. (8.16) ∂H ∂p

= T

∂ðV
dpÞ ∂p

=V T

ð8:18Þ

8.3 Thermodynamics: Basis Principles

321

and the partial differential is ∂H ∂T

= cp

ð8:19Þ

p

The first derivative of the enthalpy function after the variable p, i.e., the partial indicates that the thermodynamic potential is the volume V, differential ∂H ∂p T

which is a state variable. The first derivative of the enthalpy function after the indicates that the thermodynamic variable T, i.e., the partial differential ∂H ∂T p potential is the heat capacity cp, which is a material property. → Attention While a differential is usually

written, a partial differential is written ∂G ∂x

Example: The expression

means “partial differential from G with

dG dx is ∂G ∂p T

respect to p at constant temperature T”. Let us now consider the dependence of the free enthalpy G on pressure p and temperature T. Here, too, we want to describe the dependence of p and T as thermodynamic potentials. The differential of Eq. (8.15) is dG = dH - T
dS - S
dT

ð8:20Þ

dH = T
dS þ Vdp

ð8:21Þ

dG = T
dS þ Vdp - T
dS - S
dT

ð8:22Þ

dG = Vdp - S
dT

ð8:23Þ

with

we get

i.e.

if we write the p-T dependence of dG as partial differentials, this is dG =

∂G ∂p


dp þ T

∂G ∂T


dT

ð8:24Þ

p

By comparing of Eq. (8.23) with Eq. (8.24) we see ∂G ∂p

=V T

ð8:25Þ

322

8

Thermal Properties

and -

∂G ∂T

=S

ð8:26Þ

p

The partial differential of G, i.e., the first derivative, provides the potential of G with respect to p or T. The partial differential of the free enthalpy G with respect to pressure is identical to the volume V of the system. The negative partial differential of the free enthalpy G with respect to temperature is identical to the entropy of the system. Let us compare again with the case of mechanical position energy. There, the potential was the mechanical force that drives the body into equilibrium. Therefore, V and S take on the role of “mechanical force” with respect to G, which drives the system towards thermodynamic equilibrium. As a simple case, let’s consider a food that emits water vapor and is on its way to equilibrium with its ambient atmosphere. We leave the temperature constant at room temperature. If we express the volume of water vapor by the volume of an ideal gas, we obtain for the potential ∂G ∂p

=V = T

R
T p

ð8:27Þ

dp p

ð8:28Þ

or dG = R
T
or dG = R
T
d ln p

ð8:29Þ

The change in free enthalpy in this simple system can be calculated after measuring the water vapor pressure. If we consider that water vapor is not an ideal gas, we can replace the pressure p with the so-called fugacity f of the water vapor. dG = R
T
d ln f

ð8:30Þ

The dimensionless relative fugacity is the so-called activity of a substance. In case of water, we have already know this quantity as water activity aW (refer Chap. 1). Up to this point, we have focused on the variables p and T. In the case of chemical reactions, dG is dependent on the amount of substance n in the reactants and products, and is generally dependent on ni. The partial differential of free enthalpy with respect to the amount of substance ni is again a thermodynamic potential called chemical potential μ. ∂G ∂ni

= μi

ð8:31Þ

p,T,njðj ≠ iÞ

The chemical potential describes the potential of a substance to undergo a reaction and thereby to release reaction energy.

8.4 Heat capacity

323

G n f μ S aW

Gibbs energy in J substance in mol fugacity in Pa chemical potential in J
mol-1 entropy in J
K-1 water activity

In this section, we have seen the physical significance of the first derivative of enthalpy and free enthalpy. In Sect. 8.5, we will see that the second derivative of these functions is also physical quantities.

8.4

Heat capacity

Heat capacity of a material is a thermal property that indicates the ability of the material to hold and store heat. It can be quantified by specifying the amount of heat that is needed to raise the temperature by a specified amount. Mathematically, it is the quotient of heat ΔQ divided by change in temperature ΔT: C=

ΔQ ΔT

ð8:32Þ

C=

dQ dT

ð8:33Þ

For infinitesimal changes

When heat capacity is defined only in this way, it will also depend upon the mass or size of the material sample, and serves as a property only of the specific sample size measured. For this reason, we normally measure and report the heat capacity bases on a common unit of mass. When we do this, we call it the specific heat capacity. Sometimes this property is called specific heat of the material but this should be avoided because dQ/dm = q is specific heat. C m

ð8:34Þ

1 dQ dq
= m dT dT

ð8:35Þ

c= c=

In order to help better understand heat capacity, let us assume we wish to determine how much heat is needed to raise the temperature of 1 L of water at 21 °C up to 23 °C (by 2 degrees K). When we do this, we measure the heat required to be 8.36 kJ. When heat is added to a system like this (liter of water), the water molecules experience an increase in their kinetic energy. They move in both rotational and translational motion at faster rates. If we insert our finger (or a thermometer) into this

324 Table 8.6 Heat capacity terms at constant pressure and volume

8

Thermal Properties

p = const.

V = const.

C p = dH dT cp = m1
dH dT dh cp = dT

CV = cV = cV = -1

dU dT 1 dU m
dT du dT

heat capacity in J
K specific heat capacity in J
kg-1
K-1 heat in J specific heat in J
kg-1 temperature in K mass in kg enthalpy in J specific enthalpy in J
kg-1 internal energy in J specific internal energy in J
kg-1

C c Q q T m H h U u

liter of water, we can sense this increased thermal energy level by a warming sensation on our finger, and a rise in the reading of the temperature scale of the thermometer. Therefore, we use temperature as a measure of increased thermal energy. In this case, the temperature increase was ΔT = 2 K. Example Calculation of the specific heat capacity of water. 1 kg of water was supplied with 8.36 kJ of heat, after which the temperature of the water rose by 2 K: 1 ΔQ
m ΔT 8:36 kJ cp = 1 kg
2 K cp = 4:18 kJ
kg - 1
K - 1

cp =

Often in thermodynamics, it is necessary to distinguish between internal energy terms in which displacement work is present or not present. When there is no displacement work, the volume of the system remains constant (V = const., dQ = dU ), and the subscript V is used with the heat capacity term. When displacement is present, then the pressure of the system remains constant ( p = const., dQ = dH ), and the subscript p is used with the heat capacity term. Table 8.6 summarizes these designations.

8.4 Heat capacity

8.4.1

325

Ideal Gases and Ideal Solids

For ideal gases and solids, the movement of molecules in response to thermal energy can be predicted from theory. Therefore, the heat capacity of such ideal substances can also be predicted from theory. In an ideal gas, the molecules are free to have translational motion in the three directions of three-dimensional space (back and forth, side-to-side, and up and down). Thus, we say, they have three degrees of freedom for translational motion, f = 3. In addition to translational motion, molecules are also free to have rotational motion. Gas molecules consisting of only two atoms, such as oxygen, O2, or nitrogen N2, are bonded linearly. This makes them look like a rigid body which can rotate about two axes. This gives them two more additional degrees of freedom. Thus, they have a total of five degrees of freedom, f = 5. In the case of solids, the molecules are fixed in place, and can have no translational or rotational motion. Therefore, they have no translational or rotational degrees of freedom. However, they are free to oscillate in three directions. Thus, they have six degrees of freedom in response to thermal energy. Table 8.7 lists the degrees of freedom for atoms or molecules in some simple ideal systems. In the mechanical model of an ideal gas, the kinetic energy is equal to the thermal energy: f 1 m
v2 = kT 2 2 E=

ð8:36Þ

f kT 2

ð8:37Þ

The internal energy U for N particles (atoms or molecules) is f U = N
kT 2

ð8:38Þ

N f f
k
NA
T = n
R
T NA 2 2

ð8:39Þ

f R f U = n
M


T = m
Rs
T 2 M 2

ð8:40Þ

U=

With the heat capacity for constant volume Table 8.7 Degrees of freedom for atoms and molecules in simple, ideal systems One atom gas Two-atom gas molecule Molecules in a solid

Translational 3 3 –

Rotational – 2 –

Oscillatory – 0 6

Total f=3 f=5 f=6

326

8

CV =

Thermal Properties

dQ dU = dT dT

ð8:41Þ

we get f C V = m
Rs 2

ð8:42Þ

or f R 2 s

ð8:43Þ

dQ dH dU p
dV dU m
Rs
dT = = þ = þ dT dT dT dT dT dT

ð8:44Þ

cV = In the isobaric case we get Cp = So, we see

C p = CV þ m
Rs

ð8:45Þ

cp = c v þ R s

ð8:46Þ

or

k NA m v N U M R Rs f p V Cp CV cp cV

Boltzmann’s constant Avogadro’s constant mass in kg velocity in m
s-1 number of particles internal energy in J molecular mass in kg
mol-1 universal gas constant in J
K-1
mol-1 specific gas constant in J
K-1
kg-1 degrees of freedom pressure in Pa volume in m3 heat capacity ( p = constant) in J
K-1 heat capacity (V = constant) in J
K-1 specific heat capacity ( p = constant) in J
kg-1
K-1 specific heat capacity (V = constant) in J
kg-1
K-1

In the case of solids without very high pressure, it is cp ≈ cV

ð8:47Þ

8.4 Heat capacity

327

Table 8.8 Theoretical and experimental values of specific heat capacity for selected systems (from [5]) Substance He N2 Air Pb

f 3 5 5 6

Rs (kJ
kg-1
K-1) 2.078 0.297 0.287 0.0401

cp (theory) (kJ
kg-1
K-1) 5.195 1.039 1.0045 0.1203

cp (exp.) (kJ
kg-1
K-1) 5.23 1.03 1.005 0.129

with f = 6, we have: cp ≈ 3
Rs

ð8:48Þ

This is called the rule of Dulong-Petit for the heat capacity of solids. Table 8.8 illustrates how theoretical values based on thermodynamic considerations and experimental data for heat capacity fit. From Table 8.8 we can see that the agreements for estimates are quite sufficient, especially in the area of gases, which can be approached as ideal. In the case of solids, however, there are also numerous major deviations from the Dulong-Petit rule at room temperature. To estimate the heat capacity of solid foods, the following method should preferably be used. It is based on an additivity of the heat capacities of the material components.

8.4.2

Heat Capacity of Real Solids

If specific heat capacities of the chemical constituents of a substance and its composition are known, the heat capacity of the substance can be roughly estimated by adding up the heat capacities of the individual components: cp =

xi
cp,i

ð8:49Þ

mi m

ð8:50Þ

i

with xi = cp xi cp,i mi m

specific heat capacity in J
K-1
kg-1 mass fraction of component i specific heat capacity of component i in J
K-1
kg-1 mass of component i in kg total mass in kg

Table 8.9 lists the values for specific heat capacities of food components.

328

8

Table 8.9 Some data for specific heat capacities of food constituents [6]

Water Carbohydrates Proteins Fats Ash (Minerals) Ice Non-fat solids (from animal) Non-fat solids (from plants)

Thermal Properties cp in kJ
kg-1
K-1 ≈ 4.2 ≈ 1.4 ≈ 1.6 ≈ 1.7 ≈ 0.8 ≈ 2.1 ≈ 1.34 . . . 1.68 ≈ 1.21

Example Calculation of the heat capacity of a food The composition of a food product is given as follows: Mass fraction xi in% (m/m) 84 12 3 1

Water Fat Protein Minerals

Solution: The heat capacities of the individual components are added to the total heat capacity. cp kJ
kg - 1
K - 1 = cp = 3:8 kJ
kg

8.5

-1


K

0:84
4:2 þ 0:12
1:7 þ 0:03
1:6 þ 0:01
0:8

-1

Classification of Phase Transitions

When we look at a simple phase transition, such as when a substance undergoes freezing from a liquid to a solid phase, then we normally see significant changes in enthalpy, entropy and volume. When we start in the liquid phase toward transition to the solid phase, we will observe these changes at the phase transition point, which is a point on the p-T-diagram (phase diagram) of the material. The chemical potential of both phases is equal at the transition point, but the enthalpies and entropies of the different phases (solid and liquid) are different. In other situations, we can observe phase transitions where a solid is melting to become a viscous liquid without sudden change in enthalpy. This type of phase transition is called a glass transition, and is observed when we deal with

8.5 Classification of Phase Transitions

329

non-crystalline materials. For example, formulations with solid carbohydrates like confectionery or powders can exhibit glass transitions [7]. According to Ehrenfest phase transitions are called first-order (n = 1) phase transitions when the first derivative of Gibbs’ enthalpy over temperature has a discontinuity (a “jump” in the curve). In the same manner, a phase transition is of second order (n = 2) when the discontinuity appears in the curve for the second derivative of Gibbs’ enthalpy over temperature. Let us consider a simple example of a liquid–solid phase transition upon cooling of a material. Then, the thermal effect observed during this transition will be the release of an enthalpy ΔtrsGm which is the difference in the enthalpy of the liquid phase G00m and the solid phase G0m . Δtrs Gm = G00m - G0m = 0

ð8:51Þ

The partial derivate over temperature is: ∂G00m ∂Δtrs Gm = ∂T ∂T

p

∂G0m ∂T

= - S00m þ S0m = - Δtrs Sm ≠ 0

ð8:52Þ

p

with the help of the second law of thermodynamics this means for the phase transition enthalpy ΔtrsHm T
Δtrs Sm = Δtrs H m ≠ 0

ð8:53Þ

From Sect. 8.4 we know Cp is the derivative of H over T, so it is the second derivative of ΔtrsGm over temperature. Having this in mind, let us now look at the graphs in Fig. 8.3. Let us imagine moving along on a curve in the diagram from higher to lower temperature (in the direction from liquid to a solid state). In the upper left figure, we see the G-T-graphs of the liquid phase (2) and the solid phase (1), which meet at the transition temperature. Upon decreasing the temperature, the system will normally change at Ttrs from curve 2 to curve 1 to reach a low energy state. Plotting the derivative of these curves, we get the H-T-curve immediately below (middle left), which has a distinct discontinuity (“jump”) at the transition temperature Ttrs. This is a first-order phase transition. Let us now look at the graphs for second-order transitions shown on the right side in Fig. 8.3. In the upper right graph, we can see that, at the transition temperature Ttrs, the G-T-curve of phase 2 and phase 1 do not intersect, but are tangent to each other, and simply touch at a point of tangency. Upon decreasing the temperature, we will enter the transition from phase 2 to 1. The first derivative is shown in the graph immediately below (middle right). In this case, the first derivative curve, which is H, simply changes direction abruptly at the transition point Ttrs, but shows no discontinuity or “jump.” The second derivative (bottom right) does show such a discontinuity “jump” at Ttrs: This transition is a second-order phase transition. A glass transition is a typical example of a second-order phase transition. In a thermal analysis experiment,

330

8 first order phase transition (n = 1)

Thermal Properties

second order phase transition (n = 2)

2

2 1

1

Gp,m

Gp,m 1

2 1

2

Ttrs

Ttrs

T

Hm

T

Hm

Ttrs

Ttrs

T

c p,m

T

c p,m

Ttrs

Ttrs

T

T

Fig. 8.3 Classification of phase transitions after Ehrenfest, temperature scan Table 8.10 Classification of phase transition based on the derivatives of Gibb’s enthalpy

1. Derivative of von G over T 2. Derivative of von G over T

∂Δtrs Gm ∂T p 2

= - Δtrs Sm

∂ Δtrs Gm ∂T 2 p

=-

Δtrs cp,m T

First order phase transition ≠0

Second order phase transition =0

≠0

≠0

these orders of transition can be identified by having a “jump” in the heat capacity Cp,m, but not in the enthalpy ΔHm. Table 8.10 summarizes that. Ehrenfest’s classification based on the derivative of Gibbs’ enthalpy generally formulated is: A phase transformation of a pure substance of n-th order has an unsteadiness of the n-th derivative over temperature of Gibbs’ enthalpy.

8.5 Classification of Phase Transitions

∂ G0m ∂T n

331

n

∂ G00m ∂T n n

≠ p

p

Ehrenfest’s classification of phase transitions works analogously with the derivative of Gibbs’ enthalpy over pressure instead of temperature: ∂ G0m ∂pn n

∂ G00m ∂pn n

≠ T

T

We had already learned about the first derivative of Gibbs’ enthalpy with respect to pressure when using the thermodynamic potential for the water activity of a food (refer Eq. 8.25) ∂Δtrs Gm ∂p

= Δtrs V m T

The second derivative of Gibbs’ enthalpy over pressure is 2

∂ Δtrs Gm ∂p2

= - V m
Δtrs κ T

So the order of a phase transformation can be determined experimentally by a temperature scan or by a pressure scan. In a temperature scan, one will pay attention to jumps in enthalpy and heat capacity, in a pressure scan to jumps in molar volume and compressibility κ T (Fig. 8.4). C c Q T p H h U u G S κT V

heat capacity in J
kg-1
K-1 specific heat capacity in J
kg-1
K-1 heat in J temperature in K pressure in Pa enthalpy in J specific enthalpy in J∙kg-1 internal energy in J specific internal energy in J∙kg-1 Gibbs’ enthalpy in J entropy in J∙K-1 isothermal compressibility in K-1 volume in m3

332

8

Thermal Properties

Fig. 8.4 Classification of phase transitions after Ehrenfest, pressure scan

Indices trs m n

transition molar order

With these excursions into the thermodynamics of phase transitions, which are not simply freezing/melting or evaporating/condensing, we want to leave this section and return to thermal properties. We might keep in mind that glass transitions, as well as some solid–solid transitions are related to the quality and the stability of the food substance in which they occur [8]. There are two approaches through which to obtain data for constructing phase diagrams: One approach is by scanning the temperature (thermal analysis) and observing the related quantities that result from the scan. The other approach is to study materials under an isothermal pressure scan.

8.6

Heat Transfer in Food

Spontaneous heat transfer always takes place from a region of higher temperature to a surrounding region of lower temperature. There are four mechanisms by which heat can transfer: radiation, conduction, convection, and phase transitions. In

8.6 Heat Transfer in Food

333

Table 8.11 Mechanisms of heat transfer Mechanism Heat radiation Heat conduction Heat convection Phase transitions

Description Electromagnetic radiation with wavelengths from 1 μm to 1 mm Transport of heat by excited molecules “bumping” against each other Transport of heat being carried by a flowing fluid Uptake/release of latent heat

Example Broiling, grilling, infra-red heat lamps, solar heating. Solid being heated at one end, warming opposite end Flowing hot water or air in central home heating system Condensing of water vapor

Table 8.11 they are listed and described briefly. In this chapter section and those that follow, we will discuss each of these mechanisms and the thermal properties that are needed in each case.

8.6.1

Heat Radiation

Heat radiation is electromagnetic radiation with frequencies below that of visible light. It is also called infrared radiation. All bodies with a temperature above 0 K emit heat by radiation. As with all heat transfer mechanisms, heat will flow from the body at higher temperature to the one at lower temperature. However, there is no need for the bodies to contact each other, nor is there any need for any substance to exist between the two bodies. Therefore, heat radiation can occur in a perfect vacuum and over great distances. For example, this is how we receive heat from the sun. For heat radiation to occur, the bodies must simply be able to “see” each other. The Stefan–Boltzmann law allows us to calculate the heat flow under radiation: Q_ = A
ε
σ
T 4 ε σ T A

ð8:54Þ

emissivity of a body Stefan–Boltzmann constant thermodynamic temperature of the body in K area which is emitting or absorbing radiation in m2

When we have two bodies of different temperatures T1 and T2, each one is emitting heat radiation according to Stefan–Boltzmann’s law, and at the same time they are receiving radiation from the other body. So there is a net heat flow of:

334

8

Thermal Properties

ΔQ_ 12 = A
C12
T 42 - T 41

ð8:55Þ

heat flow from/to body 1 to/from body 2 in J∙s-1

ΔQ_ A T1 T2 C12 ε1 ε2

area emitting radiation in m2 temperature of body 1 in K temperature of body 2 in K radiation exchange factor in W
K-4
m-2 emissivity of body 1 emissivity of body 2

The heat radiation exchange factor C12 is a quantity combining the emissivity of the bodies as well as their geometries (shape factors) and relative surface absorptivity. If, for example, we have two parallel plates with identical geometry and surface absorptivity emitting/absorbing heat radiation, then the heat radiation exchange factor would consist of the following expression, C12 [9]: C 12 =

1 ε1

σ þ ε12 - 1

ð8:56Þ

8.6.1.1 Emissivity The emissivity of a material gives an indication of its ability to emit electromagnetic radiation (in this case, heat radiation). A corollary property is absorptivity, which indicates the ability of a material to absorb heat radiation. When a material is in thermal equilibrium, we can assume that its emissivity and absorptivity are equal. This is known as Kirchhoff’s Law of thermal radiation. An ideal black body is defined as a body having maximum absorptivity, with a value for absorptivity of ε = 1. An ideal reflective body is incapable of absorbing any heat radiation, and has a value for absorptivity of ε = 0. All real bodies are known as “grey bodies,” and have values for absorptivity between zero and one. If we know the emissivity ε of a material body and its surface temperature, we can calculate the energy lost from the body caused by radiation heat transfer. Likewise, if we do not know the temperature but can measure the quantity of radiant heat energy emitted, we can calculate the surface temperature of the body. This is the principle for measuring surface temperatures by infra-red thermometry (IR thermometers, see also Chap. 16). If we want to tabulate values for the emissivity of materials, the surface condition of the material must be specified. Thus, a polished metal surface can have ε below 0.05 while the corroded surface of the same material has a value of ε > 0.75. With the help of suitable coatings, the emissivity of materials can be controlled [10]. This is especially important and useful with packaging materials. The emissivity of liquid water is around ε = 0.95 and unpackaged foods also have values of this magnitude [11].

8.6 Heat Transfer in Food

8.6.2

335

Heat Conduction Transfer

Heat transfer by conduction was first mentioned in Chap. 6 as an example of other diffusion-like transport phenomena that can be described with the same type of mathematical equation as that used to describe molecular diffusion through a material substance. When we first approach the study of heat conduction, it is important to make a distinction between heat conduction under steady state conditions (temperatures remain constant at any point over time while heat is flowing) and unsteady state, or transient, conditions (temperatures at any point change over time while heat is flowing). In this section, we will limit our study to steady state heat conduction. This is the situation encountered most frequently in food processing involving the heating and cooling of liquid food products through heat exchangers and holding tubes. These heating and cooling systems operate under steady state conditions because product and heat exchange medium temperatures at the entrance and exit of these systems remain constant over time while conduction heat transfers at a steady (constant) rate. Most heat exchangers used in food processing operations are either a plate-type (made up of flat plates) or tubular-type (made up of cylindrical tubes). In a plate heat exchanger, liquid product flows through a narrow space between two parallel stainless-steel plates. On the other side of these plates, a heat exchange medium at a different temperature is flowing (e.g., hot water or condensing steam). This situation causes heat to flow from the high-temperature side to the low-temperature side causing the temperature of the cool incoming product to rise as it exchanges heat with the hot heat exchange medium fluid on the other side. In a tubular heat exchanger, the product flows through a narrow cylindrical tube to exchange heat with a flowing fluid heat exchange medium on the other side of the tubular wall. For this reason, we will first consider heat conduction across a flat plate “wall” barrier. Then, we will consider heat conduction across a cylindrical “wall” barrier. Heat transfer through a material can occur in all three directions (dimensions) of space. However, in heat exchanger applications, the temperature gradient across the thin stainless steel wall drives the heat transfer predominantly in the one direction crossing the wall, and there is little or no temperature gradient (potential) causing heat to travel along the metal wall material in either of the other two directions. Therefore, we can limit our analysis of heat conduction to the simplest case of one-dimensional steady state heat conduction across a flat plate wall.

8.6.2.1 One-Dimensional Steady State Heat Conduction Across a Flat Plate Let us consider heat travelling across a solid flat plate by conduction like in Fig. 8.5 we can express the quantity of heat flow by Fourier’s first law:

336

8

Fig. 8.5 Temperature profile across a solid flat plate

Thermal Properties

O

T0 T1 T

G

x

Table 8.12 Sign of temperature gradient and resulting heat flow in Fig. 8.5 Case x>0 T1 > T0 x>0 T1 < T0

Temperature gradient Positive

Description T is increasing with increasing x

Negative

T is decreasing with increasing x

Q_ dT = -λ
A dx

ð8:57Þ

Q_ T - T1 =λ
0 δ A

ð8:58Þ

T - T1 Q_ = A
λ
0 δ

ð8:59Þ

with the symbols of Fig. 8.5

so

dT dx

temperature gradient in K
m-1

λ Q_

heat conductivity in W
K-1
m-1 heat flow in W

Q_ A

heat flow density in W
m-2

The negative sign in Fourier’s law is necessary in order to have a positive heat flow in response to a negative temperature gradient. This is because heat will only flow from high to low temperature, which is “down hill,” or having a negative slope. Table 8.12 shows the type of responding heat flow to positive and negative temperature gradients.

8.6 Heat Transfer in Food

337

The heat transferred can be calculated by integration over time: dQ Q_ = dt

ð8:60Þ

or ðT 1 - T 0 Þ
dt δ

ð8:61Þ

λ A

ðT 0- T 1 Þ
d t δ

ð8:62Þ

dQ = - A
λ
it is t

Q= t=0

The λ is the thermal conductivity of the material of the solid wall is the same. The reciprocal quantity is a resistance to heat conduction, it is called specific thermal resistivity. Materials designed to minimize heat conduction must have low thermal conductivity (e.g., materials for thermal insulation on buildings or pipelines). Materials with high thermal conductivity are suitable for achieving rapid heat transfer in heat exchangers for heating or cooling.

8.6.2.2 Three-Dimensional Steady State Heat Conduction In the most general case, when temperature gradients exist in all three dimensions of space, heat will transfer through a material in all three directions. In this situation, Fourier’s law must be expressed in the form of a partial differential equation to include terms for the temperature change in each of the three directions x,y,z: Q_ ∂T ∂T ∂T = -λ
, , A ∂x ∂y ∂z

ð8:63Þ

or Q_ ∂ ∂ ∂ , , T = - λ
grad T = -λ
A ∂x ∂y ∂z

ð8:64Þ

or with the abbreviation ∇T = grad T =

∂ ∂ ∂ T , , ∂x ∂y ∂z

ð8:65Þ

shortly Q_ = - λ
∇T A where ∇ is called Nabla or dell operator in matrix algebra.

ð8:66Þ

338

8

Fig. 8.6 Temperature profile across a multilayer solid flat wall

Thermal Properties

O3

O1 T0 T

T1

G1

T2

G2

x

T3

G3

8.6.2.3 One-Dimensional Steady State Heat Conduction Across Multiple Layers Figure 8.6 illustrates the case of conduction heat transfer occurring across a flat wall that is made up of several layers of different materials, with each material having a different thermal conductivity λ and different layer thickness δ. We can calculate the total heat flow through the wall with help of the concept of additive resistors, refer to Chap. 6. In this case, the mathematical expression for the quantity of heat flow can be derived as follows. Because the heat flowing through all three layers is the same: Q_ 1 = Q_ 2 = Q_ 3

ð8:67Þ

Q_ 1 Q_ 2 Q_ 3 = = A A A

ð8:68Þ

with T0 - T1 =

Q_ 1 δ1
A λ1

T - T0 Q_ 1 = -λ
1 A δ1

ð8:69Þ

and T1 - T2 =

Q_ 2 δ2
A λ2

T - T1 Q_ 2 = - λ2
2 A δ2

ð8:70Þ

8.6 Heat Transfer in Food

339

and T2 - T3 =

Q_ 3 δ3
A λ3

Q_ 3 T - T2 = - λ3
3 A δ3

ð8:71Þ

By adding the temperature differences from Eqs. (8.69–8.71) we get the total temperature difference Δ T Δ T = T0 - T3 =

Q_ δ δ δ
1þ 2þ 3 λ1 λ2 λ3 A

ð8:72Þ

So, we get for heat flow density: Q_ = A

1 δ1 λ1

þ

δ2 λ2

þ δλ33


ΔT

ð8:73Þ

Writing a more general expression for the steady state case with n layers we have: for the heat flow density Q_ A

=

1

n

i=1

δi λi

for the heat flow Q_ = A
n1
Δ T


ΔT

i=1

δi λi

for the heat quantity Q = A
n1
Δ T
t i=1

δi λi

When we have different sizes of areas for each layer, we write for the heat flow Q_ =

1 n i=1

δi λi
Ai


ΔT

ð8:74Þ

A comparison of Eq. (8.74) with the related expressions in Table 7.7 shows that the term in the denominator is equal to the sum of the heat resistances R of each layer. The heat conduction resistance R results from the reciprocal thermal conductivity and the geometric ratio of each layer Aδ : R=

δ λ
A

ð8:75Þ

The concept of addition of resistances can also be used here in the case of steady state multilayer conduction of heat. We can write this in a more abbreviated way as follows: n i=1

δi = λi
A i

n

Ri = Rges i=1

With Rtotal First Fourier’s Law gets the appearance of Ohm’s Law:

ð8:76Þ

340

8

Thermal Properties

1
ΔT Rtotal

ð8:77Þ

Δ T = Rtotal
Q_

ð8:78Þ

Q_ = Respectively

λ δ Q_

heat conductivity in W
K-1
m-1 thickness in m heat flow in W

Q A R i n

heat in J area in m2 heat resistance in K
W-1 layer index total number of layers

8.6.2.4 One-Dimensional Steady State Conduction Across a Single Layer Cylindrical Wall Figure 8.7 illustrates the case when heat must transfer across a cylindrical wall, such as when tubular heat exchangers are used. In this case, the same expression for Fourier’s law still applies except that the total wall thickness must be expressed as the difference between outer and inner tube radii, and the incremental distance along the tube wall thickness must be expressed as radius r. Expressing the Fourier law for an infinitesimally thin cylinder wall Q_ dT = -λ
A dr

ð8:79Þ

we get the heat flow by integration over the cylindrical wall dr: Fig. 8.7 Temperature profile across a single-layer cylindrical wall, schematic

T0

T1 r0

r1

8.6 Heat Transfer in Food

341

dT = T1

r1

dT = -

Q_ Q_ dr = dr A
λ 2π
l
r
λ r1

Q_ Q_ dr = 2π
l
r
λ 2π
l
λ

r0

T0

ð8:80Þ

dr r

ð8:81Þ

r0

T1 - T0 = -

Q_ r ln 1 2π
l
λ r 0

ð8:82Þ

so 2π
l
λ
ðT 0- T 1 Þ Q_ = ln rr10

ð8:83Þ

An alternative calculation for the heat conducted in Fig. 8.7 is to use an average area Am. The value of Am can be calculated by use of an average radius rm. For the simplest case in a thin-walled tube, we can use the arithmetic mean between outer and inner diameter for the average radius rm rm =

r1 þ r0 2

ð8:84Þ

But, let us first calculate a general average radius rm between r0 and r1. Equation (8.79) is going to be: -

λ Q_ Q_ =
ðT 1- T 0 Þ = A m 2 π r m
l ðr 1 - r 0 Þ

ð8:85Þ

For the heat flow, we get: 2 π
rm
l
λ
ðT 0- T 1 Þ Q_ = ðr 1 - r 0 Þ

ð8:86Þ

Comparing this with Eq. (8.83) provides: 2π
l 2 π
rm
l
λ
ðT 0- T 1 Þ =
λ
ðT 0- T 1 Þ ln rr10 ðr 1 - r 0 Þ

ð8:87Þ

so rm =

r1 - r0 ln rr10

ð8:88Þ

So, now we have an algorithm to calculate rm for all type of tubes. This type of average radius rm is called the logarithmic mean radius. It can be used for all cases of

342

8

ra 2˜r i I

II

Thermal Properties

r a  2˜ r i III

Fig. 8.8 Thick walled tube (I) and thin walled tube (III). II is borderline between I and III

tubes. In case of a thin-walled tube, we learned we could use the arithmetic average. But how can we decide what is a thin-walled or thick-walled tube? Figure 8.8 illustrates that a thin-walled tube is a tube with outer radius smaller than twice the inner radius (ra < 2 ri). The opposite case is a thick-walled tube when the outer radius is greater than twice the inner radius (ra > 2 ri), and we call a tube thick walled. Between both cases we have the case in which the outer radius is precisely twice the inner radius (ra = 2 ri), which presents a borderline situation, and either approach can be used. At this point, we have learned that usage of the arithmetic mean is an approximation, and usage of the logarithmic mean is exact. But how big is the error we might get by using the approximation from the arithmetic mean instead of the exact result from the logarithmic mean? Let us compare: We calculate for case II in Fig. 8.8 Logarithmic mean r m = raln-rari

Arithmetic mean rm =

r a þri 2

2 ri - ri 2r ln r i

rm =

2 r i þr i 2

ri ln 2

r m = 32 r i rm = 1.5
ri

ri

with ra = 2ri is rm =

i

rm =

rm = 1.44
ri

i - 1:44
r i Therefore, relative error is:Δrmrm = 1:5
r1:44 = 0:06 1:44 = 4%
r i This means that if, for reasons of simplification, the arithmetic mean value for the mean radius I m is used instead of the logarithmic mean, the value calculated in this way for the radius of thin-walled pipes is a maximum of 4% below the correct value. The thinner the pipe wall, the smaller the difference between arithmetic and logarithmic mean. In the case of thick-walled pipes, the thicker the pipe wall, the greater the difference, which is always greater than 4%. The approximation “arithmetic mean instead of the logarithmic mean” is therefore suitable for thin-walled tubes and

8.6 Heat Transfer in Food

343

rather not suitable for thick-walled pipes. Considerations of this kind allow the quantification of the error caused using an approximation. By knowing the error quantitatively, better decisions can be made. There are cases in which an error of 4% is not tolerable, but also cases in which, for example, due to other uncertainties, an error as great as 10% can be tolerated. For the cylindrical, one-dimensional, steady state, single-layer case we get: Heat flow density Q_ Am

=

λ r a - ri


ðT i- T a Þ

Heat flow Q_ = Am
ra -λ ri
ðT i- T a Þ

Heat Q = Am
ra -λ ri
ðT i- T a Þ
t

8.6.2.5 One-Dimensional Steady State Conduction with Multi-layer Cylindrical Wall Figure 8.9 illustrates the case of conduction heat transfer occurring across a cylindrical wall that is made up of several layers of different materials, with each material having a different thermal conductivity λ and different layer thickness δ. In this case, the general mathematical expression for the quantity of heat flow can be expressed the same as in the case for a flat plate wall, refer to Eq. (8.74). Starting with the equation from the flat plate multilayer case: Q_ =

1 n i=1

δi λi
Ai


ΔT

ð8:89Þ

The values of the areas Ai depend on the inner and outer radius of each layer. Therefore, they are not the same for all the layers. They must to be added separately. The denominator of Eq. (8.89) is the sum of all n heat resistances. n i=1

δi = λi
Ai

n

ð8:90Þ

Ri i=1

This is the sum of the n involved thermal conduction resistances Ri. Again, we have a case in which to use the concept of adding resistances. Fig. 8.9 Multilayer cylindrical solid wall

r0

r1

r2

344

8

Thermal Properties

However, the areas Ai depend on their respective radii ri and therefore are not the same size for all layers. For this reason, the areas Ai must be included individually in calculating of the heat flow. With the designations rn and rn - 1 for outer and inner radii and rm, n for the average radius of each layer, and with Eq. (8.88) in the form of rm =

rn - rn - 1 ln rnr-n 1

ð8:91Þ

we get Q_ =

1

rn - rn - 1 λn
2 π rm,n
l

2π l


ΔT =

rn - rn - 1 λn
rm,n


ΔT

ð8:92Þ

the denominator alone is: rn - rn - 1 = λn
r m,n

ðr n - r n - 1 Þ
ln rnr-n 1 = λ n
ðr n - r n - 1 Þ

ln rnr-n 1 λn

ð8:93Þ

So, for the heat flow we get: Q_ =

2π l ln r rn

n-1


ΔT

ð8:94Þ

λn

Written as sum of heat conduction resistances Ri n

1
2π
l i=1

ln rnr-n 1 = λn

n

Ri = Rtotal

ð8:95Þ

i=1

we get again ΔT Q_ = Rtotal

ð8:77Þ

Recall that in addition to heat transfer by radiation and conduction, we also have heat transfer by convection and phase transition as alternative mechanisms by which to transfer heat (Table 8.11). So, to calculate heat transfer in general we have more properties and parameters yet to study, which are only mentioned briefly here.

8.6.3

Convection Heat Transfer

Convection heat transfer occurs when heat energy is carried along by a moving fluid in contact with a solid surface. The fluid can be either liquid or gas. For example, we feel warm when we enter a heated room from the cold outdoors because the surface of our body encounters warm heated air that circulates in the room. Likewise, the air in the room receives heat when it encounters the heated surface of a metal radiator.

8.6 Heat Transfer in Food

345

The inside metal walls of the radiator become heated when hot water flows in contact with the inner surface of these walls. The means by which heat transfers from the flowing hot water to the inside metal surface of the radiator, as well as from the outer metal surface of the radiator to the circulating air in the room, is heat transfer by convection. In the case of convection heat transfer, the fluid experiencing heating or cooling also moves. This movement may be due to the natural buoyancy effect of decreased density with increased temperature (natural convection), or it may be caused artificially by imparting mechanical energy to the fluid, such as with pumps or blowers for liquids and gases, respectively (forced convection). Determining the rate of heat transfer from convection is complicated because of this fluid motion. When a fluid is flowing over a solid surface, shear stresses occur in the fluid near the surface because of the viscous properties of the fluid. Molecules at the surface try to attach themselves to the surface while neighboring molecules are trying to pass them by within the bulk fluid flow. This causes a velocity profile to develop near the surface. The fluid next to the surface does not move but sticks to it, while neighboring fluid flow is slowed down by the friction of trying to pass the stationery molecules. Therefore, fluid velocity gradually increases with distance away from the surface until the region of bulk fluid flow is reached, where the fluid velocity is all the same at the maximum. This region is known as a boundary layer. Along with the velocity profile, a temperature profile develops in this boundary layer near the wall. This is because the rate of convective heat transfer from a fluid to a solid surface depends in part on the relative velocity of the fluid in contact with the surface. Since this relative velocity is near zero at the surface because of the sticky viscous effects, transfer of heat is also poor. This means the surface does not readily sense the temperature in the bulk fluid flow, and the temperature will change with distance away from the surface as it moves from surface bulk fluid temperature. Therefore, this invisible boundary layer region near the surface acts as though it were an additional layer of insulating material, adding further resistance to the heat transfer between the fluid and the surface (Fig. 8.10).

8.6.3.1 Heat Transfer Coefficient Mathematically, we can account for this convective heat transfer resistance by assuming it represents another layer of material across which heat must transfer in the expression for steady state heat conduction from Fourier’s first law. We can do this by assigning a value for the “thermal conductance” of this invisible boundary layer (Fig. 8.10), which is known as the surface heat transfer coefficient α. Then we could express the simple case of heat transfer across such a boundary layer from a hot surface temperature Ts to a cool bulk fluid temperature T1 in the following way:

346

8

Fig. 8.10 Convection heat transfer. Showing boundary layer where velocity and temperature profiles exist near wall surface, 1 hot wall, 2 boundary layer

Thermal Properties

1 2 velocity

TS

temperature

Fluid Flow

Q_ = α
A
ðT S- T 1 Þ Q_ α A TS T1

ð8:96Þ

heat flow in J
s-1 heat transfer coefficient in W
m-2
K-1 area in m2 temperature of wall in K temperature of bulk fluid in K

It is important to note that the heat transfer coefficient is a numerical value that represents the overall “thermal conductance” of the invisible boundary layer. Its value will depend on the combined effects of the physical, thermal and viscous properties of the fluid in contact with the surface, relative velocity of the fluid at the surface, as well as system geometry at the contacting surface, among other things. Therefore, the heat transfer coefficient is not a material property, and cannot be looked up in a handbook. It is a parameter (coefficient) in heat transfer equations that depends on conditions of the process system under study. Normally, it is determined experimentally for heat exchanger systems by running experiments under controlled conditions, where all temperatures, surface areas, material properties and flow rates are known, and the heat transfer coefficient is the only unknown. There are also various approaches for attempting to estimate values for the heat transfer coefficient under different specified sets of heating and cooling conditions and surface contact geometries. These approaches can be found in references devoted more completely to heat transfer and fluid mechanics [12, 13].

8.6 Heat Transfer in Food

347

8.6.3.2 Overall Heat Transfer Coefficient The convective boundary layers are only part of the multi-layer systems across which heat must transfer in more realistic heating and cooling situations. Recall our radiator for heating the room. The air in the room receives heat when it encounters the heated surface of the metal radiator. The inside metal walls of the radiator become heated when hot water flows in contact with the inner surface of these walls. Let us take a close look at just a small section of radiator wall that we could imagine as a flat plate (thermal conductivity λ and thickness δ) with hot water flowing along the inside surface, and cool room air flowing along the outside surface. In this case, we have a boundary layer on the inside caused by the flowing hot water represented by a heat transfer coefficient α1. We also have a boundary layer on the outside caused by the flowing cool air represented by a heat transfer coefficient α2. This combination of heat transfer 1, heat conduction and heat transfer 2 can be characterized by a single parameter called overall heat transfer coefficient k. To understand the idea of the overall heat transfer coefficient let us look again in the concept of adding resistors. The heat has to pass resistances of boundary 1, of the flat plate and of boundary 2 one after the other. That means we can add the resistances like in a serial circuit to get the total resistance against heat flow: Rtotal = R1 þ Rλ þ R2

ð8:97Þ

The heat transfer resistance R is the reciprocal of the heat transfer resistance coefficient α. The heat conduction resistance R is the reciprocal of the quotient of heat conductivity λ and thickness δ of the solid material. So we can write Rtotal =

1 δ 1 þ þ α1 λ α2

ð8:98Þ

So Rtotal is the total resistance against heat flow causes by serial circuit of boundary 1, plate and boundary 2. The reciprocal of the total resistance is the overall heat transfer coefficient k. Rtotal =

1 k

ð8:99Þ

so 1 1 δ 1 þ þ = k α1 λ α2

ð8:100Þ

respectively k=

1 1 α1

þ δλ þ α12

ð8:101Þ

To calculate the heat flow through several heat resistances, we can use the overall heat transfer coefficient k in an equation analogous to Fourier’s first law:

348

8

Fig. 8.11 Multilayer heat transfer with convection on both sides. Example: Material 2 is steel, 1 is rust and 3 is paint. The overall heat transfer coefficient k combines α1, α2, λ1, λ2, λ3,δ1,δ2 andδ3 in a single parameter

Thermal Properties

D

D2

1

hot water

O1

O2

O3

G1

G2

G3

Q_ = A
k
ΔT

cold air

ð8:102Þ

heat flow in J
s-1

Q_ A α k ΔT λ δ

area in m2 heat transfer coefficient in W
m-2
K-1 overall heat transfer coefficient in W
m-2
K-1 temperature difference in K thermal conductivity in W
m-1
K-1 thickness in m

Let us imagine further that the radiator wall has a layer of rust on the inside and a layer of paint on the outside in addition to the metal core between these rust and paint layers. Each of these material layers would have its own thermal conductivity and thickness. Then we would have a multilayer situation consisting of five layers. This would be a situation similar to that shown earlier in Fig. 8.6 but with two additional boundary layers with their respective heat transfer coefficients on each side, as shown in Fig. 8.11. So the total resistance is Rtotal = R1 þ Rλ1 þ Rλ2 þ Rλ3 þ R2

ð8:103Þ

i.e. Rtotal =

1 δ1 δ2 δ3 1 þ þ þ þ α1 λ1 λ2 λ3 α2

ð8:104Þ

8.6 Heat Transfer in Food

349

the overall heat transfer coefficient therefore is k=

1 1 α1

þ

δ1 λ1

þ

δ2 λ2

þ δλ33 þ α12

ð8:105Þ

For an n-layer solid the overall heat transfer coefficient is k=

1 αouter þ 1

n i=1

δi λi

1 þ αinner

ð8:106Þ

Process engineers often use Eqs. (8.106) and (8.102) for heat transfer calculations involving heat exchangers. Often it is much too difficult to estimate or measure the exact thermal conductivity and thickness of various material layers in a system, especially if they consist of poorly defined rust and scale deposits or fouling layers. It is also difficult to correctly characterize the fluid boundary layers from convective heat transfer in order to estimate convective surface heat transfer coefficients. The overall heat transfer coefficient k considers all these factors. It is most often determined experimentally for heat exchanger systems by running experiments under controlled conditions, where all temperatures, surface areas, and flow rates are known, and the overall heat transfer coefficient k is the only unknown. For more details, see hydrodynamics and process engineering [6, 14–16]. In the field of building construction and thermal insulation, there are regulations and standardized measurement methods for the heat transfer coefficient, too [17–19].

8.6.4

Heat Transfer by Phase Transition

We learned earlier from our review of basic thermodynamics that when we look at a simple phase transition, such as when a substance undergoes evaporation from a liquid to a vapor, freezing from a liquid to a solid, or condensing from a vapor to a liquid, then we normally see significant changes in enthalpy, entropy and volume. These are caused by the change of enthalpy ΔH in the form of latent heat of fusion or latent heat of vaporization. These enthalpy changes involve significant quantities of heat transfer to occur in support of the phase change. Because of these significant quantities of latent heat energy transferred during phase change, process engineers prefer to use condensing steam (water vapor) whenever practical as a heat exchange medium in heat exchanger systems. Example When 1 kg of saturated steam at 100 °C condenses onto the cool metal surface of a heat exchanger, it immediately gives up its latent heat of vaporization as the enthalpy changes from the value at saturated vapor (2676 kJ
kg-1) to that (continued)

350

8

Thermal Properties

of saturated liquid (419 kJ
kg-1). This represents a net heat transfer of (at normal pressure) 2257 kJ
kg-1. If instead, we used 1 kg of liquid hot water at 100 °C as the heat exchange medium, it would probably experience a change in temperature of about 30 K, and the heat given up would be the sensible heat loss. We would get Q = cp
ΔT = 4.2 kJ
K
kg-1
30 K = 126 kJ
kg-1 and thus less than 1 10 of the heat of condensing water vapor. Clearly in the case of condensing water vapor, the amount of heat transferred from the same quantity of steam or water is two orders of magnitude greater for condensing steam (phase transition) than when water is used alone without any phase change. The enthalpy change occurring during condensation corresponds to the enthalpy change occurring during evaporation (with the opposite sign). Therefore, the heat released during condensation can be calculated from the enthalpy of evaporation: jQsteam j = m
Δhvap Q h Δhvap m cp T

ð8:107Þ

heat in J specific enthalpy in J
kg-1 specific evaporation enthalpy in J
kg-1 mass in kg specific heat capacity in J
kg-1
K-1 temperature in K

In addition to the condensation of water vapor, the evaporation of water is also used for heat transfer—i.e., for cooling. In so-called adiabatic cooling, the temperature of the air is lowered by spraying water with nozzles into small droplets and let it evaporate. In addition, many power plants use the evaporation of water to cool their cooling water. The solid-liquid phase change of water has also long been used for heat transfer, e.g., in ice cellars or cold baths. Modern concepts of buildings use ice storage as a latent heat source [20]. A pumpable ice-water mixture can be used to cool buildings or products such as food [21]. Remark The enthalpy of evaporation of water at normal pressure is 2257 kJ
kg-1. Methane (CH4), a substance with a molecular weight similar to that of water, has an enthalpy of evaporation of only 510 kJ
kg-1. The cause of the high value of water lies in its molecular structure. Due to the dipolar character of the H2O molecule, numerous hydrogen bonds are formed in the liquid phase, which must be dissolved during evaporation. This is not the case with a nonpolar molecule such as CH4.

8.6 Heat Transfer in Food

351

In order to carry out heat transfer calculations involving phase change, it is necessary to know the thermodynamic properties of the material and which phase transition it is undergoing. In the case of condensing steam, we would need the thermodynamic properties of saturated steam. These properties include enthalpy, entropy and specific volume of both saturated liquid and saturated vapor as a function of temperature and pressure. These are material properties, but they depend on temperature and pressure. Therefore, they are normally published in hand books and text books in the form of tables or charts for easy reference [22]. An example of such a steam table can be found in the Appendix.

8.6.5

Thermal Conductivity

In previous chapter sections, we described conduction heat transfer through solid materials (across heat exchanger walls), and how we use Fourier’s law to derive mathematical expressions that characterizes conduction heat transfer. You will recall that the thermal conductivity λ was the singular coefficient representing material properties in Fourier’s law. In the case of conduction heat transfer, this material property is called thermal conductivity, and indicates how easily heat will pass through the material. Different materials have different thermal conductivities. This difference helps to explain why a metal spoon gets too hot to hold in your hand when stirring a boiling liquid in a pan, but a wooden spoon of the same size and dimensions does not. We have already been using the thermal conductivity as the coefficient λ in Fourier’s first law: Q_ dT = -λ
A dx

ð8:108Þ

The reason different materials have different thermal conductivities is because they have different chemical and physical compositions. Recall that heat is conducted through solid materials because the increasing kinetic energy imparted to the molecules at the point where the heat is entering the material excites them to oscillate at greater speeds and amplitudes. This excitation is driven by higher electron mobility within the atomic structure of each molecule, and causes the molecules to experience increased “bumping” into neighboring molecules. This molecular “bumping” propagates along the material, and further continues this process of heat transmission (conduction) along the material substance. Therefore, thermal conductivity must depend on both molecular structures at the atomic electron level (the chemical composition), as well as the physical lattice structure by which the molecules are held in place within the material substance (physical structure). In the case of liquid and gaseous substances, heat convection occurs in parallel in addition to heat conduction. In solids where convection cannot take place, heat can only be transported by heat conduction. For this reason, we first consider solid substances. These can be foodstuffs and materials for packaging technology, e.g., polymers or materials for mechanical engineering, e.g., stainless steel.

352

8

Thermal Properties

8.6.5.1 Solids In metal solids, much of the molecular excitation that occurs in response to heat is due to the relatively high mobility of the electrons in metals. Therefore, this electron mobility accounts for the fact that in metals, electron conductivity is the major contributing mechanism for the relatively high thermal conductivity found in metals. The ratio of electric conductivity to thermal conductivity is a linear function of temperature (Wiedemann–Franz-law): λ =L
T σ λ σ L T

ð8:109Þ

thermal conductivity in W
K-1
m-1 electric conductivity in S
m-1 Lorenz-coefficient (for metals at room temperature is L ≈ 2.4
10-8 V2
K2) temperature in K

Disturbances of crystal structure such as degradations and defects reduce the thermal conductivity. Therefore Aluminum alloys or stainless steel have a lower thermal conductivity than pure aluminum or pure iron. Among the metals, silver and copper have the highest values for thermal conductivity. Selection of a material for critically functioning components or equipment such as heat exchangers is always made as a compromise between mechanical and thermal properties, corrosion resistance and price. For this reason, stainless steels are often chosen for food processing equipment over pure metals. When packaged food is heated for the purpose of preservation (refer Table 8.1), the heat transfer to the core of the food is also influenced by the thermal conductivity of the packaging material. Therefore, different processes are necessary for glass or plastic packages than for sheet metal packages. Even with non-metallic materials and polymers, thermal conductivity decreases with decreasing crystallinity of the material. In solid foods, thermal conductivity is strongly determined by their water content. Empirical calculation equations for the thermal conductivity of solid foods therefore contain the water content of the material as a key variable. Water in solid foods is more or less immobilized (refer to Chap. 1). For this reason, heat transport in solid foods does not take place by convection but by heat conduction. Example We have made a jelly from a fruit juice (water content 90% wb) by adding gelatin (The water content is also 90% wb). We put a glass of jelly and a glass of the original fruit juice in the refrigerator and record the core temperature of both samples. It turns out that the temperature in the juice drops faster. The juice seems to transport the heat better. (continued)

8.6 Heat Transfer in Food

353

Explanation: In addition to heat conduction, in the juice heat convection also occurs, i.e., the liquid in the glass is mixed by natural convection as increasing density of the cold juice at the surface causes it to sink to the bottom while the less dense warmer juice at the center rises upward. In jelly, heat is transported by heat conduction only. Both samples have the same geometry and the same mass. The only difference is that in the jelly sample there is a gel network that immobilizes the water molecules making it behave as a solid body. In the Annex values for the thermal conductivity of some foods and materials can be found. For further values see, e.g. [23–26]. Since gases have a much lower thermal conductivity than liquid phases, the value of the thermal conductivity of food drops sharply during the formation of foam. Definition Foam is a disperse system. The dispersed phase is gaseous. Foamed foods are produced by creating gas bubbles in the food matrix. Examples are whipped cream, ice cream, sugar foam but also baked goods. The gas pores are usually filled with air, but also with N2O (whipped cream) or CO2 (by baking soda or yeast in baked goods). For foamed materials air content is therefore another variable for the calculation of thermal conductivity. If a material contains pores, a decrease in thermal conductivity is observed with increasing porosity. The pore radius distribution also plays a role in affecting thermal conductivity. A distinction is made here between open-pore systems, in which the pores have connection to the ambient atmosphere and closedpore systems. In open-pored systems, there is a clear dependence of the thermal conductivity on the relative humidity of the environment, as this determines the surface cover and capillary condensation, i.e., the water content in the pores (refer Chap. 1). For these reasons, solid materials for thermal insulation often consist of closed-porous, non-crystalline polymers with high porosity. In isotropic materials, the physical properties are the same in all spatial directions. For non-isotropic materials, the physical properties depend on the spatial direction. In the case of thermal conductivity, this is observed with fibrous materials based on plants or animals. As a rule of thumb for meat, the thermal conductivity parallel to the fiber is 7–10% higher than perpendicular to the fiber (Fig. 8.12).

354

8

Fig. 8.12 Thermal conductivity of water (H2O), butter (B) and turkey meat (T). T: upper curve indicates λ parallel to fiber, lower curve λ perpendicular to meat fiber [6]

Thermal Properties

H2O

O

T

B

0 -/ qC

-10

10

Definition A material is called isotropic if the physical properties are the same in all spatial directions. Anisotropic materials are non-isotropic, i.e., some physical properties, e.g., optical or mechanical properties depend on the spatial direction. Fibers and fibrous materials are examples of anisotropic materials.

8.6.5.2 Multilayer Solids The thermal conductivity of solids which consist of several layers with different thermal conductivities can once again be calculated using the concept of additive resistors again. Heat conduction resistivity is the reciprocal of thermal conductivity. In the case of series connection, thermal conduction resistances are added to the total resistance. If the thermal conductive resistances are parallel to each other, not the resistances R are added, but their reciprocal size R1, the so-called heat conductance is added to get the total heat conductance Gtotal. In Table 8.13 the cases are listed side by side. In Chap. 7 we already learned about the analogous mathematical treatment of steady state heat conduction and conduction of electricity. To get used to work Table 8.13 Total resistance to heat conduction at serial connection and parallel connection Serial connection Rtotal = Ri i

Parallel connection 1 Rtotal

= Gtotal

Gtotal =

addition of resistances

addition of conductances

Rtotal =

Gtotal =

i

di Ai


λ1i

Gi i

i

Ai d i λi

when A1 = A2 = A3... and d1 = d2 = d3... it is 1 1 λi λtotal =

λtotal =

addition of reciprocal thermal conductivities

addition of thermal conductivities

i

λi i

8.6 Heat Transfer in Food

355

Table 8.14 Analogy of heat conduction electrical conduction: terms Heat conduction heat conductivity λ in W
K-1
m-1 1 λ = Rheat
Ad

Electrical conduction Electrical conductivity κ in S
m-1 κ = 1ρ = R1el
Ad

= λ
ΔT Fourier’ s first law d Heat conductance in W
K-1 1 A 1 Rheat = λ
d Rheat = Gheat

I A

Q_ A

= κ
ΔU d

Ohm’ s law

Electrical conductance in S 1 1 A 1 Rel = ρ
d Rel = Gel

Heat conduction resistivity in K
m
W-1 1 1 A λ = Rheat
d

Electrical resistivity in Ω
m 1
Ad ρ = Rheat

Heat conduction resistance in K
W-1 Rheat = 1λ
Ad Rheat = ΔT Fourier’ s first law Q_

Electrical resistance in Ω Rel = ρ
Ad Rel = UI Ohm’ s law

Q_ λ A d ΔT Rheat Gheat κ ρ U I Rel Gel xi

heat flow in W specific heat conductivity in W
K-1
m-1 area in m2 thickness in m temperature difference in K heat conduction resistance in K
W-1 heat conductance in W
K-1 electrical conductivity in S
m-1 electrical resistivity in Ω
m voltage in V electrical current in A electrical resistance in Ω electrical conductance in S mass fraction of component i

with heat conduction resistances, it is good to recall the analogy between Ohm’s law and Fourier’s first law. Table 8.14 can help to remember the related terms. This principle can also be applied to mixtures of solids, such as foods made of several different solid ingredients with different thermal conductivities λi. For mixtures of solids the parallel connection applies, i.e., the thermal conductivities are added together and weighted according to their relative mass fractions xi. In Table 8.15 some Indicative values for λi are listed. λges =

xi
λ i

ð8:110Þ

i

If the ingredient formulation of a food is imagined to be simplified to only water and dry matter, the thermal conductivity of such composite systems can be quickly added. With the mass fractions

356

8

Table 8.15 Indicative values for the thermal conductivity of food components [6]

Thermal Properties

Air Protein Carbohydrate Fat Water, liquid Ice Non-fat dry matter from animals Non-fat dry matter from plants

λ (W
K-1
m-1) 0.025 0.20 0.245 0.15 0.55 2.21 0.26 0.22

xdm þ xH2 O = 1

ð8:111Þ

λtotal = xdm
λdm þ xH2 O

ð8:112Þ

the thermal conductivity is

Values for λdm can be found in Table 8.15. In the case of frozen foods, the water content of the food is split into a frozen fraction α and a non-frozen fraction (1 - α). Then we have λtotal = xdm
λdm þ xH2 O ð1- αÞ
λH2 O þ xH2 O
α
λice

ð8:113Þ

The determination of the frozen fraction α can be carried out experimentally, e.g. by DSC, see Sect. 8.8.2 or by NMR, see Chap. 11. Example A sample of fruit concentrate (dry matter: 47% (m/m) has at -20 °C a frozen fraction of 80% The thermal conductivity of the frozen fruit concentrate can be estimated to λtotal = xdm
λdm þ xH2 O ð1- αÞ
λH2 O þ xH2 O
α
λice λtotal = 0:47
0:22 W
K - 1
m - 1 þ 0:53
ð1 - 0:8Þ
0:55 W
K - 1
m - 1 þ 0:53
0:8
2:21 W
K - 1
m - 1 λtotal = 1:1 W
K - 1
m - 1

8.6.5.3 Temperature Dependency of Thermal Conductivity Thermal conductivity of solid food materials is only mildly dependent on temperature, and increases slightly with increasing temperature. This mild dependency is overshadowed by the effect of gross composition, such as content of water or air. However, at temperatures near the freezing point of water, solid foods with high water content will experience much stronger temperature dependency. The thermal

8.6 Heat Transfer in Food

357

conductivity of water at its freezing point undergoes a dramatic change in value as water undergoes phase transition from liquid into solid ice. Because of the orderly crystalline structure of solid ice, its thermal conductivity is much greater than that of liquid water. Therefore, while water, or food with high water content, is freezing, the thermal conductivity will become a strong function of the ice fraction α present in the water-ice mixture, as shown in Fig. 8.12. mass fraction of dry matter in kg
kg-1 mass fraction of water in kg
kg-1 frozen fraction in kg
kg-1 thermal conductivity of dry matter in W
K-1
m-1 thermal conductivity of water in W
K-1
m-1 thermal conductivity of ice in W
K-1
m-1

xdm xH2 O α λdm λH2 O λice

8.6.5.4 Liquids In liquids, heat transfer occurs mostly by convection, the mechanism by which heat is carried along with the flowing fluid. So, it is difficult to measure pure thermal conductivity in liquids. Pure heat conduction can only take place in liquids if they can be completely prevented from flowing in some way, such as when they are formed into a gel. In the case of water, the thermal conductivity of water can be estimated as follows [9]: λH2 O W
K - 1
m - 1 = 0:100 þ 0:00166
T ðKÞ

ð8:114Þ

thermal conductivity of water in W
K-1
m-1 temperature in K temperature in °C

λH2 O T ϑ

More accurate values are provided by the following polynomial equations [6]: for water λ mW
K - 1
m - 1 = 568:96 þ 188
ϑ ð ° CÞ - 8:2
10 - 3
½ϑð ° CÞ2 þ 6:02
10 - 6
½ϑð ° CÞ3 for dairy products with a fat content xf between 3 and 62% λ mW
K - 1
m - 1 = 0:5406 - 0:0055
xf

with xf in%ðm=mÞ

358

8

Thermal Properties

for dairy products with a fat content xf between 62 and 100% (m/m): λ mW
K - 1
m - 1 = 0:2309 - 0:00051
xf

with xf in%ðm=mÞ

8.6.5.5 Gases In the case of gases, molecules are not fixed in place, but are free to have translational motion as they become excited with increased kinetic energy from heat input. Therefore, heat conduction in gases evidences itself by the propagation of kinetic energy of the molecules as they “bump” into each other in response to receiving heat energy. This transmission of kinetic energy occurs at a faster rate with increasing velocity of the molecules. Therefore, thermal conductivity in gases increases with increasing temperature, and with decreasing molecular weight of the gas molecule. This is why we use low molecular weight gases like water vapor and helium for heating and cooling when we want high thermal conductivity, and we use heavier molecular weight gas like Xenon for thermal insulation when we want low thermal conductivity. The thermal conductivity of air at atmospheric pressure in the temperature range from 0 °C to 500 °C can be estimated from the following empirical expression: λ mW
K - 1
m - 1 = 0:701 þ 0:0629
T ðKÞ More exact values for air at different relative humidity can be found in reference [13]. Recall that thermal conductivity in gases will increase as molecules are “bumping” against each other with increased kinetic energy. At the same energy level, this “bumping” will occur more frequently when the molecules are located more closely together. Therefore, thermal conductivity in gases will also increase as the mean distance between molecules decreases. At low pressures or at low concentrations of gas molecules in a given space, the mean distance between molecules can be very large. When such a gas is confined to limited volume of space, such as when held in a closed tank or vessel, this mean distance between molecules may extend beyond the interior space of the enclose tank or vessel. When this occurs, the molecules are forced to be closer together than they would normally be, and pressure develops. As pressure increases the molecules are forced to be closer together. Therefore, in the case of such gases, thermal conductivity will also increase with increasing pressure. For this reason, the thermal conductivity in a vacuum is very low and essentially non-existent. That is why thermally insulated bottles are sometimes lined with a glass-enclosed partial vacuum space for very effective thermal insulation. The temperature dependency of thermal conductivity in gases is shown in Fig. 8.13. We take advantage of this low thermal conductivity in a vacuum by carrying out certain food processes under vacuum, such as in freeze drying. The vacuum in the freeze dryer does not permit any heat conduction from the gaseous phase to reach the

8.6 Heat Transfer in Food

359

Fig. 8.13 Pressure dependency of thermal conductivity of gases, schematic

O

p

food product. Similarly to the thermally insulated vacuum bottle, thermally insulated windows are made of two panes of glass just 1 or 2 cm apart trapping a “dead” space of partial vacuum between them to serve as very effective thermal insulation.

8.6.5.6 Apparent Thermal Conductivity In systems containing both solids and liquids, heat will transfer by convection as well as conduction (and radiation in some cases). In these types of systems, the overall rate of heat transfer is greater than if it were to occur only by pure conduction. If we assume this overall heat transfer is coming only from conduction, then we observe an “apparent” thermal conductivity that is greater than the pure thermal conductivity of any of the component substances in the system. Let’s imagine a pile of vegetable bulk materials. It contains numerous cavities filled with air. These cavities are connected to the surrounding atmosphere, this is called an open-pore system. With a temperature difference between the environment and the vegetables, heat is transported by both heat conduction and convection of the air in the cavities. The convective part increases with increasing temperature gradient and with increasing pore diameter. Natural convection has a predetermined direction, it is opposite to gravity. As a result, the apparent thermal conductivity is direction-dependent. Figure 8.14 shows the different apparent thermal conductivities of a bulk material pile in different spatial directions. The pure thermal conductivity

Fig. 8.14 Increase of apparent thermal conductivity in a bulk good with increasing temperature gradient: Heat going from bottom to top (1), from top to bottom (2). Thermal conductivity of air, alone, is independent of direction (3)

O

1 2

3

T

360

8

Thermal Properties

of air, i.e., without any convective component, has only a low-temperature dependence and shows no directional dependence.

8.6.6

Thermal Diffusivity

Recall at the beginning of the chapter, we mentioned the need to distinguish between steady state and unsteady state (transient) heat transfer. In steady state heat conduction, the temperature gradient is constant with time, and we used Fourier’s first law to describe it mathematically (Fig. 8.15). In transient (unsteady state) heat conduction, the temperature gradient is not constant and varies as a function of time (Fig. 8.16). We can mathematically describe unsteady state heat conduction with Fourier’s second law, Eq. (8.116). The coefficient a in Fourier’s second law is known as the thermal diffusivity of a material, and it plays a similar role to the thermal conductivity in his first law. Fourier’s second law for transient heat

Fig. 8.15 Steady state heat conduction: temperature gradient is constant over time

-2

-1

x

Fig. 8.16 Transient heat conduction: temperature gradient is a function of time t

-

x

8.6 Heat Transfer in Food

361

one dimensional: dT d2 T =a
2 dt dx

ð8:115Þ

dT = a
∇2 T dt

ð8:116Þ

three dimensional:

whereas ∇2 T =

d2 T dx2

2

2

þ ddyT2 þ ddzT2

Thermal diffusivity differs from thermal conductivity because it must also account for the other physical and thermal properties (density and specific heat capacity) that govern the rate of temperature response to a given rate of heat flow. Therefore, thermal diffusivity is made up of the quotient of thermal conductivity divided by the product of density and specific heat capacity. a=

a T t x λ ρ cp a

λ ρ
cp

ð8:117Þ thermal diffusivity in m2
s-1 temperature in K time in s length in m thermal conductivity in W
K-1
m-1 density in kg
m-3 specific heat capacity in J
K-1
kg-1 thermal diffusivity in m2
s-1

Magnitude of Thermal Diffusivity: Most foods have thermal conductivities in the range of 0.2–0.6 W
K-1
m-1 (above 0 °C), and densities from 900 to 1500 kg
m-3 and heat capacities between 1.2 and 4.2 kJ
kg-1
K-1. So, the values of thermal diffusivity according to Eq. (8.117) can range from 0.02
10-6 to 0.6
10-6 m2
s-1. Very often we find values in the range 0.1
10-6–0.6
10-6 m2
s-1. More exact values can be found by laboratory experiments involving heat penetration tests.

362

8.6.7

8

Thermal Properties

Measurement of Thermal Conductivity and Thermal Diffusivity

Thermal conductivity and thermal diffusivity are material properties that govern the rate at which heat will conduct through the material. Both of these properties pertain to conduction heat transfer. Thermal conductivity is the governing property in steady state heat conduction, while thermal diffusivity is the governing property in transient (unsteady-state) heat conduction. Recall that steady state heat conduction occurs in response to a constant temperature gradient that does not change with time. Therefore, measurement techniques used to measure thermal conductivity rely on a constant temperature gradient and constant flow of heat. Techniques for measuring thermal diffusivity are carried out under a transient temperature gradient that usually decreases with time causing flow of heat to also decrease with time.

8.6.7.1 Steady State Techniques The primary advantage of measurement under steady state conditions is that experiments can be performed with relatively simple and inexpensive laboratory equipment and straight forward data analysis following Fourier’s first law. The disadvantage is that measurements must be taken under steady state conditions, and it can be very time consuming to reach a controlled steady state condition in many laboratory situations. A basic type of experiment for measuring thermal conductivity of a material involves arranging a sample of known geometry (area and thickness) so that it will be subjected to heat transfer in only one direction. Once steady state is reached (temperatures at opposite ends of the sample remain constant), it is only necessary to record the constant temperatures and measure the flow of heat. Then, thermal conductivity can be calculated from Fourier’s first Law. A schematic drawing of the laboratory apparatus used for such an experiment is shown in Fig. 8.17. A sample of material of known geometry is sandwiched on both sides of a heat source between the heat source (hot plate) and a heat sink (cold plate), with the edges thermally insulated to prevent any heat transfer in a transverse direction [27]. When we use the type of apparatus shown in Fig. 8.17 it is possible to put only the sample material of unknown thermal conductivity on one side of the hot plate, and a reference material sample of known thermal conductivity on the other side of the hot plate. This allows determination of the thermal conductivity in the sample material on a relative basis to that of the reference material, and eliminates the need to Fig. 8.17 Guarded hot plate method: measurement of thermal conductivity. P sample, C cooled plate, H heater, I thermal insulation

C

I

P

P

C

I

H

8.6 Heat Transfer in Food

363

Fig. 8.18 Two-plate technique with reference material for measurement of thermal conductivity relative to a reference material. P sample material, R reference material

Q&

dP

P

'TP

dR

R

'TR

measure heat flow, as shown in Fig. 8.18. It is only necessary to measure the steady state temperatures. Then we have: λp d p ΔT R =
λR dR ΔT p

ð8:118Þ

and λp = λR


d p ΔT R
dR ΔT p

ð8:119Þ

For measurement of thermal conductivity in liquid or fluid samples, a flat plate method is not appropriate, and a concentric cylinder method is used, as shown in Fig. 8.19. In this method the heat source is a cylindrical rod as the axis inside a hollow cylindrical tube leaving a narrow annular space in which the fluid sample is held. In order to avoid development of any natural convection, the thickness of the annular space must be kept very small (1–3 mm) depending on fluid viscosity, and the temperature differences must also be small (1–5 K).

8.6.7.2 Transient State Techniques The principle of transient measurement methods is to introduce a defined heat flow into the sample and record the resulting temporal course of the temperature. When using a measuring device as in Fig. 8.19, the material to be examined is filled into a cylinder in which an axially tensioned platinum wire of length L is located. At the beginning of the experiment (t = 0), the wire and sample are at the same temperature. After switching on the heating current Q_ with a constant heating power, the temporal course of the temperatures is recorded. In the laser flash method, a test specimen is uniformly heated on the underside by a short-term laser pulse and the resulting temperature increase on the top of the sample is measured with the help of an infrared detector [28]. The transient hot bridge method (THB) uses a setup like in Fig. 8.17 with a strip-shaped electrical conductor which is used for heating and acts simultaneously as a resistance thermometer [29].

364

8

Fig. 8.19 Concentric cylinder method for measurement of thermal conductivity of fluid sample materials, schematic. A thin layer of sample P is in the annular space between heater rod H and cooled outer cylindrical jacket C

Thermal Properties

C

P

P

C

H

8.7

Caloric Value of Foods

8.7.1

Caloric (Energy) Requirement of the Human Body

Foods contain energy, which is taken up by the body when they are consumed. This energy is used first to sustain basic metabolism of the cells in the body tissues and organs, which is necessary even when the body is at rest (basal metabolism). Next, additional energy is converted into work energy by the muscle tissues of the body in contracting and expanding for the body to undergo movement and do work. Finally, more energy is also given off as excess heat from the body after maintaining normal body temperature. This total energy requirement can be calculated from the following energy balance: EA = EM þ EN EA EM EN EB

ð8:120Þ energy taken up in J energy metabolized in J energy which is not metabolized in J energy excreted with body waste in J

Very little energy is normally excreted with body waste (urine and feces) from a healthy person because these waste materials contain very little protein, carbohydrates or fat. Also, the carbon dioxide exhaled from breathing contains no combustion energy. In the case of carbohydrates in our foods, only the digestible part, which can be absorbed from the intestinal track, is taken up as energy by the body. The remainder is called dietary fiber that is made up largely of cellulose fibers, and is excreted with the body waste. We now know that this undigested dietary fiber still plays an important role in sustaining a healthy balance of natural microbial populations in

8.7 Caloric Value of Foods Table 8.16 Power required by a human body of a 70 kg person at different activities, from [32] in [31]

365 E_ (W) 80 265 400 700 100

Type of activity Sleeping Walking at 5 km
h-1 Cycling at 15 km
h-1 Cycling at 21 km
h-1 24 h—average

the intestinal track. These cellulose fibers are further broken down by enzymes produced by intestinal microbes to provide important nutrients for sustaining microbial metabolism in the intestines. Therefore, this activity provides an important physiological effect for sustaining good health, but it does not contribute to any remarkable further uptake of energy by the body. So, looking at the energy balance once again, we can say: EA = EM þ EB

ð8:121Þ

The energy turnover E_ of the body is the energy converted per unit of time. dE E_ = dt

ð8:122Þ

The energy turnover in J/s thus has the physical meaning of power in watts. The power required by the human body E_ depends on many factors, such as age, sex, work load, surrounding temperature that may fluctuate strongly during a day. As a rule of thumb, we require a magnitude of approximately 100 W over a 24-h period as a measure of human power intake. It is remarkable that about 70% of this power is needed just to sustain basal metabolism. The basal metabolic rate, BMR of a 70 kg person is ca. 80 W [30]. The metabolic basal rate is the result of basal functions such as cell metabolism, synthesis of enzymes, hormones and other proteids, maintenance of body temperature, uninterrupted work of muscles for cardiological and respiratory functions, as well as brain function—in other words, just for “being there“. If we add the energy turnover for physical activities to the basal metabolic rate, you get the total metabolic rate, MR. The average total metabolic rate (MR) of a 70 kg person in leisure time is 110 W, while a cyclist for short periods can reach 1600 W [31]. Table 8.16 lists more examples of physical activities. Example A rule of thumb for metabolic basal rate of our body is: 1 kcal per hour for each kg body weight we carry. At a body weight of 65 kg we get E_ = 65 kg
1 kcal=hso (continued)

366

8

Thermal Properties

65
103 cal 65
103
4:18 J E_ = 65
1000 cal=h = = = 75 J
s - 1 = 75 W 3600 s 3600 s

Example Expressing “1 kcal per hour for each kg body weight” in SI units we get a further rule of thumb: 1 kcal 4:2 kJ 24
4:2 J E_ = = = ≈ 100 kJ
day - 1
kg - 1 h
kg h
kg day
kg The metabolic basal rate of for a human body is about 100 kJ per kg body weight per day.

Example The daily energy requirement E of a man with an a 24 h—average total metabolic rate of 116 W is: E = E_
t E = 116 W
24 h = 116 W
24
3600 s = 10, 022, 400 J = 10 MJ The approximate value of 10 MJ per person per day obtained in this example corresponds to around 2400 kcal day-1 and is thus at the center of frequently mentioned dietary recommendations, such as 1900–2600 kcal day-1. → Attention Colloquially, calories are often spoken of, where kilo calories (kcal) are meant. For example, the recommendation to consume 2000 calories a day through food means 2000 kcal. Therefore sometimes food calories are expressed in the unit Cal (C in upper case). Meaning: 1 Cal = 1000 cal = 1 kcal (with c in lower case).

8.7 Caloric Value of Foods

367

Example The global demand for food can be estimated by multiplying the daily energy requirement of 10 MJ daily by a population of 9 billion: E = 9
109
10 MJ = 9
109
10
106 J = 90
1015 J = 90 PJ Multiplying this daily value by 365 days results to E = 9
109
10 MJ
365 = 32
1018 J = 23 EJ The wide range of recommendations of daily energy intake through food, such as 1900–2600 kcal day-1, is due to the fact that the energy needs of humans are individually different, and factors such as physical activity, weight, age, gender must be taken into account. An individualized calculation of the energy requirement can be carried by multiplying the individual basal metabolic rate (BMR) with a factor for the physical activity level (PAL). Let us look onto this type of calculation more in detail. Basal Metabolic Rate The individual basal metabolic rate (BMR) can be measured experimentally, e.g., by calorimetry. These are extensive long-term measurements with numerous boundary conditions such as body position, inactive non-sleeping state, time distance to the last meal, etc. Alternatively there are numerous suggestions for the calculation of the BMR, e.g., based on body weight, body surface area, body mass index, BMI [33]. In Table 8.17 some equations of body weight mK are shown for the calculation of the basal daily energy requirement of adults. Further equations are listed [33] and are based on empirical data of [34]: Example A 22-year-old woman with a body weight of 65 kg has a basal metabolic rate of BMR kcal
day - 1 = 14:818
mK ðkgÞ þ 486:6 = 14:818
65 þ 486:6 = 1449:8

Table 8.17 BMR calculation equations based on body weight (examples from [33]) Males Females

Age: 18–30 years BMR (kcal
day-1) = 15.057
mK (kg) + 692.2 BMR (kcal
day-1) = 14.818
mK (kg) + 486.6

Age: 30–60 years BMR (kcal
day-1) = 11.472
mK (kg) + 873.1 BMR (kcal
day-1) = 8.126
mK (kg) + 845.6

368

8

Table 8.18 Values of PAL for different lifestyles [33]

Thermal Properties

Lifestyle Light activity Moderate activity Vigorous activity

PAL 1.4–1.7 1.7–2.0 2.0–2.4

Calculation formulas, such as those in Table 8.17 always apply to a narrowly defined group of people such as healthy, not pregnant, not breastfeeding, etc. and nevertheless have statistical uncertainties. Physical Activity The physical activity level (PAL) is a number representing the daily average of the individual activities a person performs. It results from the weighted addition of the factors of the individual activities (PAR) which are tabulated for a variety of activities in different countries [35]. Resulting values of PAL are in the range between 1.0 and 2.4 [33]. In Table 8.18 magnitudes are given for different lifestyles. The total metabolic rate of a person can be calculated from the product of individual basal metabolic rate and the factor for physical activity: E_ = BMR
PAL

ð8:123Þ

The energy requirement for a specified period of time Δt, e.g. a day, then results from E = BMR
PAL
Δt

ð8:124Þ

Example A female person with a basal metabolic rate of 1450 kcal
day-1 and a PAL of 1.5 (8 h sleep, 8 h screen work, low leisure activity). With E = BMR
PAL we get E_ = 1450 kcal
day - 1
1:5 = = 2175 kcal
day - 1 The daily energy requirement by food is 2175 kcal. A male person (50 years, 65 kg) with a basal metabolic rate of 1600 kcal
day-1 and a PAL of 1.5 has E_ = 1600 kcal
day - 1
1:5 = 2400 kcal
day - 1 . The daily energy requirement by food is 2175 kcal.

8.7 Caloric Value of Foods

369

Example For comparison: With the rule of thumb “basal metabolic rate = 100 kJ/kg
d” we get for a person with 65 kg body weight E = 65 kg
100

kJ
1 day
1:5 = 65
100
1:5 kJ kg
day

E = 9750 kJ = 2333 kcal: It can be seen that the value is close to the previously calculated values, but that gender-specific and activity-related influences cannot be taken into account in this calculation. In addition to measuring the emitted body heat (calorimetry) to determine the metabolic rate it is possible to calculate the total metabolic rate from oxygen consumption of a test person. A unit that is often used here is the metabolic equivalent, MET. Thus, the total metabolic rate for “sitting, standing” is referred to as one metabolic equivalent. The oxygen consumption during physical activities then is given as a multiple of this metabolic equivalent. Table 8.19 gives some examples.

8.7.2

Caloric Value of Food

Since the energy gain from food is mainly due to oxidation, the reaction energy, i.e. the energy supply from ingested foods, can be calculated in a simple way. The physiological caloric value of a food indicates the specific energy that humans can obtain from these foods through physiological, chemical transformations. The physiological caloric values of some groups of substances are listed in Table 8.21. The physical measurement of the caloric value of a substance is carried out calorimetrically by complete oxidation, i.e., combustion of this substance. The value determined in this way is called physical caloric value. Since the human body does not completely oxidize proteins, but excretes the contained nitrogen as urea, the physiological caloric value of proteins is slightly lower than the physical caloric value. This difference does not exist in carbohydrates, fats and alcohol, as these substances are also completely oxidized in the human metabolism, i.e., the Table 8.19 Metabolic equivalents (MET) of physical activities, examples from [31]

Physical activity Sitting, standing Driving car Walking, 4 km h-1 Running, 8 km h-1 Swimming, crawl 45 m min-1

MET 1 1.5 3.5 7.5 15

370

8

Thermal Properties

Table 8.20 Comparison of physiological and physical caloric values, from [36] Group Proteins Carbohydrates Fats Ethanol

Physiological combustion value e (kcal
g-1) e (kJ
g-1) 17.2 4.1 17.2 4.1 38.9 9.3 29.7 7.1

Table 8.21 Physiological caloric values of the main energy sources of food [4]

Physical combustion value Δhcomb (kJ
g-1) Δhcomb (kcal
g-1) 22.6 5.4 17.2 4.1 38.9 9.3 29.7 7.1

ei (kJ
g-1) 17 17 37 29 8 13

Group Proteins Carbohydrates Fats Ethanol Dietary fiber Organic acids

e (kcal
g-1) 4 4 9 7 2 3

carbon contained leaves the body as CO2. The differences between physiological caloric value and physical caloric value can be seen from Table 8.20. According to the official European Food Information Regulation [4] the physiological caloric values given in Table 8.21 are used in declaration of food composition. According to this regulation, the physiological energy value of a food must be indicated on the packaging [4]. The values to be reported are not obtained by calorimetric measurements of production samples but mathematically using the composition and data as shown in Table 8.21. The sum of the products of relative mass fraction xi and associated specific physiological caloric value ei gives the specific physiological caloric value of the food (in J/g). By multiplying by 100 we obtain the indication for 100 g of food. etotal =

xi
ei

ð8:125Þ

i

with xi =

E ei etotal

mi mtotal

ð8:126Þ

energy contained in food in J (physiological caloric value of the food) specific physiological caloric value of the component i in J∙g-1 (Atwater coefficient) specific physiological caloric value of the food in J∙g-1 (continued)

8.7 Caloric Value of Foods

mi mtotal xi Δhcomb

371

mass of component i in g mass of food in g mass fraction of component i specific enthalpy of combustion i in J∙g-1 (physical caloric value)

Remark This calculation method is also called the Atwater system, after Wilbur Atwater (1844–1907), who early worked on the combustion calorimetry of food. The specific physiological caloric values are also referred to as Atwater factors or Atwater coefficients.

Example Physiological caloric value of pea, Pisum sativum L. (hull and seed): Composition of 100 g is [37] mi (g) 75.2 6.6 0.5 12.3 4.3 0.3

Water Protein Fat Carbohydrate Dietary fiber Organic acids

xi
ei with that values we get e =

According to etotal = i

1 ð6:6 g
100 g

17:2 kJ
g - 1 þ 0:5 g
38:9 kJ
g - 1 þ 12:37 g
17:2 kJ
g - 1 þ 4:3 g
8 kJ
1 g - 1 þ 0:3 g
13 kJ
g - 1 Þshorter :e = ð6:6
17:2 kJ þ 0:5
38:9 kJ þ 100 g 1 12:3
17:2 kJ þ 4:3
8 kJ þ 0:3
13 kJÞe =
382:8 kJ = 382:8 kJ=100 100 g ge = 91:6 kcal=100 g with the official values of EU, see Table 8.21, we get a slightly different result (continued)

372

8

Thermal Properties

1 ð6:6
17 kJ þ 0:5
37 kJ þ 12:3
17 kJ þ 4:3
8 kJ þ 0:3
13 kJÞ 100 g e = 378:1 kJ=100 g = 90:2 kcal=100 g

e=

For foods with a negligible content of fiber or organic acids, the caloric value can be roughly calculated by focussing on proteins, carbohydrates and fats. Knowing the mass fractions xi these food components, respectively, their mass in 100 g of the food under consideration is first multiplied their related Atwater coefficients and then, added together to obtain the caloric value of 100 g of that food. Example Physiological caloric value of peeled potato. Composition of 100 g edible substance: mi (g) 79.8 2.1 0.1 17.7

Water Protein Fat Carbohydrates

According to etotal =

xi
e i i

0:1 g 17:7 g 2:1 g
38:9 kJ
g - 1 þ
17:2 kJ
g - 1
17:2 kJ
g - 1 þ 100 g 100 g 100 g 1 e= 2:1 g
17:2 kJ
g - 1 þ 0:1 g
38:9 kJ
g - 1 þ 17:7 g
17:2 kJ
g - 1 100 g written shorter : 1 e= ð2:1
17:2 kJ þ 0:1
38:9 kJ þ 17:7
17:2 kJÞ 100 g 1 e= ð36:1 þ 3:9 þ 304:4Þ kJ 100 g 1 e=
344:4 kJ = 344:4 kJ=100 g 100 g e = 82:4 kcal=100 g

e=

If we need the caloric value of a meal or of a certain portion that is not exactly 100 g, it is possible to directly add the products of mass mi and associated specific physiological caloric value of component i. This is how you get the caloric value of the given portion in J.

8.7 Caloric Value of Foods

373

Etotal =

mi
ei

ð8:127Þ

i

Example One serving of nougat cream contains 4.8 g of fat, 8.5 g of carbohydrates and 0.1 g of protein. With the Atwater coefficients from Table 8.21 we get the physiological caloric value of the serving to E = 4:8 g
37 kJ
g - 1 þ 8:5 g
17 kJ
g - 1 þ 0:1 g
17 kJ
g - 1 = 324 kJ It has already been pointed out that the caloric values given on the food packaging are not laboratory-measured values of the respective production batch, but computational values calculated with the Atwater system. For this purpose, the average composition of the product is used. This average composition may be subject to crop or seasonal variations. In addition, the Atwater coefficients are themselves average values for the specific physiological caloric value of main energy sources in food. Strictly speaking, not all monosaccharides, starches, amino acids, animal and vegetable proteins have the same value of 17.2 kJ
g-1 but differ slightly. In addition, the digestive system of humans works differently, i.e. not every person can get the energy from a food component that the Atwater coefficient indicates. There is evidence that the food matrix itself also influences the energetic usability of ingredients [38], which is not taken into account in the Atwater system. These limitations should be taken into account when interpreting and applying the calculated caloric values. Not only the energy demand of humans but also the energy utilization of food is subject to individual influences.

8.7.3

Measurement of Caloric (Combustion) Values

The caloric value (energy content) of a food sample can be measured experimentally in a laboratory with the use of a combustion calorimeter. The sample is placed in the combustion calorimeter and completely oxidized. The heat evolved from this oxidation reaction causes a rise in temperature that is measured and recorded, from which the caloric value can be calculated. Standard methods for measuring caloric food values with combustion calorimeters can be found in [39, 40]. In a combustion calorimeter all the protein, carbohydrate and fat are completely oxidized into NO2, CO2, and H2O. However, the human body has a different need for the nitrogen in the protein, and does not oxidize the nitrogen into NO2 as does the calorimeter. Much of the nitrogen coming from proteins in the human body is metabolized in the following way: 80–90% to urea (H2N–CO–NH2), 3–5% to ureic acid and >1 χ> >0 μ 1 ≈1 0.999991 1000 500–3000 1–1.6 χ -9.65∙10-6 -9.03∙10-6 -8.60∙10-9 1.86∙10-6 3.62∙10-3 2.08∙10-5 50–500

10.2.1 Hysteresis in Magnetization When a ferromagnetic material is placed within a magnetic field, its level of magnetization will depend on the strength of the external magnetic field. As the field strength H is increased the magnetization will also increase, but at a retarding rate as it approaches an upper saturation limit (Fig. 10.3). When the field strength is decreased, the level of magnetization will follow a different path and remain at higher levels for a given field strength in response to decreasing field strength than when responding to increasing field strength. This type of behavior is known as

Fig. 10.3 Hysteresis behavior in a ferromagnetic material, schematic. When the field strength H is reduced to zero, the magnetization M does not return to zero

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hysteresis. The amount of magnetization retained upon release of the magnetic field is known as remanent magnetization. Because of this behavior, the magnetic permeability of a material is not constant but will depend on the magnetic history of the material. Figure 10.3 shows the hysteresis curve of a ferromagnetic material. The point at which the curve intersects the magnetic field strength axis (H-axis) gives the coercive field strength. This coercive field strength is the field strength required to bring the remaining magnetization in the material (remanent magnetization) back to zero. Materials with high levels of remanent magnetization tend to retain their magnetism. These types of materials are most useful in such commercial products as magnetic memory tapes, magnetic strips, and permanent magnets in electric motors and generators. On the other hand, materials with low levels of remanent magnetization tend to change their magnetism. They are good, e.g., for write-read devices for magnetic data storage units. Classic computer hard disk drives (HDDs) are magnetic devices. They consist of rotating disks coated with a magnetic material. A read/write head writes bits by magnetizing the surface in two states called 1 or 0. One bit on the magnetic material has a dimension of about 10 nm. Reading of data takes place in reverse; the read head scans the magnetization of the surface without contact. Magnetic hard drives are increasingly being replaced by solid-state drives (SSDs). Like a field-effect transistor (FET), the SSD flash semiconductor memory can have two different states: electrically charged and uncharged. Definition Hysteresis is a change that occurs with a delay to the cause. As a result, materials can have properties that result from a previous treatment of the material. With hard magnetic materials in magnetic strips or computer hard drives, these properties are used to store information. The source used for a magnetic field can be a permanent magnetic material (magnet), or it can be an electromagnet, acting like a magnet only when energized by an electric current, which can be switched on and off (Fig. 10.4). Our planet Earth also has a magnetic field, but its strength is very small when compared to the strength of magnetic fields from magnets used in commercial industry. The magnetic field strength H has the SI unit A
m‐1; the magnetic flux density B is given in the unit (Tesla) 1 T = 1 V
s
m-2. The Earth’s magnetic field has a horizontal magnetic flux density of about 40 μT near the ground, clinical MRI examinations are carried out with about 3 T, and the magnitude of strong magnetic fields that can be produced in the laboratory is 10 T, in short pulses up to 100 T.

10.2

Magnetization

439

Fig. 10.4 Magnetic field in the vicinity of an electric current: wire (right), wire loop (left), schematic

10.2.2 Metal Separation The most obvious property of the magnetic field is the force effect of a magnet on other bodies. In material recycling, such as in food packaging, metals can be separated and sorted in this way. Sorting is carried out according to susceptibility. While ferromagnetic and paramagnetic metals are attracted to magnets, diamagnetic metals experience a repulsive force in alternating magnetic fields. This results from the magnetic field of the eddy currents induced in the diamagnetic metal. Strong alternating magnetic fields can be easily generated by rotating permanent magnets. In order to separate even small paramagnetic metal parts from the product stream, e.g., a powder, strong magnets are required at a short distance from the product. Packaged products can be tested for metallic contaminants with inductive metal detectors consisting of a transmitter coil and a receiver coil. Products being inspected are passed through the alternating electromagnetic field of the transmitter coil. The field change by each individual product is monitored with the receiver coil. A material with electrical conductivity experiences an electrical induction voltage in the variable magnetic field, which leads to eddy currents in the material. The magnetic field of these eddy currents weakens the causative field and can be detected by the receiver coil. As a result of induction, a foreign body with electrical conductivity can be detected in the product. The higher the magnetic susceptibility of the foreign body, the stronger the signal; i.e., ferromagnetic materials are more easily detected than paramagnetic and diamagnetic materials. The penetration depth of the alternating electromagnetic field depends on the frequency of the field. Detection limits are often given as the diameter of a sphere made of the material in question, which can barely be detected. Since foods themselves have a certain electrical conductivity (see Chap. 9), they themselves cause a signal in the metal detector, the so-called product effect. For foods that have a large product effect, the detection of small metal parts becomes more difficult. Inductive metal detectors are often tunnel-shaped so that bulk materials, liquids, or conveyor belts with packaged food can be passed through them. In Fig. 10.5, an arrangement of transmitter and receiver coils for a tunnel-shaped metal detector is shown schematically.

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Fig. 10.5 Transmitter (I) and receiver (II) of a three-coil arrangement, schematic

10.2.3 Lorentz Force When electrically charged particles are moving at a velocity v in a magnetic field with field strength B, they will be subjected to a force acting on them called the Lorentz force FL. The Lorentz force will increase with increasing velocity, charge, and field strength and will act in the direction that is mutually perpendicular to both the velocity of the moving particle and the magnetic field. The Lorentz force acting on a charged particle moving in a magnetic field can be expressed as follows: → →

jF L j = Q
v
B
sin∢ v ; B FL Q v B

ð10:14Þ

Lorentz’s force in N electric charge in C velocity in m
s-1 magnetic flux density in V
s
m-2

The Lorentz force is the reason why charged particles cannot travel in a straight line when passing through a magnetic field. The Lorentz forces will “push” them in a direction perpendicular to that of their initial travel direction, so that charged particles will move along a curved path instead of a straight path. In symmetrical cases, this leads to circular currents called eddy currents. Eddy currents cause heating of a material. In this way, the temperature of electrically conductive materials can be increased by induction without contact. This is the principle of so-called induction cooking.

10.2.4 Induction Cooker Under a glass-ceramic plate an electrically operated coil is placed that generates an alternating electromagnetic field of 20–100 kHz. In the bottom of a cooking pot made of an electrically conductive material, eddy currents are generated, which

10.2

Magnetization

441

Fig. 10.6 Example of the construction of induction cookware, schematic: 1. ferromagnetic stainless steel on the outside, 2. aluminum or copper in the core, 3. standard stainless steel on the inside

cause heating of the bottom of the pot. An electrically non-conductive material on the glass-ceramic plate shows no effect. The advantage of inductive heating is that the heat is generated within the bottom of the pot and does not have to get there by heat conduction (hot plate), heat transfer (hot flame), or heat radiation (infrared radiator). For this reason, the heating rate in inductive cooking is high. The electrical resistance of the material from which the bottom of the pot is made must not be so low to get heat generated by eddy currents become high. Here, stainless steel with a higher specific resistance is more advantageous than copper or aluminum which have a higher electrical conductivity. Due to the skin effect, eddy currents of highfrequency fields are predominantly located in shallow material depths. In order to effectively transport the heat generated there into the volume of the pot bottom, materials with high thermal conductivity can be encapsulated in the stainless-steel pot bottom. While inductive heating works in principle with all electrically conductive materials, a magnetic stainless steel (ferritic or martensitic stainless steel) is advantageous for induction pots: Below the Curie temperature, the periodic demagnetization of the Weiss districts in the alternating electromagnetic field of the induction cooker causes additional heat generation. It is significantly larger than in paramagnetic or diamagnetic materials where no Weiss districts exists. The high permeability of magnetic stainless steel further concentrates the electromagnetic field of the cooker in the pot, which means radiation losses are decreased. Some induction cookers even cannot be started if the magnetic permeability of the cookware used is detected as too low (Fig. 10.6). Bottom line Cookware for induction cookers is often made of several materials to obtain a combination of their properties. The bottom of a pot or pan from the outside to the inside: Ferromagnetic stainless steel on the outside ensures optimal use of the magnetic field. This is followed by a layer of aluminum or copper, sometimes both, as these metals have a higher thermal conductivity than stainless steel. On the inside there is again a layer of stainless steel, because aluminum or copper is not sufficiently corrosion-resistant against food. Cast iron can be used in place of stainless steel. It is ferromagnetic and has a slightly higher thermal conductivity than stainless steel. As an option, the inside of the cookware can be provided with a non-stick coating for ease of cleaning.

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Fig. 10.7 Generation of an Hall voltage UH, schematic

A Hall probe consists of a semiconductor platelet through which an electric current flows. A magnetic field perpendicular to the current direction leads to a Lorentz force on the charge carriers. In Fig. 10.7 the principle is illustrated: An electric current I flows through the plate perpendicular to the B field. The Lorentz force FL directs positive charge carriers in the semiconductor upward and negative charge carriers downward; this leads to a voltage UH, the Hall voltage, perpendicular to the current direction. The Hall voltage is proportional to the magnetic flux density B, so Hall probes can be used to detect magnetic fields and measure the field strength. Since electric currents are the cause of magnetic fields, currents can be measured without contacting any surface because of the Hall effect. By using a Hall probe to detect magnetic field changes, it can, for example, act as a distance sensor or as a speed sensor. Hall sensors are used in processing plants for pig detection. A piston-shaped cleaning device, which is used in pipe systems for cleaning or inspection, is colloquially referred to as a pig [8]. In magnetic inductive flowmeters (MID, also called magnetars), the Lorentz force generates an electrical voltage proportional to the flow velocity, analogous to the Hall effect. This makes it possible to measure the volume flow in a pipe. The principle of magnetic inductive flow sensors is explained in Chap. 16. The Lorentz force can also be used to direct electron beams. In this way, in electron microscopes, rays can be focused with so-called magnetic lenses. The deflection of charged particles in mass spectrometers, synchrotrons, and cyclotrons is also caused by the Lorentz force.

10.3

Magnetic Resonance

Spectroscopic methods, in which the transitions between different precession states of magnetic moments are exploited, are called magnetic resonance methods. A distinction is made between nuclear magnetic resonance (NMR), electron spin resonance (ESR), cyclotron resonance, ferromagnetic resonance (FMR), nuclear quadrupole resonance (NQR), and muon spin resonance (μSR). Let us first look at the nuclear spin resonance.

10.3

Magnetic Resonance

443

Fig. 10.8 Rotating sphere with a magnetic moment μ (magnetic dipole with north pole N and south pole S) as a simple model of an atomic nucleus with a magnetic moment

N

P

S Table 10.5 Atomic nuclei with and without magnetic moments Number of protons Odd Even Odd Even

Mass number Odd Odd Even Even

Named Odd-odd nucleus Even-odd nucleus Odd-even nucleus Even-even nucleus

Magnetic moment Yes Yes Yes No

Fig. 10.9 Adjustment options of the nuclear spin in magnetic field B. The two states differ by the energy difference ΔE

Example F 13 C 1 H 12 C 19

P

B

P

B

Atomic nuclei with an odd number of protons (atomic number) or an odd atomic mass number (the sum of neutrons and protons) have a magnetic moment. The magnetic moment of an atomic nucleus is very small; it is about a factor of 1000 less than the magnetic moment of the atomic shell. We can imagine such an atomic nucleus with the help of a simple model where a rotating sphere within which the nuclear charge has a magnetic north pole and south pole (Fig. 10.8). Atomic nuclei of the isotopes 1H, 13C, 19F, or 31P therefore have a magnetic moment. Not the atomic nucleus of 12C, which consists of six neutrons and six protons. 12C cannot therefore be investigated by NMR. In Table 10.5, these examples are listed. Like atoms, atomic nuclei can also go to higher energy levels by excitation, i.e., by energy absorption. For quantum mechanical reasons, there are only certain discrete states that can be taken. There are two possible states for the nuclear spin. Without an external magnetic field, these two states are practically the same energetically, but due to the magnetic moment of the atomic nucleus, in a → magnetic field it is a difference whether μ is pointing in the direction of the field or in the opposite direction. Figure 10.9 shows the two states; the vector

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Fig. 10.10 Energetic states of atomic nucleus spin. In a magnetic field (II), the energy difference ΔE between energy states is higher than without a magnetic field (I)

→

→

μ performs a precession movement around vector B with a characteristic frequency, the Larmor frequency. Figure 10.10 illustrates the splitting into two energy levels, which differ by ΔE. Definition Precession is the change of direction of the axis of a rotating body when external forces act on it. The part of the external force acting perpendicular to the axis of rotation causes a torque that causes the axis of rotation to precess. If the torque is constant, the axis of rotation circles around its original position with the precession frequency. This energy difference ΔE is used in magnetic resonance spectroscopy. By studying which frequencies of an electromagnetic external field a material can absorb, information about this excitation energy ΔE is obtained. As soon as an excitation frequency matches the Larmor frequency of the material, resonance occurs. In nuclear resonance, these frequencies are in the range of radio waves (30 MHz to 300 GHz). The resonance frequency of the atomic nucleus depends on its magnetic moment μ and the strength of the magnetic field B used. It results from: h
fL =2
μ
B

ð10:15Þ

to fL = h fL μ B

2
μ
B h

ð10:16Þ Planck’s constant in J
s Lamor frequency in s-1 magnetic moment in C∙m magnetic flux density in V
s
m-2

10.3

Magnetic Resonance

445

Example The Lamor frequency of a proton 1H results according to fL =

2
μ
B = B
42:5 MHz
T - 1 h

In a magnetic field with 1 T fL =

2
μ
B = 1 T
42:5 MHz
T - 1 = 42:5 MHz h

and in a magnetic field with 24 T fL =

2
μ
B = 24 T
42:5 MHz
T - 1 = 1:02 GHz h

However, the exact resonance frequency depends somewhat on the physical and chemical composition of the atomic space surrounding the nucleus. This influence is known as chemical shift of the resonance frequency. The electron clouds of an atom, as well as those of neighboring atoms and their atomic bonding, also further influence the resonance frequency slightly. These combined influences cause a measurable difference in resonant frequencies between small and large atoms and their types of bonding to other atoms. For example, we can use detection of resonance frequency to distinguish between the hydrogen nucleus in a C–H bonding and an O–H bonding. Nuclear magnetic resonance (NMR) spectroscopy is based upon the ability to make this type of distinction. NMR techniques are used to measure the resonance frequencies (absorption frequencies) of atomic nuclei. This gives us information, not only about the atom but also about the state of chemical bonding in the neighborhood of the atom. In this way, NMR spectra can help us to identify chemical groups and side groups, as well as their chemical state. This is the principle of material investigation by magnetic resonance. Definition The Zeeman effect is the splitting of an energy level into two different energy levels by a magnetic field. The Zeeman effect was discovered during the splitting of spectral lines as an effect of the magnetic field on the electrons of an atom. Since certain atomic nuclei in the magnetic field also have a splitting of nuclear spin energy levels, this is analogously referred to as the Zeeman effect of the atomic nucleus. Simplified, an NMR spectrometer consists of a magnetic field in which a sample holder is located. A transmitter coil generates electromagnetic radio waves, and a

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Fig. 10.11 Schematic of an NMR spectrometer. 1 Magnet, 2 sample holder, 3 transmitter and receiver coil

Fig. 10.12 Relaxation signal. The attenuation of the signal is called FID, free induction decay

I

0

t detector coil records the behavior of the sample, i.e., checks for resonance. Figure 10.11 shows an NMR spectrometer schematically. A distinction is made between pulsed NMR and continuous wave methods (CW). While in pulse operation the radio frequency signal is irradiated for a short moment, only, the radio frequency in CW operation is permanent. To find the resonance frequency in CW operation, we can either slowly vary the magnetic field strength (field sweep) or vary the radio frequency (frequency sweep) with a constant magnetic field. In pulse operation, the atomic nuclei are excited in the constant magnetic field with a radio pulse. The duration of the pulse is in the ms range. The direction of the radio frequency field is inclined toward the direction of the B field by 90°. The atomic nuclei absorb the “appropriate” frequencies and absorb energy. After the end of the radio pulse, the release of this absorbed energy by the atomic nuclei begins. This is called relaxation. The relaxation signals of an atomic nucleus appear at the detector coil as a cosine oscillation, the amplitude of which decreases rapidly as a result of damping. The temporal decrease of this signal is called free induction decay, FID (Fig. 10.12). From the kinetics of damping, we get information about the relaxing nuclei.

10.3

Magnetic Resonance

447

Definition Radio waves are electromagnetic waves in the range of 30 MHz and 300 GHz.

Bottom line Nuclear magnetic resonance spectroscopy (NMR) determines the frequencies that are absorbed by the atomic nuclei of a material in the magnetic field. The frequencies provide information about the gyromagnetic ratio of the nuclei, which depends on the material environment of the nucleus. The intensities of the single frequencies and the relaxation behavior provide further information. In high-resolution nuclear resonance spectroscopy (HR-NMR) the relaxation signal is analyzed to see what frequencies it contains. This is achieved by Fourier transformation of the signal. The Fourier transformation provides the frequency spectrum, i.e., the intensities of the occurring frequencies, see Fig. 10.13. If the determined intensities are not applied as a function of the frequency but as a function of the chemical shift compared to a reference value, this is called NMR spectrum in the chemical laboratory language [9]. The intensities in the NMR frequency spectrum are a measure of the number of atomic nuclei that show resonance at that frequency. The frequencies are characteristic of the atomic nucleus, i.e., for the isotope under consideration and its electronic physicochemical environment. For example, the frequencies of 13C and 12C isotope differ, as well as the frequencies of proton (1H) and deuteron (2D). The isotope ratios in food depend on their origin, e.g., composition of soil, rainwater, and animal feed. This allows conclusions to be drawn about the growing region of a food such as the origin with the help of the NMR spectrum. To authenticate food, the NMR spectrum is compared with spectra stored in databases [9, 10]. The different metabolism in C3 and C4 plants also leads to characteristic 12C/13C isotope ratios, so that conclusions can be drawn from the NMR spectrum in combination with mass spectroscopy about which plants were used [11]. In this way, e.g., glucose sugar from sugar cane and sugar beet can be distinguished from glucose sugar obtained from corn [12]. Similar applications exist for honey [13], fats and oils [14], wine [15], and other foodstuffs where fraud plays a role [16]. The spectra of NMR, in which the intensity is plotted over a frequency axis, are one-dimensional spectra (1D NMR). If we add another coupling quantity, we get spectra with two frequency axes on which the intensity axis is perpendicular, so-called two-dimensional spectra (2D NMR). With the help of multidimensional NMR spectroscopy (2D, 3D, etc.) the analytic specificity regarding the chemical environment of the atomic nuclei under consideration can be increased and thus the field of application of NMR can be expanded [10, 12, 17].

448 Fig. 10.13 HR-NMR: By Fourier transformation of the free induction decay (upper picture) we get the intensities of the resonance frequencies (picture in the middle) of a sample. Often the frequency shift related to a standard is used in such spectra (picture at the bottom)

10

I

Magnetic Properties

0

t

I

Q

I

5

4

3

2

chemical shift/ ppm

1

0

10.3

Magnetic Resonance

449

Definition Isotopes are atoms with the same atomic number but different masses. They thus have the same number of protons but a different number of neutrons in the atomic nucleus. Isotopes can be distinguished by NMR in combination with mass spectroscopy. For example, the isotope ratio of 12C and 13C can be determined in food. In low-resolution nuclear resonance spectroscopy (LR-NMR), the FID signal is not analyzed in regard to the frequencies contained, but to its shape, i.e., with regard to its attenuation, which depends on the relaxation time of the atomic nuclei. The FID signal results from a superposition of spin–lattice relaxation (relaxation time T1) and spin–spin relaxation (T2) of the atomic nuclei involved [9]. Protons (1H atomic nuclei) relax at different rates depending on the physicochemical environment of the atomic nuclei. For example, protons in solid phases relax faster than protons in liquid phases and protons in water relax faster than protons in oil. For these reasons, LR-NMR can be used to determine, e.g., fat contents, water contents, and solid–liquid ratios in the sample. In a strict sense, this type of NMR spectroscopy (so-called time domain NMR) does not belong to the spectroscopic methods, since no spectra (i.e., intensity as a function of frequency or frequency domain NMR) are obtained. Table 10.6 lists the terms. In Fig. 10.14, the FID signal of a fat sample is shown schematically. The initial loss of intensity is mainly caused by the solid-phase protons in the investigated material. The longer delay part of the signal is determined by liquid-phase protons, which have a longer relaxation time. By mathematically reconstructing these two

Table 10.6 Pulse NMR, differences between high-resolution and low-resolution NMR Synonym: Fourier transformation of FID: Evaluation of:

Fig. 10.14 FID signal of a fat sample after an NMR pulse: solid (I) and liquid phase (Il) show different relaxation curves. The measured FID (III) is the superposition of both curves

Low resolution Time domain NMR No Attenuation of FID

High resolution Frequency domain NMR Yes Frequency spectrum

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relaxation curves from the measurement curve, the solid–liquid ratio of the sample (e.g. solid fat content, SFC) can be determined [18, 19]. For the 1H-low-resolution NMR various possibilities of signal evaluation are possible, such as comparison of intensities, evaluation of the relaxation behavior, and evaluation of the spin echo. From the intensity, we get information about the number of protons. By appropriate sample preparation, it can then be achieved that this number of protons correlates with the fat content or the water content in the sample. Due to the different relaxation behavior of protons in a solid and liquid phase, solid–liquid ratio can be determined. To enhance the different relaxation behavior of solid and liquid substance, the spin–echo technique can be used. Here, after a radio pulse, a short moment is waited until most of the solid-phase protons are relaxed and then a second 90° rotated radio pulse is irradiated. The result is another decay signal, the so-called spin echo, which is caused by the nuclei with the greater relaxation time [12]. Imaging methods of nuclear resonance are referred to as MRI (magnetic resonance imaging) methods. Like in biomedical engineering, it is possible to visualize the structure of food or agricultural products. The investigation of food structure can serve in conducting quality assurance [20–23] in food processing [24–26]. While high-resolution NMR devices need more space, low-resolution 1 H-NMR (time domain NMR) devices are available as compact desktop devices [27]. Further developments include portable NMR devices for rapid quality checks; see Chap. 16.

10.3.1 Electron Spin Resonance While we have been studying effects from the magnetic moment of atomic nuclei so far, we now look at the magnetic moment that comes from unpaired electrons in an atom. Materials with atoms that have one or more unpaired electrons show the Zeeman effect in the magnetic field. This means that these atoms absorb a characteristic radio frequency in the magnetic field similar to NMR. The analysis method is called electron spin resonance (ESR) or electron paramagnetic resonance (EPR). Since chemical radicals are also compounds with an unpaired electron, they are detectable by ESR. This is used to detect irradiation of food and packaging. Highenergy electromagnetic radiation such as gamma radiation leads to the formation of radicals, which can be detected by ESR [28]. However, due to the high reactivity of radicals their concentration is not constant over time, and this must be considered when interpreting the ESR spectra. There are methods for irradiation checks of meat, fruits, vegetables, spices, cereals, oilseeds, coffee, hazelnuts, sugars, cellulose, etc. [29–35]. Since the oxidative spoilage of food is also associated with radicals, ESR can be used to assess the antioxidant capacity of foods and additives [30]. Electron spin resonance imaging (ESRI) is a method to visualize the spatial concentration distribution of the radicals [28].

References

451

Further Reading Freezing under the influence of magnetic fields Preservation process under the influence of magnetic fields Thermometry with the help of paramagnetic particles Honey, beer, spices: authentication by NMR Coffee: authentication by NMR Wine: authentication by NMR Apple: tissue characterization by MRI Automated classification of food products using 2D low-field NMR Oilseed residues—simultaneous determination of oil and water content by pulsed NMR Animal and vegetable fats and oils, determination of solid content of fat (SFC) Cheese and dairy products: metabolites by NMR Meat structure using MRI Whey protein gels, drying, MRI NMR for rheological characterization of fluids Hazelnut: irradiation detection by ESR Meat (beef, pork, chicken), irradiation detection by ESR Oxidation of food, examination by ESR Antiplastization of starch-sucrose blends—positron lifetime and NMR study

[36–38] [39] [40] [13, 41, 42] [43] [15] [20] [44] [45, 46] [19] [47] [21, 22] [48] [49] [34] [32] [30] [50]

Summary The reasons for the different behavior of materials in the magnetic field are explained in a fundamental way and illustrated based on applications such as induction cooking and metal detection. The different forms of magnetic resonance to study the composition, structure, and origin of food are explained in simple words. Imaging techniques and electron spin resonance for the detection of radicals in food are addressed. At the end of the chapter, application examples are listed, which illustrate the potential of magnetic resonance spectroscopy and may stimulate further studies.

References 1. Rodriguez AC, Sanchez-Benitez J, Sanz PD (2017) Simulation of the magnetic freezing process applied to foods. Food Eng Rev 9(4):271. https://doi.org/10.1007/s12393-017-9169-3 2. Minano HLA, Silva ACD, Souto S, Costa EJX (2020) Magnetic fields in food processing perspectives, applications and action models. Processes 8(7). https://doi.org/10.3390/ pr8070814

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3. Barbosa-Canovas GV, Schaffner DW, Pierson MD, Zhang QH (2000) Oscillating magnetic fields. J Food Sci 65(s8):86. https://doi.org/10.1111/j.1750-3841.2000.tb00622.x 4. Rodríguez Plaza AC (2017) Characterization of electromagnetic freezing in food matrixes and model food. University Polytechnika, Madrid 5. Pobell F (2007) Matter and methods at low temperatures. Springer, Heidelberg. https://doi.org/ 10.1007/978-3-540-46360-3 6. Kohlrausch F (1996) Praktische Physik Bd.3. Teubner, Stuttgart. https://doi.org/10.1002/piuz. 19870180109 7. Czichos H, Hennecke M, e.V. AVH (eds) (2008) HÜTTE - Das Ingenieurwissen. Springer, Heidelberg. https://doi.org/10.1007/978-3-540-71852-9 8. Mühlthaler W, Hiltscher G, Smits J (1999) Molchtechnik. Wiley-VCH, Weimheim. https://doi. org/10.1002/9783527625789 9. Hatzakis E (2019) Nuclear magnetic resonance (NMR) spectroscopy in food science: a comprehensive review. Compr Rev Food Sci Food Saf 18(1):189. https://doi.org/10.1111/ 1541-4337.12408 10. Cao R, Liu X, Liu Y, Zhai X, Cao T, Wang A, Qiu J (2020) Applications of nuclear magnetic resonance spectroscopy to the evaluation of complex food constituents. Food Chem:128258. https://doi.org/10.1016/j.foodchem.2020.128258 11. Zhou L, Zhang C, Qiu Z, He Y (2020) Information fusion of emerging non-destructive analytical techniques for food quality authentication: a survey. TrAC Trends Anal Chem 127: 115901. https://doi.org/10.1016/j.trac.2020.115901 12. Marcone MF, Wang S, Albabish W, Nie S, Somnarain D, Hill A (2013) Diverse food-based applications of nuclear magnetic resonance (NMR) technology. https://doi.org/10.1016/j. foodres.2012.12.046 13. Kuballa T, Brunner TS, Thongpanchang T, Walch SG, Lachenmeier DW (2018) Application of NMR for authentication of honey, beer and spices. Curr Opin Food Sci 19:57. https://doi.org/ 10.1016/j.cofs.2018.01.007 14. Guyader S, Thomas F, Portaluri V, Jamin E, Akoka S, Silvestre V, Remaud G (2018) Authentication of edible fats and oils by non-targeted 13C INEPT NMR spectroscopy. Food Control 91:216. https://doi.org/10.1016/j.foodcont.2018.03.046 15. Mascellani A, Hoca G, Babisz M, Krska P, Kloucek P, Havlik J (2021) 1H NMR chemometric models for classification of Czech wine type and variety. Food Chem 339:127852. https://doi. org/10.1016/j.foodchem.2020.127852 16. Sobolev AP, Thomas F, Donarski J, Ingallina C, Circi S, Cesare Marincola F, Capitani D, Mannina L (2019) Use of NMR applications to tackle future food fraud issues. Trends Food Sci Technol 91:347. https://doi.org/10.1016/j.tifs.2019.07.035 17. Gouilleux B, Charrier B, Akoka S, Felpin F-X, Rodriguez-Zubiri M, Giraudeau P (2016) Ultrafast 2D NMR on a benchtop spectrometer: applications and perspectives. TrAC Trends Anal Chem 83:65. https://doi.org/10.1016/j.trac.2016.01.014 18. Linke C, Guthausen G, Floter E, Drusch S (2018) Solid fat content determination of dispersed lipids by time-domain NMR. Eur J Lipid Sci Technol 120(4). https://doi.org/10.1002/ejlt. 201700132 19. DIN EN ISO 8292 (2010) Tierische und pflanzliche Fette und Öle - Bestimmung des Festanteils von Fett durch das Verfahren mit gepulster magnetischer Kernresonanz. Beuth, Berlin. https:// doi.org/10.31030/1633109 20. Defraeye T, Lehmann V, Gross D, Holat C, Herremans E, Verboven P, Verlinden BE, Nicolai BM (2013) Application of MRI for tissue characterisation of ‘Braeburn’ apple. Postharv Biol Technol 75:96. https://doi.org/10.1016/j.postharvbio.2012.08.009 21. Clerjon S, Bonny J-M (2018) NMR imaging of meat. In: Webb GA (ed) Modern magnetic resonance. Springer International, Cham, pp 1609–1628. https://doi.org/10.1007/978-3-31928388-3_131 22. Bajd F, Skrlep M, Candek-Potokar M, Sersa I (2017) MRI-aided texture analyses of compressed meat products. J Food Eng 207:108. https://doi.org/10.1016/j.jfoodeng.2017.03.026

References

453

23. Van As H, van Duynhoven J (2013) MRI of plants and foods. J Magn Reson 229:25. https://doi. org/10.1016/j.jmr.2012.12.019 24. Ezeanaka MC, Nsor-Atindana J, Zhang M (2019) Online low-field nuclear magnetic resonance (LF-NMR) and magnetic resonance imaging (MRI) for food quality optimization in food processing. Food Bioprocess Tech 12(9):1435. https://doi.org/10.1007/s11947-019-02296-w 25. Xu FF, Jin X, Zhang L, Chen XD (2017) Investigation on water status and distribution in broccoli and the effects of drying on water status using NMR and MRI methods. Food Res Int 96:191. https://doi.org/10.1016/j.foodres.2017.03.041 26. Bows JR, Patrick ML, Nott KP, Hall LD (2001) Three-dimensional MRI mapping of minimum temperatures achieved in microwave and conventional food processing. Int J Food Sci Techol 36(3):243. https://doi.org/10.1046/j.1365-2621.2001.00444.x 27. Blümich B (2016) Introduction to compact NMR: a review of methods. TrAC Trends Anal Chem 83:2. https://doi.org/10.1016/j.trac.2015.12.012 28. Leveque PP, Godechal Q, Gallez B (2008) EPR spectroscopy and imaging of free radicals in food. Isr J Chem 48(1):19. https://doi.org/10.1560/IJC.48.1.19 29. Yu LL, Cheng Z (2008) Application of electron spin resonance (ESR) spectrometry in nutraceutical and food research. Mol Nutr Food Res 52(1):62. https://doi.org/10.1002/mnfr. 200700395 30. Andersen ML, Skibsted LH (2017) ESR spectroscopy for the study of oxidative processes in food and beverages. In: Webb GA (ed) Modern magnetic resonance. Springer International, Cham, pp 1–14. https://doi.org/10.1007/978-3-319-28275-6_25-1 31. Ahn J-J, Akram K, Kim H-K, Kwon J-H (2012) Electron spin resonance spectroscopy for the identification of irradiated foods with complex ESR signals. Food Anal Method 6(1):301. https://doi.org/10.1007/s12161-012-9440-4 32. Park YD, Kim DY, Jin CH, Yang HS, Choi DS, Yook H-S, Byun M-W, Jeong IY (2011) Development of a method based on ESR spectroscopy for the identification of irradiated beef, pork, and chicken meats. Food Sci Biotechnol 20(2):367. https://doi.org/10.1007/s10068-0110051-4 33. Marchioni E (2006) ESR as a technique for food irradiation detection. In: Webb GA (ed) Modern magnetic resonance. Springer, Dordrecht, pp 1855–1860. https://doi.org/10. 1007/1-4020-3910-7_211 34. Chiappinelli A, Mangiacotti M, Tomaiuolo M, Trotta G, Marchesani G, Chiaravalle AE (2019) Identification of X-ray-irradiated hazelnuts by electron spin resonance (ESR) spectroscopy. Eur Food Res Technol 245(10):2323. https://doi.org/10.1007/s00217-019-03349-2 35. Barba FJ, Roohinejad S, Ishikawa K, Leong SY, El-Din A Bekhit A, Saraiva JA, Lebovka N (2020) Electron spin resonance as a tool to monitor the influence of novel processing technologies on food properties. Trends Food Sci Technol 100:77. https://doi.org/10.1016/j. tifs.2020.03.032 36. Otero L, Rodríguez AC, Pérez-Mateos M, Sanz PD (2016) Effects of magnetic fields on freezing: application to biological products. Compr Rev Food Sci Food Saf 15(3):646. https:// doi.org/10.1111/1541-4337.12202 37. Dalvi-Isfahan M, Hamdami N, Xanthakis E, Le-Bail A (2017) Review on the control of ice nucleation by ultrasound waves, electric and magnetic fields. J Food Eng 195:222. https://doi. org/10.1016/j.jfoodeng.2016.10.001 38. Woo MW, Mujumdar AS (2010) Effects of electric and magnetic field on freezing and possible relevance in freeze drying. Dry Technol 28(4):433. https://doi.org/10.1080/ 07373930903202077 39. Grigelmo N, Soliva-Fortuny R, Barbosa-Cánovas G, Martin-Belloso O (2011) Use of oscillating magnetic fields in food preservation. In: Zhang HQ, Barbosa-Cánovas GV, Balasubramaniam VM, Dunne CP, Farkas DF, Yuan JTC (eds) Nonthermal processing technologies for food. Blackwell, pp 222–235. https://doi.org/10.1002/9780470958360.ch16 40. Ghiron K, Litchfield B (1997) Magnetic thermometry in the aseptic processing of foods containing particulates. J Appl Phys 81(8):4321. https://doi.org/10.1063/1.364756

454

10

Magnetic Properties

41. Galvin-King P, Haughey SA, Elliott CT (2018) Herb and spice fraud; the drivers, challenges and detection. Food Control 88:85. https://doi.org/10.1016/j.foodcont.2017.12.031 42. Siddiqui AJ, Musharraf SG, Choudhary MI, Rahman AU (2017) Application of analytical methods in authentication and adulteration of honey. Food Chem 217:687. https://doi.org/10. 1016/j.foodchem.2016.09.001 43. Milani MI, Rossini EL, Catelani TA, Pezza L, Toci AT, Pezza HR (2020) Authentication of roasted and ground coffee samples containing multiple adulterants using NMR and a chemometric approach. Food Control 112:107104. https://doi.org/10.1016/j.foodcont.2020. 107104 44. Greer M, Chen C, Mandal S (2018) Automated classification of food products using 2D low-field NMR. J Magn Resonan 294:44. https://doi.org/10.1016/j.jmr.2018.06.011 45. ISO 10632 (2000) Ölsaatenrückstände - Gleichzeitige Bestimmung des Öl- und Wassergehalts Spektralphotometrisches Pulse NMR-Verfahren. Beuth, Berlin 46. DIN EN ISO 10565 (1998) Ölsamen - Gleichzeitige Bestimmung des Öl- und Wassergehaltes Verfahren mit gepulster Kernresonanzspektroskopie. Beuth, Berlin. https://doi.org/10.31030/ 7537439 47. Scano P, Cusano E, Caboni P, Consonni R (2019) NMR metabolite profiles of dairy: a review. Int Dairy J 90:56. https://doi.org/10.1016/j.idairyj.2018.11.004 48. Fan LY, Yang JX, Casali RA, Jin X, Chen XD, Mercade-Prieto R (2017) Magnetic resonance imaging (MRI) to quantify the swelling and drying of whey protein hydrogels. J Food Eng 214: 97. https://doi.org/10.1016/j.jfoodeng.2017.06.033 49. Callaghan PT (1999) Rheo-NMR: nuclear magnetic resonance and the rheology of complex fluids. Rep Prog Phys 62(4):599. https://doi.org/10.1088/0034-4885/62/4/003 50. Martini F, Hughes DJ, Badolato Bönisch G, Zwick T, Schäfer C, Geppi M, Alam MA, Ubbink J (2020) Antiplasticization and phase behavior in phase-separated modified starch-sucrose blends: a positron lifetime and solid-state NMR study. Carbohydr Polym 250:116931. https:// doi.org/10.1016/j.carbpol.2020.116931

11

Electromagnetic Properties

Electromagnetic properties of foods are all those properties that are related to interaction of the food with electromagnetic waves. Among these interactions is the absorption of a certain wavelength, which can give us information about the food material. This information can lead to the design and development of sensors for quality testing, or energy transfer to accomplish heating, as well as other uses. Applying different frequency ranges of electromagnetic radiation (Table 11.1) can lead to the formation of such energy patterns as microwaves, radar, X-ray, and ultraviolet radiation. In this chapter, we first focus on the reasons for the interaction of matter with electromagnetic rays and then discuss microwaves and terahertz waves. The application of lower frequency electromagnetic rays has already been discussed in Chap. 9 “Electrical properties” and Chap. 10 “Magnetic properties.” The range of higher frequencies (NIR, visible light, UV, and X-ray) is dealt with in the following chapter (optical properties) and in Chap. 14 on gamma radiation.

11.1

Electromagnetic Waves

Electromagnetic waves consist of an electric field E and a magnetic field B, which propagates simultaneously with a periodically changing field strength. In the propagation of the wave, the distance between two wave peaks is the wavelength λ (Fig. 11.1). The product of wavelength and frequency f is the propagation speed c of the wave. λ
f =c c λ f

ð11:1Þ propagation speed in m
s-1 wavelength in m frequency in Hz

# The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. O. Figura, A. A. Teixeira, Food Physics, https://doi.org/10.1007/978-3-031-27398-8_11

455

456 Table 11.1 Electromagnetic radiation

11

Frequency (Hz) 3∙1024 3∙1023 3∙1022 3∙1021 3∙1020 3∙1019 3∙1018 3∙1017 3∙1016 3∙1015 3∙1014 3∙1013 3∙1012 3∙1011 3∙1010 3∙109 3∙108 3∙107 3∙106 3∙105 3∙104 3∙103 3∙102 3∙101

Electromagnetic Properties

Name

γ-Radiation X-ray

Ultraviolet Light infrared (heat)

Radio waves

AC electricity

Fig. 11.1 Electromagnetic wave with wavelength λ

Electromagnetic waves propagate at the speed of light. The speed of propagation in air is close to the vacuum speed of light (c0 = 3
108 m
s-1). The propagation rate in a material (dielectric) is lower; it depends on the interaction of the material with the electromagnetic wave in question. Figure 11.1 shows that electric and magnetic field strength oscillate at a 90° angle to each other. If the field strength oscillates in a fixed plane as in Fig. 11.1, we have a linear polarized wave. Also in terms of electric material properties, there is the concept of polarization or polarizability. While we have already learned about magnetic polarizability in Chap. 10), we now want to understand the causes of electric polarization.

11.2

11.2

Electric Polarization

457

Electric Polarization

As stated earlier, materials that possess polar molecules will respond to an electric field such as microwave radiation because these polar molecules will become polarized in the presence of such an electrical field. First, we must distinguish between molecules that are permanent electric dipoles like water (H2O) from those that are temporary dipoles like hydrocarbons. When permanent dipoles are brought into the presence of an electric field, they will orient themselves to become aligned with the direction of the electric field. This is called orientation polarization. Temporary dipoles are not normally polar molecules, but they can be deformed under the influence of an electric field. This causes them to acquire dipoles and become temporarily polarized, so they will also orient themselves to become aligned with the direction of the electric field. However, they will immediately lose their polarization as soon as the electric field disappears when switched off. Polarization of both permanent and temporary dipoles depends on the strength of the external electrical field. Both together contribute to the overall polarization P of a volume V of material by the following expression: P=

p V

ð11:2Þ polarization in C
m-2 dipole moment in C
m volume in m3

P p V

The degree of polarization exhibited by a material when in the presence of an electric field can be quantified by comparing it with the degree of polarization exhibited in an empty space (vacuum) in that same electric field. Let us consider an electric displacement field D: It is D = ε
ε0
E

ð11:3Þ

D 0 = 1
ε0
E

ð11:4Þ

D - D 0 = ε
ε 0
E - ε 0
E = ε 0
E ð ε - 1Þ = P

ð11:5Þ

P = ðε - 1Þε0
E

ð11:6Þ

in a vacuum

so, the difference is:

the polarization is

458

11

Electromagnetic Properties

and with χ = ðε - 1Þ

ð11:7Þ

P = χ
ε0
E

ð11:8Þ

it is

D E ε0 ε χ

electric displacement field in C
m-2 electric field in V
m-1 electric field constant: 8.854
10-12 C
V-1
m-1 relative permittivity electric susceptibility

The relative permittivity ε and the susceptibility χ, are measures for the polarization potential of a material. The greater the polarization potential of a material the more energy it can absorb from an electric field. When the direction of the electric field alternates periodically at a given frequency, then the dipoles also alternate their orientation to follow the alternating direction of the electric field at the same frequency. Thus, the direction of polarization in the material alternates with the alternating electric field. This continuously changing orientation is not completely elastic. A significant part of the absorbed energy is lost in the form of heat that is generated when the dipoles try to reverse their direction of orientation at high frequencies. This heat is absorbed by the material and results in an increase in the temperature of the material. This is why foods become heated when placed in a microwave oven. Analogous to mechanics, in the lossless case we speak of an elastic behavior of the dipoles. The heating of a material due to energy dissipation is thus a consequence of in-elastic polarization. When distinguishing the two cases, the elastic case is characterized with χ ′ and ε′ and the inelastic case with χ ″ and ε″, the so-called loss factor. How well a material can be heated by electromagnetic absorption is determined 00 00 by the ratio εε0 and χχ 0 . These material properties play a major role in heating of food by absorption of electromagnetic radiation, the so-called dielectric heating or microwave heating. The values of these material sizes depend on the frequency of the electromagnetic radiation. To understand this let us look at the causes of electromagnetic absorption. The polarization potential of a material αtotal can be calculated as the sum of the polarization potential of the temporary (induced) and permanent dipoles αind and αpermanent, respectively. αtotal = αind þ αpermanent

ð11:9Þ

11.2

Electric Polarization

459

So, we can say the polarization is: P = N
αtotal
E

ð11:10Þ

The orientation of permanent dipoles, such as water (H2O) molecules, when placed into an external electric field, is impeded by the thermal movement of the molecules. The increasing temperature causes the permanent polarization to decrease. This decrease in permanent polarization potential as a function of temperature can be estimated as follows: αpermanent =

μ2 3kT

ð11:11Þ

With this it is: P = N
αind þ

P N αind μ k T E ε0

μ2 3k B T


E

ð11:12Þ

polarization in C
m-2 particle concentration in m-3 induced polarization potential in C2
m2
J-1 molecular dipole moment in C
m Boltzmann’s constant: kB = 1.38064852
10-23 m2
kg
s-2
K-1 temperature in K electric field in V
m-1 electric field constant: 8.854
10-12 C
V-1
m-1

Data for the induced polarization potential αind of different molecules are listed in published literature. Often instead of αind the so-called polarization volume α′ is listed. It is: α0 =

α 4π
ε0

ð11:13Þ

Also, values for the permanent dipole moment μ for different molecules are available from referenced literature. A few examples are given in Table 11.2. Table 11.2 Dipole moments and polarization potential of some simple molecules, examples from [1] H2 N2 CO2 H2O C2H5OH CH3OH

μ (C∙m) 0 0 0 6.17
10-30 5.64
10-30 5.70
10-30

μ (D) 0 0 0 1.85 1.69 1.71

αind (C2
m2
J-1) 9.11
10-41 1.97
10-41 2.93
10-40 1.65
10-40 – 3.59
10-40

α0ind (cm3) 8.19
10-25 1.77
10-24 2.63
10-24 1.48
10-24 – 3.23
10-24

460

11

Electromagnetic Properties

Sometimes the dipole moment μ is expressed in units of Debyes (D), instead of the SI units of C∙m. A Debye is D = 3.3
10-30 C∙m. The Clausius–Mossotti–Debye Eq. (11.14) describes the relationship between relative permittivity ε and polarization potential αind. It can be used to express the polarization potential of a material [1]: μ2 ε-1 N αind þ = 3kB T ε þ 2 3
ε0

ð11:14Þ

This expression has no unit, as we can see by analysis of the right side of the 2 V
m C2
m2 equation:C
m = V
C
C
m 3 J J
m2
C = 1. We can (Eq. 11.14) also write as ε-1 3 1 μ2 αind þ
=N
ε þ 2 4π 4πε0 3k B T

ð11:15Þ

by using Eq. (11.9) we get: αtotal = αind þ αpermanent = αind þ

μ2 3kB T

ð11:16Þ

With Eq. (11.15) we get ε-1 3 1
α
=N
ε þ 2 4π 4πε0 total

ð11:17Þ

with Eq. (11.13) we can abbreviate φN = N
α0total

ð11:18Þ

The quantity φN is dimensionless and represents the polarization potential caused by a particle concentration N. The molar polarization potential can be calculated by: With Eq. (11.14) φN is identical to φN =

N μ2
αind þ 3kB T 4π
ε0

ð11:19Þ

with Eq. (11.16) we get the relation between φN and the relative permittivity φN =

ε-1 3
ε þ 2 4π

ð11:20Þ

The mass-specific polarization potential φm can be calculated by dividing φN by the mass density ρ of the material. It has the SI unit m3
kg-1.

11.2

Electric Polarization

461

φm =

φN ε-1 3 =
ρ ε þ 2 4π
ρ

ð11:21Þ

In physical chemistry, instead of mass-specific polarization potential φm the molar polarization potential φn is often used instead of mass-specific quantities. We get φn by multiplication of Eq. (11.22) with the molar mass M of the material: φn =

φN ε-1 3
M
M=
ρ ε þ 2 4π
ρ

ð11:22Þ

Because of N=

N N
NA n
NA n
NA
M m
NA ρ
NA = = = = = V
NA V V
M M V
M V

ð11:23Þ

we see that multiplication by Mρ is identical to a division by the particle density N and parallel multiplication by Avogadro’s constant: M 1 =
NA ρ N

ð11:24Þ

volume-based polarization potential in m3
m-3 mole-based polarization potential in m3
mol-1 mass-based polarization potential in m3
kg-1 molar mass in kg
mol-1 density in kg
m-3 Avogadro’s constant: 6.022
1023 mol-1 particle density in m-3 number of particles volume in m3 amount of substance in mol

φN φn φm M ρ NA N N V n

11.2.1 Temperature Dependency From Eqs. (11.20) and (11.22) we can derive the following expression which indicates the temperature dependency of the polarization potential and the permittivity of a material: φm =

φN N 1 μ2 αind þ =
ρ ρ 4πε0 3k B T

ð11:25Þ

When we measure the relative permittivity ε of a material at different temperatures, we can calculate the mass-based polarization potential φm according to Eq. (11.22) and plot it versus 1/T. Then we are able to obtain the permanent dipole moment μ and the induced polarization potential αind of the material. As shown in

462

11

Electromagnetic Properties

Fig. 11.2 Temperature dependency of mass-based polarization potential by graphical evaluation

b

Mm a

1 T Fig. 11.2 the intercept, a, and the slope, b, provide us with the following expressions from Eq. (11.25): αind 4π
ε0
ρ

ð11:26Þ

μ2 12π
ε0
ρ
k B

ð11:27Þ

a= and b=

For this type of characterization of the electrical polarizability of biological materials such as food, only knowledge of the density is necessary. For questions in which the molecular weight or at least an average molecular weight M of the material in question is known, the graphical application of the molar polarizability volume φn can also be used for evaluation. Then we have φn =

M 3 ε - 1 NA 3 ε - 1

=

ρ 4π ε þ 2 N 4π ε þ 2

ð11:28Þ

it is φn =

NA μ2 N αind þ
N 4π
ε0 3k B T

ð11:29Þ

NA μ2 αind þ 4π
ε0 3kB T

ð11:30Þ

or φn =

11.2

Electric Polarization

463

then the intercept, a, and slope, b, in the φn versus T1 plot are a=

N A
μ2 N A
αind ; b= 4π
ε0 12π
ε0
kB

We can see that the polarization potential and the permittivity of a material decrease with rising temperature. The reason for this is that with increasing temperature, the random thermal movement of the molecules increases and counteracts the orientation of the permanent dipoles by the external electric field. The induced polarization potential, however, is not affected by the movement of the molecules. As a result, materials consisting of polar molecules show less polarizability at high temperatures than at low temperatures. The relative permittivity number of liquid water in a static, electric field (frequency = 0) at 100 °C is ε = 55.7 and at -35 °C it is ε = 107.7 [2]. In dielectric heating of food [3, 4] knowledge of polarizability and its temperature dependence is necessary. Remark Water that is still liquid at temperatures below 0 °C is called supercooled water. Very strong supercooling can only be achieved under laboratory conditions.

11.2.2 Frequency Dependency The energy absorbed by a molecule leads to translation, oscillation, or rotation of the molecule. Let us assume the rotation of a small molecule takes about 10-12 s. If we now increase the frequency of an electric field to over 1012 Hz, only less than 10-12 s are available per change of direction. As a result, permanent dipoles such as water molecules can no longer follow an alternating electric field from about this frequency onward. The contribution of orientation polarization to the overall polarization therefore decreases significantly when this frequency is reached. The contribution of displacement polarization, on the other hand, does not change. Figure 11.3 illustrates the course of polarizability with increasing frequency. Orientation polarization therefore plays practically no role at optical frequencies. If the frequency of the irradiated alternating field is increased even further up to the UV range, the displacement polarization is also finally eliminated (Fig. 11.3). Alternating electromagnetic fields with frequencies in the UV or X-ray range only interact with the electrons of a molecule and serve as the electronic contribution to polarization in Fig. 11.3. This means that at these and higher frequencies, the model of aligned Debye dipoles is no longer applicable. When calculating polarizability, the contribution of the permanent dipoles can be neglected in this frequency range, i.e.,

464

11

Electromagnetic Properties

Fig. 11.3 Frequency dependency of electric polarization potential [1] permanent dipoles

Dtotal temporary dipoles

electronic

radio microwave IR

f

φn =

NA μ2 αind þ 4π
ε0 3kB T

ffi

NA
α 4π
ε0 ind

vis

UV

ð11:31Þ

The measurement of polarizability with frequencies in the IR range or with frequencies of visible light provides only the proportion of polarizability based on displacement polarization. Maxwell’s equation ε = n2

ð11:32Þ

expresses the electrical polarization at frequencies in the optical range, which is determined by the refractive index n (s. Chap. 12, optical properties): φn =

NA M 3 ε-1 M 3 n2 - 1 αind =

==

2 4πε0 ρ 4π ε þ 2 ρ 4π n þ 2

ð11:33Þ

What we find here is a strong frequency dependence of the polarizability of the permittivity and refractive index of the material in question. The dependence of a physical quantity on the frequency of a wave is called dispersion. → Attention The Latin word dispergere is used with different meanings. In materials science, dispersion is a multiphase system (cf. Chap. 3, Disperse systems). Light is scattered by multiphase (“disperse”) systems. In physics, however, the term is for the frequency dependence of quantities.

11.2

Electric Polarization

465

11.2.3 Complex Permittivity Materials can absorb electromagnetic radiation elastically or inelastically. Elastic absorption is an energy storage as it occurs in elastic elements in mechanics. Inelastic absorption, on the other hand, leads to energy dissipation, i.e., to an increase in entropy, which is usually associated with heating of the material. Ideally elastically absorbed radiation can be emitted without loss like the elastic energy of an ideally mechanical spring. Inelastically absorbed radiation, on the other hand, is not emitted again, but is converted into heat that remains in the material. This inelastic part is also called the loss part. Complex permittivity ε is used to mathematically express the material behavior. It consists of a real part ε′ for the elastic behavior and an imaginary part ε″ for the inelastic behavior. It is ε = ε0 þ i
ε00 ε ε′ ε″ i

ð11:34Þ complex permittivity real permittivity imaginary permittivity p imaginary unit -1

Complex quantities have the advantage that two properties can be expressed with one size. In polar coordinates, the real part and the imaginary part are orthogonal vectors whose Pythagorean sum indicates the amount of the complex quantity: ε =

ε0 2 þ ε00 2

ð11:35Þ

We have already worked with complex quantities to describe elastic and inelastic parts of a physical property in previous chapters (Chap. 4 “Rheological properties,” Chap. 8 “Thermal properties,” and Chap. 9 “Electrical properties”). An ideal oscillator, i.e., a completely elastic and frictionless oscillator, would oscillate indefinitely long not losing energy to its environment. Real oscillators, on the other hand, lose energy to the environment, so the amplitude of the oscillation decreases over time. This is called the damping of the oscillation. The degree of damping is determined by the inelastic part ε″ (so-called loss factor). The ratio of the real part and the imaginary part is the tangent of the angle δ between both vectors. In this context, the angle δ is called dielectric loss angle and the tangent is called loss tangent. ε00 = tan δ ε0

ð11:36Þ

For a material that behaves ideally elastic, i.e., lossless, the angle is δ = 0°. In the case of a material that behaves completely inelastic, the angle is δ = 90°. For this

466

11

Electromagnetic Properties

Fig. 11.4 Frequency dependence of the permittivity of water. Real part ε′ (top) and imaginary part ε″ (bottom), according to [5]

reason, the real part is also called “in-phase component” and the imaginary part “outof-phase component.” Let us consider liquid water as our model material: Water can absorb electromagnetic radiation. Due to the dispersion of permittivity, the absorption is strongly dependent on the frequency of electromagnetic radiation. At a frequency of 100 MHz, elastic absorption predominates. At a frequency of a few GHz, inelastic absorption predominates. This leads to heating of the water and is the basis of microwave heating [5, 6]. In Fig. 11.4, the frequency dependence of the permittivity is shown. The substance data of water shown in Fig. 11.4 were determined for high-purity water with a very low electrical conductivity. If an aqueous solution has a significant electrical conductivity, the value of the imaginary permittivity increases by the so-called conductivity term σ, which in turn is frequency-dependent [2]: ε = ε0 - i
ε00 þ

σ ε0
ω

ð11:37Þ

with ω = 2π
f

ð11:38Þ

11.3

11.3

Microwaves

467

Microwaves

Electromagnetic waves in the frequency range between 300 MHz and 300 GHz are called microwaves. Since microwaves are used for navigation and telecommunications purposes (air traffic, satellites, mobile phones, radar, etc.), their use is regulated by law. In most industrialized nations, the frequencies 915 ± 13 MHz and 2450 ± 50 MHz are reserved for industrial, scientific, and medical use (regulation by ITU, International Telecommunication Union). With the relationship for the propagation speed of electromagnetic waves c=λ
f

ð11:1Þ

frequencies and wavelengths can be calculated. propagation speed in m
s-1 wavelength in m frequency in Hz

c λ f

Example What is the wavelength of microwave radiation of the frequency 915 MHz or the frequency 2.45 GHz? Solution:with c = λ
f it is λ=

c f

assuming the speed of light in a vacuum as propagation speed, it is λ=

2:99
108 m
s - 1 = 0:328 m 915
106 s - 1

λ=

2:99
108 m
s - 1 = 0:122 m 2:45
109 s - 1

or

Microwaves are basically subject to the same laws as electromagnetic waves of other frequencies, such as IR or light. Microwaves can be reflected, absorbed, and transmitted. With the help of metal reflectors, microwaves can be redirected, focused, or scattered. Materials that strongly absorb microwaves have a high degree of absorption and thus a low degree of transmission. Materials such as glass, ceramics, and plastics show little or no microwave absorption. Due to their specific electrical polarizability (low dissipation factor ε″), they have a low degree of

468

11

Electromagnetic Properties

absorption in this frequency range, but this is dependent on temperature and frequency. Figure 11.4 shows the high capability of liquid water for absorption of electromagnetic waves in the GHz range. A high value for the loss factor ε″ means a high absorption coefficient. Therefore, it is not surprising that telecommunication transmissions in this frequency range are influenced by water in the atmosphere. For example, heavy rain leads to the weakening of mobile communications (1–4 GHz), Wi-Fi (2–6 GHz), satellite signals (10–12 GHz), and other similar services based on microwaves. The attenuation of the telecommunications fields by atmospheric water is an undesired effect. However, due to the dispersion of the absorption coefficient of water, there are frequency ranges that are less attenuated by liquid water, such as radio waves or visible light.

11.3.1 Conversion of Microwaves into Heat The inelastic absorption of microwaves leads to the heating of a material. While the conversion of field energy into heat in electrical engineering is considered an “energy loss” (loss factor ε″), this effect is the desired effect in microwave heating. The electrically polarizable molecules of a material absorb energy from the irradiated field if this energy “fits.” In the case of water and similarly constructed molecules, the frequencies of molecular rotation are in the range of microwave radiation. Because the absorbent molecules release energy to neighboring molecules, it does not come to (elastic) reflection of the microwaves. Instead, there is (inelastic) dissipation of the radiation energy, which leads to a temperature increase in the material. This does not work with the frequency of radio waves, because at frequencies of a few MHz, water has a very low absorption coefficient. The proportion of absorbed radiant energy that is converted into heat depends linearly on the imaginary part of the electrical permittivity ε″ (loss factor). The power PD dissipated per unit volume is proportional to the frequency and depends on the square of the electric field strength E. It is [7] PD = 2π
ε0
f
ε00
jEj2

ð11:39Þ

or as a numerical value equation PD PD E f ε″

W
m - 3 = 55:61
10 - 12 jE j V
m - 1

2


f ðHzÞ
ε00

ð11:40Þ

dissipated power in W electric field strength in V
m-1 frequency in Hz loss factor

11.3

Microwaves

469

Example What is the volume-based heating power in a microwave oven at 2 GHz, when water is heated within it and the field strength is 100 V/m? Solution: Asssuming ε″ = 10 with PD = 2π
ε0
f
ε00
jEj2 we get PD = 2π
8:854
10 - 12 C
V - 1
m - 1
2
109 Hz
10
ð100 V
m - 1 Þ

2

PD = 55:6
10 - 12 C
V - 1
m - 1
2
1014 s - 1
V2
m - 2 PD = 11, 126:3 V
A
s
s - 1
m - 3 = 11, 126:3 W
m - 3 PD = 11:1 kW
m - 3 or in short: PD

PD

ðW
m - 3 Þ = 55:61
10 - 12 jEj ðV
m - 1 Þ

2


f ðHzÞ
ε00

ðW
m - 3 Þ = 55:61
10 - 12 ð100Þ2
2
109
10 = 55:61
10 - 12


2
1014 = 11, 122 The field strength |E| used in this example is the field strength within the absorbent material. Since this is hardly known in most cases, the performance of a microwave oven is most often determined calorimetrically. For this purpose, the temperature change ΔT of a known material is measured, which occurs during a microwave treatment with a defined duration Δt. Then it is PD = PD cp ρ ΔT Δt

ρ
cp
ΔT Δt

ð11:41Þ dissipated power in W specific heat capacity in J
kg-1
K-1 density in kg
m-3 temperature increase in K duration in s

470

11

Electromagnetic Properties

Example Water with a density of 1000 kg
m-3 and a specific heat capacity of 4.2 kJ
kg-1
K-1 shows a change in temperature from 10 °C to 40 °C within 1 min in a microwave oven. The dissipated power is PD =

ρ
cp
ΔT Δt

1000 kg
m - 3
4:2 kJ
kg - 1
K - 1
30 K = 2100 kJ
s - 1
m - 3 60 s = 2100 kW
m - 3

PD =

i.e. PD = 2:1 kW per liter In conventional heating processes, heat first must be transported to the outside of a food. The transport takes place by means of heat transfer through a wall material (e.g., cooking pot) or convective heat transfer from a heat transfer medium (e.g., hot air). From the outside of the food, the heat enters the interior of the food through heat conduction or convection (see Chap. 8, Thermal Properties). With microwave heating, the heat transport to the outside of the food is omitted. In addition, the heat is generated within the internal mass of the food and does not enter it from the outer surface only. However, it is a mistake to believe that microwave heating begins at the core of the food. The microwaves also penetrate the food from the outside and have defined penetration depths depending on the material.

11.3.2 Penetration Depth of Microwaves When microwaves penetrate a material and its polarizable dipoles absorb energy, the energy of the microwave increasingly decreases as it propagates into that material. The higher the absorption coefficient, the stronger the attenuation of the penetrating microwave. In highly absorbent materials such as liquid water, microwaves lose a large part of their energy after just a few centimeters. The distance d, after which the microwave power has decreased by absorption to 1e of its original power, is called the penetration depth z of the microwave [8]. The microwave power decreases exponentially [9]. P = P0
e - d z

ð11:42Þ

11.3

Microwaves

471

The penetration depth is 1 2

z=

z P E f ε″ ε′ c

c
2
π
f

2 ε0

ε00 2 ε0

1þ

ð11:43Þ -1

penetration depth in m power in W electric field strength in V
m-1 frequency in Hz imaginary part of permittivity (loss factor) real part of permittivity propagation speed in m
s-1

Example Consider a location inside the food at a distance d from the surface. In this example, the distance d should have exactly the calculated value of z. Then the microwave power at place d is P = P0
e - d z

P = P0
e - 1 or P 1 = = 0:368 e P0 The microwave power at location d is only 36.8% of its original power. The other 63.2% have already been absorbed by the previously irradiated material. An exponential decrease in microwave intensity along the absorption pathway means that the inner areas of a food receive a lower intensity than exterior areas. During microwave heating, the rate of temperatures increase depends on the location. An inhomogeneous temperature distribution leads to spatially inhomogeneous physical properties (e.g., permittivity), which makes it difficult to model microwave heating [10].

472

11

Electromagnetic Properties

→ Attention High microwave absorption results in a low penetration depth, while lower absorption results in a high penetration depth. This seemingly paradoxical connection is clarified when we recall the definition of the depth of penetration: It is the depth where the power has fallen to 36.8%.

Example What is the penetration depth of 2450 MHz microwaves in a raw potato? The permittivity data of the potato are ε′ = 64 and ε″ = 15. Solution: 1 2

z=

c
2
π
f

2 ε0

1þ

ε00 2 ε0

-1 1 2

z=

3
108 m
s‐1
2
π
2450
106 s‐1

2 64

1þ

15 2 64

= 0:019 m
1:0739 -1

= 0:021 m z = 2:1 cm While the penetration depth in a raw potato is 2 cm (see example), it is several meters for materials with a low microwave absorption such as glass or porcelain. This means that the microwaves show no noticeable loss of performance during the passage of a plate made of glass or porcelain. Also, solid water (ice) has low microwave absorption, i.e., a large penetration depth. → Attention Sometimes, instead of the physical penetration depth where the power has fallen to 36.8%, the penetration depth is given as the distance where the power has fallen to 50%, the so-called half-power depth. The conversion is z50% = 0.347
z36.8%

11.3

Microwaves

473

11.3.3 Microwave Heating The main differences between conventional food heating and microwave heating of food [10, 11] have already been listed in Sect. 11.3.1. The advantage of heating with microwaves is the high heating speed. Because there is no external heat transfer resistance, microwaves can be used for very fast heating processes. The microwave power can be controlled without delay. Foods with high moisture content have a high loss factor and, as a rule, are well suited for microwave heating. While liquid water shows high values for loss factor ε″ (Fig. 11.4), solid water (ice) shows very low values, and thus, low microwave absorption (Table 11.3). The composition of food is often inhomogeneous; think of pizza, quiche, chunky stews, and ready meals from several foods. This means that there is a complicated spatial distribution of electrical permittivity, which leads to an inhomogeneous temperature distribution throughout the food during microwave heating. The inhomogeneity of the material leads to a spatially different heat gain. The corresponding inhomogeneous temperature distribution in the material can further be changed over time by different thermal conductivities and heat capacities of the food components. Because of the limited penetration depth of microwaves, foods with large dimensions have a lower core temperature than small particles of the same material [12]. It is often necessary to downregulate the microwave power or to work intermittently (“on/off”) to avoid selective overheating of the food. Then a temperature equalization can be reached within the food by heat conduction. The electrolyte content of an aqueous solution affects the value of the loss factor ε″. Therefore, the electrolyte concentration is a factor that, in addition to the water content, affects microwave absorption. Materials that have a high porosity, on the other hand, can act like insulators in thermal processes. Gas-filled pores have a very low thermal conductivity (see Chap. 8, Thermal Properties) and can therefore also impede heat exchange between hot areas and less hot areas in the microwaved food. Glass, ceramics, and porcelain do not show microwave absorption due to their low loss factors ε″. The same applies to microwave-grade packaging. However, it is possible to equip packaging with materials that show high absorption in the microwave field to reach higher temperatures. These materials are called susceptors and have the purpose of bringing the food into contact with a hot surface during microwave treatment to support a thawing process or to cause browning reactions. Instead of metallic susceptors, there are also ways to make susceptor materials from biological materials [12, 14–16].

Table 11.3 Dielectric properties of solid water [13] Ice

ε′ 3.2

ε″ 0.0029

tan δ 0.0009

474

11.4

11

Electromagnetic Properties

Terahertz Waves

Electromagnetic waves with frequencies in the range between 100 GHz and 10 THz are referred to as terahertz waves. The wavelength range is between 3 mm and 10 μm. It thus covers the area of the far infrared (FIR) and extends into the area in the mid-infrared (MIR), cf. Table 11.4. For this reason, terahertz radiation is sometimes referred to as far-infrared radiation (FIR). Wavelengths in the range of λ = 0.78...1000 μm are called infrared radiation (IR). In IR spectroscopy, in addition to the wavelength, the reciprocal size or so-called wavenumber is often used. Although the SI unit of wave number is m-1, the unit cm-1 is often used. Table 11.4 illustrates the subdivision of the IR range based on wavelength, wave number, and frequency. It is ν=

1 λ

ð11:44Þ

λ ν

wavelength in m wave number in m-1

Definition Terahertz waves are electromagnetic waves in the frequency range between microwave and infrared radiation. Since terahertz radiation can excite molecular rotations, terahertz radiation can be used for molecular spectroscopy. The absorption of terahertz radiation is strongly substance specific. The absorption coefficient of polar substances such as water is high; less polar materials such as paper, cardboard, wood, plastics, and textiles show lower absorption but can be distinguished. Metals reflect terahertz radiation. For this reason, terahertz radiation is suitable for the inspection of packaged goods. The energy of terahertz radiation is much lower than, for example, that of X-rays (see Chap. 13). So, terahertz inspectors can also be used for personal control at airports as an example. The penetration depth of terahertz radiation in textiles and plastics is very high, while they hardly penetrate metals and water. Since terahertz absorption depends on the chemical bonds of the inspected materials, high-contrast images of packaged or hidden materials can be generated and it is possible to chemically Table 11.4 Subdivision of IR wavelength ranges according to ISO 20473 [17] Designation Near infrared Mid-infrared Far infrared

NIR MIR FIR

Wavelength λ (μm) 0.78–3 3–50 50–1000

Wave number ν (cm-1) 3333–12,820 200–3333 10–200

Frequency f (Hz) 1∙1014–3.8∙1014 6∙1012–1∙1014 3∙1011–6∙1012

11.5

NIR

475

identify organic solids in parallel. [18]. Terahertz imaging offers numerous possibilities for non-invasive inspection with simultaneous material identification. Compared to X-ray inspection, the distinguishability of different organic materials in terahertz inspection is high. Since terahertz waves are reflected by metals, the applications only apply to cases where there is no metallic shielding. Due to the high absorptiveness of liquid water, terahertz waves are absorbed by aqueous biological materials practically at the surface [19, 20]. In this way, the water distribution of biological surfaces can be imaged or water in thin samples of dried materials can be determined. Pulsed terahertz waves can be used to determine the properties of thin dielectric films in a very short time. As an example, it is possible to build sensors with semiconductor chips that are protein-sensitive or gene-sensitive, whose signal is derived from the terahertz absorptiveness of the chip [21].

11.5

NIR

Electromagnetic waves with frequencies in the NIR range (Table 11.4) can be absorbed by molecular bonds and are therefore suitable for spectroscopy of materials. For this purpose, the absorption spectrum is measured, which is the graphical application of the absorption over the frequency or the wavelength or the wave number. The absorption spectrum provides information on which molecular bonds occur frequently and less frequently in the material and thus allows conclusions to be drawn about the composition of the material. In addition to absorption spectra, transmission spectra are also common. Let us consider the water molecule as an example of an infrared-active molecule (Table 11.5). The stretch oscillations and the scissoring oscillation of the O–H bonds in the H2O-molecule can be excited by specific IR frequencies. If these frequencies or wave numbers occur in the absorption spectrum of a mixture of substances, this is an indication of the presence of water in the sample. Table 11.5 Examples of fundamental oscillations of the H2O molecule Designation Symmetric stretching

ν (cm-1) (absorption) 3652

Asymmetric stretching

3756

Scissoring

1596

476

11

Table 11.6 Fundamental oscillations and harmonics

Electromagnetic Properties

Designation Fundamental oscillation 1. Harmonic

Frequency f 2
f

2. Harmonic

3
f

Wavelength λ λ 2 λ 3

etc.

The wave numbers given in Table 11.5 belong to the so-called fundamental oscillations of the H2O molecule. In addition to these fundamental oscillations, molecules can also perform oscillations with the double, triple, etc. frequency, so-called harmonics. Table 11.6 shows the systematics. Definition Harmonics have an integer multiple frequency of a fundamental oscillation. In analogy to acoustics, harmonics are sometimes referred to as overtones. We have looked at H2O molecules up to this point and see that despite the simple molecular structure, several absorption frequencies exist. Larger molecules have much more absorption frequencies. The coupling of adjacent oscillations leads to additional so-called combination vibrations. Although NIR spectra have many peaks that are hardly distinguishable peaks, there are absorption frequencies that are characteristic for groups of substances; see Table 11.7. Figure 11.5 illustrates the formation of typical bands in a spectrum. Typical absorption ranges of food ingredients are listed in Table 11.8. Table 11.7 Absorption bands (middle MIR) of molecular groups [1] Alkanes

Alkenes Alkynes Aromatic compounds

Ethers Amines Aldehydes, ketones Carboxylic acids Amides

Oscillation – CH stretching and scissoring – CH2 stretching and scissoring – CH3 stretching and scissoring – CH olefin stretching – CH acetylene stretching – CH aromatic stretching – C=C – stretching – OH stretching – OH scissoring C–O stretching C–O asymmetric stretching – NH primary and secondary amine stretching – C=O stretching – OH (doublet) – C=O stretching – C=O stretching – NH stretching – NH scissoring

ν (cm-1) (absorption) 2800–3000 1420–1470 1340–1380 3000–3100 3300 3000–3100 1600 3200–3600 1300–1500 1000–1200 1000–1220 3300–3500 1700–1735 2700–2850 1720–1740 1640–1670 3100–3500 1550–1640

11.5

NIR

477

Fig. 11.5 IR spectrum of cellulose (bottom) caused the absorption of chemical bonds (top), C: fundamental oscillation, 1.O, 2.O: first and second harmonics, from [22]

1.O 2.O } C } } H2O

H2O

H2O

ROH

RCOOH

ROH RNH2

RNOH

CH

CH

CH

CH2

CH2

CH2

CH3

CH3

1100

CH3

1500

1900 O / nm

2300

1500

1900

2300

0.6 0.5 0.4 0.3

1 log 0.2 R

0.1 0 -0.1 1100

O / nm Table 11.8 Absorption ranges (NIR) of food ingredients [1] Component Water Proteins, peptides Fats Carbohydrates

Oscillation – OH stretching/combination – OH stretching – NH deformation – CH stretching – CH2 and – CH3 stretching C-O and O-H stretching/combination

λ (nm) 1920–1950 1400–1450 1560–1670 2080–2220 2300–2350 1680–1760 2060–2150

Definition NIRS stands for NIR spectroscopy, i.e., near-infrared spectroscopy. The great advantage of NIR spectroscopy is that materials can be examined without time-consuming sample preparation; even measurements in the running

478

11

Electromagnetic Properties

process are possible. For powders, however, physical parameters such as particle size, crystallinity, and bulk density slightly influence the spectrum. This must be considered during calibration. Conversely, this offers the possibility of including these physical quantities in NIR quality control [23]. Definition Hyperspectral imaging (HSI) is the integration of spectroscopy and imaging techniques. While in a conventional image a single property, such as color, is assigned to each pixel, in HSI there is a spectrum for each pixel [24]. This can be an NIR spectrum for each image pixel.

Further Reading Dielectric polarization Moisture determination in agricultural materials using dielectric properties Meat: rapid moisture determination using dielectric measurement Dielectric properties of pharmaceutical powders Honey: water content measurement using dielectric properties

[2, 5, 25] [26] [27] [28]

Microwaves Potato chips produced by microwave multiflash drying Edible microwave susceptor made by alginate and salt Microwave sterilization: comparison to conventional sterilization Monitoring of food drying and freezing microwaves More homogeneous temperature distribution using microwave guides Computational Modeling of heat transport during microwave treatment Microwave-assisted freeze drying Relationship between dielectric response and water activity Radiofrequency thawing of frozen minced fish Rice model food system: dielectric properties relevant to microwave sterilization

[29] [14] [3] [30] [31] [32] [33] [34] [35] [36]

Terahertz waves Tomatoes: pesticide residue by THz-spectroscopy Antibiotic residues by THz-spectroscopy Chocolate: foreign body detection pulsed THz spectroscopy Sausage: detection of foreign materials by THz spectroscopic imaging Cocaine detection by THz-spectroscopy Terahertz spectroscopy: agricultural applications

[37] [38] [39] [40] [18] [41, 42]

(continued)

References

Hyperspectral imaging on meat Terahertz detection of transgenic food Terahertz sensing for food and water security Identification of food by measurement of complex permittivity Terahertz phase imaging and biomedical applications

479

[43] [44] [45] [46] [47]

NIR Quality of fruit and vegetable by NIR spectroscopy Linking flavour sensory and NIRS Quality of oilseeds and edible oils by NIRS Quality of cocoa bean and cocoa bean products by NIRS Beer: vis/NIR spectroscopy during fermentation NIR-fluid bed process monitoring Pharmaceutical technology support by NIR Dairy: NIR process control Nuts: characterization by IR-spectroscopy Mushrooms: authentication and quality analysis by NIRS Maize flour: secondary biochemical components by NIR Hyperspectral imaging of oil, milk, yogurt

[48, 49] [50] [51] [52] [53] [54] [23, 55] [56] [57] [58] [59] [60]

Summary The electrical polarizability of molecules is the cause of the ability to absorb electromagnetic radiation of certain wavelengths. Polarizability is explained based on the model of the Debye dipole in simple words and described with the complex permittivity. Applications such as heating with the help of microwaves and material characterization with terahertz radiation and NIR spectroscopy illustrate the interaction between electromagnetic radiation and biological materials such as food. At the end of the chapter, examples from the literature are listed, which can be used for further studies and as suggestions for one’s own scientific work.

References 1. Atkins PW, De Paula J (2011) Physical chemistry for the life sciences, 2nd edn. W.H. Freeman, Oxford 2. Udo K, Christof H (2010) Electromagnetic techniques for moisture content determination of materials. Meas Sci Technol 21(8):82001. https://doi.org/10.1088/0957-0233/21/8/082001

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3. Soni A, Smith J, Thompson A, Brightwell G (2020) Microwave-induced thermal sterilization – a review on history, technical progress, advantages and challenges as compared to the conventional methods. Trends Food Sci Technol 97:433. https://doi.org/10.1016/j.tifs.2020.01.030 4. Meda V, Orsat V, Raghavan V (2017) Microwave heating and the dielectric properties of foods. Woodhead Publishing, pp 23–43. https://doi.org/10.1016/b978-0-08-100528-6.00002-4 5. Andryieuski A, Kuznetsova SM, Zhukovsky SV, Kivshar YS, Lavrinenko AV (2015) Water: promising opportunities for tunable all-dielectric electromagnetic metamaterials. Sci Rep 5: 13535. https://doi.org/10.1038/srep13535 6. Kaatze U (1989) Complex permittivity of water as a function of frequency and temperature. J Chem Eng Data 34(4):371. https://doi.org/10.1021/je00058a001 7. Tiwari G, Wang S, Tang J, Birla SL (2011) Analysis of radio frequency (RF) power distribution in dry food materials. J Food Eng 104(4):548. https://doi.org/10.1016/j.jfoodeng.2011.01.015 8. Schubert H, Regie M (2006) Novel and traditional microwave applications in the food industry. In: Willert-Porada M (ed) Advances in microwave and radio frequency processing. Springer, Berlin, pp 259–270 9. Hippel vAR (1995) Dielectric and waves. Artech House, Boston 10. Chandrasekaran S, Ramanathan S, Basak T (2013) Microwave food processing – a review. Food Res Int 52(1):243. https://doi.org/10.1016/j.foodres.2013.02.033 11. Guo QS, Sun DW, Cheng JH, Han Z (2017) Microwave processing techniques and their recent applications in the food industry. Trends Food Sci Technol 67:236. https://doi.org/10.1016/j. tifs.2017.07.007 12. Raaholt BW (2020) Influence of food geometry and dielectric properties on heating performance. In: Erle U, Pesheck P, Lorence M (eds) Development of packaging and products for use in microwave ovens. Woodhead Publishing, pp 73–93. https://doi.org/10.1016/B978-008-102713-4.00002-5 13. Matsuoka T, Fujita S, Mae S (1996) Effect of temperature on dielectric properties of ice in the range 5–39 GHz. J Appl Phys 80(10):5884. https://doi.org/10.1063/1.363582 14. Albert A, Salvador A, Fiszman SM (2012) A film of alginate plus salt as an edible susceptor in microwaveable food. Food Hydrocoll 27(2):421. https://doi.org/10.1016/j.foodhyd.2011. 11.005 15. Risman PO (2020) Measurements of dielectric properties of foods and associated materials. In: Erle U, Pesheck P, Lorence M (eds) Development of packaging and products for use in microwave ovens. Woodhead Publishing, pp 201–223. https://doi.org/10.1016/B978-0-08102713-4.00005-0 16. Chen FY, Warning AD, Datta AK, Chen X (2017) Susceptors in microwave cavity heating: modeling and experimentation with a frozen pie. J Food Eng 195:191. https://doi.org/10.1016/j. jfoodeng.2016.09.018 17. ISO 20473 (2007) Optik und Photonik - Wellenlängenbereiche. Beuth, Berlin. https://doi.org/ 10.31030/3123472 18. Dobroiu A, Otani C, Kawase K (2006) Terahertz-wave sources and imaging applications. Meas Sci Technol 17(11):R161. https://doi.org/10.1088/0957-0233/17/11/r01 19. Wang C, Qin J, Xu W, Chen M, Xie L, Ying Y (2018) Terahertz imaging applications in agriculture and food engineering: a review. Trans ASABE 61(2):411. https://doi.org/10.13031/ trans.12201 20. Ok G, Shin HJ, Lim M-C, Choi S-W (2019) Large-scan-area sub-terahertz imaging system for nondestructive food quality inspection. Food Control 96:383. https://doi.org/10.1016/j. foodcont.2018.09.035 21. Mickan S, Lee K-S, Lu T-M, Munch J, Abbott D, Zhang XC (2013) Double modulated differential THz-TDS for thin film dielectric characterization. Microelectron J 33:1033. https://doi.org/10.1016/S0026-2692(02)00108-8 22. Berntsson O (2001) Characterization and application of near infrared reflection spectroscopy for quantitative process analysis of powder mixtures. Doctoral thesis, KTH, Stockholm

References

481

23. Jamrógiewicz M (2012) Application of the near-infrared spectroscopy in the pharmaceutical technology. J Pharm Biomed Anal 66:1. https://doi.org/10.1016/j.jpba.2012.03.009 24. Park B, Lu R (2015) Hyperspectral imaging technology in food and agriculture. https://doi.org/ 10.1007/978-1-4939-2836-1 25. Kozhevnikov A (2010) Wideband radio-frequency device for measurements of dielectric properties of small volumes of liquids. Meas Sci Technol 21(4):043001. https://doi.org/10. 1088/0957-0233/21/4/043001 26. Wang J, Fan L, Zhou Q, Li J, Zhao P, Wang Z, Zhang H, Yan S, Huang L (2018) Rapid determination of meat moisture content using radio-frequency dielectric measurement. IEEE Access 6:51384. https://doi.org/10.1109/access.2018.2864308 27. Magee TRA, McMinn WAM, Farrell G, Topley L, Al-Degs YS, Walker GM, Khraisheh M (2013) Moisture and temperature dependence of the dielectric properties of pharmaceutical powders. J Therm Anal Calorim 111(3):2157. https://doi.org/10.1007/s10973-012-2739-y 28. Li Z, Haigh A, Soutis C, Gibson A, Sloan R (2017) Evaluation of water content in honey using microwave transmission line technique. J Food Eng 215(Suppl C):113. https://doi.org/10.1016/ j.jfoodeng.2017.07.009 29. Barreto IMA, Tribuzi G, Marsaioli Junior A, Carciofi BAM, Laurindo JB (2019) Oil–free potato chips produced by microwave multiflash drying. J Food Eng 261:133. https://doi.org/10. 1016/j.jfoodeng.2019.05.033 30. Pandey G, Vandermeiren W, Dimiccoli L, Stiens J (2018) Contactless monitoring of food drying and freezing processes with millimeter waves. J Food Eng 226:1. https://doi.org/10. 1016/j.jfoodeng.2018.01.003 31. Zhu HK, Li D, Li SJ, Wang SJ (2017) A novel method to improve heating uniformity in mid-high moisture potato starch with radio frequency assisted treatment. J Food Eng 206:23. https://doi.org/10.1016/j.jfoodeng.2017.03.001 32. Rakesh V, Seo Y, Datta AK, McCarthy KL, McCarthy MJ (2010) Heat transfer during microwave combination heating: computational modeling and MRI experiments. AICHE J 56(9):2468. https://doi.org/10.1002/aic.12162 33. Duan X, Zhang M, Mujumdar AS, Wang R (2010) Trends in microwave-assisted freeze drying of foods. Dry Technol 28(4):444. https://doi.org/10.1080/07373931003609666 34. Renshaw RC, Dimitrakis GA, Robinson JP, Kingman SW (2019) The relationship of dielectric response and water activity in food. J Food Eng 244:80. https://doi.org/10.1016/j.jfoodeng. 2018.08.037 35. Yang H, Chen Q, Cao H, Fan D, Huang J, Zhao J, Yan B, Zhou W, Zhang W, Zhang H (2019) Radiofrequency thawing of frozen minced fish based on the dielectric response mechanism. Innovative Food Sci Emerg Technol 52:80. https://doi.org/10.1016/j.ifset.2018.10.013 36. Auksornsri T, Tang J, Tang Z, Lin H, Songsermpong S (2018) Dielectric properties of rice model food systems relevant to microwave sterilization process. Innovative Food Sci Emerg Technol 45:98. https://doi.org/10.1016/j.ifset.2017.09.002 37. Suzuki T, Ogawa Y, Kondo N (2011) Characterization of pesticide residue, cis-permethrin by terahertz spectroscopy. Eng Agric Environ Food 4(4):90. https://doi.org/10.1016/S1881-8366 (11)80007-8 38. Redo-Sanchez A, Salvatella G, Galceran R, Roldós E, García-Reguero J-A, Castellari M, Tejada J (2011) Assessment of terahertz spectroscopy to detect antibiotic residues in food and feed matrices. Analyst 136(8):1733. https://doi.org/10.1039/C0AN01016B 39. Jördens C, Koch M (2008) Detection of foreign bodies in chocolate with pulsed terahertz spectroscopy. Opt Eng 47. https://doi.org/10.1117/1.2896597 40. Wang C, Zhou R, Huang Y, Xie L, Ying Y (2019) Terahertz spectroscopic imaging with discriminant analysis for detecting foreign materials among sausages. Food Control 97:100. https://doi.org/10.1016/j.foodcont.2018.10.024 41. Afsah-Hejri L, Akbari E, Toudeshki A, Homayouni T, Alizadeh A, Ehsani R (2020) Terahertz spectroscopy and imaging: a review on agricultural applications. Comput Electron Agric 177: 105628. https://doi.org/10.1016/j.compag.2020.105628

482

11

Electromagnetic Properties

42. Li B, Zhang D, Shen Y (2020) Study on terahertz spectrum analysis and recognition modeling of common agricultural diseases. Spectrochim Acta A Mol Biomol Spectrosc 243:118820. https://doi.org/10.1016/j.saa.2020.118820 43. Feng CH, Makino Y, Oshita S, Martin JFG (2018) Hyperspectral imaging and multispectral imaging as the novel techniques for detecting defects in raw and processed meat products: current state-of-the-art research advances. Food Control 84:165. https://doi.org/10.1016/j. foodcont.2017.07.013 44. Liu J, Li Z (2014) The terahertz spectrum detection of transgenic food. Optik 125(23):6867. https://doi.org/10.1016/j.ijleo.2014.08.114 45. Ren A, Zahid A, Fan D, Yang X, Imran MA, Alomainy A, Abbasi QH (2019) State-of-the-art in terahertz sensing for food and water security – a comprehensive review. Trends Food Sci Technol 85:241. https://doi.org/10.1016/j.tifs.2019.01.019 46. Shin HJ, Oh SJ, Lim M-C, Choi S-W, Ok G (2018) Dielectric traces of food materials in the terahertz region. Infrared Phys Technol 92:128. https://doi.org/10.1016/j.infrared.2018.05.022 47. Wan M, Healy JJ, Sheridan JT (2020) Terahertz phase imaging and biomedical applications. Opt Laser Technol 122:105859. https://doi.org/10.1016/j.optlastec.2019.105859 48. Nicolaï BM, Beullens K, Bobelyn E, Peirs A, Saeys W, Theron KI, Lammertyn J (2007) Nondestructive measurement of fruit and vegetable quality by means of NIR spectroscopy: a review. Postharvest Biol Technol 46(2):99. https://doi.org/10.1016/j.postharvbio.2007.06.024 49. Walsh KB, McGlone VA, Han DH (2020) The uses of near infra-red spectroscopy in postharvest decision support: a review. Postharvest Biol Technol 163:111139. https://doi.org/10.1016/ j.postharvbio.2020.111139 50. Chapman J, Elbourne A, Truong VK, Newman L, Gangadoo S, Rajapaksha Pathirannahalage P, Cheeseman S, Cozzolino D (2019) Sensomics – from conventional to functional NIR spectroscopy – Shining light over the aroma and taste of foods. Trends Food Sci Technol 91: 274. https://doi.org/10.1016/j.tifs.2019.07.013 51. Li X, Zhang L, Zhang Y, Wang D, Wang X, Yu L, Zhang W, Li P (2020) Review of NIR spectroscopy methods for nondestructive quality analysis of oilseeds and edible oils. Trends Food Sci Technol 101:172. https://doi.org/10.1016/j.tifs.2020.05.002 52. Teye E, Anyidoho E, Agbemafle R, Sam-Amoah LK, Elliott C (2020) Cocoa bean and cocoa bean products quality evaluation by NIR spectroscopy and chemometrics: a review. Infrared Phys Technol 104:103127. https://doi.org/10.1016/j.infrared.2019.103127 53. Giovenzana V, Beghi R, Guidetti R (2014) Rapid evaluation of craft beer quality during fermentation process by vis/NIR spectroscopy. J Food Eng 142:80. https://doi.org/10.1016/j. jfoodeng.2014.06.017 54. Liu R, Li L, Yin W, Xu D, Zang H (2017) Near-infrared spectroscopy monitoring and control of the fluidized bed granulation and coating processes—a review. Int J Pharm 530(1):308. https:// doi.org/10.1016/j.ijpharm.2017.07.051 55. Chavan RB, Bhargavi N, Lodagekar A, Shastri NR (2017) Near infra red spectroscopy: a tool for solid state characterization. Drug Discov Today 22(12):1835. https://doi.org/10.1016/j. drudis.2017.09.002 56. Pu Y-Y, O’Donnell C, Tobin JT, O’Shea N (2020) Review of near-infrared spectroscopy as a process analytical technology for real-time product monitoring in dairy processing. Int Dairy J 103:104623. https://doi.org/10.1016/j.idairyj.2019.104623 57. Teixeira AM, Sousa C (2019) A review on the application of vibrational spectroscopy to the chemistry of nuts. Food Chem 277:713. https://doi.org/10.1016/j.foodchem.2018.11.030 58. Meenu M, Xu B (2019) Application of vibrational spectroscopy for classification, authentication and quality analysis of mushroom: a concise review. Food Chem 289:545. https://doi.org/ 10.1016/j.foodchem.2019.03.091 59. Kahrıman F, Onaç İ, Öner F, Mert F, Egesel CÖ (2020) Analysis of secondary biochemical components in maize flour samples by NIR (near infrared reflectance) spectroscopy. J Food Meas Charact 14(4):2320. https://doi.org/10.1007/s11694-020-00479-0 60. Baiano A (2017) Applications of hyperspectral imaging for quality assessment of liquid based and semi-liquid food products: a review. J Food Eng 214:10. https://doi.org/10.1016/j.jfoodeng. 2017.06.012

12

Optical Properties

Light is electromagnetic radiation that can be detected by the human eye. This part of the electromagnetic spectrum with wavelengths between 780 and 400 nm is also known as the optical range (Table 12.1). Although the optical properties of food are thus only a part of the electromagnetic properties (Chap. 11), a separate chapter is devoted to the optical properties. It is mainly about the color of food, which can be quantified with the help of physical measurements. It is also about the polarization of light, diffraction, and refraction, which are used in quality control. In Chap. 16 (on-line sensors), some optical sensors for process analysis are presented. Applications of microwaves, terahertz waves, infrared and near-infrared have already been dealt with in Chap. 11, ultraviolet radiation follows in Chap. 13 (UV and X-ray radiation). The product of wavelength and frequency provides the propagation speed of the wave, in this case the speed of light: c=λ
f

ð12:1Þ propagation speed in m
s-1 wavelength in nm frequency in s-1

c λ f

When electromagnetic radiation such as light hits a material, phenomena such as refraction, polarization, reflection, absorption, transmission, scattering, diffraction occur. We want to go through them one by one.

12.1

Refraction

Refraction is the change in direction of wave propagation when it hits (non-vertical) an interface between media that have different propagation speeds for this wave. According to Maxwell, the propagation speed of an electromagnetic wave is given by # The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. O. Figura, A. A. Teixeira, Food Physics, https://doi.org/10.1007/978-3-031-27398-8_12

483

484

12 Optical Properties

Table 12.1 Electromagnetic radiation: visible light lies between UV and IR Frequency (Hz) 3∙1024 3∙1023 3∙1022 3∙1021 3∙1020 3∙1019 3∙1018 3∙1017 3∙1016 3∙1015 3∙1014 3∙1013 3∙1012 3∙1011 3∙1010 3∙109 3∙108 3∙107 3∙106 3∙105 3∙104 3∙103 3∙102 3∙101 3∙100

Vacuum wavelength (m) 10-16 10-15 10-14 10-13 10-12 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 105 106 107 108

Electromagnetic radiation Cosmic radiation γ-Radiation X-rays

Ultraviolet radiation (UV) Light Infrared radiation (IR) Microwaves High-frequency waves Very high frequency (VHF) Metric waves Short waves Long waves

Low frequency Alternating current Audio frequencies

c=

Wavelength comparator Atomic nucleus

H-atom

Viruses Bacteria

Chip

House Skyscraper Mount Everest

Earth diameter

1 ε0
ε
μ 0
μ

ð12:2Þ

1 ε0
μ 0

ð12:3Þ

in a vacuum it is c0 =

The ratio n of the light propagation speed c in a material with μ = 1 to the vacuum speed of light c0 thus is n= that means

c = c0

ε0
ε
μ 0 p = ε ε0
μ0

ð12:4Þ

12.1

Refraction

485

Fig. 12.1 The angle of incidence α of a beam and the angle of the reflected beam are the same α = α′. If the speed of light in material 1 and material 2 is different, then β≠α

D

D

n1 n2

E

p n= ε ε0 ε μ μ0

ð12:5Þ

electric field constant: 8.854
10-12 C
V-1
m-1 relative permittivity magnetic permeability magnetic field constant: μ0 = 4π
10-7V
s
A-1
m-1

The change in direction of propagation of a light wave as a result of the change in the propagation speed is referred to as refraction. The ratio n is called refraction index (Fig. 12.1). Let’s look at a beam of light (Fig. 12.2), which passes through the interface of material 1 and material 2. Part of the light is reflected at the interface, the rest of the light enters material 2. According to the Huygens’ principle, all illuminated points at the interface are starting points of spherical waves that propagate in all directions, including into material 2. In material 1, the light beam has the propagation speed c1, i.e. in the time span Δt it travels the distance BD. If the waves starting at these starting points in material 2 have the propagation speed c2, then the light beam travels the path AC during the same period Δt (Fig. 12.2). If the propagation speeds c1 and c2 are different, then the distances BD and AC are of different lengths, the result is a bend in the beam path. This bending of a light beam is called refraction. If the propagation speed in material 2 is lower than in material 1 as in Fig. 12.2, then refraction takes place toward the line mutually perpendicular to the interfaces, i.e. the angle of refraction β is smaller than the angle of incidence α. With

486

12 Optical Properties

Fig. 12.2 Refraction as a result of different propagation speeds in materials 1 and 2

B A

D D

C

n1 n2

E

BD = Δt
c1 AC = Δt
c2 and BD sin α = AC sin β we have sin α c1 = sin β c2 Using the ratio of propagation speed in the material and propagation speed in vacuum n=

cn c0

ð12:6Þ

n1 =

c1 c0

ð12:7Þ

n2 =

c2 c0

ð12:8Þ

so

and

we get

12.1

Refraction

487

sin α c1 = = sin β c2

c0 n1 c0 n2

==

n2 n1

ð12:9Þ

which is called Snellius’ law: sin α n1 = sin β n2

ð12:10Þ

or sin β =

n2 sin α n1

ð12:11Þ

Snellius’ law describes the relation between refraction angle β and incidence angle α. c1 c2 c0 n1 n2 β α

speed of light in material 1 in m
s-1 speed of light in material 2 in m
s-1 speed of light in vacuum in m
s-1 refraction index material 1 refraction index material 2 angle of refraction angle of incidence

If the angle of incidence α is gradually increased, the angle of refraction β increases according to Snellius’ law. When β reaches 90°, no more light enters material 2. Now all incident light is reflected at the interface. This is called total reflection. The angle of incidence at which this occurs is called critical angle of total reflection. For the sake of simplicity, we have talked about a beam of light up to this point. On closer view, we see that the extent of the refraction also depends on the wavelength or frequency of the radiation used. We had already learned about this frequency dependence of physical quantities in the section on terahertz waves (Chap. 11) as dispersion. An example of how the refractive index decreases with increasing wavelength in the case of water is shown in Fig. 12.3. The cause of the dispersion is that the wave propagation speed in any material is dependent on the frequency. The phenomenon of refraction is not limited to visible light. Even non-visible electromagnetic waves such as infrared waves, microwaves, terahertz waves, etc. are refracted at material interfaces if their propagation speeds in the materials differ on both sides of the interface. This also applies to reflection and total reflection.

488

12 Optical Properties

Fig. 12.3 Dispersion of the refractive index of water

Table 12.2 Values for the refractive indices of some materials [1] Material Vacuum Air CO2 Ethanol Water 20% (m/m) sucrose solution KBr Quartz glass Boron-silicate glass Polymethyl methacrylate (PMMA) NaCl AgBr ZnSe Diamond

Refractive index (20 °C, 103 hPa, λ = 589nm) 1 1.0003 1.0045 1.362 1.333 1.364 1.46 1.459 1.47 1.5 1.89 2.0 2.4 2.414

12.1.1 Measurement of Refractive Index In Table 12.2 examples of the refractive indices of different materials are listed. Since the refractive index depends on the wavelength of the light used and the temperature of the material, both parameters always must be specified. In the case of gases, a dependence on pressure is added. From the table it can be seen that the refractive indices differ from pure water and an aqueous sugar solution. This is exploited by storing the concentration dependence of the refractive index of aqueous sugar solutions in the form of tables or mathematical functions in order to obtain information about the composition of the solution by measuring the refractive index. A table for the sucrose-water system is listed in the appendix.

12.1

Refraction

489

Fig. 12.4 Refractive index determination. 1 Glass window, 2 sample, 3 cover, 4 incoming beam, 5 reflected beam, 6 refracted beam, 7 total reflected beam, 8 detector

3 6 2

E

8

D

7

4 5 1

Such mathematical relationships can be integrated directly into measuring instruments, so that many refractometers can display not only the refractive index of a sample but also the concentration of ingredients in solutions such as sugar. Refractive indices can be determined experimentally with the help of Snellius’ law. A metrological simplification consists in selecting 90° as the refraction angle and only looking for the associated angle of incidence α, i.e. determining the critical angle of the total reflection. In Fig. 12.4 a refractometer is shown schematically. The liquid sample (2) lies on a glass window (1) which has a known refractive index n1, the monochromatic light beam (4) penetrates the sample from below. The angle of incidence α is varied until light escapes in the 90° direction, i.e. at the detector (8). Then is: sin α sin αG = sin 90 ° sin β

ð12:12Þ

sin αG n = 2 sin 90 ° n1

ð12:13Þ

n2 = n1
sin αG

ð12:14Þ

With the known refractive index n1 of the refractometer glass window, the refractive index of the sample can be easily determined. α β n2

angle of incidence angle of refraction refractive index of the sample (continued)

490

12 Optical Properties

refractive index of the refractometer glass critical angle of refraction

n1 αG

Laboratory refractometers work with a few drops of sample fluid that are formed into a thin film and deliver the refractive index in seconds. With continuous cuvettes refractive indexes of the flowing liquids can be obtained on-line. In-line refractometers can be used to control industrial processes. By experimentally determining the refractive index, a material can be quickly characterized. With the help of tabulated refractive index values, the sucrose concentration of foods such as beverages can be determined quickly and easily.

12.2

Polarization

Light waves emitted by a normal light source oscillate in all directions transversely to the direction of wave propagation. With a polarization filter P1 in the beam path, we can ensure that only light from a single vibration plane is transmitted. The light now present is called polarized light. With a second polarization filter P2 in the beam path, the polarization direction of the light can be detected: If the filter P2 is rotated by 90° compared to P1, no light is transmitted at all. At smaller and larger angles, some of the light passes through and when the filter P2 is in the polarization direction of the light, maximum brightness occurs. The filter P2 is also called an analyzer and P1 the polarizer. Definition Linearly polarized light refers to light whose electric field vector oscillates in one spatial direction exclusively. If we bring an optically transparent material into the beam between the polarizer and the analyzer and find that the maximum brightness occurs at a different angle, then we are dealing with the so-called optically active material. Optical activity refers to materials that rotate the angle of polarized light by a certain angle. The direction of this angular change (to the right or to the left) and the extent of the change are characteristics of the substance, so optical rotation can be used in analytics. The metrological determination of the optical rotation of a substance is called polarimetry. For this purpose, a solid sample or a cuvette with the aqueous solution of the sample is placed in the beam path of a polarimeter and the angle of rotation of the solution is determined (Fig. 12.5). Materials that change the angle of rotation to the right (with a view into the light source) are referred to as positively rotating (+) or right-turning materials, materials that change the angle of rotation to the left as negatively rotating (-) or left-turning. The extent of the angular change is proportional to the length of the radiated material and is specific to the substance under

12.2

Polarization

491

Fig. 12.5 Rotation of the polarization plane, schematic. In a linearly polarized wave that propagates in the x direction, the electric vector oscillates in the z-direction (2), during the passage of an optically active material (1), the direction of oscillation changes. The difference compared to the initial direction is the angle of rotation α

observation. The material-specific rotation is called the specific rotation αS. In the case of solutions, a proportionality to the concentration c of the solution is added. The length of the cuvette filled with the solution then is the length l. In the case of a solid sample: α = αs
l α αs l

ð12:15Þ

rotation angle of the polarization plane in ° specific rotation in °
m-1 thickness in m

In the case of a liquid sample, Biot’s law applies: α = αs
l
c

ð12:16Þ

There are different units for specific rotation, depending on which concentration unit is used (%, mol
l-1, g
l-1, g
cm-3, kg
m-3, etc.). In SI units it is: α αs l c

angle of rotation of the polarization plane in rad specific rotation in rad
m2
kg-1 thickness (cuvette length) in m concentration in kg
m3

Specific rotation depends on the wavelength of the radiation used, this phenomenon is called rotational dispersion. In addition there is a dependency on temperature and, in the case of gaseous substances, also on pressure. Therefore, these parameters must always be specified when measuring rotation angles. In tables, often values are given for sodium light λ = 589 nm and ϑ = 20 ° C. The optical activity of solids such as quartz is bound to the crystal structure, i.e. it disappears when the material melts. Optical activity can also be caused by molecular structure. For example, it occurs in carbon compounds containing at least one carbon atom with four different substituents. Such a carbon atom is also called an

492 Table 12.3 Stereoisomers of lactic acid

12 Optical Properties

(+)-Lactic acid D-lactic acid Dextrorotatory lactic acid Positive specific rotation

(-)-Lactic acid L-lactic acid Levorotatory lactic acid Negative specific rotation

asymmetric carbon atom. In such a molecular structure, there are two different possibilities that arrange four substituents around the carbon atom. This results in two different molecules that behave like image and mirror image. These two molecules are called enantiomers. Since they rotate polarized light in different ways—one to the right and the other to the left—a distinction can be made between D-enantiomer and L-enantiomer. D (Lat. dexter) means right, L (Lat. leavus) means left. Table 12.3 lists the terms used for the enantiomers of lactic acid. Example We think both of our hands are identical. The number of fingers, number of bones and joint is the same for every hand. This can be illustrated by imagining you have both hands stretched out in front of you with palms facing downward. The two hands are presumably identical to each other. But when you place one hand on top of the other, you will see that the thumb and fingers do not align with each other. The thumb of one hand is aligned with the small finger of the other. You will realize that you cannot bring your two hands to cover by no rotation. It is because they are structured like image and mirror image. We have two equal hands, but concerning its structure they are different. This shall help us to understand what is a molecule with an asymmetric carbon atom. It also exists in two different structures like image and mirror image. These two molecules are also called chiral molecules (the Greek word χειρ means “hand”).

Bottom Line Materials that rotate the oscillation plane of polarized light are optically active substances. The measurement of the angle of rotation is used for identity and purity testing as well as for concentration determinations. A 1:1 mixture of the D-enantiomer and the L-enantiomer of a substance is called a racemate. In a racemate, the rotations of the two components cancel each other out, i.e. a racemate seems to be not optically active. By separating the enantiomers, two optically active substances with opposing specific rotations can be obtained. The value of the polarimetric angle of rotation of such a substance gives information about the purity of the substance.

12.2

Polarization

493

In many substances, one of the two enantiomers is preferred in nature. The reason for this may be, for example, that the chemical formation process runs via a natural enzyme that is also chiral. Physiological processes are often adjusted to this, i.e. often only one of the two enantiomers is effective, e.g. as a flavoring substance or as a pharmacologically active ingredient, while the other enantiomer is ineffective or even toxic. The L-amino acids asparagine, tryptophan, tyrosine, and isoleucine, for example, have a bitter taste, while their D-forms are characterized as sweet [2]. The racemates produced during the chemical synthesis of such substances must therefore be separated at great expense or alternative manufacturing processes with higher enantiomer purity must be found [3]. Polarimetry can be used as a physical technique for determining purity. With the help of combinations of polarimeters and CCD cameras, many samples can be examined simultaneously (polarimetric imaging) [4]. Examples of food ingredients that have optically active isomers are carbohydrates such as fructose, glucose, sucrose, galactose, mannose, lactose, maltose, dextrin, carboxylic acids such as lactic acid, tartaric acid, malic acid, and amino acids such as proline, alanine and flavorings such as pinene, limonene, and 3-3-methylthiobutanal [2, 5–7]. We started the polarimetry section by describing linearly polarized light. In linearly polarized light, the electric field vector of the monochromatic radiation oscillates with constant frequency and amplitude unchanged in time in only one plane. When the direction of the electric field vector changes in time and describes a circular path around the direction of wave propagation, we speak of this as circularly polarized light. In the general case, the amplitude can also be variable in time, with the result that the electric field vector does not rotate circularly but elliptically around the direction of wave propagation. We now have elliptically polarized light. Since this effect is not limited to light, it is called elliptically polarized radiation. One can imagine elliptically polarized radiation as the addition of two linearly polarized waves perpendicular to each other, between which there is a phase difference. At a phase difference of 90°, the ellipse becomes a circle, i.e. circularly polarized radiation, at a phase difference of 0, the ellipse becomes a line, i.e. linearly polarized radiation. There are two options for the direction of rotation of the electric field vectors of elliptically polarized radiation: right-turning and left-turning. With this understanding, the rotation of the polarization plane described above by an enantiomer mixture in a cuvette is the splitting of the linearly polarized radiation into two oppositely circularly polarized waves, which propagate with the same frequency but different propagation velocities. The different propagation velocities result from different interactions of the radiation with the two enantiomers. At the end of the passage through the cuvette with the solution, both circularly polarized waves add up again to form a linearly polarized wave, resulting in a changed angle of rotation compared to the incident radiation [8].

494

12 Optical Properties

Definition Ellipsometry is a non-destructive, noninvasive optical technique based on the polarization of light, in which change in amplitude and direction of oscillation of polarized light by a material is measured. Concentrations of optically active substances thus can be determined polarimetrically [9]. Polarimeters for sugar determination are also called saccharimeters. Through the polarimetric observation of reactions or enzymatic reactions, reaction kinetic data or enzyme activities can also be determined indirectly from the change in sugar concentration [10, 11]. The appearance of a time-varying angle of rotation is called mutarotation. Here, the angle of rotation of an aqueous solution changes until an enantiomer equilibrium with an equilibrium rotation angle is set in the solution. The position of equilibrium and the kinetics of equilibrium adjustment are dependent on temperature, concentration, and pH. The specific rotation itself can be dependent on concentration and even the choice of solvent can affect the optical rotation. If a layer of an optically inactive substance is placed between the polarizer and the 90° twisted analyzer, this arrangement is impermeable to light, it appears dark. When the substance becomes optically active by applying an external electric field, light can pass through the arrangement, it appears bright. In this way, we can switch a pixel on and off in a display. Electro-optical displays of this type, which work with a thin layer of a liquid crystalline layer, are called liquid crystal displays, LCD. There is also a magneto-optical effect: By applying a magnetic field in the direction of propagation direction of the linearly polarized electromagnetic wave, a dark (optically inactive) arrangement becomes bright. The cause lies in a precession motion of the oscillating charges around the magnetic field vector with the Larmor frequency (cf. nuclear resonance in Chap. 10). It leads to a change in the propagation velocities of right-circular and left-circular waves and thus to optical activity [8]. We learned chirality is the cause of the optical activity of molecules. As examples, we looked at molecules with one chirality center. However, there are also more complicated molecules. In addition to central chirality (molecules with one chirality center), there is axial chirality (molecules with a chirality axis), planar chirality (molecules with a chirality plane), and helicity (molecules in a helical pattern such as DNA). Reactions in which chirality changes can be investigated with femtosecond lasers [12].

12.3

Reflection, Absorption, and Transmission

When light hits an interface of two materials, normally a part of the light is reflected. For directed reflection, the angle of reflection is equal to the angle of incidence. The non-reflected part of the light enters the material behind the interface and is absorbed or transmitted there.

12.3

Reflection, Absorption, and Transmission

495

If unpolarized light hits an interface of two transparent dielectric materials, then for the directed reflection the angle of reflection is equal to the angle of incidence. If there is an angle of 90° between the angle of incidence and the angle of reflection, the reflected wave consists of linearly polarized light. This angle of incidence is called the Brewster angle αB, it depends on the refractive indices of the materials: In accordance with Snellius’ law (1.10) from the condition αB þ β = 90 °

ð12:17Þ

n sin αB = 2 n sinðαB - 90Þ 1

ð12:18Þ

sin αB n = tan αB = 2 cos αB n1

ð12:19Þ

follows

i.e.,

or αB = arctan

n2 n1

ð12:20Þ

For this reason, reflected light is at least partially linearly polarized depending on the angle of incidence. Reflections therefore can be filtered away with polarization filters. On the other hand, incident linearly polarized light is completely, partially, or not at all reflected, depending on the direction of polarization [8]. Reflectance is the ratio between incident and outgoing radiation. Smooth surfaces lead to directed reflection, while rough surfaces cause diffuse reflection. The gloss of smooth surfaces is a consequence of the reflections of directed reflection. The smoother a surface is, the lower the proportion of diffuse reflection. Therefore, by measuring the ratio of directed and diffuse reflection, the gloss of a body can be quantified (glossmeter). Emulsions are white for the viewer, although the main constituents, e.g. water and oil, are not white. During the production of the disperse phase, in the form of small oil droplets, the white color occurs. The cause again lies in the reflection. The (white) light of our illumination is reflected by the oil droplets, so we see dispersed oil droplets white. If, on the other hand, we look at the oil before making the emulsion, we see a yellowish colored liquid. Here, the light penetrates the liquid oil phase, is partially absorbed, and leaves the oil phase again in the direction of the eye of the observer. Absorption depends on the thickness of the radiated layer (Lambert-Beer law). In photometers, the layer thickness of the sample is kept constant using cuvettes and the Lambert-Beer law is used to determine the concentration by taking the absorption of a liquid sample [8].

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12 Optical Properties

Scattering and Diffraction

Scattering is the deflection of radiation from its original direction resulting from interaction with a scattering center. For example, sunlight is scattered on the molecules of the atmosphere. If, as is the case with the molecules of air, the scattering centers are smaller than the wavelengths of the radiation, this is called Rayleigh scattering. By the way, the Rayleigh scattering on the oxygen molecules is also responsible for the blue of the sky and the diffuse scattered radiation ensures that the sky appears bright during the day even when we do not look directly into the sun. Particles in the atmosphere such as water droplets, ice and dust particles also scatter part of the incident sunlight. If the particle dimensions are in the magnitude of the wavelength of the radiation, Mie scattering occurs, with larger dimensions Fraunhofer diffraction. Diffraction is the change in direction of light waves because of interference on small objects. While we are usually not surprised when sound waves come “around the corner,” we find it surprising when light not only moves straight ahead, but like sound in many spatial directions at the same time. To understand this, we use the model of a wave that moves from a point into a space (spherical wave). When we shine light through a tiny circular aperture, we observe colored, bright, and dark patterns on a screen behind the aperture, which we can only explain by detaching ourselves from the model of the beam and using the wave model. Using monochromatic light for this experiment, concentric bright and dark circles appear on the screen. They are a consequence of the interference of waves which, according to Huygens’ principle of elementary waves, are exiting from the aperture. In spatial directions where a wave maximum meets a wave minimum (negative interference) it is dark, in directions where wave maxima add up (positive interference), it is bright. These brightness patterns are created by diffraction, which is why they are also called diffraction patterns. If the diameter of the aperture is reduced, an increase in the radius of the concentric rings is observed. In this way it is possible to measure the size of the diffraction pattern. This principle is used in particle size determination by means of laser diffraction [13, 14] (Chap. 3). The laser light diffracted on tiny particles creates bright circles as interference patterns, the diameter of which is used to calculate the radius of the diffracting particles. If a sample consists of particles of different sizes, the interference pattern produces circles with different diameters, the intensity of which is a measure of the number of particles with this diameter. In this way, laser diffraction devices can record complete particle size distributions in a second. The measurement can also be performed on moving particles, which expands the possibilities of industrial process analytics [15]. By means of laser diffraction, dispersed particles of all kinds can be measured quickly and easily, e.g. in emulsions, suspensions, aerosols, dusts [16]. While we can determine particle sizes with the help of Mie scattering and Fraunhofer diffraction, we also use the scattered light of illuminated materials, e.g. for spectroscopic quality assessment [17] and determination of color.

12.5

12.5

Colorimetry

497

Colorimetry

Colorimetry is the measurement and quantification of color. Colors are familiar to us as visual sensations that arise from the impact of certain electromagnetic radiation on the retina of the eye. The color sensed by a human viewer depends on physical parameters, such as • • • • •

illuminance type of light illumination angle observation angle surface properties

but also, from the individual color vision ability of the viewer, his color perception, and his linguistic description of the optical sensation. The goal of colorimetry is to get an objective, quantitative description of color that is unique and can therefore be used in technical systems.

12.5.1 Color as a Result of Selective Absorption Visible light is electromagnetic radiation with wavelengths in the range of 400–800 nm. Larger wavelengths (IR) and smaller wavelengths (UV radiation) are invisible to humans [18]. Within the visible area, further subdivisions can be made. The visible spectrum can be split into several different colors (“colors of the rainbow”). A rough classification is shown in Table 12.4. By mixing light of different wavelengths, any color can be produced. This is called additive color mixing. On the other hand, filtering out certain wavelengths leads to a new color of the remaining light. This is called subtractive color mixing. When we direct a beam of white light at a red grape berry, the anthocyanins in the fruit absorb some of the light. Since the anthocyanin dyes prefer to absorb energy around 500 nm (green), the reflected light lacks this green component and therefore appears red to us. We recognize that the color of the grape berry is caused by selective absorption of visible electromagnetic radiation. In this example we have considered the reflection, we can look at the transmission analogously. To do this, we shine white light on a cuvette with a sample of the liquid red grape juice and look at the light coming out on the back of the cuvette. The transmitted light lacks the green wavelengths because of the anthocyanin dyes and therefore it appears red. The extent of selective absorption depends on the concentration of the absorbent substances and the layer thickness of the sample. This behavior is described by Table 12.4 Wavelength ranges of visible light

Red Yellow Blue

700–770 nm 570–590 nm 400–475 nm

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12 Optical Properties

Lambert-Beer law. It says that the extinction E (the reciprocal transmittance) is proportional to the concentration c and the layer thickness d. I = I 0
e - μ
x

ð12:21Þ

with fixed layer thickness (cuvette length) d I = I 0
e - k
c
d

ð12:22Þ

i.e., ln

I0 =k
c
d I

ð12:23Þ

or E=k
c
d x d I I0 μ E k c

ð12:24Þ penetration depth in m thickness in m intensity at x intensity at x=0 absorption coefficient extinction extinction coefficient concentration

In this way, with known extinction coefficient and defined layer thickness, the concentration of dissolved substances can be determined conveniently and quickly in this way. This is used in photometry and spectrophotometry. In addition to visible wavelengths, other wavelengths are used, such as ultraviolet. In colorimetry we distinguish between self-luminous bodies and non-self-luminous bodies. The grape berry mentioned above is an example of non-self-luminous bodies whose color is created by illumination and selective absorption (subtractive color mixing). Example Subtractive color mixing: The dye of a text highlighter absorbs a certain wavelength from the incident light, so that the reflected light appears colored to us. The dye of a yellow highlighter absorbs blue, so that the remaining mixture of visible wavelengths appears yellow. We can say the dye subtracts blue radiation from the light. If we mark over the same highlighted area with several different highlighters, further wavelengths are subtracted, and the overall picture becomes darker until it finally appears black.

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499

Self-luminous bodies emit visible electromagnetic radiation with a characteristic spectrum. When the frequencies of self-luminous bodies are superimposed, colors are created. This is additive color mixing. Example Additive color mixing: Let’s imagine a blue, a green, and a red LED that we can turn on separately. We try different combinations. The color of the resulting light is created by adding the individual wavelengths of the LEDs we have switched on. When all three LEDs are lit at the same time, white light is produced by the superimposition of blue, green, and red. This is the principle of LED displays.

12.5.2 Physiology of Color Vision There are two types of light-sensitive receptors on the retina of the human eye. The so-called rods are light-dark-sensitive, the cones are sensitive to colors. Of the cones there are three different types, the photopigments of these types have their absorption wavelengths at 420 nm (blue), 535 nm (green), and 565 nm (red). Put simply, there are cones for red, green, and blue wavelengths. Due to the interaction of these three sensor types, we can perceive numerous colors. These colors are created by additive color mixing in the human eye. The International Commission on Illumination (Commission Internationale de l’Eclairage, CIE) developed a graphic representation of all colors perceptible by humans in the form of a so-called color triangle (Fig. 12.6). Fig. 12.6 Chromaticity diagram: Inside the horseshoe-shaped triangle are all colors which can be observed. Colors with maximum brilliance are on the horseshoe curve. Point E in the middle is zero brilliance (white)

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12 Optical Properties

The maximum saturated spectral colors are applied and placed in sequential order according to their wavelength. The result is a horseshoe-shaped spectral curve, the vertices of which are formed by violet, green, and red. The line drawn from violet to red (the purple line) leads to the formation of the triangle. All colors which can be perceived by a so-called normal observer are in this triangle [19], as well as colors that do not occur in the spectrum of the sun (rainbow colors) such as purple. The triangle is designed in such a way that the additive mixture of any two-color shades creates a mixed color that lies on the connecting line in the color triangle. For example, the mixture of red and green gives the color impression yellow or the mixture of red and blue gives a purple hue. An additive mixture of colors leading to point E provides the color impression white. Spectral colors, the mixture of which yields white, are called complementary colors.

12.5.3 Terminology In Terminology of the Commission Internationale de l’Eclairage, CIE [20] a point in the chromaticity diagram (Fig. 12.6) is called hue or color stimulus specification. Red, green, blue with defined monochromatic values for the wavelength are used as reference color stimuli (Table 12.5). A color with color stimulus specification E can be represented as a linear combination of the reference color stimuli: →

→

→

→

E =r
R þ g
G þ b
B

ð12:25Þ

The reference color stimuli R, G, and B have the role of unit vectors, while r, b, g represent the contributions with which the reference color stimuli are present in the additive color mixture.

12.5.3.1 Color as a Vector For technical purposes, we use the idea that color is a point in a three-dimensional color space [22]. The vector pointing to this point from the coordinate origin is the color vector. It has three components: • hue • brilliance • brightness Table 12.5 Wavelengths of reference color stimuli [21]

R (red) G (green) B (blue)

λ/nm 700.0 546.1 435.8

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Fig. 12.7 Color vector in polar coordinate system, L brightness axis

L

D

a

b

a

b Fig. 12.8 Representation of a color with a two-dimensional vector. The angle α indicates the hue, the length of the vector d stands for the brilliance

If you draw the vector in a three-dimensional coordinate system, denote the vertical axis as brightness L, and mark the horizontal plane with the coordinates a and b, you get a representation as in Fig. 12.7. The direction of the color vector results from the values of its coordinates L, a, and b. The values of a and b indicate the hue and brilliance (see below), while L indicates the brightness. The graphical representation can be simplified by temporarily ignoring the L-axis. Then you get an image as in Fig. 12.8 where the location of the color is now specified by a two-dimensional vector with the coordinates a and b. The length of this vector d is called color brilliance with

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Fig. 12.9 Representation of a color in the L a b system

d=

a 2 þ b2

ð12:26Þ

The direction of the vector d in the horizontal plane is determined by the ratio of b over a, which is the tangent of the hue angle α tan α =

b a

ð12:27Þ

Often a representation in which the colors are indicated in the horizontal plane is used for easier orientation as shown in Fig. 12.9. In all these graphical representations, the color vector is determined by these three components: brightness, hue, and brilliance. Let’s look again at the horseshoe-shaped curve in Fig. 12.6. If we mentally add a brightness axis perpendicular to the color plane to this diagram and insert the coordinates a and b instead of x and y, you get representations like in Figs. 12.8 and 12.9. The distance of a color point from the white point in Fig. 12.6 indicates the brilliance of the color. Colors have maximum brilliance on the horseshoe-shaped curve. As the color point moves closer to the white point, the color becomes paler, i.e. less brilliant. When a color point is located directly in the white point, it appears white, i.e. uncolored. The white point also has a brightness value and appears white, light gray, dark gray, or black depending on the brightness. If the Lab system is used to specify a color, the position of the color in the color plane can already be read from the value range of components, a and b. Positive values of a stand for red, and

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Colorimetry

503

Table 12.6 Lab color values, reading examples

L 0 100 L L L L

a a a -70 +70 0 0

b b b 0 0 -70 +70

Description Completely dark (black) Completely bright Green shades Red shades Blue shades Yellow shades

negative values of a stand for green, while positive values of b stand for yellow and negative values of b stand for blue. Table 12.6 gives some reading examples. About 200 hues can be distinguished, about 20–25 brilliance levels and about 500 brightness levels. There are thus 2 million possibilities of combining hue, brilliance, and brightness in color vision. Bottom Line A color can be clearly quantified by specifying its vector in color space with the three independent color coordinates: brightness, hue, and brilliance.

Definition The Lab system, or CIE Lab system, is the specification of a color as a vector in the color space with the coordinates L, a, and b (CIE LAB Color Space). In addition to the described color systems, there are numerous other systems for the film and media industry, manufacturers of lamps, paints, coatings, printers, and displays [23, 24]. Color Difference An advantage of using vectors to describe the color is the calculation of the color difference or the so-called color distance [25]. In the Lab system, color difference ΔE of two colors is the Pythagorean sum of the differences in their L, a, b-values: ΔE =

ðL2 - L1 Þ2 þ ða2 - a1 Þ2 þ ðb2 - b1 Þ2

ð12:28Þ

ΔL2 þ Δa2 þ Δb2

ð12:29Þ

i.e., ΔE =

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12 Optical Properties

Example Two products have their individual color values of L, a, and b L1 = 80, a1 = 56, b1 = - 31 and L2 = 81, a2 = 54, b2 = - 29. The color difference is ΔE =

ΔL2 þ Δa2 þ Δb2

p ΔE = 12 þ 22 þ 22 = 9 ΔE = 3 Automatic sorting systems make use of on-line color measurements that are programmed with the help of such color differences.

12.5.3.2 Visual Color Measurement Methods In visual assessment, two objects are referred to as identical in color if no color difference can be detected when viewed under the same lighting conditions with the naked eye. Defined color standards are used to help specify the color of an object by selecting the standard which best matches the color of the object (color match). Color standards are used in numerous industry-specific color charts. For liquids the American and European pharmacopeias recommend aqueous solutions of CaCl3 (pink), FeCl3 (yellow), and CuSO4 (blue) and their mixtures as color matching fluids. The preparation and dilution of these color reference solutions is described on a table in the Annex. Applications A common application of color measurement lies in automated quality assessment. In this way, ripe and immature, uniformly and unevenly colored agricultural products can be distinguished [26] or the quality of processed and fresh food indicated [27, 28]. Further Reading Refraction Pharmaceutical solids: refractive index measurement Winemaking: temperature dependence of grape must refractive Peanut oil: refractive index and density measurements for determining oleic and linoleic acid Glucose syrup: determination of dry matter content by refractive index Qualitative identification of food by terahertz complex refractive index mapping Fishery products: Application of refraction index measurement Seawater: refractive index and salinity

[29] [30] [31] [32] [33] [34] [35]

(continued)

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Colorimetry

505

Refractive index of aqueous solutions of Na-acetate, Na-carbonate, Na-citrate with and without glycerol Wine: Effect of ethanol, glycerol, glucose on refraction index Urine: Photonic crystal-based biosensor for the detection of glucose concentration Fiber optic refractive index sensors

[36] [37] [38] [39, 40]

Polarimetry Urine: noninvasive glucose detection by computer-based polarimeters Polarimetric examination of enzymatic reactions with L-DOPA (Parkinson’s disease drug) Imaging polarimetry for enantiomer screening Ellipsometry-based biosensing of immunoglobulins

[9] [10] [4] [41]

Reflection, Absorption, Transmission Fruits: characterization by time-resolved reflectance spectroscopy, TRS Fruits and Vegetables: gloss measurement Brewster angle microscopy on monolayers Spectral photometer for color measurement

[42, 43] [44] [45, 46] [47]

Scattering and Diffraction Particle sizing by laser diffraction Particle sizing: comparing sedimentation and laser diffraction Particle sizing by dynamic light scattering (DLS) ISO 22412 Polarized light and laser scatter imaging for inspection of packagings Dynamic light scattering (DLS) for food quality evaluation Robotic optical inspection of fruits and vegetables

[13, 14, 48] [49] [50] [51] [52] [53, 54]

Colorimetry Electronic eye for food sensory evaluation Beer: Color measurment CCD technique for image analytical color measurement Hibiscus beverages: Colorimetry Banana: Color model for quality assessment Wheat sorting by color Simple method for analyzing the color of food by digital camera Tracking bioactive compounds by color Meat: Quality and color Powders: Colorimetric determination of mixture quality Carotenoids in corn: Color difference estimation

[55] [56, 57] [58] [59] [60] [61] [62] [63] [64–66] [67] [68]

(continued)

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Inactivation of microorganisms by blue LEDs LED illumination for agriculture

[69] [70]

Imaging Chocolate: Microstructural Imaging Fermented dairy products: Imaging by confocal laser microscopy Hyperspectral imaging on food Meat: Hyperspectral imaging for quality assessment Apple: Dry matter content by NIR hyperspectral imaging Coffee: Quality sorting by hyperspectral imaging Imaging of double emulsions Food structure by atomic force microscopy (AFM) Microstructure of food by light microscopy

[71] [72, 73] [74–76] [77, 78] [79] [80] [81] [82–84] [85]

Summary The visual appearance of food depends on optical phenomena such as absorption, reflection, scattering, gloss, and color. In this chapter, physical causes of refraction, diffraction, absorption, and transmission are explained in a concise and comprehensive way, and how these phenomena stem from the interaction between electromagnetic radiation and matter. They are illustrated by numerous examples. Causes for optical activity in food ingredients are explained and techniques such as polarimetry and ellipsometry are presented. Color is represented as a vector in different color systems and compared with visual sensation. At the end of the chapter, examples of applications are listed that can be used for further studies.

References 1. Fraden J (2016) Handbook of modern sensors physics, designs, and applications. https://doi. org/10.1007/978-3-319-19303-8 2. Zawirska-Wojtasiak R (2006) Chirality and the nature of food authenticity of Aroma. Acta Sci Pol Technol Aliment 5:21 3. Maier NM, Franco P, Lindner W (2001) Separation of enantiomers: needs, challenges, perspectives. J Chromatogr A 906(1):3. https://doi.org/10.1016/S0021-9673(00)00532-X 4. Gibbs PR, Uehara CS, Nguyen PT, Willson RC (2003) Imaging polarimetry for high throughput chiral screening. Biotechnol Prog 19(4):1329. https://doi.org/10.1021/bp025729l 5. Mosandl A, Hener U, Fuchs S (2000) Natürliche Duft- und Aromastoffe — Echtheitsbewertung mittels enantioselektiver Kapillar-GC und/oder Isotopenverhältnis-massenspektrometrie. In:

References

507

Günzler H (ed) Analytiker-Taschenbuch. Springer, Heidelberg. https://doi.org/10.1007/978-3642-57180-0_2 6. Fanali C, D’Orazio G, Gentili A, Fanali S (2019) Analysis of enantiomers in products of food interest. Molecules 24(6). https://doi.org/10.3390/molecules24061119 7. D’Orazio G, Fanali C, Asensio-Ramos M, Fanali S (2017) Chiral separations in food analysis. TrAC Trends Anal Chem 96:151. https://doi.org/10.1016/j.trac.2017.05.013 8. Meschede D (2006) Gerthsen Physik. Springer, Berlin. https://doi.org/10.1007/978-3-66245977-5 9. Sutrisno YA (2017) Noninvasive and painless urine glucose detection by using computer-based polarimeter. Mater Sci Eng 202:012030. https://doi.org/10.1088/1757-899X/202/1/012030 10. Carroll A, Dick J, Selph L, Murphy M (2017) Using polarimetry to examine the kinetics of enzymatic reactions with the Parkinson’s disease drug L-DOPA. In: Research & Creative Activity Symposium, Montgomery, AL. Alabama State University. https://doi.org/10.1016/j. ijbiomac.2013.01.031 11. Honig W, Moshudis E, Oette K (1981) Polarimetric determination of alpha-amylase activity (author’s transl). J Clin Chem Clin Biochem / Zeitschrift fur klinische Chemie und klinische Biochemie 19(10):1057 12. Baykusheva D, Zindel D, Svoboda V, Bommeli E, Ochsner M, Tehlar A, Wörner HJ (2019) Real-time probing of chirality during a chemical reaction. Proc Natl Acad Sci 116(48):23923. https://doi.org/10.1073/pnas.1907189116 13. Alexander M, Dalgleish DG (2006) Dynamic light scattering techniques and their applications in food science. Food Biophys 1(1):2. https://doi.org/10.1007/s11483-005-9000-1 14. ISO 13320 (2020) Partikelgrößenanalyse - Partikelmessung durch Laserlichtbeugung. Beuth, Berlin. https://doi.org/10.31030/2333155 15. Ruscitti O, Franke R, Hahn H, Babick F, Richter T, Stintz M (2008) Zum Einsatz der Partikelmesstechnik in der Prozessintensivierung. Chem Ing Techn 80(1-2):191. https://doi. org/10.1002/cite.200700168 16. Xu R (2015) Light scattering: a review of particle characterization applications. Particuology 18:11. https://doi.org/10.1016/j.partic.2014.05.002 17. Mollazade K, Omid M, Tab FA, Mohtasebi SS (2012) Principles and applications of light backscattering imaging in quality evaluation of agro-food products: a review. Food Bioprocess Technol 5(5):1465. https://doi.org/10.1007/s11947-012-0821-x 18. Udayakumar N (2014) Visible light imaging. In: Manickavasagan A, Jayasuriya H (eds) Imaging with electromagnetic spectrum: applications in food and agriculture. Springer, Berlin, pp 67–86. https://doi.org/10.1007/978-3-642-54888-8_5 19. DIN EN ISO/CIE 11664 (2020) Farbmetrik - Teil 1: CIE farbmetrische Normalbeobachter. Beuth, Berlin. https://doi.org/10.31030/3091836 20. DIN EN ISO/CIE 11664 (2020) Farbmetrik - Teil 3: CIE-Farbwerte. Beuth, Berlin. https://doi. org/10.31030/3092071 21. Kohlrausch F (1996) Praktische Physik Bd.1. Teubner, Stuttgart. https://doi.org/10.1007/978-3322-87205-0 22. DIN 5033 (2017) Farbmessung - Teil 1: Grundbegriffe der Farbmetrik. Beuth, Berlin. https:// doi.org/10.31030/2705354 23. Parkin A (2016) Colorimetry. In: Parkin A (ed) Digital imaging primer. Springer, Berlin, pp 269–281. https://doi.org/10.1007/978-3-540-85619-1_16 24. Lübbe E (2013) Farbempfindung, Farbbeschreibung und Farbmessung. Springer Vieweg, Wiesbaden. https://doi.org/10.1007/978-3-8348-2228-4 25. ISO/CIE 11664-6 (2014) Farbmetrik - Teil 6: CIEDE2000 Formel für die Farbdifferenz. Beuth, Berlin 26. Kang SP (2011) Color in food evaluation. In: Gliński J, Horabik J, Lipiec J (eds) Encyclopedia of agrophysics. Springer, Dordrecht, pp 138–141. https://doi.org/10.1007/978-90-481-3585-1_ 236

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12 Optical Properties

27. Culver CA, Wrolstad RE (2008) Color quality of fresh and processed foods. American Chemical Society; Distributed by Oxford University Press, Washington, DC. https://doi.org/ 10.1021/bk-2008-0983 28. Wu D, Sun D-W (2013) Colour measurements by computer vision for food quality control – a review. Trends Food Sci Technol 29(1):5. https://doi.org/10.1016/j.tifs.2012.08.004 29. Mohan S, Kato E, Drennen JK, Anderson CA (2019) Refractive index measurement of pharmaceutical solids: a review of measurement methods and pharmaceutical applications. J Pharm Sci 108(11):3478. https://doi.org/10.1016/j.xphs.2019.06.029 30. Jiménez-Márquez F, Vázquez J, Úbeda J, Sánchez-Rojas JL (2016) Temperature dependence of grape must refractive index and its application to winemaking monitoring. Sens Actuators B Chem 225:121. https://doi.org/10.1016/j.snb.2015.10.064 31. Davis JP, Sweigart DS, Price KM, Dean LL, Sanders TH (2013) Refractive index and density measurements of peanut oil for determining oleic and linoleic acid contents. J Am Oil Chemists Soc 90(2):199. https://doi.org/10.1007/s11746-012-2153-4 32. ISO 1743 (1982) Glucose syrup - determination of dry matter content- refractive index method. Vernier, Switzerland 33. Shin HJ, Choi SW, Ok G (2018) Qualitative identification of food materials by complex refractive index mapping in the terahertz range. Food Chem 245:282. https://doi.org/10.1016/ j.foodchem.2017.10.056. Epub 2017 Oct 13 34. Mathew S, Raman M, Kalarikkathara PM, Rajan DP (2019) Techniques used in fish and fishery products analysis. In: Fish and fishery products analysis. Springer, Singapore. https://doi.org/ 10.1007/978-981-32-9574-2_5 35. Tengesdal ØA (2012) Measurement of seawater refractive index and salinity by means of optical refraction. University, Bergen 36. Bagheri M, Kiani F, Koohyar F, Khang NT, Zabihi F (2020) Measurement of refractive index and viscosity for aqueous solution of sodium acetate, sodium carbonate, trisodium citrate, (glycerol + sodium acetate), (glycerol + sodium carbonate), and (glycerol + trisodium citrate) at T = 293.15 to 303.15 K and atmospheric pressure. J Mol Liq 309:113109. https://doi.org/10. 1016/j.molliq.2020.113109 37. Shehadeh A, Evangelou A, Kechagia D, Tataridis P, Chatzilazarou A, Shehadeh F (2020) Effect of ethanol, glycerol, glucose/fructose and tartaric acid on the refractive index of model aqueous solutions and wine samples. Food Chem 329:127085. https://doi.org/10.1016/j.foodchem.2020. 127085 38. Robinson S, Dhanlaksmi N (2016) Photonic crystal based biosensor for the detection of glucose concentration in urine. Photonic Sens 7. https://doi.org/10.1007/s13320-016-0347-3 39. Wu Y, Liu B, Wang J, Wu J, Mao Y, Ren J, Zhao L, Sun T, Nan T, Han Y, Liu X (2021) A novel temperature insensitive refractive index sensor based on dual photonic crystal fiber. Optik 226:165495. https://doi.org/10.1016/j.ijleo.2020.165495 40. Li J (2020) A review: Development of novel fiber-optic platforms for bulk and surface refractive index sensing applications. Sens Actuators Rep 2(1):100018. https://doi.org/10.1016/j.snr. 2020.100018 41. Li K, Wang S, Wang L, Yu H, Jing N, Xue R, Wang Z (2017) Fast and sensitive ellipsometrybased biosensing. Sensors (Basel, Switzerland) 18(1):15. https://doi.org/10.3390/s18010015 42. Torricelli A, Spinelli L, Contini D, Vanoli M, Rizzolo A, Eccher Zerbini P (2008) Timeresolved reflectance spectroscopy for non-destructive assessment of food quality. Sens Instrumen Food Qual 2(2):82. https://doi.org/10.1007/s11694-008-9036-2 43. Kim M, Lee K, Chao K, Lefcourt A, Jun W, Chan D (2008) Multispectral line-scan imaging system for simultaneous fluorescence and reflectance measurements of apples: multitask apple inspection system. Sens Instrumen Food Qual 2(2):123. https://doi.org/10.1007/s11694-0089045-1 44. Mizrach A, Lu RF, Rubino M (2009) Gloss evaluation of curved-surface fruits and vegetables. Food Bioprocess Technol 2(3):300. https://doi.org/10.1007/s11947-008-0083-9

References

509

45. Pawluczyk R (1990) Modified Brewster angle technique for the measurement of the refractive index of a DCG layer. Appl Opt 29:589. https://doi.org/10.1364/AO.29.000589 46. Daear W, Mahadeo M, Prenner EJ (2017) Applications of Brewster angle microscopy from biological materials to biological systems. Biochim Biophys Acta Biomembr 1859(10):1749. https://doi.org/10.1016/j.bbamem.2017.06.016 47. Fernandes GM, Silva WR, Barreto DN, Lamarca RS, Lima Gomes PCF, Petruci JFS, Batista AD (2020) Novel approaches for colorimetric measurements in analytical chemistry – a review. Anal Chim Acta 1135:187. https://doi.org/10.1016/j.aca.2020.07.030 48. Lee Black D, McQuay MQ, Bonin MP (1996) Laser-based techniques for particle-size measurement: a review of sizing methods and their industrial applications. Prog Energy Combust Sci 22(3):267. https://doi.org/10.1016/S0360-1285(96)00008-1 49. Ferro V, Mirabile S (2009) Comparing particle size distribution analysis by sedimentation and laser diffraction method. J Agric Eng. https://doi.org/10.4081/jae.2009.2.35 50. ISO 22412 (2017) Particle Size Analysis – Dynamic Light Scattering (DLS). Beuth, Berlin 51. Barnes M, Dudbridge M, Duckett T (2012) Polarised light stress analysis and laser scatter imaging for non-contact inspection of heat seals in food trays. J Food Eng 112(3):183. https:// doi.org/10.1016/j.jfoodeng.2012.02.040 52. Rao P, Yu Z, Han H, Xu Y, Ke L (2019) Dynamic light scattering for food quality evaluation. In: Zhong J, Wang X (eds) Evaluation technologies for food quality. Woodhead Publishing, pp 535–557. https://doi.org/10.1016/B978-0-12-814217-2.00020-2 53. Zhang B, Gu B, Tian G, Zhou J, Huang J, Xiong Y (2018) Challenges and solutions of opticalbased nondestructive quality inspection for robotic fruit and vegetable grading systems: a technical review. Trends Food Sci Technol 81:213. https://doi.org/10.1016/j.tifs.2018.09.018 54. Li S, Luo H, Hu M, Zhang M, Feng J, Liu Y, Dong Q, Liu B (2019) Optical non-destructive techniques for small berry fruits: a review. Artif Intell Agric 2:85. https://doi.org/10.1016/j.aiia. 2019.07.002 55. Xu C (2019) Electronic eye for food sensory evaluation. In: Zhong J, Wang X (eds) Evaluation technologies for food quality. Woodhead Publishing, pp 37–59. https://doi.org/10.1016/B9780-12-814217-2.00004-4 56. Koren D, Hegyesné Vecseri B, Kun-Farkas G, Urbin Á, Nyitrai Á, Sipos L (2020) How to objectively determine the color of beer? J Food Sci Technol 57(3):1183. https://doi.org/10. 1007/s13197-020-04237-4 57. Fengxia S, Yuwen C, Zhanming Z, Yifeng Y (2004) Determination of beer color using image analysis. J Am Soc Brew Chem 62(4):163. https://doi.org/10.1094/ASBCJ-62-0163 58. Mendoza F, Dejmek P, Aguilera JM (2006) Calibrated color measurements of agricultural foods using image analysis. Postharvest Biol Technol 41(3):285. https://doi.org/10.1016/j. postharvbio.2006.04.004 59. Camelo-Mendez GA, Vanegas-Espinoza PE, Escudero-Gilete ML, Heredia FJ, Paredes-LopezO, Del Villar-Martinez AA (2018) Colorimetric analysis of hibiscus beverages and their potential antioxidant properties. Plant Food Hum Nutr 73(3):247 60. Nannyonga S, Bakalis S, Andrews J, Mugampoza E, Gkatzionis K (2016) Mathematical modelling of color, texture kinetics and sensory attributes characterisation of ripening bananas for waste critical point determination. J Food Eng 190:205. https://doi.org/10.1016/j.jfoodeng. 2016.06.006 61. Pearson T, Brabec D, Haley S (2008) Color image based sorter for separating red and white wheat. Sens Instrumen Food Qual 2(4):280 62. Yam KL, Papadakis SE (2004) A simple digital imaging method for measuring and analyzing color of food surfaces. J Food Eng 61(1):137. https://doi.org/10.1016/s0260-8774(03)00195-x 63. Sant’Anna V, Gurak PD, Ferreira Marczak LD, Tessaro IC (2013) Tracking bioactive compounds with colour changes in foods – a review. Dyes Pigments 98(3):601. https://doi. org/10.1016/j.dyepig.2013.04.011

510

12 Optical Properties

64. Ijaz M, Li X, Zhang D, Hussain Z, Ren C, Bai Y, Zheng X (2020) Association between meat color of DFD beef and other quality attributes. Meat Sci 161:107954. https://doi.org/10.1016/j. meatsci.2019.107954 65. Faustman C, Suman SP (2017) The eating quality of meat: I—Color. In: Toldra F (ed) Lawrie’s meat science, 8th edn. Woodhead Publishing, pp 329–356. https://doi.org/10.1016/B978-0-08100694-8.00011-X 66. Warner R (2014) Measurement of meat quality | Measurements of water-holding capacity and color: objective and subjective. In: Dikeman M, Devine C (eds) Encyclopedia of meat sciences. Academic Press, Oxford, pp 164–171. https://doi.org/10.1016/B978-0-12-384731-7.00210-5 67. Shenoy P, Innings F, Lilliebjelke T, Jonsson C, Fitzpatrick J, Ahrné L (2014) Investigation of the application of digital colour imaging to assess the mixture quality of binary food powder mixes. J Food Eng 128:140. https://doi.org/10.1016/j.jfoodeng.2013.12.013 68. Nkhata SG (2020) Total color change (ΔE) is a poor estimator of total carotenoids lost during post-harvest storage of biofortified maize grains. Heliyon 6(10):e05173. https://doi.org/10. 1016/j.heliyon.2020.e05173 69. Hyun J-E, Lee S-Y (2020) Blue light-emitting diodes as eco-friendly non-thermal technology in food preservation. Trends Food Sci Technol 105:284. https://doi.org/10.1016/j.tifs.2020.09.008 70. Gupta SD (2017) Light emitting diodes for agriculture. Springer, Singapore. https://doi.org/10. 1007/978-981-10-5807-3 71. Rousseau D (2016) Microstructural imaging of chocolate confectionery. In: Sozer N (ed) Imaging technologies and data processing for food engineers. Springer International, Cham, pp 311–333. https://doi.org/10.1007/978-3-319-24735-9_10 72. Ercili-Cura D (2016) Imaging of fermented dairy products. In: Sozer N (ed) Imaging technologies and data processing for food engineers. Springer International, Cham, pp 99–128. https://doi.org/10.1007/978-3-319-24735-9_4 73. Lorén N, Langton M, Hermansson AM (2007) Confocal fluorescence microscopy (CLSM) for food structure characterisation. In: McClements DJ (ed) Understanding and controlling the microstructure of complex foods. Woodhead Publishing, pp 232–260. https://doi.org/10.1533/ 9781845693671.2.232 74. Liu YW, Pu HB, Sun DW (2017) Hyperspectral imaging technique for evaluating food quality and safety during various processes: a review of recent applications. Trends Food Sci Technol 69:25. https://doi.org/10.1016/j.tifs.2017.08.013 75. Baiano A (2017) Applications of hyperspectral imaging for quality assessment of liquid based and semi-liquid food products: a review. J Food Eng 214:10. https://doi.org/10.1016/j.jfoodeng. 2017.06.012 76. Park B, Lu R (2015) Hyperspectral imaging technology in food and agriculture. https://doi.org/ 10.1007/978-1-4939-2836-1 77. Feng CH, Makino Y, Oshita S, Martin JFG (2018) Hyperspectral imaging and multispectral imaging as the novel techniques for detecting defects in raw and processed meat products: current state-of-the-art research advances. Food Control 84:165. https://doi.org/10.1016/j. foodcont.2017.07.013 78. Antequera T, Caballero D, Grassi S, Uttaro B, Perez-Palacios T (2021) Evaluation of fresh meat quality by Hyperspectral Imaging (HSI), Nuclear Magnetic Resonance (NMR) and Magnetic Resonance Imaging (MRI): a review. Meat Sci 172:108340. https://doi.org/10.1016/j.meatsci. 2020.108340 79. Ma T, Li X, Inagaki T, Yang H, Tsuchikawa S (2018) Noncontact evaluation of soluble solids content in apples by near-infrared hyperspectral imaging. J Food Eng. https://doi.org/10.1016/j. jfoodeng.2017.12.028 80. Nansen C, Singh K, Mian A, Allison BJ, Simmons CW (2016) Using hyperspectral imaging to characterize consistency of coffee brands and their respective roasting classes. J Food Eng 190: 34. https://doi.org/10.1016/j.jfoodeng.2016.06.010

References

511

81. Bernewitz R, Guthausen G, Schuchmann HP (2016) Imaging of double emulsions. In: Sozer N (ed) Imaging technologies and data processing for food engineers. Springer International, Cham, pp 69–98. https://doi.org/10.1007/978-3-319-24735-9_3 82. Morris VJ (2007) Atomic force microscopy (AFM) techniques for characterising food structure. In: McClements DJ (ed) Understanding and controlling the microstructure of complex foods. Woodhead Publishing, pp 209–231. https://doi.org/10.1533/9781845693671.2.209 83. Gunning AP, Morris VJ (2018) Getting the feel of food structure with atomic force microscopy. Food Hydrocoll 78:62. https://doi.org/10.1016/j.foodhyd.2017.05.017 84. Middendorf D, Bindrich U, Mischnick P, Juadjur A, Franke K, Heinz V (2016) Atomic Force Microscopy study on the effect of different lecithins in cocoa-butter based suspensions. Colloids Surf A Physicochem Eng Aspects 499:60. https://doi.org/10.1016/j.colsurfa.2016.03.057 85. Gunning PA (2013) Light microscopy: principles and applications to food microstructures. In: Morris VJ, Groves K (eds) Food microstructures. Woodhead Publishing, pp 62–95. https://doi. org/10.1533/9780857098894.1.62

13

UV and X-rays

Frequencies above the visible range of the electromagnetic spectrum are invisible to the human eye. The wavelength ranging from 380 nm to 1 nm is called ultraviolet radiation, wavelengths shorter than 1 nm are called X-rays. When UV radiation hits a material, the valence electrons are energetically excited. For this reason, UV spectroscopy is also called electron spectroscopy. X-rays, on the other hand, are absorbed by internal electrons. The propagation speed c of electromagnetic waves is the product of frequency f and wavelength λ. c=λ
f

propagation speed in m s-1 wavelength in nm frequency in s-1

c λ f

13.1

ð13:1Þ

UV

There are different perspectives on the subdivision of the ultraviolet range. Physically, it is divided into near UV (380–200 nm), far UV (200–10 nm), and extremely far UV (1–31 nm) (Table 13.1). Regarding the filtering of the Sun’s UV radiation through the Earth’s atmosphere, the distinction is made between UV-A (380–315 nm), UV-B (315–280 nm), and UV-C (280–100 nm). The UV-A range includes the wavelengths that are hardly absorbed by the atmosphere and thus reach the Earth’s surface. Wavelengths below 315 nm are largely absorbed by atmospheric ozone and wavelengths below 280 nm by atmospheric oxygen. Laboratory experiments with UV below a wavelength of 200 nm must therefore be carried out under vacuum, which is why the range of 200 to 1 nm is also called vacuum UV (Fig. 13.1).

# The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. O. Figura, A. A. Teixeira, Food Physics, https://doi.org/10.1007/978-3-031-27398-8_13

513

514

13

UV and X-rays

Table 13.1 Electromagnetic radiation, UV, and X-ray Frequency in Hz 3 ∙ 1021 3 ∙ 1020 3 ∙ 1019 3 ∙ 1018 3 ∙ 1017 3 ∙ 1016 3 ∙ 1015 3 ∙ 1014 3 ∙ 1013 3 ∙ 1012

Designation X-ray

Ultraviolet (UV) Light infrared (IR)

Vacuum wavelength in m 10-13 10-12 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4

Fig. 13.1 Classification of ultraviolet radiation (UV)

Because the UV absorption of materials provides information about valence electrons and thus about molecular properties, UV radiation is used in analytical spectroscopy. UV radiation allows electrons to change their energetic level. For example, from the lowest unoccupied molecular orbital (LUMO) in a molecule to the highest occupied molecular orbital (HOMO) [1]. Spectrometers that use both visible and UV frequencies are called UV/vis spectrometers. UV radiation can also lead to fluorescence of molecules, which is used in fluorescence spectroscopy. Numerous sensors are based on the interaction of analytes with UV radiation [2]. In addition, UV radiation is used to disinfect water and food. Between 200 and 300 nm, most proteins, amino acids, and nucleic acids absorb UV radiation. With sufficiently high energy intake, this can lead to molecular structural changes that lead to the loss of original function. For this reason, it is possible to inactivate microorganisms with short-wave UV radiation, as well as viruses and bacteriophages [3–6]. With short-wave UV radiation, bacteria in food can be inactivated and food shelf lives increased [4, 7–9]. UV radiation can be used to disinfect food surfaces or work surfaces and machines in processing plants. In addition, gases or liquids with low layer thickness can be treated. In the field of drinking water treatments, there are numerous UV applications, some in combination with other inactivation processes [10–18]. The application of pulsed short-wave UV [19–22] along with powerful LEDs with specific frequency spectrum is promising [3, 7].

13.2

13.2

X-ray

515

X-ray

Electromagnetic radiation with wavelengths in the range of 10-11 m to 10-9 m or frequencies of 1013 Hz to 1017 Hz is called X-rays. X-rays with wavelengths smaller than 10-10 m are called hard X-rays, with wavelengths larger than 10-10 m soft X-rays. Radiation with wavelengths below 10-11 m can be described as ultrahard X-rays and radiation with wavelengths above 10-9 m as ultrasoft X-rays. Conceptually, the transition from ultrasoft X-rays to UV radiation is fluid (Fig. 13.2). The energy of X-ray photons is several orders of magnitude higher than the energy of visible light photons. The energy of an X-ray photon is proportional to the frequency of the radiation. It can be expressed in electron volts (eV) instead of Joules (J). The conversion is: 1 eV = 1.609 ∙ 10-19 J. It is E=h
f E h f

ð13:2Þ

energy in J or eV Planck’s constant h = 6.62607015 ∙ 10-34 J s frequency in s-1

Example Wavelength and photon energy of X-rays having a frequency 1018 Hz is with c = λ
f it is λ=

c 3
108 ms - 1 = 3
10 - 10 m = 30 nm = f 1018 s - 1

and with E = h
f the energy is E = h
f = 6:6
1034 J s
1018 s1 = 6:6
1016 J = 4:1 eV By absorbing X-rays, ions can be generated in the absorbent material. X-rays are therefore assigned to ionizing radiation. Due to their high penetration depth, X-rays can be used to elucidate the structure of materials. X-ray diffraction can be used to elucidate the lattice structures of crystalline materials. Amorphous solid materials that can be formed during the drying of food [23, 24] can be distinguished from crystalline materials by powder X-ray diffractometry [25].

516 Fig. 13.2 Categories of X-rays

13

100Å

10Å

ultra weak 108m

weak 109m

1Å

UV and X-rays

0.1Å

hard 1010m

0.01Å

ultra hard

1011m

1012m

13.2.1 X-ray Absorption The main absorption mechanisms for X-rays are the photo effect, elastic and inelastic scattering as well as pair formation. Pairing is the formation of electron–positron pairs, which occurs with X-ray photons with energies above a few MeV. X-ray diffraction is used to elucidate the structure of solids, while the photoelectric effect is the main cause of absorption in biological materials. In the photo effect, the intake of the energy of an X-ray photon leads to the release of an electron from the hit atom. At a sufficiently high X-ray frequency, internal electrons of an atom are also released. Fluorescence or the appearance of Auger electrons occurs when the electron places have been filled up by external electrons. Both can be used spectroscopically to determine the composition of the absorbent material. The Lambert-Beer law applies to the attenuation of X-rays by the photo effect. Accordingly, the intensity decreases exponentially with the depth of penetration: I = I 0
e - μx I I0 μ x

ð13:3Þ intensity at x intensity at x = 0 absorption coefficient penetration depth in m

The absorption coefficient μ for X-rays depends on the number density and crosssection of the absorbing atoms and thus strongly on the atomic number of the absorbing material. It can be shown that the absorption coefficient is proportional to the third power of the atomic number [26]. Example Lead shows about 30 times greater X-ray absorption than iron. So a 1 mm thick layer of lead has the same shielding effect as a 3 cm thick layer of iron. If we compare the substances on the basis of their density and atomic numbers, it (continued)

13.2

X-ray

517

becomes apparent that the third power of the atomic numbers of both materials also differs by a factor of about 30. The density of lead, on the other hand, is not 30 times but only about 40% higher than that of iron: Density in g∙cm-3 Atomic number Z Z3

Lead 11.3 82 551,368

Iron 7.9 26 17,576

The different absorption coefficients of materials are used in X-ray diagnostics and X-ray inspection. Here, the X-ray transmission of a fan-shaped beam is detected by suitable detectors. Depending on the absorption coefficient and the layer thickness of the radiated material, an image with different shades of gray is created in the detector. The grayscales can be technically converted into colors for better visual interpretation. Attention The rule of thumb “the higher the density of a material, the higher its X-ray absorption” must not be taken too strictly and certainly not misunderstood as a linear relationship. There are numerous examples that contradict this rule of thumb, e.g. Beryllium has a lower X-ray absorption than potassium, although its density is twice as high.

13.2.2 Imaging Techniques With the help of X-ray imaging techniques, shapes of materials can be detected. In this way, e.g. by X-ray inspection, suspicious objects in baggage or freight containers can be detected. In the presence of strong absorption differences, materials can also be recognized or distinguished from the surrounding matrix, e.g. glass, wood, bones in food. For this reason, X-ray scanners are used in food production to inspect packaged foods. Applications range from X-ray systems for entire trucks to fast production line scanners for ready-packed food. Contrasts in the generated images, i.e. the different X-ray absorption of adjacent areas, are evaluated. High contrasts occur with large differences in atomic number Z, i.e. when there is a glass, bone, or metal in a food. Atomic numbers in food are in the range of atomic numbers of the main components, i.e. carbon (Z = 6), oxygen (Z = 8), hydrogen (Z = 1), and nitrogen (Z = 7), while the atomic number of calcium in the bones is Z = 20, of the silicon in mineral glass is Z = 14, and the atomic number of metallic materials is even higher. Materials that consist of atoms like those of food will show similar X-ray absorption and, therefore, will form only a faint contrast to food. They are therefore not easy to identify. These include materials such as wood, fruit kernels, insects, and plastics such as polypropene, which have a similar element composition to the surrounding food matrix. Therefore, X-ray scanners cannot

518

13

UV and X-rays

differentiate certain material combinations. X-ray scanners have numerous advantages over metal detectors in contaminant detection. While metal detectors have a high sensitivity to metals with high magnetic susceptibility, sensitivity to metals with low high magnetic susceptibility (Al, Cu, brass) is lower and is further reduced by foods with high electrical conductivity (refer Chap. 9). X-ray scanners for detection of foreign bodies do not have these limitations. In addition to steel and all non-ferrous metals, they can also be used to detect non-metallic impurities such as fragments of glass, stone and bone, and some plastics. Contaminant detection is also possible in aluminum-packed foods. Parallel to the inspection of a food product for foreign bodies, the completeness of a product can be determined from the X-ray image and also the layer thickness of the food on the basis of the grayscale. In this way, complete packs can be distinguished from incompletely filled ones. With appropriate calibration, the setpoint of the mass of the packaged food can be checked from the grayscale X-ray image [27]. Definition CT stands for computed tomography. Tomography comes from Greek. tóμoς (section) and Greek γράφειν (write) and refers to the production of crosssectional images. X-ray CT is a standard procedure of medical diagnostics. In a medical CT technique a fan-like X-ray beam rotates around the body to be examined. In this way, numerous transmission images from different directions are obtained, which are combined to form a cross-sectional image of the body. By slowly moving the body through the CT machine, three-dimensional representations of the inside of the body can be created. To increase the X-ray absorption of certain areas, solutions with a high atomic number are administered, so-called contrast agents like iodine-containing solutions. With the help of computed tomography on small objects (micro-CT, μCT), microstructures of food can be examined [28] and food processes can be investigated [29, 30]. In this chapter, we have become acquainted with ways to characterize food by image analysis. In addition to microscopy with electromagnetic waves and electron beams, there is acoustic spectroscopy (see Chap. 15) and nuclear magnetic resonance (see Chap. 10). A comprehensive comparison of the different analytical applications can be found in [31]. Further Reading Ultraviolet Solar water disinfection: effect of UV and temperature Fruits and vegetables: microbial inactivation by UV Corona-Virus: inactivation by UV-LEDs Tomato: microbial and color effects during UV-C treatment Lettuce: microbial and sensory effects during UV-C-treatment

[32, 33] [5, 9, 34] [3] [35] [36]

(continued)

References

Carrot juice: comparison of UV-C and thermal treatment UV-LED for inactivation of pathogens UV absorption of DNA, RNA, and proteins E. coli and L. plantarum: UV-C-sensitivity Tomato processing: inactivation of microorganisms by UV-C

519

[37] [4, 6, 7, 38] [39] [40] [41]

X-rays Identification of X-ray-irradiated hazelnuts by electron spin resonance (ESR) Density of foods by X-ray Imaging Lactose: Crystal structure by X-ray diffraction Coffee beans: Micro X-ray tomography (μCT) during roasting Frozen food: μCT on ice crystals Bread: three-dimensional quantitative by μCT Apple: pore space structure by μCT Meat: structure by μCT Foams: structure by μCT Milk: analysis of chemical elements by X-ray fluorescence Cacao: analysis of chemical elements by X-ray fluorescence Corn: kernel hardness and density by μCT X-ray imaging methods for internal quality evaluation of agricultural produce

[42] [27] [43–45] [29] [46] [47] [48] [30] [49] [50] [51] [52] [53]

Summary Ultraviolet radiation can be used for disinfection. In this chapter, basic concepts and the differences between atmospheric and technical UV are clarified in concise words. At the end of the section, application examples are listed from which the potential of future UV-LEDs can be seen. In the electromagnetic spectrum, the field of X-rays adjoins the short-wave UV. The basic features of the absorption of X-rays are presented and applications of X-ray scanners and imaging methods are described. At the end of the section, numerous application examples from the field of micro-CT are listed, which can be used for further studies and as suggestions for your own scientific work

References 1. Hinderer F (2020) UV/Vis-Absorptions- und Fluoreszenz- Spektroskopie. Springer, Heidelberg. https://doi.org/10.1007/978-3-658-25441-4

520

13

UV and X-rays

2. Fraden J (2016) Handbook of modern sensors physics, designs, and applications. https://doi. org/10.1007/978-3-319-19303-8 3. Gerchman Y, Mamane H, Friedman N, Mandelboim M (2020) UV-LED disinfection of Coronavirus: wavelength effect. J Photochem Photobiol B Biol 212:112044. https://doi.org/ 10.1016/j.jphotobiol.2020.112044 4. Koutchma T, Popović V, Green A (2019) Overview of ultraviolet (UV) LEDs technology for applications in food production. In: Koutchma T (ed) Ultraviolet LED technology for food applications. Academic, pp 1–23. https://doi.org/10.1016/B978-0-12-817794-5.00001-7 5. Baykuş G, Akgün MP, Unluturk S (2021) Effects of ultraviolet-light emitting diodes (UV-LEDs) on microbial inactivation and quality attributes of mixed beverage made from blend of carrot, carob, ginger, grape and lemon juice. Innovative Food Sci Emerg Technol 67: 102572. https://doi.org/10.1016/j.ifset.2020.102572 6. Souza VR, Koutchma T (2021) Repair mechanisms of UV-induced damage of microorganism in foods. In: Knoerzer K, Muthukumarappan K (eds) Innovative food processing technologies. Elsevier, Oxford, pp 385–397. https://doi.org/10.1016/B978-0-08-100596-5.22943-0 7. Koutchma T, Popović V (2019) UV light-emitting diodes (LEDs) and food safety. In: Koutchma T (ed) Ultraviolet LED technology for food applications. Academic, pp 91–117. https://doi.org/10.1016/B978-0-12-817794-5.00005-4 8. Popović V, Koutchma T, Pagan J (2021) Emerging applications of ultraviolet light-emitting diodes for foods and beverages. In: Knoerzer K, Muthukumarappan K (eds) Innovative food processing technologies. Elsevier, Oxford, pp 335–344. https://doi.org/10.1016/B978-0-08100596-5.22667-X 9. Barut Gök S, Gräf V, Stahl MR (2021) Engineering aspects of UV-C processing for liquid foods. In: Knoerzer K, Muthukumarappan K (eds) Innovative food processing technologies. Elsevier, Oxford, pp 171–181. https://doi.org/10.1016/B978-0-08-100596-5.23000-X 10. Xu R, Gong L, Wang H, Zhang G (2020) Disinfection effect of short-wave ultraviolet radiation (UV-C) on ASFV in water. J Infect 80(6):671. https://doi.org/10.1016/j.jinf.2020.02.021 11. Wang C-P, Chang C-S, Lin W-C (2020) Efficiency improvement of a flow-through water disinfection reactor using UV-C light emitting diodes. J Water Process Eng 40:101819. https:// doi.org/10.1016/j.jwpe.2020.101819 12. Zou X-Y, Lin Y-L, Xu B, Cao T-C, Tang Y-L, Pan Y, Gao Z-C, Gao N-Y (2019) Enhanced inactivation of E. coli by pulsed UV-LED irradiation during water disinfection. Sci Total Environ 650:210. https://doi.org/10.1016/j.scitotenv.2018.08.367 13. Wen G, Deng X, Wan Q, Xu X, Huang T (2019) Photoreactivation of fungal spores in water following UV disinfection and their control using UV-based advanced oxidation processes. Water Res 148:1. https://doi.org/10.1016/j.watres.2018.10.028 14. Li X, Cai M, Wang L, Niu F, Yang D, Zhang G (2019) Evaluation survey of microbial disinfection methods in UV-LED water treatment systems. Sci Total Environ 659:1415. https://doi.org/10.1016/j.scitotenv.2018.12.344 15. Ayoub GM, Malaeb L (2019) Solar water disinfection: UV radiation transmittance of various solar reactor tubes. Energy Procedia 157:498. https://doi.org/10.1016/j.egypro.2018.11.214 16. Nyangaresi PO, Qin Y, Chen G, Zhang B, Lu Y, Shen L (2018) Effects of single and combined UV-LEDs on inactivation and subsequent reactivation of E. coli in water disinfection. Water Res 147:331. https://doi.org/10.1016/j.watres.2018.10.014 17. Nocker A, Shah M, Dannenmann B, Schulze-Osthoff K, Wingender J, Probst AJ (2018) Assessment of UV-C-induced water disinfection by differential PCR-based quantification of bacterial DNA damage. J Microbiol Methods 149:89. https://doi.org/10.1016/j.mimet.2018. 03.007 18. Li G-Q, Wang W-L, Huo Z-Y, Lu Y, Hu H-Y (2017) Comparison of UV-LED and low pressure UV for water disinfection: photoreactivation and dark repair of Escherichia coli. Water Res 126: 134. https://doi.org/10.1016/j.watres.2017.09.030

References

521

19. Izmirlioglu G, Ouyang B, Demirci A (2020) Utilization of pulsed UV light for inactivation of Salmonella Enteritidis on shelled walnuts. LWT 134:110023. https://doi.org/10.1016/j.lwt. 2020.110023 20. Garvey M, Rowan NJ (2019) Pulsed UV as a potential surface sanitizer in food production processes to ensure consumer safety. Curr Opin Food Sci 26:65. https://doi.org/10.1016/j.cofs. 2019.03.003 21. Pedrós-Garrido S, Condón-Abanto S, Clemente I, Beltrán JA, Lyng JG, Bolton D, Brunton N, Whyte P (2018) Efficacy of ultraviolet light (UV-C) and pulsed light (PL) for the microbiological decontamination of raw salmon (Salmo salar) and food contact surface materials. Innovative Food Sci Emerg Technol 50:124. https://doi.org/10.1016/j.ifset.2018. 10.001 22. Keklik NM, Krishnamurthy K, Demirci A (2012) Microbial decontamination of food by ultraviolet (UV) and pulsed UV light. In: Demirci A, Ngadi MO (eds) Microbial decontamination in the food industry. Woodhead Publishing, pp 344–369. https://doi.org/10.1533/ 9780857095756.2.344 23. Mendonsa N, Almutairy B, Kallakunta VR, Sarabu S, Thipsay P, Bandari S, Repka MA (2020) Manufacturing strategies to develop amorphous solid dispersions: An overview. J Drug Deliv Sci Technol 55:101459. https://doi.org/10.1016/j.jddst.2019.101459 24. Thakral S, Terban MW, Thakral NK, Suryanarayanan R (2016) Recent advances in the characterization of amorphous pharmaceuticals by X-ray diffractometry. Adv Drug Deliv Rev 100:183. https://doi.org/10.1016/j.addr.2015.12.013 25. Figura LO, Epple M (1995) Anhydrous α-lactose A study with DSC and TXRD. J Therm Anal 44(1):45. https://doi.org/10.1007/BF02547132 26. Meschede D (2006) Gerthsen Physik. Springer, Berlin. https://doi.org/10.1007/978-3-66245977-5 27. Kelkar S, Boushey CJ, Okos M (2015) A method to determine the density of foods using X-ray imaging. J Food Eng 159:36. https://doi.org/10.1016/j.jfoodeng.2015.03.012 28. Schoeman L, Williams P, du Plessis A, Manley M (2016) X-ray micro-computed tomography (μCT) for non-destructive characterisation of food microstructure. Trends Food Sci Technol 47: 10. https://doi.org/10.1016/j.tifs.2015.10.016 29. Frisullo P, Barnaba M, Navarini L, Del Nobile MA (2012) Coffea arabica beans microstructural changes induced by roasting: an X-ray microtomographic investigation. J Food Eng 108(1):232. https://doi.org/10.1016/j.jfoodeng.2011.07.036 30. Frisullo P, Laverse J, Marino R, Nobile MAD (2009) X-ray computed tomography to study processed meat microstructure. J Food Eng 94(3–4):283. https://doi.org/10.1016/j.jfoodeng. 2009.03.020 31. Metilli L, Francis M, Povey M, Lazidis A, Marty-Terrade S, Ray J, Simone E (2020) Latest advances in imaging techniques for characterizing soft, multiphasic food materials. Adv Colloid Interf Sci 279:102154. https://doi.org/10.1016/j.cis.2020.102154 32. Castro-Alférez M, Polo-López MI, Marugán J, Fernández-Ibáñez P (2017) Mechanistic modeling of UV and mild-heat synergistic effect on solar water disinfection. Chem Eng J 316:111. https://doi.org/10.1016/j.cej.2017.01.026 33. Lawrie K, Mills A, Figueredo-Fernandez M, Gutierrez-Alfaro S, Manzano M, Saladin M (2015) UV dosimetry for solar water disinfection (SODIS) carried out in different plastic bottles and bags. Sens Actuators B Chem 208:608. https://doi.org/10.1016/j.snb.2014.11.031 34. Pala ÇU, Toklucu AK (2012) Effects of UV-C light processing on some quality characteristics of grape juices. Food Bioproc Tech 6(3):719. https://doi.org/10.1007/s11947-012-0808-7 35. Song H-J, Choi D-W, Song KB (2011) Effect of aqueous chlorine dioxide and UV-C treatment on the microbial reduction and color of cherry tomatoes. Hortic Environ Biotechnol 52(5):488. https://doi.org/10.1007/s13580-011-0043-6 36. Lee JH, Chun HH, Oh DH, Park J, Won M, Song KB (2012) Sensory and microbiological qualities of romaine lettuce and kale affected by a combined treatment of aqueous chlorine dioxide and ultraviolet-C. Hortic Environ Biotechnol 53(5):387. https://doi.org/10.1007/ s13580-012-0013-7

522

13

UV and X-rays

37. Riganakos KA, Karabagias IK, Gertzou I, Stahl M (2017) Comparison of UV-C and thermal treatments for the preservation of carrot juice. Innovative Food Sci Emerg Technol 42:165. https://doi.org/10.1016/j.ifset.2017.06.015 38. D’Souza C, Yuk H-G, Khoo GH, Zhou W (2017) Light-emitting diodes in postharvest quality preservation and microbiological food safety. In: Dutta Gupta S (ed) Light emitting diodes for agriculture: smart lighting. Springer Singapore, Singapore, pp 191–235. https://doi.org/10. 1007/978-981-10-5807-3_9 39. Lavilla M, Lasagabaster A, Martínez-de-Marañón I (2019) Impact of ultraviolet processing on food composition. In: Roohinejad S, Koubaa M, Greiner R, Mallikarjunan K (eds) Effect of emerging processing methods on the food quality: advantages and challenges. Springer International Publishing, Cham, pp 173–196. https://doi.org/10.1007/978-3-030-18191-8_8 40. Müller A, Pietsch VL, Schneele JS, Stahl MR, Greiner R, Posten C (2015) Effect of temperature and pH value on the UV-C sensitivity of Escherichia coli and Lactobacillus plantarum. LWT Food Sci Technol 64(2):699. https://doi.org/10.1016/j.lwt.2015.06.013 41. Abadias M, Colás-Medà P, Viñas I, Bobo G, Aguiló-Aguayo I (2021) Application of an innovative water-assisted ultraviolet C light technology for the inactivation of microorganisms in tomato processing industries. Food Microbiol 94:103631. https://doi.org/10.1016/j.fm.2020. 103631 42. Chiappinelli A, Mangiacotti M, Tomaiuolo M, Trotta G, Marchesani G, Chiaravalle AE (2019) Identification of X-ray-irradiated hazelnuts by electron spin resonance (ESR) spectroscopy. Eur Food Res Technol 245(10):2323. https://doi.org/10.1007/s00217-019-03349-2 43. Fan F, Roos YH (2015) X-ray diffraction analysis of lactose crystallization in freeze-dried lactose-whey protein systems. Food Res Int 67:1. https://doi.org/10.1016/j.foodres.2014.10.023 44. Fan F, Roos YH (2015) X-ray diffraction analysis of lactose crystallization in freeze-dried lactose–whey protein systems. Food Res Int 67:1. https://doi.org/10.1016/j.foodres.2014. 10.023 45. Wijayasinghe R, Bogahawaththa D, Chandrapala J, Vasiljevic T (2020) Crystallization behavior and crystal properties of lactose as affected by lactic, citric, or phosphoric acid. J Dairy Sci 103(12):11050. https://doi.org/10.3168/jds.2020-18375 46. Mousavi R, Miri T, Cox PW, Fryer PJ (2005) A novel technique for ice crystal visualization in frozen solids using X-ray micro-computed tomography. J Food Sci 70 47. Falcone PM, Baiano A, Zanini F, Mancini L, Tromba G, Dreossi D, Montanari F, Scuor N, Nobile MAD (2005) Three-dimensional quantitative analysis of bread crumb by X-ray microtomography. J Food Sci 70(4):E265. https://doi.org/10.1111/j.1365-2621.2005.tb07182.x 48. Mendoza F, Verboven P, Mebatsion HK, Kerckhofs G, Wevers M, Nicolai B (2007) Threedimensional pore space quantification of apple tissue using X-ray computed microtomography. Planta 226(3):559. https://doi.org/10.1007/s00425-007-0504-4 49. Lim KS, Barigou M (2004) X-ray micro-computed tomography of cellular food products. Food Res Int 37(10):1001. https://doi.org/10.1016/j.foodres.2004.06.010 50. Pashkova GV, Smagunova AN, Finkelshtein AL (2018) X-ray fluorescence analysis of milk and dairy products: a review. TrAC Trends Anal Chem 106:183. https://doi.org/10.1016/j.trac. 2018.06.014 51. Oliveira LB, dos Santos WPC, Teixeira LSG, Korn MGA (2020) Direct analysis of cocoa powder, chocolate powder, and powdered chocolate drink for multi-element determination by energy dispersive X-ray fluorescence spectrometry. Food Anal Method 13(1):195. https://doi. org/10.1007/s12161-019-01565-w 52. Guelpa A, du Plessis A, Kidd M, Manley M (2015) Non-destructive estimation of maize (Zea mays L.) Kernel hardness by means of an X-ray micro-computed tomography (μCT) density calibration. Food Bioprocess Tech 8(7):1419. https://doi.org/10.1007/s11947-015-1502-3 53. Kotwaliwale N, Singh K, Kalne A, Jha SN, Seth N, Kar A (2014) X-ray imaging methods for internal quality evaluation of agricultural produce. J Food Sci Technol 51(1):1. https://doi.org/ 10.1007/s13197-011-0485-y

Radioactivity

14

Nuclides (also: atomic nuclei) are characterized by their number of protons (atomic number) and neutrons. Numerous proton–neutron combinations cause nuclides to be unstable and convert into other nuclides by emitting characteristic radiation. This property of spontaneous conversion is called radioactivity. Of the chemical elements with numbers from 1 to 112, more than a thousand different nuclides are known, the vast majority is unstable, i.e. radioactive. The nuclides with an atomic number above 83 are consistently unstable. Definition A nuclide is an atomic nucleus with a specific number of protons and neutrons. Unstable nuclides are called radionuclides. Unstable nuclides found in nature lead to the natural radioactivity of materials. In 40 the field of biological materials, 14 6 C and 19 K belong to the naturally occurring radionuclides. Therefore, foods made from plant or animal raw materials have a measurable natural radioactivity. In addition, technical radionuclides such as 137 55 Cs can occur in food as unwanted contamination. Example The designation 14 6 C identifies the nuclide with atomic number 6 and atomic mass number 14. The nuclide contains 6 protons (atomic number) and 8 neutrons (6 + 8 = 14).

# The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. O. Figura, A. A. Teixeira, Food Physics, https://doi.org/10.1007/978-3-031-27398-8_14

523

524

14

Radioactivity

Definition 13 Isotopes are nuclides with the same atomic number, e.g. 12 6 C and 6 C. Since both nuclides have the same electron configuration, they both behave like the same chemical element, in this example here carbon. From Greek: ισo, τóπoς, same place. In order to be able to estimate the natural radiation exposure emanating from food, we start this chapter with a look at different types of radioactive decay.

14.1

Types of Radiation

The emission of particles consisting of 2 neutrons and 2 protons is called α-radiation. These so-called α-particles ejected during decay are positively charged particles with comparatively low speed and reach. Emission of electrons and positrons is referred to as β-radiation. β-particles have a speed in the order of magnitude of the speed of light and correspondingly a long reach. γ-radiation consists of quanta of high energy, it is electromagnetic radiation with frequencies above X-rays. In addition, there is the emission of neutrons from the atomic nucleus, the so-called neutron radiation (Table 14.1). Radioactive substances that predominantly emit α-radiation are called αradiators, analogously one speaks of β-radiator and γ-radiator. Since incoming rays produce secondary particles as they pass through the material, radioactive decay often results in a mixture of different types of radiation. In addition to the types of radiation shown in Table 14.1, in connection with artificial radioactivity emission of protons, positrons and neutrinos can occur. The reach of radiation depends on interaction with the material in question. Since α-particles are highly ionizing to their immediate environment, they have only a very short reach. β-particles can form secondary ions at a greater distance, depending on Table 14.1 Types of radioactive radiation Designation α-radiation β-radiation

Description Emission of slow He nuclides Emission of fast electrons or positrons

γ-radiation

Emission of high-energy γ-quanta

Neutron radiation

Emission of fast or slow neutrons

Electric charge 2+ 1or 1+ 0

0

Example α (Radium), 4.9 MeV Reach in air: some centimeters β 40 19 K 1.4 MeV Reach in air: some meters Wavelength < 10 pm, few keV to few MeV per quant Reach in air: some hundreds m

14.1

Types of Radiation

525

their energy or velocity. γ-quants have a speed close to that of light or X-rays and, depending on their energy, have an extremely long reach. Reach of the different types of radiation thus depends on the absorption ability of the environment for the respective type of radiation. Reaches of respective types of radiation in air, water, metal, or biological tissue are completely different. For characterization of γ-radiation or related radiation protection materials, the so-called penetration depth of radiation can be specified as the length after which its intensity has fallen to 1/e = 36.7%. Or the length after which the intensity has fallen to 50% (so-called half-value depth).

14.1.1 Activity and Decay Radioactive transformation of a nucleus occurs spontaneously, i.e. without any external influence. Therefore, the temporal prediction of decay in nuclei cannot be carried out for individual cases, but only with statistical methods for a large number of atomic nuclei. If we look on a large number of nuclei, we see that the decay rate is proportional to the number of existing nuclei. Gradually, this number of remaining nuclei decreases and so does the rate of decay and intensity of radioactive radiation. Since radioactive radiation is produced during each individual decay, substances radiate more strongly when many nuclei decay per second. A high rate of activity is expressed by specifying a high decay constant λ. Substances with a low decay constant, on the other hand, decay very slowly, therefore show only a comparatively low level of radioactive radiation and it takes a very long time for their activity to become immeasurably small. Activity has the unit Becquerel, 1 Bq = 1 s-1. An activity of 103 Bq means that 1000 decays take place per second. With the assumption that the decay rate dN dt is proportional to the number of existing nuclei N, the decay law is -

dN =λ
N dt

ð14:1Þ

with the decay constant λ=

- dN dt N

ð14:2Þ

The activity is thus A= If we integrate the decays over time,

dN =N
λ dt

ð14:3Þ

526

14 N

Radioactivity

t

dN =λ dt

dt

ð14:4Þ

t=0

N0

i.e., - ðln N - ln N 0 Þ = λ
ðt- 0Þ

ð14:5Þ

or ln

N = -λ
t N0

ð14:6Þ

we get an exponential relationship: N = N 0
e - λ
t N N0 A λ T1/2 m M n NA

ð14:7Þ

number of atomic nuclei initial number of atomic nuclei activity in Bq decay constant in s-1 half-life in s, d, y mass in kg molar mass in kg∙mol-1 amount of substance in mol Avogadro’s constant NA = 6.02214076 ∙ 1023 mol-1

The half-life T is the time after which half of all originally existing nuclei of the material under consideration have decayed. This is also the time after which the activity has decreased to half the value of the initial activity. It results from: N=

N0 2

ð14:8Þ

to - T 1=2
λ = ln

N 1 = ln N0 2

ð14:9Þ

that is T 1=2 =

ln 2 λ

Table 14.2 shows some examples of half-lives.

ð14:10Þ

14.1

Types of Radiation

Table 14.2 Examples of decay constants λ and halflives T1/2 of nuclei [1]

527

Nucleus 40 19 K 14 6 C 226 88 Ra 137 55 Cs 90 38 Sr 131 53 I

T1/2 1.25 ∙ 109 a

λ/Bq 1.8 ∙ 10-17

5730 a

3.8 ∙ 10-12

1580 a

1.4 ∙ 10-11

30 a

7.3 ∙ 10-10

28.1 a

7.8 ∙ 10–10

8.05 d

1 ∙ 10-6

Substances with high activity, i.e. having a high decay constant λ, therefore have a short half-life. Conversely, substances with low activity have a long half-life. A quantity derived from the activity is the so-called specific activity. It is the ratio of the activity A of a substance and its mass m. In the case of an isotope-pure sample of the mass m with N radioactive nuclei, the specific activity can be calculated as follows: a=

NA A λ
N λ
n
NA =λ
= = n
M M m m

ð14:11Þ

The specific activity has the SI unit Bq
kg-1. If materials are mixtures of radioactive and non-radioactive isotopes, the quotient of activity and total mass of the mixture is referred to as specific activity. Example 13 Bq
kg-1 .This The specific activity of the nuclide radium 226 88 Ra is 3.7 ∙ 10 10 means that 1 g of this substance has an activity of 3.7 ∙ 10 Bq. The former unit of activity Curie (Ci) corresponds exactly to this value: 1 g of the nuclide 10 226 88 Ra has the activity 1 Ci = 3.7 ∙ 10 Bq Radioactive conversions are associated with a high energy release, which often is not seen because it occurs over long periods of time (Table 14.2). Example The decay of 1 g radium, including all radioactive derivatives, leads to a heat dissipation to the environment of 0.2 W. Over the course of a half-life of 1580 years, this corresponds to a heat amount of Q = 0:2 W
3600 s
24
365
1580 = 107 kJ: For comparison: The combustion of 1 g of coal provides 32 kJ, the calorific value of 1 g of sucrose is 17 kJ (Chap. 8).

528

14

Radioactivity

14.1.2 Measurement of Ionizing Radiation (a-, b-, g-) The detection and quantitative determination of radioactive radiation is carried out by recording the effects generated by the radiation on ionization detectors which determine the number of ions generated. To distinguish between α-, β-, and γ-radiation, the radiation can be filtered before entering an ionization detector. For this purpose, electric and magnetic fields are used. The Coulomb force of an electric field causes a deflection of electrically charged α-particles and β-particles, but does not affect γ-quanta (Fig. 14.1). In the electric field, electrons (β-) and positrons (β+) can also be separated. The Lorentz force of a magnetic field supports the differentiation according to charge number, sign of charge, and particle speed (Fig. 14.2). Another possibility is selective absorption with suitable materials. While αradiation can be shielded by a sheet of cardboard, a rough distinction between βand γ-radiation can be made with metal plates of suitable thickness, depending on the energy of the radiation. As a result of the interaction of γ-rays with matter, the photo effect or the Compton effect occurs. Both are associated with the release of an electron from the atomic shell of the absorbent material. These so-called secondary ions can be used to detect γ-radiation. Another effect is the so-called pair formation, where an electron–positron pair is formed when a γ-quant hits a material. The reverse effect is called pair annihilation, here an electron–positron pair recombines to a γ-quant. Definition Wave-particle dualism: γ-radiation is an electromagnetic wave with typical frequency and wavelength. The radiation occurs quantized, a quantum has the energy E = h
f. These γ-quants behave like high-energy particles.

Fig. 14.1 Separation of radioactive radiation in the electric field

J ,n

E



E

S

14.1

Types of Radiation

529

Fig. 14.2 Deflection of radioactive radiation in the magnetic field perpendicular to the direction of movement, schematic. Low-weight electrons β--and positrons β-+ are more strongly deflected than heavier α-particles. γ-quants and neutrons (n) are not affected

14.1.2.1 Counter Tubes In a mixture of noble gases such as Helium, Neon, and Argon, ions are produced by inciting radioactive radiation. In the strong electric field of the counter tube these primary ions generate further ions (secondary ions) which cause a short-term measurable electric pulse via the photoelectric effect. If the electrical field in the counter tube is not too high, the number of detected pulses is proportional to the number of primary ions. A counter tube operated in this way is called a proportional counter tube. A well-known design is the Geiger-Müller counter tube. 14.1.2.2 Semiconductor Detectors In semiconductor detectors, the ions generated by radioactive radiation lead to countable voltage pulses. For example, ultrapure Germanium or pn-junctions in Lithium-doped Germanium are used for γ-radiation and Li-doped silicon detectors for X-rays. With semiconductor detectors it is possible to determine the energy of the γ-quants. This makes it possible not only to detect nuclear decays and to count them across the board, but also to get an energy spectrum of the radiation. Figure 14.3 shows such a spectrum schematically. With the help of such spectra, conclusions can be drawn about the type and source of the radiation. If radioactive contamination of food is found, the clarification of the type and origin of the radionuclides is an essential question. In scintillation detectors, the radiation generates a light pulse. The light emission is amplified and counted. As scintillators, crystals are used (e.g., anthracene for β-radiation, ZnS/Ag for

530

14

Radioactivity

Fig. 14.3 Energy spectrum of radioactive radiation: Intensity of the absorbed γquants with respect to their energy E

I

E Table 14.3 Isotopes of potassium and their halflives T1/2 [2]

Isotope 39 19 K 40 19 K 41 19 K 42 19 K 43 19 K

Natural prevalence in % 93.2581

T1/2 Stable

0.0117

1.25 ∙ 109 a

6.7302

Stable

0

12.36 h

0

22.3 h

α-radiation, NaI/Tl for γ-radiation). Lithium containing glasses are used as scintillator material for slow neutrons. Liquid scintillators can be mixed with the sample and measured in cuvettes. Because of low absorption losses between sample and scintillator, this is advantageous for low-energy radiation (e.g., 14C).

14.1.3 Natural Radioactivity 41 Natural potassium consists mainly of the stable isotopes 39 19 K and 19 K and to a very 40 42 43 small extent of the unstable, i.e. radioactive, isotopes 19 K, 19 K, 19 K. In Table 14.3, the natural prevalence of K- isotopes is listed. Because of the short half-life of 42 19 K and 43 K these isotopes have already completely decayed on the planet. Since they are 19 hardly newly formed, their occurrence is given as 0% in Table 14.3. The metabolism of humans needs Potassium. The Potassium requirement can be completely covered by food. The radioactive natural isotope 40K enters our body by intake of Potassium. Due to a balance between the intake of such natural radionuclides from food or air and their biological excretion, the value of natural radioactivity in the human body is largely constant, also in animals, plants, and foods.

14.1

Types of Radiation

531

Example Specific activity of Potassium: Natural Potassium consists of 0.01117% radioactive isotope 40K with a half-life of 1.28 billion years. The number N of atoms in a 1 g potassium (molecular weight M = 39.098 g mol-1) is N = n
NA =

1g
6:022
1023 atoms mol - 1 = 1:54 39:098 g mol - 1


1022 atoms Of these, 0.0117% are 40K isotopes, i.e. Nð40 KÞ = 0:000117
1:54
1022 = 1:8
1018 The decay constant of 40K is: λ=

ln 2 ln 2 = = 1:72
10 - 17 s - 1 T 1=2 1:277
109 a

The activity of one gram thus results in: A=λ
N A = 1:72
10 - 17 s - 1
1:80
1018 = 31:0 s - 1 The specific activity of Potassium according to this approximation a = 31 Bq
g-1 is essentially caused by the isotope 40K. This estimation neglected the fact that only about 90% of the 40K isotopes emit β-radiation and pass into 40Ca. About 10% of the 40K isotopes are converted to 40Ar by β+ -emission and electron capture. The Potassium content of the human body depends on age, gender, and diet. It is about 2 g per kg body weight. A more precise calculation is possible by means of the following empirical formulas:

Men Women

Potassium content c in g per kg body weight c = 2.38658 - (0.00893
age) c = 1.9383 - (0.00675
age)

Example A 30-year-old woman has a Potassium content of c = 1.9383 - (0.00675
30) = 1.7g Potassium per kg body weight.

532 Table 14.4 Activity of natural radionuclides in humans [4]

Table 14.5 specific 40K activity a of some materials, examples [5, 6]

14

Nuclide K 14 C 87 Rb 210 Pb 210 Bi 210 Po 3 H 7 Be Rn decay products 40

Material Human body Milk Milk powder Fruits, vegetables Fruit juice concentrate Hazelnuts Honey Instant coffee powder Potash fertilizer

Radioactivity

Activity in Bq 4000 3800 650 > imaginary part (e.g., when real part is nearly zero) Imaginary part = - (real part) See φ = 0°

Appendix A

603

A.3.1 Complex Physical Quantities Physical quantities can also be complex quantities which consist of a real part and an imaginary part. Such a physical quantity is called an apparent quantity, which consists of an effective component (real part) and a loss component (imaginary part). Figure A.5 illustrates a complex physical quantity with the help of these terms. Remark If we calculate "real" (i.e., without using imaginary numbers), we implicitly assume that the imaginary part of the complex quantity is zero and thus the apparent quantity is identical to the real quantity. To help us work with some of the complex properties used in this book, some examples are presented here for the purpose of illustration. We should also recognize that different scientific disciplines or fields of engineering may use their own special terms for the real and imaginary parts, but the mathematics are the same.

A.3.2 Electrical Properties In electric circuits with alternating current, we have real resistances and imaginary resistances, called active resistance and reactive resistance. The apparent quantity— the apparent resistance—is called the impedance, as shown in Table A.17 and A.18. resistance in Ω admittance in S capacitance in F inductance in H angular frequency in s-1

Z, R Y, G C L ω

Fig. A.5 Trigonomical representation of a physical quantity

y titi an u q nt re a p ap

imaginary axis

M

real axis

loss component

effective component

604

Appendix A

Table A.17 Terms of complex electric resistance Impedance Z

= =

Effective resistance R

+ +

Reactive resistance i
ωL -

1 ωC

Table A.18 Terms of complex electric conductance Admittance Y

= =

Admittance G

+ +

Susceptance i
ωC -

1 ωL

There are cases, in which the reactive resistance is zero, e.g. if the capacitance is zero or the angular frequency is ω = 0. The latter is the case in direct current circuits. We see that all observed resistances are also active resistances, such as ohmic resistances, and we do not have to deal with complex quantities. On the other hand in alternating fields, impedance and resistance is not the same, and we have to take into account both real and imaginary parts of the complex quantities. In Chap. 11 we learned about the complex permittivity.

A.3.3 Rheology By means of mechanic oscillation, it is possible to differentiate between elastic and non-elastic (viscous) properties of a material. The observed behavior can be described as a complex quantity (e.g., complex modulus), which includes both properties, elastic and viscous. Then the elastic properties are represented by the real part and the inelastic (viscous) properties by the imaginary part. In this sense the non-elastic, viscous behavior of the sample is understood as the loss part. A behavior like this can be easily described by the loss angle φ or by the loss tangent, tanφ. It is important to avoid confusion in this regard. For describing rheological behavior, it is also possible to use the apparent viscosity as a complex property which consists of real (viscous) parts and imaginary (elastic) parts. Also sometimes it is useful to use the complex compliance instead of the complex elasticity. In Chap. 4 we learned about complex compliance and complex viscosity. When further study of complex rheological properties is needed, it may be helpful to remember to refer to this appendix section once again.

A.3.4 Heat Flow Calorimetry Oscillation tests in Thermal Analysis are referred to as temperature-modulated tests (MDSC). These tests result in a complex heat flow which is composed of a real part and an imaginary part. The real part is caused by the heat capacity of the sample; it is a reversible heat flow. The imaginary part is a non-reversible heat flow, in other words the loss component. It may be caused by chemical reactions in the sample. In

Appendix A

605

an analogous way a complex heat capacity can be defined. In Chap. 8 we learned about complex heat flow and complex heat capacity.

A.4

Greek Letters (Fig. A.6)

To exercise the notation of Greek letters Fig. A.7 will be helpful: Fig. A.6 Greek block letters and their names

A α B β Г γ Δ δ E ε Z ζ H η Θ T,I ι K κ Λ λ M μ N ν Ξ ξ O ο Π π P ρ Σ σ T τ Y υ Ф φ X χ Ψ ψ Ώ ω

a b g d e z e th j k l m n x o p r s t y ph ch ps o

Alpha Beta Gamma Delta Epsilon Zeta Eta Theta Jota Kappa Lambda Mu Nu Xi Omikron Pi Rho Sigma Tau Ypsilon Phi Chi Psi Omega

606

Appendix A

Fig. A.7 Greek script

A.5

Conversion Chart: Temperature (Table A.19)

Table A.19 conversion of temperatures Given quantity in:

Required quantity in: °C °C 1 °F °R °K

ϑ/°C=5/9
(ϑ/ °F-32) ϑ/°C=5/9
(ϑ/ °R-491.67) ϑ/°C=T/ K-273.15

°F ϑ/°F=9/5
ϑ/ °C+32 1 ϑ/°F= ϑ/° R-459.67 ϑ/°F=9/5
T/ K-459.67

°R ϑ/°R=9/5
ϑ/°C +491.67 ϑ/°R=ϑ/°F +459.67 1 ϑ/°R=9/5
(T/ K-273.15) +491.67

°R degree Rankine, °C degree Celsius, °F degree Fahrenheit, K Kelvin

K T/K=ϑ/°C +273.15 T/K=(5/9
(ϑ/°F32))+273.15 T/K=5/9
(ϑ/° R-491.67)+273.15 1

Appendix A

607

A.6 Sugar Conversion Chart: Concentration, Density, Refraction (Table A.20)

Table A.20 conversion table for sugar content, i.e. solid matter (degree Brix, degree Oechsle, degree Baumé, Klosterneuburg degrees) Sucrose concentration °Bx 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

Relative density d20/20 1.00389 1.00779 1.01172 1.01567 1.01965 1.02366 1.02770 1.03176 1.03586 1.03998 1.04413 1.04831 1.05252 1.05677 1.06104 1.06534 1.06968 1.07404 1.07844 1.08287 1.08733 1.09183 1.09636 1.10092 1.10551 1.11014 1.11480 1.11949 1.12422 1.12898 1.13378 1.13861 1.14347 1.14837 1.15331

Degree Oechsle °Oe 4 8 12 16 20 24 28 32 36 40 44 48 53 57 61 65 70 74 78 83 87 92 96 101 106 110 115 119 124 129 134 139 143 148 153

Degree Baumé °Be 0.56 1.12 1.68 2.24 2.79 3.35 3.91 4.46 5.02 5.57 6.13 6.68 7.24 7.79 8.34 8.89 9.45 10.0 10.55 11.10 11.65 12.20 12.74 13.29 13.84 14.39 14.93 15.48 16.02 16.57 17.11 17.65 18.19 18.73 19.28

Klosterneuburg degrees 1.3 2.0 2.8 3.6 4.4 5.2 5.9 6.9 7.7 8.5 9.3 10.1 11.0 11.9 12.7 14.4 14.3 15.2 16.0 16.9 17.7 18.6 19.5 20.3 20.9 21.9 22.8 23.6 24.4 25.3 26.1 26.8 27.8 28.7 (continued)

608

Appendix A

Table A.20 (continued) Sucrose concentration °Bx 36 37 38 39 40

A.7

Relative density d20/20 1.15828 1.16329 1.16833 1.17341 1.17853

Degree Oechsle °Oe 158 163 168 173 179

Degree Baumé °Be 19.81 20.35 20.89 21.43 21.97

Klosterneuburg degrees -

Material Data

Here are some helpful material data. For further physical data of foods, see [3–10] (Table A.21). Table A.21 Properties of water: Density ρ, dynamic viscosity η, refraction index at 589 nm nD, permittivity ε, saturation steam pressure pv, enthalpy of evaporation Δhvap, and surface tension σ at temperature ϑ ϑ °C 0 5 10 15 20 25 30 40 50 60 70 80 90 100

ρ kg m-3 999.9 1000.0 999.7 999.1 998.2 997.1 995.7 992.2 988.1 983.2 977.8 971.8 965.3 958.4

η mPa s 1.787 1.519 1.307 1.139 1.002 0.890 0.798 0.653 0.547 0.466 0.404 0.355 0.315 0.282

nD 1.3346 1.3346 1.3343 1.3338 1.3333 1.3329 1.3323 1.3309 1.3293 1.3275 1.3255 1.3231 1.3209 1.3182

ε 87.9 85.9 84.0 82.1 80.2 78.4 76.6 73.2 69.9 66.8 63.8 60.9 58.2 55.6

pv kPa 0.610 0.872 1.228 1.705 2.34 3.17 4.24 7.38 12.33 19.92 31.16 47.34 70.1 101.3

σ mN m-1 75.6 74.8 74.1 73.3 72.6 71.8 71.0 69.4 67.7 66.0 64.3 62.5 60.7 58.9

Appendix A

609

A.7.1 Density (Table A.22) Table A.22 Polynomic equations to calculate the density [5] Water

ρ = 1000.22 + 1.0205 × 10-2ϑ - 5.8149 × 10in kg.m-3 ϑ + 1.496 × 10-5ϑ3 ρ = ρWasser + 4.039
dm + 1.273 × 10-2dm2 + 9.62 × 10-5dm3 in kg.m-3 ρ = 1040.7 - 0.2665
ϑ - 2.3 × 10-3ϑ2 in kg.m-3 -3 2 -1 in kg.m-3 ρ = 2.3 × 10 ϑ - 2.665 × 10 ϑ + 1040.7 - xf(-4.81
-5 2 -3 10 ϑ + 9.76 × 10 ϑ + 1.011) dry matter in % (m/m) density in kg.m-3 temperature in °C fat content in % (m/m) 3 2

Whey UHT milk Milk and cream dm ρ ϑ xf

Viscosity of water [11] 484:3726

η=mPa
s = 0:0318
eϑþ120:2202 Interfacial Data (Tables A.23, A.24, A.25, A.26, A.27, A.28, A.29, and A.30) Table A.23 Surface tension data of milk [5] Temperature in °C 0 10 20 30 40 50 60 70 80 90 100

Surface tension σ / N m-1 Skim milk 0.0557 0.0536 0.0515 0.0497 0.0479 0.0462 0.0446 0.0432 0.0418 0.0406 0.0395

Whole milk 0.0515 0.0507 0.0500 0.0493 0.0488 0.0483 0.0478 0.0474 0.0471 0.0469 0.0467

610

Appendix A

Table A.24 Surface tension σ of food at 20°C, examples from [5] σ / N m-1 0.0492 0.0566 0.060 0.069 0.0513 0.0489 0.0630 0.0485 0.0318

Material Cream (30 %) Whey (TS 5 %) Egg white Skim milk permeate(0.1 % protein) Skim milk retentate(5 % protein) Butter milk (bm) bm permeate bm retentate Butter fat 99.9 %, 30 °C Table A.25 Heat conductivity of frozen food, examples from [12] Product Strawberry Fruit juices Mashed potatoes Cucumbers Salmon Filet of codfish Beef Fat pork Low fat pork Bacon

λ/W K-1 m-1 Fresh 0.49 0.56 0.49 0.54 0.50 0.54 0.48 0.37 0.50 0.19

Frozen 1.12 2.10 1.10 1.26 1.17 1.20 1.40 0.72 1.55 0.27

Table A.26 Heat conductivity of food, examples from [12] Food Milk Mashed banana Butter (17% H2O) Codfish Codfish Codfish Vegetable Grain (loose) Potatoes Margarine Fruit Beef (78.5 % H2O) Beef (78.5 % H2O) Beef (78.5 % H2O) Beef (78.5 % H2O) Beef fat (7% H2O)

λ/W K-1 m-1 0.55 0.56 0.2 0.44 1.22 1.37 0.3 × 0.6 0.15 0.55 0.2 0.35 × 0.55 0.48 1.06 1.35 1.57 0.2

ϑ /°C 20 16 20 0 -10 -20 20 20 20 20 20 0 -5 -10 -20 0 (continued)

Appendix A

611

Table A.26 (continued) Food Beef fat (7% H2O) Beef fat (7% H2O) Beef fat (7% H2O) Pork (76.8 % H2O) Pork (76.8 % H2O) Pork (76.8 % H2O) Pork (76.8 % H2O) Pork fat Pork fat Starch (loose) Sodium chloride (solid) Sugar (solid) Sugar (loose) Ricinus oil Water Air

λ/W K-1 m-1 0.21 0.23 0.26 0.2 0.8 0.99 1.29 0.19 0.29 0.15 7 0.6 0.15 . . . 0.35 0.181 0.596 0.026

ϑ /°C -5 -10 -20 0 -5 -10 -20 0 -20 20 20 20 20 20 20 20

Table A.27 Thermal diffusivity of food at 20°C, examples a 106/m2 s-1 0.100 0.134 0.086 0.09 0.147 0.15 0.14 0.095 0.156 0.092 0.150 0.115 3.2 0.2

Food Ethanol Beer Butter Fat Fish Low fat meat Vegetable Grain (loose) Potatoes Margarine Oranges Olive oil Mineral salt Sugar (loose)

Table A.28 Heat conductivity λ and thermal diffusivity a of some materials at ambient temperature Material Silver Copper Aluminum Glass Special steel Bronze

λ/W K-1 m-1 407 384 220 0.1 × 1.0 8 × 16 62

a × 106/m2 s-1 176 107 94.6

18.6 (continued)

612

Appendix A

Table A.28 (continued) Material Iron Glass Quartz glass Graphite Constantan Brass Steel (St37) V2A-steel Titanium PS PVC EPS (styrofoam) Plexiglass PA PE Concrete Porous concrete Earth Window glass Gypsum Chalky sandstone Marble Porcelain Sandstone Chamotte Plaster Brick Brick wall Asphalt Roofing paper Oak/beech Fir Gum Cork Granulated cork Mineral faber Plant faber Asbestos sheet Slate

λ/W K-1 m-1 74 1 1.36 169 23 80 × 220 45 15 22 0.15 0.16 0.036 0.19 0.26 0.4 1.3 0.3 0.4 × 1.3 0.8 0.5 1.1 2.8 0.9 1.9 0.5 × 1.2 0.8 0.5 0.8 0.7 0.2 0.17 × 0.31 0.14 × 0.26 0.13 - 0.24 0.05 0.036 0.04 0.05 0.7 2.1

a × 106/m2 s-1 16.2 0.61

14.7

0.66

0.6 0.47

1.0 × 1.3 0.33 × 0.69 0.27 0.55

0.11 0.1

Appendix A

613

Table A.29 IR emissivity ε of some materials [13] Material Ideal black body Aluminum, raw Aluminum, polished Aluminum, oxidized Aluminum, anodized Brick Chromium polished Copper, oxidized Copper, polished Leaves, green Ice Silicon rubber Human skin Snow Water White paper Wood

ε 1 0.06 – 0.07 0.05 0.11 0.7 0.9 0.1 0.6 – 0.7 0.02 0.88 0.96 0.94 0.93 – 0.96 0.85 0.96 0.92 0.93

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0 1.33299 1.33442 1.33587 1.33732 1.33879 1.34026 1.34175 1.34325 1.34477 1.34629 1.34783 1.34937 1.35093 1.35250 1.35409 1.35568 1.35729 1.35891 1.36054 1.36219 1.36384 1.36551 1.36720 1.36889 1.37060

0.1 1.33313 1.33456 1.33601 1.33747 1.33893 1.34041 1.34190 1.34341 1.34492 1.34644 1.34798 1.34953 1.35109 1.35266 1.35424 1.35584 1.35745 1.35907 1.36070 1.36235 1.36401 1.36568 1.36737 1.36907 1.37078

0.2 1.33327 1.33471 1.33616 1.33761 1.33908 1.34056 1.34205 1.34356 1.34507 1.34660 1.34813 1.34968 1.35124 1.35282 1.35440 1.35600 1.35761 1.35923 1.36087 1.36252 1.36418 1.36585 1.36754 1.36924 1.37095

0.3 1.33342 1.33485 1.33630 1.33776 1.33923 1.34071 1.34220 1.34371 1.34522 1.34675 1.34829 1.34984 1.35140 1.35298 1.35456 1.35616 1.35777 1.35940 1.36103 1.36268 1.36434 1.36602 1.36771 1.36941 1.37112

0.4 1.33356 1.33500 1.33645 1.33791 1.33938 1.34086 1.34235 1.34386 1.34537 1.34690 1.34844 1.34999 1.35156 1.35313 1.35472 1.35632 1.35793 1.35956 1.36120 1.36285 1.36451 1.36619 1.36788 1.36958 1.37129

0.5 1.33370 1.33514 1.33659 1.33805 1.33952 1.34101 1.34250 1.34401 1.34553 1.34706 1.34860 1.35051 1.35172 1.35329 1.35488 1.35648 1.35810 1.35972 1.36136 1.36301 1.36468 1.36635 1.36804 1.36975 1.37147

0.6 1.33385 1.33529 1.33674 1.33820 1.33967 1.34116 1.34265 1.34416 1.34568 1.34721 1.34875 1.35031 1.35187 1.35345 1.35504 1.35664 1.35826 1.35989 1.36153 1.36318 1.36484 1.36652 1.36821 1.36992 1.37164

0.7 1.33399 1.33543 1.33688 1.33835 1.33982 1.34131 1.34280 1.34431 1.34583 1.34736 1.34891 1.35046 1.35203 1.35361 1.35520 1.35680 1.35842 1.36005 1.36169 1.36334 1.36501 1.36669 1.36838 1.37009 1.37181

Table A.30 Refraction index (20 °C, 589 nm) of aqueous sucrose-solution. The concentration in °Bx is noted in left column [14] 0.8 1.33413 1.33558 1.33703 1.33849 1.33997 1.34146 1.34295 1.34446 1.34598 1.34752 1.34906 1.35062 1.35219 1.35377 1.35536 1.35697 1.35858 1.36021 1.36186 1.36351 1.36518 1.36686 1.36855 1.37026 1.37198

0.9 1.33428 1.33572 1.33717 1.33864 1.34012 1.34160 1.34310 1.34461 1.34614 1.34767 1.34922 1.35077 1.35234 1.35393 1.35552 1.35713 1.35875 1.36038 1.36202 1.36368 1.36535 1.36703 1.36872 1.37043 1.37216

614 Appendix A

25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51

1.37233 1.37407 1.37582 1.37758 1.37936 1.38115 1.38296 1.38478 1.38661 1.38846 1.39032 1.39220 1.39409 1.39600 1.39792 1.39986 1.40181 1.40378 1.40576 1.40776 1.40978 1.41181 1.41385 1.41592 1.41799 1.42009 1.42220

1.37250 1.37424 1.37599 1.37776 1.37954 1.38133 1.38314 1.38496 1.38679 1.38865 1.39051 1.39239 1.39428 1.39619 1.39812 1.40006 1.40201 1.40398 1.40596 1.40796 1.40998 1.41201 1.41406 1.41612 1.41820 1.42030 1.42241

1.37267 1.37441 1.37617 1.37793 1.37972 1.38151 1.38332 1.38514 1.38698 1.38883 1.39070 1.39258 1.39447 1.39638 1.39831 1.40025 1.40221 1.40418 1.40616 1.40817 1.41018 1.41222 1.41427 1.41633 1.41841 1.42051 1.42262

1.37285 1.37459 1.37634 1.37811 1.37989 1.38169 1.38350 1.38532 1.38716 1.38902 1.39088 1.39277 1.39466 1.39658 1.39850 1.40044 1.40240 1.40437 1.40636 1.40837 1.41039 1.41242 1.41447 1.41654 1.41862 1.42072 1.42283

1.37302 1.37476 1.37652 1.37829 1.38007 1.38187 1.38368 1.38551 1.38735 1.38920 1.39107 1.39277 1.39485 1.39677 1.39870 1.40064 1.40260 1.40457 1.40656 1.40857 1.41059 1.41262 1.41468 1.41675 1.41883 1.42093 1.42305

1.37320 1.47494 1.37670 1.37847 1.38025 1.38205 1.38386 1.38569 1.38753 1.38939 1.39126 1.39135 1.39505 1.39696 1.39889 1.40084 1.40280 1.40477 1.40676 1.40877 1.41079 1.41283 1.41488 1.41695 1.41904 1.42114 1.42326

1.37337 1.37511 1.37687 1.37865 1.38043 1.38223 1.38405 1.38588 1.38772 1.38958 1.39145 1.39333 1.39524 1.39715 1.39908 1.40103 1.40299 1.40497 1.40696 1.40897 1.41099 1.41303 1.41509 1.41716 1.41925 1.42135 1.42347

1.37354 1.37529 1.37705 1.37882 1.38061 1.38241 1.38423 1.38606 1.38790 1.38976 1.39164 1.39352 1.39543 1.39734 1.39928 1.40123 1.40319 1.40517 1.40716 1.40917 1.41120 1.41324 1.41530 1.41737 1.41946 1.42156 1.42368

1.37372 1.37546 1.37723 1.37900 1.38079 1.38259 1.38441 1.38524 1.38809 1.38995 1.39182 1.39371 1.39562 1.39754 1.39947 1.40142 1.40339 1.40537 1.40736 1.40937 1.41140 1.41344 1.41550 1.41758 1.41967 1.42177 1.42390

(continued)

1.37389 1.37564 1.37740 1.37918 1.38097 1.38277 1.38459 1.38643 1.38827 1.39014 1.39201 1.39390 1.39581 1.39773 1.39967 1.40162 1.40358 1.40557 1.40756 1.40958 1.41160 1.41365 1.41571 1.41779 1.41988 1.42199 1.42411

Appendix A 615

52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76

0 1.42432 1.42647 1.42863 1.43080 1.43299 1.43520 1.43743 1.43967 1.44193 1.44420 1.44650 1.44881 1.45113 1.45348 1.45584 1.45822 1.46061 1.46303 1.46546 1.46790 1.47037 1.47285 1.47535 1.47787 1.48040

Table A.30 (continued)

0.1 1.42454 1.42668 1.42884 1.43102 1.43321 1.43542 1.43765 1.43989 1.44216 1.44443 1.44673 1.44904 1.45137 1.45371 1.45608 1.45846 1.46085 1.46327 1.46570 1.46815 1.47062 1.47310 1.47560 1.47812 1.48065

0.2 1.42475 1.42690 1.42906 1.43124 1.43343 1.43565 1.43787 1.44012 1.44238 1.44466 1.44696 1.44927 1.45160 1.45395 1.45631 1.45870 1.46109 1.46351 1.46594 1.46840 1.47086 1.47335 1.47585 1.47837 1.48091

0.3 1.42497 1.42711 1.42928 1.43146 1.43365 1.43587 1.43810 1.44035 1.44261 1.44489 1.44719 1.44950 1.45184 1.45419 1.45655 1.45893 1.46134 1.46375 1.46619 1.46864 1.47111 1.47360 1.47610 1.47862 1.48116

0.4 1.42518 1.42733 1.42949 1.43168 1.43387 1.43609 1.43832 1.44057 1.44284 1.44512 1.44742 1.44974 1.45207 1.45442 1.45679 1.45917 1.46158 1.46400 1.46643 1.46889 1.47136 1.47385 1.47653 1.47888 1.48142

0.5 1.42539 1.42754 1.42971 1.43190 1.43410 1.43631 1.43855 1.44080 1.44306 1.44535 1.44765 1.44997 1.45230 1.45466 1.45703 1.45941 1.46182 1.46424 1.46668 1.46913 1.47161 1.47410 1.47661 1.47913 1.48167

0.6 1.42561 1.42776 1.42993 1.43211 1.43432 1.43654 1.43877 1.44102 1.44329 1.44558 1.44788 1.45020 1.45254 1.45489 1.45726 1.45965 1.46206 1.46448 1.46692 1.46938 1.47186 1.47435 1.47686 1.47938 1.48193

0.7 1.42582 1.42798 1.43015 1.43233 1.43454 1.43676 1.43900 1.44125 1.44352 1.44581 1.44811 1.45043 1.45277 1.45513 1.45750 1.45989 1.46230 1.46473 1.46717 1.46963 1.47210 1.47460 1.47711 1.47964 1.48218

0.8 1.42604 1.42819 1.43036 1.43255 1.43476 1.43698 1.43922 1.44148 1.44375 1.44604 1.44834 1.45067 1.45301 1.45537 1.45774 1.46013 1.46254 1.46497 1.46741 1.46987 1.47235 1.47485. 1.47736 1.47989 1.48244

0.9 1.42625 1.42841 1.43058 1.43277 1.43498 1.43720 1.43944 1.44170 1.44398 1.44627 1.44858 1.45090 1.45324 1.45560 1.45798 1.46037 1.46278 1.46521 1.46766 1.47012 1.47260 1.47510 1.47761 1.48015 1.48270

616 Appendix A

77 78 79 80 81 82 83 84 85

1.48295 1.48552 1.48811 1.49071 1.49333 1.49597 1.49862 1.50129 1.50398

1.48321 1.48578 1.48837 1.49097 1.49359 1.49623 1.49889 1.50156 1.50425

1.48346 1.48604 1.48863 1.49123 1.49386 1.49650 1.49915 1.50183 1.50452

1.48372 1.48629 1.48889 1.49149 1.49412 1.49676 1.49942 1.50210 1.50479

1.48398 1.48655 1.48915 1.49175 1.49438 1.49703 1.49969 1.50237 1.50506

1.48423 1.48681 1.48941 1.49202 1.49465 1.49729 1.49995 1.50263 1.50533

1.48449 1.48707 1.48967 1.49228 1.49491 1.49756 1.50022 1.50290 1.50560

1.48475 1.48733 1.48993 1.49254 1.49517 1.49782 1.50049 1.50317 1.50587

1.48501 1.48759 1.49019 1.49280 1.49544 1.49809 1.50076 1.50344 1.50614

1.48526 1.48785 1.49045 1.49307 1.49570 1.49835 1.50102 1.50371 1.50641

Appendix A 617

618

A.8

Appendix A

Color Test Solutions

In conformity with the European Pharmacopoeia [15], the five so-called color standard solutions can be prepared from mixtures of three color stock solutions (yellow: FeCl3-solution; red: CoCl2-solution; blue: CuSO4-solution). The five resulting colors are brown, brownish-yellow, yellow, greenish-yellow, and red. Color test solutions can then be prepared from defined dilution levels carried out in six steps. The color stock solutions are very stable and keep their quality indefinitely. However, the color standard solutions and the color test solutions are not stable and must be freshly prepared at the time of use. There are two established methods by which to compare a sample color with a color test solution that are described as follows: Method I:

Method II:

2 cm3 of each, sample and test solution, are filled in colorless reagent bottles with 12 mm inside diameter and observed under diffuse daylight in a horizontal orientation to a white background. 10 cm3 of each, sample and test solution, are filled in colorless reagent bottles with 16 mm inside diameter, and observed under diffuse daylight in a vertical orientation to a white background.

The advantage of method II is that results are more reproducible because of the greater depth of the solutions. It is advisable to use a so-called Nessler-cylinder instead of the reagent bottles. These are translucent, colorless cylinders with 16 mm inside diameter and flat, translucent bottom.

A.8.1 Preparation of Color Stock Solutions Yellow color stock solution: A quantity of 46 g ferric chloride (FeCl3) is dissolved in ca. 900 ml of a mixture of 25 ml hydrochloric acid and 975 ml water and diluted with this mixture to 1000.0 ml. After analysis to confirm solution composition, the solution is further diluted with sufficient hydrochloric acid-water mixture until there are 45.0 mg FeCl3
6H2O in 1 ml of the solution. Analysis of content A quantity of 10.0 ml of yellow color stock solution and 15 ml water, 5 ml hydrochloric acid and 4 g potassium iodide are mixed in a 200-ml Erlenmeyer flask with glass stopper. Immediately, the flask is closed and placed under darkness for about 15 min. After the addition of 100 ml water the resulting iodine solution will be titrated with 0.1 N-sodium thiosulfate solution. At the end of the titration, 10 drops starch-solution has to be added. 1 ml 0.1 N-sodium thiosulfate solution conforms to 27.03 mg FeCl3
6H2O. Red color stock solution: A quantity of 60 g cobalt(II)-chloride (CoCl2) is dissolved in ca. 900 ml of a mixture of 25 ml hydrochloric acid and 975 ml water and diluted with this mixture to

Appendix A

619

a total of 1000.0 ml. After the analysis of contents, the solution will be diluted with sufficient hydrochloric acid-water-mixture, until there is 59.5 mg CoCl2
6H2O in 1 ml of the solution. Analysis of content A quantity of 5.0 ml of red color stock solution and 5 ml hydrogen-peroxide and 10 ml of a 30 percent (m/V) solution of sodium hydroxide are mixed in a 200-ml Erlenmeyer flask with glass stopper, and heated to low boiling for about 10 min. After cooling down to room temperature, the solution is mixed with diluted sulfuric acid and 2 g potassium iodide, and the flask is immediately closed and shaken slowly to dissolve the iodide. The resulting brownish colored iodine solution is titrated with 0.1 N-sodium thiosulfate solution until the color is pink. Near to the end of the titration, 10 drops of starch-solution R are added. 1 ml 0.1 N-sodium thiosulfate solution conforms 23.79 mg CoCl2
6H2O. Blue color stock solutions: A quantity of 63 g cupric sulfate (CuSO4) is dissolved in ca. 900 ml of a mixture of 25 ml hydrochloric acid and 975 ml water and diluted with this mixture up to 1000.0 ml. After the analysis of contents, the solution is diluted with sufficient hydrochloric acid-water-mixture, until there is 62.4 mg CuSO45H2O in 1 ml of the solution. Analysis of contents: A quantity of 10.0 ml blue color stock reference solution and 50 ml water, 12 ml diluted acetic acid, and 3 g potassium iodide are mixed in a 200-ml Erlenmeyer flask with glass stopper. The resulting iodine solution is titrated with 0.1 N-sodium thiosulfate solution until the solution is blue colored. Near to the end of the titration, 10 drops of starch-solution R are added. 1 ml 0.1 N-sodium thiosulfate solution conforms 24.979 mg CuSO4
5H2O.

A.8.2 Color Standard Solutions The nomenclature for the 5 color standard solutions and the color test solutions are given in Table A.31. Table A.31 Color standard solutions [15] Color standard solutions B (brown) BG (brownishyellow) G (yellow) GG (greenishyellow) R (red)

Yellow color stock solutions ml 3.0 2.4

Red color stock solutions ml 3.0 1.0

Blue color stock solutions ml 2.4 0.4

Hydrochloric acid 1% (m/V) ml 1.6 6.2

2.4 9.6

0.6 0.2

0.0 0.2

7.0 0.0

1.0

2.0

0.0

7.0

620

Appendix A

With these 5 color standard solutions, the color test solutions are mixed like shown in Table A.32 Table A.32 Color test solutions [15] Color test solutions B Color standard Color test solution solution B in ml 1.50 B1 B2 1.00 0.75 B3 B4 0.50 0.25 B5 0.10 B6 Color test solutions BG Color test solution Color standard solution BG in ml 2.00 BG1 BG2 1.50 BG3 1.00 BG4 0.50 0.25 BG5 BG6 0.10 Color test solutions G Color test solution Color standard solution G in ml 2.0 G1 G2 1.50 1.00 G3 0.50 G4 G5 0.25 0.10 G6 Color test solutions GG Color test solution Color standard solution GG in ml 0.5 GG1 0.30 GG2 0.17 GG3 0.10 GG4 0.06 GG5 GG6 0.03 Color test solutions R Color test solution Color standard solution R in ml 2.00 R1 R2 1.50 1.00 R3 R4 0.75 R5 0.50 0.25 R6

Hydrochloric acid 1 % (m/V) in ml 0.50 1.00 1.25 1.50 1.75 1.90 Hydrochloric acid 1 % (m/V) in ml 0.00 0.50 1.00 1.50 1.75 1.90 Hydrochloric acid 1 % (m/V) in ml 0.00 0.50 1.00 1.50 1.75 1.90 Hydrochloric acid 1 % (m/V) in ml 1.50 1.70 1.83 1.91 1.94 1.97 Hydrochloric acid 1 % (m/V) in ml 0.00 0.50 1.00 1.25 1.50 1.75

Appendix A

621

In the US pharmacopeia color standard solutions are referred to as color matching fluids and are given letter designations A through T, as shown in Table A.33. Table A.33 Color Standards according to USP XII [16] Matching fluid A B C D E F G H I J K L M N O P Q R S T

Parts of Cobalt Chloride CS 0.1 0.3 0.1 0.3 0.4 0.3 0.5 0.2 0.4 0.4 0.5 0.8 0.1 0.0 0.1 0.2 0.2 0.3 0.2 0.5

Parts of Ferric Chloride CS 0.4 0.9 0.6 0.6 1.2 1.2 1.2 1.5 2.2 3.5 4.5 3.8 2.0 4.9 4.8 0.4 0.3 0.4 0.1 0.5

Parts of Cupric Sulfate CS 0.1 0.3 0.1 0.4 0.3 0.0 0.2 0.0 0.1 0.1 0.0 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.0 0.4

Parts of water 4.4 8.5 4.2 3.7 3.1 3.5 3.1 3.3 2.3 1.0 0.0 0.3 2.8 0.0 0.0 4.3 4.4 4.1 4.7 3.6

Literature

1. BIPM - On the revision of the International System of Units (SI) (2018) Bureau International de Poids et Mesures, Versailles 2. Kurzweil P (2013) Das Vieweg Einheiten-Lexikon: Begriffe, Formeln und Konstanten aus Naturwissenschaften, Technik und Medizin. Vieweg, Wiesbaden. https://doi.org/10.1007/9783-322-92920-4 3. Database of Physical Properties of Food Food Properties Awareness Club. http://www.nelfood. com 4. Rahman S (2009) Food properties handbook. CRC Press, Boca Raton, FL. https://doi.org/10. 1201/9781420003093 5. Kessler HG (2002) Food and bio process engineering: dairy technology. A. Kessler, München 6. Panagiotou NM, Krokida MK, Maroulis ZB, Saravacos GD (2004) Moisture diffusivity: literature data compilation for foodstuff. Int J Food Properties 7:273–299. https://doi.org/10. 1081/JFP-120030038 7. Saravacos GD, Maroulis ZB (2014) Transport properties of foods. CRC Press, Boca Raton, FL. https://doi.org/10.1201/9781482271010 8. Krokida MK, Panagiotou NM, Maroulis ZB, Saravacos GD (2001) Thermal conductivity: literature data compilation for foodstuffs 2001. Int J Food Prop 4:111–137. https://doi.org/10. 1081/JFP-100002191 9. Berk Z (2018) Physical properties of food materials. Food process engineering and technology, pp 1–29. https://doi.org/10.1016/B978-0-12-812018-7.00001-4 10. Arana I (2016) Physical properties of foods - novel measurement techniques and applications. CRC Press, Boca Raton, FL. https://doi.org/10.1201/b11542 11. Gröber E, Grigull U (1963) Die Grundgesetze der Wärmeübertragung. Heidelberg, Berlin. https://doi.org/10.1007/978-3-662-29015-6 12. Tscheuschner HD (2016) Grundzüge der Lebensmitteltechnik. Behrs, Hamburg 13. Fraden J (2016) Handbook of modern sensors physics, designs, and applications. https://doi. org/10.1007/978-3-319-19303-8 14. De Whalley HCS (1964) ICUMSA methods of sugar analysis. International Commission for uniform methods of sugar analysis. Elesvier, Amsterdam 15. Europäsches Arzneibuch. Dt. Apotheker Verlag, Eschborn 16. United States Pharmacopeia USP. United States Pharmacopeial Convention, Rockville, MD

# The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. O. Figura, A. A. Teixeira, Food Physics, https://doi.org/10.1007/978-3-031-27398-8

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Index

A Absolute temperature, 312 Absorbent, 23 Absorption, 495 range, 476 spectrum, 475 Absorptivity, 334 Accuracy, 576 Acoustics, 539 characterization, 545 information, 545 spectrum, 545 technology, 546 Activation energy, 183, 299 Activity, 2, 322, 525, 532 coefficient, 7 Additive color mixing, 497, 499 Additive resistors, 285, 295, 305, 339, 343, 347, 354 Adhesion, 267, 269 energy, 267 Adiabatic, 319 demagnetization, 434 mode, 393 Admittance, 404, 419 Adsorbate, 23 Adsorption, 7, 12, 18, 271 equilibrium, 11 isotherm, 22, 271 Aerosol, 102 Agglomeration tendency, 424 Air, 63 classification, 140 density, 63 Alcoholometer, 85 α-radiation, 524 Alternating DSC (ADSC), 387 Alternating electric field, 458

Amorphous, 42, 515 Ampere, 585 Amphiphilic, 260 Analyte, 572 Analytical sieving, 134 Andrade, 183 Angle of rotation, 492 Angular velocity, 159 Anion, 403 Anisotropic, 151, 354 electrical conductivity, 415 Anisotropy, 407 Anomaly, 71 Antibody, 571, 573 Anti-ferromagnetism, 434 Antigen, 571 Antioxidant capacity, 450 Apparent mass, 62 Apparent thermal conductivity, 359 Apparent viscosity, 179 Approximation, 260 Aptamer, 573 Apta-sensor, 573 Archimedes, 79 Areometer, 84 Arithmetic mean, 104 Arrhenius, 34, 37, 183, 299 At-line, 553 Atomic angular momentum, 432 Atomic nucleus, 443 Atomic number, 443, 517 Attenuation, 470 Atwater, 370, 373 Audibility, 544 Audible, 223 Auger electron, 516 Austenitic steel, 433 Authentification, 447

# The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. O. Figura, A. A. Teixeira, Food Physics, https://doi.org/10.1007/978-3-031-27398-8

625

626 Avogradro, 62 Avoidurpois, 591

B Balance, 60, 61, 65 Ball drop measurement, 216 Ball viscosimeter, 215 Basal metabolic rate (BMR), 365, 367 Basal metabolism, 364 Basic unit, 585 Baumé, 85 Becquerel, 525 Bending fracture test, 243 Bending oscillator, 560 Bending test, 243 Best-before, 43 Best by, 44 β-radiation, 524 Bi-disperse, 105 Bilayer, 263 Binding heat, 8 Bingham, 167, 173 Bio-detection, 572 Biopolymer, 298 Bio-recognition, 572, 573 Bio-sensitive, 571, 572 Bio-sensor, 551, 571, 572, 575 Biot, 491 Black body, 334 Blaine, 141 Blood cell, 134 Boltzmann, 312 Boltzmann factor, 597 Bonding, 12, 540 Bostwick, 216 Boundary layer, 260, 345 Bound water, 1, 8, 13 Bq, 525 Brewster angle, 495 Brightness, 500, 502, 503 Brilliance, 500, 502, 503 Brown, 133 Brunnauer, Emmet, Teller (BET), 20, 39 equation, 29 model, 22 one-point, 32 surface, 141 Bubble, 252 Bubble point pensiometer, 279 Bulk module, 152 Bulk modulus, 147 Buoyancy, 62, 77, 79, 82, 86, 88, 273, 379

Index Buoyancy correction, 62 Burgers, 191 Burgers element, 238

C Cal, calorie, 318, 366 Calibration, 61, 62, 562 Caloric value, 364–374 Calorimeter, 392 Calorimetry, 367, 604–605 Candela, 585 Cannon–Fenske, 215 Capacitance, 404, 567, 572 Capacitive balance, 65 Capacitive reactance, 404, 416, 418 Capacitive sensor, 567 Capacitor, 567 Capillary absorption, 14 Capillary adsorption, 17 Capillary condensation, 15, 17 Capillary method, 274 Capillary pressure, 250, 253 Capillary viscometer, 214 Capsule, 263 Carnot, 314 Casson, 180, 181 Cataphoretic effect, 412 Cation, 403 Cavitation, 539, 546 Cell disintegration index (CDI), 415 Cell growth, 570 Celsius, 313 Central value, 103 Centrifuge separation, 90 Charge number, 405 Checkweigher, 560 Chemical potential, 2, 322, 328 Chemical shift, 445 Chemisorption, 8, 572 Chemo-sensitive, 572 Chemo-sensor, 551, 572, 575 Chewing noise, 545 Chiral, chirality, 492 Chirality, 494 Ci, 527 CIE, 499, 500 Circularly polarized light, 493 Clausius-Clapeyron, 35 Clausius-Mosotti-Debye, 460 Closed loop, 558 Closed pore, 15 Closed-pore system, 353

Index Coating, 334 Coaxial cylinder system, 193 Cohesion, 267, 269 Cohesion energy, 267 Cold plasma, 426 Colloid, 261, 265 Color, 483, 497, 500, 571 difference, 503 match, 504 perception, 497 plane, 502 space, 500, 503 standard, 504, 618 triangle, 499 vector, 501 Colorimetry, 497 Combustion, 369 calorimeter, 373, 392 energy, 364 Complementary color, 500 Complex conductance, 419 Complex elasticity module, 241 Complex electrical resistance, 404 Complex heat capacity, 390 Complex heat flow, 390 Complex interfacial elasticity, 280 Complex number, 212, 390, 597, 602 Complex permittivity, 465 Complex physical quantity, 212 Complex quantity, 419, 465, 603 Complex refractive index, 132 Complex resistance, 419 Complex shear modulus, 213 Complex viscosity, 211 Compliance, 190, 211, 604 Composite film, 295 Compressibility, 72, 151 Compression, 147, 151, 226 modulus, 542 test, 231 Compressive stress, 226 Compton effect, 528 Computed tomography (CT), 518 Concave, 250, 253 Condensing, 349 Conductance, 289 Conductive heating, 403 Conductive resistance, 354 Conductivity, 289 Conductometry, 420 Cone-plate system, 195, 200 Cones, 499 Consistency coefficient, 177, 178, 181

627 Contact angle, 266, 272, 273, 279 Contamination, 565 Continuous phase, 101 Continuous wave (CW), 446 Contour analysis, 276, 277 Control, 558 Controlled shear rate (CSR), 195 Controlled shear stress (CSS), 195 Convection, 344, 357 Convex, 252, 253 Cookware, 441 Coriolis force, 563 Couette, 194 Coulomb force, 403, 528 Counter tube, 529 Creep, 189 Creep test, 230, 232, 236 Crispness, 539 Critical micelle formation concentration (cmc), 261 Critical micelle temperature (cmt), 261 Critical temperature, 256 Curie, 527 Curie temperature, 378, 432, 433, 441 Curie transition, 377 Curved interfaces, 250

D Damper, 188 Damping, 446, 465 Deborah, 239 De Brouckere diameter, 128 Debye, 460 Debye-Falkenhagen effect, 413 Debye length, 423 Decay, 450, 525 Decay constant, 527 Decibel (dB), 541 Decile, 106 Degradation reaction, 41 De Haven, 176 Density, 59, 64, 560, 609 Density gradient column, 89 De number, 239 Depletion, 263, 271 Derived unit, 585 Desiccator method, 48 Desinfection, 514 Desorption, 8, 12, 17, 374 Desorption isotherm, 22 Detection limit, 576 Diamagnetic material, 433, 437

628 Diamagnetism, 433–435 Dielectric, 456 Dielectric heating, 458 Dietary fibre, 364 Differential, 321 Differential scanning calorimeter (DSC), 356, 379, 382 Diffraction, 496 Diffuse layer, 423 Diffuse reflection, 495 Diffusion, 285 Diffusion coefficient, 287 Dilatancy, 169 Dilatant, 169, 177 Dipole moment, 459 Dispersant, 136 Disperse, 101, 255 Disperse phase, 101 Dispersion, 101, 420, 464, 487, 496 Dispersion of permittivity, 420 Displacement polarization, 463 Displacement work, 316 Dissipation, 465 Dissipation factor, 467 Distribution density, 103 Distribution density function, 111, 138 Distribution function, 103, 591 Distribution sum, 138 Distribution sum function, 110, 138 DNA, 573 Doppler effect, 546, 567 Dose, 533 Dose equivalent, 533 Drag force, 139, 140, 412 Drop, 254, 265 shape, 265 viscometer, 216 volume tensiometer, 278 volumeter, 279 Droplet, 16, 101, 249, 252, 279 Dry basis (db), 24 Drying, 13, 38, 39, 42, 45, 374 DSC oven, 380 DTG, 375 Dulong-Petit, 327 Du Noüy, 272 Dynamic dewpoint isotherm (DDI), 49 Dynamic isopiestic method (DIM), 49 Dynamic light scattering (DLS), 132, 133 Dynamic mechanical analysis (DMA), 213, 241 Dynamic vapour sorption (DVS), 49 Dynamic viscosity, 158, 164

Index E Ear drum, 540 Earth, 60, 61, 438 Eddy current, 439 Ehrenfest, 329, 330 Elastic, 146 absorption, 466 body, 153 case, 458 oscillator, 465 Elasticity, 72, 148, 150, 228, 239, 539 Electrical capacitance, 416 Electrical conductance, 405 Electrical conductivity, 305, 403, 405, 422, 466, 518 Electrical double layer, 424 Electrical nose, 575, 576 Electrical polarizability, 416, 462, 467 Electrical polarization, 464 Electrical potential, 423 Electrical properties, 403 Electrical resistance, 134, 403, 405, 559 Electrical resistivity, 404 Electrical voltage, 405 Electric bilayer, 423 Electric cell perforation, 420 Electric current, 403 Electric field, 403, 455 Electric field strength, 403 Electric polarization, 456 Electrolyte, 407, 408, 473 Electromagnetic absorption, 458 Electromagnetic detection, 564 Electromagnetic induction, 565 Electromagnetic radiation, 455, 483, 497 Electromagnetic spectrum, 483 Electromagnetic wave, 455, 513 Electron, 304, 524, 528 beam, 536 capture, 535 conductivity, 352 spectroscopy, 513 Electronic tongue, 575 Electron paramagnetic resonance (EPR), 450 Electron spin resonance (ESR), 442, 450–451, 535 Electro-optical, 494 Electro-perforation, 426 Electrophoretic analysis, 413 Electrophoretic effect, 413, 422 Electrophoretic mobility, 424 Electro-resistive sensor, 567

Index Electrozone counter, 134 Ellipsoid particle, 113 Ellipsometry, 494 Elliptical polarized radiation, 493 Elliptically polarized light, 493 Ellis, 176 Elongation, 226 Emissivity, 334 E-modulus, 150, 153 Emulsion, 101, 263 Enantiomer, 492 Encapsulation, 263 Energy, 249, 314 Enthalpy, 317, 384 of adsorption, 34 change, 349 Entropy, 319, 322, 465 Entropy of sorption, 38 Enzymatic tanning, 41 Eötvös, 256 Equivalent conductivity, 408, 414 Equivalent diameter, 113, 133, 134, 136, 140 Equivalent number, 408 ESR spectroscopy, 535 Euclid, 155 Euklid, 183 Euler, 600 eV, 515 Evaporation, 8, 12, 349 Exergetic, 38 Extensional viscosity, 226 Extinction, 498

F Fahrenheit, 314 Failure strain, 227 Failure stress, 227 Far infrared radiation (FIR), 474 Feces, 364 Feret, 118 Ferrimagnetism, 434 Ferrites, 434 Ferritic steel, 441 Ferromagnetic, 377, 433, 441 materials, 433, 437 steel, 441 Ferromagnetism, 433 Ferrous material, 433 Ferry, 176 Fibrous, 353 Fick, 287, 305 Field sweep, 446 Filling level, 570 Film, 291

629 Film former, 267 First law of thermodynamics, 318 First order transition, 329 Fixed point, 314 Flavoromics, 577 Floating method, 88 Flocculation, 424 Flow, 226 behavior index, 177 exponent, 177, 178 sensor, 563 Fluid, 9 Fluidity, 165 Fluorescence, 516 Flux, 287 Foam/foaming, 90, 101, 353 Foodomics, 577 Food safety, 311, 573 Form factor, 118 Fourier, 305, 335, 337, 339, 345, 347, 351, 355, 360, 390 Fourier transformation, 447 Fraction, 109 Fracture, 224 strain, 242 stress, 242 test, 242 Fraunhofer, 132 Fraunhofer diffraction, 496 Free energy, 319 Free enthalpy, 262, 319, 322 Free induction decay (FID), 446, 449 Free water, 8, 13, 38 Frequency, 455 Frequency sweep, 213, 446 Freundlich, 39 Freundlich-model, 18, 20 Friction, 140 Frozen fraction, 356 Fugacity, 3, 7, 322 Functionalized surface, 574 Fundamental oscillation, 476

G γ-quant, 528 Gamma radiation (γ-radiation), 450, 524 Gas adsorption method, 22 Gas chromatography, 535 Gas constant, 75 Gates, Gaudin and Schuhmann (GGS), 129, 130 Gauss, 107 Geiger-Müller counter, 529 Gel electrophoresis, 413

630 Gibbs, 37, 329, 331 Gibbs energy, 262, 319 Gibbs enthalpy, 38 Glass transition, 42, 185, 328, 332, 386, 387, 391 Glass transition temperature, 43 Glassy state, 43, 186 Gloss, 495 Glossmeter, 495 G-modulus, 150, 153 Gravitational acceleration, 60 Gravitational force, 60 Gravity, 60 Gray, 533 Guggenheim, Anderson, de Boer (GAB), 23, 40 equation, 32 model, 23 parameter, 33 Gy, 533 Gyromagnetic ratio, 447

H Habitus, 118 Hagen-Poiseuille, 214 Half-life, 526 Half-power depth, 472 Half-value depth, 525 Hall probe, 442 Haptic, 223 Harmonics, 476 Hearing, 223 Heat, 304, 314, 458, 465 capacity, 323, 325, 327, 361, 390, 605 conductance, 354 conduction, 335 exchanger, 335, 337, 349, 352 flow, 347, 604 flow calorimetry, 379 radiation, 333 transfer, 311, 332, 345, 470 transfer coefficient, 345, 346 transfer resistance, 345 treatment, 311 Heating, 470 Heinz, 180 Helmholtz energy, 319 Helmholtz potential, 423 Hencky, 238 Henry, 298 Henry-Gesetz, 20 Henry´s Law, 20 Heraclitus, 145, 240

Index Herschel-Bulkley, 180 Heywood factor, 126 Hook, 183, 189 Hooke, 147, 148, 150, 155, 158, 212, 231 Hookean, 148, 232, 239 Höppler, 216 Hue, 500, 502, 503 Human body, 531, 533 Human ear, 540, 543, 544 Human eye, 483, 497, 499 Human viewer, 497 Humidity, 570 Huygens, 485, 496 Hydration, 263 Hydrogen bond, 11 Hydrometer, 84 Hydrophilic, 260 Hydrophobic, 260 Hydrostatic balance, 79, 80, 82 Hygroscopicity, 17, 27, 28 Hyper sound, 540 Hyperspectral imaging (HSI), 478 Hysteresis, 17, 18, 22, 437, 438

I Ideal gas, 3, 74, 325, 597 Ideal solution, 3 Image analysis, 108, 117 Imaginary, 132 number, 597 part, 212, 465 permittivity, 466 unit, 419, 597 Impedance, 403, 418, 422, 570, 572, 603 Impedance-based index, 420 Impedance measurement, 415, 420 Inactivation bacteria, 514 microorganisms, 539 Incompressible, 155 Inductance, 404, 417 Induction, 439 cooker, 426 cooking, 440 time, 422 Inductive metal detector, 439 Inductive reactance, 404, 417, 418 Inelastic absorption, 466, 468 In-elastic case, 458 Infrared-active, 475 Infrared radiation (IR), 333, 474 Infra-red thermometry (IR thermometer), 334

Index

631

Infrasound, 539 In-line sensor, 551, 552 In-phase, 466 Inspection, 475 Integral mean, 119–121, 123 Integrating sensor, 558 Interface, 9, 247 Interface-active, 260 Interfacial elasticity, 280 Interfacial energy, 249, 264 Interfacial phase, 247 Interfacial tension, 86, 248, 256, 272, 279 Interference, 496 Intermediate moisture, 47 Internal energy, 314 International Office of Cocoa, Chocolate and Sugar Confectionery (IOCCC), 182 International system de unite, 60 International temperature scale (ITS), 314 Interval limit, 109 Intrinsic viscosity, 187 Ion, 403, 407 Ionization, 425 Ionization detector, 528 Ionizing radiation, 515, 533 Irradiation, 534–536 Isentropic, 319 Isentropic coefficient, 542 Isobaric, 317 Isometric, 112 Isoperibole, 393 Isosteric sorption enthalpy, 36 Isotope, 443, 447, 449, 524 Isotropic, 151–153, 353

Lactometer, 85 Lambert-Beer law, 495, 516 Lambert-Beer’s, 497 Laminar, 139, 166 Langmuir, 20, 39, 269, 271, 279 Langmuir-Blodgett film, 269 Langmuir model, 19 Laplace, 252, 253, 276, 279 Larmor frequency, 444, 494 Laser, 131 diffraction, 108, 131, 496 diffractometer, 132 Latent heat, 314, 349, 350 LED, 514 Lenz´s law, 433 Lewicki, 39 Life time, 255, 279 Light, 483, 484, 497 Light scattering, 133 Linearly polarized light, 490 Liquid crystal display (LCD), 494 Liter, 591 Log-normal-distribution, 130 Lorentz force, 440, 442, 528, 564 Loss, 465 factor, 465, 468, 473 tangent, 465 Loudness, 543 Loudness level, 543 Low angle laser light scattering (LALLS), 131 Low-resolution nuclear resonance spectroscopy (LR-NMR), 449 Lyophilic, 260 Lyophobic, 260

K Kelvin, 254, 312, 585 Kelvin body, 156 Kelvin element, 188, 189, 237 Kelvin`s equation, 15 Kelvin/Vogt, 191 Kilogramm, 585 Kinematic viscosity, 165 Kinesthetic, 223 Kinetic energy, 312 Kirchhoff, 334 K-modulus, 153 Kohlrausch, 408 Krafft, 262 Kurtosis, 107

M Magmeter, 564 Magnetar, 442 Magnetic dipole, 431 Magnetic field, 431, 435, 455, 564 Magnetic history, 438 Magnetic inductive flowmeter, 442 Magnetic-inductive flow sensor, 564 Magnetic moment, 434, 443, 450 Magnetic momentum, 432 Magnetic permeability, 431, 435, 436 Magnetic polarity, 433 Magnetic polarization, 433, 434 Magnetic properties, 431 Magnetic refrigeration, 434–435 Magnetic resonance, 442, 445 Magnetic resonance imaging (MRI), 450 Magnetic resonance spectroscopy, 444 Magnetic stainless steel, 441

L Lab system, 502, 503

632 Magnetic susceptibility, 436, 437, 439, 518 Magnetization, 377, 431, 436 Magnetocaloric effect, 434 Magneto-optical effect, 494 Margules, 198, 199 Martensitic steel, 441 Martin, 118 Mass, 59 Mass density, 59 Mathematical model, 40 Maxwell, 483 Maxwell body, 156 Maxwell-Boltzmann, 13, 595 Maxwell element, 188, 234 Maxwell equation, 464 MDSC, 387, 391, 604 Mechanical oscillation, 562 Median, 104, 125, 138 Mesopore, 15 Metabolic equivalent (MET), 369 Metabolic rate (MR), 365 Metabolomics, 576 Metal detector, 439, 518, 565 Metal separation, 439 Metal solids, 352 Meter, 585 Metrology, 575 mho, 406 Micelle, 261, 271 Microbalance, 563 Microbalance sensor, 574 Micro-cantilever, 574 Microcrystalline cellulose (MCC), 51 Micro-CT, 518 Micro-Electro-Mechanical Systems (MEMS), 560, 563, 573 Microemulsion, 264 Microorganisms, 570 Micropore, 15 Microscopic, 131 Microwave, 467 absorption, 473 heating, 466, 468 oven, 458, 469 packaging, 473 MID, 442 Middle fifty, 107 Mid infrared radiation (MIR), 474 Mie, 132 Mie scattering, 496 Miscibility, 247 Mist, 101 Mode, 104, 125, 138

Index distribution, 105 Model, 23, 40, 258 Modulus of elasticty, 151 Mohr-Westphal balance, 82 Moisture, 1, 570 Moisture content, 23 Mol, 585 Molecular imprintable polymer (MIP), 573 Mono-disperse, 105 Monolayer, 12, 13, 22 mass, 22 moisture content, 30 sorption enthalpy, 37 Monomolecular, 260, 269 Multilayer, 12, 13, 20, 285, 292, 295, 343, 348 Multiple emulsion, 265 Multi-sensor, 575 Must weight, 86 Mutarotation, 494

N Nano-dispersion, 265 Nanoemulsion, 265 Nanoparticle, 134 Nanopore, 15 Natural radionuclide, 530 Near-infrared (NIR), 475, 571 Near-infrared spectroscopy (NIRS), 477 Néel temperature, 434 Negative adsorption, 271 Neutron, 443, 523, 535 Newton, 155, 176, 183, 189, 231, 232 Newton element, 235 Newtonian, 232 Newtonian fluid, 164 Nitrogen adsorption, 141 Noise, 544 Non-enzymatic browning, 41 Non-isotropic, 353 Non-Newtonian, 173 Non-Newtonian fluid, 166 Non-reversing heat flow, 390 Non-wetting, 267 Nuclear decay, 529 Nuclear magnetic resonance (NMR), 356, 442, 445, 447 spectrometer, 445 spectroscopy, 449 spectrum, 447 Nuclear spin, 443 Nuclear spin resonance, 442 Nuclide, 523

Index Number, 598 Number distribution, 103, 121

O Oe, 86 Oechsle, 86 Off-line, 553 Ohm, 305, 339, 355, 406 Ohmic heating, 426 Ohmic resistance, 404, 418 Ohm’s law, 405 Online sensor, 551, 554 Open loop control, 558 Open pore, 15 Open-pore system, 353 Optical activity, 490 Optical properties, 483 Optical rotation, 490 Organoleptic, 575 Orientation polarization, 457, 463 Oscillating air, 540 Oscillating shear stress, 208, 241 Oscillating strain, 240 Oscillating tube, 90 Oscillation rheometer, 213 Ostwald, 255 Ostwald-de-Waele, 176, 177, 199 Ostwald-Faktor, 178 Out-of-phase, 466 Oven drying, 48 Overall heat transfer coefficient, 347, 349 Overflowing cylinder, 279 Overpressure, 252 Overrun, 90 Overtone, 476 Oxidation, 41 Oxidation stability index (OSI), 422 O/W emulsion, 263

P Packaging, 292, 303 Packaging material, 285, 292 PAR, 368 Parallel resistors, 290 Paramagnetic material, 432, 437 Paramagnetism, 431, 432 Partial differential, 321 Partial integration, 385 Partial pressure, 3 Particle, 102 shape, 118

633 size, 102, 111, 131, 140, 539 size analysis, 108 size distribution, 129, 496 sizing, 496 Pascal, 155, 183 Peeling test, 227 Pendant drop, 276 Penetration depth, 426, 470, 472–474, 525 Penetration test, 227 Percentage, 25 Percentile, 106 Perception of sound, 545 Permanent dipole, 457, 463 Permanent polarization, 459 Permeability, 285, 288 Permeability coefficient, 298 Permeation, 289, 298, 299 Permittivity, 416, 458, 463, 473, 570, 604 Phase, 247 change, 349 shift, 209, 391 transition, 328, 332, 357 Photo effect, 516 Photoelectric effect, 529 Photometer, 495 Photometry, 498 Photon, 515 Photon correlation spectroscopy (PCS), 132, 133 Physical activity, 365, 368 Physical activity level (PAL), 367, 368 Physical caloric value, 369, 373 Physical constants, 585 Physical equivalent diameter, 114 Physikalisch-Technische-Bundesanstalt (PTB), 61 Physiological caloric value, 370, 373 Physiological energy, 370 Physisorption, 8, 572 Pickering, 265 Piezoelectric, 567 Piezoelectric effect, 545 Piezoresistive, 559 Plan sieving, 134 Planet, 60, 61 Plasma, 425, 426 Plasmon resonance, 571 Plastic, 148, 155, 173, 180 Plastic flow behavior, 170 Plasto-viscoelastic, 191 Plate capacitor, 416 Plate heat exchanger, 335 Plate-plate system, 195, 202

634 Plate-type, 335 Plato, 113 Poise, 166 Poiseuille, 166 Poisson, 150, 154, 155 Polarimeter, 490, 494 Polarimetry, 493 Polarizability, 420 Polarizable component, 416 Polarization, 490 potential, 458, 460, 463 volume, 459 Polyacrylamide gel electrophoresis (PAGE), 413 Polydisperse, 105 Pore, 9, 17 radius, 15 shape, 15 Porosimetry, 17 Porosity, 14, 473 Positron, 524, 528 Potassium, 530 Potential, 286, 320 Potential gradient, 403 Pound, 591 Powder, 22, 27, 101 Power compensation, 382 Power-law, 129, 177 Precession, 444 Precession frequency, 444 Precision, 575, 576 Prefix, 588 Preservation, 45, 536 Pressure, 147 Pressure scan, 331 Primary ion, 529 Process analytics, 555 Process automation, 553 Process control, 553, 557 Process monitoring, 555 Propagation speed, 455 Proton, 443, 449, 523 Pseudoplastic, 167, 168, 177 Pulsed NMR, 446 Pycnometer, 77 Pythagorean sum, 419

Q Quality factor, 533 Quantile, 106 Quantity, type of, 107, 121 Quartile, 106

Index Quartile distance, 107 Quartz crystal microbalance (QCM), 563 Quasi-elastic light scattering (QELS), 133

R Racemate, 492 Rad, 533 Radiation, 535 Radical, 535 Radioactivity, 523 Radionuclide, 523 Rainbow, 500 Random deviation, 64, 65 Raoult, 4, 40 Rate of shear, 160 Rayleigh scattering, 496 Reactance, 403, 418 Real part, 212, 465 Real-time, 553 Receptor, 572 Reference weight, 61, 63 Reflectance, 495 Reflection, 494 Reflective body, 334 Refraction, 483, 571, 572 Refraction index, 485 Refractive index, 464, 488 Refractometer, 490, 571 Refractometric sensor, 571 Reiner, 176 Relative density, 66, 77, 82 Relative fugacity, 322 Relative humidity, 1 Relaxation, 446, 449 test, 234 time, 189, 234 Rem, 533 Remanent magnetization, 438 Repeatability, 576 Reproducibility, 576 Residue, 136 Resistance, 289, 603 Resistive balance, 65 Resistivity, 290 Resistor, 290 Resolution, 576 Resonance, 560, 572, 575 Resonant frequency, 90 Resonator, 563 Retardation time, 191 Reversible heat flow, 390 Rheology, 145, 224, 604

Index Rheometer, 193 Rheopectic, 170 Rheopexy, 170, 173 Rigid, 155 Rigidity, 153 Ring method, 273 RNA, 573 Roasting, 374 Rods, 499 Rosin, Rammler, Sperling and Benett (RRSB), 130 Rotational dispersion, 491 Rupture, 148 strain, 242 test, 242

S Saccharimeter, 85, 494 Safety, 292 Sampling, 131 Saturated steam, 351 Sauter diameter, 127 Scattering, 496 Scattering, of light, 131 Scissoring oscillation, 475 Screening aid, 135 Searle, 194 Second, 585 law of thermodynamics, 319 order transition, 329 Secondary ion, 529 Sedimentation, 139 Selective absorption, 497 Selectivity, 573, 576 Self-luminous, 498 Sensitive heat, 312, 314 Sensitive layer, 572, 573, 575 Sensitivity, 573, 576 Sensomics, 577 Sensor, 552 Sensorial analysis, 575 Sensory, 223 Sensory quality, 223 Serial resistors, 290 Sessile drop, 276 Shape, 276 Shape of particles, 118 Shear, 146 activation energy, 183 angle, 153, 161 force, 161 modulus, 147

635 oscillation, 213 rate, 159 speed, 160 strain, 152 stress, 153, 161 test, 227 velocity, 160 Shear-thinning, 168 Shelf life, 38, 39, 41, 43, 292, 534 Shelf stable, 47 SI, 60, 585 Siemens, 406 Sieve analysis, 108, 136 Sieve mesh size, 113 Sievert, 533 Sieving, 113, 134 Sisko, 176 Size class, 103, 109, 134, 136 Skewness, 107 Snellius, 487 Sodium dodecyl sulfate (SDS), 413 Solid fat content (SFC), 450 Solid foam, 102 Solid surface, 7 Solvent, 136 Sorbate, 301 Sorbent, 300 Sorption, 8 enthalpy, 34, 36, 39 equilibrium, 13 isoster, 49 isotherm, 13, 14, 18, 27, 28, 48 Sorptive, 12 Sound, 539 intensity, 541 level, 541 power, 540 Span, 107 Specific activity, 527 Specific electrical conductivity, 406 Specific electrical resistance, 405 Specific gas constant, 75 Specific gravity, 66 Specific heat, 323 Specific heat capacity, 323 Specificity, 575, 576 Specific polarization potential, 461 Specific quantity, 69 Specific rotation, 491 Specific sorption enthalpy, 34 Specific surface area, 14, 125 Specific viscosity, 187 Specific volume, 67

636 Spectrophotometry, 498 Speed of light, 456, 483 Speed of sound, 539, 541 Spherical, 102, 113 Sphericity, 119 Spinning drop, 277 Spoilage, 41, 44, 292 Spray, 101 Spread, 103 Spreading, 266 Spreading pressure, 267 Spring, 188 Stability, 576 Stainless steels, 352, 433 Stalagmometer, 278 Standard, 51 climate, 301 deviation, 121 state, 299 volume, 288 Statistical moment, 119, 121, 125, 141 Staudinger, 187 Steady state, 14, 285, 292, 300, 335 Stefan-Boltzmann, 333 Steiner-Steiger-Ory, 176 Stern layer, 423, 425 Stern´s potential, 423 Sticky, 27, 42 Stokes, 139, 140, 166, 215 Stokes-Einstein, 133 Strain, 146, 238 gauge, 559 retardation, 236 Stress, 153, 226 relaxation test, 232 test, 230 Stress-strain diagram, 147, 227 Stretch oscillation, 475 Strong electrolyte, 409 St. Venant, 155, 183 Submersion technique, 86 Subtractive color mixing, 497, 498 Sum distribution, 110 Surface, 9, 247 Surface-active, 260, 271 Surface plasmon resonance (SPR), 571 Surface pressure, 268 Surfactant, 260 Susceptance, 404 Susceptibility, 458 Susceptor, 473 Suspension, 101 Sv, 533

Index Systematic deviations, 64, 65 System International, 585 Szyszkowski, 261

T Tactile, 223 Temperature, 312, 324, 606 coefficient, 421 gradient, 336 modulated DSC, 387 scan, 331 Temporary dipole, 457 Tensile stress, 226 Tensile test, 227, 231 Tension, 147 Terahertz absorption, 474 Terahertz radiation, 474 Terahertz waves, 474 Tercile, 106 Texture, 223 analysis, 224, 545 test, 232 Texture profile analysis (TPA), 244 Thermal analysis (TA), 374 Thermal conductance, 346 Thermal conductivity, 304, 337, 351, 354–356, 362 Thermal degradation, 376 Thermal diffusivity, 360, 361 Thermal energy, 312, 324 Thermal expansion, 69 Thermal expansion coefficient, 71 Thermal insulation, 337, 353, 358, 359 Thermal process operations, 311 Thermal properties, 311 Thermal resistivity, 337 Thermodynamic activity, 5 Thermodynamics, 314, 318 Thermodynamic temperature, 312 Thermogravimetry (TG), 374–379 Thickness measurement, 535 Thick-walled tube, 342 Thin walled tube, 342 Thixotropic, 169 Thixotropy, 173 Throwing sieving, 134 Time constant, 558 Tomography, 518 Total resistance, 291 Transient measurement, 363 Transient state, 300, 335, 360 Transition temperature, 329

Index Transmission, 497 Transparent, 495 Transport equation, 286, 287, 304, 305 Transport process, 285, 287 Transverse strain, 154 Troy, 591 True density, 78 True mass, 62 Trueness, 575, 576 Tubular heat exchanger, 335 Tubular-type, 335 Turnover plot, 386 Two-platen model, 160

U Ubbelohde, 215 Ultracentrifuge, 90 Ultrasonic, 135, 141 flow meter, 566 puls, 567 waves, 545 Ultrasonic attenuation spectroscopy (UAS), 141 Ultrasonography, 546 Ultrasound, 539, 545 Ultraviolet radiation, 513 Uncertainty, 260 Uniaxial, 145, 226, 227 Uniformity index, 107 Unsteady state, 335, 360 Urine, 85, 364 UV spectroscopy, 513 UV/vis spectrometer, 514

V Vacuum, 62, 358, 457 Vacuum UV, 513 van der Waals, 11 Vapor pressure, 1 Vapour pressure reduction, 17 Variance, 121 Vesicle, 263 Vibrating tube, 90 Vibration sieving, 134 Vibration technology, 562 Virus, 573 Viscoelastic, 156, 187, 232, 234, 238 Viscoelasticity, 239 Viscometer, 193 Viscoplastoelastic, 239 Viscosity, 239, 564, 604 Viscosity function, 165 Viscous, 155, 157, 226

637 Viscous body, 231 Vogel, 184

W Water activity, 1, 41, 44, 322 Water activity standard, 50 Water binding, 374 Water content, 1, 24, 25 Water vapor, 301 Water vapor permeability, 301 Water vapour, 3 Waveguide, 571 Wavelength, 455, 474 Wave number, 474 Weak electrolyte, 409 Weighing, 65, 559 Weight, 60 Weighted mean, 119 Weiss, 441 Weiss region, 433 Wet basis (wb), 24 Wet screening, 135 Wetting, 267, 269, 280 angle, 273 tension, 269 Whipping, 90 Wiedemann-Franz, 352 Wilhelmy, 272 Williams-Landel-Ferry (WLF), 185, 186 Windhab, 182 W/O emulsion, 264

X X-ray absorption, 517 X-ray diffractometry, 515 X-ray imaging, 517 X-ray inspection, 517 X-ray photon, 516 X-rays, 513 X-ray scanner, 517

Y Yield, 148 point, 170, 171 stress, 167, 170, 180 Young, 147, 150, 269

Z Zeeman effect, 445, 450 Zeta potential, 424, 425