The physics of phase transitions

The Physics of Phase Transitions P Papon J Leblond P.H.E Meijer The Physics of Phase Transitions Concepts and Applications Translated from the French by S.L Schnur With 180 Figures Second Revised Edition ABC Pierre Papon Jacques Leblond Paul H.E Meijer Catholic University of America Department of Physics Washington, DC 20064, USA E-mail: MEIJER@cua.edu École Supérieure de Physique et de Chimie Industrielles de Paris (ESPCI) Laboratoire de Physique Thermique 10 rue Vauquelin 75005 Paris, France E-mail: pierre.papon@espci.fr jacques.leblond@espci.fr Translator S.L Schnur Concepts Unlimited 6009 Lincolnwood Court Burke, VA 22015-3012, USA Translation from the French language edition of Physique des transitions de phases, concepts et applications by Pierre Papon, Jacques Leblond and Paul H.E Meijer, Second Edition c 2002 Editions Dunod, Paris, France This work has been published with the help of the French Ministère de la Culture – Centre national du livre Library of Congress Control Number: 2006923230 ISBN-10 3-540-33389-4 2nd Edition Springer Berlin Heidelberg New York ISBN-13 978-3-540-33389-0 2nd Edition Springer Berlin Heidelberg New York ISBN-10 3-540-43236-1 1st Edition Springer Berlin Heidelberg New York ISBN-13 978-3-540-43236-4 1st Edition Springer Berlin Heidelberg New York This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable for prosecution under the German Copyright Law Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typesetting: by the authors and techbooks using a Springer LATEX macro package Cover design: 2nd Editon, eStudio Calamar, Pau/Spain Printed on acid-free paper SPIN: 11735984 56/techbooks 543210 Foreword We learned in school that matter exists in three forms: solid, liquid and gas, as well as other more subtle things such as the fact that “evaporation produces cold.” The science of the states of matter was born in the 19th century It has now grown enormously in two directions: (1) The transitions have multiplied: first between a solid and a solid, particularly for metallurgists Then for magnetism, illustrated in France by Louis N´eel, and ferroelectricity In addition, the extraordinary phenomenon of superconductivity in certain metals appeared at the beginning of the 20th century And other superfluids were recognized later: helium 4, helium 3, the matter constituting atomic nuclei and neutron stars There is now a real zoology of transitions, but we know how to classify them based on Landau’s superb idea (2) Our profound view of the mechanisms has evolved: in particular, the very universal properties of fluctuations near a critical point – described by Kadanoff’s qualitative analysis and specified by an extraordinary theoretical tool: the renormalization group Without exaggerating, we can say that our view of condensed matter has undergone two revolutions in the 20th century: first, the introduction of quantum physics in 1930, then the recognition of “self-similar” structures and the resulting scaling laws around 1970 It would be na¨ıve to make too much of these advances: despite all of this sophistication, we are still very unsure about certain points – for example, the mechanism governing superconducting oxides or the laws of the glass transition However, a body of doctrines has been formed, and it is an important element of scientific culture in the 21st century This knowledge is generally expressed solely in works dedicated to only one sector The great merit of the book by Drs Papon, Leblond and Meijer is to offer a global introduction, accessible to students of physics entering graduate school I notice with pleasure the addenda of this new edition on Bose-Einstein condensates, on colloids, etc The panorama is broad and VI Foreword will stimulate the interest of the young public targeted here: this book should guide them soundly I wish it great success Paris, France January 2006 P.G de Gennes Preface to the Second Edition This book takes up and expands upon our teachings on thermodynamics and the physics of condensed matter at the School of Industrial Physics and Chemistry and Diplˆ ome d’Etudes Approfondies in Paris and at the Catholic University of America in Washington D.C It is intended for graduate students, students in engineering schools, and doctoral students Researchers and industrial engineers will also find syntheses in an important and constantly evolving field of materials science The book treats the major classes of phase transitions in fluids and solids: vaporization, solidification, magnetic transitions, critical phenomena, etc In the first two chapters, we give a general description of the phenomena, and we dedicate the next six chapters to the study of a specific transition by explaining its characteristics, experimental methods for investigating it, and the principal theoretical models that allow its prediction The major classes of application of phase transitions used in industry are also reported The last three chapters are specifically dedicated to the role of microstructures and nanostructures, transitions in thin films, and finally, phase transitions in large natural and technical systems Our approach is essentially thermodynamic and assumes familiarity with the basic concepts and methods of thermodynamics and statistical physics Exercises and their solutions are given, as well as a bibliography In this second edition, we have taken into account new developments which came up in the states of matter physics, in particular in the domain of nanomaterials and atomic Bose-Einstein condensates where progress is accelerating We have also improved the presentation of several chapters by bringing better information on some phase transition mechanisms and by illustrating them with new application examples Finally, we would we like to thank J F Leoni who assisted in the preparation of the manuscript and the drawings and diagrams and Dr S L Schnur who put much effort into translating the book as well as Dr J Lenz and F Meyer from Springer-Verlag who provided helpeful advice in publishing the book We are also grateful to our colleague Prof K Nishinari, from Osaka City University, for his valuable comments on our manuscript Paris, France Paris, France Washington, D.C., U.S.A., January, 2006 Pierre Papon Jacques Leblond Paul H.E Meijer Contents Thermodynamics and Statistical Mechanics of Phase Transitions 1.1 What is a Phase Transition? 1.2 Thermodynamic Description of Phase Transitions 1.2.1 Stability and Transition – Gibbs–Duhem Criterion 1.2.2 Phase Diagrams 1.2.3 Thermodynamic Classification of Phase Transitions 1.3 General Principles of Methods of Investigating Phase Transitions 1.3.1 Calculation of Thermodynamic Potentials and Quantities 1.3.2 Equation of State 1.3.3 Dynamic Aspects – Fluctuations 1.4 The Broad Categories of Phase Transitions 1.4.1 Transitions with a Change in Structure 1.4.2 Transitions with No Change in Structure 1.4.3 Non-Equilibrium Transitions 1.5 The Major Experimental Methods for Investigation of Phase Transitions 1.6 The Broad Categories of Applications of Phase Transitions 1.7 Historical Aspect: from the Ceramics of Antiquity to Nanotechnologies Problems Dynamics of Phase Transitions 2.1 A Large Variety of Mechanisms 2.2 Nucleation 2.2.1 The Diffusion Phenomenon – Fick’s Law 2.2.2 Diffusion Coefficient and Activation Energy 2.2.3 Nucleation of a New Phase 2.2.4 Nucleation Rate 2.2.5 Global Phase Transformation – Avrami Model 2.3 Spinodal Decomposition 2.3.1 Thermodynamics of Spinodal Decomposition 1 4 13 17 18 22 22 25 26 28 29 30 31 32 35 37 37 38 38 39 40 46 51 55 56 X Contents 2.3.2 Experimental Demonstration – Limitation of the Model 2.4 Structural Transition 2.4.1 Dynamics of a Structural Transition – The Soft Mode 2.4.2 Martensitic Transformation 2.5 Fractals – Percolation 2.5.1 Fractal Structures 2.5.2 Percolation and Gelation 2.6 Dynamics of Phase Transitions and Properties of Materials 61 64 64 66 67 67 72 75 Phase Transitions in Liquids and Solids: Solidification and Melting 3.1 Ubiquitous Phenomena 3.2 Characterization of the Phenomena 3.2.1 Thermodynamic Characterization 3.2.2 Microscopic Approach 3.2.3 Delays in the Transition: Supercooling–Superheating 3.2.4 Methods of Observation and Measurement 3.3 Melting 3.3.1 The Lindemann Model 3.3.2 The Role of Defects 3.3.3 Melting and Surface of Materials 3.4 Solidification 3.4.1 Theoretical Approach to Crystallization with Intermolecular Potentials 3.4.2 Case of Colloids 3.4.3 Crystallization and Melting of Polymers 3.5 Crystallization, Melting, and Interface 3.5.1 Surface Melting 3.5.2 Size Effect on Small Particles 3.5.3 The Special Case of Ice 3.6 Very Numerous Applications 3.6.1 Melting – Solidification in Metallurgy 3.6.2 Molding of Polymers 3.6.3 Production of Sintered Ceramics 97 104 106 111 111 114 114 117 118 120 121 Phase Transitions in Fluids 4.1 The Approach with Equations of State 4.2 The Liquid–Gas Transition in Simple Liquids 4.2.1 Van der Waals Equation of State 4.2.2 The Law of Corresponding States 4.2.3 Behavior Near the Critical Point 4.3 Thermodynamic Conditions of Equilibrium 4.3.1 Liquid–Gas Equilibrium 4.3.2 Maxwell’s Rule 125 125 127 127 128 130 132 132 133 79 79 80 80 82 84 86 90 90 92 95 96 Contents XI 4.3.3 Clausius–Clapeyron and Ehrenfest Equations 4.4 Main Classes of Equations of State for Fluids 4.4.1 General Principles 4.4.2 One–Component Fluids 4.4.3 Variants of the van der Waals Equation 4.5 Metastable States: Undercooling and Overheating 4.5.1 Returning to Metastability 4.5.2 Drops and Bubbles Formation 4.6 Simulation of Phase Transitions 4.6.1 Principles 4.6.2 Molecular Dynamics 4.6.3 Monte Carlo Method 4.7 Mixture of Two Components 4.7.1 Conditions of Phase Equilibrium in a Binary Mixture 4.7.2 Systems in the Vicinity of a Critical Point 4.7.3 Equation of State of Mixtures 4.7.4 Mixtures of Polymers or Linear Molecules 4.7.5 Binary Mixtures far from the Critical Point 4.7.6 Supercritical Demixing 4.7.7 Tricritical Points 134 135 135 136 137 139 139 139 140 140 141 143 145 145 146 147 152 155 158 159 The Glass Transition 5.1 Glass Formation 5.2 The Glass Transition 5.2.1 Thermodynamic Characteristics 5.2.2 Behavior of the Viscosity 5.2.3 Relaxation and Other Time Behaviors 5.3 The Structure of Glasses 5.3.1 Mode Coupling Theory 5.3.2 Industrial Applications 5.3.3 Models for Biological Systems 165 165 168 168 171 173 173 176 183 185 Gelation and Transitions in Biopolymers 6.1 The Gel State and Gelation 6.1.1 Characterization of a Gel 6.1.2 The Different Types of Gels 6.2 Properties of Gels 6.2.1 Thermal Properties 6.2.2 Mechanical Properties 6.3 A Model For Gelation: Percolation 6.3.1 The Percolation Model 6.4 Biopolymers Gels 6.4.1 An Important Gel: Gelatin 6.4.2 Polysaccharide Gels 6.4.3 Modeling of the Coil ⇔ Helix Transition 189 189 189 190 192 192 193 196 197 200 200 203 204 B Percus–Yevick Equation 395 (b) (a) S(k) g(r) l l r k Fig B.1 Pair correlation function (a) and structure function (b) for a liquid in normal conditions second contribution to h(r 12 ) is an average over all positions of the intermediate molecule r : h(r 12 ) = c(r 12 ) + c(r 13 ) h(r 32 )dr (B.11) This equation in fact defines function c(r) It can be solved by taking the Fourier transform and using the properties of the convolutions H(k) = c(k) + ρ H(k) c(k) (B.12) that is: H(k) = c(k) − ρ c(k) (B.13) Substituting H(k) in (B.11), we then have: kT ∂ρ ∂p T = + ρ H(0) = 1 = − ρ c(0) − ρ c(r)dr (B.14) We must still determine c(r) This can be done by taking an approximation that describes short-range interactions Here c(r) is the difference between the function gtotal (r) corresponding to the effective interaction represented by potential w(r)(g(r) = e−βw(r) ), and the indirect interaction which is the remainder of the potential when the direct interaction is eliminated: c(r) = gtotal (r) − gindirect (r) = e−βw(r) − e−β[w(r)−u(r)] (B.15) This is the Percus–Yevick approximation for function c(r) Using the Ornstein-Zernicke method, we can then solve the system of equations Assuming y(r) = eβu(r) g(r), we can rewrite (B.9): y(r 12 ) = + ρ [e−βu(r13 ) − 1]y(r)13 )h(r 32 )dr (B.16) This is the Percus–Yevick equation As h = g −1 = eβu y −1, this equation determines y(r) and thus g(r) It was solved for hard spheres The equation for p and thus the equation of state is not the same as a function of whether it 396 B Percus–Yevick Equation is determined with the compressibility (B.9) or by direct calculation utilizing the equation for the virial (Problem 4.1): ρ2 p =ρ− kT 6kT ∞ ru(r)g(r)4πr2 dr (B.17) The Carnahan–Starling equation is established using the first method C Renormalization Group Theory Take the Ising Hamiltonian H0 for a magnetic system H0 = −J0 σi σj − µ0 H0 i,j σi (C.1) i The free energy F is the sum of the singular part near the critical point FS and the regular part FR corresponding to nonmagnetic lattice contributions It is written (7.74): e−F/kT = e−βH1 e−FR1 /kT (C.2) (Ω1 ) after taking only one out of two spins in the summation Σ, where (Ω1 ) represents the new configurations H1 has the same form as H0 but with half the spins and with new values of exchange constant J1 and applied field H0 Introducing the reduced variables K = J/kT and b = µ0 H0 /kT , we have: K1 = f (K0 ) and b1 = g(b0 ) (C.3) This process is iterated several times, doubling the scale with each operation, and a sequence of new variables is obtained: K1 , K2 b1 , b2 The critical point will be a fixed point corresponding to variables KC and bC so that: KC = f (KC ) and bC = g(bC ) (C.4) because no scale change can modify the effective Hamiltonian any further For finding TC , we operate in the vicinity of this temperature after n renormalization operations Kn = f (Kn−1 ) and KC = f (KC ), that is: δKn = KC −Kn = f (KC )−f (Kn−1 ) = f (KC −Kn−1 ) = f δKn−1 (C.5) At TC , by definition KC = J0 /kTC (renormalization no longer alters the form of the Hamiltonian) Moreover, bn = g bn−1 and at the critical point, H = bC = (C.5) is rewritten: Kn = KC − δKn = KC − f δKn−1 knowing that (C.6) 398 C Renormalization Group Theory δK0 = KC − K0 = J0 /kTC − J0 /kT = Kn = J0 [1 − f n ε]; kTC bn = g n J0 ε kTC (C.7) µ0 H0 kTC After n renormalizations, if d is the dimension of the system and n is the dimension of order parameter σ, the total number of remaining spins for N initial sites is N/(2d )n The free energy by spins Fsn is a function of reduced variables Kn and bn and thus of (f n )ε and g n H0 , with ε = (TC − T )/T As J0 , TC , and H0 are constants, we have: Fsn 2dn = F (f n ε, g n H) dn (C.8) d p d q If we put λ = , f = (2 ) , g = (2 ) , we find an equation similar to (7.69): λFsn = F (λp ε, λq H) (C.9) We have a scaling law here which allows calculating the critical exponents A Hamiltonian representing the energy, which has a Landau-type form, is then introduced: H= tP (x)2 + u4 P (x) + a|∇P (x)|2 + dd x (C.10) where t, u4 , and a are functions of T , but where t does not necessarily have the form given it in the classic Landau expansion (t can be different from (T − TC )) Application of renormalization represented by operator R results in an effective Hamiltonian 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under strong compression, in Rep Prog Phys., 59, 29–90 (1996) Houghton, J T.: The Physics of Atmosphere, (Cambridge University Press, Cambridge 1995) Leggett, A J.: Bose–Einstein Condensation in the Alkali Gases, in Reviews of Modern Physics, 73, 2, 307–356, (2001) Lecoffre, Y.: La Cavitation, (Herm`es, Paris 1994) Peixoto, J P., Oort, A H.: Physics of Climate, (American Institute of Physics, New York, 1992) Ross, M.: Matter under extreme conditions of temperature and pressure, in Rep Prog Phys., 48, 1–52 (1985) Sadhal, S S., Avyaswamy, P S., Chang, J N.: Transport Phenomena with Drops and Bubbles, (Springer–Verlag, Heidelberg 1997) Trenberth, K E (ed.): Climate System Modeling, (Cambridge University Press, Cambridge 1995) Whalley, P B.: Boiling, Condensation, and Gas-Liquid Flow, Clarendon Press, Oxford 1987) Index Acentric factor 130 Aerogels 211 Aerosol 328, 366 Alloy 10, 34, 67, 119, 217, 236, 246 Amorphous 106, 116, 119, 168, 246, 330 Antiferroelectric 231, 233 Antiferromagnetism 222 Arrhenius 171, 182 Avrami 52–54, 108 BCS theory 239 Bergeron process 366 Bethe lattice 138 Bethe model 225 Binary – alloy 12, 55 – mixture 10, 70, 145, 155 – solution 19 Binodal 134 Birefringence 262, 273, 320 Boiling 371 Born criterion 92 Bose–Einstein condensation 292, 295, 301, 355 Bragg–Williams 21, 220 Broken symmetry 26 Bubble 307, 370 – magnetic 317, 319 Cahn–Hilliard equation 60 Carnot cycle 368 Cavitation 370 Ceramic 32, 121, 247, 309, 312, 331 Chirality 253, 255, 258, 288 Clapeyron equation 80, 135 Climate 358, 365 Coexistence – curve 21 – line 135 Coherence length 239, 243 Coil 201, 205 Colloidal 29, 68, 70, 104, 189, 305, 324, 327 Composites 121 Compressibility 15, 19 Compressibility factor 136, 137 Cooper pair 239, 302, 320 Correlation length 24, 25 Corresponding states 128, 242 Critical – exponent 199, 229, 230, 236, 241, 243, 245 – lines 147, 151 – opalescence 23, 30, 131 – radius 43, 44 Crystallisation 104, 108, 185, 306 Curie–Weiss law 225 Debye theory 173 Defect 92, 309, 322, 344 Demixing 126, 147, 185 – supercritical 158 Dendrites 51, 68, 70, 71, 118, 306 Density functional 98 Diamagnetic 236 – anisotropy 261 Diamagnetism 221, 276 Diamond anvil cell 90, 348 Dielectric constant 234, 235, 274 Differential thermal analysis 30, 86 Diffraction – neutrons 101, 112 Diffusion 37, 38, 71 Diffusion coefficient 40 Dislocation 92 406 Index Distillation 12 DNA 185, 209, 210, 332 Domain 317 – magnetic 316 Droplets 140, 364 Ehrenfest 13, 15, 34, 135 Einstein model 91 Elasticity 275 Electrets 167, 230 Emulsion 324, 326 Equation of state 22, 125, 131 – Berthelot 137 – Carnahan–Starling 137, 155, 396 – Dieterici 138 – mixtures 147 – Peng–Robinson 138 – Redlich–Kwong 137 – Soave–Redlich–Kwong 138 – van der Waals 22, 34, 127, 136 – virial 136 Equilibrium – metastable 6, 55 – stable – unstable Ergodic 176, 178 Eutectic 52, 374 Extrusion 120 Ferrimagnetism 222 Ferroelectric 230, 232, 247 – domain 320 – fluid 287 Ferroelectricity 230, 258, 287 Ferromagnetic 28 Ferromagnetism 28, 221, 222 Fick’s law 38, 49 Film 247, 291, 306, 326, 335 – Langmuir 337 – Langmuir–Blodgett 338, 344 – liquid 374 – polymer 344 Flocculation 29, 327, 328 Fluctuation 22, 58, 242, 244 Fokker–Planck equation 49 Fractal 68, 71, 73, 306 Frenkel theory 39, 92 Frost 366 Frustration 229 Fullerene 3, 331, 353 Gas hydrates 375 Gel 70, 73, 189 – chemical 191 – physical 191, 203 – silica 191 Gelatin 191, 200, 209 Gelation 37, 68, 72, 116, 189 Geomaterials 353 Gibbs Duhem criterion 5, 18 Gibbs–Thomson effect 114 Glass 50, 62, 165 Grain boundaries 45, 112, 122, 306, 307, 314 Grains 306 Greenhouse effect 365, 367 Hall–Petch effect 330 Hard sphere 96, 98, 142, 173 Heat engine 367 Heisenberg – model 318 Heisenberg model 222 Helimagnetism 222 Helium – 294, 296, 297 – fountain effect 295 – four 160, 291, 293 – mixture 160 – sound propagation 300 – three 160, 291, 301–303 Helix 200, 205 Holes 92, 95 Hydrogen bond 84, 115, 182, 191, 204 Ice 2, 112, 114, 116, 366 Inertial confinement 355 Interfacial 114 Interfacial energy 309, 318 Irradiation 3, 93, 119 Ising model 34, 143, 215, 223 Isotherm 125, 128, 133 Kadanoff 34, 131, 242, 244 Kauzmann paradox 94, 170 Kevlar 290 Kohlrausch–Williams–Watts 173, 182 Landau Index – expansion 221 – model 234, 238, 285 Landau–de Gennes model 270, 273, 285 Landau–Ginzburg theory 238, 239, 242 Latent heat 2, 13, 33, 80, 176, 359, 364, 374 Lattice model 19 Legendre transformation 5, 18 Lennard–Jones 82, 104, 129, 143, 173 Lever rule 12, 156 Levitation 119 Lindemann model 90, 95, 104 Liquid crystal 2, 26, 212, 251, 252 – cholesteric 255, 287 – discotic 259 – nematic 253, 260, 286 – order parameter 260, 269, 270 – smectic 256, 257 Liquidus 10 Lyotropic 290 Magnetic susceptibility 19, 217, 229, 277 Magnetism 33 Magnetization 17, 19, 22, 215, 316 Magnetostriction 246 Magnets 246 Maier–Saupe theory 263, 269, 285 Markov process 49, 144 Martensite 62, 67, 75, 118, 309, 311 Martensitic transformation 66, 75, 311 Maxwell’s rule 133 Meissner effect 236, 321 Melting 83, 88, 90 Mesomorphic 26, 251, 290 Metamagnetic 160 Metastability 6, 20, 139 Metastable 56, 84, 168, 268 Micelle 327 Microcrystalline 174 Microcrystallites 41, 55 Microstructures 37, 51, 75, 106, 182, 305 Mode-coupling theory 176, 179 Molding 120 Molecular dynamics 22, 64, 141 407 Molecular field 34, 138, 222, 229, 235, 242 Monte Carlo method 22, 64, 143, 350 Mott – model 93 – transition 351 Nanomaterials 76, 305, 331, 332 Non-equilibrium thermodynamics 192 Nuclear Magnetic Resonance 30, 34, 229, 246, 247, 262 Nuclear reactor 368, 369, 374 Nucleation 3, 22, 37, 38, 40, 139 – boiling 371 – heterogeneous 44, 45, 364, 370 – rate 46, 47 Order parameter 17, 34, 199, 218, 238 Ornstein–Zernike equation 25, 100, 393 Overheated 139 Pair correlation function 394 Paramagnetism 221 Percolation 72, 73, 116, 197 Percus–Yevick equation 102, 395 Peritectic 11 Perovskite 27, 231, 247, 354 Phase diagram Phase transition 1, 3, 6, 31, 367 Phases Piezoelectricity 33, 230 Plasma 4, 347, 355 Plasma crystals 106 Plastic crystal 28, 252 Point – azeotropic 156 – bubble 132, 146 – condensation 363 – critical 6, 125, 130, 229 – dew 132, 146, 152, 362 – eutectic 10, 66, 309 – λ 160, 294 – Leidenfrost 371 – multicritical 15 – supercritical 158 – tricritical 9, 159, 343 – triple 6, 82, 126, 146 Poisson equation 363 408 Index Polarization 230, 233–235, 282 Polycrystalline 45, 121, 305, 330 Polyelectrolites 203, 210 Polymer 62, 106, 108, 110, 120, 152, 185, 192, 200 Polysaccharides 191, 200, 203 Porcelain 121, 313, 316 Premelting 111, 112 Protein 185, 186, 200, 290 Pyroelectric 247 Pyroelectricity 230 Quasicrystal 311 Quenched 311 Quenching 55, 62, 67, 75, 118, 167, 184, 312 Rankine cycle 368 Renormalization group theory 34, 131, 244, 245 Retrograde condensation 152 Richard rule 80 Scaling laws 34, 131, 242 Scattering – light 25, 30, 62 – neutrons 25, 30, 62, 115, 229 – Raman 66, 115 Shear modulus 195 Sintering 121, 122, 312–314 Soft mode 65, 233 Sol 189 Sol–gel method 211 Solidification 83, 96, 98, 103, 104, 374 Solution – regular 20 – solid 10 Spherulites 51, 106, 109, 192, 306, 309 Spin glass 228 Spinodal 6, 20, 86, 134 Spinodal decomposition 37, 56, 61, 246 States Steel 11, 62, 66, 75, 118, 309 Stokes–Einstein law 39, 176 Structure function 24, 61, 70, 175, 394 Supercavitation 371 Superconductivity 236, 247 Superconductor 320 – high temperature 240, 241, 321, 322 Supercooled 43, 84, 117, 166, 168, 182, 194 Supercooling 44, 85, 94, 366 Supercritical extraction 159 Superfluid 160, 291 Superfluidity 292, 301 Superheated Superheating 86, 95, 369, 371 Surface melting 95, 112 Surfactants 159, 325 Symmetry 17, 25 Symmetry breaking 16, 82, 96, 216 Syneresis 192, 210 Technological applications 31, 118, 183, 209, 245, 286, 326, 329, 367, 374 Temperature – Bose–Einstein condensation 297 – critical 1, 220, 296 – Curie 222 – gelation 192 – glass transition 106, 169, 170 – N´eel 222 – superfluid transition 297 – transition isotropic liquid-nematic 267, 272 Thixotropy 212 Tisza model 298 Tokamak 4, 355 Transition – superfluid A–superfluid B 302 – boiling 371 – coil–helix 201, 208, 209 – conformational 204 – displacive 26, 231 – ferroelectric–paraelectric 27, 230 – ferromagnetic–paramagnetic 222, 225 – first order 13, 16, 17, 204 – Frederiks 280, 281, 284 – glass 3, 26, 29, 106, 165 – helix–coil 204 – insulator-metal 351, 352 – isotropic liquid–nematic 263 – Kirkwood–Alder 105 – Kosterlitz–Thouless 324 – liquid crystal 26 – liquid–gas 367 Index – liquid–solid 111 – metal-insulator 28 – multicritical 15 – nematic–smectic A 284 – order–disorder 218 – order-disorder 26, 231, 234 – plastic crystal-crystal 28 – second order 15, 16, 217, 343 – second-order 221 – sol–gel 29, 68, 189 – solid–liquid 103 – solid–solid 47 – structural 26 – structural 64 – superconducting 28, 236 – superfluid 29, 293 Turnbull 85 Two-phase system 305, 324, 367, 369, 374 Universality Undercooled Zeldovitch–Frenkel equation 139 409 34, 241 Vacancies 40 van der Waals force 149, 191, 263, 328 Variance Virial coefficients 136 Virial expansion 22 Viscosity 195, 199, 292, 294 Vitreous 165 Vogel–Tammann–Fulcher 172, 177 Volmer model 41, 47 Vortex 300, 321, 323 Water 84, 114, 116, 159, 184, 325, 335, 358 Weiss 219, 223, 317 Wetting 114 Wilson theory 244 Xerogels 211 49 [...]... case of a liquid like water, the intervention of hydrogen bonds between the molecules explains the abnormal properties of this liquid (for example, its density maximum at 4◦ C and the fact that the density of the solid phase is lower than the density of the liquid phase) In general, phase transitions are a central problem of materials science: the relationship between the macroscopic properties and the. .. Investigating Phase Transitions For a material that can undergo a phase transition, three types of questions can be investigated: the conditions of the phase change (temperature, 18 1 Thermodynamics and Statistical Mechanics of Phase Transitions pressure, magnetic field, etc.); the behavior of physical quantities in the vicinity of the transition; the new properties of the material when it undergoes a phase. .. see the very wide variety of phase transitions that can be encountered with different types of substances and materials involving a large number of properties and phenomena The study of phase transition phenomena and their applications is the subject of this book We will consider the applications of phase transitions to technical and natural systems in each chapter of the book as a function of their... these transitions, we continuously pass from one phase to another without being able to really speak of the coexistence of the two phases: at the liquid/gas critical point, the liquid phase can no longer be distinguished from the gas phase (their densities are strictly equal) These very different thermodynamic behaviors can be demonstrated experimentally by directly or indirectly studying the thermodynamic... the symmetry groups of the two phases are such that none is strictly included in the other: they are always first-order (with latent heat) in Ehrenfest’s sense; • transitions for which an order parameter can be defined and for which the symmetry group of the least symmetric phase is a subgroup of the symmetry group of the most symmetric phase If the order parameter is discontinuous at the transition, it... points with coordinates GA and GB , and the slope of the tangent is equal to the difference in chemical potentials µA − µB of B and A in the mixture This is a general property of phase diagrams In fact, phase diagrams especially have the advantage of allowing us to discuss the conditions of the existence and thus the stability of multiphasic systems as a function of thermodynamic variables such as temperature,... with the liquid If the phase rule permits determining the regions of stability for the different phases of a system (for example, a binary alloy), it nevertheless does not allow calculating the fractions of liquid and solid phases in equilibrium This can be done with the so-called lever rule, illustrated for the binary copper–nickel alloy in Fig 1.9 If x designates the concentration of copper in the. .. observed in the transition At point C, on the contrary, we continuously pass from the liquid phase to the vapor phase: there is neither latent heat nor density discontinuity We can thus schematically say that two types of phase transitions can exist: transitions with latent heat on one hand, and transitions without latent heat on the other hand This is a thermodynamic classification More generally, the physicist... corresponds to the G(x) diagrams at 800 K for phases β and f.c.c.; they have a common tangent at a and b The chemical potentials µAg and µM g are identical for these two phases, which are thus in equilibrium Gi Gm is the free enthalpy of formation of phase β of composition xi from pure solids The segments of the curves in dashed lines correspond to metastable phases β and f.c.c The concentration of Mg is... discontinuity of state variables such as the density, for example This is the situation encountered at the critical point of the liquid/gas transition and at the Curie point of the ferromagnetic/paramagnetic transition The thermodynamic characteristics of phase transitions can be very different Very schematically, there are two broad categories of transitions: 1.1 What is a Phase Transition? 3 those associated