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Vladimir P. Skripov and Mars Z. Faizullin Crystal-Liquid-Gas Phase Transitions and Thermodynamic Similarity Related Titles Baidakov, V. G. Mazenko, G. F. Explosive Boiling of Superheated Equilibrium Statistical Cryogenic Liquids Mechanics approx. 400 pages with approx. 105 figures 630 pages and approx. 15 tables 2000 2006 Hardcover Hardcover ISBN 0-471-32839-1 ISBN 3-527-40575-5 Senatskommission zur Beurteilung von Sinaiski, E., Lapiga, E. J. Stoffen in der Landwirtschaft (ed.) Separation of Multiphase,Multi- Transient Phenomena in Multi- component Systems phase and Multicomponent approx. 550 pages with approx. 240 figures Systems and 22 tables Research Report 2006 395 pages with 241 figures and 10 tables Hardcover 2000 ISBN 3-527-40612-3 Softcover ISBN 3-527-27149-X Schmelzer, J. W. P. Nucleation Theory and Applica- tions 472 pages with 157 figures 2005 Hardcover ISBN 3-527-40469-4 Mazenko, G. F. Fluctuations,Order,and Defects 540 pages 2003 Hardcover ISBN 0-471-32840-5 Kostorz, G. (ed.) Phase Transformations in Materials 724 pages with 392 figures and 33 tables 2001 Hardcover ISBN 3-527-30256-5 Vladimir P. Skripov and Mars Z. Faizullin Crystal-Liquid-Gas Phase Transitions and Thermodynamic Similarity WILEY-VCH Verlag GmbH & Co.KGaA The Authors All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and Vladimir P. Skripov publisher do not warrant the information contained in Institute of Thermal Physics of the Ural Branch these books, including this book, to be free of errors. of the Russian Academy of Sciences Readers are advised to keep in mind that statements, [email protected] data, illustrations, procedural details or other items may inadvertently be inaccurate. Mars Z. Faizullin Institute of Thermal Physics of the Ural Branch Library of Congress Card No.: of the Russian Academy of Sciences applied for [email protected] British Library Cataloguing-in-Publication Data Consultant Editor A catalogue record for this book is available from the British Library. Jürn W. P. Schmelzer University of Rostock Bibliographic information published by Physics Department Die Deutsche Bibliothek [email protected] Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>. © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim All rights reserved (including those of translation into other languages). No part of this book may be repro- duced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Printing Strauss GmbH, Mörlenbach Binding J.Schäffer Buchbinderei GmbH, Grünstadt Printed in the Federal Republic of Germany Printed on acid-free paper ISBN-13: 978-3-527-40576-3 ISBN-10: 3-527-40576-3 Contents Foreword VII 1 Introduction 1 1.1 BasicAimsandMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 StatesofAggregation.PhaseDiagramsandtheClausius–ClapeyronEquation 2 1.3 MetastableStates. RelaxationviaNucleation . . . . . . . . . . . . . . . . . 3 1.4 PhaseTransformationsinaMetastablePhase. HomogeneousNucleation . . 6 2 Liquid–GasPhaseTransitions 11 2.1 BasicFact: ExistenceofaCriticalPoint . . . . . . . . . . . . . . . . . . . 11 2.2 MethodofThermodynamicSimilarity . . . . . . . . . . . . . . . . . . . . 19 2.3 SimilarityNeartheCriticalPoint:TheChangeofCriticalIndices . . . . . . 22 2.4 New UniversalRelationships for Liquid–VaporPhase Coexistence in One- ComponentSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4.1 Correlation Between Pressure and Densities of Liquid and Vapor AlongtheSaturationCurve . . . . . . . . . . . . . . . . . . . . . . 27 2.4.2 CorrelationBetweenCaloricPropertiesandDensitiesofLiquidand VaporAlongtheSaturationCurve . . . . . . . . . . . . . . . . . . 30 2.4.3 Correlation Between Surface Tension and Heat of Evaporation of NonassociatedLiquids . . . . . . . . . . . . . . . . . . . . . . . . 35 2.4.4 One-ParameterCorrelationfortheHeatofEvaporationofNonasso- ciatedLiquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3 Crystal–LiquidPhaseTransitions 47 3.1 TheBehavioroftheCrystal–LiquidEquilibriumCurveatHighPressures . . 47 3.2 Experimental Methods of Investigation of Melting of Substances at High Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3 ApplicationofSimilarityMethodsforaDescriptionofMelting . . . . . . . 54 3.4 The Extension of the Melting Curve into the Range of Negative Pressures andtheScalingofThermodynamicParameters . . . . . . . . . . . . . . . . 58 3.5 InternalPressureinaLiquidAlongtheEquilibriumCurveswithCrystaland Vapor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.6 Stability of ThermodynamicStates and the Metastable Continuationof the MeltingCurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Crystal–Liquid–GasPhaseTransitionsandThermodynamicSimilarity.VladimirP.SkripovandMarsZ.Faizullin Copyright©2006WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim ISBN:3-527-40576-3 VI Contents 3.7 TheBehavioroftheViscosityofaLiquidAlongtheCoexistenceCurvewith theCrystallinePhase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.8 TheBehaviorofVolumeandEntropyJumpsAlongtheMeltingCurve . . . 94 3.9 TheSurfaceTensionofSimpleLiquidsAlongtheMeltingCurve . . . . . . 97 3.10 CorrelationsBetweenThermodynamicPropertiesCharacterizingMelting. . 103 3.11 MeltingandCrystallizationofSmallParticles . . . . . . . . . . . . . . . . 116 3.11.1 ThermodynamicAspects . . . . . . . . . . . . . . . . . . . . . . . 116 3.11.2 KineticAspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4 PhaseTransitionsinSolutions 125 4.1 GeneralizedClausius–ClapeyronEquationforSolutions . . . . . . . . . . . 125 4.2 ApplicationoftheGeneralizedClausius–ClapeyronEquationforthePlotof thePhaseDiagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.3 ThermodynamicCorrelationsforPhase-SeparatingSolutions . . . . . . . . 138 4.4 ExperimentalStudiesofPhase-SeparatingSolutions . . . . . . . . . . . . . 141 4.5 ThermodynamicSimilarity of Phase-SeparatingBinary Solutions with Up- perCriticalDissolutionTemperature . . . . . . . . . . . . . . . . . . . . . 145 4.6 Thermodynamic Similarity of Phase-Separating Binary Solutions with LowerCriticalDissolutionTemperature . . . . . . . . . . . . . . . . . . . 150 4.7 ConcludingRemarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 A Appendices 157 A.1 ListofSymbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 A.2 SuperscriptsandSubscripts . . . . . . . . . . . . . . . . . . . . . . . . . . 159 References 161 Index 173 Foreword Thismonographiswrittenbytwooutstandingspecialistsinthefieldoftheexperimentaland theoreticalanalysisoffirst-orderphasetransitions,AcademicianProf.VladimirP.Skripovand Dr.MarsZ.Faizullin.Itpresentsforthefirsttimeacompleteoverviewontheresearchofboth authorsonthecomparativeanalysisof solid–liquidandliquid–vaporphasetransitions, their similaritiesanddifferenceswithspecialemphasisontheaspectsofthermodynamicsimilarity. Prof. Skripov has been the founder of an outstanding school of research in the above mentioned field; for several decades he worked as the director of the Institute of Thermal PhysicsoftheUralBranchoftheRussianAcademyofSciencesinEkaterinburg,Russia. His resultsarepublishedinfourmonographsandmorethan300journalpublications.Inparticular, IwouldliketomentionhismonographMetastableLiquids(Nauka,Moscow,1972),published in the English version by WILEY in 1974, which is still highly popular among scientists dealing with the processes of boiling of liquids. Prof. Skripov is presently apppointedas a CouncilloroftheRussianAcademyofSciences. Dr. Faizullin has been engaged in the problems of thermodynamic similarity in liquid– vaporandcrystal–liquidphasetransitionsformorethantwo decades. He defendedbothhis PhDandDScthesesonrelatedtopics.Heistheauthorofmorethansixtyarticlesinscientific journals and one monograph. Presently, he is deputy director of the Institute of Thermal PhysicsoftheRussianAcademyofSciencesinEkaterinburg,Russia. The results of their long-standinghighly original investigations have been presented by theauthorsanddiscussedwithmuchinterestinseveraloftheresearchworkshopsNucleation Theory and Applications at the Bogoliubov Laboratory of Theoretical Physics of the Joint InstituteforNuclearResearchinDubnanearMoscoworganizedbytheeditorofthepresent bookregularlyeachyearsince1997. Firstaccountsoftheresultsarepublishedinthework- shop proceedings (Nucleation Theory and Applications, Dubna 1999 and 2002) and in the monographNucleationTheoryandApplicationspublishedbyWILEY-VCHin 2005. Itisa real pleasure to have the opportunityto presentnow the extendedEnglish translation of the monographoftheauthorspublishedinRussianin2003.Ibelieve,thepresentmonographcan beofsimilarimpactontheresearchinthefieldoffirst-orderphasetransitionsasthealready citedmonographofVladimirP.SkripovpublishedbyNaukain1972andbyWILEYin1974. Inthetitleofthebooktheterm“gas”butnot“vapor”isused. Itisconvenientforades- ignationofthethreestatesofaggreationofmatter: crystal,liquid,andgas. Butindiscussing thecoexistenceofthedifferentphasesitispreferabletousetheterm“vapor”butnot“gas”. Insuchaway,itisdoneinthisbook. Crystal–Liquid–GasPhaseTransitionsandThermodynamicSimilarity.VladimirP.SkripovandMarsZ.Faizullin Copyright©2006WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim ISBN:3-527-40576-3 VIII Preface Finally, I would like to acknowledge the valuable assistance of Dr. Irina G. Polyakova (St.Petersburg,Russia),Dr.AlexanderS.AbyzovandAndrewA.Abyzov(Kharkov,Ukraine) inthepreparationofthebookforpublication. Rostock(Germany)&Dubna(Russia),August2005 JürnW.P.Schmelzer Preface Inthismonograph,awidespectrumofthermodynamicaspectsoffirst-orderphasetransitions isanalyzed. Herebytheanalysisisextendedbeyondtherangeofphasecoexistenceofstable phases in order to incorporatephase coexistence of metastable states and an analysis of the limitsofstabilityofthemetastablycoexistingphases. Thisextensionoftheanalysisallowed us to arrive at a variety of previously unknown relationships reflecting the thermodynamic similarity of different one-componentsubstances in the phase transitions crystal–liquid and liquid–vapor. Thisapproachisthenextendedtothedescriptionofphaseequilibriaofbinary solutionswithupperandlowercriticaldissolutionpoints. Asalreadymentioned,oneofthespecificcharacteristicsofthismonographconsistsinthe extensionof theanalysisof liquid–vaporand crystal–liquidphasecoexistenceto metastable states. Insomerespects,thesetwoclassesofphasetransformationprocessesbehavesimilarly; however,thereexistavarietyoffeaturesdistinguishingthesetransformations.Forexample,a coexistenceofliquidandvaporcanoccuratpositivepressures,exclusively. Withanincrease oftemperature,thephasecoexistenceisterminatedatthecriticalpointwherethebothcoex- istingphasesbecomeidentical.Thethermodynamicpropertiesofthefluidatthecriticalpoint mayserveasscalingparametersallowingustoestablishthesimilarityofdifferentsubstances withrespecttoliquid–vaporphasetransitions. Thecriticalpointissimultaneouslycharacter- izedbytheapproachoftheboundaryofstabilityofthefluid.Incontrast,themeltingcurvesdo notexhibitsuchhigh-temperaturelimitofphasecoexistence.Inaddition,liquid–crystalphase equilibriamaybepreservedalsoatnegativepressures(i.e.,ifbothphasesareexposedtosome tensilestress). Themeltingcurvehasametastablecontinuationtolowertemperaturesbeyond the triple point of crystal–liquid–vaporequilibriumcoexistence. In the case of crystal–melt phasecoexistence,scalingparametersmaybechosenestablishedbyemployingcharacteristic parametersobtainedfortheasymptoticlimitT → 0. Withanincreaseofthetensilestrength (oranincreaseoftheabsolutevalueofthenegativepressuretowhichthesystemisexposed), thedegreeofmechanicalstabilityofbothliquidandcrystalcoexistingphasesisdecreasedbut thestabilityboundariesarenotnecessarilyreached. This monographis basically a translation of the Russian version of the book published recently (V. P. Skripov, M. Z. Faizullin: Phase Transitions Crystal–Liquid–Gas and Ther- modynamic Similarity, Fizmatlit, Moscow, 2003 (in Russian)). However, some additional Preface IX paragraphsareadded. InSection2.3,adiscussionoftheuniversalbehaviorofdifferentsub- stancesintheapproachoftheliquid–vaporcriticalpointisgiven. Thisanalysisisconnected withtheexperimentallyobservedcrossoverfromclassicalmean-fieldtoanonclassicalbehav- ior. In Section 3.5, estimates of the magnitude of the internal pressure in metals are given, inSection3.6thelimitsofthermodynamicstabilityofthecondensedphasesofalkalimetals arediscussedandinSection3.10acomputationofthechangeoftheentropyatanisothermal homophaseexpansionofsolidmetallicsubstancesisperformed. Finally,theauthorswouldliketoexpresstheirdeepgratitudetoProf.J.W.P.Schmelzer, who performedmuch work in the preparation of the English version of our monographfor publication. Ekaterinburg(Russia),August2005 VladimirP.Skripov MarsZ.Faizullin 1 Introduction 1.1 Basic Aims and Methods Theproblemsofthethermodynamicandkineticdescriptionofequilibriumphasetransitions offirstorderarediscussedinvariousoriginalpublicationsandmanuals. Itmayseemthatthe commonly employed general approach to the different forms of phase transitions based on Gibbs’s theory is quite sufficient for mostcases of applicationand further detailed analyses arenotrequired.Ontheotherhand,itiseasytonoticethatcrystal–liquidphaseequilibriaand phase transitions(thetermsphase transitionand phase equilibriumare oftenemployedhere withasimilarmeaningexceptforthecaseswhenthetransformationkineticsisstudied)arenot as thoroughlyanalyzedas comparedto the liquid–vaporphase transition. In the latter case, the existence of a critical point in the coexistence of two fluid phases defines characteristic scales ofthermodynamicvariables(volume,temperature,pressure, entropyandenergy)and allowsonetointroducetheconceptsofcorrespondingstatesandthermodynamicsimilarityof varioussubstances. Forthecrystal–liquidphasetransitionsofsimplesubstancesthesituationisdifferent.The meltinglineswerefoundnottocontainwiththeincreaseoftemperatureafundamentalsingu- larityliketheliquid–vaporcriticalpoint. Thisfeaturemakesimpossiblethenaturalchoiceof somescalingparameterssimilartotheliquid–vaporphasetransition. Butonecanimplement anotherapproachtotheproblem,whichisbasedonthelow-temperatureasymptoticbehavior of themeltinglinesof substancesofnormaltype. Sucha procedurerequiresoneto include intothethermodynamicconsiderationthebehavioroftherespectivephasesatmetastablecon- ditions. The mere fact of considering systems at such conditions represents one of the dis- tiguishing features of the presentbook. Here the problem of similarities and differencesof crystal–liquidandliquid–vaporphase transitionsin single-componentsystems is the central problemunderconsideration. Inthisanalysis, muchattentionisdevotedtotherevelationof thethermodynamicsimilarityinthebehaviorofdifferentsubstancesatthephasetransitions. The reason for this is that similarity concepts exhibit the very general deep properties of a classofeffectsretainingsomeparticulardifferencesin behaviorin otherparticularrespects. Themorecompleteistheunderstandingofthenatureoftheeffectsconsideredthemorecom- pletely and clearly the similarity in the behaviorcan be exhibited. In the present work, the mentionedconnectionisclearlydemonstratedinapplicationbothtoliquid–vaporandcrystal– liquidphasetransitionsfirstinapplicationtothebehaviorofone-componentsystems. Inthe final chapter, the analysis is then extended to crystal–liquid–vapor and liquid–liquid phase equilibriaintwo-componentsystemsanalyzedfromthesamepointofview. Thepresentworkiswrittenemployingbasicallytheframeworkofphenomenologicalther- modynamics. Theapplicationofstatistical-mechanicalapproaches,e.g.,byutilizingGibbs’s Crystal–Liquid–GasPhaseTransitionsandThermodynamicSimilarity.VladimirP.SkripovandMarsZ.Faizullin Copyright©2006WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim ISBN:3-527-40576-3