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Long-Qing Chen Thermodynamic Equilibrium and Stability of Materials Thermodynamic Equilibrium and Stability of Materials Long-Qing Chen Thermodynamic Equilibrium and Stability of Materials Long-QingChen DepartmentofMaterialsScienceandEngineering DepartmentofMathematics DepartmentofEngineeringScienceandMechanics MaterialsResearchInstitute ThePennsylvaniaStateUniversity UniversityPark,PA,USA ISBN978-981-13-8690-9 ISBN978-981-13-8691-6 (eBook) https://doi.org/10.1007/978-981-13-8691-6 ©SpringerNatureSingaporePteLtd.2022 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Preface Thisbookonthermodynamicsofmaterialsisintendedasatextbookforjunior/senior undergraduatestudentsandfirst-yeargraduatestudentsinmaterialsscienceandengi- neering and related fields as well as a reference for researchers who would like to refreshtheirunderstandingofbasicthermodynamicsandneedtoderivethermody- namicrelationsfortheirresearch.Imadeaconsciousefforttoexplainthethermo- dynamic concepts and quantities using plain languages. The focus of this book is onthefundamentalsofthermodynamicsandtheirapplicationstogenericthermody- namicsystemsandprocessesratherthanspecificmaterialssystems.Areaderafter finishing the book is expected to achieve a high-level fundamental understanding of thermodynamics and acquire the analytical skills of applying thermodynamics to determining materials equilibria under different thermodynamic conditions and computingdrivingforcesformaterialsprocesses. Oneofthepurposesforwritingthisbookistosharemyownunderstandingofther- modynamicsthroughnearly30yearsofteachingthermodynamicsofmaterialsatthe PennsylvaniaStateUniversity.Myunderstandingofthermodynamicshasprimarily been based on reading “Scientific Papers of Josiah Willard Gibbs” which laid the foundationofthermodynamicsandillustratedthetruemathematicalrigorandbeauty ofthermodynamics.1Ialsobenefitedgreatlyreadinganumberofmodernthermody- namicsbooks,includingThermodynamicsandanIntroductiontoThermostatistics by Herbert B. Callen2, Thermodynamics and Its Applications by Michael Modell andRobertC.Reid3,ModernThermodynamics:FromHeatEnginestoDissipative StructuresbyDilipKondepudiandIlyaPrigogine4,TheDynamicsofHeatbyHans 1J. Willard Gibbs, “On the Equilibrium of Heterogeneous Substances,” Transactions of the ConnecticutAcademy,III.pp.108–248,October1875–May1876,and343–524,May1877–July 1878. 2HerbertB.Callen,“ThermodynamicsandanIntroductiontoThermostatistics,”SecondEdition, Wiley,1985. 3MichaelModellandRobertC.Reid,“ThermodynamicsanditsApplications,”SecondEdition, PrenticeHallPTR,1983. 4DilipKondepudiandIlyaPrigogine,“ModernThermodynamics:FromHeatEnginestoDissipa- tiveStructures,”FirstEdition,JohnWiley&Sons,1998. v vi Preface U.Fuchs5,andPhaseEquilibria,PhaseDiagramsandPhaseTransitionsbyMats Hillert6. There are several existing excellent textbooks specifically devoted to applica- tionsofthermodynamicstomaterialsscienceandengineering,e.g.,Introductionto ThermodynamicsofMaterialsbyDavidR.GaskellandDavidE.Laughlin7,Ther- modynamicsofMaterialsbyDavidV.Ragone8,andThermodynamicsinMaterials SciencebyRobertDeHoff9.Thenaturalquestionwouldbe“whyanothertextbook onthermodynamics?”Itrytoanswerthisquestionbyemphasizingseveral,whichI believe,ratheruniquefeaturesofthisbook: (cid:129) EntropySandchemicalpotentialµ,consideredbymanyastwoofthemostdiffi- cultandabstractconceptstofullycomprehendinthermodynamics,areintroduced veryearlyinChap.1aspartofthedefinitionsandterms,emphasizingtheanalog ofchemicalpotentialµtotemperatureT andpressurepasaformofpotential, andtheanalogofStovolumeV andamountofchemicalsubstanceNasatypeof matter.Allsevenbasicthermodynamicvariables(U,S,V,N,T,p,µ)associated withasimplethermodynamicsystemasdefinedbyGibbsareseparatedintothree categories:energy(U),matter(S,V,N),andpotential(T,p,µ),emphasizingnot onlytherelationsbutalsothedifferencesamongthethreetypesofthermodynamic quantities:energy,matter,andpotential. (cid:129) Thediscussiononthefirstlawofthermodynamicsisprimarilyfocusedonopen systems rather than closed systems as it was done in essentially all modern textbooks on thermodynamics of materials. Open systems, by definition, allow exchangesofallthreekindsofmatter(entropy,volume,andchemicalsubstance) and thus the exchanges of the three forms of energy (thermal, mechanical, and chemical)betweenasimplesystemanditssurrounding.Oneofthebenefitsfor discussingthefirstlawofthermodynamicsusingopensystemsistheearlyintro- ductionanddiscussionofchemicalpotentialintermsofchemicalenergyorthe Gibbsfreeenergyperunitamountofmatterorthederivativesofdifferentther- modynamic potential energy functions with respect to the amount of substance underdifferentthermodynamicconditions. (cid:129) Thesecondlawofthermodynamics,anotherconceptoftenperceivedtobedifficult in thermodynamics, is quantified by directly connecting the amount of entropy producedtotheamountofthermodynamicenergydissipatedordrivingforce,and thusallowingonetoquantitativelycalculatetheamountofentropyproducedin any given irreversible thermodynamic process using thermodynamic quantities 5Hans U. Fuchs, “The Dynamics of Heat: A Unified Approach to Thermodynamics and Heat Transfer,”SecondEdition,Springer-VerlagNewYork,2010. 6M. Hillert, “Phase Equilibria, Phase Diagrams and Phase Transformations,” Second Edition, CambridgeUniversityPress,Cambridge,2008. 7DavidR.GaskellandDavidE.Laughlin,“IntroductiontotheThermodynamicsofMaterials,” SixthEdition,CRCPressLLC,2018. 8DavidV.Ragone,“ThermodynamicsofMaterials,”VolumeIandII,JohnWiley&Sons,Inc., 1995. 9RobertDeHoff,“ThermodynamicsinMaterialsScience,”CRCPressLLC,2006. Preface vii associated with the system of interest. In the existing literature, the connection betweenentropyincreaseandenergydissipationisoftendiscussedinirreversible thermodynamics10. (cid:129) Thebookfullyembraces,ratherthanshiesawayfrom,themathematicalbeauty and rigor of Gibbs thermodynamics throughout the book by emphasizing the importanceoffundamentalequationofthermodynamicsfromwhichallthermo- dynamic properties of a material can be derived. However, a reader with basic first-yearundergraduatecalculusskillsisexpectedtobeabletogetthroughthe book without too much difficulty. It offers detailed descriptions of the step-by- stepproceduresforcomputingallthethermodynamicpropertiesfromthefunda- mentalequationofthermodynamicsortheconstructionoffundamentalequations of thermodynamics in different alternative functional forms based on a set of common, experimentally measurable thermodynamic properties. The emphasis ofthechapteronthebriefintroductiontostatisticalthermodynamicsisalsoonthe connectionbetweenthemicroscopicinteractionsinasystemandthefundamental equationofthermodynamicsassociatedwiththesystem. (cid:129) By far, the most emphasized concept of this book is chemical potential and its applicationstomaterialsscienceandengineering.Notonlytheconceptofchem- icalpotentialisintroducedattheverybeginningofthebooktogetherwithtemper- atureandpressure,butalsoessentiallyalltheapplicationsofthermodynamicsare discussed based on the concept of chemical potentials, e.g., phase equilibria in single-componentsystemsaswellasbinaryormulticomponentsolutions,chem- icalreactionequilibria,andlatticeandelectronicdefectsincrystals,andmulti- physicsproblemsincludingcouplingamongchemistry,mechanics,andelectricity. OnecanevendiscusschemicalreactionequilibriawithoutusingthetermofGibbs free energy. The thermodynamic driving force for the reaction 2A+B = A B 2 can be perfectly described by the difference between the chemical potential of theinitialstate(thereactants)withchemicalpotentialµr =2µ +µ (inthe A2B A B state of mixture of A and B) and the chemical potential of the final state (the product)µp = µ (intheformofacompoundormolecule,A B).Itisthe A2B A2B 2 chemicalpotentialthatdeterminesthestabilityofchemicalspecies,compounds, andphasesandtheirtendencytochemicallyreacttoformnewspecies,transform tonewphysicalstates,andmigratefromonespatiallocationtoanother.There- fore,itisthechemicalpotentialdifferencesorgradientsthatdriveessentiallyall materialsprocessesofinterest. (cid:129) The book avoids or at least minimizes the use of terms such as molar Gibbs freeenergyorpartialmolarGibbsfreeenergybecausemolarGibbsfreeenergy ofamaterialorpartialmolarGibbsfreeenergyofacomponentispreciselythe chemicalpotentialofthecorrespondingmaterialorthecorrespondingcomponent regardlessofwhetherthematerialisasingleormulticomponentsystem.Thereis acommonmisconceptionintheliteraturethatthestatement,“molarGibbsfree energyofasystemisequaltoitschemicalpotential,”ortheequation,G/N=µ, 10SybrenR.deGrootandPeterMazur,“Non-EquilibriumThermodynamics,”DoverPublications, Inc.,New.York,1984. viii Preface whereGisGibbsfreeenergy,N isnumberofmolesofchemicalsubstance,and µisitschemicalpotential,isonlytrueforsingle-componentsystems.Potentials areuniforminahomogeneoussystematequilibrium,andtheyarealsouniform withinaheterogeneoussystemifthecorrespondingtypesofmatterareallowed tofreelyflowortransportacrosstheinterfacesamongthedifferenthomogeneous regions regardless of the external thermodynamic conditions. Molar Gibbs free energyofasystematagivenoverallcompositionisuniforminbothhomogeneous andheterogeneoussystemsinwhichallthechemicalspeciesareallowedtofreely redistributeacrosstheinterfacesbetweenthedifferenthomogeneous regionsor phases and the whole system is in thermodynamic equilibrium. Therefore, the molarGibbsfreeenergyofamulticomponentsystemisapotential,thechemical potential.ThispointwasalsopointedoutintheoriginalpaperbyGibbsaswellas inthebookbyHillert.AccordingtoGibbs,forthepurposeofdefiningchemical potential,“anychemicalelementorcombinationofelementsingivenproportions may be considered a substance, whether capable or not of existing by itself as a homogeneous body.” One can define the chemical potential of a component by“choosingasoneofthecomponentsthematterconstitutingthebodyitself,” so that the molar Gibbs free energy of a system “may always be considered as a potential.” According to Hillert, “Actually, one can define a component with the same composition as the whole system. The chemical potential of such a componentisequaltoG …”whereG isthemolarGibbsfreeenergy. m m (cid:129) Another feature for this book is the rather extensive discussion on the thermodynamics of electron systems. Existing thermodynamics books in materials are largelyfocusedonthethermodynamicsofphaseequilibria,andthereisminimal coverage of thermodynamics of electronic systems. The concept of chemical potentialorelectrochemicalpotentialofelectronicdefectsisextremelyimportant forunderstandingthethermodynamicsofmanyprocessesinvolvingelectronssuch aselectronictransportinthermoelectricsandphotovoltaicsandelectrochemical reactionsinbatteries,fuelcells,andcorrosion,etc.Thisbookdiscussesthecontri- butionsofelectronstothermodynamicsbytreatingthemanadditionalchemical, chargedspecies. Itshouldbeemphasizedthatafewtopicsthatwereemphasizedinmanyofthe existingthermodynamicstextbooksarenottreatedindetailinthisbook.Forexample, althoughtheconceptofelectrochemicalpotentialisintroduced,anditsapplications inenergyconversionsarebrieflydiscussed,thediscussionsonthermodynamicsof electrolytesolutionsarenotincluded.Furthermore,theconceptofinterfacialther- modynamics is touched upon in a number of sections in the book, but there is no separatechapterdevotedtothistopic.Inaddition„thewholebookhasbeenfocused onheterogeneousbulksystemsconsistingofseveralhomogeneousregions,andthe thermodynamics of diffuse interfaces is not discussed. Finally, the book does not includeanytablesofthermodynamicdatasinceitisnowmucheasierforastudent todigoutdatadirectlyfromInternetsources. Foradoptingthebookasatextforanundergraduatecourseinthermodynamicsof materials,thefollowingsectionsmaybeskipped:2.4.6–2.4.10,2.5.4–2.5.7;3.2,3.3, Preface ix 3.5–3.6;Chap.4;5.3–5.6;6.3.3–6.3.6;7.8–7.9;8.3–8.5;9.6.3;10.1.2,10.2.1,10.2.4, 10.3,10.4.3–10.4.5;11.9–11.10;12.8,12.10,12.14,12.24,12.26,12.28,12.29. I would like to acknowledge the help and contributions from many students in thermodynamics classes over the years, colleagues, current and former graduate students,andpostdocsforproof-readingthedifferentversionsofbookmanuscript. In particular, I would like to thank Prof. Jiamian Hu (University of Wisconsin), Prof. Kasra Momeni (University of Alabama), Dr. Yanzhou Ji (Penn State), Prof. YuhongZhao(NorthUniversityofChina),Prof.ZijianHong(ZhejiangUniversity), Mr. Carter Dettor (Penn State), Mr. Erik Furton (Penn State), Dr. Chengchao Hu (Liaocheng University), Mr. Haowei Zhang (SDIC Unity Capital Co), Ms. Tina J. Chen (UC Berkeley), Prof. John Mauro (Penn State), Prof. Yi Wang (Penn State), Ms.SandraElder(PennState),andmanyothers. ThesolutionstotheexercisequestionswereprimarilyworkedoutbyDr.Yuhui HuangfromZhejiangUniversitywhovisitedPennStateformorethanayearasa visitingscholar,andaseparatesolutionmanualwillbepublishedandavailableasa companiontothistextbook. Finally,Iwouldliketoexpressmysinceregratitudetomywife,Shuet-funMui, forherpatienceandunderstandingduringthepreparationofthebookmanuscript. UniversityPark,USA Long-QingChen [email protected] Contents 1 ThermodynamicSystemandItsQuantification .................. 1 1.1 Introduction ............................................ 1 1.2 ThermodynamicSystems ................................. 2 1.3 ThermodynamicVariables ................................ 4 1.3.1 InternalEnergy .................................. 5 1.3.2 Entropy ........................................ 7 1.3.3 Volume ........................................ 7 1.3.4 AmountofChemicalSubstance .................... 8 1.3.5 PotentialsWithinaSystem ........................ 9 1.4 Densities ............................................... 13 1.5 ExtensiveandIntensiveVariables .......................... 15 1.6 ConjugateVariablePairs ................................. 16 1.7 Classical, Statistical, and Nonequilibrium Thermodynamics ........................................ 17 1.8 Exercises .............................................. 18 2 FirstandSecondLawsofThermodynamics ..................... 21 2.1 ThermodynamicStatesandStateVariables .................. 21 2.2 ThermodynamicProcesses ............................... 23 2.2.1 Spontaneous,orNatural,orIrreversibleProcesses .... 24 2.2.2 ReversibleProcesses ............................. 24 2.3 ThermodynamicSystems ................................. 25 2.4 FirstLawofThermodynamics ............................ 26 2.4.1 First Law of Thermodynamics for Isolated Systems ........................................ 30 2.4.2 First Law of Thermodynamics for Closed Systems ........................................ 30 2.4.3 FirstLawofThermodynamicsforReversible ProcessesinOpenSystems ........................ 32 2.5 SecondLawofThermodynamics .......................... 36 2.5.1 QuantifyingSecondLawofThermodynamics ....... 38 2.5.2 IsolatedSystems ................................. 40 xi