第14周transport properties谁来回答这个大问题

(Non-euilibriumThermodZHAOSHIXITel:E-mail:GraduateSchoolatShenzhen,TsinghuaEmotion些细部重新装到一起。‐‐‐‐‐‐‐‐‐‐‐‐‐‐阿尔文托夫勒以Progogine以Progogine为首的布鲁塞尔学派又经过多年的努力，在对理论——耗散结构理论。这一理论于1969年由Progogine在一他因此获得Nobel化学奖。《从混沌到有序》伊.普里戈金，伊.Thischapterwilldescribethreefurthercrystalpropertiesrepresentedbytensors,namely,thermalandelectricconductivity,andthermoelectricity.Sincetheyareconcernedwithtransportprocessesandthermodynamicallyirreversiblephenomena,thesepropertiesdonotfitintotheschemeofreversibleeffectssetupinabovechapter.Itisthereforemoreconvenientandlogicaltotreatthemseparatelyhere.1.Thermal1.Thermal1.1thethermalconductivityandresistivityhhkiihk ijxThethermalconductivitykijisasecond-ranktensor,and[kij]isasymmetricaltensor.kijk k31Referredtoitsprincipalkkk3323T 0h1k1x,k0T10k33h2k hkT,2333(2)(2)hk ijxTWheretherijarefunctionsofthekij,therijrelatetwovectorsandhenceformasecond-ranktensor,thethermalresistivitytensor.Theresistivityisthereciprocaloftheconductivity.rk Buttherelationbetweentheindividualcomponentsof[kij]and[rij]isnot,ingeneral,oneofsimplereciprocity.Thus,forr k12andr12refertoquitedifferentk12givestheresultofmeasuringthex1componentoftheheatflowinanexperimentinwhichthetemperaturegradientisalongx2;thush 122r12,ontheotherhand,referstoanexperimentinwhichthex1componentoftemperaturegradientismeasuredwhilethetotalflowofheatisalongx2;Tr121Sincekijkji,itiseasilyprovedfromtherrk WhenWhenthesymmetricaltensor[rij]isreferredtoitsprincipalaxes,whosedirectionsarethesameasthoseoftheconductivitytensor,wehaveTrh,Trh,Tr123 r11k1,r21k2,r31Theresistivityellipsoid,whoseequationreferredtotheprincipalaxesrxrxrx2 1 2 3Thushassemi-axesofthelengthsinverselyproportionaltothoseoftheconductivityellipsoid.2.ElectricalTheformalanalysisoftheconductionofelectricityinanisotropiccrystalsissimilartothatoftheconductionofheat.ThefundamentalequationisthegeneralizedformofOhm’sLaw:Wherejiisthecurrentdensity,σijistheelectricalconductivitytensor,ФispotentialandEkistheelectricfieldEiik ρikistheelectricalresistivitytensor.Theresistivitymatrixisthereciprocaloftheconductivitymatrix.。。TherateofJouleheatingTherateofJouleheatingoftheconductorisexpressedbythescalarproductofthecurrentdensityandthefield.Inacrystal,therefore,theheatproducedinunittimeandunitvolumeisWherejisthemagnitudeofthecurrentdensityandρistheresistivityinthedirectionofthecurrent.jE jj3.Theconductionofheatandtheconductionofelectricityincrystalsweretreatedinlasttwopartsastwoseparateprocesses.Thiswaspossiblebecausewewereconcernedwithsituationswhereonlyoneoftheprocessesoccurredatatime.However,whenbothprocessesoccurredtogethertheyinterferedwithoneanother,theresultsofthisinterferencebeingobservedasthephenomenaofthermoelectricity.Inthepresentsectionweformulatethebasicequationswhichgovernthermoelectricityincrystalsandweshowhowtheequationsleadtothevariousobservedeffects.Itisfirstnecessarytodiscussthermoelectricityinisotropicconductors.3.13.1ThermoelectriceffectsinisotropicTherearethreethermoelectriceffectsinisotropicThethermoelectrice.m.f.(Seebeckeffect塞贝克效应).Ifacircuitismadeoftwodifferentmetalsaandb,thejunctionsaremaintainedatdifferenttemperatures,ane.m.f.issetupinthecircuit.Ifacondenser(电容器）isinsertedintheconductora,forinstance(Fig.14.1)itbecomes托马斯约翰塞贝克[1]（也有译做“西伯克”）1770年生于说，科学家们的眼睛让奥斯特（电磁学的先驱）ThePeltierheat.Whencurrentisallowedtoflowacrossajunctionbetweentwodifferentmetalsitisfoundthatheatmustbecontinuouslyaddedorsubtractedatthejunctioninordertomaintainitstemperatureconstant,theheatisproportionaltothecurrentflowingandchangessignwhenthecurrentisreversed.WewriteQabab WhereQabistherateatwhichheatisabsorbedatthejunctionwhenacurrentJpassesfrommetalatometalb.andПabisthePeltiercoefficient,whichdependsonthenatureoftheconductorsandthetemperature.TheThomsonheat.whenacurrentTheThomsonheat.whenacurrentflowsinawire,ofhomogeneousmaterialandofconstantcross-sectionbutwithanon-uniformtemperature,heatmustbesuppliedtokeepthetemperaturedistributionsteady.Theheatthatmustbesuppliedinunittimetoanelementofthewireinwhichthetemperaturerise,inthedirectionofthecurrent,isdTisdQ WhereτistheThomson三个热电效应（thermoelectriceffect）Therearetworelationsbetweenthemagnitudesofthethermoelectriceffects(i),(ii),(iii)aboveknownastheThomsonrelations.Wenowderivethem,usingthemethodsofirreversiblethermodynamicsoutlinedinthelastsection.SupposethejunctionsareattemperaturesTandT+ΔTintwolargeheatreservoirsAandB,andthatapotentialdifferenceΔφisestablishedacrossthecondenserC,thesidenearertoBbeingatthehigherpotential.Thecondenserissupposedtohavenoheatcapacityandthewiresaandbaresupposedtobethermally(andelectrically)insulated.DerivationoftheThomsonTheequationsTheequationsconnectingtheflowofelectricityj1,andofheatj2,with“force’X1=-gradφ,andX2=-1/TgradT,jLXLXj1L11X1L12X2 21 222WiththeOnsagerForthecircuitheretheequationstaketheJLLT 12THLT 22Tje1Th ikkikTxkiki ikTxkT4.The4.TheOnsager’sreciprocalTheheatconductivitykijinanisotropiccrystalsisasymmetricalsecond-rankkijkThisassumptionisjustifiablebynomeansobvious.Itasserts,forexample,that,ifatemperaturegradientinthex1directionproducesacertainheatflowinthex2direction(givenbyk21),thesametemperaturegradientappliedinthex2directionwouldgivenaheatflowofpreciselythesamemagnitudeinthex1direction(givenbyk12).Toseewhatisinvolvedintheassumptionitishelpfulfirsttodiscoverwhattheconsequenceswouldbeif(4)didnothold.ΔT—x1directionh—x2ΔT—x2directionh—x1Firstsplit[kij]intoasymmetricalandananti-symmetrical k31 k31 000k330k23kk0 k3323Thepresenceofsymmetryinthecrystalwilltendtoimposerestrictionsonthisschemeofcoefficients.Bywayofillustrationwetakethecaseofacrystalhavinga2-,3-,4-,or6-foldaxis(eitherrotationorinverse)paralleltox3.ifatemperaturegradientsisestablishedparalleltox3,theheatflowmustalsobeparalleltox3,bysymmetry.Hence1→2，2→-1，3k23k31x1 0k00k33K12=-LarsOnsager(November27,1903–October5,1976)wasaNorwegian-bornAmericanphysicalchemistandtheoreticalphysicist,winnerofthe1968NobelPrizeinChemistry.HeheldtheGibbsProfessorshipofTheoreticalChemistryatYale进入挪威诺尔格斯工学院主修化学工程，2年至9年，9年移居美国，年~939年诺贝尔化学奖。昂萨格在1萨格名誉工学博士学位，以弥补当年的失误 Thisisageneralprinciplewhichappliestoalltransportphenomena—underwhichtermmaybeincludedtheconductionofheat,theconductionofelectricity,andthetransportofmatterthattakesplacebydiffusion.Inthermodynamics,theOnsagerreciprocalrelationsexpresstheequalityofcertainratiosbetweenflowsandforcesinthermodynamicsystemsoutofequilibrium,butwhereanotionoflocalequilibriumexists.Theolderthermodynamicshasbeenmoreaccuratelydescribedas“thermostatics”;incontrast,Onsager’stheoryisessentiallythethermodynamicsofirreversibleTheequationgoverningtheflowofanelectriccurrentinanisotropicjMaybejL 1J1representsaflux(ofcharge)andX1representsaInInthesamewaythefluxofheatisproportionaltotheforceontheheataccordingtotheequationjL 2Wherethefluxj2istheheatflow,formerlydesignatedbyh,andtheforceX2=-1/TgradT.thereasonforthefactor1/Twillbereferredtolater(Tistheabsolutetemperature).Equations(23)and(24)wouldbecorrectastheystandifthetwoprocessesoftransportofelectricityandofheattookplaceindependentlyofoneanother.But,ingeneral,thisconditionisnotsatisfied;forwhenthetwoprocessesoccursimultaneouslyinacircuitcomposedoftwodifferentmetalstheyinterfere,asisshownbytheappearanceofthevariousthermoelectriceffects.Ingeneralitisnecessarytousethemorecomprehensiveequations.jLXLXj1L11X1L12X2 21 222Inwhicheachofthefluxesislinearlyrelatedtoboththeforcesinsteadofonlytooneofthem.Onsagerstatesthatinsuchacase,providedthefluxesandforcesarecorrectlychosen.L12=L21Itmaybeverifiedthat,withourchoiceofdefinitionsforthej’sandX’sthedimensionsofL12andL21arethesame.InmoreInmorecomplicatedsituations,wheretheremorethantwofluxesandforces,equations(25)aregeneralizedjL i,j1,2,, ijandOnsager’sPrincipleassertsL Itisimportanttonoticethatthej’sandX’sin(25)and(27)mustbecorrectlychosenbeforeOnsager’sPrinciplecanbeapplied.By“cor