W

Wagnerenhancementfactor701●defectsinsolidsWagnerCarl(Reproducedfrom[ix])(May,,Leipzig,Germany–Dec.,,Göttin-gen,Germany)WagnerstudiedchemistryattheUniver-sitiesofMunichandLeipzigandobtainedhisPh.D.fromLeipzigUniversitywhereheworkedunderLeBlanc.From–hewasscientificassistantatthePharma-ceuticalInstituteoftheUniversityofMunich.InhebecamePrivatdozentforAppliedChemistryinMunich.From–hewasattheUniversityofBerlin,from–PrivatdozentattheUniversityofJena,wherehebecameextraordinaryProfessorin.From–hewasinHamburg,andfrom–extraordi-naryProfessorofPhysicalChemistryattheTechnischeHochschuleDarmstadt.ThereheworkedasfullProfes-sorfrom–.From–hewasScientificAd-visoratSubofficeRocket,FortBliss,Texas,USA.FromtohewasVisitingProfessoratMIT,USA,from–ProfessorofMetallurgyatMIT,andfrom–DirectorattheMax-Planck-InstituteofPhys-icalChemistry,Göttingen,Germany[i].CarlWagnerisoneofthefoundersofsolid-stateelectrochemistry,andhiscontributionstosolidstatechemistryandcor-rosionarefundamental,bothwithrespecttothermo-dynamicsandkinetics(transportprocessesandreac-tionrates).TogetherwithSchottkyheledthebasisofthethermodynamictreatmentofdefectsinsolids[ii,iii],heexplainedtheformationofdoublesalts(e.g.,spinells)[iv],explainedthetarnishingofmetalswiththehelpofdefecttheory[v],andhepioneeredtheun-derstandingofcorrosionintermsofmixedpoten-tialformation[vi].WiththehelpofdefecttheoryhewasabletoexplaintheconductionmechanismthatisoperativeintheNernstlamp[vii].Hecontributedtomanyotherfieldsofresearch,e.g.,heatconduction.AttheendofhislifeWagnerpublishedaveryinter-estingphilosophicalbookaboutthemethodsofscien-tificandtechnicalresearch[viii].Seealsothefollowingentries.Refs.:[i]SchmalzriedH()Bunsenmagazin():;[ii]WagnerC,SchottkyW()ZphysChemB:;[iii]SchottkyW,UlichH,Wag-nerC()Thermodynamik.Springer,Berlin;[iv]WagnerC()ZphysChemB:;[v]WagnerC()ZphysChemB:;[vi]WagnerC,TraudW()ZElektrochem:;[vii]WagnerC()Naturwiss:;[viii]WagnerC()Methodendernatur-wissenschaftlichenundtechnischenForschung.B.I.-Wissenschaftsverlag,Mannheim;[ix]MartinM()Bunsenmagazin():FSWagnerenhancementfactor—describesusuallytherelationshipsbetweentheclassicaldiffusioncoefficient(self-diffusioncoefficient)ofchargedspeciesiandtheambipolardiffusioncoefficient.Thelatterquantityistheproportionalitycoefficientbetweentheconcentra-tiongradientandthesteady-statefluxofthesespeciesunderzero-currentconditions,whenthechargetrans-feriscompensatedbythefluxesofotherspecies(elec-tronsorothersort(s)ofions).Theenhancementfac-torsshowanincreasingdiffusionratewithrespecttothatexpectedfromamechanisticuseofFick’slaws,duetoaninternalelectricalfieldacceleratingtransferoflessmobilespecies[i,ii].Inasystemwithtwochargecarriers(iands),thisrelationshipcanbeexpressedusingthetransportnumbers(t),activities(a),concentrations(c)andself-diffusioncoefficients(D)[iii]˜DisDi=tiDsDi∂lnas∂lncs+ts∂lnai∂lnciwhere˜Disistheambipolardiffusioncoefficient.SeealsoWagnerfactor.WhenthetransportofionicspeciesBzischarge-compensatedbythetransferofelectroniccarriers(e)[ii,iii]˜DBzeDBz=te∂lnaB∂lncBzwithtebeingtheelectrontransferencenumber.Seealsoelectronicdefectsandelectrolyticpermeability.Sometimes,thistermisalsousedforanalysisofrela-tionshipsbetweenthemacroscopicdiffusioncoefficients702Wagnerequation●W(predictedbyFick’slawsfromthetotalconcentrationgradient)andmicroscopicdiffusioncoefficients,takingintoaccountthetransportviaandoccupationofpartic-ularcrystallographicpositions[iv].Seealso:defectsinsolids,diffusion:determinationinsolids.Refs.:[i]WagnerC()ZphysChemB:;[ii]ChebotinVN()Chemicaldiffusioninsolids.Nauka,Moscow;[iii]YokotaI()JPhysSocJap:;[iv]WagnerC()ZphysChemB:VKWagnerequation—denotesusuallyoneoftwoequa-tionsderivedbyWagnerforthefluxofchargedspeciesBzunderanelectrochemicalpotentialgradient,andfortheelectromotiveforceofagalvaniccellwithamixedionic–electronicconductor[i–v]:JBz=−χBz(zF)[∇μBz+(zF)∇φ]E=−zFνBμBμB(−te)dμBwhereχBzisthepartialconductivityofthesespecies,teistheelectrontransportnumber,andμBandμBarethechemicalpotentialsofneutralcomponentBatthecellelectrodes.Thefirstequationshowsthatthefluxofchargecarriersinanelectrochemicalsystemisproportionaltothepartialconductivityandthermodynamicdrivingforce,namelytheelectrochemicalpotentialgradient(seeOnsagerequation).WhenthechemicalpotentialofBzparticlesisconstant,thisequationtransformsintoOhm’slaw.Whentheelectricalpotentialisconstant,thisequationmaybetransformedintothefirstFick’slawusingtheNernst–Einsteinequation.Thefirstequationcanbewritteninamoregeneral-izedformaccountingforthetemperaturegradient:JBz=−χBz(zF)'T∇#μBzT$+(zF)∇φ+uBzT∇T*whereuBzisthetransportedheatofBzspecies.Animportantformofthesecondformula,whichisalsodenotedastheWagnerequation,canbeobtainedbyintegration:E=(−¯te)ETwhereETisthetheoreticalNernstemf(electromo-toricforce),thetransportnumberisaveragedforagivenchemicalpo