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Do cathedralglasses flow? Edgar DutraZanottoa) Departml!//1of lvlaleria/s EI/gineering. Federal VI/il'ersit;; of SWJ Car/os. !3565-905. Sao Car/os-SP. Bra:il (Received23April 1997:accepted17October1997) A generalbdief amongmembersof thescientificcOlnm~nityis thatglassarticlescanbebent irreversiblyandthattheyflowat ambienttemperature.This mythis mostlybasedon widespread storiesthatstained-glasswindowsofmedievalcathedralsarethickerinthelowerpans.In thispaper I estimatethetimeperiodsrequiredfor glass(0 flowanddeformatordinarytemperatures.using calculatedviscositycurvesfor severalmodemandancientglasscompositions.The conclusionis thatwindowglassesmaynowatambienttemperatureonlyoverincrediblylongtimes.whichexceed thelimitsof humanhistory. @ /998 ,~lIIenClll/ Associatiol/ o( Phy.,ics TClIdu:rs. I. I.\'TRODUCTION Is glass (/ liql/id or i.l'iT I/OT?While teachingmaterials sciene::andtechnologycoursesoverthelasttwodecades.I ha\'ebeenaskedthatquestionby studentsandcolleagueson se\'eraloccasions.bec:ll1setheyhadheardthat800-year-old stained-glasswindows of 12thcenturycathedralswere thickerin theirIowa part.whichsuggestsadownwardflow of glassatroomtemperature.It is interestingtonotethatno oneknewthesourceof thisinfonnation. At first I thought.thattheabove-describedinterpretation wasa Brazilianmyth.however.I laterdiscoveredthatacol- leaguehadalsoheardthatstoryin Argentina.I A refereeof the AmericanJournalof Physicsconfinnedthatthesame storyis widespreadin theUSA. Thenarrativeis alsoempha- sizedin at leastoneAmericanjournal.z Additionally,a text- bookof materialsscience.3andeventheprestigiousEncy- clopediaBritannica..!alludetothisphenomenon,statingthat glasspiecesbentovera periodof severalmonthsatordinary temperatureswill notreturntotheiroriginalshape.Although somescientistsmayknowthetruthor couldinferit by using simplearguments.to myknowledgethereareno published calculationson thesubject.Thusit appearsthatthealleged flcw of ancientwindowglasses.or moregenerally,of the pennanentdefonnationof glassat roomtemperature.is a quiteuniversalconceptandthereforemeritsclarification. TnthiscommunicationI usesimpleconceptsof physicsto uemonstratethat typical window glasses,which contain KzO-NazO-CaO-MgO-AlzOrSiOz anda certainamount of impumies.mayflowappreciablyatordinarytemperatures only in in;";..:essibletimes.over geologicalperioci~,no; withinthelimitsof humanhistory. II. THEORY Viscous t1uids easily show detectable relaxation phenomena-thechangeof any measurablepropertywith timefollowinga perturbation.To a goodapproximation,a numberof relaxationprocessesmaybedescribedbya modi- fied Maxwell express;on.a stretchedexponentialequation havingthe1'01111[7([)=PIIl't-/3t1n(e.g..0.5<{3<I forstress relaxationinglasses).wherePo is theinitialvalueof agivcn pr,'peny.p(T) is therelaxedvalueafteranelapsedtime.I. and "r is the characteristicrelaxationtime.Thus. whenT =T. p\T) has relaxed to approximately 60%-37% of its originalvalue.dependingonthemagnitudcof {3. The viscositycoefficientor simplyviscosity.TJ.of a liquid is onemeasureof (herelaxationtimeT.It givesanapproxi- mationtotherateof structuralchangeandthedependenceof thisratcon temperature.The \'iscosityis relatedto anaver- agerelaxationtime,(TI. for bulkthennodynamicproperties by (T)=CTJ. (I) whereC is someconstantthatdependsonthepropertybeing studied-enthalpy,volume,stress,etc.For a shearrelaxation time.C is theinverseof theinfinitefrequencyshearmodu- lus.G-x;.Equation(I) indicatesthatall bulkstructuralrelax- ationprocessesof theliquidhave,ontheaverage,thesame temperaturedependence,andexperimentallythisseemstobe thecase.sAn approximatederivationof Eg. (I) is givenin theAppendix.For typicalcompositionsof windowglasses. G-x;is about30 GPa from theabsolutezero to the glass transitionrange.6 Theviscosity,however,variessignificantlywithcomposi- tion andtemperature.As for severalotherthennallyacti- vatedprocesses,the viscositycould, in principle,be de- scribedby anArrhenius-typeexpression: TJ(T)=TJoexp(E,,/kT), (2) whereE" is theactivationenergyfor viscousflow, TJois a constallt.k is theBoltzmanr.constant.andT is theabsolute temperature.However,asthestructuresof mostglassesvary withtemperature,theactivationenergiesalsovary.andthll" onecannotuseanArrhcnianexpressiontocalculatethevis- cosity. Fxceptionshaveonly beenrpportedfor very few (pure)networkglassfonningoxides.suchas SiOz, GeO~. andP20Sglass,for whichthestructureis temperatureinde- pendent.Hence,theviscosityversustemperature<"Jrves01 mostglassesareusuallydescribedby an empiricalexpres- sionof theVogel-Fulcher-Tamman(VFT) type: log TJ=A+B/(T-To). t31 whereA. B. andTo arcempiricalparameterswhichdepend solelyon theglasscomposition. PhysicalinsightintoEq. (3) is givenby thefree-volumc theorysummarizedin Appendix2. whichassumesth:ltthe "now units"(groupsof moleculesthatnowsimultaneously) cannotjump if the volumeof neighboringvacanciesi~ smallerthantheirownvolume.Sucha situationis observed at a characteristictemperatureTo wheretheviscositytend~ to infinity. Ta!'>l.:I. Typical compositions(wt%) and VFr parameters'oj winJuw ~l;1:o:ses. 'Y dlow glassof (heGatienCathedral.Tours(France). III. CALCULATIONS AND RESULTS Table I showstypicalcompositionsof bothmodemand medievalwindowglasses.While thecompositionsof the formerare relativelyunifonn.thecompositionsof anciem glassesvaryenomlOusly.asreponedin Ref.7, whereabout 350 glasseswereanalyzed.In general,medievalwindow glasseshavea higherlevelof impurities,suchas ironand manganese,andarepotassiumrich,whilecontemporarywin- dowglassesarericherin sodium. An imponamcharacteristicof glassesis thattheydo not haveany microstructuralfeatures,suchas crystalphases, grainboundaries.pores,etc.,whichdependonbothprocess- ing conditionsand chemicalcomposition.Hence,several bulk propeniesof glasses.e.g., thennalexpansioncoeffi- cient,density,refractiveindex,and viscosityare additive functionswhichsolelydependonthechemicalcomposition. Therefore,numericalcoefficientswhichrelatea givenprop- eny to theglasscompositioncanbeempiricallydetennined. Indeed,handbooks6andconunercialsoftwareareavailable, whichareextensivelyusedby theglassindustryto estimate severalpropeniesfroma knowledgeof theglasschemical analysis.Oneof themostsuccessfulandwidelyusedproce- du~s is dueto Lakatosetal.,8whichrelatesthecontentof K:P, Na20, CaO, MgO, A1203,andS102in a glassto its Vogel-Fulcher-TanunannparametersA, B, andTo. The VFT constantsof Table I werecalculatedusingthe Lakatosfonnulas,8neglectingtheeffectof minorimpurities. This ;Jrocedurewill pr~bablyleadto sliglillyoverestimated valuesof viscosity.However,it will nothaveanysignificant effecton the "order of magnitudc"calculationspresented here.For instance,for a yellowpotashglassof theGatien Cathedral.Tours(France),theestimatedVFT constantsare: A =-4.:22.B=5460.9.andTu=196.30c. Theviscosity(in Pa s) can beobtainedby Eq. (3).The calculatedviscosity curves of a typical contemporaryglass, two medieval glasses.anda Ge02glassareploucdin Fig. I. Despitebeingcapableof describingquitewell thetem- pcraturedependenceof flowresistanceoverscveraldecadcs in viscosity.fromthemcltingrange(1400-15000c) to the glass transitionrange(T~-550-600°CL a complicating factorariseswith Eq. (3) becauseTu(180-360°C) is wcll abovcroomtcmperaturc.T". Hcnce.Eq. (3)predictsan in- finitelylargeviscosityatthattcmperature.Evidently,a dif- ferentflow mcchanismcouldoccurbclowTu (for instance. Am. J. Phy'" V..I. hI,. 'I" 5. \by "/<IX .. . soda-modem ...soda-old - potash-old.Ge02 o 800 . I 400 600 Temperature,oC Fig. I. ViscosilYof Jiffer':nI glass.:sasa function..f l.:mperature. smallmoleculesor evenindividualatomsinsteadof mol- eculesor "flow units" maydiffuseindependently)andthus another,unavailable.equationshou!':!beusedtoestimatethe relaxationtimesat roomtetr.perature.In spiteof this fact.onecanestimatewhatthenecessarytemperaturewouldhave to beto observesignificantviscousflow in a timespanof a fewcenturies.By usingEqs.(I) and(3)andthedatafor the Frenchglass,oneconcludesthatit is necessaryto heata typicalmedievalglasstoapproximately414°C toobservea significantflow in800years. A numericalcalculationof therelaxationtimeat Ta, al- beitapproximate,canbemadeby referringto theviscosity of Ge02glass,whichhasanequivalenttransitiontempera- tureto windowglass(Fig. I). For Ge02glass,theviscosity canbedescribedby anArrhenius-typeequation,andhence. onecanextrapolatetheviscositycurvedownto roomtem- peraturetoestimateTJ(Ta)'By substitutingTJ(Ta)in Eq. (I), onehasa lowerboundfor therelaxationtimeof window glassbecause,on decreasingtemperature,theviscosityof Ge02doesnotriseasquicklyas thatof windowglass(Fig. I). The viscosityof Ge02glassmaybeadequatelydescribed by Eq. (3) withA= -9.94 andB= 17962 (TJ[Pa s]; T[K]) and TO=0.9Therefore.the predictedrelaxationtime for Ge02atroomtemperatureis 1032yr, Hence,therelaxation period(characteristicflowtime)of cathedralglasseswould beevenlonger.In fact,thatperiodis wellbeyondtheageof the Universe (- 1010yr)! Onemightarguethattheimpuritiesof medievalglasses (whicharenottakenintoaCCO'tntby theLakatosfonnulas) couldlowertheviscosityto levelswhich would leadto a muchfasterflowthananticip~ted.However,evenassuming a plausibledecreasein theviscosityof oneor twoordersof magnitude.thatwouldnotaltertheconclusionsof theprevi- ouscalculations.Additionally.theeffectof weatheringand leachingof themedievalglasswindowsduringseveralccn- turiesmightseemimponant;however.thatprocessonly leadstoasuperficialchemicalauackandonlydiminishesthe glassshineandtransparencybuthasnosignificanteffecton viscosityandotherbulkpropenies. Experimentalevidencetoreinforcctheidcaof largerclax- atiuntimesat roomtemperatureis thc factthatglassvases fromthousandsofyearsagorcmainundefonnedin museums aroundtheworld.Thepossibilitythatsomecathedralglasses arethickerattheirbottommaybeexplainedby thefactthat ancicntwindowglasseswereblownintocylindcrsthatwcre Modem Medievalglasses SiO 73.2 45-75 a:O 13.4 O.I-IS CaO 10.6 1.0-25 AI03 1.3 0.8-2.0 K:O 0.8 2.0-25 :>.tgO 0.7 0.8-:;.0 Fe:OJ 0.1 0.3-2.1 :>.tnO ... 0.3-2.3 P:O; ... 2.5-10 ..\ -2.6 -..a...J B *077.7 5460.9' To 254.7 196.3' ((I) A 0- Fig.2. Schem3!icbeh3\"iorof thederon113tionYt1.1of 3 viscoclastic m3[C- rial under constantsuess. splitandtlattenedmanually.Hence,thepieceswerenotuni- fonn in thicknessandsomelowerpanscouldbethickerthan theupperparts. IV. CONCLUSION As aresultof thepreviousdiscussions.itcanbeconcluded thatmedievalandcontemporarywindowglassescannotflow at roomtemperaturein humantimescales! ACKNOWLEDGMENTS I amindebtedtoProfessorF. A. B. Coutinho,ProfessorL. N. Oliveira,andProfessorJ. S. Sintrafor stimulatingdiscus- sionson thesubject.Thanksarealsodueto CNPq andto PRONEX for financialsuppon. APPENDIX 1. Empirical theoryof viscoelasticit.y Assumingthe Burgessmodelof viscoelasticity,Slet us considera creepexperimentin whicha constantshearstress TOis suddenlyappliedto a viscousliquid.The experimen- tallyobservedbehaviorof thedefonnationy(t) undercon- stantstressconsistsof threepans,schematicallyshownin Fig. 2: y(t)/70=[I/Gx.+h(t)+tl 1]], (AI) whereI/Gx referstotheinstantaneouselasticresponse(seg- ment4.B in Fig. 2),h(t) istherecoverabledelayedelasticity [11(0)=0.h(:x»=constant,segmentBC]. andthethirdtenn relers to the iIfecoverat>leviscousdeform..1lion(segment CD). If thestressis suddenlyreleased,instantaneousrecov- ery occurs(segmentDE) followedby a relaxationperiod representedby segmentEF andgovernedby therelaxatlOli function</J(t).The relaxationfuncti:)nis an intrinsicprop- eny of thematerialunderstudy.The right-handsideof Eq. (A I) is definedas thecomplianceof the Burgessmodel. J(t) . For longtimes,theelasticresponseandthedelayedelas- ticityvanish.andthus (A2) whichddlnes theshearviscosity,1].for Newtoniantluids suchasoxideglasses. Thechangein timeof somepropertyof aglass.duetothe imposedchangeof a variable(temperature,stress.etc.)can .-\m. J. Phys.. Vol. 66. No.5. M3Y 1<)<)11 C~calculatedif one knows the'viscoelasticfunction~- ,;;:laxationmodulusG(t) andthematerialcomplianceJ ((). An expressionwhichrelatesG(t) andJ(t) in thecaseof a constantshearstressTOis givenby TO=y(O)G(t)+ !:G(t-tl)(JYIJt')dtl. (A3) wherey(t) is givenby Eq. (A I) whichcanberewritten<.!, y(t)=70J(t), whereG(t) is therelaxationmodulus[G(t) =G.r.</J(t): and </J(1)is theshearrelaxationfunction];whichnormallyhasall exponentialfonn. Expression(A3) is extensivelyusedby rheologistsandis discussedinseveraltextbooks.e.g..Ref.5. It is knownasthebasicequationof linearviscoelasticity. FromEq. (A4), theshearstrainrateis givenby Jyl iif =TOJJ 1Jt. 1.~5i Substituting(A4) and(AS) into(A3) onehas 1 =J(O)G(t)+ fGlt-t')JJIJt1dt" This equationrelatesthe viscoelasticfunctions:relaxation modulusG(t), andcomplianceJ(t). FromEq. (AI), JJIJt= 1/1]+JII 1Jt. IA.7) SubstitutingJ(O)= lIGx and(A7) intoEg. (A6): I=G(t)/Gx.+ !:GU-tl)[(lI7])+JhIJt')]df" (AS) Thus 1=<f>(t)+{Gx.17])!:</J(t-t')dt' +Gx.J~<f>(t-t') X[JhIJt']dt'. (A.9) In the limit t~:x>,thefirstand thirdtennsvanish.and therefore. 7]= Gx. fox <f>(t)dt. (A 10) This is a ratherextraordinaryresult,becausethe:;~lear\'is- cositycanbedetenninedsimplyasan integralovertimeof thestressrelaxationfunction</J(t). SomeinsightcannowbegainedintoEq. (1): TJ=(T)/C. (All) Thena comparisonof Eqs.(AIO)and(All) showthat C= l/G-x;and (T)= (X </J(t)dt.Jo (AI2) so (r) is thetimeaverageof the shearrelaxationfunction~ 2. The free-volumemodelof viscosity Most clementsandcompoundswhenmoltenhavea ,'is. cosityaboutthesameasthatof water(10-2 Pas). On cool- ingthemelt,crystallizationoccursveryrapidlya littlehelow thefreezingpointTr. Thereare,however,a few materials EdgarD. Z' oth> ! I E .E I ~ . '" 0. '" E ~o> r T<;r. Temperature_ Fig. 3. R~Ia!iunb~lw~~nth~glassy.liquid.andsolidSlal~S. whichfonu meltswhichareconsiderablymor':'!iscous.The hioh viscositvindicatesthattheatomsor ffivle-:uk.>in the m~ltarenot~oeasilymovedreIati\'eto oneanotherby ap- pliedstresses.On coolingbelowthefreezingpoint.crystal- lizationdoesoccur,butata significantlylowerratethanin thematerialsof thefirstgroup.Theprocessof crystallization involvesstructuralchanges.i.e..therearrangementof atoms relativeto oneanother.In simpletenus,therelativelyhigh viscosityof themeltandthelow rateof crystallizationare bothconsequencesof thegreaterresistanceto atomicrear- rangementencounteredin thesematerials. If thecrystaIlizationrateis low enough,it is possibletogo oncoolingthemeltbelowthefreezingpointwithoutcrystal- lizationtakingplace.As themeltcools.itsviscositycontin- uesto increase.This viscousliquidbelowthefreezingpoint is asupercooledliquid.Thus.strictlyspeaking,it is incorrect to referto it asaglass.Furthercoolingresultsin theviscos- ity risingto suchahighvaluethatthemechanicalproperties of thematerialarecloselysimilartothoseof anidealelastic solid.Theviscositywill thenbeatleast1012_1013Pas.This solidmaterialis a glass. The volume-temperaturediagramshownin Fig. 3 is use- ful in discussingthetransfonuationfroma supercooledliq- ufd to a gl:tSs.If the meltcrystallizeson cooling,this is usuallyaccompaniedby a markedincreasein densityatthe meltingpoint,Tf' No suchchangeoccursif themeltsuper- cools.The volumedecreasesalongthelinebe.Thedecrease in volumeon coolingis duepartlyto thedecreasingampli- tudeof atomicvibfations.andpartlytochan.!!esin thestruc- t:.:~eof themeltwhichreSl'1tin it becomingmorecompactas thetemperaturefalls. At temperaturesnearTf thesestruc- turalchangescanoccurveryrapidlyandwill appeartooccur instantaneou"lyfollowinganychangein thetemperatureof thematerial.As theviscosityincreaseswithfallingtempera-ture,thestructuralchangesoccur increasinglyslowlyuntil eventuallytheviscositybecomessohighthatnosuchfurther changesarepossiblein laboratorytimescales.A decreasein slopeis thenfoundin the V vs T curve(pointe). With a furtherfall of temperature.thedecreasingvolumeis dueal- mostentirelyto thedecreasingamplitudeof theatomicvi- brations. The temperatureat whichthechangein slopeoccursis calledthetransfonuationtemperatureorglasstransitiontem- perature,T~. Only belowT~is it correctto describethe Am. J. P!IY'.. Vul hI,. :,"".5. :\I.IY "J'll! materialas a glass.The changefromsupercooledliquidto glass,whichmaybeconsideredastakingplaceat thistem- perature,is nota suddenone,noris Tga well-definedtem- peraturefor anyparticularglass.Indeedthetenu"transfor- mationrange"is usedmorefrequentlythan"transfonuation temperature."Thetemperatureatwhichthechangeinslope occursis foundto decreaseas the rateof coolingis de- creased.Also,if theglassisheldatihetemperatureT. a little belowTg, its volumedecreasesslowlyuntil it reachesa pointon thedottedline. which is an extrapolationof the contractioncurveof the supercooledliquid. The rateof changeof volumedecreasesasthedottedlineis approached. i.e., as the structureof theglassapproachesan "equilib- rium" configurationcharacteristicof thesupercooledmeltat the temperatureT. This equilibriumconfigurationhas a lowerfreeenergythanotherliquidlikestructuresorconfigu- rations,butit is not,of course,thatatTangementof molecules whichhasthelowestpossiblefreeenergyatthetemperatUre T (thecrystallinearrangement).However.at temperatUres significantlybelowT~. therateatwhichtheliquidlikeglass structUrecanchangeis inverselyproportionaltotheviscosity andis veryslow.assh~wnpreviouslyin thispaper. CohenandTurnbulll developedafree-volumemode!of viscousflowbasedontheideathatflowoccursbymovement of molecules(flow units)into voidsof a sizegreaterthan somecriticalsize.Thatis, themoleculesrattlearoundin the cagecreatedbysurroundingmolecules,untildensityfluctua- tionscreatea holelargeenoughfora moleculetojumpinto. The freevolume(Vf) is somewhatvaguelydefined,but it representsroughlythespacenotoccupiedby thecorevol- ume(vo) of themolecules.The viscositycanbewrittenas TJ=TJoexp(ovO/vf)' Vf/VO=IT(al- ag)dT',To (Al3) (AI4) where0 is a constantcloseto unit,To is thetemperature wherevf=O, andal andag arethethenualexpansionco- efficientsof supercooledliquid and glass, respectively (a,>ag), shownin Fig. 3. This equationreducesto the VFT equationif al-ag is constant. "Eleclronicmail:dedz@powcr.u..scar.br 'EduardoMari, "Los mitesdelvidrio:' C~ramicay Cristal107.29(1991). !MichaelWysession.uThe inner working. of thc carth,U Am. Sci. 83. 134-147(1995). JW. O. Fellcrs.Mal.,rial~Sci.,nc.,.resling an': I'mperri.,s(Premice-Hall. EnglewoodCliffs. NJ. 1990).p. 2O-t. .£lIcyciopediaBrirallnica-uViscosily" (WilliamBemon.Chicago.1966). Vol. 23.p. 198. sSlcvenBrawcr.Relaxalionill ViscousUquidsandGlasse.f(AmericanC~- mmicSociclY.Columbus.Ohio. 19851,p.24. "NarotanP. BansalandRobertH. Duremus.J{t/I,db()()kof Gla.fsProperlie.f (Academic.Orlando.1986). 7W. Muller. M. Torge.and K. Adam. "Ratio of CaO/K20>2 as an cvi- ,kncc of a specialRheinishIypeof mediaevalslainedglass:' Glaslech. Ber.-GlassScience& Technology67 C!I.~5-4!1(994). "1'.Lakalos.L..G. Johansson.andB. Simmingskold."ViscosilY tempera. Illrerelalionsin (he~IasssYSIemSiO:.AI:O,-Na!O-K!O-CaO-MgO in the c",npositiunrangcuf lechnicalglasses'"GlassTechnulogy13(21.88-95 (1972). "EdgarD. :lanouo. "lsulhemlal and AdiabaticNuclealionin Glass'" J. Non.Crys!.Solids89.361-370 (1987). "'DavidTurnbulland Morrel Cohen. "On Ihe free-volumemodelof Ihe li4uid-glasstransition:'1.Chem.Phys.52 (6). 3031!(1970). Ed~ar 0, Zanolll' 395 Do cathedralglassesjiow?-Additional remarks EdgarD.Zanottoa) . VitreousIHaterialsLilboralOry-LaMaV.Departmelllof MaterialsEngineering-DE:Ha.Fcc/alii Un;,'er.riry of Soo Carlos-UFSCar. lJ655-905.Sao CarlosSP. Bra:il PrabhatK. Gupta Depllrt/llelllof MaterialsScience& Engineering.TireOlrioSllltl!UIl;,'ersity.204I CollegeNo(/{l.Columbus. Olrio432/0 (Received31August1998:accepted15September1998) We presenta revisedestimateof therelaxationtimeatroomtemperatUrefora windowglassbased on extrapolatedisostructuralviscositydata.This estimate.whileseveralordersof magnitUdeless (hanthe previousestimateof Zanotto[Am. J. Phys. 66 (5). 392-395 (\998)], supportshis conclusionthatwindowgla.>scannotflowat roomtemperatUrein humantimescales.@ 1999 Al1Ier;cw:"''iodatiunl~"PllysicsTt.'llcJrers. I. INTRODUCTION Oneof us.Zanotto.'r~:t>ml'..examinedthevaliditvof the widely held (bUt not by a1l2)-beliefthatbecause~edieval cathedralglasswindowsarethickeratthebottomthanatthe top,windowglassflowsslowly(overhundredsof years)at roomtemperatUreundertheinfluenceof gravity.using the Maxwellrelaxationtime: 7\T)= 1](T)/G(T), where1]is theviscosityat temperatureT andG theinfinite frequencyshearmodulus,to estimatethetimefoi'"glassto flow noticeably,anda valueof the"equilibrium"viscosity (of thesupercooledmelt)extrapolatedto roomtemperature, Tr. Zanottoarguedthat7(Tr) is at least1032years.He con- cludedthatflow of glasscouldnotbethecauseof theob- servedthicknessvariationin cathedralwindows. Therewasan immediateresponsetothearticlefromboth thescientificcommunityandsciencemagazines,seefor in- stanceRefs.3-6. Scientiststhroughouttheworldmadenu- merousquestionsandremarks.However,themostrelevant of thesewasa commentby P. Gupta,thesecondauthorof thepresentarticle.thattheuseof "equilibrium"viscosity ~nlygivesanupperboundfor T(Tr). Thereforethequestion whetherthe windowglass flows at roomtemperaturere- mainsunresolved. Guptaalso pointedout thattheuseof the isostructural viscosity(i.e.. the viscosityof the glassystatewherethe structureis frozen)-insteadof theequilibriumviscosity- extrapolatedto Tr shouldgivea morerealisticestimateof T(T,). In thiscommunication.wereportsucha revisedesti- mateof T(T,>usingi<;ostructuralviscositydatafor thewin- dow glasscomposition.We showthat,eventhoughthere- vised;-(Tr) is severalordersof magnitudelessthanthevalue estimatedby Zanotto.his conclusionthatcathedralwindow glassdocsnotflowatroomtemperaturestill remainsvalid. II. DISCUSSION A. Isostructural\'iscosity Followinga jumpin temperalUreatconstantpressure.the propcrticsof a viscousliquid\:ontinueto changewith time e\'enafterthermalequilibriumhasbeenrca\:hed.This slow \:hangeis knownasstructuralrelaxationandreflectsthetime requiredfor thestrU\:lUreto rearrangeintoitsnew"equilib- rium" configuration.The averagestructuralrelaxationtime. ~hO ..\on, J. Ph\',. (.7131.Mard, I')')') (I) Ts' increases\'cryrapidlywithdecreasein thetemperature in thesupercooledliquidstate.As a consequence,\\'hena liquidis cooledatsomeconstantrate.if. itsstructureremains in equilibriumwithinthetimescaleof observation(chara\:- terizedby- IIq) for T> Tf!if). thefictivetemperature(see AppendixA). The structUrefallsoUtof equilibriumand is frozenfor T<Tl q) whereT, becomeslargerthantheob- servationtime.For typicalcoolingratesusedin glassform- ing, TJq) is approximatelygivenby theglasstransition temperature,Tg (temperaturefor whichtheequilibriumvis- cosityis IOt2Pas). Thepropertiesof asupercooledliquidassumetheir"equi- librium" valuesaboveTJq) and"isostroctural"valuesbe- low TJq) andshowtransitionbetweenthetwoin thevicin- ity of TJq). For example.the first-orderthermodynamic properties,suchasdensity,showachangeinslopewhilethe second-orderpropertiessuchasheatcapacityshowdisconti- nuities.The equilibriumand isostructuralbehaviorsof the viscosityareshownin Fig. 1for awindowglasscomposition (seeTableI). The isostructuraldataweremeasuredby Ma- zurinetaL.7neartheglasstransitiontemperatUreTg' Figure I showsthattheequilibriumviscosityof windowglassdi- vergesata temperatureaboveTr whiletheisostructuralvis- cosityremainsfiniteatroomtemperature.B. Extrapolationof thc isostructuralviscosityto the roomtcmperaturc In orderto calculateT(Tr}, it is necessaryto extrapolate the isostructuralviscosity from the temperaturesin the neighborhoodof Tg (about820K) whereexperimentalmea- surementsweremade.downtoroomtemperature.Mazurin's measurementsindicatedthattheisostructuralviscosityfol- lowsan Arrheniustemperaturedependence.This was con- firmedby Scherer,swhoalsoexaminedthestructuralrelax- ationdatain glasses.SchereralsoconcludedthattheAdam- Gibbs theory'9(AppendixB) providesthe mostreasonable descriptionforthetemperaturedependenceof boththeequi- libriumandtheisostructuralviscosities. Ac\:ordingto theAdam-Gibbsmodeltheviscosityis '1='IIIexp[AITS,.], whereA and1]11arcconstants.The configurationalentropy. S". i~given by Fig. \. Temperaturevariationsof theequilibriumandisostructuralviscosi- tiesof thewindowglassbasedonEq. HI. The parametersarethosereponed by Scherer(Ref. 121andTj= d16K. J Tf Sc(Tf)= (:1C:plT)dT. To Here/::1cp is thedifferencein heatcapacitybetweentheequi- libriumliquidandthefrozenglass,To is theKauzmanntem- peraturedefinedsuch thatSc(To)=O.Equation(3) shows thatthefictivetemperature,Tf' governstheconfigurational entropyof thefrozenstate. As shownrecentlyby RichertandAngeIl,lO/::1cp,in the vicinityof Tg, is well approximatedby BIT, whereB is a constant.Equations(2) and(3), then,leadto 17=17oexp[QI(T(l-ToITf)], (4) whereQ=AToIB. For equilibriumsupercooledliquid, Tf =T and Eq. (4) reducesto the Vogel-Fulcher-Tamman equation(describedin Ref. I). For theisostructuralstate,the fictivetemperature,Tf' is constantandEq.(4)reducestothe Arrheniusequation.The validityof theAdam-Gibbsmodel hasbeenwell-established.II To useEq. (4), one needsto establishthevaluesof the param~rsQ, To, and170for thewindowglass.Fortunately, Schererl2hasdeterminedthevaluesof the~eparametersby carefullyanalyzin~the volume(density)relaxationdataof HaraandSuetoshi3 in a soda-lime-silicaplategla~s(com- positiongivenin Table I) aswell asMazurin'sisostructural viscositydatain termsof Eq. (4).Th.: valuesof theparam- ~tersasreportedby Schererfortheglassused.inRef. 13are: 710=9X 10-6Pas, Q = 14900K andTo=436K. As is clearfromFig. I, thevalueof theisostructuralvis- cosityat roomtemperaturedependson thefictivetempera- ture.The highertheTf' theloweris thevalueof isostruc- tural 71at T,. Therefore,we needto establishthe fictive temperatureof thecathedralglass.Sincethecathedralwin- dowswereannealedafterforming,thefictivetemperatureof thecathedralwindowsmustbe lessthantheannealingtem- perature(generallytakenas the temperaturefor whichthe ~quilibriumviscosityis 1012.4PasI4).To beontheconserva- tive side.we assumeTJ equalto T~, wheretheequilibrium viscosityis 1012Pas. For theHaraami Suetoshicomposi- tion. thiscorrespondsto a temperatureof 816K. Taking T,=300K. Tf=816K. andC(Tr)=30GPaI5 in Eqs. (I) and (4), one obtains:,(Tf)-2X 1023years.This 261 Am. J. I'hys..Vol. 67.No.3. Ylarch I'J'J'J Table1.Chemicalcomposition(wt%) of medievalglassesandof window glassesusedby HaraandSuetoshi,HS (Ref. 13)and by Mazurin etal. (Ref. 7). (na=notavailable.) (3) value,althoughseveralordersof magnitudelessthanthe originalestimateof Zanotto,still impliesthatthe dimen- sional variationsof the cathedralglasswindowsare not causedbycoldflowof glass. Thereremainsthequestionasto whatis thecauseof the (suggested)dimensionalnonuniformitiesin cathedralglass windows.Wenowknowthatit is notbecauseof theflowof glass.Wespeculatedin Ref. 1 thatanciemwindowglasses wereblownintocylindersthatweresplitandflattenedmanu- ally. Hence,thepieceswerenotuniformin thicknessand somelowerpartscouldbethickerthantheupperparts.An- otherpossibility,mentionedby Hares,16is that window glassesmadeby thecrownprocesshad "a thicknessthat decreasedwith increasingdistancefromthecenter." It is quitepossiblethatthecathedralwindowmakersinstalledthe cut up windowpanesinstinctivelywith thickersideat the bottom.17 III, CONCLUSION The revisedestimateshowsthatwindowglasswill only flow appreciablyatroomtemperatureif onewaitsuntil the "SecondComing"! ACIGI/OWLEDGMENTS EDZ acknowledgesencouragementandcriticalcomments of ProfessorFranciscoA. B. Coutinho.ProfessorLuis N. Oliveira,andProfessorJose F. Perezof FAPESP. He also acknowledgesfundingby CNPq and ?RONEX (Brazil). PKG acknowledgesmeaningfuldiscussionswith Professor Arun K. Varstneyaof AlfredL'i.iverslly,NY. APPENDIX A: FICfIVE TEMPERATURE (Tf) OF A GLASS TooI1Sintroducedtheconceptof fictivetemperatureto characterizethenonequilibriumstructureof a glass.It is de- finedas thetemperaturewherethestructureof the corre- spondingequilibriumliquid (normalor supercooled)is the sameasthatof thegivenglass.Sometimes.Tf is referredto as thestructuralor theconfigurationaltemperature.Glasses of thesamecompositionshowingdifferentvaluesof a prop- ertyhavedifferentfictivetemperatures.The fictivetempera- tureof a glassis determinedbyitshistoryof formationfrom theliquidstate. The notionthatthestructureof a glasscanbecharacter- izedbyasingleparametersuchasTf is anapproximateone. Only for an idealizedhistorywhena liquid is cooledv~ry NOlesandDlscussiuns 261 HS Mazurin Medieval 5iO! 7\.6 72.7 45.0-75.0 AI!O} \.6 1.3 0.8-2.0 -Isostructurall CaO 7.9 8.6 \.0-25.0 ylgO 3.8 3.4 0.8-8.0 :\a!O 13.7 13.6 0.1-18.0 K!O 0.5 0.4 2.0-25.0 iT = 436K =3001 TiO! 0.3 na na 0 Fe!O) 0.1 na 0.3-2.1]r , SO} 0.3 na na 20 30 40 104rr,K' 80 70 60 fJ) 50 oj 0-_ 40 '? -;; 30 0 - 20 10 0 10 slowly down to a tcmp~r;;.tur.:TI .md is ~::~nrapidly quenchedto theroom temp~rature..:;u..thestru.::ureof the glassbedescribedby Tf . In general,oneparameteris notsufficienttodescribethe structureofaglass.Thisisevidencedbythefactthat,forthe sameglass,differentpropertiesshowdifferentfictivetem- peratures.For thisreason.Narayanaswamyl9redefinedfic- tivetemperatureof a propertyp.asfollows: Pg(T)=Pe(T J.)+ IT (iJp/;JT)..dT. (Ai)Tip)' Here the subscriptg refersto theisostructural(i.e., glassy) state,andthe subscripte refersto the (equilibrium)super- cooledliquid.For typicallaboratorycoolingrates,thevalues of TIp) for differentpropertiesaresomewhatdifferentbut all are close to theglasstransitiontemperature.Tg, where theviscosityis 1012Pas. APPENDIX B: THE ADAM-GIBBS MODEL TheAdam-Gibbsmodelis basedontheideathatrelax- ationoccursby theinternalcooperativerearrangementof independentregionsof II molecules.As the temperatUre drops,movementof onemoleculedistUrbsanincreasingly largernumberof itsneighbors.AdamandGibbs9assumed thatthebarrierto rearrangementis proportionalto n, and determinedthetemperatUredependenceof n in termsof the configurationalentropy,Sc' Theirresultfortheviscosityis 7]=7]0exp[~,uIn(w*)/TScJ. (BI) where7]0is aconstant,6.,uis thepotentialbarrierpermol- eculehinderingrearrangement,w* isthenumberofconfigu- rationsavailabletothesmallestgroupof atomsthatcanun- dergoacooperativerearrangement(w*-2). ')Home-page:http://www.nit.ufscar.brllamav IEdgarD. Zanono. "Do cathedralglassesOow?" Am. J. Phys. 66. 39:!- 395(1998).Erratum:The stretchedexponentialequationon page392of this paper should read: pet) =Po exp[ -(tl1'Y'). 2Roy G. Newton, "Fact or fiction? Can cold glass Oow under its own weightandwhathappensto stainedglasswindows?," GlassTechnol. 37. 143(1996). )The AmericanInstituteof Physics."Do camedralglassesflow?," Phvs. News Update370 (May 1998). - .Erik Stosktad, "Camedral glass myth shattered,"Sci. Now 12 May (1998). sJeffrey Hetch. "That's anothermythshattered."New Sci. 16 May. 25 (1998). 6CorinaWu. "Analysis shanersglassmym." Sci. News 153,May 30. 3~1 (1998). 70legV. Mazurin. Y. K. Startsev.andS. V. Stoljar. "Temperaturedepen- denceof viscosityof glass-formingsubstancesatconstantfictivetempera- tures," J. Non-Crysl. Solids 53, 105-114(1982). sGeorgeW. Scherer."Useof theAdam-Gibbs equationin theanalysisof structuralrelaxation,"J. Am. Ceram.Soc.67 (7),504 (1984). 9GeroldAClamand Julian H. Gibbs. "On thetemperaturedependenceof cooperativerelaxation propenies in glass-formingliquids." J. Chem. Phvs.43. 139-143 t1965). lOR.'RichenandC. A. Angell. "Dynamicsof gla.~sforming liquids. V. On the link betweenmoleculardynamicsand configurationalentropy," J. Chem.Phvs.108(21).9016(998). "Yan BO!li~gaandPascalRichet. "SilicatemeltstruclUralrela.~ation:Rhe- ology.kinetics,and Adam-Gibbs theory,"Chem.Geol. 12S:119(1996;. 12GeorgeW. Scherer. "Volume relaxationfar from equilibrium." J. Am. Ceram.Soc.69, 374-381 (1986). IJMorihisa Ham and S. SuelOshi."Density changeof glassin thetransfor- mationRange," Rep. Res.Lab.. Asahi GlassCo. Ltd. 5, 126-135 (19551. 14JerzyZarzycki. Glassesand the VitreousState(CambridgeU.P.. Cam- bridge.1991),p. 267. ISNarottanP. BansalandRoben H. Doremus.Handbookof GlassProperties (Academic,New York. 1986),p. 322. 16GeorgeB. Hares, "3500 yearsof glassmaldng"in CommercialGlasses. editedby D. C. Boyd andJ. F. MacDowell (ACerS. 1986). 17Arun K. Varshneya(personalcommunication). 18AnhurQ. Tool, "Variations causedin theheatingcurvesof glassby heat treatment,"J. Am. Ceram.Soc. 14.276 (1931). 19See,for example,GeorgeW. Scherer,Relaxationin Glassesand Compos- ites(Wiley. New York. 1986),p. 116. WHAT DO WE WANT FROM PHYSICS? Physicistshavealwaysclaimedthat their scienceis ethicallyneutral.But in recentyears, philosopherscf science-particularlyfeministphilosophers-havechallengedthisclaim.Knowl- edge,theysay.is notneutral.butalwaysthefruit of someintention,whetherconsciouslyrecog- nizedor not....Ratherthanleavingphysiciststo telluswhattheywanttodo andjusthandingover themoneyto do it, asa societywe mustbe involvedin decidingwhatwe wantfromphysicsand whatpurposeswe wantit to serve.We mustconsciouslymoveit backto a moresociallyrespon- siblegrounding. MargaretWenheim,Pythagoras'Trousers-God. Physics.and theGenderWars (RandomHouse.New York. 1995),pp. 251-252. :!6:! Am. J. Phys.. Vol. 67. No.3. 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