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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. March 1999 NOlesandDiscussions 262

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