Measuring the dynamic thermal expansivity of molecular liquids near the glass transition

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by  Kristine Niss

The dynamicthermalexpansivity of liquidsnearthe glass transition.
Kristine Niss, Ditte Gundermann, Tage Christensen, and Jeppe C. Dyre
DNRF Centre Glass and Time, IMFUFA, Department of Sciences,
Roskilde University, Postbox 260, DK-4000 Roskilde, Denmark
(Dated:January15,2013)
BasedonpreviousworksonpolymersbyBaueretal. [Phys,Rev. E(2000)],thispaperdescribesa
capacitativemethodformeasuringthedynamicalexpansioncoefficientofaviscousliquid. Dataare
presentedfortheglass-formingliquidtetramethyltetraphenyltrisiloxane(DC704)intheultraviscous
regime. Comparedto the method of Bauer et al. the dynamical range has been extendedby mak-
ingtime-domainexperimentsandbymakingverysmallandfasttemperaturesteps. Themodelling
2
of the experimentpresentedin this paper includes the situation where the capacitor is not full be-
1
causetheliquidcontractswhencoolingfromroomtemperaturedowntoaroundtheglass-transition
0
temperature,whichisrelevantwhenmeasuringonamolecularliquidratherthanpolymer.
2

n
a Theglasstransitionoccurswhentheconfigurationaldegreesoffreedomofaliquidarefrozenin. Belowtheglass
J transitiontemperature,T ,onlyisostructuralcontractiontakesplaceastemperatureisdecreasedfurther. Themea-
g
8 suredthermalexpansioncoefficient,α(andheatcapacity,c )arethereforelowerintheglassthanintheequilibrium
p
1 liquid. Thischangeofthethermalexpansioncoefficient(andtheheatcapacity),isprobablythemostclassicalsigna-

tureoftheglasstransition,andafigureillustratingthischange(seeFig. 1)isalmostinevitablythestartingpointof
]
t introductorytalksortextsontheglasstransition(eg. Ref. 1,2).
of The change in the heat capacityat the glass transition ∆cp = cp,liq−cp,glass has been studied extensively and is
s widely believed to play a role for the dynamics of liquids close to the glass transition. The change in expansion
t. coefficient∆αp =αp,liq−αp.glasshasreceivedlessattention,butisofsimilarimportance. Thisisseen,forinstance,in
a
m theliteraturerelatedtothePrigogine-Defayratio,adimensionlessnumbercharacterizingtheglasstransition3–7.
Theglassisanout-of-equilibriumstateandthereforethevaluesofthethermodynamicderivativesarenotrigor-
-
d ously well defined. They depend on cooling rate and also on the time spent in the glassy state. Contrary to this
n the linearresponseof the metastableequilibriumliquid state iswelldefined andhistoryindependent7. Thelinear
o expansioncoefficientofaviscousliquidclosetoitsglasstransitionisdynamic,thatistime(orfrequency)dependent
c
withshorttimesgivingalow(glass-like)value,α ,whilelongtimesgiveahigherliquidvalue,α . Thediffer-
[ p,fast p,slow
encebetweenthesetwolevels∆α = α −α thusgiveswell-definedinformationontheconfigurational
p,lin p,slow p,fast
4 partoftheexpansioncoefficient. Likewise∆c =c −c iswelldefined.
p,lin p,slow p,fast
v The relaxation between the fast and the slow response takes place on a certain time scale which is temperature
4
dependent. Consideredinthiswaythemeasurementof theexpansioncoefficientcanbeviewedasatypeof spec-
0
troscopy, which gives both a relaxationtime and a spectralshape analogous to other methods like dielectric spec-
1
troscopyormechanicalspectroscopy. Thestudyofthetemperaturedependenceofrelaxationtimesandofthespec-
4
. tral shape of differentresponse functions is vital for understanding the viscous slowing down. There is a general
3
beliefthatthe liquid hasarelaxationtime, which isfairlywelldefinedindependentofprobe,butalsosuggestions
0
1 thatdifferentprocessesmaydecouplefromeachotheratlowtemperatures8.
1 There are good scientific reasons to study the dynamic linear expansion coefficient, but almost no data of this
: typearetobefoundinliterature.Thetime-dependentexpansioncoefficientcanbefoundbystudyingthechangein
v
volumeasafunctionoftimeafteratemperaturestep. Suchvolumerelaxationexperimentsareveryclassicinglass
i
X scienceandstillimportant9–12. However, volume relaxationexperimentsaretraditionallyperformedasnon-linear
r agingexperiments, i.e.,withlargeamplitudesinthe temperaturejump. Thistypeofexperimentgivesinformation
a
ontherelaxationoftheconfigurationaldegreesoffreedom,buttheexpansioncoefficientanditscharacteristictime
scale cannot be determined, because the results depend on the amplitude and sign of the temperature jump. For
sufficientlysmalltemperaturestepsthisisnotthecase,definingthelinearresponseregime.
The only linear dynamic data we are aware of were reported about a decade ago by Bauer et al.13,14 followed
by a paper by Fukao and Miyamoto15. These papers reported frequency-domain measurements on thin polymer
films, performed with temperature scans at a couple of fixed frequencies, covering 1.5 decade of the dynamics.
The measurements were pioneering, but 1.5 decade is not very much for studying relaxation in viscous liquids,
because the relaxation is extremely temperature dependent and quite “stretched”, which means that even at one
fixedtemperaturetherelaxationcoversseveraldecades.
The technique developed by Bauer et al. is based on a principle where the sample is placed in a parallel plate
capacitorsuchthatitisthesamplethatmaintainsthespacingbetweentheplates. Thusachangeinsamplevolume
in response to temperature change leads to a change of the capacitance. This principle is also used in the present
work. Theadvantageofthistechniqueisthatcapacitancecanbemeasuredwithhighaccuracyanditisthisaccuracy
2
V
liquid
glass
α
p
liquid
glass
T
T
g
FIG.1: Illustrationofthe temperaturedependenceofvolumeand expansioncoefficientofaliquidinthe vicinityofthe glass-
transition. Uponcoolingtheexpansivitydecreasesabruptlyattheglasstransition. Thisgivesrisetoakinkinthetemperature
dependenceofthevolume.Thesefeaturesaretheoriginalsignaturesoftheglasstransition.
3
whichmakeslinearexperimentspossible.
The use of sample-filled capacitors for measuring an expansion coefficient is not unique and it has been done
by others before and after Bauer et al. (see eg. Ref. 16–19) in capacitative scanning dilatometry, i.e. working in a
temperature ramping mode. Capacitativescanning dilatometry has to our knowledge never been used on simple
liquids. It is particularlyusefulfor studying thin polymer films because the signal gets better with a thin sample.
The technique has been used for determining the glass-transition temperature for example as a function of film
thickness17,18 or asa function of cooling rate16. The main focusof these papersis on the temperaturedependence
oftheexpansioncoefficient,whilelittleattentionisgiventotheabsolutevalues. Therehavebeennostudiesofthe
dynamicssincethepioneeringworkofBauerandnoattemptstoextendthedynamicalrange.
Tothebestofourknowledgetherearenomeasurementsofthedynamiclinearexpansioncoefficientofmolecular
liquids. Thereporteddatafromscanningdilatometryandnon-linearvolumerelaxationarealsomainlyforpolymers,
while data on molecular liquids is relatively scarce. This may be due to the higher technological importance of
polymers. It is probably also related to the fact that working with molecular liquids requires other experimental
conditions,meaningthattechniquesdevelopedforpolymersarenotalwaysdirectlyapplicabletoliquids.
This paper gives a description of an experimental method developed for measuring the dynamical expansion
coefficient of a viscous. As mentioned, the principle is based on the capacitive technique by Bauer et al.13,14. The
methodismodifiedinthreerespectscomparedtotheworkofBaueretal.: 1)Themodellingtakesintoaccountthe
situation where the capacitor is not full, which is relevant when measuring on a molecular liquid rather than on
a polymer. 2) The experiment is performed in the time domain using a very fast temperature regulation, which
gives a dynamical range of more than four decades. 3) The sensitivity is enhanced by using a capacitance bridge
withaveryhighresolution. Thismakesitpossibletomeasuretheresponsefollowingverysmalltemperaturesteps,
ensuringthattheresponseisclosetoperfectlylinear. Asanapplicationofthetechniquethepaperpresentsdataon
theglass-formingliquidtetramethyltetraphenyltrisiloxane(DC704)intheultraviscousregime.
I. RESPONSEFUNCTIONSWITHCONSISTENTDIMENSIONS
Inalinearresponseexperiment,theresponseofasystemtoanexternalperturbationisstudied. Iftheperturbation
issmalltheoutputisassumedtobelinearlydependentontheinput. Theformalismtodescribethisiswellknown.
However,differentformulationscanbeused,andtheversionusedinthisworkwhenconvertingthemeasuredtime-
domain response to the frequency-domain response function is maybe not the most common one. The formalism
usedherehasthe advantagethatthetime-domainresponsefunction andthefrequency-domainresponsefunction
havethesamedimensionandthereisnodifferentiationinvolvedwhentransformingbetweenthetwo. Thissection
givesasummaryoftheresponsefunctionformalismusedincludingacomparisontothestandardformalism.
Thefundamentalassumptionisthattheoutputdependslinearlyontheinput. Themostgeneralstatementisthat
thechangeininputdI(t′)attimet′leadstoacontributioninoutputdO(t)attimet:
dO(t)= R(t−t′)dI(t′). (1)
Itishereassumedthatthechangeinoutputonlydependsonthetimedifference(t−t′). Causalityimpliesthat
R(t)=0fort <0. (2)
IntegratingonbothsidesofEq. (1):
t
O(t)= R(t−t′)dI(t′),
−∞
Z
dI(t)
andsubstitutingt′′ =t−t′andwriting I˙(t)=
dt
0
O(t) = − R(t′′)I˙(t−t′′)dt′′.
∞
Z
Changingt′′tot′:
∞
O(t)= R(t′)I˙(t−t′)dt′. (3)
Z0
4
IftheinputisaHeavisidefunction:
0 fort≤0
I(t) = I H(t)= I
0 0 1 fort>0
(cid:26)
then
∞
O(t)= I R(t′)δ(t−t′)dt′ = I R(t), (4)
0 0
Z0
anditisseenthatR(t)istheoutputfromaHeavisidestepinput.
Linear response canalsobe studied in the frequencydomain. In the case of a harmonic oscillating input I(t) =
I0ei(ωt+φI), the outputO(t) = O0ei(ωt+φO) will be a periodic signal with the same frequency ω, but there will be a
phaseshiftoftheoutputrelativetotheinput. FromEq. (3)theoutputis
∞
O(t) = R(t′)iωI eiφIeiω(t−t′)dt′
0
Z0 ∞
= I eiωteiφIiω R(t′)e−iωt′dt′
0
Z0
= I(t)R(ω),
whereR(ω)isthefrequencydomainresponsefunction,whichisgivenbytheLaplacetransformofR(t)timesiω:
∞
R(ω)=iω R(t′)e−iωt′dt′.
Z0
The linear response relation is often expressed in an alternative formulation where the linearity assumption is ex-
pressedby
t
O(t) = µ(t−t′)I(t′)dt′,
−∞
Z
whereµissometimescalledthememoryfunction,butitisalsosometimescalledtheresponsefunction. Theuseof
thewordresponsefunctionforµ(t)issomewhatinconvenientbecauseithasadifferentdimensioncomparedtothe
frequency-domainresponsefunctionR(ω). Substitutingagain(t′′ = t−t′)andchangingt′′tot′
∞
O(t) = µ(t′)I(t−t′)dt′.
Z0
ApplyingaHeavisideinputagain
∞
O(t) = µ(t′)I H(t−t′)dt′
o
Z0
t
= I µ(t′)dt′. (5)
0
Z0
FromEq. (4)and(5)wehave
t
R(t)= µ(t′)dt′,
Z0
andtherefore
dR(t)
=µ(t). (6)
dt
Inthememoryfunctionformalismthefrequencydomainresponseisagainfoundbyinsertingaharmonicoscillating
input. Inthiscasetheresultbecomes
R(ω)= ∞µ(t′)e−iωt′dt′ = ∞ dR(t)e−iωt′dt′,
Z0 Z0 dt
wherethelastequalitycomesfrominsertingEq. (6). ThisexpressionisformallyequivalenttoEq. (22)whichcanbe
shownbyintegrationbypartsandbyinvoking R(t = 0) = 0. However,whenconvertingdatainpracticeEq. (22)
hastheadvantagesthatdifferentiationofthetimedomaindataisavoided.Itisalwaysgoodtoavoiddifferentiation
ofnumericaldatabecauseitintroducesincreasednoise. Moreover,ifweintroducean“instantaneous”responsein
termsof R(t → 0) 6= 0correspondingto veryshort timeswhere we cannot measuresthe time dependenceof the
response,thenthisinformationwouldbelostbydifferentiation.
5
II. PRINCIPLE,DESIGNANDPROCEDURE
Themethodrequiresthatthereisasimplerelationbetweensampledensityanddielectricconstant. Thedielectric
constantingeneralhastwocontributions: atomicpolarizationandrotationalpolarization20. Theatomicpolarization
is due to the displacement of the electron cloud upon application of a field. This contribution is governed by the
microscopic polarizability of the molecule, x (usually called α, but α is reservedfor the expansivity in this paper).
The atomic polarizability can be assumed to be temperature and density independent in the relevant range. This
means that the desired simple relation between density and dielectric constant can be obtained when the atomic
polarizationistheonlycontribution.
Therotationalpolarizationisduetorotationofthepermanentdipolesinthesample. Thiscontributionisrelevant
whentheliquidhasapermanentdipolemomentandmainlyatfrequencieslowerthanorcomparabletotheinverse
relaxation time of the liquid. The rotational contribution gives the dielectric signal monitored in standard dielec-
tricspectroscopy. Therotationalpolarizationistemperature-,density-andfrequency-dependent,anditistherefore
non-trivial to relate the density to the dielectric constant when rotational polarization is present. Therefore, in ca-
pacitativedilatometryitisacontributiononewouldliketoavoid. Itissometimesassumedthatthehighfrequency
plateauvalue of the dielectric constant measured in dielectric spectroscopy contains only atomic polarization and
thatitcorrespondstothesquareofrefractionindexn2. However,thereisalsofast(“glass-like”)contributiontothe
rotationalpartofthepolarization. Thefastrotationalcontributionwilldominateoverthegeometriceffectsevenat
highfrequenciesifthesamplehasahighdipolemoment. ThiswasdemonstratedinRef. 21. Tominimizetherota-
tionalcontributiontwothingsaredone: 1)Onlyliquidswithverysmalldipolemomentarestudied-i.e. liquidsin
whichtheatomicpolarizationisdominantatallfrequenciesandtemperatures. 2)Theseliquidsareonlystudiedat
frequenciesmuchhigherthantheinverserelaxationtime.Inthedatareportedinthispaperthemeasuringfrequency
is10kHzandtherelaxationtimeis100secondsormore.
Thecellisacapacitormadeofcircularcopperplatesof1cmdiameterand1mmthickness,witha50µmspacing.
Theseparationiskeptbyfour0.5mmx0.5mmand50µmthickKaptonspacers.Thespacingbetweenthecapacitor
platesisfilledwiththesampleliquid. Thethinspacingresultsinareasonablylargedielectricsignal(emptycapac-
itanceis14pF)despitethesmallsize. Thethinspacingmoreovermakesitpossibletoheatorcoolthesamplefast,
eventhoughtheheatdiffusioninthesampleliquidisslowcomparedtotheheatdiffusioninthecopperplates.
The cell is integrated with a microregulator, which is a tiny temperature regulator based on an NTC-thermistor
(placed in the lower cupper plate of the capacitor-cell), a Peltier-element acting as a local source of heating and
cooling,andananalogPID-control. Theintegratedcellandmicroregulatorareplacedinourmainkryostat.Withthis
setupthetemperatureofthesamplecanbechangedbystepsofupto2Kwithinlessthan10sandthetemperature
canbe kept stable is within a fewmicro Kelvin over daysand weeks. The cell is shown in Fig. (2) and the whole
systemofthemainkryostatandthemicroregulatorisdescribedindetailinRef. 22.
The principle of the experiment is to make an “instantaneous” step in temperature and subsequently measure
the capacitance at a fixed frequency as a function of time. From the capacitance we calculate the time-dependent
expansion coefficient. In order for the temperature step to be “instantaneous” compared to the time scale of the
relaxationweneedtherelaxationtimetobe100sorlonger. Thismeansthatthemeasurementsareperformedator
below the conventional glass-transition temperature. Nevertheless, it is important to emphasize that, the liquid is
inequilibriumwhentheexperimentisperformedbecausewewaitatleastfiverelaxationtimeswheneverstepping
toanewtemperaturebeforemakingameasurement. Themeasurementsthemselvesalsomustbecarriedoutover
five relaxation times in order to obtain the relaxation curve all the way to equilibrium. All together, it takes days
and sometimes even weeks to take a spectrum at a given temperature. This means that the experiment would be
impossiblewithoutthestabletemperaturecontrolensuredbythemicroregulator.
Therelaxationtimeofviscousliquidsclosetotheglasstransitionisextremelytemperaturedependent. Wethere-
fore need to make small temperature steps in order for the measured response to be linear. This means that the
changeinvolumeandtherebythemeasuredcapacitanceisverysmall,therelativechangesincapacitancedC/Care
of order 10−4. We use an AH2700AAndeenHagerling ultra-precisioncapacitancebridge, which measurescapac-
itance with an accuracy of 5 ppm and true resolution of 0.5 attoFarad in the frequency range 50 Hz-20 kHz. The
capacitanceismeasuredeverysecondat10kHz.
The sample used is liquid atroom temperatureand the capacitor is filled by letting the liquid suck in using the
capillaryeffect. Completefillingischeckedbymeasuringthecapacitancebeforeandafterfilling,comparingtothe
measureddielectricconstantmeasuredatthesametemperaturewithalargercapacitor(whichiseasytofill).
6
PEEK posts
Electrode pin
NTC thermistor
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
Dielectric cell (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) + Microregulator
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
Peltier element
− Temperature
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
Copper base (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) Control System
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
DC current
FIG. 2: Schematic drawing of the dielectricmeasuring cell with the microregulator. The liquid is depositedin the 50 µm gap
between the discs of the dielectric cell. The Peltier element heats or cools the dielectric cell, depending on the direction of
the electrical current powering the element. The current is controlled by an analog temperature-control system that receives
temperaturefeedbackinformationfromanNTCthermistorembeddedinonediscofthedielectriccell(reproducedfromRef.23).
.
III. GEOMETRYANDBOUNDARYCONDITIONS
In order to model the relation between the measured change in capacitance and the expansion coefficient some
assumptionsmustbemaderegardingthebehavioroftheliquidduringtheexperiment. Inthissectionwedescribe
theseassumptionandtheargumentsonwhichtheyarebased.
Thecapacitorisfilledcompletelyatroomtemperaturewithalowviscositymolecularliquid. Themeasuringtem-
peratures(closetoandbelowtheconventionalglass-transitiontemperature)aretypicallyaround100degreesbelow
roomtemperatureforthesetypesofliquids. Thecoolingmakestheliquidcontractintheradialdirectionbecausethe
distance betweenthe platesis maintained by the spacers(which have a much smaller expansion coefficient). This
has the consequence that the capacitor is not completely filled at the temperatures where the measurements take
place. Thisgivesrisetoadifferencecomparedtothemeasurementsdoneonpolymersinearlierwork13,14,adiffer-
encewhichistakenintoaccountwhencalculatingtherelationbetweentheexpansioncoefficientandthechangein
capacitanceinthefollowingsection.
Theliquidcontracts/expandsradiallyaslongasithaslowviscosity,butthesituationchangeswhentheliquidgets
ultraviscous. Athighviscositiestheliquidgetsclampedbetweentheplatesduetothesmalldistancebetweenthem.
Thishastheconsequencethattheliquidcannolongercontract/expanduponcooling/heating byflowingradially,
but will contract/expand vertically and pull/push the plates changing the distance between them. This effect is
the basisforthe measurement, becausethe verticalexpansionmakesthe capacitancechange, andwe calculatethe
expansionfromthechangeincapacitance.
The distance between the plates is kept by the Kapton spacers at high temperatures (and long times) when the
sample liquid flows. However, at times where the sample cannot flow, it is the sample, not the Kapton spacers
whichdeterminesthedistance. ThisistruebecauseKaptonhasastiffness24 ofthesameorderofmagnitudeasthe
sample(intheGPa-range),butonlytakesupapproximately1%oftheareabetweentheplates.
The temperature change gives rise to an internal pressure, which is released by pressure diffusion via viscous
flow. Thecharacteristictime τ oftheradialflowbetweentwoplatesoffixeddistancel canbeestimatedbythe
flow
followingargument. Atemperaturestepof∆Tinitiatesaninternalpressure∆p= K α ∆Tintheliquid. Thiscreates
T p
a radial flow that eventually discharges the surplus volume ∆V = ∆Tα πR2l. Although the volume flows in the
p
radialdirectionwemayasacrudeestimationtakethevolumevelocity,V˙,ofplanarPouiseuilleflow25V˙ = ∆p Wl3,
12ηL
where L(thedimensioninthedirectionoftheflow)canbetakenas R,andW (thedimensionperpendiculartothe
flow) canbetakenas2πR. Thecharacteristicdischargeflowtime thenbecomes τ low = ∆V = 6 η (R)2. Thebulk
f V˙ KT l
modulusandtheshearmodulusareofthesameorderofmagnitude. ItfollowsthattheMaxwellrelaxationtimeis
roughlygivenbyτ ≃ η/K andthatτ ∝ (R/l)2τ . Intheexperimentwehavel=50µmand R = 5mmfrom
M T flow M
whichitfollowsthattheradialflowtimeistenthousandtimeslongerthantheMaxwelltime. Thealpharelaxation
timeisroughlygivenbytheMaxwelltime,theflowtimewillbemorethantendayswhenthealpharelaxationtime
is one hundred seconds. This means that the liquid canbe considered asradiallyclamped in the regionwe study
(where all relaxationtimes are longer than 100 seconds). The transition between the radialflow and the clamped
situation can be seen in dielectric constant when it is measured as a function of temperature, and the observed
7
behaviorisconsistentwiththeaboveestimate.
Theexpansioncoefficientwestudywiththeboundaryconditionsdescribedaboveisnottheconventionalisobaric
expansion coefficient, α = 1 ∂V , because the liquid is clamped in two directions and only free to move in one
p V ∂T
p
direction. We call this expansion(cid:12)coefficient the longitudinal expansion coefficient, in analogy to the longitudinal
(cid:12)
modulus,(anothernameforitcou(cid:12)ldbetheiso-areaexpansioncoefficient). Itisexpressedbyα = 1 ∂V = 1 ∂l ,
l V ∂T l ∂T
A A
whereAistheconstantareaandlisthedimensionwhichisfreetorespondtothetemperaturechange.(cid:12)Thelongit(cid:12)u-
(cid:12) (cid:12)
dinalexpansioncoefficientisrelatedtotheisobaricexpansioncoefficientαp viathefollowingrelation (cid:12) (cid:12)
1
α (ω)= α (ω).
l 1+ 4G(ω) p
3KT(ω)
Where G is the shear modulus and K is the isothermal bulk modulus, which are both dynamic i.e. frequency or
T
timedependentasarethethermalexpansioncoefficients.
Fromthisexpressionweseethatα issmallerthanα ,exceptatlowfrequencies(longtimes,orhightemperatures)
l p
where G = 0 which implies α = α . This expression for the longitudinal expansion coefficient is given (but not
l p
derived)inanotherequivalentformintermsofthePossoin’sRatioinRefs. 13,17,26andcanbederivedfromrow3
ofEq. (53)inRef. 27. Alsonotethatthereisatotallackofstandardnotation. Baueretal. use α tonotethelinear
p
expansioncoefficient,whichisthequantityoftenusedtoexpressvolumeexpansionofsolids. Thatistheirα is1/3
p
ofourαp. ThelinearexpansioncoefficientiscalledαLbyWallaceetal.26,whileFukaoetal.17callitα∞. Thequantity
wecallthelongitudinalexpansioncoefficientα isdenotedα (CAforclampedarea)byBauer,α byWallaceand
l CA N
α byFukao(nfornormal).
n
IV. RELATINGTHEMEASUREDCHANGEINCAPACITANCETOα
l
A. Derivingtherelation
Inthemeasurementweperformasmalltemperaturestep δT andsubsequentlymeasurethecapacitanceC asa
m
functionoftime. Fromthemeasurementswefindthetimedependentquantity 1 ∆Cm(t). Inthefollowingsection
Cm ∆T
we show that this quantity is proportional to the expansion coefficient, α (t), with a proportionality constant that
l
dependson ǫ∞ and the degree of filling of the capacitor, f, but not on the geometrical capacitanceor the distance
betweentheplates.
The startingpoint isthatthe onlycontribution tothe high-frequencydielectricconstant, ǫ∞, isthe atomicpolar-
izability(Sec. II).WemoreoverusetheLorentzfield20 fromwhichitfollowsthatdielectricconstantisgivenbythe
Clausius-Mossottirelation:
ǫ∞−1 n
= x,
ǫ∞+2 3ǫ0
where x is the polarizability of a single molecule, n is the number density of molecules, and ǫ is the vacuum
0
permeability.
Moreover,weassumethatwehaveaparallelplatecapacitorwhichispartiallyfilledwithadielectricliquid. The
degreeoffillingisdenotedby f andthemeasuredcapacitanceisgivenby
Aǫ Aǫ
Cm = fǫ∞ 0 +(1− f) 0 =[fǫ∞+(1− f)]Cg (7)
l l
whereC = Aǫ0 isthegeometricalcapacitanceoftheemptycapacitoratthegiventemperature.
g l
Thederivativewithrespecttotemperatureisnowgivenby
dCm =[fǫ∞+(1− f)] dCg +Cgfdǫ∞. (8)
dT dT dT
Hereitisassumedthattheliquiddoesnotcontractradiallyatthetemperatures(andonthetimescale)weconsider
(see Sec. III), thus df/dT = 0. The next step is to calculate dCg and dǫ∞ under the assumption that the area is
dT dT
8
constant. This was done by Bauer13,14. For completeness we include a detailed derivation as an Appendix. The
resultis
dǫ∞
dT = −K(ǫ∞)αl (9)
whereK(ǫ∞)isgivenbyK(ǫ∞) = (ǫ∞−1)(ǫ∞+2)/3and
dC
g
= −C α . (10)
dT g l
InsertingEq. (9)and(10)inEq. (8)yields
dC
dTm =[fǫ∞+(1− f)](−Cgαl)−CgfK(ǫ∞)αl
= −Cg[fǫ∞+(1− f)+ fK(ǫ∞)]αl.
InsertingCg = Cm/[fǫ∞+(1− f)]anddividingbyCmleadsto
1 dCm fǫ∞+(1− f)+ fK(ǫ∞)
= − α . (11)
Cm dT fǫ∞+(1− f) l
Isolatingfinallyα gives
l
α = − fǫ∞+(1− f) 1 dCm
l fǫ∞+(1− f)+ fK(ǫ∞)Cm dT
1 dC
m
αl = P(f,ǫ∞)) , (12)
C dT
m
where
fǫ∞+(1− f)
P(f,ǫ∞)= −
fǫ∞+(1− f)+ fK(ǫ∞)
B. Theabsolutevalueofα
l
The determination of α and also the uncertainties of the measured value depend on determining correctly the
l
proportionalityconstantP(f,ǫ∞). Inordertodosoweneedtodeterminetherelevantvaluesof f andǫ∞. Tofind f
weusetheexpansioncoefficientandtofindthedielectricconstantǫ∞weusethemeasuredemptycapacitancealong
withthemeasuredfullcapacitance.
The high-temperature expansion coefficient is found28 to be 0.7*10−3 K−1; atlow temperatureswe find29 that it
isaround0.5*10−3 K−1 inthelongtimelimit. Weuse0.6*10−3 K−1 asanaveragevalue,andfindfromthisthatthe
degreeoffillingis f = 0.95iftheliquidisassumedtocontractradiallydownto213Kwheretherelaxationtimeis
100s. Thechoiceofexpansioncoefficientintherange0.5-0.7*10−3K−1andfinaltemperaturesintherange210-215K
makes f changewith±1%. Theeffectofchanging f withinthisrangeleadsonlyto±0.5%changesinP(f,ǫ∞).
IsolatingthedielectricconstantfromEq. (7)gives:
C −C (1− f)
m g
ǫ∞ = (13)
fC
g
Fromthisitisseenthattheuncertaintyin f alsogivesanuncertaintyinǫ∞,andthisactuallyhasagreaterimpacton
theuncertaintyof P thanthe directeffectoftheuncertaintyon f. Includingthiseffect,the uncertaintyin P dueto
uncertaindegreeonfillingisstillonly±1%.
In order to determine ǫ∞ from Eq. (13) we need to know the geometric capacitance, Cg. This is found from
measurements on the empty capacitor at the measuring temperature. We estimate that the uncertainty is ±2% on
C . This estimate is made by comparingmeasurements madeon the capacitor afterassembling itanew. The total
g
uncertaintyonǫ∞ isroughly±3%,whichleadstoanuncertaintyonPof±2%.
9
Altogether the uncertainty on P(f,ǫ∞) and therefore on the absolute value of αl is about ±3%. It should be
emphasized that this uncertainty has no effect on the shape or the time scale of the measured relaxation. This is
so aslong aswe stick to linear experiments. For largertemperaturesteps therewill be (atleastin principle) some
second-ordereffectsmakingP(f,ǫ∞)changeduringtherelaxationbecauseofthechangeinǫ∞.
Inthemodellingoftheconnectionbetweenmeasuredchangeincapacitancetoα wehavenotconsideredtheradial
l
expansion of the electrode plates. Including this (in the simples possible way) gives rise to an extra additiveterm
1 ǫ0dA inEq.(11).Thesizeofthistermwillbegivenbythelinearexpansioncoefficientoftheelectrodes.Theyarein
Cm l dT
thiscasemadeofcopper,whichattherelevanttemperaturehasalinearexpansionofapproximately15×10−6K−1.
The total measured change in the capacitance is about 50-100 times bigger, thus the effect is small. However, the
timedependenceisdifferentthereforeitcouldberelevanttoincludethiseffectinthefuture. Alternativelywealso
considershiftingtoanelectrodematerialwithanevensmallerexpansioncoefficientinordertoavoidtheeffectall
together.
ItshouldbekeptinmindthatwehaveusedtheLorentzfield. Thisisanimportantassumptionandtheuseofan
otherlocalfield,whenconnectingdensitywiththedielectricconstantwillchangetheresult. Usingthemacroscopic
Maxwell field, will yield the same everywhere, except for K(ǫ∞) in Eq. (9) which will be given by KMax(ǫ∞) =
(ǫ∞ −1) instead of the KLor(ǫ∞) = (ǫ∞ −1)(ǫ∞ +2)/3. This leads to a 20 % increase in P and the calculated
numerical value of α . Again we stress that using another local field will change the absolute values, but will not
l
changethetimescaleorshapeofthemeasuredrelaxation.
Whilenoneoftheabove-mentionedthingsaffectthetimescaleorthespectralshapeofthemeasuredrelaxation,the
temperaturedependenceofǫ∞ couldinprincipleaffectthetemperaturedependenceofthecalculatedαl. However,
thiseffectisnegligibleoverthe6degreerangestudiedintheworkandPwillbeconsideredconstant.
To summarize, the problems discussed in this section can lead to an unknown temperature- and frequency-
independentscalingofallthemeasuredα -values.
l
C. Theshapeoftherelaxationcurve
In the following we describe the measuring protocol in detail and describe a correction made on the data. We
moreoverusethistogiveanestimateoftheuncertaintyontheshapeoftherelaxationcurvesreported.
Amainissueis,ofcourse,thefirstpartofthemeasuringcurvewherethetemperaturegetsinequilibrium. Fig. (3)
showsdetailsofasingletemperaturestep. Itisclearlyseenhowthetargettemperatureisachievedwithinlessthan
10s,correspondingtoacharacteristictimeof2s.
Fig. (4)a)showsatypicalsetoftemperaturesteps: aseriesofupanddownjumpsaremadeatthesametempera-
ture,withvariableamplitude.
Fig. (4)b)showstherawmeasuredcapacitancecorrespondingtothetemperaturestepsinFig. (4)a). Twothings
areworthnoticing. Firstweseetheexpectedriseincapacitancewhentemperatureisdecreased. Secondly,weseea
long time driftofthe equilibrium level. Atlowtemperatureswherethe liquid cannotcontractradiallyitcontracts
vertically. Comparingmeasurementsonthe emptycapacitorwithliquid filled measurementsweestimate thatthe
expansion coeficient of the liquid is roughly 10 times larger than that of the Kapton spacers. This means that the
liquid compresses the Kapton. However, on very long times it will be the Kapton which dominates (becuase the
liquidflows)andtheKaptonwillthereforeslowlyrelaxandpresstheelectrodesapart. Webelievethatthiseffectis
whatleadstothelongtimedriftseeninFig. (4)b). Thedriftissubtractedbeforetreatingthedata,asillustratedin
Fig. (4)c)andFig. (5).
We make both up jumps and down jumps intemperatureand the subtraction of the drifthasan opposite effect
onthetwo. Wecanthereforecheckthatthesubtractionismadecorrectlybycomparingupjumpsanddownjumps.
ThisisillustratedinFig. (6). Thesuperpositionofdataobtainedinupanddownjumpsalsodemonstratesthatthe
experimentislinearandgivesageneralestimateofhowprecisethedeterminationofthecurveshapeis.
Thecomparisonofupanddownjumpsmoreoverservestoguaranteethatthestepsarelinear. Therelaxationtime
isstronglytemperaturedependentwhentheliquidisclosetotheglasstransition,andthestepsthereforehavetobe
verysmallinordertomaintainlinearbehavior. Smallerstepscanbemadeaswell, andtheshapeoftherelaxation
ismaintained,butthecurvestartstogetnoisybecausethesignalisverysmall. Whenwemakelargertemperature
steps, we beginto get typicalnon-linear agingbehavior. Thatis, the relaxationisslower for downjumps thanfor
upjumpswhenthefinaltemperatureisthesame. Thesetupisactuallywellsuitedfornonlinearexperimentsalso;
becauseoftheextremelyhighresolutionofthemeasuredquantitywegetverywell-definedcurvesandcanclearly
seethenonlinearbehavioralreadyatstepsof1degree. Weplantousethesetupforthesetypesofstudies,aswell,
butfocusinthispaperonthelinearresults.
10
209.2
209.15
]
K
209.1
[

T
209.05
209
0 10 20 30 40 50
t [s]
FIG. 3: Zoom on the temperature monitored in the NTC-bead in the lower capacitor plate during the first 40 seconds of at
temperaturestep.
209.2
a) 35.770 b) 35.770 c)
209.1 35.768 35.768
35.766 35.766
T [K]209.0 ′C [pF]35.764 ′C [pF]35.764
35.762 35.762
208.9
35.760 35.760
208.8
0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5
t [s] x 105 t [s] x 105 t [s] x 105
FIG.4: a)Exampleoftemperatureprotocolatonetemperature. Aseriesofupanddownjumpsaremadeatthesamereference
temperaturewithdifferentamplitude. ThetemperaturesshownarethosemeasuredwiththeNTC-beadinthelowercapacitor
plate. Noticethatthesmallestjumpsare0.01K.b)Themeasuredcapacitance(bluepoints). Noticethattherelativechangesin
capacitance (dC/C)forthe smalljumps islessthan 10−4 and canstillbe measuredprecisely. Thereisalongtime driftinthe
measured capacitance, the dashed line illustrates this background driftand this slope is subtracted of the data before further
treatment.
c)Themeasuredcapacitanceaftersubtractionofthedrift.
V. DATA
Fig. (7)shows theexpansioncoefficientasafunctionoftimeatfourdifferenttemperatures. Thedataareshown
for steps made with ≈0.1 K, exceptthe data at 211K which are taken with a temperaturestep of ≈0.01 K. This is
whythereismorenoiseonthisdataset.