Direct Test of Supercooled Liquid Scaling Relations

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by  Tina Hecksher

Direct Test of Supercooled Liquid Scaling Relations
T. Hecksher,1,a) D. H. Torchinsky,2,a) C. Klieber,3,a) J. A. Johnson,3,a) J. C. Dyre,4 and Keith A. Nelson3
1)Glass and Time, IMFUFA, Dept. of Science and Environment, Roskilde University, DK-4000 Roskilde,
Denmark
2)Department of Physics, MIT, Cambridge, MA 02139, USA
3)Department of Chemistry, MIT, Cambridge, MA 02139, USA
4)DNRF Centre Glass and Time, IMFUFA, Dept. of Sciences, Roskilde University, DK-4000 Roskilde,
Denmark
Diversematerialclassesexhibitpracticallyidenticalbehaviorwhenmadeviscousuponcoolingtowardtheglass
transition, suggesting a common theoretical basis. The first-principles scaling laws that have been proposed
todescribetheevolutionwithtemperaturehaveyettobeappropriatelytestedduetotheextraordinaryrange
oftimescalesinvolved. Weusedsevendifferentmeasurementmethodstodeterminethestructuralrelaxation
kinetics of a prototype molecular glass former over a temporal range of 13 decades and over a temperature
7
1 range spanning liquid to glassy states. For the material studied, our results comprise a comprehensive
0 validation of the two scaling relations that are central to the fundamental question of whether supercooled
2 liquid dynamics can be described universally. The ultrabroadband mechanical measurements demonstrated
n have fundamental and practical applications in polymer science, geophysics, multifunctional materials, and
a other areas.
J

5
The extraordinary slowing down of viscous liquid dy- uid in order to examine the universal alpha and beta re-

] namics upon cooling toward the glassy state plays a key laxationprocesseswithoutcomplicationsfromadditional
t
f role in myriad contexts including polymer processing, dynamics.
o survival of living organisms in extreme cold, amorphous
A variety of empirical models9–14, as well as the first-
s
metal synthesis, and many others. Glass-forming liquids
. principles mode-coupling theory (MCT)15,16, have been
t display a wide array of common features, despite quite
a developedinanefforttorationalizetheuniversalfeatures
m different chemistry ranging from high-temperature cova-
of supercooled liquid dynamics. In its idealized form,
lentlybondedglassformerstovanderWaalsliquidsthat
- MCT predicts for the local density dynamics a critical
d typically form glasses below room temperature1–6.
temperature T at which there is a transition from er-
n Viscoelastic relaxation behavior derives from two dis- c
godic behavior above T to nonergodic behavior below,
o tinct and sequential processes common to all glass form- c
c corresponding to arrest into a metastable glassy state.
ing liquids. The primary or “alpha” structural relax-
[ The theory predicts a scaling law referred to as “time
ation dynamics, which dictate the time scales for molec-
temperaturesuperposition”(TTS).TTSimpliesthatthe
1 ulardiffusionandflow, arenon-exponentialintime, typ-
alpha relaxation spectrum retains the same width and
v
ically extend over several decades of time scales at a sin-
0 shape as the temperature is changed, even as the fre-
gletemperature,andshiftdramaticallyfrompicoseconds
1 quency range varies widely. A wide range of practices in
at high temperatures and low viscosities to many sec-
3 polymer processing, rheology, aging, and other areas are
1 onds as the sample is cooled and the glassy state is ap- based on TTS17–21, but it has never been tested directly
0 proached. This behavior gives rise to broad loss peaks
for mechanical properties across most of the time scales
. in elastic compliance spectra, covering an extended fre-
1 spanned. TTSisconsistentwithearlierheuristicdescrip-
0 quency range at any temperature and shifting from gi- tionsofsupercooledliquidsextendingfromhightemper-
7 gahertz to millihertz frequencies as the temperature is
ature all the way to the glass transition temperature.
1 lowered (see Fig. 1). In addition to the temperature-
Mode-coupling theory also predicts distinct power-law
: dependent alpha relaxation dynamics for reorganization
v scalingofalphaandbetarelaxationprocessesattemper-
i ofintermoleculargeometries,fastlocalrearrangementsof atures above T , with the dynamic exponents connected
X moleculeswithinexisting“cage”geometries,theso-called c
through arithmetic relations reminiscent of those associ-
r “beta” relaxation processes, result in a higher-frequency
a atedwithothercriticalphenomena. Thus,theseemingly
feature in the loss spectrum. This is the simplest sce-
disparaterelaxationprocesses,separatedbydecades,are
nario, which appears to obtain when structural relax-
predicted to be intimately related.
ation is slowed down through obstruction among neigh-
boring molecules but not through extensive networks as The density dynamics of supercooled liquids could be
in hydrogen-bonded liquids or entanglements as in poly- characterized through measurement of the temperature-
mers. In those more complicated cases, additional pro- dependent elastic compliance spectrum, but due to vari-
cesses between these two can be observed7,8. For the ousexperimentalchallengesthishasneverbeendoneover
present study, we chose a van der Waals molecular liq- anywherenearthefullfrequencyrangeofinterestinasin-
glematerial. Ultra-broadbanddielectricmeasurements22
and depolarized light scattering spectra23 clearly show
the alpha and beta relaxation features, but these tech-
a)Theseauthorscontributedequally. niques measure the dynamics of molecular dipoles or po-
2
(a) ing and neutron scattering spectra (see, e.g.,24–28), but
β: α: notthroughdirectmeasurementofdensitydynamicsover
n 1
o a sufficient frequency range.
ati β
el 0.8 t−a
r
r
co 0.6 −tb
o I. EXPERIMENTS AND RESULTS
ut
y a 0.4 α
sit high T low T We used seven complementary methods, based on six
en 0.2 experimental setups, to compile ultrabroadband longi-
D
tudinal compliance spectra for the glass-forming liquid
0
10−1310−1210−1110−1010−910−810−710−610−510−410−3 tetramethyl tetraphenyl trisiloxane (sold commercially
Time (s) as DC704). Four of the methods were photoacoustic
(b)
techniques29–33 through which acoustic waves in the fre-
J’’10−1 low T high T quency range 1 MHz–100 GHz were generated and de-
e, tected optically. The corresponding acoustic wavelength
c α α α α α
n
plia10−2 αtriemlaex−atteiomnp sepraetcutrreum β raarnegceomwapsar∼ab2l0e ntoms–a2mmpmle.siLzoesn,gseor afocrouloswticerwfarevqelueenngctihess
m scaling
o in the range ∼ 1 mHz–100 kHz, dynamical mechanical
s c ω−b ωa analysis methods involving piezo-ceramics that shear or
os10−3 αspectral wing βspectral wing compress the entire sample34–36 were used. The mea-
h. l poswcearli−nlgaw poswcearli−nlgaw surements spanned more than 13 orders of magnitude
c
Me in frequency with less than three decades of gaps, yield-
10−1403 104 105 106 107 108 109 1010101110121013 ing structural relaxation dynamics in the temperature
Frequency (Hz) range 200–320 K. Our earlier measurements in the low
andhigh-frequencyranges34,37lackedsufficientfrequency
coverage for comprehensive testing of the scaling predic-
FIG. 1. Schematic depictions of primary (alpha) and fast tions, but the gap was bridged by the newly developed
(beta) relaxation processes in glass-forming liquids. Alpha NanosecondAcousticInterferometry(NAI)method. The
andbetarelaxationdynamicsin(a)thetimedomain(density- experimental methods and data analysis through which
density correlation function) and (b) the frequency domain we determined the compliance spectra and the elastic
(elasticcompliancelossspectrum)attemperaturesabovethe
modulus (the inverse of the compliance) are described in
critical temperature T predicted by mode-coupling theory
c the online Supporting Information33.
(MCT).Time-temperaturesuperpositionisillustratedasthe
Figures 2(a) and 2(b) present the complex longitudi-
alpha relaxation spectral shape and width remain constant
nal modulus and compliance spectra of DC704 in the
whilethefrequencyrangeshiftsbymanydecadesasthetem-
perature is varied. Dynamic critical exponents a and b de- frequency range of approximately 10−3-1011 Hz. The
scribe the asymptotic power-law scaling of the relaxation ki- imaginary parts of the spectra have gaps in the ranges
netics predicted by MCT. At temperatures near T there is of approximately 104-106 Hz and 1-10 GHz. In order
c
a clear separation between alpha and beta relaxation time to facilitate visualization of the temperature-dependent
scales, with beta power-law relaxation of the density toward trends, themodulusandcompliancespectrashowonlya
atime-independent“plateau”valuefollowedeventuallybyal- subset of the data collected.
pha power-law relaxation kinetics. In the frequency domain,
The primary temperature dependence, observed
the relaxation spectral wings show corresponding power-law
clearly in both real and imaginary components of the
frequencydependencesaroundaminimuminthecompliance
modulus spectra, is the movement of the alpha relax-
loss spectrum. The drawings in (a) show a schematic rep-
ationpeakacrossaboutninedecadestolowerfrequencies
resentation of the beta relaxation among multiple potential
energyminima(thedifferentlocationssampledbythecentral asthetemperatureisreducedtowardtheglasstransition
moleculecoloredgreen)withinpre-existingintermolecularge- temperatureTg ≈210K.At anygivenfrequency ω0, the
ometriesor“cages”(formedbythered-coloredmolecules)and real part of the modulus M(ω ,T) increases as the sam-
0
alpha relaxation involving larger-scale rearrangement of the ple is cooled, reflecting the stiffening of the liquid as it
intermolecular geometries to permit molecular diffusion and approaches the glassy state. At any given temperature
flow. T , M(ω,T ) plateaus at frequencies above those of the
0 0
alpharelaxationspectrum,sinceatsuchhighfrequencies
the liquid cannot undergo structural relaxation during
larizabilities whose coupling to density is complicated the acoustic oscillation period, resulting in a solid-like
and potentially temperature dependent. The key fea- response.
tures of supercooled liquid structural relaxation dynam- Thecompliancespectrashowanalogousbehaviorupon
ics that might reveal universal behavior at least within cooling: movement of the alpha relaxation spectrum
some classes of glass-forming materials (e.g., organic van across many decades and a decrease in the real part at
derWaalsliquids)havebeentestedthroughlightscatter- low temperatures and high frequencies. The beta relax-
3
(a) 7
′M[GPa] 6354 215K 220K 225K 230K 235K 248K256K 27259K0K mpliance (J’/J)∞1100−01 J’’∝Jω’/J1∞ α J’’∝ω0β.3 J’’/J’’max 1100−01 Normalized com
2 p
co lia
GPa]1100−10 Normalized 10−2 222222122334505058KKKKKK 10−2maxnce (J’’/J’’
[ 256K )
′′M 10−3 275K J’’∝ω−0.5 10−3
10−2
10−410−2 100 102 104 106 108 10101012
ω/ω
max
10−3
10−310−210−1100 101 102 103 104 105 106107 108 109101010111012
frequency[Hz]
(b) FIG. 3. Master plot of the compliance demonstrating time-
0.55 PSG+PBG PBGres.NAI ISTS TDBSPUI
temperature superposition. The individual traces are scaled
0.5 and shifted according to the fit values. The low-frequency
]0.45 DC704 331150KK sideoftheimaginarypartfollowsalinearpower-law,J(cid:48)(cid:48) ∝ω1
−1GPa00.3.45 332200985005KKKK (icdsa.sThehdelfiunlel)c,ucrovrersegspivoenadninaglyttoicemxopdoenlefinttisablalosendg-otnimaenkailnpehta-
[ 0.3 228705KK relaxationspectrumthatfollowsanω−0.5 high-frequencyde-
′J0.25 226684KK cay. The ω0.3 power-law also shown here corresponds to the
256K
0.2 224486KK value found to fit the beta relaxation spectral wing near the
0.15 224442KK minimum at 248 K. (See Eq. 2 and Fig. 5)
10−1 224306KK .
234K
232K
230K
228K
−1]10−2 222222654KKK F(t/τα(T))describestherelaxationdynamicsatalltem-
Pa 222220KK peratures, with the temperature dependence contained
G 218K
[10−3 221165KK only in the values of the characteristic relaxation time
′′J 221140KK τα(T). InthatcasethealpharelaxationspectrumJ(cid:48)(cid:48)(ω)
displays analogous behavior. In order to test TTS, we
10−4 normalizedtheimaginarycompliancespectrabythepeak
heights and the real spectra by the high-frequency lim-
10−310−210−1100101102103104105106 107108109101010111012
frequency[Hz] iting value J , and shifted both by amounts that made
∞
the peak frequencies in J(cid:48)(cid:48)(ω) coincide. The results are
shown in Fig. 3. The alpha relaxation features super-
FIG. 2. (a) Longitudinal modulus spectra of DC704. Solid pose well across the entire temperature regime studied.
lines are guides to the eye. (b) Elastic compliance spectra of
The imaginary parts have the characteristic asymmetric
DC704. Thetechniquesusedinthedifferentfrequencyranges
shape found for most glass-forming materials: the low-
are indicated.
frequencysidesofthealpharelaxationspectrafollowthe
Maxwell behavior, J(cid:48)(cid:48)(ω)∝ω1, while the high-frequency
sides follow a power-law J(cid:48)(cid:48)(ω) ∝ ω−1/2. The shifting
ation feature is observed most clearly in the imaginary
ofthespectratomakethealphapeaksoverlapseparates
parts of the modulus and compliance functions, which
thebetarelaxationfeatures(whichallappearatthehigh-
both rise at high frequencies.
frequency range in Fig. 2) on the scaled frequency axis.
The scaled plots highlight the increase in separation be-
tween alpha and beta relaxations upon cooling.
II. SCALING ANALYSIS
Figure 4(a) shows the values of the characteristic re-
laxation time τ of the alpha process as a function of
α
Time-temperature superposition states that for the temperature determined from fits to the stretched ex-
alpha relaxation dynamics, the normalized density au- ponential function exp(−(t/τ )n) with a temperature-
α
tocorrelation function can be written as Φ(t,T) = independent stretching exponent n and strongly tem-
F(t/τ (T)), i.e., a single temperature independent form perature dependent τ values. The stretching exponent
α α
4
was fixed at n = 0.5, which at high frequencies corre- features26,47. This was the case for our results below
sponds to an ω−1/2 decay of the imaginary part of the 248 K and thus MCT can be meaningfully tested in the
compliance4,38,39. The observed non-Arrhenius temper- temperature range 240–248 K.
ature dependence is typical of glass-forming liquids and
Figure 5 shows the region around the minima between
is associated generally with a complex energy landscape
thealphaandbetafeaturesofthecompliancelossspectra
rather than a single activation energy for all relaxation J(cid:48)(cid:48)(ω) at temperatures 240 K and 248 K. As a final test
processes2–6,40. The relaxation time data were fit to the
Eq. (2) was compared to the data with fixed exponent
MCT power-law prediction15
values. Only the minimum frequency ω and the min-
min
imum compliance loss J(cid:48)(cid:48) positions of each curve were
τ =τ [T /(T −T )]γ (1) min
α x c c adjusted to fit the data. The exponent value b = 0.50
was determined from the high-frequency wing of the al-
with τ and γ as free parameters. This prediction
x pha relaxation spectra evident in the TTS plot (Fig. 3).
is for temperatures above T (taken as 240 K as de-
c Fromthisthepredictedvalueoftheexponentacouldbe
scribed below), although as suggested heuristically we
calculatedusingEq.(3). Thisyieldedthevaluea=0.28,
find that TTS applies also below 240 K. The full set of
whichshoulddescribethelow-frequencywingofthebeta
τ (T) values was fitted to the commonly used empirical
α relaxation spectra. The resulting prediction completely
Vogel-Fulcher-Tammann (VFT) function41,42 τ (T) =
α determines theshapeof thecurveswhichgo through the
τ exp[DT /(T −T )].
0 0 0 data well within the experimental scatter.
The MCT interpolation formula43
J(cid:48)(cid:48) (cid:34) (cid:18) ω (cid:19)−b (cid:18) ω (cid:19)a(cid:35)
J(cid:48)(cid:48)(ω)= min a +b (2)
a+b ω ω
min min
III. CONCLUSIONS
connects the two relaxation features around the mini-
mum between. This expression was fitted to the mea-
Our measurements provide the first ultra-broadband
sured spectra in the region around the minimum. The
mechanical relaxation spectra of a glass-forming liquid
fitted values for a and b are shown in Fig. 4(b). The un-
reachingthehighfrequenciesofthebetarelaxationspec-
certaintyofthefitsincreasesdramaticallyasthetemper-
trumandextendingtothelowfrequenciesofthealphare-
ature increases due to the merging of the two processes
laxation spectrum even as T is approached. The results
g
making a distinction of the separate processes difficult.
give direct experimental support for time-temperature
TheMCTpredictsarelationbetweenthetwoexponents
superpositionofthealpharelaxationspectrumaboveand
according to which
below the MCT critical temperature (240 K), the va-
lidity of which is often assumed in empirical studies of
Γ(1−a)2 Γ(1+b)2
λ = =λ = (3) glass-forming liquids and in modeling used for practical
a Γ(1−2a) b Γ(1+2b) applications. Our results also permitted calculation of
the dynamic critical exponents of MCT above T , yield-
where Γ(z) is the gamma-function defined as Γ(z) = c
(cid:82)∞xz−1e−xdx. TherelationistestedinFig.4(c),which ing results consistent with predictions that relate the al-
0 pha and beta relaxation dynamics. In continuing work,
shows that the data are consistent with this prediction
we are filling in the frequency gaps of just under three
at temperatures above 240 K, but clearly breaks down
decadeswithinthemorethan13decadesthatthepresent
below. We take this to define the MCT critical tem-
measurements cover. We are also extending the mea-
perature. The value of the alpha relaxation time at
surements to higher frequencies in order to improve the
T , τ (240 K) ≈ 0.1 µs, is typical for T , below which
c α c reliability of our tests. We also have measured shear re-
idealized MCT breaks down16 due to the onset of ther-
laxation dynamics across a wide frequency range37, and
mally assisted hopping44–46 through which alpha relax-
filling in the frequency gaps for these measurements will
ation continues to occur.
permit comparison between longitudinal and shear dy-
Figure 4(d) compares the alpha relaxation critical ex-
namics that may show distinct temperature-dependent
ponent γ determined in Fig. 4(a) based on T = 240 K
c behavior48.
to the MCT prediction that connects the power-law ex-
ponents to γ through the relation15 DC704 is the first sample for which mechanical relax-
ationdynamicshavebeenmeasuredacrossthefrequency
γ =1/(2a)+1/(2b). (4) rangewehaveexplored. Additionalglass-formingliquids
mustbeexaminedinordertoassessthegeneralityofTTS
TheaboveanalysisindicatesthatMCTisvalidattem- and other MCT predictions. Our results demonstrate
peraturesabove240Kforthestudiedliquid. Forthefits that access is now available to the extraordinarily wide
tobeusefulasatestoftheMCTpredictions,theremust frequency range needed for such comprehensive tests of
be sufficient separation between the alpha and beta re- supercooled liquids and a wide range of partially dis-
laxation spectra as T → T from above that the a and ordered materials including relaxor ferroelectrics, block
c
b exponents can be associated distinctly with the two copolymers, and many others.
5
Temperature[K] 1.5
400 350 300 275 250 225 200 xp. 1 ab (b)
101 (a) aw e 0.5
erl
10−1 VFT−fit to all points ow 0
P
MCT−fit to dotted points -0.5
[s] 10−3 2 (c )
Jntime,τα10−5 λλ/ab 01 MCT prediction
o
ati 10−7
x
a -1
el
R 10−9 6 (d)
b
2
1/4
10−11 Tc Tg 1/2a+2 MCT-fitted γ
=
10−13 γ0
2.5 3 3.5 4 4.5 5
220 240 260 280
1000/T [K−1] Temperature [K]
FIG.4. (a)ThecharacteristicalpharelaxationtimeplottedasafunctionofinversetemperaturewheretheArrheniusequation
gives a straight line. The glass transition temperature is at Tg ∼= 210 K. The VFT equation was fitted to all data points (full
line) yielding logτ =−15.0±0.8, D =6±1 and T =183±5 K. The MCT critical temperature T =240 K was identified
0 0 c
from(b)asthetemperaturebelowwhichMCTpredictionsclearlybreakdown;theMCTpower-lawfittothehigh-temperature
points (T > T ) (marked by central black dots) yielded the following fit parameters: log(τ /s) = −12.0±0.5, γ = 2.7±0.4.
c x
(b) Values of powerlaw exponents a and b over a wide temperature range, determined as fitting parameters to Eq. (3). (c)
Test of the MCT prediction in Eq. (3) that λ = λ . The test is useful over the 240–248 K temperature range. (d) The
a b
calculated exponent γ as a function of temperature. The green dotted line shows the value of obtained from fitting Eq. (1) to
the relaxation times (in (d)) and the green area gives the estimated uncertainty on this value.
IV. MATERIALS AND METHODS
J′′(ω)=J′m+′nb b ωωmn + ωωmn −b
(cid:20) (cid:16) (cid:17) (cid:16) (cid:17) (cid:21) A. Overview
b=0.50(fromTTS)
248K =0.28(MCTpredictionfromb)
) The seven different measurement methods and the
1
a− frequency ranges that they cover are summarized in
P
G Fig. S1; detailed descriptions of the techniques and
(
J′′ data collected from them are discussed in the Support-
240K ing Information33. The techniques include three low-
10 2 frequency methods involving piezo-ceramics that shear
−
orcompresstheentiresamplequasi-statically34–36,49 and
fourhigher-frequencymethodsutilizingshortlaserpulses
toexciteandsubsequentlydetectacousticwavesinanir-
108 109 1010 1011 radiated region30–32,50–53.
Frequency(Hz)
Thetwolowest-frequencymethodsdeterminethecom-
plex frequency-dependent bulk modulus K(ω) and shear
modulusG(ω)directly,whereω istheangularfrequency.
FIG. 5. Data around the minimum of the imaginary part
The longitudinal modulus is then given by M(ω) =
of the compliance function fitted to the MCT power-law re-
lation, Eq. 2, without adjustable shape parameters (critical M(cid:48)(ω)+iM(cid:48)(cid:48)(ω) = K(ω)+(4/3)G(ω). The four meth-
exponents a and b) at sample temperatures of 240 K and ods covering MHz-GHz frequencies determine acoustic
248 K. The exponent b = 0.50 was determined from the parameters. The complex longitudinal modulus is given
low-temperature data using time-temperature superposition; from the acoustic data as M(ω,T) = ρ(T)(c (ω,T))2
L
Eq. 3 then yielded a=0.28. where ρ(T) is the temperature-dependent density and
c (ω,T) the complex frequency-dependent longitudinal
L
sound velocity.
In order to determine the modulus as a function of
temperature from the sound velocity and damping rate,
thethermalcontractionofthesamplemustbeaccounted
6
the sample was observed to become slightly opaque at
Picosecond Ultrasonic Interferometry
thecoldesttemperatures. Thisislikelyduetophasesep-
Time Domain Brillouin Scattering
aration of dissolved impurities from the base liquid and
Impulsive Stimulated Scattering disappeared when the liquid was reheated. The prob-
Nanosecond Acoustic Interferometry lem was overcome in subsequent experiments by mixing
thesamplewithanhydrousMgSO ,combinedwithheat-
PBG resonance 4
ing under vacuum before filtration. This approach was
Piezo-electric Bulk modulus Gauge
adopted for the NAI, TDBS, and PUI techniques. Com-
Piezo-electric Shear modulus Gauge
parison of data both with and without treatment by the
1mHz 1Hz 1kHz 1MHz 1GHz 1THz dryingagentshowednodifferenceintheacousticparam-
eters. The DC704 samples never crystallized during the
FIG. 6. Survey of mechanical spectroscopic techniques. At
course of our experiments.
lowfrequencies(frommHzupto∼10kHz)thepiezo-electric
The transducer techniques utilize home-built closed-
ShearmodulusGauge(PSG)andthepiezo-electricbulkmod-
cycle cryostats capable of keeping the temperature con-
ulus Gauge (PBG) were used to measure the shear and bulk
moduli. Analysis of the overtones in the PBG provided ac- stant within a ∼ 5 mK58, while the NAI and ISS mea-
cess to some data points in the 100−500 kHz region. In the surements were performed in a commercially available
low MHz range, nanosecond acoustic interferometry (NAI) cold-finger cryostat. In these measurements, tempera-
was used to probe the frequency dependence of the longi- ture sensing was provided by a factory calibrated plat-
tudinal sound speed and attenuation rates directly. In the inum resistors immersed in the liquid a few millimeters
higherMHzrange,impulsivestimulatedscattering(ISS)was away from the optical beams. The TDBS and PUI tech-
usedtomeasurelongitudinalsoundvelocitiesanddampingat
niques were performed in a commercial sample-in-vapor
specifiedacousticwavevectors. Finally,time-domainBrillouin
cryostat and the temperature was monitored at a posi-
lightscattering(TDBS)andpicosecondultrasonicinterferom-
tion a few millimeters away from the sample.
etry(PUI)wereusedtomeasurelongitudinalacousticspeeds
We did not carry out any calibration of temperatures
and attenuation in the MHz-GHz frequency ranges shown.
between the cryostats of the different labs. However, the
low-frequency part of the longitudinal spectrum was ob-
for. Using literature data of the thermal expansion coef- tainedasthesumoftwoindividualmeasurements,which
ficient α=7.2×10−4[K−1]54,55 and assuming this quan- were carried out in the same experimental set-up (same
tityistemperatureindependent,thefollowingexpression cryostat, same electronics). We estimate that the uncer-
for the temperature dependence of the density can be taintyontheabsolutetemperatureislessthantheoverall
derived37. noise in the high-frequency methods.
ρ(T )
ρ(T)= ref
1+α (T −T ) ACKNOWLEDGMENTS
p ref
(5)
1.07 [kg/m3]
=
1+7.2×10−4 [K−1](T−298[K]) The work at MIT was supported in part by National
Science Foundation Grant No. CHE-1111557 and De-
Once the modulus is obtained, the complex longitudinal partment of Energy Grant No. DE-FG02-00ER15087.
complianceisgivenbyJ(ω)=1/M(ω)=J(cid:48)(ω)−iJ(cid:48)(cid:48)(ω). The work at Roskilde University was sponsored by the
MCT relations are predictions for the mechanical sus- DNRF Grant no 61.
ceptibility χ(ω) to which J(ω) is proportional. Strictly
speaking, χ(ω) is related to the density autocorrelation 1Kauzmann, W. (1948) The nature of the glassy state and the
behavior of liquids at low temperatures. Chemical Review 43,
function and therefore to the bulk compliance, not the
219–256.
longitudinal compliance. However, the difference be- 2Angell, C. A. (1995) Formation of glasses from liquids and
tweenK(ω)andM(ω)(andthusalsothebulkandlongi- biopolymers. Science267,1924–1935.
tudinalcompliances)isnegligiblebecausetheshearmod- 3Debenedetti, P. G. (1996) Metastable liquids : concepts and
ulus is considerably smaller than the longitudinal mod- principles. (Princeton,N.J.: PrincetonUniversityPress).
4Dyre,J.C. (2006)Solidityofviscousliquids.IV.densityfluctu-
ulus, and their frequency-dependent dynamics are very
ations. Phys. Rev. E74,021502.
similar at both low34,56,57 and high48 frequencies. 5Roland,C.M. (2010)Relaxationphenomenainvitrifyingpoly-
mersandmolecularliquids. Macromolecules43,7875–7890.
6Berthier, L & Biroli, G. (2011) Theoretical perspective on the
glass transition and amorphous materials. Rev. Mod. Phys. 83,
B. Sample Preparation
587.
7Johari, G. P & Goldstein, M. (1970) Viscous Liquids and the
ForthePSG,PBG,andISSmeasurements,DC704was Glass Transition. II. Secondary Relaxations in Glasses of Rigid
obtained from Sigma Aldrich and used without purifica- Molecules. J. Chem. Phys.53,2372.
8Johari,G.P. (1982)Effectofannealingonthesecondaryrelax-
tion. For the ISS measurements, the liquid was trans-
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1
Supplementary Material: Direct Test of Supercooled Liquid Scaling Relations.
I. EXPERIMENTAL METHODS
Overview over the seven different measurement methods and the frequency ranges that they cover are summarized
in Fig. S1; detailed descriptions of the techniques and data collected from them are discussed in the subsequent
sections below. The techniques include three low-frequency methods involving piezo-ceramics that shear or compress
the entire sample quasi-staticallyS1–S4 and four higher-frequency methods utilizing short laser pulses to excite and
subsequently detect acoustic waves in an irradiated regionS5–S11.
Picosecond Ultrasonic Interferometry
Time Domain Brillouin Scattering
Impulsive Stimulated Scattering
Nanosecond Acoustic Interferometry
PBG resonance
Piezo-electric Bulk modulus Gauge
Piezo-electric Shear modulus Gauge
1mHz 1Hz 1kHz 1MHz 1GHz 1THz
FIG.S1. Surveyofmechanicalspectroscopictechniques. Atlowfrequencies(frommHzupto∼10kHz)thepiezo-electricShear
modulus Gauge (PSG) and the piezo-electric bulk modulus Gauge (PBG) were used to measure the shear and bulk moduli.
AnalysisoftheovertonesinthePBGprovidedaccesstosomedatapointsinthe100−500kHzregion. InthelowMHzrange,
nanosecond acoustic interferometry (NAI) was used to probe the frequency dependence of the longitudinal sound speed and
attenuation rates directly. In the higher MHz range, impulsive stimulated scattering (ISS) was used to measure longitudinal
sound velocities and damping at specified acoustic wavevectors. Finally, time-domain Brillouin light scattering (TDBS) and
picosecondultrasonicinterferometry(PUI)wereusedtomeasurelongitudinalacousticspeedsandattenuationintheMHz-GHz
frequency ranges shown.
A. Low-frequency methods
Thelow-frequencymethodsmeasuremechanicalmodulidirectly. Thesetechniquesdonotmeasurethelongitudinal
modulus, butthebulkandshearmoduli. Intheisotropiccase(e.g., inaliquid)thereareonlytwouniquemechanical
moduli, and the longitudinal modulus M is given in terms of the bulk (K) and shear (G) moduli as: M =K+4/3G.
PSG PBG
Electrodes
(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) Liquid (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
Piezo electric ceramics
FIG.S2. SchematicdrawingofthePSGandPBG.InthePSGthreeelectrode-coveredpiezo-electricceramicdiscsaremounted
in a layer construction; the liquid is loaded into the gaps between the discs. In the PBG, a spherical shell of piezo-electric
ceramicmaterialisfilledwiththeliquidviaaholedrilledintheceramics. Aliquidreservoir(notshownhere)isattachedabove
the hole, allowing extra liquid to be drawn in as the temperature is lowered.
Both the piezo-electric shear modulus gauge (PSG) and the piezo-electric bulk modulus gauge (PBG) methods are
based on the piezo-electric effect, i.e., the conversion of electrical to mechanical energy. These methods have been
documented in detail by Christensen and Olsen in Refs. S3 and S2, respectively. In the following we briefly sketch
the steps in modeling of the devices, which allow us to deduce a mechanical modulus from the electrical data.
2
1. Piezo-electric Shear Modulus Gauge
The PSG is constructed of three electrode-covered piezo-electric ceramic discs mounted in a layered construction,
which prevents unwanted bending of the discs and further has the advantage that it can be mapped mathematically
to a one-disc system involving a fixed wallS3.
The liquid is loaded into the 0.5 mm gap between the discs (Fig. S2). Depending on the polarity of the discs
as compared with the direction of an applied electric field, the discs expand or contract in the radial direction.
Electrically, the middle disc is connected in parallel with the two outer discs in series as shown in Fig. S2. Here, the
smalldotsindicatethepolarityofthepiezo-electricdiscs; thuswhenanelectricfieldisapplied,themiddlediscmoves
in opposition to the two outer discs. With this construction the gap between the discs is field free, and the liquid is
subjected to a purely mechanical perturbation.
Thecapacitanceofeachdiscdependsonitsstrainstate,soiftheliquidispartiallyclampingthedisc(thushindering
its motion), the measured capacitance is lower than that of freely moving discs. By a precise measurement of the
electrical capacitance of the PSG one can obtain the stiffness of the liquid in contact with the disc. In other words,
knowing the exact relationship between the two, we can convert the electric impedance into the shear modulus.
Theelasto-electriccompliancematrixdescribestheconnectionbetweenthecomponentsofthestressσ andstrain
ij
(cid:15) tensors and the electrical field of the piezo-electric material. The equations describing a axially polarized ceramic
ij
can be split into four independent parts, the relevant components of which can be reduced to the following
    
σ c c −e (cid:15)
rr 11 12 13 rr
σφφ=c12 c11 −e13(cid:15)φφ (S1)
D e e εS E
z 13 13 33 z
where c and c are elastic constants of the ceramic, (cid:15)S is the dielectric constant, and e is the coupling constant.
11 12 33 13
The measured capacitance C of the disc can be found by integrating the charge density D and dividing by the
m z
voltage
Q (cid:82)r02πrD (r)dr
C = = 0 z , (S2)
m U ξE
z
where the charge density D is given by Eq. (S1) and ξ is the thickness of the disc. D depends both on the strain
z z
state and the applied electrical field E . Evaluating this integral it is found that the capacitance is a function of the
z
radial displacement at the edge of the disc u (r )
r 0
C =Au (r )+B (S3)
m r 0
where A and B are known constants. It remains to determine the displacement at the edge of the disc u (r ) as a
r 0
functionofrigidityoftheliquid. Thedisplacementu isfoundbysolvingtheradialequationofmotion,whichreduces
r
to
c (cid:0)r2(u(cid:48)(cid:48))+u(cid:48) −u (cid:1)−σ r2 =−ω2r2ρu (S4)
11 r r r l ξ r
where the prime indicates the derivative with respect to r, ξ is the thickness of the disc and σ is the tangential stress
l
that the liquid exerts on the disc. σ is by definition proportional the shear modulus of the liquid σ = G(ω)u /d,
l l r
where d is the thickness of the liquid layer (or equivalently the distance between the discs), which is the quantity
relevant to determining the relaxation.
Figure S3 shows the measured capacitance of the empty (black trace) and liquid-filled (blue trace) PSG. At high
temperatures there is no influence from the liquid at these frequencies and the two spectra are identical. The
resonances in the spectrum are mechanical resonances of the discs. At lower temperatures, the shear modulus of the
liquidincreasesandpartiallyclampsthediscs. Thisisobservedasadropinthecapacitancebelowthefirstresonances.
Wewillrefertotherangeoffrequenciesbelowthefirstresonance(<100kHz)ofthesystemasthequasi-static region.
The liquid also influences the positions of the overtones as compared with the spectrum of the empty device. In the
quasi-static region, the shear modulus is found via the described inversion procedure. The inverted data, i.e., the
inferred complex shear modulus, are shown in Fig. S4.
2. Piezo-electric Bulk Modulus Gauge
The PBG, which was also depicted in Fig. S2, consists of a spherical shell of a piezo-electric ceramic material
polarized in the radial direction. The shell is covered by an electrode material both on the inside and the outside.
3
] 15
F
n 300K
[
C′10
5
] 15
F
n
[ 226K
C′10
5
1 2 3 4 5 6
log(freqency/Hz)
FIG. S3. Raw data of the empty (black) and liquid-filled (blue) PSG at two different temperatures. At 300 K the presence of
the liquid does not affect the signal because the liquid is quite fluid. At 226 K the liquid partially clamps the discs, which is
manifested as a drop in capacitance in the quasi-static region and a shift of the resonances in the high-frequency region.
DC704
] 1
a
P
G
[0.5
′
G
0
210K
Pa]0.3 216K 222K 228K
G
[0.2
′
′
G0.1
0
-3 -2 -1 0 1 2 3 4
log(freqency/Hz)
FIG. S4. Real and imaginary parts of the shear modulus of DC704 at four temperatures approaching T .
g
Applying an electric field to the capacitor, which these electrodes constitute, deforms the ceramic (expanding or
contracting depending on the direction of the field) and effectively changes the inner volume of the sphere.
A liquid inside the shell will oppose this deformation and thus change the measured capacitance. The difference in
capacitancebetweentheempty,freelymovingshellandthepartiallyclampedshellcanberelatedtothebulkmodulus
of the liquid. The deformation is radial. An analysis of forced vibrations in a visco-elastic sphere shows that in the
low-frequency (quasi-static) region of the measurement this corresponds to an isotropic compression of the liquid,
while at high frequencies it is a mixture of bulk and shear deformations.
In order to be able to fill the PBG with liquid, a hole is drilled in the shell. A tube is attached over the hole.
Filling this tube, as well as the entire shell, allows the PBG to draw in extra liquid when the liquid in the shell
contracts during cooling. Thus the liquid volume is constant throughout the duration of the measurement, i.e., at all
temperatures.
ThemodelingofthePBGissomewhatsimplerthanthatofthePSGsinceonecanassumethatthethicknessofthe
ceramicisnegligibleinthedirectionofitsmotion. Thuswecanexpressthemodelintermsofanelectricalequivalent
diagram, shown in Fig. S6, where the conversion from electrical to mechanical energy is modeled by a transducer T .
r
On the electrical side of the diagram there is a capacitor C which corresponds to the actual capacitor constituted by
1
thetwoelectrodes. Onthemechanicalside,thecapacitorC modelstheelasticpropertiesoftheceramic,theinductor
2
L models the inertance, and the resistor R models the friction. The “black box” C is the liquid capacitance which
liq
is what we want to determine.