Thermalization Calorimetry: A simple method for investigating glass transition andcrystallization of supercooled liquids

Bo Jakobsen,∗ Alejandro Sanz, Kristine Niss, Niels Boye Olsen, Ib H.

Pedersen, Torben Rasmussen, Tage Christensen, and Jeppe C. DyreDNRF centre “Glass and Time”, IMFUFA, Department of Sciences,

Roskilde University, Postbox 260, DK-4000 Roskilde, Denmark

We present a simple method for fast and cheap thermal analysis on supercooled glass-formingliquids. This “Thermalization Calorimetry” technique is based on monitoring the temperature andits rate of change during heating or cooling of a sample for which the thermal power input comes fromheat conduction through an insulating material, i.e., is proportional to the temperature differencebetween sample and surroundings. The monitored signal reflects the sample’s specific heat and issensitive to exo- and endothermic processes. The technique is useful for studying supercooled liquidsand their crystallization, e.g., for locating the glass transition and melting point(s), as well as forinvestigating the stability against crystallization and estimating the relative change in specific heatbetween the solid and liquid phases at the glass transition.

Keywords: Thermalization, calorimetry, supercooled liquid, glass transition, crystallization

I. INTRODUCTION

Einstein is often quoted for stating that if you have justone choice of measurement, go for the specific heat. Thepoint is that this quantity by integration determines theentropy, the free energy, etc, i.e., all of the relevant ther-modynamics. Indeed, calorimetry is a standard methodthroughout the scientific, technical, and industrial world,which has been commercially available in many differentversions for more than 50 years [1–5]. Modern calorime-try is highly sophisticated and measures with unprece-dented accuracy and, even more importantly, on smallerand smaller samples [6–8]. Thus nanocalorimetry is nowbecoming routinely available, which is absolutely essen-tial for expensive samples, e.g., in modern molecular bi-ology or advanced material science [7–9].

At the same time there is, however, a need for “quickand dirty” calorimeters in some branches of physics andchemistry in which fast answers are needed, whereas ac-curacy is less important for initial investigations. Ourgroup Glass and Time at Roskilde University [10] hasfor many years conducted research into viscous liquidsand the glass transition. “What is the glass transitiontemperature?” is the first question one asks when inves-tigating a new liquid. Here accuracy is not important,but speed, flexibility, and simplicity are. This paperpresents our simple “Thermalization Calorimetry” (TC )method, which allows for studying glass-forming super-cooled liquids and their approach to the glass transitionduring cooling, possible crystallization, as well as meltingof glass or crystal upon heating.

The TC method is based on recording the tempera-ture as a function of time for a system that is slowlycooled or heated with a heat current that depends lin-early on the temperature of the sample because it de-rives from heat conduction to the surroundings. The TC

∗ boj@dirac.ruc.dk

technique is simpler to implement than traditional dif-ferential scanning calorimetry (DSC) because the linear-ity between heat current and temperature is obtained bypassive insulation of the sample and no heating device isrequired.

In TC changes in specific heat, exothermic, and en-dothermic process, etc, are observed by plotting the rate

at which the temperature changes with time, dT (t)dt , ver-

sus the temperature, T (t). Such a plot reveals a lot ofthermodynamic information about the sample, its glasstransition, crystallization, etc. The TC method was in-spired by simple experiments conducted more than 40years ago by our colleague J. Højgaard Jensen [11] fordemonstrating the glass transition in propanols. Sincethen the TC method under the nickname “The Red Box”has been a workhorse in Glass and Time. TC is now ourstandard tool for initial studies of new glass-forming liq-uids by which we locate the glass-transition temperatureand investigate the liquid’s stability against crystalliza-tion. Furthermore, TC turned out to be excellent forteaching purposes, and it has been used for numerousstudent projects at all levels from high school to mastersproject, in which its simplicity and ease of adaptation hasallowed for studies of samples ranging from amorphousdrugs to candy.

The paper is structured as follows: Section II intro-duces the technique and how to interpret TC data. Ex-amples of applications to glass transition and crystal-lization are given in Sec. III. Finally Sec. IV presentsa short discussion of pros and cons for the techniques,and perspectives on adaptation to further applications.The Appendix gives details on the experimental proto-cols (Appendix A), on the technical implementation ofthe technique (Appendix B), as well as a discussion of asimple Tool-type [12] model of the glass transition thatis useful also for teaching purposes (Appendix C).

arX

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5

2

1

1

T (t)

liquid-N2T0,cool

(a)

TT0,cool T0,heat

0 (R0Cliq)−1

(R0Cglass)−1

dTdt

PassiveCooling

PassiveHeating

(b)

Tg,TC

PassiveCooling

PassiveHeating

T (t)T0,heat

P(t) = − 1R0

(T (t) − T0)

FIG. 1. Illustration of the two typical modes of operation ofthe TC technique and the expected

(T, dT

dt

)-traces for a glass

forming liquid.a) The experimental setup. The gray region is in both casesthe insulating material, the red line and dot illustrate thethermocouple inserted in the liquid (marked by blue). Thebasic assumption is that the thermal power entering the sam-ple, P (t), is proportional to the difference between the sam-ple temperature, T , and the temperature of the surroundings,T0, with the proportional constant 1/R0 in which R0 is thethermal resistance. Left part illustrates a slow heating of apre-cooled glassy sample using a block of insulation material.T0 is here the (room) temperature outside the block. Rightpart illustrates slow cooling of a sample placed in an insulat-ing container inserted directly into the cooling agent (whichis normally liquid nitrogen). T0 is here the temperature ofthe cooling agent, i.e., 77 K.b) Expected

(T, dT

dt

)-traces for the two protocols applied to a

glass-forming liquid with constant specific heat in the liquid(Cliq) and the glass (Cglass). The curves “break” at the glasstransition temperature, Tg,TC, due to the change in specificheat between the equilibrium supercooled liquid and the glass.The dashed lines are guides to the eye, extending to (T0, 0) inthe two cases. See the main text section II for details on theinterpretation.

II. THE THERMALIZATION CALORIMETRY(TC) TECHNIQUE

Figure 1 illustrates the two ways we normally use theTC technique. As mentioned in the Introduction, the TCmethod is based on heating or cooling via a passive, insu-lating material; this results in a heat current that is pro-portional to the temperature difference between sample

and surroundings. During this process the temperatureand its time derivative are recorded. In the following webriefly describe the technique and the two protocols nor-mally used for studying glass-forming molecular liquids.The TC technique can be adapted to many different con-ditions and protocols, some of which are discussed in Sec.IV.

The most commonly used protocol for characterizingglass-forming molecular liquids is a “Fast quench” to theglassy state followed by the “slow heating” of thermaliza-tion. Alternatively, the liquids may also be studied under“slow cooling”, and subsequent “slow heating” (exam-ples of all combinations are given in Sec. III A). Figure1 illustrates the “slow heating” and “show cooling” ar-rangements. For details on the experimental protocolsassociated with these experiments see the Appendix A.

In all experiments the liquid is positioned in a smallglass sample test tube (8 mm in diameter and 40 mm inlength), and a thermocouple-based thermometer is sub-merged in the liquid for measuring the temperature. Liq-uid nitrogen (l-N2) provides an excellent cooling agentfor studying molecular glass-forming liquids, and room-temperature likewise is a convenient heating tempera-ture. A fast quench is performed by dipping the testtube into a jar of boiling liquid l-N2. For the slow heat-ing, from l-N2 temperature to room temperature, thepre-cooled sample is subsequently quickly transferred toa block of expanded polystyrene (EPS) — pre-cooled toestablish the temperature gradient — with ≈ 5 cm wallthickness. Such a block gives typical heating rates of afew tenths of a Kelvin per second (≈ 10K/min). For“slow cooling” similar cooling rates can be obtained byinsulating the test tube from the liquid nitrogen by afew millimeters of insulating material. The temperaturerange and cooling/heating rates can easily be modifiedby using other cooling/heating agents and insulating ar-rangements.

The implementation of the TC technique has changedover time in our lab. Two types of implementation exist,one version that is based on analog electronics for differ-entiation and averaging of the data (used for the majorityof data presented in this paper), and a more recent digi-tal version in which the differentiation is performed afterdigitization of the T (t) signal. The implementations arediscussed and compared in the Appendix B.

Interpretation of(T, dT

dt

)-graphs

The data analysis of the TC technique is always basedon plotting

(T, dTdt

)-graphs. The basis of the interpre-

tation is that the rate at which the sample changes its

temperature, dT (t)dt , is inversely proportional to the spe-

cific heat of the sample at a given heat current input,P (t). This can immediately be seen from the definition

3

of the specific heat:

∆T =1

CQ (1)

dT (t)

dt=

1

C

dQ

dt=

1

CP (t), (2)

where ∆T is the temperature increase after introducingsome amount of heat, Q. In the long time limit, this holdsfor the macroscopic sample investigated here (neglectingheat diffusion in the sample).

A fundamental assumption of the analysis is that theheat current into the sample is proportional to the dif-ference between the environment temperature, T0, andsample temperature:

P (t) = − 1

R0(T (t) − T0) (3)

where R0 is the thermal resistance in the experiment.Combining the above equations leads to the following

differential equation:

dT (t)

dt= − 1

R0C(T (t) − T0). (4)

This is easily solved analytically for constant R0C [13],but that is not the route we will take for analyzing data.

Instead, direct information is obtained by plottingdTdt as a function of T (see Fig. 1). For constant C

the graph is a straight line with slope − 1R0C

ending at

(T = T0,dTdt = 0). The instantaneous value of − 1

R0Cat a

given temperature can be found as the slope of the secantfrom the point to T = T0,

dTdt = 0.

The ability to directly monitor changes in specific heatas a function of temperature is particularly useful for in-vestigating glass-forming liquids, because a clear signa-ture of the glass transition is a change in specific heat, asillustrated on Fig. 1. This change of specific heat is theresult of the “freezing in” of the structural degrees of free-dom when the liquid dynamically falls out of equilibriumto form a glass [14]. This means that the glass transitionhas the appearance of a second-order phase transition,but as is well known the transition is a dynamic phe-nomenon, the transition is not sharp, and the transitiontemperature changes (logarithmically) with cooling rate[15].

Exothermic and endothermic processes likewise give avery clear TC signal because they change dT

dt .

III. APPLICATIONS

The TC technique is useful for studying different as-pects of supercooled liquids. In the following we presentexamples of how the technique can be used for locatingthe glass-transition temperature and how it can be usedfor studying crystallization.

A. Analysis of data from two glass-forming liquids

Two typical molecular glass-forming liquids,tetramethyl-tetraphenyl-trisiloxane (DC704) andglycerol, have been investigated [16]. The former hasbeen studied in great detail by the Glass and Timegroup by a number of techniques [17–21]. The latter isa standard example of a glass-forming liquid.

1. Measurements

Figure 2 shows(T, dTdt

)-traces from measurements on

the two liquids. The figure includes data for “slow reheat-ing after fast quench” and temperature cycles consistingof a “slow cooling” followed by a “slow reheating”.

The heating curves start in equilibrium at liquid ni-trogen temperature (77.3 K). They first show a sharpincrease in dT

dt deriving from the transfer of the sam-

ple to the insulating block. A small dTdt overshoot is of-

ten observed, depending on how fast this transfer is per-formed and how well the insulating block has been pre-equilibrated to the sample and room temperature (seethe Appendix A). However, the trace quickly becomesapproximately linear. The cooling curves likewise startin equilibrium at room temperature with a fast decreasein dT

dt until equilibrium cooling conditions have been es-tablished.

A clear signature of the glass transition (at 215 Kfor DC704 and 194–195 K for glycerol) is seen duringboth heating and cooling as changes in the slope of the(T, dTdt

)-trace (see also Fig. 1). It is seen that the sig-

nature of the glass transition is very robust, also whenchanging cooling protocol, and that cooling and heatingcurves are consistent.

On the cooling curves small sharp peaks close to theglass transition are often seen, e.g., at ≈ 205 K forDC704. We believe that this stems from crack forma-tion in the sample as it solidifies. On the cooling curvefor glycerol a problem often encountered in cooling canbe observed. The

(T, dTdt

)-trace on the low temperature

side of the glass-transition does not “point to” the (T0, 0)point. We attribute this to the rather simple insulatingarrangement used; if cooling curves are needed for ac-curately evaluating the relative change in specific heat(as discussed below) a better arrangement can be con-structed; however, the identified glass-transition temper-atures is not changed by this.

2. Analysis

Even though the glass transition is well resolved inthe

(T, dTdt

)-traces, it covers a substantial temperature

range. In order to give an experimentally well-definednumber we define the experimental TC glass-transitiontemperature, Tg,TC, to be the temperature of the localminimum on the heating curve.

4

−0.6

−0.4

−0.2

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0.2

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100 150 200 250 300

(a)

Tg,TC

−0.4

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Tg,TC

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340

500

600

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1100

1200

150 160 170 180 190 200 210

(dT/d

t)[K

/s]

T [K]

DC704

Slow coolingSlow heating (after slow cooling)

Slow heating (after quench cooling)

(dT/d

t)[K

/s]

T [K]

Glycerol

(R0

C) h

eatin

g[s

]

(R0

C) c

oolin

g[s

]

T [K]

(R0

C) h

eatin

g[s

]

T [K]

FIG. 2. Glass transition studied by TC. Measurements on two typical glass forming liquid, (a) DC704 and (b) glycerol [16].The dark red curves show heating after a fast quench to liquid nitrogen temperature. The light red and blue curves show a cycleof slow cooling to liquid nitrogen temperature (blue curves) followed by slow heating to room temperature (light red curves).The glass transition is clearly seen in both cooling and heating. The inserts show the apparent specific heat represented as theR0C value from Eq. 4 (notice the different y-axis for the heating and cooling R0C curves).

The Tg,TC values are given in table I together with Tgand characteristic times at T = Tg,TC from studies ofthe equilibrium shear modulus (τG), dielectric (τε), andspecific heat (τCl

) [20, 22]. It is seen that the Tg,TC is atslightly higher temperature than the traditionally definedTg of τ = 100 s. In fact, the glass-transition temperatureis never uniquely defined; different experimental meth-ods has different characteristic time scales at the sametemperature — therefore different Tg.

The apparent specific heat, represented as R0C can befound from the data using Eq. 4, and is shown in theinsets of Fig. 2. The change in specific heat between theliquid and glass is clear in this representation. The abovediscussed experimentally determined Tg,TC correspondsto the maximum of the observed overshoot in the spe-cific heat. From the R0C data an estimate of the relativechange in specific heat between the glass and liquid canbe found; these data are given in Table I together withliterature data. It is seen that the TC technique pro-vides a good first estimate of this quantity. Consistencybetween the cooling and heating curves is observed (forDC704).

A significant difference between the heating curves of

quenched and slowly cooled glycerol is observed. Thisdifference can be rationalized based on the difference inthermal history and hence the glasses formed. However,depending on the properties of the liquid studied the sig-nature of thermal history will be more or less visible, asis seen when comparing the DC704 and glycerol data.A simple model that capture the essence of glass forma-tion, including these differences in fast and slow cooling,is presented in the Appendix C.

B. Glass-transition temperature of glycerol-watermixtures

Liquid-liquid mixtures in general and alcohol-water so-lutions in particular have been widely explored, not onlyin fundamental studies, but also for their potential ap-plicability [24]. Here we take as a case study a fam-ily of glycerol-water mixtures. Glycerol-water is a well-known miscible system in which the crystallization abil-ity of water is highly suppressed. For this reason, manyauthors have used this mixture to address fundamentalquestions of the physics of pure liquid water connected

5

Tg τ at Tg,TC∆CCliq

Tg,TC Tg,Cl Tg,ε Tg,G τCl τε τG TC Litt.

DC704 215 K 211.5 K 210 K 208 K 4.8 s 1.4 s 0.3 s 0.2 0.2 [19]

Glycerol 194–195 K 186 K 180 K 1.9 s 0.03 s 0.3 0.4–0.5 [23]

TABLE I. Glass transition temperature (Tg) from TC and three dynamic response functions (shear modulus, G, dielectricconstant, ε, and specific heat, Cl), characteristic timescale (τ) of the same three response functions at Tg,TC, and relative

change in specific heat between glass and liquid (∆C/Cliq =Cliq−Cglass

Cliq). The values are based on the TC data shown on Fig.

2 and from measurements[20, 22] of the three dynamic quantities on the thermo-viscoelastic (metastable) equilibrium viscousliquid. Tg from TC is found as described in the main text, from the local minimum on the heating curve. For the three responsefunctions Tg is defined as the temperature where the characteristic time is 100 s.a τ at Tg,TC is found from the same dynamicaldata for the three response functions. ∆C/Cliq from TC is found from the data shown on the insert of Fig. 2 and is comparedto values from literature.a The timescales are defined as τ = 1/(2πνlp), where νlp is the frequency at maximum loss in isothermal measurements of the frequency

dependent complex response functions. Data on the characteristic timescale as function of temperature have been extrapolated downto τ = 100 s by linear extrapolation of (T, log τ) data.

to its anomalies. Currently, glycerol-water mixtures arereceiving much interest due to their utilization as biopro-tective agents of human cells and proteins [25].

Four glycerol-water mixtures (100, 90, 80 and 72 %)volume fraction of glycerol) were prepared by mixinganhydrous glycerol (purity ≥ 99.5 %) with distilled anddeionized pure water (Arium 611r). Estimates of theglass-transition temperature of pure water give valueslower than those corresponding to pure glycerol (the ex-act glass-transition temperature of water is not knownas water crystallizes by spontaneous homogeneous crys-tallization in the supercooled regime before reaching theglass transition (see, e.g. [26])). One expects that mis-cible mixtures of glycerol-water are characterized by amonotonic decrease of the Tg when the water content

increases. Figure 3(a) shows the(T, dTdt

)-traces from

slow heating of initially quenched glycerol-water mix-tures. The signature of the glass transition shifts towardslower temperatures with the water content, as expected.Figure 3(b) shows how the values of Tg become lowerwhen the volume fraction of glycerol decreases.

C. Crystallization and melting

Besides studying the glass transition, the TC techniqueis well suited for studying the stability of a supercooledliquid against crystallization and locating the meltingpoint of the crystal.

Most systems are easily supercooled to form a glasswhen cooled fast enough. Some systems, however, re-crystallize during heating at similar rates from the glassystate. The transformation of supercooled liquids intocrystals is of great interest for its implications in funda-mental science as well as industry [29]. In pure systems,at sufficiently low temperatures the thermodynamicallystable form is crystalline. When liquids are supercooledand situated between the melting temperature (Tm) andthe glass-transition temperature (Tg), these are suscepti-

0

0.1

0.2

0.3

0.4

0.5

100 150 200 250 300

(a)

170

175

180

185

190

195

200

70 75 80 85 90 95 100

(b)

(dT/d

t)[K

/s]

T [K]

Glycerol-water mixtures

72 vol% Glycerol80 vol% Glycerol90 vol% Glycerol

100 vol% Glycerol

T g,T

C

Glycerol contents [vol%]

FIG. 3. Glass-transition temperature of glycerol-water mix-tures studied by the TC technique. a)

(T, dT

dt

)-traces for

glycerol-water mixtures with varying water contents. b)Glass-transition temperature, Tg,TC, as a function of glycerolcontent (given as the volume fraction).

ble to crystallization. This transition of a metastable liq-uid into a crystal is controlled by several factors like therate of nucleation and growth of the crystallites, viscos-ity, and the heating or cooling rate to which the systemis subjected.

6

Tm Tg,TC

1-Octyl-1-methylpyrrolidinium TFSI 260 K 194–195 K

Diethylene glycol 260 K (262.7 K [27]) 178–181 K

Acetone 176.5–177.5 K (178.8 K [28])

TABLE II. Melting, Tm, and glass-transition temperature, Tg,TC, determined from TC on three substances [16] (raw TC datashown on Fig. 4). Numbers in parenthesis are from the literature.

Data on three liquids with different crystallization be-havior is presented in the following. These liquids are:1-octyl-1-methylpyrrolidinium TFSI (OctPyr-TFSI) (aroom-temperature ionic liquid), diethylene glycol (a typi-cal molecular glass-forming liquid), and acetone (a small-molecule solvent) (see [16] for details on the samples).Figure 4 shows TC data for the crystallization of theseliquids and Table II summarizes the melting and glasstransition temperatures derived from TC.

Figure 4(a) shows a heating curve of the initiallyquenched OctPyr-TFSI. The glass transition is first (go-ing from low to high temperature) seen at 194–195 K, fol-lowed by crystallization of the supercooled liquid around230 K, which is, of course, an exothermic process as seenin the large increase in dT

dt . Finally, the melting of thecrystal is observed at 260 K as an endothermic process,seen as the sharp dip in dT

dt down to 0.

Figure 4(b) shows a similar curve for diethylene gly-col, which has a very pronounced glass transition, buta much smaller exothermic signal form the crystalliza-tion and not so well-defined a melting. This is the typi-cal behavior for molecular liquids. Nevertheless, the TCtechnique gives a good estimate of the melting point anda simple test of the stability of the supercooled liquidagainst crystallization.

As a final example of a crystallizing liquid data on ace-tone are presented in Fig. 4(c), showing

(T, dTdt

)-traces

for a fast cooling and subsequent slow heating. Formost molecular liquids, crystallization is suppressed dur-ing cooling, even at moderately slow cooling rates (thehere used rate of ≈ 0.5 K/s), but for acetone this is notthe case, since acetone unavoidably crystallizes aroundits freezing point. On cooling we observe an exothermictransition around 178 K that corresponds to the crystal-lization point of pure acetone. On slow reheating of thecrystal we observe an endothermic peak around 178 Kthat confirms the melting of crystalline phase of ace-tone. At slightly higher temperatures a narrow exother-mic peak emerges followed by a region of small peaks.The origin of this is unclear, but the observation resem-bles additional crystallization and melting events.

Finally, a weak signal is observed on heating (evenmore weak on cooling) at 130 K, which resembles a glasstransition, however, in the crystalline phase. Acetoneis known to have low-temperature thermal transitions[30], the origin of which has been debated during thelast decades (e.g. [28, 31]). It has been proposed that anorder-disorder transition or more complex phase transi-

tions exists [28, 30], but it has also been argued thatvariation in the strength of the electrostatic interactionsbetween polar groups along the crystalline lattice causedthese transitions [32]. Dielectric spectroscopy on thecrystal shows that a dielectric process exists in the crys-talline phase [33], showing that some dynamical processesare still active in the crystal, indicating a plastic crys-talline behavior. Altogether this demonstrates that TCalso can be used for “quick and dirty” investigations ofcomplex phenomena.

Stability against crystallization is a complicated prob-lem, and the above sketched experiments are by no meansmeant to represent all different ways crystallization canoccur. However, due to the simplicity of the TC setup,different types of crystallization experiments can be per-formed ad hoc, e.g., by keeping the liquid at isothermalconditions above Tg and studying whether excess heat isgenerated by crystallization in a following heating.

IV. SUMMARY AND OUTLOOK

We have presented the general principle of thermaliza-tion calorimetry (TC), a simple technique for investigat-ing supercooling, glass transition, and crystallization ofliquids. The key idea is to monitor

(T, dTdt

)-curves for a

system in which the thermal input power is proportionalto the temperature difference to the smaple surround-ings. Changes in specific heat (as at the glass transitiontemperature), exothermic, and endothermic processes allhave a clear signal in the

(T, dTdt

)curves.

Uses of the TC technique for locating the glass-transition temperature, studying stability against crys-tallization, locating melting point, and studying complexcrystallization behavior have been demonstrated. TheTC technique should not be seen as a way to determineTg and Tm precisely, which in any case is not possible forTg (see table I and [20]), but as a quick way of getting agood estimate of such quantities as well as, e.g., ratios ofspecific heat between solid and liquid.

An advantage of the TC technique is that it can eas-ily be adopted to different experimental situations. Bychanging the temperature of the environment and the in-sulation used, different temperature regimes and differentcooling-rate regimes can be explored. An easy adapta-tion is to use an oven as outer temperature, in this wayallowing for studying substances with high Tg and Tmas, e.g., many substances used in food has (as sugar and

7

0

0.2

0.4Tg,TC

Tm

Cryst.

0

0.2

0.4

(b) Diethylene glycol

Tg,TC

Tm

Cryst.

−0.4

−0.2

0

0.2

0.4

100 150 200 250 300

(c) Acetone

Tm

Cryst.

dT/dt

[K/s

](a) 1-octyl-1-methylpyrrolidinium

TFSIdT

/dt

[K/s

]dT

/dt

[K/s

]

T [K]

FIG. 4. Crystallization, melting, and glass transition stud-ied on three samples [16] by the TC technique (melting andglass-transition temperatures are given in Table II). a) andb)

(T, dT

dt

)-trace for slow heating after fast quench showing

glass transition, crystallization from the supercooled liquid,and melting. c)

(T, dT

dt

)-trace for slow cooling and subsequent

slow heating, showing crystallization in cooling and meltingin heating (see the main text for a discussion of further fea-tures).

cocoa butter). By changing the thermometer type, theTC technique can also be adopted to lower temperatureswhere thermocouplers are less efficient.

TC can be realized in a variety of ways, since all that

is needed is a recording of T and dTdt . Initially, we used

analog electronics for the computations, with the lateraddition of a build-in ADC (Analog to Digital Converter)for recording the signal. With the development of goodand cheap high resolution ADC’s and small microcon-trollers, it has become possible to make a fully digitalversion with a high degree of flexibility, which can easilybe built for, e.g., educational purposes.

Altogether, the TC technique is useful for studies ofglass-forming liquids as a test of new liquid, as well asfor teaching purposes for instance in student projects.

ACKNOWLEDGMENTS

All students and staff at IMFUFA which have workedwith and improved the TC-technique over the last 40years are gratefully acknowledged for helping in makingthe TC-technique a standard (workhorse) technique inour lab. The centre for viscous liquid dynamics “Glassand Time” is sponsored by the Danish National ResearchFoundation via grant DNRF61.

Appendix A: Experimental details

This Appendix provides details on the experimentalprotocols used for TC measurements. Some sources ofuncertainty are also discussed.

The liquid is in all experiments positioned in a smallglass tube (8 mm in diameter and 40 mm in length). Thethermocouple is fed through a hole in a soft silicon plug,which allows for a tight sealing when placed in the beaker(see Fig. 5). It is important that the thermocouple junc-tion does not touch the beaker; the thermocouple shouldbe positioned as close as possible to the middle of thebeaker. Liquid nitrogen (l-N2) is used as cooling agent.The experiment normally takes place at room tempera-ture.

For slow heating a block of expanded polystyrene withapproximately 5 cm wall thickness is used as insulatingmaterial (see Fig. 5). With these dimensions heatingrates of a few tenths of Kelvin per second (≈ 10K/min)are typically obtained.

In our realization of the slow cooling protocol, the in-sulation consists of a simple test tube lined with paper,by which cooling rates of order 0.5 K/s are obtained. Asin the case of heating, the cooling rate can easily be ma-nipulated by changing the insulation material.

The experimental protocol for slow heating after initialquench is as follows:

• The sample beaker with liquid is quench cooled di-rectly in l-N2, until thermalized.

• The block of expanded polystyrene is precooled byplacing a beaker filled with l-N2 in the block, for

8

FIG. 5. Photo of TC setup. Shown are the sample withthermometer (the thermocouple) (also shown enlarged in theinsert), the insulation block (of expanded polystyrene) usedfor slow heating, the test tube, which filled with l-N2, is in-serted in the insulation block before the heating experimentfor precooling the insulation block to establish the temper-ature gradient. Finally, the analog TC box (known as the“Red box”) is seen. The analog TC box (see Appendix B)normally communicates the digitized

(T, dT

dt

)measurements

to an attached PC over USB. However, analog representationof the

(T, dT

dt

)signal is also available.

long enough time to establish a stationary temper-ature gradient through the insulation block (withour dimensions, approximately 5 minutes).

• Finally, the precooling beaker is removed, the sam-ple beaker transferred from the l-N2 to the blockof expanded polystyrene, and a plug inserted. Thisshould be done as quickly as possible in order tominimize unintended heating of the sample andpolystyrene block.

•(T, dTdt

)is recorded during the thermalization of the

sample.

A number of sources of uncertainties can be identified:

• For slow heating experiments, a too short precool-ing time or too slow transfer of the sample to theblock of expanded polystyrene lead to deformationof the first part of the heating curve, as Eq. 3 willnot hold initially.

• The sample might be rather inhomogeneous af-ter quench cooling (it will normally fracture intopieces), leading to inhomogeneous conversion be-tween the phases.

• Due to the size of the sample gradients will exist.

• For slow cooling it is often observed that the(T, dTdt

)-trace does not point to (T0, 0) in the low

temperatures regime (as in Fig. 2(b). This is at-tributed to inhomogeneous insulation of the sam-ple.

Appendix B: Details of implementation of the TCtechnique

The TC technique can be implemented in differentways since the idea is simply to monitor T (t) with sometype of thermometer and acquire

(T, dTdt

). Two major

versions of the technique have been developed over theyears at Roskilde University: An analog based and a dig-itally based version. In the following the two methods aredescribed briefly [34] and compared (see also Fig. 6 for aschematic representation of the two implementations).

Common to both version is that a “J” type thermo-couple is used as thermometer. It has a reasonable sensi-tivity in the temperature range from room temperature(50 µV/K) to liquid nitrogen temperature (20 µV/K). Athermocouple has the advantage that the thermal “mass”of the thermometer itself is very small, and that it is me-chanically robust and cheap. Both implementation use amicrocontroller for delivering the results to an attachedPC in which the data are analyzed and visualized usingthe Matlab software package.

The analog implementation, which was developed atRoskilde University approximately 40 years ago, is basedon analog electronics (see Fig. 6(a)). It utilizes high qual-ity operation amplifiers (op-amp) for analog amplifica-tion, differentiation, and low-pass filtering of the ther-mocouple signal, and has analog reference point temper-ature compensation. Originally the analog output sig-nal proportional to

(v, dvdt

)(where v is the thermovolt-

age) was visualized by the use of an “XY-plotter”. Cur-rently, the analog signal is digitized using a high reso-lution analog-to-digital converter (ADC) connected to amicrocontroller that communicates the results to an at-tached PC. In the control and visualization software thethermovoltage (v(t)) and the time derivative of this (dv

dt )are converted to actual temperature (T ) and temperaturerate (dT

dt ). The data presented in this paper are obtainedutilizing this version of the TC technique.

The analog version has very good resolution and lownoise. However, it also requires the ability to buildcustom-made electronic circuits, which might be out ofrange for some applications (e.g. for use in high schoolphysics education). The digital version takes advantageof the fact that cheap high-quality microcontrollers andADC’s have become available in recent years.

Our digital TC-technique implementation utilizes theLTC2983 “Multi-Sensor High Accuracy Digital Temper-ature Measurement System” chip (Linear Technology).The LTC2983 has the advantage that thermocouples(and other commonly used types of thermometers) canbe directly connected to the chip; the signal is measuredby a high quality 24bit ADC. Furthermore, the LTC2983has multiple input channels, allowing for easy digital “ref-erence junction compensation” by having a second ther-mometer measuring the temperature of the “referencejunction”.

The LTC2983 is available on a demo board for which“daughter boards” for different thermometers (including

9

USB/serial

∝ v (t) + vref

∝ v (t) + vref

∝ dv (t)dt

Analog TC box PC + Matlab

Digital TC box

USB/serial

(v , t)

(Tref, t)

(T , t)

(v + vref, dvdt )

PC + Matlab

(T , dTdt )

ConversionT = TJ(v + vref)dTdt = dTJ

dvdvdt

+Visualization

Visualization

DigitizationAD7714

+MicrocontrollerLPC 1768

Conversion&

Ref. junctioncompensation

T = TJ(v + vJ(Tref))

Numerical differentiationTaken over “running blocks” of N(T , t) data points:

T = 1/N∑N

i=1 Ti

t = 1/N∑N

i=1 ti

dTdt

∣∣∣∣t

=∑N

i=1(Ti − T )(ti − t)∑Ni=1(ti − t)2

−

+

Vin

C

R

Vout

Vout(t) = −RCdVin(t)

dt

Analog differentiatorEquivalent (at low frequencies) to an“ideal op-amp differentiator”:

AnalogRef. junction

compensation+

Amplification

v (t)

(a)

(b)

v (t)

DigitizationThermovoltage &

T Ref. junctionLTC2983

+Microcontroller

&Time stamp

LinduinoTM One

FIG. 6. Schematic representation of the two major version of the TC-technique developed at Roskilde University. v is thethermovoltage, vref the thermovoltage associated with the reference junction temperature relative to 0 ◦C, TJ(v) the conversionfunction[35] between (reference point compensated) thermovoltage and temperature (and vJ(T ) the inverse function). Analogquantities are designated as time dependent (e.g. v(t)), and digitized quantities as pairs (e.g. (v, t)). a) The original analogbased TC-technique implementation consisting of an “Analog TC-box” (see Fig. 5) which performs analog signal processing anddifferentation of the thermovoltage. The analog signals are digitized by a high quality ADC and the results are communicatedto an attached PC by a microcontroller, and is finally converted to temperature for visualization and analysis. b) DigitalTC-technique implementation consisting of a “Digital TC-box” (see main text for details) which digitizes the thermovoltageand reference point temperature and communicates the measurements together with time stamps via a microcontroller to theattached PC. Averaging, differentiation, and visualization is done in software on the PC.

thermocouplers) are available, which easily connects toan microcontroller [36]. These three boards (LCT2983demo board, thermocoupler daughter board, and micro-controller) together provides a fully TC-system.

The LTC2983 is configured to give the raw digitizedthermovoltage, not performing any automated referencejunction compensation (since this introduces additionalnoise in the temperature signal) and not using the inter-nal table for converting thermovoltages to temperature(this was found not to be very good at low tempera-tures). Including time for communicating results to thePC, the measuring frequency is ≈ 5.8 Hz. The tempera-ture of the “reference junction” is measured every ≈7 s;however, for the presented data the “reference junction”temperature was to a good approximation temperature

independent — the “reference junction temperature” wasestimated based on the data taken at liquid nitrogentemperature. Besides controlling the LTC2983 and com-municating with the PC the microcontroller also timestamps all measurements.

The(T, dTdt

)data is found by calculating mean

and linear-regression slope over “running blocks” of N(t, T (t)) data points. To obtain data of a compara-ble quality to the one from the analog version of TC-technique, the averaging and slope calculations is doneover blocks of N = 60 points (corresponding to averagingover ≈ 10 s). With this block size the standard derivationon the slope becomes of order (1–2) × 10−3 K/s. LargerN smears our the

(T, dTdt

)trace and smaller N introduces

more noise.

10

00.10.20.30.40.50.60.7

100 150 200 250 300

(dT/d

t)[K

/s]

T [K]

DC704

Digital versionAnalog version

FIG. 7. Comparison of the analog and digital versions ofthe TC technique applied to a slow heating of an initiallyquenched sample of the glass-forming liquid DC704.

(T, dT

dt

)traces are given from both the analog and digital version ofthe TC technique.

Figure 7 shows a comparison on the data obtainedby the analog and digital version of the TC technique,demonstrating that both give excellent results.

If shorter averaging time is desired (for e.g. monitoringrelatively fast processes), a simple amplifier can easily beadded to the the digital TC implementation. Utilizinga high quality op-amp in a standard inverting amplifierconfiguration [37], the necessary averaging is reduced toN = 20 points (corresponding to ≈ 3.3 s) for noise levelin the slope of order (1–2) × 10−3 K/s.

Appendix C: A simple model for understanding theTC signal of a glass-forming liquid

In this section we briefly discuss a simple model thatcan be used for understanding the results from the TCtechnique when applied to glass-forming liquids. Themodel is based on the old understanding of Tool and oth-ers [12] — that structure may be quantified in terms ofa so-called fictive temperature [15]. This is the temper-ature at which the liquid’s structure is identical to thatof the glass. Thus for the metastable supercooled liquidthe fictive temperature, Tf , is the actual temperature,whereas below the glass transition Tf is constant, equalto the glass transition temperature. For annealing justbelow the glass transition the structure changes and thefictive temperature moves slowly towards the annealingtemperature.

Figure 8(a) shows an electrical network analog of themodel and Fig. 8(b) shows typical heating and coolingcurves obtained by solving the model. In the networkrepresentation voltages correspond to temperatures andelectrical currents to heat currents. Specific heat is repre-sented as capacitors and thermal resistance as resistors.

This model is the simplest possible model for the spe-cific heat of a glass-forming liquid (it predicts a Debye-

R0I0Rs(T )

Is

Ci

Ii

Cs

T Tf

T0

(a)

−2

−1.5

−1

−0.5

0

0.5

1(b)

1

1.2

1.4

1.6

1.8

2

1

1.5

2

2.5

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

(dT/d

t)/(T

g/τ

g)

T f/T

gFast coolSlow heat after fast coolSlow coolSlow heat after slow cool

Cap

p/C

i

T/Tg

FIG. 8. A simple model for understanding the TC signal froma glass-forming liquid as described in appendix C (Eq. C1 andC2). a) Electric network representation of the model, volt-ages represents temperatures, electrical currents representsthermal currents, resistors represents thermal resistance, andcapacitors represents specific heat. (see main text for details).b) Results from model calculations by simple Euler integra-tion using the function shown in Fig. 9 and the parametersgiven in Ref. [38]. Capp is the apparent specific heat as wouldbe observed by the TC technique.

11

function [ T array , Tf array , dTdt array , . . .t a r ray , C app array ] = . . .

TC Model Euler ( T in i t , T f i n i t , tau 0 , . . .C,Ta ,T0 , dt , t end )

% Al l va r i a b l e are in dimensionless uni t s

%ca l cu l a t e time arrayt a r r ay =0:dt : t end ;

% i n i t i a l i z e output arraysT array=zeros ( s ize ( t a r r ay ) ) ;Tf ar ray=zeros ( s ize ( t a r r ay ) ) ;dTdt array=zeros ( s ize ( t a r r ay ) ) ;C app array=zeros ( s ize ( t a r r ay ) ) ;

% use i n i t i a l condi t ionsT=T in i t ;Tf=T f i n i t ;

% Euler in t eg ra t i on of the modelfor k=1: length ( t a r r ay )

% ca l cu l a t e tau s inc lud ing% a high temperature cut o f ftau s=max( tau 0 ∗1e−3,exp(Ta . ∗ ( 1 . /T−1)) ) ;

% ca l cu l a t e the time de r i v a t i v e sdTdt=(T0−T) . / tau 0−(T−Tf ) .∗C./ tau s ; % dT/dtdTfdt=(T−Tf ) . / tau s ; % dTf/dt

% update temperaturesT=T+dTdt∗dt ;Tf=Tf+dTfdt∗dt ;

% update apparent s p e c i f i c heat% ( normalized to C i )C app=−(T−T0 ) . / dTdt/ tau 0 ;

% store r e s u l t s for outputT array (k)=T;Tf array (k)=Tf ;dTdt array (k)=dTdt ;C app array (k)=C app ;

end

FIG. 9. Matlab/octave code for numerical integration of themodel given by Eq. C4 and illustrated in Fig. 8(a).

type equilibrium frequency-dependent specific heat atvariance with experiments). The specific heat can beseparated into two parts: First, there is an instanta-neous contribution, Ci, which is able to take up heatwithout structural rearrangement of the molecules; thiscorresponds to the high-frequency limit of the dynamicspecific heat. Secondly, there is a structural contribution,Cs.

The major characteristic of a glass-forming liquid isthat the time scale separating the instantaneous andstructural contributions to relaxation phenomena in-creases dramatically when decreasing the temperature.This is implemented in the model by the temperature-dependent resistor Rs(T ), which in this simple model istaken to be Arrhenius (in general molecular glass-formingliquids have non-Arrhenius behavior (e.g. [39])):

Rs(T ) = R∞eTa/T . (C1)

From the network it is seen that when the fictive tem-perature (Tf) is identical to the actual temperature of thesystem (T ) the system is in equilibrium and there is no

current in the Rs resistor (Is), corresponding to no ex-change of energy between instantaneous and structuraldegrees of freedom.

In order to model the results of TC experiments thetotal model also includes the outer environment tempera-ture, T0, and the thermal resistance to the surroundings,R0, (according to Eq. 3).

The model can be expressed as a system of two non-linear coupled differential equations involving the sampletemperature, T , and the fictive temperature, Tf :

dT (t)

dt=I0 − IsCi

=T0 − T

R0Ci− T − TfRs(T )Ci

(C2a)

=T0 − T

τ0− T − Tf

τs(T )C (C2b)

dTf(t)

dt=

IsCs

=T − TfRs(T )Cs

=T − Tfτs(T )

, (C2c)

where τ0 = R0Ci is the characteristic time of the TC-technique, τs(T ) = Rs(T )Cs the characteristic relaxationtime of the specific heat (separating instantaneous and

structural degrees of freedom), and C = Cs/Ci the ratiobetween structural and instantaneous components to thespecific heat.

The structural characteristic time can be rewritten as:

τs(T ) = CsRs(T ) = CsR∞eTa/T = τge

Ta(1/T−1/Tg)

(C3)

where Tg is the glass-transition temperature and τg is thecorresponding relaxation time.

By choosing τg and Tg as basic the time and tempera-ture, the set of equations can be brought to dimensionlessform:

dT

dt=T0 − T

τ0− T − Tf

τs(T )C (C4a)

dTf

dt=T − Tf

τs(T )(C4b)

τs(T ) = eTa(1/T−1) (C4c)

The model described by Eq. C4 can be numerically in-tegrated by simple Euler integration. The Matlab func-tion shown in Fig. 9 performs the integration, and Fig.8(b) shows results from such model calculations.

The model is seen to qualitatively capture the shapesof the

(T, dTdt

)-traces, both on cooling and heating. Fur-

thermore, the model is seen to capture the difference be-tween slow heating after slow and fast cooling, compareFig. 8(b) and 2(b).

12

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[34] Details, schematics and software associated with the TC-implementations are available on request.

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[36] The boards used from Linear Technology for the digi-tal TC-implementation are: DC2209 (main demo cir-cuit containing the LTC2983), DC2212 (ThermocoupleDaughter Board), and DC2026 (LinduinoTM One, Ar-duino compatible microcontroller).

[37] The amplifier utilized for the digital TC implementation,is based on the Analog Devices AD8571 op-amp. This isa single supply (5 V), ultra low noise, zero drift op-amp.The op-amp was powered directly from the 5 V supplyon LTC2983 demo board. A standard inverting amplifierconfiguration was used, with a passive low-pass filter onthe input and 255.5 times amplification).

[38] Model parameters used for the integration of Eq. C4 to

give the results presented in Fig. 8(b) are: Troom = 2,

Tcool = 0.2, C = 1, Ta = 20, τ0,fast = 0.5, τ0,slow =3. For cooling, the system is assumed to be initial inequilibrium (T = Tf = Troom). The final values from thecooling is used as initial condition for the heating. Theintegration was performed using the function presentedin Fig. 9 using a time step size of 0.0001.

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