Thermalization Calorimetry: A simple method for investigating glasstransition and crystallization of supercooled liquids

Bo Jakobsen,1, a) Alejandro Sanz,1 Kristine Niss,1 Tina Hecksher,1 Ib H. Pedersen,1 Torben Rasmussen,1 TageChristensen,1 Niels Boye Olsen,1 and Jeppe C. Dyre1

DNRF centre “Glass and Time”, IMFUFA, Department of Science, Roskilde University, Postbox 260,DK-4000 Roskilde, Denmark

(Dated: 27 January 2016)

We present a simple method for fast and cheap thermal analysis on supercooled glass-forming liquids. This“Thermalization Calorimetry” technique is based on monitoring the temperature and its rate of change duringheating or cooling of a sample for which the thermal power input comes from heat conduction through aninsulating material, i.e., is proportional to the temperature difference between sample and surroundings.The monitored signal reflects the sample’s specific heat and is sensitive to exo- and endothermic processes.The technique is useful for studying supercooled liquids and their crystallization, e.g., for locating the glasstransition and melting point(s), as well as for investigating the stability against crystallization and estimatingthe relative change in specific heat between 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 years1–5. Modern calorime-try is highly sophisticated and measures with unprece-dented accuracy and, even more importantly, on smallerand smaller samples6–8. Thus nanocalorimetry is nowbecoming routinely available, which is absolutely essen-tial for ultra fast measurements and expensive samples,e.g., in modern molecular biology or advanced materialscience7–9.

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, and accuracyis less important for initial investigations. Our groupGlass and Time at Roskilde University10 has for manyyears conducted research into viscous liquids and theglass transition. “What is the glass transition temper-ature?” is the first question one asks when investigat-ing a new liquid. Here accuracy is not important, butspeed, flexibility, and simplicity are. This paper presentsour simple “Thermalization Calorimetry” (TC ) method,which has proven to be an excellent tool for initial andpreliminary investigations of glass-forming liquids.

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

a)Electronic mail: 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 processes, are observed by plotting the rate at

which the temperature changes with time, dT (t)dt , versus

the temperature, T (t). Such a plot reveals a lot of ther-modynamic information about the sample, its glass tran-sition, crystallization, etc. The TC method was inspiredby simple experiments conducted more than 40 yearsago by our colleague J. Højgaard Jensen11 for demon-strating the glass transition in propanols. Since thenthe TC method (nicknameed “The Red Box”) has beena workhorse in Glass and Time. TC is now our stan-dard tool for initial studies of new glass-forming liquidsby which we locate the glass-transition temperature andinvestigate the liquid’s stability against crystallization.Furthermore, TC turned out to be excellent for teachingpurposes. It has been used for numerous student projectsat all levels from high school to masters project, in whichits simplicity and ease of adaptation has allowed for stud-ies of samples ranging from amorphous drugs to candyand amber.

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 appendices give details on the experimental proto-cols, Appendix A, on the technical implementation ofthe technique, Appendix B), as well as a discussion of asimple Tool-type12 model of the glass transition that isuseful also for teaching purposes, Appendix C.

2

1

1

T (t)

liquid-N2T0,cool

(a)

TT0,cool T0,heat

0

(R0Cglass)−1

dTdt

PassiveCooling

PassiveHeating

(b)

Tg,TC

PassiveCooling

PassiveHeating

T (t)T0,heat

P(t) = − 1R0

(T (t) − T0)

(R0Cliq)−1

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). The ba-sic assumption is that the thermal power entering the sample,P (t), is proportional to the difference between the tempera-ture of the surroundings, T0, and the sample temperature,T , with the proportional constant 1/R0 in which R0 is thethermal resistance of the insulating material. Left part illus-trates slow passive cooling of a sample placed in an insulatingcontainer inserted directly into the cooling agent (which isnormally liquid nitrogen). T0 is here the temperature of thecooling agent, i.e., 77 K. Right part illustrates a slow passiveheating of a pre-cooled glassy sample using a block of insula-tion material. T0 is here the (room) temperature outside theblock.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 change slope andshow a “jump” at the glass transition temperature, Tg,TC,due to the change in specific heat between the equilibriumsupercooled liquid and the glass. The dashed lines are guidesto the eye, extending to (T0, 0) in the two cases. See Sec. IIfor details on the interpretation.

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 sampleand 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 “slow cooling” ar-rangements. For details on the experimental protocolsassociated with these experiments, see 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 to measure the temperature. Li-quid nitrogen provides an excellent cooling agent forstudying most molecular glass-forming liquids, and room-temperature likewise is a convenient heating tempera-ture. A fast quench is performed by dipping the test tubeinto a jar of boiling liquid nitrogen. For the slow heating,from liquid nitrogen temperature to room temperature,the pre-cooled sample is subsequently quickly transferredto a block of expanded polystyrene (EPS) — pre-cooledto establish the temperature gradient — with ≈ 5 cmwall thickness. This combination of sample size and in-sulation block gives typical heating rates of a few tenthsof a Kelvin per second (≈ 10K/min). For “slow cooling”similar cooling rates can be obtained by insulating thetest tube from the liquid nitrogen by a few millimetersof insulating material. The temperature range and cool-ing/heating rates can easily be modified by using othercooling/heating agents and insulating arrangements.

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 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 R0C13, but

that is not the route we will take for analyzing data.Instead, direct information is obtained by plotting

dTdt 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(T, dTdt

)point can be found as the slope of the

secant from 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 glass14. 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, and the transition temperature changes (loga-rithmically) with cooling rate15.

Exothermic and endothermic processes likewise give avery clear TC signal because they dramatically changethe overall

(T, dTdt

)trace by respectively a large increase

in dTdt and an approach to dT

dt = 0.

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 (see Ref. 16 for detailson the samples). The former has been studied in greatdetail by the Glass and Time group by a number oftechniques17–21. The latter is a standard example of aglass-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 (seeAppendix A). However, the trace quickly becomes ap-proximately linear. The cooling curves likewise start inequilibrium at room temperature with a fast decrease indTdt until equilibrium cooling conditions have been estab-lished.

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 affected 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-transition

4

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

100 150 200 250 300

(a)

Tg,TC

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

100 150 200 250 300

(b)

Tg,TC

500

550

600

650

700

750

800

850

200 205 210 215 220 225 230220

240

260

280

300

320

340

500

600

700

800

900

1000

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 (seeRef. 16 for details on the samples). The dark red curves show heating after a fast quench to liquid nitrogen temperature. Thelight red and blue curves show a cycle of slow cooling to liquid nitrogen temperature (blue curves) followed by slow heating toroom temperature (light red curves). The glass transition is clearly seen in both cooling and heating. The inserts show theapparent specific heat represented as the R0C value from Eq. 4 (notice the different y-axis for the heating and cooling R0Ccurves).

temperature, Tg,TC, to be the temperature of the localminimum on the heating curve.

The Tg,TC values are given in table I together with Tgand characteristic times at T = Tg,TC from studies20,22 ofthe equilibrium shear modulus (τG), dielectric (τε), andspecific heat (τCl

). It is seen that the Tg,TC is at slightlyhigher temperature than the traditionally defined Tg ofτ = 100 s. In fact, the glass-transition temperature isnever uniquely defined; different experimental methodshas different characteristic time scales at the same tem-perature — 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 with lit-erature data. It is seen that the TC technique provides agood first estimate of this quantity. For DC704 the cool-

ing and heating curves give consistent results, this couldnot be checked for glycerol due to the problems with the(T, dTdt

)-trace endpoint for the cooling measurement, as

discussed in Sec. III A 1.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 Appendix C.

B. Glass-transition temperature of glycerol-water mixtures

Liquid-liquid mixtures in general and alcohol-watersolutions in particular have been widely explored, notonly in fundamental studies, but also for their poten-tial applicability24. Here we take as a case study a fam-

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 measurements20,22 of the three dynamic quantities on the thermo-viscoelastic (metastable) equilibrium viscous liquid.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.

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 connectedto its anomalies. Currently, glycerol-water mixtures arereceiving much interest due to their utilization as biopro-tective agents of human cells and proteins25.

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. Ref. 26)). One expects thatmiscible mixtures of glycerol-water are characterized bya monotonic 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 fun-

damental science as well as industry29. 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-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.

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 Ref. 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 for OctPyr-TFSIwhich was initially quenched in liquid nitrogen. Goingfrom low to high temperature the glass transition is firstseen at 194–195 K, followed by crystallization of the su-percooled liquid around 230 K, which is, of course, anexothermic process as seen in the large increase in dT

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

dtdown to 0.

Figure 4(b) shows a similar curve for diethylene glycol,which has a very pronounced glass transition, but a muchsmaller exothermic signal form the crystallization and anot so well-defined melting. This is the typical behav-ior for molecular liquids. Nevertheless, the TC techniquegives a good estimate of the melting point and a sim-ple test of the stability of the supercooled liquid againstcrystallization.

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

(T, dTdt

)-traces

for a slow cooling and subsequent slow heating. For

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 substances16 (raw TC datashown on Fig. 4). Numbers in parenthesis are from the literature.

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 glyc-erol content (given as the volume fraction of glycerol). Errorbars are estimated from the shape of the local minimum inthe

(T, dT

dt

)-trace defining Tg,TC.

most 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 at the cooling rates we can realize.On cooling we observe an exothermic transition around178 K that corresponds to the crystallization point ofpure acetone. On slow reheating of the crystal we ob-serve an endothermic peak around 178 K that confirmsthe melting of crystalline phase of acetone. At slightlyhigher temperatures a narrow exothermic peak emerges

followed by a region of small peaks. The origin of thisis unclear, but the observation resembles additional crys-tallization 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 transitions30,the origin of which has been debated during the lastdecades (e.g. Ref. 28 and 31). It has been proposedthat an order-disorder transition or more complex phasetransitions exists28,30, but it has also been argued thatvariation in the strength of the electrostatic interactionsbetween polar groups along the crystalline lattice causedthese transitions32. Dielectric spectroscopy on the crys-tal shows that a dielectric process exists in the crystallinephase33, showing that some dynamical processes are stillactive in the crystal, indicating a plastic crystalline be-havior. Altogether this demonstrates that TC also canbe used for “quick and dirty” investigations of complexphenomena.

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 subsequent 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 sample surround-ings. Changes in specific heat, glass transition, exother-mic, and endothermic processes all have a clear signal inthe

(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 Ref. 20), but as a quick way of getting

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 studiedon three samples16 by the TC technique (melting and glass-transition temperatures are given in Table II). a) and b)(T, dT

dt

)-trace for slow heating after fast quench showing glass

transition, crystallization from the supercooled liquid, andmelting. c)

(T, dT

dt

)-trace for slow cooling and subsequent

slow heating, showing crystallization in cooling and meltingin heating. See the main text, Sec. III C, for a discussion offurther features.

a good estimate of such quantities as well as, e.g., ratiosof specific heat between solid and liquid.

An advantage of the TC technique is that it can eas-ily be adopted to different experimental situations. By

changing 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 Tm as,e.g., many substances used in food science (as sugar andcocoa butter). By changing the thermometer type, theTC technique can also be adopted to lower temperatureswhere thermocouples are less efficient.

TC can be realized in a variety of ways, since all thatis needed is a recording of T and dT

dt . Initially, we usedanalog 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 initial stud-ies of glass-forming liquids, as a test of new liquid, as wellas for 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 is used as cooling agent. Theexperiment normally takes place at room temperature.

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.

8

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

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

• The block of expanded polystyrene is precooled byplacing a beaker filled with liquid nitrogen in theblock, for long enough time to establish a stationarytemperature gradient through the insulation block(with our dimensions, approximately 5 minutes).

• Finally, the precooling beaker is removed, the sam-ple beaker transferred from the liquid nitrogen De-war to the block of expanded polystyrene, and aplug inserted. This should be done as quickly aspossible in order to minimize unintended heatingof the sample and polystyrene 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 distortion ofthe first part of the heating curve, as Eq. 3 will nothold 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 (significant) gradientswill 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.

• The heat capacity of the sample beaker will influ-ence the measured specific heat.

Appendix B: Details of implementation of the TC technique

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-ital based version. In the following the two methods aredescribed briefly34 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

FIG. 5. Photo of TC setup. Shown are the sample withthermometer (the thermocouple) (also shown enlarged in theinsert), the insulation block of expanded polystyrene used forslow heating, the test tube, which filled with liquid nitrogen,is inserted in the insulation block before the heating experi-ment for precooling the insulation block to establish the tem-perature 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.

(50 µV/K) to liquid nitrogen temperature (20 µV/K). Athermocouple has the advantage that the heat capacityof 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-

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 conversionfunction35 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 Appendix B 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.

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 (includ-ing thermocouples) are available, which easily connectsto a microcontroller36. These three boards (LCT2983demo board, thermocouple 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 temperatureindependent — 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 time

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.

stamps 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.Figure 7 shows a comparison on the data obtained

by the analog and digital version of the TC technique,demonstrating that both give excellent and consistent re-sults.

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 ampli-fier configuration37, the necessary averaging is reducedto N = 20 points (corresponding to ≈ 3.3 s) for noiselevel in the slope of order (1–2) × 10−3 K/s.

Appendix C: A simple model for understanding the TCsignal 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 andothers12 — that structure may be quantified in terms ofa so-called fictive temperature15. This is the tempera-ture 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, equal

R0

I0

Rs(T )

Is

Ci

Ii

Cs

T TfT0

(a)

Insulation Liquid

−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

g

Fast 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) Electrical 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 implementation shown in Fig. 9 and the pa-rameters given in Ref. 38. Capp is the apparent specific heatas would be observed by the TC technique.

11

function [ T array , T f a r ray , dTdt ar ray , . . .t a r r a y , C app array ] = . . .

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

% A l l v a r i a b l e are i n dimensionless u n i t s

%c a l c u l a t e t ime ar rayt a r r a y =0: d t : t end ;

% i n i t i a l i z e output ar raysT ar ray=zeros ( size ( t a r r a y ) ) ;T f a r r a y =zeros ( size ( t a r r a y ) ) ;dTd t a r ray=zeros ( size ( t a r r a y ) ) ;C app array=zeros ( size ( t a r r a y ) ) ;

% use i n i t i a l cond i t i onsT= T i n i t ;Tf= T f i n i t ;

% Euler i n t e g r a t i o n o f the modelfor k =1: length ( t a r r a y )

% c a l c u l a t e tau s i n c l u d i n g% a high temperature cut o f ftau s=max( tau 0∗1e−3,exp ( Ta . ∗ ( 1 . / T−1 ) ) ) ;

% c a l c u l a t e the t ime d e r i v a t i v e sdTdt =(T0−T ) . / tau 0−(T−Tf ) . ∗C . / tau s ; % dT / d tdTfd t =(T−Tf ) . / tau s ; % dTf / d t

% update temperaturesT=T+dTdt∗dt ;Tf=Tf+dTfd t∗dt ;

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

% sto re r e s u l t s f o r outputT ar ray ( k )=T ;T f a r r a y ( k )= Tf ;dTd t a r ray ( 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).

to 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.

The specific heat can be separated into two parts:First, there is an instantaneous contribution, Ci, whichis able to take up heat without structural rearrangementof the molecules; this corresponds to the high-frequencylimit of the dynamic specific heat. Then there is a struc-tural contribution, Cs.

The major characteristic of a glass-forming liquid is

that 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. Ref. 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 nocurrent 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.

12

This model is the simplest possible model for the spe-cific heat of a glass-forming liquid (it predicts a Debye-type equilibrium frequency-dependent specific heat atvariance with experiments). The model is seen to qual-itatively capture the shapes of the

(T, dTdt

)-traces, both

on cooling and heating. Furthermore, the model is seento capture the difference between slow heating after slowand fast cooling, compare Fig. 8(b) and 2(b).

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diffusion pump oil: tetramethyl-tetraphenyl-trisiloxane. Glyc-erol (propane-1,2,3-triol) is a standard Flukar R© glycerol. 1-octyl-1-methylpyrrolidinium TFSI is the ionic liquid 1-octyl-1-methylpyrrolidinium bis(trifuoromethanesulfonyl)imide acquiredfrom Solvionic. Diethylene glycol is standard quality. Acetone isstandard laboratory solvent quality.

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

35M. C. Croarkin et al., “Temperature-electromotive force refer-ence functions and tables for the letter-designated thermocoupletypes based on the ITS-90,” (NIST, 1993) Chap. Revised Ther-mocouple Reference Tables: Type J.

36The boards used from Linear Technology for the digital TC-implementation are: DC2209 (main demo circuit contain-ing the LTC2983), DC2212 (Thermocouple Daughter Board),and DC2026 (LinduinoTM One, Arduino compatible microcon-troller).

37The amplifier utilized for the digital TC implementation, is basedon the Analog Devices AD8571 op-amp. This is a single supply(5 V), ultra low noise, zero drift op-amp. The op-amp was pow-ered directly from the 5 V supply on LTC2983 demo board. Astandard inverting amplifier configuration was used, with a pas-sive low-pass filter on the input and 255.5 times amplification).

38Model parameters used for the integration of Eq. C4 to give theresults 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 in equilibrium (T = Tf = Troom). The finalvalues from the cooling is used as initial condition for the heating.The integration was performed using the function presented inFig. 9 using a time step size of 0.0001.

39T. Hecksher, A. I. Nielsen, N. B. Olsen, and J. C. Dyre, NaturePhys. 4, 737 (2008).