The Deborah number, relaxation phenomena andthermally stimulated currents

Joaquim J. Moura-Ramos* and Nata lia T. Correia

Centro de QuõÂmica-FõÂ sica Molecular, Complexo I, IST, Av. Rovisco Pais,

1049-001 L isboa, Portugal. E-mail: mouraramos@ist.utl.pt

Receivved 4th September 2001, Accepted 19th October 2001First published as an Advvance Article on the web 5th December 2001

In the ®eld of the science of ¯ow and deformation of matter (rheology), the so-called Deborah number plays animportant role, since it describes the in¯uence of time on the observed ¯ow properties. The technique ofthermally stimulated depolarisation currents (TSDC) constitutes a dielectric technique whose experimentalresult is an electric current ¯ow that is the manifestation of a dipole relaxation. In the present work we use the

concept of the Deborah number to describe and interpret a TSDC experiment. We show that, in the linearheating ramp of a TSDC experiment, the Deborah number decreases as the temperature increases, in such a waythat it reaches a value of unity at the temperature of maximum intensity of the experimental peak. This

particular feature is the origin of a new procedure, proposed here, for evaluating the fragility of a glass-formingliquid.

Introduction

In her famous song of praise to God, after the victory over theCanaanites, the prophetess Deborah sang: `` the mountains¯owed before the Lord ''.1 Two important concepts are implicitin this statement. The ®rst is that everything ¯ows (Heraclitus),even the mountains. Secondly, the reason why the mountains¯owed before the Lord, and not before the humans, is thathumans, with their short lifetime, cannot see them ¯owing,while the time of observation of God is in®nite. The secondconcept implicit in the song of Deborah has to do with therelationship between the timescale of an event and that of itsobservation. The ratio between the timescale of the pheno-menon and that of the observation method,

D � timescale of the observed

timescale of the observer� relaxation time

observation time�1�

is thus a dimensionless quantity. It is well known in rheology,namely in viscoelasticity, and is called the Deborah number.1

The magnitude of the Deborah number, D, provides interest-ing indications. If our time of observation (the timescale of ourexperimental technique) is very large or, conversely, if therelaxation time of the material under study is very short, we seethe sample ¯owing and the material behaving as a liquid. Incontrast, if the relaxation time of the material is larger than thetimescale of the observer, the material behaves as a solid. TheDeborah number thus appears as a fundamental quantity thatbrings solids and liquids under a common concept. The`` everything ¯ows '' statement of Heraclitus simply corre-sponds to the limit of an in®nitely small relaxation time or anin®nite time of observation.The depolarisation process of a previously polarised sample

is a relaxation process, where the sample tends to reach anequilibrium state characterised by a random distribution of thedipole orientations. As a consequence, an electric depolarisa-tion current ¯ows in the sample, which is the manifestation ofa slow dipole relaxation. This is the basis of the technique ofThermally Stimulated Depolarisation Currents TSDC. In thepresent work we will use the concept of Deborah number inorder to describe and understand the evolution of a TSDC

experiment. As a consequence of the presented discussion, anew procedure will be suggested to evaluate the fragility indexof a glass-forming system.

Experimental

The experimental TSDC data used in the present work werepreviously reported.2±5 TSDC experiments were carried outwith a TSC=RMA 9000 instrument (TherMold Partners,Stamford, CT, USA) covering the temperature range betweenÿ170 and �400 �C. The explanation of the physical basis ofthe TSDC technique, and of the nature of the information itprovides, can be found in several publications e.g. ref. 3 and 6.

Discussion

The Deborah number and the glass transition

In order to study relaxation processes, it is usual to modify thetimescale of the experimental probe and=or the timescale of thesystem under study. To modify the timescale of the observer,i.e. of the experimental technique, we can change the frequencyof an applied periodic perturbation (dielectric relaxationspectroscopy, temperature modulated di�erential scanningcalorimetry), the heating rate of a scanning ramp (conven-tional di�erential scanning calorimetry, thermally stimulatedcurrents), or the observation time of a time-dependentproperty (isothermal depolarisation or demagnetisation of apreviously polarised or magnetised sample). The modi®cationof the timescale of the system can be done by changing thevalues of the state variables, namely the temperature and=orthe pressure.Let us consider a temperature scanning experiment where

the measurement of a given variable (heat ¯ux, depolarisationcurrent) is performed along a linear temperature ramp of rater� dT=dt. The observation time in such an experimentbecomes shorter as the rate of the ramp, r, increases. Theobservation timescale can thus be considered as the reciprocal

DOI: 10.1039/b107984k Phys. Chem. Chem. Phys., 2001, 3, 5575±5578 5575

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of the heating rate of the experiment, 1=r� dt=dT, and itrepresents the mean time interval that is spent in each one-degree temperature interval. The timescale of the system, onthe other hand, can be de®ned in the temperature domain asthe rate of variation with temperature of a characteristicrelaxation time, ÿdt=dT. The Deborah number is de®ned as:

D�T � � ÿdtdT

ÿ �dtdT

ÿ �:

�2�

It should be noted that the minus sign guarantees that thetimescale is a positive quantity (given that the relaxation timedecreases with increasing temperature), and that the physicalunits of the two timescales (of the observer and of the system)are identical. Since the activation energy at temperature T isde®ned as

Ea�T � � Rd ln t�T �d1=T

� �T

�3�

eqn. (2) can be written as

D�T � � Ea�T �RT2

rt�T � �4�

where R is the ideal gas constant. The glass transition is akinetic event that corresponds to the transformation, oncooling, of the metastable supercooled liquid into a glassystate. Below the glass transition temperature we have the glass,characterised by high values of D. Above the glass transitiontemperature we have the liquid (or the rubber state in the caseof amorphous polymers, or the liquid crystalline phase in thecase of liquid crystalline polymers), characterised by lowvalues of D. Given the kinetic nature of the glass transition, theglass transition temperature, Tg , depends on the thermalhistory of the sample, and a hysteresis behaviour is oftenobserved in the transition range. In this context, it is oftenconsidered that, at the glass transition temperature, there is acrossover between the timescale of the experimental probe andthat of the studied system, so that the two timescales becomeequal, and the value of the Deborah number is one. TheDeborah number thus appears as a relevant concept in the

physics of the glass transition. Since the quantityEa�Tg�RT2

gis often

found to be not far from 0.5,7 it comes out from eqn. (4) thatfor r� 4 �C minÿ1 we have t(Tg)� 30 s and, for r� 1 �C minÿ1,t(Tg)� 120 s. That is why the glass transition temperature isoften conventionally de®ned as the temperature at which themolecular motions slow on cooling to t� 100 s.

The Deborah number and thermally stimulated currents

TSDC is a dielectric experimental technique where the depo-larisation current of a previously polarised sample is recordedin a constant rate heating scan.3,4 The TSDC technique ishighly sensitive, and provides the possibility of enhancing theprocess of interest by selecting adequate polarisation condi-tions. The experimental output of a TSDC experiment is thedepolarisation current intensity as a function of temperature,whose graphical representation is a current peak. Since thedepolarisation current density (current intensity per unit area),is the rate of decrease of the polarisation, we have

J�T � � I�T �A� ÿ dP�T �

dt�5�

where J(T ) is the current density at temperature T (or at timet) of the constant rate heating ramp (depolarisation step), P(T )is the remaining polarisation at temperature T (or at time t),and A is the e�ective area of the electrodes. Thermal sampling(TS), or fractional polarisation, is a procedure often used inTSDC to probe only those dielectric relaxations which areactivated in a narrow temperature interval, in a given

polarisation time.3,4 The TS procedure allows the polarisationof speci®c segments of a complex global relaxation or, other-wise stated, it permits the experimental deconvolution of aglobal, heterogeneous relaxation process into its individual,component relaxation modes.4 The analysis of the TS results isbased on the following equation:

J�T � � P�T �t�T � �6�

where P(T ) is the remaining polarisation at temperature T,and t(T ) is a temperature-dependent relaxation time, char-acteristic of the mode of motion under consideration. Eqn. (6)admits that the rate of decrease of the polarisation at eachtemperature of the heating ramp (the experimental currentdensity) is proportional to the distance to equilibrium, i.e. toP(T ) (to the number and strength of the relaxing entities), andalso proportional to some rate constant, or to the reciprocal ofsome relaxation time, t(T ), characteristic of the studiedmotional process. Eqn. (6) describes a thermally stimulatedprocess where temperature and time are related byT (t)� T0� rt, and the relaxation time is temperature depen-dent given that the depolarisation occurs during a constantrate heating process. In this context, eqn. (6) can be written as

dP�T �dT

� ÿ 1

r

P�T �t�T � : �7�

where P(T ) is the remaining polarisation at temperature T (orat time t). Taking the derivative of eqn. (6), and taking eqn. (7)into account, leads to:

d J�T �dT

� ÿ 1

rÿ dt�T �

dT

� �P�T �t2�T � : �8�

From eqn. (8) we conclude that, at the temperature of max-imum intensity of the TS peak (T� Tm), we have

dt�T �dT

� �T�Tm

� ÿ 1

r� ÿ dt

dT�9�

or

D�Tm� � ÿdt�T �dT

h iT�Tm

dtdT

ÿ � � 1 �10�

Eqn. (10) is a fundamental equation of the TSDC technique. Itimplies that the Deborah number is unity at the maximum of aTS peak. It thus indicates that, at the temperature of maximumintensity of a TS peak, Tm , there is a crossing between thetimescale of the depolarisation process, de®ned by the tem-perature derivative of the relaxation time, and the timescale ofthe TS experiment, de®ned by the reciprocal of the heatingrate. In fact, the temperature dependence of the relaxation timeis such that it decreases as the temperature increases. If T< Tm ,the rate of decrease of the relaxation time with temperature is,in modulus, higher than 1=r, while if T > Tm , it is lower than1=r. If the depolarisation that occurs along the constant rateheating ramp obeys eqn. (6), the two timescales are equal atT� Tm . In a thermal sampling experiment the Deborah num-ber should be equal to one at the temperature of maximumintensity, Tm , of the TS peak. From eqn. (4) we will have:

Ea�Tm�RT 2

m

rt�Tm� � 1:0: �11�

Eqn. (11) should be followed for any TS peak, for the second-ary relaxations as well as for the main relaxation, ora-process. A thermal sampling experiment probes a specificmode of motion. Below the Tm , of the corresponding peak,the mode of motion is `` frozen '', and the sample behaves as asolid with respect to this motional process. Above Tm the modeof motion is `` free '', and the sample is liquid with respect to it.

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Fig. 1 shows an experimental TS peak obtained on the glasstransition relaxation of cyanoadamantane (continuous line),8 asubstance showing a solid rotator (plastic) phase whichtransforms by cooling in a reorientational glassy state. Thepeak in Fig. 1 is a TSDC signature of the transformation of thereorientational glass into the plastic phase. It is, in fact, a TSpeak in the glass transition region, whose temperature locationis Tm� ÿ 90.6 �C, and that was obtained with a heating rater� 8 �C minÿ1.The dotted line in Fig. 1 is the representation of the Deborah

number as a function of temperature for this particularrelaxation process, calculated from eqn. (4). It can be observedfrom Fig. 1 that the Deborah number is unity at Tm , that itgoes to zero at high temperatures, and that it strongly increaseswith decreasing temperature. Below TM� ÿ 90.6 �C the reor-ientational motions in the positional crystal are essentiallyfrozen, the sample behaves as an orientational glass, andD> 1. Above TM� ÿ 90.6 �C the molecular rotations are`` free '', and D< 1.

The Deborah number and the fragility index

The Deborah number is de®ned as the ratio between theresponse time and the observation time, and it is unity at thecrossover between the two timescales. In the particular case ofthe glass transition relaxation, the TS peak with higherintensity in the glass transition region has a maximum currentintensity at a temperature, TM , that is the glass transition

temperature provided by the TSDC technique.3,4 Eqn. (4) canalso be rewritten at the temperature TM of this particular TSpeak as

D�TM� � Ea�TM�RT 2

M

rt�TM� � 1:0: �12�

The values of TM , Ea(TM) and t(TM) obtained from TSDCdata must be such that eqn. (12) is obeyed, so that this equa-tion is useful to check the consistency of the TSDC results.The activation energy at the glass transition temperature,Ea(TM), is an important parameter of the glass transition re-laxation. Namely, it is used in the calculation of the fragilityindex of a glass-forming system, m, a fundamental concept inglass state physics, which is defined as:9

m � d log t�T �d�TM=T �

� �T�TM

� Ea�TM�2:303RTM

�13�

Combining eqn. (12) and (13) the fragility index can be ex-pressed as:

m � TM

2:303 rt�TM� : �14�

Eqn. (14) corresponds to a new procedure to evaluate the fra-gility of a glass-forming system from TSDC data. The relaxa-tion time at TM , t(TM), is easily obtained from the rough TSdata using eqn. (6). On the other hand, in order to obtain theactivation energy at TM , Ea(TM), the ln t(T ) versus 1=T dataobtained from eqn. (6) must be fitted with an appropriate equa-tion, in order to determine the slope at TM . The uncertaintiesassociated with the determination of the activation energy atTM are thus higher than those associated with the determina-tion of the relaxation time at TM , so that eqn. (14) is more ad-vantageous than eqn. (13) for the determination of the fragilityindex of a glass-forming system. Table 1 shows the values of thefragility index calculated from eqn. (14) on the basis of TSDCdata, together with values reported in the literature.Given that the values of the fragility index published in the

literature, and obtained by di�erent experimental techniques,present a considerable scatter, the agreement between our valuesand those of the literature is good from both qualitative andquantitative points of view. It is however to be noted that thevalue presented here for the fragility of maltitol is signi®cantlylower than that we proposed previously,2 also based on TSDCdata. A recent careful treatment of the early experimental data,allowed us to minimise the in¯uence of a conductivity tail whichappeared in the vicinity of the glass transition peak of maltitol,and is the origin of the reported di�erence. According to ourresults, maltitol appears thus as a stronger liquid when com-pared with sorbitol, while some results in the literature suggestthat they have similar fragilities. This particular problem needs

Fig. 1 Thermally sampled peak of higher intensity in the glasstransition region of 1-cyanoadamantane (continuous line). The ex-periment was carried out with a polarising ®eld of 400 V mmÿ1, apolarisation temperature of ÿ102 �C, a width of the polarisationwindow of 3 �C and a heating rate of 8 �C minÿ1. The dotted linedescribes the temperature variation of the Deborah number.

Table 1 Some features of the TS peaks of the glass transition of some glass-forming systems. TM is the temperature of maximum intensity of theTS-TSDC peak with higher intensity in the glass transition region. The relaxation time at TM , the activation energy at TM , and the Deborah num-ber at TM are t(TM), Ea(TM) and D(TM) respectively. The values of the fragility index, m, were calculated using eqn. (13) and (14), and are comparedwith values taken from the literature, m(lit.). All the TSDC experiments were performed with a heating rate of 4 �C minÿ1, except for cyanoada-mantane where it was 8 �C minÿ1

TM=�C t(TM)=s

Ea(TM)=kJ molÿ1 D(TM)

m(eqn. (13))

m(eqn. (14)) m(lit.)

1-Cyanoadamantane ÿ90.6 25.4 75.6 0.92 22 23 3510

Glycerol ÿ79.9 25.5 182 1.00 49 49 5310

Maltitol 46.1 35.3 364 1.01 59 59 7211

Indomethacin 42.3 31.2 386 0.97 64 66 7712

Sorbitol ÿ0.8 24.6 370 0.98 71 72 7211

Salol ÿ47.6 17.7 328 0.92 76 83 7310

m-Toluidine ÿ81.8 13.3 311 0.91 85 94 7913

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further clari®cation. Moreover, the literature values of the fra-gility index ofmaltitol, indomethacin, sorbitol and salol are verysimilar, while the values obtained from eqn. (13) and (14) aresigni®cantly di�erent, allowing a very clear discrimination ofthese di�erent glass formers.

Conclusions

The Deborah number, D, is a familiar concept in mechanics ofcontinuous media, or rheology, but is unfamiliar in physicalchemistry and in chemical physics. However, it provides aconceptually rich and interesting description of relaxationphenomena. We used the Deborah number concept to describea TSDC experiment. In this context, we showed how to cal-culate the temperature variation of the Deborah number, D,associated with a TSDC peak, and concluded that, at themaximum of the peak, the value of D should be unity. Thisindicates that, at the temperature of maximum intensity, Tm ,of the peak, there is a crossover between the two timescales:that of the experiment and that of the motional process understudy. The importance of this occurrence was underlined, andit was used to suggest a new procedure to determine the fra-gility index of a glass former from TSDC data. The values ofthe fragility index obtained from this procedure are in rea-sonable agreement with the values published in the literature.

Acknowledgement

The authors are indebted to Dr George Collins (TherMoldPartners, L. P.) for reading and commenting on the manuscript.

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