Effect of cooling rate on enthalpy and volume relaxation of polystyrene

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NOC 12666 No. of Pages 11, Model 5+

20 June 2007 Disk UsedARTICLE IN PRESS

www.elsevier.com/locate/jnoncrysol

Journal of Non-Crystalline Solids xxx (2007) xxx–xxx

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OF

Effect of cooling rate on enthalpy and volumerelaxation of polystyrene

J. Hadac a, P. Slobodian a,*, P. Rıha b, P. Saha a,R.W. Rychwalski c, I. Emri d, J. Kubat a,c

a Polymer Centre, Tomas Bata University in Zlın, T.G.M. 275, 76272 Zlın, Czech Republicb Institute of Hydrodynamics, Academy of Sciences of the Czech Republic, Pod Patankou 5, 16612 Prague, Czech Republic

c Department of Materials and Manufacturing Technology, Chalmers University of Technology, SE-41296 Gothenburg, Swedend Center for Experimental Mechanics, University of Ljubljana, Cesta na Brdo 49, 1000 Ljubljana, Slovenia

Received 31 January 2007; received in revised form 14 May 2007

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PAbstract

This paper presents the results of measurements of physical aging on polystyrene with a narrow molecular weight distribution (Mw/Mn = 1.03). The evolution of the aging process was followed by recording the relaxed enthalpy and the accompanying decrease in vol-ume, using differential scanning calorimetry and mercury-in-glass dilatometry, respectively. The measurements were carried out aftercooling the sample at constant rate to the aging temperature. The cooling rate varied between 0.0037 and 1 �C/min. The aging data werefitted with the KWW and the TNM functions. The differences in the time scales of enthalpy, h, and volume, v, relaxation were relativelysmall. From the h(v) graphs the derivative dh/dv was found to amount to about 1.8 GPa, a value significantly exceeding the predictions ofthe thermodynamic model containing an additional internal variable.� 2007 Published by Elsevier B.V.

PACS: 61.20.Lc; 64.70.Pf; 65.60.+a

Keywords: Glass transition; Polymers and organics; Calorimetry; Enthalpy relaxation; Thermodynamics; Stress relaxation; Structural relaxation
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R1. Introduction

Cooling of amorphous polymers or other glass-formersbelow their glass transition range leads to a metastableglassy state. An obvious consequence of this is an ongoingstructural consolidation. In the simplest case, that is whenthe temperature is kept constant, this consolidation resultsin easily measurable changes in a number of physicalparameters of the material until a state of equilibrium isreached. The time dependence of the volume of a sampleunder such conditions may illustrate this. When stoppingthe cooling procedure below the glass transition tempera-

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0022-3093/$ - see front matter � 2007 Published by Elsevier B.V.

doi:10.1016/j.jnoncrysol.2007.05.017

* Corresponding author. Tel.: +420 57 603 1350; fax: +420 57 603 1444.E-mail address: [email protected] (P. Slobodian).

Please cite this article in press as: J. Hadac et al., J. Non-Cryst. So

ture, Tg, and keeping the temperature constant, oneobserves a continued decrease in volume until it reaches avalue lying on the extrapolated liquidus line in the vol-ume–temperature (v,T) graph. Given enough time theglassy branch of this graph is thus transformed into theextrapolated liquidus branch. Normally, this is the caseonly at relatively low levels of undercooling. At still lowertemperatures the rate of the consolidation process becomesimmeasurably low. Time dependence of the volume is butone example of the phenomenon known as physical aging.Another quantity frequently appearing in aging studies isthe enthalpy, that is to say the amount of heat leavingthe sample after a cooling procedure. However, aging pro-cesses are not confined to samples subjected to coolingwithin the glassy or Tg regions, they appear also whenthe temperature is changed in the opposite direction

lids (2007), doi:10.1016/j.jnoncrysol.2007.05.017

mailto:[email protected]

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2 J. Hadac et al. / Journal of Non-Crystalline Solids xxx (2007) xxx–xxx



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(temperature up-jump). When the material is in equilib-rium before a temperature increment both volume andenthalpy increase, although obeying different kinetic lawsthan in the cooling case. When the material is not in equi-librium before the temperature change, memory effects areobserved such as an increases in volume and enthalpy fol-lowed by a decreases, or a simple continuation of a decreas-ing trend if the material was not consolidated long enoughbefore a temperature up-jump. Despite of the vast numberof publications on these subjects, dealing not only with vol-ume and enthalpy, but also including other properties, likemechanical properties, there appears to be no consensusregarding the basic mechanisms underlying the transitionprocess and its kinetics. For reviews the reader is referredto Refs. [1–4].

Changes in volume and enthalpy during the equilibra-tion of undercooled polymers belong to the most inten-sively studied subjects in this field. Normally the focus ison the kinetics of the aging process, particularly withregard to the behavior upon reheating wholly or partiallyequilibrated samples, where distinct maxima in thermalexpansivity and heat capacity are observed when passingthe Tg region [1,4–6].

Direct comparisons of the kinetics of volume, v, andenthalpy, h, relaxations are rare. This is also the case whenconsidering possible physical significance of the derivativedh/dv, which appears to remain constant during the majorpart of the aging process. Adachi and Kotaka [7] foundthat the value of this derivative for PS amounts to about2 GPa, and assumed that it is related to the enthalpyneeded to create free volume. Similar results have beenreported by Oleinik [8] for PS, who carried out both tem-perature down-jumps and up-jump experiments findingfor dh/dv about 30% higher values in the latter case. Olei-nik finds dh/dv to be about 1–2 GPa, noting that this pointstoward a mechanism involving nucleation and disappear-ance of vacancies. Simon et al. studied the aging processfollowing T down-jumps in polyetherimide [9] and PS[10] and found the ratio dh/dv to be 1.8 and 1.2 GPa,respectively. Cowie et al. [11] measured the h(v) relation-ship in poly(vinyl acetate). In their case dh/dv reached amaximum whereafter it decreased to zero at a rate depend-ing on the aging temperature.

Typically, the initial step in aging experiments carriedout in the cooling mode is a down-jump in temperature(quench). In dilatometric measurements, where the volumeof the sample amounts to 2–3 cm3, a sudden change in T isnot possible to realize. This is due to the low thermal con-ductivity of common polymers leading to relatively longequilibration times. The initial part of the aging processthus remains virtually unknown. In this respect there is abasic difference between volumetric measurements andmeasurements of enthalpy, where samples of a few milli-grams are sufficient for the T-scans in common DSC equip-ment. Imagining the quench being carried out bytransferring a MIG device (mercury-in-glass dilatometry)and a DSC sample from a thermostatic bath to one with


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a lower temperature provides a tangible illustration of thedissimilar conditions to be taken into account. The impactof this fact is discussed in [12] where the authors analyzethe influence of the thermal histories on simultaneous vol-ume and enthalpy relaxation.

In this paper, we present the results from enthalpy andvolume relaxation measurements performed on PS calori-metric and dilatometric samples with same thermal history,cooled at constant rate from equilibrium at above Tg to theaging temperature Ta. To our knowledge, such experimentshave not been described before. The data presented belowrelate both to varying cooling rates and aging tempera-tures. During the aging process the volume of the sampleis monitored using the MIG method. The change inenthalpy is determined from the area of the endothermicpeaks appearing on the temperature scans during reheatingof samples aged a predetermined period of time. From thevolume and enthalpy results the derivative dh/dv is evalu-ated and found to be around 1.8 GPa. Possible implica-tions of these findings are discussed below. The KWWfunction (stretched exponential) yielded a fair degree ofagreement with the observed v(t) and h(t) kinetics.

2. Experimental

The material used was polystyrene, PS140, from Frei-burger Materialforschungszentrum, Freiburg i. Br., Ger-many. Its average molecular weight Mw was 145 kg/mol,the polydispersity ratio Mw/Mn = 1.03. The volumetricmeasurements were carried out using mercury-in-glass(MIG) dilatometry according to ASTM standard D864-52, following practical details given in [13]. The PS speci-men for the measurements was prepared by compressionmoulding and subsequent cutting and grinding to a barwith rounded edges. The cross-section of the bar was about6 · 6 mm, its volume about 3 cm3. The total volume of thedilatometer was about 6 cm3. The specimen was annealedabove Tg at T = 120 �C for 1 h in order to erase possibleeffect of internal stresses and orientation originating inthe preparation process. It was placed in the glass bulb ofthe dilatometer which was then sealed and filled with mer-cury (purity 99.995%) under a vacuum of about 2 Pa. Thetemperature was controlled by immersing the dilatometerin a precision (±0.01 �C) thermostatic bath (Julabo HP-4). The absolute values of the specific volume measuredby the dilatometer were determined by calibration at115 �C using volume–temperature isobars determined bya piston-die type apparatus (PVT 100 analyzer, SWO Poly-mertechnik GmbH, Krefeld, Germany). Before starting thecooling to the aging temperature Ta = 92 �C, in some cases90 and 94 �C, the sample was annealed at 115 �C for15 min. Time zero for collecting of relaxation data wasset at the moment the aging temperature was reached. Allexperiments at varying cooling rate were carried out usingthe same sample/dilatometer arrangement.

While the MIG-dilatometry is a direct method, yieldingthe whole v(t) dependence in real time, the measurement of


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0.96

0.97

fic

volu

me

v [c

m3 /

g]

α=2.56x10-4K-1

α=7.02x10-4K-1

Tβ=72ºC

Tg=97.4˚C

J. Hadac et al. / Journal of Non-Crystalline Solids xxx (2007) xxx–xxx 3



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the h(t) dependence can be done only in an indirect way.This method is based on a procedure where enthalpychanges are evaluated from the amount of heat releasedby the sample to its surroundings during the aging process.This quantity is determined from the endothermic peakappearing around Tg on a DSC scan upon reheating thespecimen after a certain period of aging. The area of thepeak is calculated by integration starting from the instru-ment baseline. By subtracting the peak area measured onan unaged sample one obtains the amount of heat, that isto say the enthalpy, given off during the aging process[14]. This procedure has to be repeated for every point onthe h(t) curve. In this study a set of 8 mg PS specimensenclosed in standard aluminum pans were placed in thethermostatic bath for annealing, cooling and aging at Ta.The specimens were enclosed in Al foil in order to avoidcontamination with the silicon oil of the bath. In thisway the DSC samples were given the same thermal historyas the dilatometric one. At the end of the aging period theywere taken out and placed with the Al foil removed in theDSC apparatus and subjected to a T up-scan in a DSC 1Pyris, Perkin–Elmer, at a heating rate of 10 �C/min. Thetemperature and heat flow were calibrated using indiumstandard. During this manipulation (removing of Al foiland transferring into the DSC device) the samples tempo-rarily attained ambient temperature.

3. Results and discussion

3.1. Glass transition temperature

Before presenting the aging results we show in Fig. 1 thedependence of the specific volume of the PS specimen ontemperature at different cooling rates ranging from 1.0 to

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80 90 100 110Temperature T [ºC]

0.952

0.956

0.96

0.964

0.968

Spec

ific

vol

ume

v [c

m3 /g

]

Cooling rate [ºC/min]1.00.50.20.080.020.0037

Fig. 1. Dependence of specific volume on temperature for PS duringcooling at different cooling rates.


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0.0037 �C/min. In the liquid state, that is above Tg, theexperimental points arrange themselves along a commonstraight line yielding a coefficient of thermal expansional = 7.02 · 10�4 K�1 independent of the cooling rate. Attemperatures below the Tg range the v(T) dependence againbecomes linear, implying a constant value of the coefficientag applying in the glassy state. The Tg value is defined asthe intersection of the liquidus with the v(T) line of theglass. At closer look at the v(T) graph a secondary transi-tion is found around 20 �C below Tg. The temperature ofthis transition, Tb, is determined in the same way as Tg.A graphical representation of the v(T) dependence mea-sured at a cooling rate of 0.5 �C/min is shown in Fig. 2,where both Tg and Tb are marked. The numerical valuesof these quantities, together with the corresponding a val-ues, are given in Table 1. The standard deviations in valuesof Tg and Tb and cooling rate 0.5 �C/min were calculatedfrom a set of cooling experiments and these values werealso used as a typical values for the other rates. Thermalexpansion coefficients above, al, and below, ag, Tg and ab

below Tb are calculated from the linear parts of volume–temperature records in T-intervals typically 112–102 �C

20 40 60 80 100 120 140

Temperature T [˚C]

0.94

0.95Spec

i

α=2.19x10-4K-1

Fig. 2. Dependence of specific volume on temperature during cooling at0.5 �C/min.

Table 1Parameters from volumetric analysis of PS by mercury-in-glassdilatometry

q [�C/min]

Tg [�C] Tb

[�C]al · 10�4

[K�1]ag · 10�4

[K�1]ab · 10�4

[K�1]

1.0 97.8 ± 0.2 75 ± 1 7.02 ± 0.08 2.67 ± 0.11 2.32 ± 0.070.5 97.4 ± 0.2 72 ± 1 2.56 ± 0.09 2.19 ± 0.070.2 96.8 ± 0.2 73 ± 1 2.59 ± 0.07 2.12 ± 0.060.08 95.7 ± 0.2 71 ± 1 2.56 ± 0.08 2.11 ± 0.120.02 95.3 ± 0.2 69 ± 1 2.50 ± 0.10 2.17 ± 0.110.0037 93.1 ± 0.2 70 ± 1 2.43 ± 0.11 2.05 ± 0.08


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0 1 2 3 4 5Relaxation time, log ta [min]

0

1

2

3

Exc

ess

enth

alpy

, δH [

J/g]

Cooling rate [˚C/min]10.50.10.020.0037TNM modelKWW model

Fig. 4. Enthalpy relaxation at 92 �C after cooling at different rates. Solidlines KWW function, dashed lines TNM model.




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for al, 84–76 �C for ag and some 64–45 �C in the case of ab.The standard deviations of the thermal expansion coeffi-cients were calculated as lower and upper bounds com-puted with the help of Student t distribution, probabilityof error a = 0.005. As can be seen both Tg and Tb seemsto slightly decrease at lower cooling rates. This is in agree-ment with dilatometric data obtained by Greiner andSchwarzl [15] but in contrast with the findings of Goldbachand Rehage [16] who found an opposite trend. The remark-able closeness of the two transitions in PS, and the largedifference in their intensity appear to contribute to thesediscrepancies. We have included here a remark about Tb

not only for the sake of completeness but also in order toillustrate the quality of the volumetric recordings. Sincethe secondary transition does not appear to play any rolein the present context, it will not be taken into account inthe following presentation.

3.2. Isothermal relaxation

The kinetic features of the volume consolidation follow-ing constant rate cooling from above Tg to Ta = 92 �C aredisplayed in Fig. 3, where the cooling rates range from 1 to0.0037 �C/min. The corresponding enthalpy relaxationgraphs are shown in Fig. 4. The volume change in Fig. 3is expressed as dv = (v � v1)/v1, the change in enthalpy,Fig. 4, as dh = Dh � Dh1, where v1 and Dh1 relate to val-ues measured after reaching equilibrium. As can be seenboth quantities approach this state at largely similar times,irrespective of the cooling rate. On the other hand lowerrates result in a decrease of the initial values of both dv

and dh. Also the width of the underlying spectrum of relax-ation times is distinctly reduced. The experimental data inFigs. 3 and 4 were fitted using the KWW function

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0 1 2 3 4 5Relaxation time, log (ta/min)

0

4

8

12

16

20

Rel

ativ

e ex

cess

vol

ume,

δ [

x104 ]

Cooling rate [˚C/min]10.50.10.020.0037TNM modelKWW model

Fig. 3. Volume relaxation at 92 �C after cooling at different rates. Solidlines KWW function, dashed lines TNM model.


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amy–Moynihan) model [1,4,17,18]. The sets of volume orenthalpy relaxation curves were fitted all together to theTNM model. The cooling history was including. Also thedata of Figs. 5 and 6, showing the enthalpy and volumerelaxation after cooling at 1 �C/min to 94, 92 and 90 �Cwere fitted in the same way. In general, the agreement withthe measured results was reasonable with both the methodsused. It should be remarked that more complex processesassociated with the kinetics around Tg cannot be describedby a plain KWW function. Due to the scatter found in the

0 1 2 3 4 5 6Relaxation time log (ta/min)

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5

10

15

20

25

δ v R

elat

ive

exce

ss v

olum

e [x

104 ]

Relaxation temperature Ta

90˚C92˚C94˚CTNM modelKWW model

Fig. 5. Volume relaxation at 90, 92 and 94 �C. Solid lines KWW function,dashed lines TNM model, cooling rate 1 �C/min.


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0 1 2 3 4 5 6Relaxation time, log (ta/min)

0

1

2

3

4

Exc

ess

enth

alpy

Δh

[J/g

]

Relaxation temperature Ta

90˚C92˚C94˚CTNM modelKWW function

Fig. 6. Enthalpy relaxation at 90, 92 and 94 �C. Solid lines KWWfunction, dashed lines TNM model, cooling rate 1 �C/min.




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graphs, especially with regard to the enthalpy data, itappears to be of little use to make any further commenton the quality of the fits. The fitting parameters are sum-marized in Tables 2 and 3 for the KWW function andthe TNM model, respectively.

With regard to the fits using the TNM model a fewremarks may be in place. We have included this type of fit-ting since it has become a customary procedure to emulatedifferent kinds of aging data. In the simple case treatedhere, that is a simple relaxation after cooling to Ta, thereis apparently no tangible advantage of introducing such amodel compared to a plain KWW function. As well known

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Table 2Enthalpy and volume relaxation parameters from KWW equation

q [�C/min] Volume

b s [103 s] dv0 · 104 Bv · 104

1 0.44 24 17.1 6.480.5 0.47 33 14.8 5.860.1 0.58 45 10.5 5.200.02 0.64 85 6.6 3.560.0037 0.84 105 4.3 3.06

T [�C]

90 0.40 210 20.5 6.5892 0.44 24 17.1 6.4894 0.57 3 7.9 3.77

Table 3Enthalpy and volume relaxation parameters from TNM model

Experimental set up Volume

x b DH/R [kK]

1, 0.5, 0.1, 0.02, 0.0037 K/min 0.33 0.58 14790, 92, 94 �C 0.36 0.44 147


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the TNM model is based on a KWW exponential, wherethe relaxation time s is assumed to depend on the activa-tion energy, H*, and the fictive temperature, Tf, in the fol-lowing way

sðT f ; T Þ ¼ A expxH �

RTþ ð1� xÞH �

RT f

� �: ð1Þ

In this modified Arrhenius law A is the pre-exponential fac-tor, and x (0 < x < 1) the non-linearity parameter takinginto account the structural and thermodynamic contribu-tion to s (e.g. [19]). In our case the apparent activation en-thalpy was calculated from the variation of Tg with thecooling rate, q, according to the relation [20]

d ln qdð1=T gÞ

¼ �H �

R: ð2Þ

The value of H*/R obtained in this way was 147 ± 38 kK,which is relatively high when compared with similar datareported for PS [10,21–26]. This appears to be related tothe monodispersity and comparatively high molecularweight of our sample, both factors known to increase thequantity in question [21–24]. In Ref. [10] H*/R values of146 kK and 100.4 kK were obtained from enthalpy andvolume measurements, respectively, for PS with Mw/Mn = 2.4 and Mw = 221 kg/mol. Adapting data obtainedby Greiner and Schwarzl [15], who measured the coolingrate dependence of Tg for PS with Mw = 417 kg/mol andMw/Mn = 1.98, one arrives at 106 ± 5 kK. Giordanoet al. [25] find H*/R = 80 ± 2 kK for PS with Mw =230 kg/mol (no Mw/Mn ratio is specified). Chen and Wang[26] give H*/R = 175 ± 17.5 kK for PS with Mn = 200 kg/mol and a polydispersity lower than 1.06. Finally in [27]one can find 385 kK for Mn = 104 kg/mol and Mw/Mn

1.04–1.10. It may also be mentioned that the parameter

Enthalpy

[cm3/cm3] b s [103 s] dh0 [J/g] Bh [J/g]

0.48 58 2.7 1.100.57 33 2.2 1.020.6 65 1.6 0.810.78 98 1.1 0.710.85 163 0.6 0.43

0.43 222 3.7 1.290.48 58 2.7 1.100.51 8 1.6 0.67

Enthalpy

ln(A) [s] x b DH/R [kK] ln(A) [s]

�391 0.58 0.65 187 �499�391 0.56 0.48 187 �499


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b, describing the shift in Tg with a changing cooling rateaccording to

T g ¼ T g0 þ b logqq0

� �ð3Þ

was found to be 1.9 ± 0.4 �C per decade of q, which is morethan 0.7 ± 0.3 �C reported in [27] and less than2.9 ± 0.1 �C as given in [15]. Again the monodispersity ofthe PS used is likely to explain this difference. The remarksabout H*/R have been included here because of their con-nection with the pre-exponential factor A in Eq. (1). Nor-mally an Arhenius type equation describes a thermallyactivated process where 1/A is the frequency with whicha potential barrier is negotiated. The rate of the flow pro-cess, here measured by 1/s, is then given by the Boltz-mann-factor. For aging experiments involving relativelysmall departures from Tg, that is small values of Tf–Tg,Eq. (1) simplifies to

s ¼ A expH �

RT f

� �; ð4Þ

where there is no partitioning via x, and where it makes lit-tle difference whether Tf or T is used [28]. Eq. (4) showsthat the exceedingly low values of A, commonly found inapplications of the TNM fitting scheme, are but a compen-sation for the high values of H*/R amounting to severalhundreds kK. Using the simplified Eq. (4) it can be easilyverified that in our case ln A values around �400 resultfrom the measured variation of Tg with the cooling rate,Eq. (2). The fitting of our results using a software basedon TNM model produced even lower values, Table 3. Sim-ilar A figures can be found in [29] dealing with aging of PEIand PEI-based composites. Critical remarks regarding theTNM model can be found in [30,31], with the latter refer-ence clearly stating that the inverse value of A should beclose to the frequencies encountered in thermal motion ofsolids.

On the whole, our v(log t) and h(log t) graphs appear tochange their shape with the cooling rate in a similar way asstress relaxation curves of samples deformed to a certainstrain level using widely differing strain rates, the resultbeing a diminishing width of the observed s-spectrum.

3.3. The enthalpy–volume relation

In a number of papers [32–37] dealing with physicalaging of several amorphous polymers and amorphous sele-nium our interest was centered on the relation betweenrelaxed enthalpy and volume. These studies included bothtemperature down-jumps experiments as well as thosewhere the temperature was changed in the opposite direc-tion. In both cases the Dh/Dv plot was reasonably linear.In certain cases deviations were observed in the initialand final stages of the process. We found that the slopeDh/Dv was similar in magnitude to the inverse value ofthe compressibility at corresponding temperatures.

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In this paper, we present Dh vs. Dv plots calculated fromaging data where Ta was reached by cooling at a constantrate instead of using a step-wise change in temperature.These results are presented in Fig. 7(a)–(e) displaying theenthalpy–volume separately for each cooling rate. Withthe exception of Fig. 7(e) where the scatter of the experi-mental points precludes any reasonable evaluation, onefinds a high degree of linearity between the relaxingenthalpy and the accompanying volume change. A summa-rizing view of the data of Fig. 7(a)–(d) is presented inFig. 7(f), where Fig. 7(e) has been excluded. The slopeDh/Dv is close to 1.8 GPa. As a supplement to the abovedata we show in Fig. 8 the h(v) plots obtained after coolingat 1 �C/min to 90 �C and 94 �C. As can be seen the resultsare largely similar to those reported above for Ta = 92 �C.

An important aspect of the h(v) plots shown in Figs. 7and 8 are possible differences in the time scales of theunderlying enthalpy and volume relaxation kinetics. Whenthe relaxation rate is the same in both cases, then theh(log t) and v(log t) graphs should coincide, provided theyhave been properly normalized with regard to the initialstate. The question whether the kinetics of the two pro-cesses is the same, or whether there are reproducible differ-ences between them appears to remain contentious. Simonet al. [9] compare the time scales for the relaxation ofenthalpy and volume, as well as creep and dynamicmechanical and dielectric parameters as reported by vari-ous authors for a number of polymeric glasses. However,these data provide no reliable information in this matter.In a later publication Simon et al. [10] find for PS thatthe normalized rate of approach to equilibrium is the samefor a 0.5 �C down-jump, while for a 4.6 �C jump toTa = 97 �C the enthalpy relaxed slower than the volume,even though h and v reached equilibrium at the same timein both cases.

When discussing the time scales of the relaxation of h

and v, one should keep in mind the large scatter of enthal-pic data and the somewhat arbitrary nature of the initialvalues h0 and v0 used to construct the normalized log t-plotson which the comparison is based. There are also differ-ences between normally used calorimetric and dilatometricmethods with regard to the thermal history to which thesamples have been subjected. Without going into detailsof the time scale discussion we compare in Fig. 9 the nor-malized KWW fits to the experimental points shown inFigs. 3, 4 and 7. As can be seen, there is a distinct tendencyof the enthalpic curves to be shifted towards longer times.On the other hand, there is a high degree of synchroniza-tion when equilibrium is approached. This indicates thatat least a part of the differences between enthalpy and vol-ume may be eliminated by renormalizing these plots. Thiswould be justified by the simple fact, that the true initialvalues of h and v remain uncertain due to the history pre-ceding the initial stage of the measurements.

One could compare this with stress relaxation experi-ments where the recording of the decaying stress beginsafter an arbitrary time and where the rate and extent of


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0 0.0004 0.0008 0.0012Excess volume Δv[cm3/g]

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J/g]

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Ka=1.87±0.17GPa

q [ºC/min] Ka[GPa]0.02 1.86±0.45

q [ºC/min] Ka [GPa]0.1 1.71±0.22

q [ºC/min] Ka [GPa]1.0 2.01±0.24

q[ºC/min] Ka[GPa]0.0037 1.02±1.23

q [ºC/min] Ka[GPa]0.5 1.88±0.15

a b

c d

e f

Fig. 7. Plots of h vs. v after cooling to 92 �C, constructed from Figs. 3 and 4. Parts (a)–(e) represent data measured at cooling rate 1, 0.5, 0.1, 0.02 and0.0037 K/min. Part (f) represents summarized data from the parts (a)–(d).




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Cthe initial deformation are not known, that is where thebeginning of the relaxation curve is missing. In such cases,proper evaluation of the results is not possible. This is, forinstance, the case with fitting a KWW function to graphswith arbitrary h0 or v0 values. For this reason we do notascribe such fittings any particular importance in the pres-ent context. They were used here only to facilitate theassessment of possible time scale differences between thetwo quantities.

It does not seem meaningful to discuss this question anyfurther, considering the basic differences in the methodsused to measure v and h, and the considerable scatter indata relating, in the first place, to the latter quantity. Itmay be added that the h(v) graphs extracted from the timeevolvement of the two quantities appear to be reasonably


linear, see Figs. 7 and 8. Furthermore they point to the ori-gin, which would not be the case were the two time scalessignificantly different, at least in the more advanced stagesof the process.

In a series of papers from these laboratories the slope ofthe h(v) graphs was found to be of similar magnitude as theinverse value of the compressibility in the higher end of theTg region. The data presented above seem to support sucha notion. To illustrate this we show in Fig. 10 the variationof the bulk modulus (inverse of the compressibility)extracted from pvT data measured by the confining liquidtechnique as reported in [38,39]. The characteristics of thePS samples used were as follows: Mw = 9, 35 and 110 kg/mol with Mw/Mn = 1.06 in Zoller and Walsh [38], andMw = 210 kg/mol in Schmidt et al. [39]. Also included in


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0 0.0003 0.0006 0.0009

Ka=1.82±0.18GPa

Ka=2.01±0.24GPa

Ka=1.69±0.37GPa

a b c

Fig. 8. Plots of h vs. v after cooling to 90, 92 and 94 �C, constructed from Figs. 5 and 6. Parts (a)–(c) represent data measured at 90, 92 and 94 �C, coolingrate 1 �C/min.




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Fig. 10 are our data on the PS grade used in the agingexperiments. While in [38,39] the measurements were car-ried out in the isothermal mode, we used a floating-pistontype device (PVT 100 Analyzer, SWO PolymertechnikGmbH, Krefeld, Germany) where the temperature waschanged at constant pressure (isobaric mode). Isobars atatmospheric pressure and the corresponding K-values wereextrapolated using Tait’s equation of state on which thesoftware of the equipment was based [40]. As can be seenboth methods yield largely similar results.

As follows from the literature cited above the values ofthe ratio dh/dv cover a range of 1–2 GPa. For PS Simonet al. [10] report 1.2 GPa, which is similar to the resultsobtained by Takahara et al. [41]. The cause of the discrep-ancy between these data and those measured by us is notknown. Nevertheless, in both cases are the values of dh/dv distinctly higher than the predictions of the thermody-namic model using an additional internal variable in orderto emulate the irreversible character of the aging process[42,43]. The internal variable can be interpreted in termsof the commonly used fictive temperature, Tf, or alterna-tively, a fictive pressure, pf, formally accounting for the vol-ume changes accompanying the consolidation process.Using Tf and pf to express the quantity of heat, q,exchanged between the aging sample and its surroundingsat constant T and p one obtains [42,43]

ohov

� �T ;p

¼ Dcp

vDa¼ T

DaDj

; ð5Þ

with Dcp, Da, and Dj denoting the differences between li-quid and glass in the specific heat capacity, coefficient ofthermal expansion and the compressibility, respectively.

Eq. (5) applies to equilibrium conditions where the affin-ity (driving force) (og/on)T,p = 0 (g – Gibbs free energy, ninternal variable defining the degree of advancement ofthe process). Baur [44] provides an explicit description ofthe approach to equilibrium accounting for the contribu-tion of (og/on)T,p. Using the calculations in [44] one finds


ED

PRoh

ov

� �T ;p

¼ v� ve

veDjþ T

DaDj

; ð6Þ

where v is the volume at a given time, and ve the equilib-rium volume. Due to the small volume changes normallyencountered in aging measurements, the first rhs. term ofEq. (6) remains negligible in comparison with the secondone during the whole process. One may note that the firstrhs. term defines a pressure identical with the fictive pres-sure of Davies and Jones [42,43]. According to Baur [44]the derivative (oh/ov)T,p can also be expressed as

ohov

� �T ;p

¼ T f � TT

Dcp

veDaþ Dcp

veDa: ð7Þ

Here we find again that the first rhs. term does not contrib-ute significantly to (oh/ov)T,p. This is due to the low valuesof (Tf � T)/T normally measured. The results of the origi-nal approach of Davies and Jones [42,43] and that of Baur[44], where the dissipation of g is taken into account, arethus practically equivalent.

From Eq. (5) the Prigogine-Defay ratio P ¼ DcpDj=TvðDaÞ2 ¼ 1 is obtained. Originally P was calculated start-ing from the pressure dependence of the transition temper-ature of second order transitions, where both phases are inequilibrium, and where no dependence of any quantity ontime is involved. The notion that the glass transition can betreated in a similar way is commonly accepted when inter-preting both dh/dv and dTg/dp = Dj/Da data. Within thisformalism the latter relation also applies to dTf/dpf. Itmay be of interest to note that the P ratio can be extractednot only from the equilibrium Eq. (5), containing entropicterms only, but also from the first rhs. terms of Eqs. (6) and(7) relating to the dissipation of g, when setting Da =(v � v1)/ve(Tf � T), which is but a linearization of dv/v = Da Æ dTf.

When applying the above formalism of standard ther-modynamics extended with an internal variable to includetime dependent changes we find that the measured dh/dv


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0 2 4Relaxation time log (t/min)

1K/min

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0

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0.02 K/min

volumeenthalpyKWW fit

Fig. 9. Comparison of normalized values of KWW fits of volume and enthalpy taken from Figs. 3 and 4. Parts (a)–(e) relate to rates 1, 0.5, 0.1, 0.02 and0.0037 �C/min.




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Cvalues significantly exceed the predictions of Eqs. (6) or (7).For the quantities entering these equations we obtainedDj = 0.27 GPa�1, Dcp = 0.32 J g�1 J�1, and Da = 4.35 ·10�4 K�1, in fair agreement with literature data found foramorphous polymers [45]. This yields TDa/Dj = 0.59 GPaand Dcp/vDa = 0.75 GPa, thus well below the recorded dh/dv of around 1.8 GPa.

A possible explanation of the findings that the constantof proportionality between h and v is significantly largerthan TDa/Dj may be sought in internal work dissipatedduring the consolidation process. Such an option has beenproposed by Niewenhuizen in [46,47], dealing with discrep-ancies relating to the Prigogine-Defay ratio. In our casethis would imply a non-negligible contribution of dissi-pated free energy to the measured enthalpy. Since our pro-


cess takes place at constant T and p, the following standardrelation applies

Dh ¼ Dg þ TDs: ð8Þ

In an irreversible process no work is extracted. The dissipa-tion of Dg produces an additional amount of heat, TDs’,which must be removed in order to keep the temperatureconstant. The quantity Dg is necessarily negative; its dissi-pation thus enhances the exothermic character of the agingprocess under consideration, where Dh < 0. It follows fromEq. (8) that an exothermic process is enthalpy driven, touse the terminology of reaction kinetics. This means thatthe enthalpy, which is the amount of heat measured bythe calorimetric device, is larger than the TDs term. Inexperiments involving a T up-jump, where the process is


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40 80 120 160 200Temperature T [ºC]

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k m

odul

us K

[M

Pa]

This work (Mw=145kg/mol, Mw/Mn=1.03)

(Mn=9kg/mol, Mw/Mn=1.06) [36]

(Mw=35kg/mol, Mw/Mn=1.06) [36]

(Mw=110kg/mol, Mw/Mn=1.06) [36]

(Mw=210kg/mol) [37]

Fig. 10. Temperature dependence of the bulk modulus of PS measured inthe Tg region. Data from Zoller and Walsh [38] and Schmidt et al. [39](Confining liquid technique) compared with our results obtained using afloating piston device.




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endothermic, we find the opposite situation dominated bythe entropic contribution.

In the internal variable model described above, the irre-versible production of entropy due to the dissipation ofpotential energy amounts to TDs = pfdv. This follows bothfrom the original treatment of Davies and Jones [42,43] andfrom Eq. (6), and amounts to saying that the dissipatedenergy equals the energy due to the elastic volume changeproduced by the fictive pressure pf.

Basically, the h(log t) and v(log t) plots relating to agingspecimens subjected to a T down-jumps are linear over sev-eral decades of time. This may not be clearly evident in theresults presented above, the explanation being the slowapproach to the aging temperature. In this respect, theaging process is reminiscent of stress relaxation in bothpolymers and metals, as well as of various recovery phe-nomena. Within the linear log(t) range the rate of changeof the measured quantity, in our case _h and _v, dependsexponentially on the quantity itself, that is

_h � expðC � hÞ; ð9Þ

with a similar relation applying to v. In this way a naturallink is established to the concept of thermally activatedprocesses, where the activation energy (=enthalpy) dependslinearly on the relaxing quantity, with the entropy of acti-vation included in the pre-exponential factor. A typicalexample of this type of behavior is stress relaxation where,again, the rate is determined by a linear decrease of theactivation energy with the momentary stress [48].

An important conclusion following from Eq. (9) relatesto the physical significance of the measured enthalpyreleased by the sample during aging as a quantity directlyassociated with the height of the effective potential barrier


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in the thermal activation process This, in turn, provides alink to the well-known semi-empirical relations betweenthe activation quantities and the bulk properties of thematerial in question. Originally such relations have beenformulated for crystalline solids where phenomena like dif-fusion can be described in terms of mobile point defects[49,50]. In [51] Eby finds a good agreement between theempirical relation

DS� ¼ 4aDH � ð10Þ

and experimental data recorded in dynamic mechanicalmeasurements on a number of polymers. Eq. (10) was ob-tained by combining the following relations [49,50]

DV � ¼ ja

DS� ð11Þ

and

DV � ¼ cjDH �; ð12Þ

where for polymers c = 4, and as in Eq. (10), the starredquantities relate to the activated state. For reviews discuss-ing the physical background of these relations the readermay consult Ref. [52].

We note that while Eq. (11) corresponds to (os/ov)T inequilibrium thermodynamics, Eq. (12) does not. It may suf-fice here to mention that the constant c in Eq. (12) is relatedto the Gruneisen parameter. Not without interest in thiscontext is a recent publication by Ingram et al. [53] onion transport in polymer electrolytes. The proportionalitybetween H* and V* is interpreted in terms of a ‘new’ mod-ulus of elasticity, the value of which is found to be 2.4 GPafor PEO, 2.1 GPa for PPO, and 5.0 GPa for a molten mix-ture of calcium and potassium nitrates.

It appears that the proportionality between H* and V*

enjoys a fair degree of acceptance in the interpretation ofthermally activated processes in solids of varying structureand composition. Since we measure the change in H* (as h),at least within the range of the logarithmic time law, andsince the relaxed h and v are proportional to each other,it seems plausible to ascribe the constant of proportionalitylinking these quantities a similar meaning as in Eq. (12).The experimental results reported here and by otherauthors certainly do not contradict such a notion. This,however, does not exclude the possibility that this constant,which we find to have a value similar to 1/j may also berelated to other moduli. Whatever the proper meaning ofthis constant may be, it appears difficult to explain theexperimental results solely in terms of changes in configu-rational entropy.

4. Conclusions

From the results presented above the following conclu-sions can be drawn.

1. When the rate of cooling of the PS sample from equilib-rium above Tg to Ta is decreased, the remaining width of


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the spectrum of relaxation times becomes smaller, dueto, formally speaking, relaxation of the faster processesduring the cooling stage. This is reflected in the corre-sponding changes of the exponent b in the KWW func-tion fitted to the experimental data. In this respect thedata presented above are reminiscent of stress relaxationmeasurements following different strain rates.

2. With regard to the time scales of the relaxation ofenthalpy and volume no substantial differences werefound. In agreement with this finding was the fair linear-ity of the h(v)-plots.

3. The slope of the h(v)-plots, that is dh/dv, was found tobe about 1.8 GPa. There was no dependence on thecooling rate at which Ta was reached. Similar valueshave been reported by other authors.

4. The value of dh/dv was about twice as high as the valuepredicted by the model based on standard thermody-namics extended with an internal variable (order param-eter). Numerically, our dh/dv data were similar to theinverse value of the compressibility of PS at Ta as deter-mined by pvT-measurements in both the isobaric and theisothermal modes. Qualitatively the h(v) dependence issimilar to the empirical relation connecting the corre-sponding activation quantities (Keyes [50]).

Acknowledgements

The authors wish to acknowledge the Ministry of Edu-cation CR (Grant number MSM 7088352101) for thefinancial support. This work was also financially supportedby the Grant Agency of the Academy of Sciences of theCzech Republic (Grant No. IAA2060401) and the Instituteof Hydrodynamics fund AV0Z20600510. The authorswould also like to thank Dr Pavel Cicmanec from theDepartment of Physical Chemistry, University of Pardub-ice, for providing the software for the TNM calculations.

Special thanks are due to Dr H. Baur, author of thebook mentioned in Ref. [44], for highly valuablediscussions.

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https://www.researchgate.net/publication/231708764_Physical_Aging_in_Polyvinyl_acetate_2_Relative_Rates_of_Volume_and_Enthalpy_Relaxation?el=1_x_8&enrichId=rgreq-aa517bff4acee1954e58acd175c28c6a-XXX&enrichSource=Y292ZXJQYWdlOzIyMjAxMDkwNztBUzo5OTkyNjE1NTQ2NDcwOEAxNDAwODM1NjYxMzE5

https://www.researchgate.net/publication/231708764_Physical_Aging_in_Polyvinyl_acetate_2_Relative_Rates_of_Volume_and_Enthalpy_Relaxation?el=1_x_8&enrichId=rgreq-aa517bff4acee1954e58acd175c28c6a-XXX&enrichSource=Y292ZXJQYWdlOzIyMjAxMDkwNztBUzo5OTkyNjE1NTQ2NDcwOEAxNDAwODM1NjYxMzE5

https://www.researchgate.net/publication/231690260_Effects_of_Annealing_and_Prior_History_on_Enthalpy_Relaxation_in_Glassy_Polymers_6_Adam-Gibbs_Formulation_of_Nonlinearity_Macromolecules?el=1_x_8&enrichId=rgreq-aa517bff4acee1954e58acd175c28c6a-XXX&enrichSource=Y292ZXJQYWdlOzIyMjAxMDkwNztBUzo5OTkyNjE1NTQ2NDcwOEAxNDAwODM1NjYxMzE5

https://www.researchgate.net/publication/231690260_Effects_of_Annealing_and_Prior_History_on_Enthalpy_Relaxation_in_Glassy_Polymers_6_Adam-Gibbs_Formulation_of_Nonlinearity_Macromolecules?el=1_x_8&enrichId=rgreq-aa517bff4acee1954e58acd175c28c6a-XXX&enrichSource=Y292ZXJQYWdlOzIyMjAxMDkwNztBUzo5OTkyNjE1NTQ2NDcwOEAxNDAwODM1NjYxMzE5

https://www.researchgate.net/publication/231689313_Structure-dependent_enthalpy_relaxation_at_the_glass_transition_of_polystyrenes?el=1_x_8&enrichId=rgreq-aa517bff4acee1954e58acd175c28c6a-XXX&enrichSource=Y292ZXJQYWdlOzIyMjAxMDkwNztBUzo5OTkyNjE1NTQ2NDcwOEAxNDAwODM1NjYxMzE5




Effect of cooling rate on enthalpy and volume relaxation of polystyrene

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