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Thermochimica Acta 532 (2012) 172– 175

Contents lists available at ScienceDirect

Thermochimica Acta

journa l h o me page: www.elsev ier .com/ locate / tca

tructural and vibrational relaxations at spontaneous aging in glass�

iroshi Kobayashia,∗, Haruyuki Takahashib

National Institute of Advanced Science and Technology (Retired), 8-1-309, Tomooka, Nagaokakyo 617-0843, JapanGraduate School of Science and Engineering, Ibaraki University, 4-1-12, Nakanarusawa, Hitachi 316-8511, Japan

r t i c l e i n f o

rticle history:vailable online 7 May 2011

eywords:lass transitionntropypontaneous aging

a b s t r a c t

The entropy change of intermediate-range orders (IROs) through the glass transition is discussed on thebasis of the theories developed by Adam–Gibbs and Prigogine. It is estimated that the time dependence ofthe configurational entropy of glass is zero, but that of its vibrational entropy shows a constant decreasewith the smallest change, while maintaining a constant fictive temperature and an isostructural state.It is shown that the spontaneous aging of well annealed glass shows a vibrational relaxation with along relaxation time, while the annealing and physical aging of glass show a structural relaxation with

ntermediate range orderime symmetryano-emergence

a short relaxation time. The results indicate that the volume of glass decreases at a constant time ratethrough spontaneous aging at a constant temperature. It is emphasized that the existence of the constantconfigurational entropy below the glass transition temperature is the basic feature of glass forming.The nanoorder-size of the IROs in glass determines the residual entropy. The time symmetry in glassis spontaneously violated. The glass transition is a nano-emergence through the abrupt decrease in the

ooled

fluctuations in the superc

. Introduction

Understanding the glass transition is to understand how glassan be made by cooling. There are two views in explaining thelass state. One is that the glass state is halfway along a longrocess to the crystal state, because glass has an extremely longelaxation time, and therefore, a very large Deborah number. Thisrocess depends on the time used for measurements, because sim-le molecules can become glass by virtue of a high cooling rate. Thether is that the glass state is the 4th state after gas, liquid and crys-al. This view depends on the fact that there is a residual entropyn glass at 0 K but no entropy in crystal at 0 K according to the thirdaw of thermodynamics.

Academic research on the glass transition was begun in the920s by Gibson and Giauque [1], and Simon and Lange [2]. Theyeasured the heat capacity and entropy changes with temperature

or glycerol by the calorimetric method and reported an abruptecrease in its heat capacity at the glass transition temperatureTg). In 1958, McLaughlin and Ubbelohde [3] discussed the non-rrhenius temperature dependence of the viscosity of supercooled

iquids using a model of “packed clusters” in ortho-terphenyl (OTP).n 1965, Adam and Gibbs [4] explained the non-Arrhenius propertyf the relaxation time in supercooled liquids using the model of

� This paper was presented at the special session in Honor of Prof. S. Seki and Prof.. Suga in ICCT-2010 held at Tsukuba, August 1–6, 2010.∗ Corresponding author.

E-mail address: hkoba@mx2.canvas.ne.jp (H. Kobayashi).

040-6031/$ – see front matter © 2011 Elsevier B.V. All rights reserved.oi:10.1016/j.tca.2011.04.029

liquid state.© 2011 Elsevier B.V. All rights reserved.

“cooperatively rearranging regions (CRRs)” of molecules and poly-mer segments.

We measured the viscosity–temperature relationship in awide temperature range of supercooled liquids and glasses, andproposed the idea that the glass transition is due to the self-organization of intermediate range orders (IROs) which havethe size of a few nanometers in the glass state [5–7]. In thispaper, we propose that glass is a physical state different froma halfway approaching to a crystal state on the basis of thediscussions on the entropy change through the glass transitionof the IROs in the framework of Adam–Gibbs and Prigoginetheories.

2. Extended Adam–Gibbs theory

Following the Adam–Gibbs theory [4], the configurational relax-ation time �(T) of a glass-forming material at the temperature Tis

�(T) = �0 exp[

z∗��

kT

], (1)

where �0 is a constant, z* is the number of molecules in a CRR, whichis the lower critical limit to the size of the cooperative region (tran-sitionable ensemble) that can yield nonzero transition probability,

�� is the potential energy hindering the cooperative rearrange-ment per a molecule, and k is the Boltzmann’s constant.

When Avogadro’s number is Na, s∗c is the critical configura-

tional entropy corresponding to the ensemble of z* molecules

H. Kobayashi, H. Takahashi / Thermochimica Acta 532 (2012) 172– 175 173

Cpe

Cpg

Cpv

T2Tf’T g T m

Hea

t Cap

acity

Fc

(e

S

wa

�

�

wc

S

≈

wimpgi

S

wb

i[(ve

S

=

Fig. 2. (a) Temperature dependence of the configurational entropy (Sc) of

ig. 1. Heat capacity diagram of the liquid (equilibrium), glass (nonequilibrium) andrystal (vibrational) states.

s∗c = k ln W ≥ k ln 2, W: number of configurational states of thensemble) and Sc is the molar configurational entropy, we have

c =(

Na

z∗

)s∗

c, (2)

here at T � Tg (z* = 1), S∗c = (Na)s∗

c and at T � Tg (z* = Na), S∗c = s∗

cnd sc* increases monotonically with the size of the ensemble.

(T) = �0 exp[

Na��s∗c

kTSc

], (3)

(T) = �0 exp[

C

TSc

], (4)

here C = Na��s∗c/k. Here, it is assumed that the interaction of a

ooperative region with its environment is weak.The following equations are deduced.At T ≥ Tg,

c =∫ T

Tg

�Cp

TdT +

∫ Tg

T2

�C ′p

TdT, (5)

�Cp lnT

T2, (6)

here Sc = 0 at T = T2. �Cp is the difference between the heat capac-ties of the supercooled liquid and crystal states at Tg ≤ T ≤ Tm (Tm:

elting point). Here, it is assumed that �Cp is independent of tem-erature. �Cp

′ is the difference between the heat capacities of thelass and crystal states at T2 ≤ T ≤ Tg. Here, it is assumed that �Cp

′

s equal to �Cp to obtain a simple form of Eq. (6).At T ≤ Tg,

c ≈ �Cp lnTf

T2, (7)

hich is constant and Tf is the fictive temperature, which shoulde called the Tool temperature.

These situations are shown in Fig. 1, in which Tf′ is the limit-

ng fictive temperature described in the paper of Moynihan et al.8]. Tf = T at T � Tg and Tf = Tf

′ at T � Tg. As the exact value of �Cp′

= Cpg − Cpv at T2 ≤ T ≤ Tg) is not constant (shown in Fig. 1), the exactalue of Sc is calculated as follows, where Sc in Eq. (7) at Tf = Tf

′ isqual to the residual entropy S0 at T = 0.

0 =∫ Tg (Cpg − Cpv)

TdT, (8)

T2∫ Tg

T ′f

(Cpe − Cpv)T

dT, (9)

polystyrene. (b) Temperature dependence of the number of molecules (z*) in anIRO of polystyrene.

where Cpg is the measured heat capacity of the glass state, Cpe isthe heat capacity of the equilibrium state and Cpv is the vibrationalheat capacity of the glass state and is the same as that of the crystalstate.

Agarwal et al. [9] reported that the 1100 cm−1 IR absorptionband in the IR reflection spectra in silica glasses is directly corre-lated with the glass fictive temperature, which reaches a constantintensity when the sample is well annealed. They presented amethod for determining the fictive temperature of silica glassesusing the IR spectroscopy.

Using Eqs. (6) and (7), we can estimate the temperature depen-dence of Sc for polystyrene over a wide temperature range, asshown in Fig. 2(a). The data used were taken from Ref. [4]. Forpolystyrene, �Cp = 27 J K−1 mol−1, Tf ≈ Tg = 373 K, T2 = 311 K. Here,we assumed that the values of Tf for glass with low Tg are mostlydistributed below Tg. We further assume that the intersegmentalmotion, which is the primary relaxation event for the polymer, has3 segments. If we assume two internal rotation on one axis (back-bone C–C bond) for one segment, then ln W ≥ ln 23 = 2.08. We canalso obtain the temperature dependence of z* from Eqs. (2), (6)and (7), as shown in Fig. 2(b). Here, s∗

c reaches its maximum andSc reaches its minimum, as the temperature reaches to Tg. Then,z* reaches its maximum and is constant at T ≤ Tg. In Fig. 2(b), itis assumed that s∗

c = k ln 23. This indicates a fixed size of theIROs, in which the molecules undergo a limited rearrangement.It is also assumed that the limited rearrangement below T2 is inthe frozen-in state because of its extremely long relaxation time,which realizes a true glass. The infinitely slow cooling would givethe relation of T2 = Tf

′ = Tg in Fig. 1. The constant Sc and z*, which give

a constant activation energy in the relaxation time-temperaturerelationship following Eq. (3), are explained by the freezing of theIROs. Constant values of Sc and z* were experimentally observed

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n some glassy molecules by Yamamuro and colleagues [10,11]hrough calorimetric measurements; they were in good agreementith our above estimations.

Pauling [12] describes that the observed entropy of H2O crystalt low temperature provides strong support for its particular struc-ure, which owes its existence to hydrogen bonds between water

olecules. He supposed that at low temperature, H2O crystal isn a frozen-in disordered state and has a finite residual entropy.e calculated that the arrangement of protons in H2O crystal pro-uce a residual entropy of R ln(3/2) = 3.37 J K−1 mol−1, which has

good agreement with an experimental value of 3.41 J K−1 mol−1

etermined by Haida et al. [13]. The residual entropy of H2O crystaletermined by the arrangement of protons is constant but that oflass determined by a limited rearrangement of molecules in theROs decreases with lowering a cooling rate.

At T > Tg, the interactions between the IROs are weak, while theooperative interactions between molecules in the IRO are strong.t T = Tg, the interactions between the IROs suddenly becometronger, and then, the IROs freeze, where z* is constant and deter-ines S0 following Eq. (2). Tf

′ includes the effect of the strongnteractions between IROs besides that of the constant z*. It is sup-osed that the occurrence of these strong cooperative interactionsetween IROs would be related to the decrease of the fluctuations inhe supercooled liquid state suggested by the work of Mizubayashit al. [14]. This phenomenon is considered to be a nano-emergence,hat is, a self-organization of the IROs [7]. The nanoorder-size ofhe IROs determines the residual entropy which is a characteristicroperty of glass.

. Evaluation of entropy with Prigogine theory

In this section, we discuss the glass transition phenomenaonsidering the entropy change of the IROs from a microscopiciewpoint. Following Nicolis and Prigogine [15], the total entropyeneration dS is expressed as

S = dSi + dSe, (10)

here dSi is the internal entropy generation (dSi ≥ 0 according tohe second law of thermodynamics), and dSe is the external entropyeneration (dSe = dQ/T, Q: calorific value obtained from the envi-onment). dS is the entropy generation of glass which determineshether the glass state is equilibrium (dS = 0) or nonequilibrium

dS ≥ 0). It is supposed that dS = 0, dSi = 0 and dSe = 0 for the equi-ibrium state but dS = 0, dSi > 0 and dSe < 0 for the nonequilibriumnd stable state. dSi is constructed from the change in the fluctua-ions of the chemical potentials in glass and the generation of newlements in it. dSi cannot be measured by an experimental method.

In the case of glass-forming materials,

Se = dSc + dSv, (11)

here dSc is the amount of configurational entropy generation andSv is that of vibrational entropy generation. Sc represents the netntropy of the glass or supercooled liquid state and is the differenceetween the entropy of the glass or supercooled liquid state andhat of the crystal state, and Sv represents the entropy of the crystaltate. Sv is constructed from the contributions of the intermolecularnd intra-molecular vibrations in crystal. Here, it is assumed thathe vibrational heat capacity of the glass or supercooled liquid states equal to that of the crystal state.

From the phenomenological consideration of the IROs, on theasis of the proceeding discussions, the quantity dS is expected to

ave different values in the following three temperature regions.

In the supercooled liquid state where

> Tg, dS = 0. (12)

imica Acta 532 (2012) 172– 175

This is due to a free rearrangement of molecules in the IROs.Thus, the liquid is in an equilibrium and stable state.

At the glass transition where

T = Tg, dS < 0, (13)

where the configurational heat capacity exhibits an abruptdecrease, which originates from the freezing of the IROs. Thus, thematerial is in a fluctuating state. Mizubayashi et al. [14] reportedthat a resonant collective motion of many atoms excited by elec-tropulsing in Zr–Cu metallic glass causes the transformation of theglass state to the crystalline state at a temperature much lower thanthe thermal crystallization temperature. This means that there areshort-range order fluctuations in the supercooled and glass states,which are coupled with the enhanced motion of atoms with thethreshold values in initial discharge currents that causes the ather-mal crystallization of metallic glasses. It is assumed that the originof the abrupt decrease in the heat capacity is the suppression ofthese fluctuations, which are related to the configurational entropy,and there is no fluctuation in the crystal state.

For a glass sample after the sufficiently rapid cooling, its entropyincreases with time, dS > 0. It is assumed that the entropy of glassfinally reaches a constant value, dS = 0, because the well annealedglass has an extremely long relaxation time.

In the glass state where

T < Tg, dS = 0, (14)

where dSc = 0 according to Eq. (7), which indicates a time-independent Sc as well as a temperature-independent Sc, anddSi = −dSv ≥ 0 according to Eqs. (10), (11) and (14). Prigoginedemonstrated, in his nonequilibrium linear thermodynamic theory,that in the nonequilibrium and stable state, dSi reaches a minimumand is constant because dSi ≥ 0 and d(dSi)/dt = 0. This is the fourthlaw of thermodynamics. This means that dSv < 0, and Sv showsa linear time decrease with the smallest change and has a longvibrational relaxation time. This state is realized by a limited rear-rangement of molecules in the IROs that shows a constant valuein the temperature- and time-dependences of Sc determined bythe fictive temperature Tf. Thus, the glass is in the non-equilibriumand stable state. Here, it is assumed that in the time interval formeasurements, the dynamic process of glass can be discussedon the basis of the nonequilibrium linear thermodynamic theorydeveloped by Prigogine, because of its extremely long structuralrelaxation time (on the order of 104 years). The frequency of latticevibrations in glass is lower than that in crystal because the den-sity of glass is lower, and the anharmonicity effect in glass is largerbecause glass is disordered. It is inferred that these properties ofglass are the origins of the time decrease in Sv in the glass state. Itis supposed that the fluctuations of the chemical potentials in glassbecome equilibrium according to the second law of thermodynam-ics, accompanied by the decrease of the volume of glass. Physicalaging in polymeric and inorganic glasses shows the evolution of thethermodynamic state towards equilibrium through volume recov-ery after the temperature jump (up-jump or down-jump) from theinitial temperature [16]. Chen [17] measured the structural relax-ation in PdNiP metallic glass over a wide temperature range fromwell below to above Tg by the calorimetric method. He reported thatthe long annealing time decreases the configurational enthalpy,which means a decrease in the volume of glass undergoing struc-tural relaxation. These results predict that some physical propertiesrelated to Sv of glass would decrease with time upon spontaneousaging, maintaining a constant fictive temperature and an isostruc-

tural state showing the vibrational relaxation. Here, spontaneousaging means the aging in the natural state at a constant temperatureof the well annealed glass with a constant fictive temperature. Anexperiment on measuring the volume change of glass at a constant

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H. Kobayashi, H. Takahashi / The

emperature over a long time is desired, because dSv is the changen the vibrational energy proportional to the volume of glass.

Suga and Seki [18] reported the entropy diagram of isopentanebtained by calorimetry, in which the temperature dependences ofhe entropies of its glass and crystal states are shown. Their resultseveal a constant Sc and a linear temperature decrease in Sv at T < Tg.

We consider that the physical properties in the glass state shoulde treated as those of the nonequilibrium and stable state by theonequilibrium linear thermodynamic theory, but that the glassransition should be treated as an emergence-phenomenon, thats, a self-organization of the IROs, by the nonequilibrium nonlinearhermodynamic theory developed by Prigogine. It is well knownhat the diffusion phenomena in glass are treated by the linearheory, but the harmonic generations of sound waves in solids arereated by the nonlinear theory.

It is supposed that nanomaterials of important topics in recentcience and technology would have the same mechanism as an IROn glass to make it stable and generate a new material in a nonequi-ibrium and stable state. It is considered that nanomaterials havewn relaxation times in their time processes, similar to glass. If theelaxation time of a material is short, it is not applicable to elementsf nanodevices.

. Conclusions

The evaluations of the configurational and vibrational entropieshrough the glass transition on the basis of Adam–Gibbs andrigogine theories indicated that the time- dependence of the con-gurational entropy of glass is zero and that of the vibrationalntropy shows a linear decrease with the smallest change, retain-ng an isostructural state. This result means that the volume of glassonstantly decreases in spontaneous aging, maintaining a constantctive temperature. Glass shows vibrational relaxation as well astructural relaxation. The size of the IROs in glass determines theesidual entropy, which is related to Tf of glass. These states are real-zed by a limited rearrangement of molecules in the IROs. The statef gas, liquid, and crystal are determined as potential minima ofheir free energies, but the state of glass is determined as a nonequi-

ibrium and stable state shown by Prigogine’ s theory [15]. It isroposed that the glass state is a physical state different from therystal state, that is, glass is the 4th state after gas, liquid and crystal.lass radiates heat through a decreasing volume. This means that

[

[

imica Acta 532 (2012) 172– 175 175

the time symmetry in glass is violated spontaneously. The glasstransition is a nano-emergence through the abrupt decrease in theshort-range order fluctuations in the supercooled liquid state.

References

[1] G.E. Gibson, E.F. Giauque, The third law of thermodynamics. Evidence from thespecific heats of glycerol that the entropy of a glass exceeds that of a crystal atthe absolute zero, J. Am. Chem. Soc. 40 (1923) 93–104.

[2] F. Simon, F. Lange, Zur frage der entropie amorpher substanzen, Z. Phys. 38(1926) 227–236.

[3] E. McLaughlin, A.R. Ubbelohde, Cluster formation in relation to the viscosity ofmelts in the pre-freezing region, Trans. Faraday Soc. 54 (1958) 1804–1810.

[4] G. Adam, J.H. Gibbs, On the temperature dependence of cooperative relaxationproperties in glass-forming liquids, J. Chem. Phys. 43 (1965) 139–146.

[5] H. Kobayashi, H. Takahashi, Y. Hiki, Temperature dependence of the viscositythrough the glass transition in metaphosphate and polystyrene, Mater. Sci. Eng.A 442 (2006) 263–267.

[6] H. Kobayashi, H. Takahashi, Viscosity–temperature relationship of glasses withlow melting points obtained by wide-range measurement, J. Ceram. Soc. Jpn.116 (2008) 855–858.

[7] H. Kobayashi, H. Takahashi, Temperature dependence of intermediate rangeorders in the viscosity–temperature relationship of supercooled liquids andglasses, J. Chem. Phys. 132 (2010) 104504–104511.

[8] C.T. Moynihan, A.J. Easteal, M.A. Debolt, J. Tucker, Dependence of the fictivetemperature of glass on cooling rate, J. Am. Ceram. Soc. 59 (1976) 12–16.

[9] A. Agarwal, K.M. Davis, M. Tomozawa, A simple IR spectroscopic method fordetermining fictive temperature of silica glasses, J. Non-Cryst. Solids 185 (1955)191–198.

10] S. Takahara, O. Yamamuro, T. Matsuo, Calorimetric study of 3-bromopentane:correlation between relaxation time and configurational entropy, J. Phys. Chem.99 (1995) 9589–9592.

11] O. Yamamuro, I. Tsukushi, A. Lindqvist, S. Takahara, M. Ishikawa, T. Matsuo,Calorimetric study of glassy and liquid toluene and ethylbenzene: thermody-namic approach to special heterogeneity in glass-forming molecular liquids, J.Phys. Chem. B 102 (1998) 1605–1609.

12] L. Pauling, The structure and entropy of ice and of other crystals with somerandomness of atomic arrangement, J. Am. Chem. Soc. 57 (1935) 2680–2684.

13] O. Haida, T. Matsuo, H. Suga, S. Seki, Calorimetric study of the glassy stateX. Enthalpy relaxation at the glass-transition temperature of hexagonal ice,J. Chem. Thermodyn. 6 (1974) 815–825.

14] H. Mizubayashi, T. Takahashi, K. Nakamoto, H. Tanimoto, Nanocrystallinetransformation and inverse transformation in metallic glasses induced by elec-tropulsing, Mater. Trans. 48 (2007) 1665–1670.

15] G. Nicolis, I. Prigogine, Self-Organization in Nonequilibrium Systems, JohnWiley & Sons, Inc., New York, 1977.

16] G.B. Mckenna, On the physics required for prediction of long term performance

of polymers and their composites, J. Res. NIST 99 (1994) 169–189.

17] H.S. Chen, On mechanisms of structural relaxation in a Pd48Ni32P20 glass, J.Non-Cryst. Solids 46 (1981) 289–305.

18] H. Suga, S. Seki, Thermodynamic investigation on glassy states of pure simplecompounds, J. Non-Cryst. Solids 16 (1974) 171–194.