Physical aging and molecular mobility of amorphous polymers

Journal of Non-Crystalline Solids 353 (2007) 3871–3878 www.elsevier.com/locate/jnoncrysol

Physical aging and molecular mobility of amorphous polymers S. Etienne

a,b,* ,

N. Hazeg c, E. Duval d, A. Mermet d, A. Wypych

d,e,f

, L. David

f

a

d

Laboratoire de Physique des Mate´riaux, UMR CNRS 7556, Ecole Supe´rieure des Mines, Parc de Saurupt, 54042 Nancy cedex, France b EEIGM, 6 rue Bastien Lepage, Nancy Universite´, 54010 Nancy cedex, France c Laboratoire d’Energe´tique et de Me´canique The´orique et Applique´e, UMR CNRS 7563, BP 239, 54500 Vandoeuvre les Nancy cedex, France Laboratoire de Physico Chimie des Mate´riaux Luminescents, UMR CNRS 5620, Universite´ de Lyon, Universite´ Lyon 1, 69622 Villeurbanne cedex, France e Department of Molecular Physics, Technical University of Lodz, Zeromskiego 116, 90-924 Lodz, Poland f Laboratoire des Mate´riaux Polyme`res et Biomate´riaux/IMP, UMR CNRS 5223, Universite´ de Lyon, Universite´ Lyon 1, 15, bd Latarjet, 69622 Villeurbanne cedex, France Available online 27 August 2007

Abstract The classical approaches of the slow dynamics of liquid glass formers and glasses are recalled. Actually, it is well acknowledged that several features are common to glasses and glass forming liquids. For example, the quasi universal occurrence of secondary b slow (or socalled Johari–Goldstein) relaxation and its correlation with the a relaxation process has to be included in any physical model. This ingredient is now introduced in the coupling model proposed by Ngai. However, recent experimental data, mainly dealing with physical aging, show that some aspects remain unsolved and many questions are still to be answered.  2007 Elsevier B.V. All rights reserved. PACS: 64.70.Pf; 82.35.Lr; 81.40.Cd; 62.40.+i Keywords: Raman scattering; X-ray diffraction; Glass transition; Mechanical, stress relaxation; Polymers and organics; Calorimetry; Enthalpy relaxation; Glass transition; Fragility; Structural relaxation; Viscoelasticity; Viscosity

1. Introduction 1.1. Historical aspects The description of molecular mobility in glass forming liquids and glasses is an old story devoted to the liquid glass transition, and a lot of papers have been published on this topic. One of the numerous issues of these descriptions is the understanding of the phenomenon of physical aging of glasses that is closely related to molecular mobility in amorphous matter. The classical way to define the liquid glass transition is concerned with the variation of the liquid glass former vis*

Corresponding author. Address: Laboratoire de Physique des Mate´riaux, UMR CNRS 7556, Ecole Supe´rieure des Mines, Parc de Saurupt, 54042 Nancy cedex, France. Tel.: +33 3 83 36 83 14; fax: +33 3 83 36 83 36. E-mail address: [email protected] (S. Etienne). 0022-3093/$ - see front matter  2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2007.04.035

cosity as a function of temperature. On cooling, the molecular motions in the liquid slow down: the mean transition time s increases and as a result, the viscosity g increases drastically as the temperature approaches Tg (see for example Fig. 1). Usually, the liquid glass transition is considered to occur at a temperature Tg, where the characteristic time is about 100 s. According to the Maxwell equation s = g/ G1 (where G1 is the shear modulus at infinite frequency), this means that the viscosity g at Tg is about 1012.3 Pa s and 1011.3 Pa s for oxide glasses and amorphous polymeric systems, respectively. Obviously, in the case of polymeric systems, it is the ‘local’ viscosity resulting from segmental motions which to be taken into account, and not that associated to viscous flow. Considering the different descriptions of the glass former behavior proposed in the literature, it appears clearly that the early approaches were based on empirical descriptions of the viscosity or mean relaxation time while recent theories are focused on the microscopic aspect of the molecular

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Log 10(η /poise)

10 9 8 7 6 5 4 3 2 400

600

800

1000

1200

Temperature (˚C) Fig. 1. Viscosity of a lead silicate glass former liquid as a function of temperature. Approximate molar composition: SiO2 77.3%; PbO 10.4%; K2O 11.9%. The liquid glass transition Tg is 460 C. Full circles: experimental values. Solid line: best fit with a Vogel Fulcher Tamman equation g = g0 exp[B/(T  T0)] taking g0 = 8.5 · 102 poises, B = 9866 K and T0 = 173 C.

dynamics. Let us recall briefly these approaches, that are commonly regarded as main steps of the historical development, namely (i) the empirical equation Vogel–Tamman– Fulcher [1–3], (ii) the free volume theory of Doolittle [4] lately transformed into the well-known Williams Landel Ferry (WLF) equation [5], the model of Cohen and Turnbul [6] and (iii) the cooperative relaxation theory of Adam and Gibbs [7]. Although experimental curves are well fitted by empirical equations (see Fig. 1), it was early recognized that the mean relaxation time s (i.e. the viscosity g) is not the only one pertinent parameter: it means that the overall time domain or frequency domain response of glasses and glass former liquids cannot be ignored. In such a way several equations, like the famous Kohlrausch Williams Watts (KWW) [8,9] (time domain) or Havriliak–Negami equation [10] (frequency domain), with more or less empirical origin, have been proposed (see for example the review by Harrison [11] and Havriliak and Havriliak [12]). 1.2. Recent approaches A great effort in recent research works has focused on the microscopic aspect of the fast dynamics at T > Tg where fast (cage effect) and a processes are taken into account in the mode coupling theories (MCT) (see for example the work of Go¨tze and coworkers [13–15]), and generalized-MCT including thermodynamically activated processes in order to explain the two step (the so-called b fast and a regime, respectively) decay function.

Recent molecular dynamics simulations, showing the relation between local structure and dynamics, appear to be particularly attractive. Leonforte et al. [16] conclude on a non-affine displacement field, i.e. a non-homogeneous spatial distribution of elastic constants of the glassy network at the nanometer scale, in agreement with inelastic light and neutron scattering recently reported [17]. Any model of dynamic behavior of amorphous matter must necessarily include this aspect, among others. The present review addresses the molecular mobility in the liquid glass transition range and more particularly the ‘slow dynamics’. As similar degrees of freedom are responsible for the main (or a) relaxation and physical aging phenomenon, the effect of thermal history has to be carefully examined experimentally, as a test for these concepts. In addition, the secondary relaxation, attributed to localized molecular rearrangements (so-called b slow process) is observed in almost all the glasses and liquid glass forming systems. Thus, this b process and its possible coupling with the a process is another ingredient to be included into any theory describing the response of amorphous matter in the frequency domain (or alternatively in the time domain), together with the spatial heterogeneity at the nanometer scale, as quoted above. Actually, the effect of local environment on molecular dynamics has been considered by several authors. For example more than 40 years ago, Glarum proposed a model [18] to explain the asymmetrical shape of the a relaxation spectrum in amorphous matter. The problem of molecular rearrangements localization has been intensively studied by Johari and Goldstein who conclude on nonhomogeneous nature of relaxation distribution, i.e. the b process is assumed to occur in ‘islands of mobility’ [19–22]. Cooperativity is expected in ‘complex systems’ like glasses and glass forming systems. In order to take into account the interactions among degrees of freedom at the microscopic scale, several physical approaches have been proposed, namely the coupling model (CM) introduced by Ngai [23]. Two ingredients are at the basis of the CM model, namely (i) a preliminary event with a characteristic time s0 and (ii) non-linear interactions. At short time t < tc, non-linear interactions are not yet active and the correlation function is a simple exponential exp(t/s0). After tc, as non-linear interactions develop, the relaxation rate is slowed down and the correlation function u(t) changes into the stretched exponential form (KWW function) with characteristic time s* and exponent 1  n. This model yields a set of predictive equations: (i) the relaxation function and (ii) the characteristic relaxation time  t 1n uðtÞ ¼ exp   ; ð1Þ s 1  1n s0 ; ð2Þ s ¼ t c tc where n is the coupling parameter (0 < n 6 1, n = 0 if there is no non-linear interactions and the KWW exponent is

S. Etienne et al. / Journal of Non-Crystalline Solids 353 (2007) 3871–3878

equal to 1) and the microscopic time tc is taken equal to 2 ps (from neutron scattering experiments). The coupling model as proposed by Ngai is applied with success since 1979 to various relaxation phenomena occurring in complex systems like ionic motions in glasses, segmental dynamics in polymers and more generally to relaxation processes related to the liquid glass transition. Similar equations were derived from an alternative model based on (i) fluctuations of disorder, or so-called ‘structural defects’, and (ii) correlated motions or hierarchical molecular rearrangements (HCM). In HCM model, the preliminary event for the a relaxation is supposed a priori to be the b relaxation process (Johari–Goldstein relaxation) (see for example Ref. [24]). If both models (CM and HCM) yield similar equations, the procedures and hypotheses used to extract the parameters are not identical. For example, tc is an adjustable parameter in the HCM model while it is taken equal to 2 ps in the CM model. As a result, different basic definitions of parameters raise questions about the actual microscopic nature of the preliminary event and the value of the parameter n. Actually, it is not unreasonable to state that the cooperativity exponent n is connected to the structural state of the glass, but also to the ‘fragility’ of the glass former liquid. Since the pioneering work of Johari and Goldstein [19–22] and Struik [25], several aspects of physical aging remain unclear despite the great efforts as reported in the abundant literature devoted to this topic (see for example Ref. [26]), namely (i) what is the origin of the relation between n, i.e. cooperativity (fragility) and aging effects? and (ii) what is the effect of structural state on the b relaxation and its relation to the a relaxation and (iii) even the sensitivity of the b relaxation to aging is under debate, some authors claiming the absence of any effect of aging on the secondary process [27]. Thus, it appears necessary to clarify the effects of the structural state on the slow dynamics taking into account a and b relaxation processes with possible coupling. To yield this goal, advantage is taken of recent and reliable experimental features support. So doing, the effect of thermomechanical history is demonstrated via selected examples involving physical aging, quenching and plastic deformation. Furthermore, a modeling is expected to be consistent not only with experimental observations about the relaxation processes themselves, but also with what is known about the structure of non-crystalline matter at the nanometer scale.

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together with Small Angle X-ray Scattering (SAXS) and Differential Scanning Calorimetry (DSC), respectively. 2.1. Materials Recent experiments are reported on various amorphous polymers like poly(methyl methacrylate) (PMMA), atactic polystyrene (PS), polycarbonate (BPA-PC or PC), poly(ethylene naphthalene 2,6 dicarboxylate) (PEN) (see Table 1). 2.2. Techniques 2.2.1. Relaxation spectroscopy The mechanical relaxation spectra presented in this review were obtained by means of a high resolution low frequency mechanical spectrometer working in forced torsion mode, described elsewhere, see e.g. Ref. [28]. The strain amplitude did not exceed 105 in order to avoid any effect on the structure of the sample induced by the measurement. 2.2.2. LFRS The depolarized LFRS spectra where recorded with a high resolution five grating monochromator (Dilor). The specimen was illuminated by a laser beam at a wavelength k = 514.5 nm. The scattered intensity I(x) was observed at a scattering angle of 90 in HV polarization mode. The normalized scattered intensity was calculated according to I n ðxÞ ¼

IðxÞ ; ½nðxÞ þ 1x

ð3Þ

where n(x) is the Bose factor. The frequency domain investigated makes it possible to deduce the vibrational density of states (VDOS) below 200 cm1, thus allowing to capture the boson peak which appears as an excess with respect to the Debye regime. It has been proposed that the amorphous network exhibits a non-homogeneous structure at the nanometer scale: domains (mean size D) with closely packed molecules or atoms are separated by loosely packed zones. According to this model, spatial fluctuations of the elastic constants are responsible for the boson peak [29]. Within this frame, the frequency xbp of the boson peak yields the domains size D according to vt xbp  0:8 ; ð4Þ D where vt is the velocity of transverse sound waves.

2. Experimental details

Table 1 Characteristics of polymers referred to in this work

The dynamics of slow (glass transition) and sub-Tg relaxation processes, the short/medium range (up to nanometer scale) structure and the thermodynamic state (enthalpy, entropy) are closely interrelated. These relations can be assessed by mechanical (or dielectric) relaxation spectroscopy, Low Frequency Raman Scattering (LFRS)

Material

Mn (g/mole)

Ip

Tga (K)

PC PS PMMA

15 600 93 000 120 000

1.85 2.58 1.6

418 368 384

Ip = polydispersity index. a Assessed by DSC (mid point, heating rate10 K/min).

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2.2.3. SAXS The small angle X ray scattering data were recorded at ESRF (Grenoble, France), on D2AM (BM2) beam line. The SAXS patterns of glassy polymeric specimens in various states were collected in situ during heating in a cryofurnace, at an incident energy of 16 keV and with short exposure times (few seconds). The q-range calibration was performed with silver behenate. The monitoring of sample attenuation was performed with two photo-multiplicators located before and after the sample, and located above 8 lm thick polyimide (kapton) foils that reflect a negligible fraction of the incident and transmitted beam intensities, respectively. The limit value I(q = 0), which yields the mean density fluctuations, was obtained after subtraction of the empty cell response and extrapolation of the intensity values at the scattering vector q = 0 [30]. 2.2.4. DSC The specific heat Cp was measured by means of differential scanning calorimetry (Perkin Elmer DSC 7). The thermograms were recorded with a heating rate 10 K/min. 3. Experimental features: selected examples 3.1. Effect of thermal and mechanical history on the structural state LFRS spectra were taken after thermal quenching and plastic deformation. Such spectra were compared to the spectra exhibited by annealed specimens taken as references. The effect of thermal quenching is shown in Fig. 2. It appears clearly that the VDOS observed on PMMA in

the region of the boson peak is increased after thermal quenching. These effects are thermo-reversible i.e. they are erased by annealing at temperature not far above Tg. This LFRS observation is in agreement with inelastic neutron scattering experiments [17]. Another information is extracted from Fig. 2: the size D of the closely packed domains is about 4 nm according to Eq. (4). PMMA [31] and PC [32] specimens subjected to pure shear plastic deformation show similar behavior: again an increase of the VDOS in the region of the boson peak is observed. Both thermal quenching and plastic deformation induce an increase of the disorder. It is deduced that the boson peak is increased when the state of disorder is increased. This disorder resulting from quenching or plastic deformation is related to enthalpy or configurational entropy level which can be quantitatively assessed by calorimetry [33–35]. Within the frame of the model of heterogeneous structure of glasses [29], this means that spatial fluctuations of the elastic constants at the nanometer scale are increased. SAXS patterns taken on PMMA samples (Fig. 3) (i) subjected to plastic deformation and (ii) thermally quenched and compared with a well annealed specimen show interesting features: thermal quenching induces dynamic density fluctuations (estimated by the value I(q = 0)), while plastic deformation induces also dynamic and additional static density fluctuations as revealed by the scattered intensity excess at low temperature. This static component is attributed to long range fluctuations, whereas the dynamic component is attributed to fluctuations in electron density that originate from molecular mobility associated with the a and b relaxation processes.

70 5

I n [a.u.]

50 40

120

4

110

3 2

100

1 0

30

50

100

150

200

Wavenumber [cm-1]

20

I(q=0) [a.u.]

In REF - I n AGED [a.u.]

60

90 80 70 60

10 0 50

100

150

200

Wavenumber [cm -1] Fig. 2. Comparison of LFRS normalized intensity of a PMMA reference sample (quenched state, solid line) and after aging below Tg during 10 months at room temperature and at 70 C during 3 weeks (dashed line). The inset shows the difference curve: aged specimen subtracted from reference specimen. Data from Ref. [50].

50 200 220 240 260 280

300 320 340 360

380 400 420

Temperature (K) Fig. 3. SAXS intensity extrapolated to scattering vector q = 0, for PMMA specimens after different thermomechanical histories:

– annealed state (empty circles); – thermally quenched (empty squares); – plastically deformed (full circles).

S. Etienne et al. / Journal of Non-Crystalline Solids 353 (2007) 3871–3878

3.2. Effect of thermal and mechanical history on the slow dynamics It is well-known that the molecular mobility in the region of a relaxation is increased by both plastic deformation and thermal quenching [24,36–39] and is depressed on annealing below Tg. The effect of plastic deformation was studied in detail on PC [40]. The specimen was plastically deformed by fast rolling at room temperature and then allowed to cool down to the liquid nitrogen temperature. Fig. 4 shows the dynamic behavior observed on successive heating scans with increasing maximum temperature, each scan being followed by fast cooling. This procedure, classical in solid state physics, allows to stress the kinetic aspect of thermal annealing. It is observed that the a relaxation outset of the freshly deformed specimen (first scan) is lowered by approximately 100 K in comparison with an annealed specimen. Obviously, the molecular mobility is drastically enhanced by several orders of magnitude. Annealing and structural relaxation take place on heating, and the effect of plastic deformation is gradually erased: the shoulder appearing initially at room temperature is shifted to higher α relaxation

3 2

10

0

6 5 4

tan(ϕ) G' [GPa]

3 2

10 -1 6 5 4

Physical Aging

β relaxation

3 2

10 -2 100

200

300

400

Temperature (K) Fig. 4. Loss factor deduced from the dynamic shear modulus of PC measured at constant frequency (0.1 Hz) on successive heating scans (1 K/ min) after plastic deformation. The real part of the shear modulus is shown for the last scan for clarity. Data from Ref. [40]. Loss factor: (d):First scan, (s):second scan, (+):third scan, (·):fourth and final scan. Real part of the shear modulus:(–·–): final scan.

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temperatures on successive heating scans and completely disappears. In addition, the b relaxation is slightly changed. In fact, the high temperature side of the b relaxation peak increases on aging. This is a demonstration of a concomitant evolution of the a and b relaxation processes. Such effects were also described in closer relation to the modeling in the HCM frame in the case of amorphous PEEK [41] as a result of a gradual increase of the relaxation time through the increase of the correlation parameter n with annealing (i.e. ascribed to a increase of chain segment packing order, or alternatively to a decrease of disorder). An other interesting feature is the fast aging even at room temperature (i.e. far below Tg) for PC [40] and other amorphous para-substituted polyaryls [41]. The high sensitivity of the a process to the structural state and the fragility index seem to correlate in PC, but also in PEEK, PES, PPS which are aromatic fragile glass formers. This correlation between fragility index and aging effects strength have already observed in many systems, for example in side chain liquid crystal polymers [42]. Another attractive aspect is the effect of long term aging not far below Tg on the low frequency dynamics. Fig. 5 displays the evolution of the shear dynamic modulus of a PS specimen. In a first step, the quenched specimen is aged at temperature Ta (Tg-15 C = 80 C) and its behavior is studied in isothermal condition by successive frequency scans as a function of aging time ta up to 21 days (the duration of each frequency scan is kept much lower than ta). After such a long term physical aging, a JG like b-slow relaxation develops and becomes resolved on the high frequency of the a process (Fig. 5(a)) for ta = 21 days. This effect of long term physical aging of PS is also obvious on the low temperature tail of the a relaxation peak by measurement at constant frequency (1 Hz) and temperature scan up to 125 C (heating rate 1 K/min). The first temperature scan was performed on the specimen previously aged 21 days at 80 C and was followed by quenching and a second heating scan up to 125 C. The comparison of the two scans is instructive: the loss factor tanu maximum (b peak) is erased by heating above Tg (rejuvenation) while tanu is increased in the temperature range between the a and b processes (Fig 5(b)). In agreement with other authors [43–46], the effect of aging leads to a better separation of the a and b relaxation peaks as result of a higher sensitivity of cooperative relaxation processes to aging. Considering the complexity of the effect of aging on viscoelastic spectra, the effect of physical aging was recorded on PMMA specimens according well controlled isothermal procedures, with variable aging temperatures Ta and aging duration ta after thermal quenching. Fig. 6(a) demonstrates that both a and b relaxation characteristics evolve on aging below Tg. In order to separate the effects on both relaxation processes, experiments are to be carried out at very low frequencies [47–49]. PMMA samples aged in the same controlled conditions were analyzed by calorimetry. DSC thermograms were recorded on heating as a function of aging time ta. Fig. 6(b) shows the result for an aging

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a

0.10

0.1

5340 min 6810 min 8250 min 14230 min 27190 min 28900 min 30300 min 31440 min

0.08 0.07 -4

10

10

α

Tβ

Aging times at 80˚C

0.09

b

tan(ϕ )

20

| G*|( a. u .)

tan(ϕ )

1

10

Tg Ta

-3

10

-2

10

-1

1

F (Hz)

50

100

Temperature(˚C)

Fig. 5. Effect of long term physical aging on the low frequency dynamic shear modulus of polystyrene (a) measured at constant temperature (80 C) and frequency scans at a function of aging time at 80 C. (b) measured at constant frequency (1 Hz) and temperature scan (heating rate: 1 K/min): after 21 days of aging at 80 C (solid circles) and subsequent rejuvenation due to fast cooling from 125 C (open circles). Data from Ref. [52].

Fig. 6. Effect on physical aging on PMMA at 363 K after thermal quenching. The aging times are reported in the inset (a) on the mechanical loss factor tan(u) (measured at 1 Hz, heating rate 1 K/min) (b) on the specific heat (heating rate 5 K/min).

temperature Ta = 363 K. This experimental investigation shows that the enthalpy decreases drastically when the aging temperature Ta is not far below Tg, as already observed by Wypych et al. [50]. As ta increases, the evolution of the endothermic peak indicates that the final state corresponds to a deeper energy minimum, in direct connection with the slowing down of the dynamics in the glass transition range. Actually, Fig. 6(a) shows that additional effects exist at lower temperature in the range of the b relaxation process, which illustrates the high sensitivity of the low frequency mechanical relaxation spectroscopy technique. 4. Discussion The selected examples presented in the previous section, together with the documented literature, tell us that several ingredients must enter definitively any model of relaxation processes observed in glasses and in liquid glass formers, in order to take into account the overall data, namely: (a) the quasi universal occurrence of a b slow process (JG relaxa-

tion or more localized conformational changes), (b) the a relaxation is described with a good approximation by a KWW relaxation function, (c) the inhomogeneous structure of the glassy network, (d) the boson peak strength (i.e. the fluctuation of elastic constants) decreases as the fragility index increases and (e) the physical aging effects on molecular mobility increase as the fragility index increases. Indeed, the coupling model (CM) is attractive taking into account points (a) and (b), regarding the JG relaxation process (characteristic time sb) as the preliminary event (s0 identified to sb) for the a relaxation. Thus a connection is provided between a and b relaxation processes via the set of Eqs. (1) and (2) recalled above. Furthermore, a scaling relation between the apparent activation energy of the glass transition process Ea, the activation energy of the b process Eb and the coupling parameter n [51] is included. Similar explanation was provided by the HCM model [24]. The CM model appears consistent with experimental data in many materials and in particular in amorphous polymers like e.g. PMMA, PET. Actually, the correlation between

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characteristics of the b relaxation generally results in a more stretched relaxation function. Ignoring this distribution may lead to underestimate the value of n. There is no reason that the thermal activation parameters of sb determined in the glassy regime should be valid in the metastable equilibrium state. The assignment of the observed secondary relaxation as the genuine preliminary event of the a relaxation in some cases is controversial [53,54]. The secondary relaxation process is not observed in the case of very strong glass formers, like silica (see Fig. 7). This is explained by Eqs. (1) and (2) assuming the coupling parameter n equal to zero, i.e. the a and b relaxation processes are merging (sa is identical or very close to sb) [55]. In some cases, it is claimed that b relaxation manifests as an excess wing in the high frequency tail of the a process [43–46]. Generally, the b and a relaxation processes are well resolved. Then, it is to be stressed that it remains unclear why the localized rearrangements, the extension of which is the a process, gives rise to a b relaxation well resolved experimentally by frequency and/or temperature scanning. The evolution of correlation parameter n and fragility index m with nanostructure has to be explained invoking microscopic arguments. Other questions are asked: why the aging effects increase when fragility increases [42]? and why the more distinguished the a and b relaxation processes the higher the fragility index m [56]? The coupling model, taking into account cooperativity, and characteristics of secondary and main relaxation is an attractive proposal [57], but since many and important aspects remain unsolved, more research work is requested. In particular, one of the essential challenge is to relate the parameters involved in the CM or HCM descriptions to macroscopic and measurable properties. Obviously, the mean density which is often related to free volume is not

0.06

33

0.05 32

0.03

31

tan(ϕ)

0.04

G ' (GPa)

a and b relaxation processes is not obvious for example in the case of PC. It was proposed elsewhere in the frame of the HCM [40] that in PC sa and sb are connected via Eqs. (1) and (2) taking a rather high value of n approaching 0.8, the b relaxation being the process observed near 160 K at 1 Hz, a very low temperature compared to Tg. This calculation was performed by analyzing the very high frequency tail of the a process. In addition, only the high temperature side of the b relaxation peak is affected by aging at room temperature. In the frame of the CM model, assuming tc = 2 ps, would lead to a secondary relaxation process where the mechanical or dielectric loss factor goes trough a minimum (see Fig. 4). In both cases, fitting with the CM or HCM models is not yet satisfactory. Similar problems are found to explain experimental results on PEN [52] where assignment of the true b relaxation as the preliminary event of the a relaxation is not obvious (the primary event of the a process corresponds to the so-called b* process that exhibits cooperativity, a feature that is unusual for an ‘elementary or primary motion’). The connection between a and b motions in PEN were more extensively explored by the study of long term physical aging effects at Tg-15 K, that could be explained considering the inhomogeneous structure of the amorphous network (point c), assuming the conversion of part of the a rearrangements of limited extension into secondary relaxation events exhibiting cooperative character, as proposed elsewhere [40,52]. This explanation is consistent with the evolution of the boson peak intensity on aging as shown by low frequency Raman scattering (Fig. 2) and inelastic neutron scattering [17,50]. This explanation is also consistent with the energy landscape model [40,52]. The evolution of the thermodynamic and structural state (Fig. 6(b)) is also explained within the energy landscape model: on long term aging at temperature not far below Tg, the system falls into deep minimum with high energy barriers and the characteristic time of the a process is monotonously increased, whereas the relaxation time of the b* motions evolves with time in a more complex way. The relation between fragility index, boson peak intensity and physical aging, (namely point (c) and (d)) are generally explained within the frame of inhomogeneous nature of the amorphous network and compatible with a description with the concept of energy landscape. However, an interpretation at the microscopic level is still to be found.

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0.02

5. Concluding remarks

30 0.01

It has been demonstrated that the coupling model is consistent with the main aspects of the dynamics in liquid glass formers and glasses. However, some questions are still unsolved. Among them, it is worth to underline the following: It is difficult to extract reliable values of parameters (in particularly the cooperativity parameter n) because of distribution. Using the set of Eqs. (1) and (2) ignoring the spatial distribution of cooperativity parameter n and

29 600

0.00 700

800

900

1000

1100

Temperature(˚C) Fig. 7. dynamic shear modulus of silica measured on heating (1 K/min) at three frequencies: Solid line: 1 Hz - - - - -: 0.1 Hz –Æ–Æ–Æ–Æ-: 0.01 Hz No b relaxation peak is observed. The positive temperature coefficient of the real component G 0 of the elastic modulus is observed as usually observed for silica. The a relaxation process is responsible for decrease of G 0 at high temperature. Data from Ref. [55].

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the pertinent parameter [24,58]. Rather, the fluctuation of density or fluctuation of cohesive energy at the nanometer scale could be related to the dynamics in a more general way, valid for glass formers effects both in the liquid and in the glassy states. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

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