Physical Aging of Glasses: The VFT Approach : The VFT Approach [1 ed.] 9781616680022, 9781607413165

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Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved. Physical Aging of Glasses: The VFT Approach : The VFT Approach, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved. Physical Aging of Glasses: The VFT Approach : The VFT Approach, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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PHYSICAL AGING OF GLASSES: THE VFT APPROACH

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Physical Aging of Glasses: The VFT Approach Jacques Rault 2009. ISBN: 978-1-60741-316-5

Physical Aging of Glasses: The VFT Approach : The VFT Approach, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

MATERIALS SCIENCE AND TECHNOLOGIES SERIES

PHYSICAL AGING OF GLASSES: THE VFT APPROACH

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

JACQUES RAULT

Nova Science Publishers, Inc. New York

Physical Aging of Glasses: The VFT Approach : The VFT Approach, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material.

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Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA

Rault, Jacques. Physical aging of glasses : the VFT approach / Jacques Rault. p. cm. Includes index. ISBN H%RRN 1. Glass--Mechanical properties. 2. Glass--Heat treatment. 3. Enthalpy. I. Title. TA450.R328 2009 620.1'4492--dc22 2009006381

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CONTENTS

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Preface

vii

Chapter 1

Introduction

1

Chapter 2

Properties of the Different Motions α and β in Glass Former Materials

5

Chapter 3

Model of Aging: The VFT Relaxation Function

17

Chapter 4

Volume and Enthalpy Recovery

31

Chapter 5

Mechanical Properties

71

Chapter 6

Conclusion

95

Annexe A.

The Grüneisen Parameter

99

Annexe B.

The Multiple Glass Transitions

103

References

107

Index

111

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PREFACE The aging properties (volume, enthalpy and mechanical properties) of fragile glasses are reviewed and interpreted in the framework of the Vogel-Fulcher-Tamann-Hess (VFT) law which is a consequence of the coupling of the cooperative and individual motions, α and β. In the equilibrium melt state the relaxation times of these motions verify the relation τα ~ (τβ) 1/n where n ~ T-T0 is the Kohlrausch exponent (inverse of the number of individual motions participating to a cooperative motion) which extrapolates to ∝ at the Vogel temperature T0 . In the non equilibrium glassy state we show that the same relation applies but the Kohlrausch exponent n(T’) is now a function of the equivalent temperature T’, temperature of the equilibrium melt which would have the same specific volume. The relaxation time τα (T,T’) is then dependent on the aging time in the glassy state, on the heating and cooling rates, on the pressurization and depressurization rates. The differential VFT relaxation equations (VFTRE) giving the evolution of V and H of glasses under atmospheric and high pressures are given. The experimental non linear and non exponential relaxation curves during isothermal and non isothermal aging (cooling and heating) are explained by this generalized VFT equation. The glass properties (V, H) during aging and/or during temperature scans (or/and pressurization) depend only on the parameters of the liquid state (WLF parameter; C1, C2, the Vogel and merging temperatures T0 and T*) and the thermal expansion (or/and compressibility) coefficients. The solutions of the VFT-RE are compared to the KWW functions (stretched and compressed exponentials). The relation between the parameters (nk, τk) of the KWW function and the relaxation time τα (T,T’) of the VFT-RE is given. It is shown that the KWW function does not give good agreement with the experimental relaxation curves of Kovacs, and that the KWW parameters have no straightforward meaning. The memory effects (amplitude and memory time) observed by Kovacs, Struik and Adachi et al are explained by this model, without any adjustable parameters; relaxation of glass formers materials, after an up T-jump (and during heating), involves the two processes: β at small time and α at long time. The various glass temperatures (always defined arbitrarily) in these non equilibrium systems and deduced from the VFT-RE are compared to the Deborah glass temperature. We emphasize that the relation between the volume and enthalpy relaxations in these non equilibrium systems can be deduced from the Grüneisen relation, and that the effects of pressure can be deduced from the linear variation of the temperatures T0 and T* with P and from the “fan structure” of the isotherms and isobars. Finally the mechanical properties, modulus, yield stress, creep and stress relaxation, are interpreted in the framework of this generalized VFT model. The evolution of the mechanical properties can be fitted with the KWW function or the Struik-Andrade law. It is shown that the relaxation time τ and the

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viii

Preface

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Kohrausch exponent n describing these evolutions vary inversely with the aging time, temperature and stress. The observed correlation law, n log τ ~ Constant, observed below Tg by yield creep and stress relaxation is a consequence of the generalized VFT law.

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Chapter 1

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INTRODUCTION Most of materials (metallic, mineral, organic) which do not crystallize follow the VogelTamann-Fulcher-Hess (VFT) law : during cooling the viscosity increases, then at a certain temperature during the time te of experiment (of observation) the material has solid like properties; at low temperature the material is frozen. The temperature of transition between the two different “states”, in equilibrium and out of equilibrium is called Tg, the glass transition temperature. Since the pioneer woks of Tool[1,2] in 1946, Douglas[3] in 1947, Kovacs[4,5,6,7] between 1950 and 1963 and many others authors (see for example references in the review papers of Kovacs [7] and Hodge[8]), it is well recognized that Tg depends on the method of measurement : cooling and heating rates at constant pressure, application of the pressure at constant temperature, time of observation or frequency of measurement, aging time of the material at low temperature before heating, thermal history (memory effects) etc... In 1978 Struik[9] in his book gave a general description of the properties of glasses aged (annealed ) below Tg ; creep, modulus and specific volume vary in a similar manner with the aging time. Most of the authors in the pass (1960-1980) interpreted the glass transition in term of free volume, see references in the Kovacs review7 and in the book of Ferry[10]. Since the last two decades several authors pointed out that these models could not explain several experimental facts, in particular the effects of pressure. Also since 1970 a great number of authors have compared the glass transition to a second order transition (see references in the book of Haward[11] and in a recent review of McKenna[12]), in fact the Ehrenfest relations characteristic of this last transition are not verified, unless if order parameters (without any evident physical meaning) are introduced. During the same decades, following the work of Narayanaswamy[13], different authors have given phenomenological relaxation models, the references can be found and review papers [12,13] and books of Scherer[14] and Donth[15]. At present time there is no unified model to explain (without adjustable parameter) all the kinetic aspects of the glass transition. If we want to explain the variations of Tg with the various methods of measurement and experimental conditions we must relate this phenomenon to the properties of the different cooperative and elementary motions existing in the glass and in the liquid. A kinetic approach of the glass transition must be based principally on :

a) The VFT law which gives the dependence of the relaxation time of the cooperative α motions with the temperature (and pressure) in the liquid state (at equilibrium). b) The properties of the individual motions, generally called the β process. Physical Aging of Glasses: The VFT Approach : The VFT Approach, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

2

Jacques Rault

c) On the thermo dynamical parameters of the glass former material: the expansion coefficients ( α g and α l ) and heat capacities ( C g and C ) of the solid and liquid l “states”.

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The aim of this book is to show that the VFT law (verified above Tg) explains the variation of volume and enthalpy of glass former materials during cooling and heating, during aging at constant and varying pressure in the glass state, during pressurization at constant temperature. In others words we want to show that the kinetics of aging of glasses (volume, enthalpy and mechanical properties) are ruled by the viscoelasticity of the liquid measured above Tg and extrapolated below Tg. Most of the experimental works quoted in this book concern polymer glasses; the principal reason is that, because of the low values of their Tg, these glasses have been intensely studied. The various types of glasses (mineral, organic, metallic behave in similar manner), here we are mostly interested in fragile (VFT) glasses and not in strong (Arrhenius) glasses. The paper is organized as follow : In section 2 we recall the properties of the different motions appearing in melt and in glass former materials (crystallizing or not crystallizing). The effects of temperature and pressure on the cooperativity of the α motions are recalled. The Grüneisen relations between thermal expansion heat capacity and compressibility in glass and in melt are compared this will permits to give the relation between volume and enthalpy in glasses during aging. In section 3 the model of cooperative α motions above Tg is recalled[16]. We propose a modified VFT relaxation equation (VFT-RE) which describes the evolution of the volume and enthalpy of glass former materials below Tg. In these non equilibrium systems the relaxation time τα (T,T’) is dependent on the aging time ta, the equivalent temperature T’(ta) (different from the Tool temperature) is defined. The general properties (non linearity, incubation and final relaxation times) of the non analytic solutions of this VFT relaxation equation are described and compared to common analytic relaxation functions; distribution of exponentials, stretched and compressed exponentials. In section 4 we compare the predictions of the VFT-RE with the experimental results of Kovacs, Struik, and others authors on physical aging of glasses. We analyze the kinetics of aging (V and H) of glasses during isothermal and non isothermal thermal treatments; aging during the 4 elementary temperature jumps according to the procedure of Kovacs, aging during the two sequential temperature jumps leading to the memory effects, aging during cooling or heating at constant rate and finally aging of glasses under pressure (at constant and varying pressurization rate). The solutions of the VFT-RE involve only the characteristic parameters of the VFT law applied to the liquid state (or equivalently the constant C1 and C2 of the WLF relation). The forms of the experimental curves V(T) and H(T), non monotonic variations, width of the transition, variation of the various characteristic temperatures Tgi with the thermal and aging treatment are explained by this model. The Deborah criteria for estimate the onset of the Tg transition is discussed and compared to the VFT-RE predictions. In section 5 we explain the mechanical properties of polymer glasses in the framework of the generalized VFT law. The evolution of the properties (modulus, yield stress, stress relaxation modulus, creep at short and long time) as function of stress, aging time, glass formation can be described by the KWW-Andrade laws. It is show that the relaxation time τ and the Kohlrausch exponent n verify the generalized VFT law, n log τ ~ Constant, predicted

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Introduction

3

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by the model. The importance of the glass preparation on the mechanical properties and the parallel evolution of volume recovery and creep is pointed out. In section 6 we recall the principal conclusions and we emphasize that the glass transition can be interpreted as a pure kinetic transition. Prospective are suggested. In annexe A we show that the Grüneisen parameters of the melt and glass states of a series of glass former materials are of the same order of magnitude. One stress that the Grüneisen law linking the volume compressibility and heat capacity of a material can be deduced from the Keyes and Lawson experimental law (compensation law) linking the volume activation and the activation energy of the diffusion processes. In annexe B we give two examples of glass former materials (polymers) which present two different types of glass transitions.

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Chapter 2

PROPERTIES OF THE DIFFERENT MOTIONS α AND β IN GLASS FORMER MATERIALS 2.1. COOPERATIVITY OF THE β

MOTIONS , ORIGIN OF THE VFT LAW

By spectroscopic methods (mechanical , dielectric, NMR etc..) the relaxation times of collective (cooperative) and individual motions have been widely studied since half a century. [10,15, 17,18] The most important peak of internal friction (loss peak, called alpha peak) is due to cooperative motion involving a number nb of individual motions. In most glasses (fragile glasses according to the Angell [19] classification) the frequency of the α motions follows the VFT law. In these materials various peaks called β , γ, δ peaks of weaker intensity (I T > Tg. Dixon and Nagel [26] , by calorimetry spectroscopy on OTP , were the first authors who noted in 1988 that the Kohlrausch exponent has the scaling form n ~ 1/T . A great number of works, since that time, show that n~1/w decreases with T (see references in ref.16). We have shown that the temperature dependence of the KWW exponent can be put on the form:

n(T ) =

1 T * T − T0 = ⋅ nd T T * −T0

(1c)

n(T ) =

1 T − T0 = n d T * −T0

(1d)

At the extrapolated Vogel temperature T0 (often called the Kauzman temperature and noted TK) viscosity and relaxation time τα diverge. As shown by the author [16] these two relations are not very different and fit with the same accuracy the experimental results observed in the viscoelastic domain (T >Tg) of fragile glasses, polymers and minerals like KCN[27]. Recently Soldera et al[28] by simulation found that the Kohlrausch exponent of atactic PMMA verifies rel.1d. The number x of elementary β motions participating to a cooperative α motion is infinite at T0 and 1 at the merging temperature T*, therefore we postulate that x=1/n. The probability P(α)∼ 1/τα of such a motion is (P(β))x, P(β ) ∼ 1/τβ being the probability of a β motion. The relaxation time τβ obeys the Arhenius law :

τ β = τ 0* exp ( Eβ / RT) . Physical Aging of Glasses: The VFT Approach : The VFT Approach, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(2a)

8

Jacques Rault The frequency

τ 0* is of the order of 10-14 s-1 . The relaxation time τα has then the form :

⎛ τ β ⎞1 / n τα (T) = τ 0* ⎜ ⎟ ⎝τ 0 ⎠

(3a)

The relaxation time of the α motions is then :

τ α = τ *0 exp

Eβ n (T ) RT

(3b)

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The activation energy of the α motions is Eα = Εβ / n. Ngai[29] gave a similar analysis, we suggest that the term β transition (not necessary the first secondary transition below Tg, see further) should be restricted to the precursory (elementary) motions involved in the α transition. Ngai and Paluch [30] analysing the properties othe various secondary transitions, noted that the β transition (the Johari and Goldstein secondary transition) play an important role in the glass transition phenomenon. In figure 2 we schematize the variations with T of the volume and enthalpy of a glass and the corresponding relaxation time and Kohlrausch exponent of the material which would be in equilibrium (extrapolated liquid curve), between the Tg and the Vogel temperature TO. These extrapolated curves will permit to define the equivalent temperature, T’, of the glass (see futher). The aim of this paper is to show that rel.3 apply below Tg if the Kohlrausch exponent n=1/x, describing the cooperativity of the α motions, is correctly defined. Above Tg at equilibrium (Tg < T < T* ) rel.1 and 3 give the α relaxation time[16]:

log τα = log τ 0* +

Eβ T * −T0 2.3 R T (T − T0 )

log τ α = log τ *0 +

Eβ T * −T0 2.3 R T * (T − T0 )

(n ~ T )

;

;

(n ~ −1/T )

(4a)

(4b)

Rel.4b leads to the VFT and WLF laws :

τ α = τ *0 exp log

B (VFT) T − T0

τα T − Tg = − C1 ( WLF) τg T − Tg + C2

(5a )

(5b)

Rel.4a would lead a similar expression fitting the experimental results with the same accuracy. This relation would explain why in many glass formers the log τα(1/T) and log

τβ(1/T) curves become tangent at the merging temperature T* (Figure1). Physical Aging of Glasses: The VFT Approach : The VFT Approach, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Properties of the Different Motions α  and β  in Glass Former Materials

9

Figure 2. Variation of the volume, enthalpy, relaxation times log τ, Kohlrausch exponent n of a glass former material as function of T. The sample is cooled from the melt state. The system begins to freeze at Tg. At TA the glass A densifies after an incubation time τ0i given by the VFT law, applied to the equilibrium melt of same volume (point A0). At B after an aging time ta the relaxation time of the α motions is dependent on T and T’ the equivalent temperature, the Kohlrausch exponent of the α motions is n(T’). During aging the relaxation time τ and n are given by rel.9a.

In the following we discuss the aging properties near Tg (see below, the equivalent temperature T’ will be always near Tg), for simplicity reasons only rel.5b and its first order approximation:

C log τα / τ g ≈ − 1 (T − Tg) ; Tg-15 °C < T < Tg C2 Physical Aging of Glasses: The VFT Approach : The VFT Approach, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(5c)

10

Jacques Rault

will be often used; aging properties are mostly studied between Tg and Tg-15 °C. It must be noted that below Tg at the equilibrium (at the end of aging) these relations should apply. By calorimetry the author[31,32] has shown that the equilibrium time of isotropic and oriented polymer glasses (PS, PC, PVAc) and Triphenylethene follows the VFT law in the glass state between Tg and Tg-15°C, this time between 100 s and one month can be easily measured. There is no reason to believe that at lower temperature this law does not apply. The main question is : What is the relaxation time of the α motions below Tg during aging, the glass former material being out of equilibrium ? The most popular techniques to study the kinetics of physical aging are dilatometry and Differential Scanning Calorimetry (DSC). Before analyzing the volume and enthalpy data of various authors it is necessary to recall the relation between thermal expansion, capacity and compressibility of solids and liquids.

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2.2. RELATION BETWEEN VOLUME AND ENTHALPY: THE GRÜNEISEN PARAMETER We must note a great difference between the science of crystalline and amorphous solids; in crystalline solids the frequencies and energies of the lattice modes are well known, in amorphous solid there no discrete lattice modes and no theory predicts the properties of the different vibration modes (energy and distribution of the frequencies) as function of temperature, pressure, thermo-mechanical history and nature of the glass former material. Only a few authors have studied by dielectric spectroscopy the intensity the α and β dielectric loss in polymers (and in organic materials) containing dielectric moments (see references in the books of Donth[15] and Kremer and Schönhals[17]). If the different peaks α and β of the dielectric spectrum can be deconvoluted then the total intensity of each peak gives an estimate of the density of vibration modes (number of individual and collective motions per second). In figure 1 the dielectric strength Δεα and Δεβ of the α and β peaks are reported (maximum of the loss peak, not the integrated intensity), these two peaks vary in a opposite manner with the temperature. If the integrated intensities Iα and Iβ of the two deconvoluted peaks follows the same trends we conclude that the total intensity Iα+ Iβ varies weakly with temperature ; in liquid (melt at equilibrium) the two types of motions α and β are coupled. Thermal expansion in crystalline solids is associated with the anharmonic terms of the potential energy of the crystal. As suggested by Zimann[33], in default of adequate information about anharmonic terms… we may assume phenomenologically that the frequency of the lattice modes is a function of volume. Such a assumption should hold in solid amorphous materials. As reported by this author for simplicity one may suppose that any change of volume gives rise to the same relative change of frequency mode (discrete lattice mode in crystal and distribution of the α and β modes in glasses) :

Δν ν

= −γ

ΔV V

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(6a )

Properties of the Different Motions α  and β  in Glass Former Materials

11

γ is the Grüneisen constant. To simplify in glass former materials one can assume that only there are two type of motions (α and β), it is evident that others motions appearing at low temperature (γ and δ ) are not involved in the processes of aging and mechanical deformation near Tg. This relation applied to the α motion suggests that the relaxation time τα is dependent mostly on the volume and not on the temperature, for this reason in the following we introduce the concept of equivalent temperature. The total free energy F(V) of the crystal and amorphous solids is the sum of two terms: the potential energy ½( 1/κ) (ΔV/V)2 associated with the compressibility κ of the solid and the sum of the free energies Ē(T,V)=Σ Ēi(T,νι ) of all the lattice modes. Using the Bose Einstein mechanical statistic this second term can be calculated (for crystals) as function of νi and therefore as function of V , taking into account rel.6a . The condition of minimum energy is found by differentiating F(V) with respect to volume. The dilatation is then:

_ ΔV = γ κ E (T) V and the thermal expansion coefficient

(6b)

α , derivative of the dilatation with temperature, is then _

proportional to the heat capacity at constant volume CV = E (T) /dT . The difference in heat capacities at constant pressure, Cp , and volume, Cv , is relatively small for crystalline and amorphous polymer solids[34], 1 < Cp / Cv Tg) and out of equilibrium (glass T > Tg) are of the same order of magnitude, this tends to confirm that dilatation or contraction affects the different modes (α and β) of vibration in the same way (rel.6a). Assuming that the compressibility is constant, this is a coarse approximation, integration of rel.6c gives:

ΔV (T ) ΔV (t a ) = κl γl ; = κg γg ΔH(T ) ΔH(t a )

(6d)

The first relation tells us that V and H present similar variations with T as it is well verified, Figure2 . Physical Aging of Glasses: The VFT Approach : The VFT Approach, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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12

Jacques Rault

Figure 3. Volume V, Kohlrausch exponent n and relaxation time τ of a glass former material under pressure P0 and P. The material is supposed to be obtained either in the amorphous state or in the crystalline state; the melting temperature Tm and the merging temperature T* of the α and β relaxation curves coincide. The main effect of pressure is to shift differently the Vogel and the merging temperatures (T0 and T*).

The second relation (of rel.6d) tells us that if densification occurs during aging at constant temperature then the enthalpy changes in the same way. Kubat et al[36] have shown that this second relation can be used for calculating the apparent bulk modulus of glassy PS and PVAc during aging. It is to be noted that these relations explain why the discontinuity of the V(T) and H(T) curves appear at the same temperature Tg (if measured at the same cooling rate).

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Properties of the Different Motions α  and β  in Glass Former Materials

13

We stress that these rel.6 rest on several assumptions which have not been discussed up to know. In the following we will see that in fact that the compressibility of the solid and melt states vary with P and T. The main interest of the above relation is to permit the comparison of the effect of aging of glasses measured by dilatometry and calorimetry in a small domain of temperature and pressure.

2.3. EFFECT OF PRESSURE, RELATION BETWEEN ACTIVATION ENERGY AND ACTIVATION VOLUME Since 1957 a great number of works concerns the effect of pressure on the glass transition of mineral and organic materials. One will find in ref.[37,38] recent reviews on the dynamics of glass forming liquids under hydrostatic pressure. Floudas in its review[37] reports various works confirming that the dielectric strength of the α and β processes are interrelated via the conservation law Δε = Δεα + Δεβ ∼ constant. Increasing the pressure shifts weakly the

β process but more importantly the α process. At constant pressure when temperature increases the strength of the α process increases at the expense of the faster β process (as represented in Figure1). The effects of pressure on the volume enthalpy, relaxation time and Kohlrausch exponent are schematized in Figure3. Pressure has weak effect on the β motions. Accurate spectroscopic measurements have shown that the relaxation time τ β can be put on the form:

[

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τ β = τ *0 exp (Eβ + v β* P) /RT The activation volume

v β*

]

(7a)

is independent of T, typically its values for polymers[39] is

20 to 50 cm3/mol. The relaxation time τ α is then:

τ α (T ,P)

= τ *0

E β* + v β* P Eα + v α* P * exp = τ 0 exp n(T, P ) RT RT

(7b)

The Kohlraush exponent is still given by rel.1 but now the characteristic temperatures T0 and T* are function of P. The activation volume and energy of the β and α motions in pure polymers and in polymers containing spin probes have been measured as function of temperature and pressure (see references in the book of Kovarsky[39]). At Tg for several glass former materials the relation :

Vα* (Tg ) = Vβ* / n g ;

Eα (Tg) = E β / n g ; ng = n(Tg,P)

(7c)

has been verified at atmospheric pressure, ng been between 0.2 and 0.3 at the arbitrarily temperature Tg measured at frequency 10-2 Hz. This suggests that the activation parameters are temperature dependent via the Kohlrausch exponent n(T) : Physical Aging of Glasses: The VFT Approach : The VFT Approach, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

14

Jacques Rault

V * (T ) = V * / n(T ) ; Eα (T) = E β / n (T ) α β

(7d)

By dielectric spectroscopy on Polychloroethylene (containing ESR probes) Naoki et al[40] found that the Eα and V * follow the same temperature variations as it is indicated in α the above relation. When the temperature decreases from T* to Tg the activation parameters * (Tg) and V * (Tg) . Rel.7c,d confirm again that the α and β increase from Eβ and V * to Eα α β processes are interrelated as it is expressed in rel.7b.

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Generally V * / E is of the order of 10-10 Pa-1, therefore in the domain of applied β β pressure, 0 to 108 Pa (103 atmospheres), the term ( V * / E ) P is negligible. From rel.7b we β β draw the important conclusion: For different values of the couple T and P which give the same relaxation time, the values of the Kohlrausch n(T,P) and therefore of the width W of the dispersion curve log τ(ν) are the same. Ngai et al [41] reported recently that the dielectric loss peaks of several glass formers are superimposable for different values of T and P if the relaxation deduced from the peak are the same. This important property confirms that the Kohlrausch exponent in the modified VFT law (rel.7b) is related to the width of the dispersion curve (rel.lb). In crystallizable glasses T* is very near the melting temperature Tm, also it has been noted that the pressure dependence are similar, dTm/dP = dT*/dP ∼ 20 °C per kilo Atmosphere (0.2 K/MPa) in most polymer glasses (see reference in ref.15-18,38-40). In figure 3 we give a schematic view of the variation of V (or H), n and log τ versus temperature. It must be noted that the V(T) curves for different pressures are divergent this indicates that the thermal expansion coefficient α l of the melt is a function of P . This dependence is generally described by the Tait relation[11,37,38]. For pressure below 1400 bar, we will see that the dependence of the specific volume with P and T (of polymers) can be described by a simpler relation, based on the “fan shapes” of the isobars and isotherms. In the figure we have represented the V(T) curves of the supercooled melt and glass in bold lines and of those of the crystalline material in dashed lines (for two different pressures). It must be noted two important properties which permit to deduce (geometrically) the glass temperature of a crystallizable glass under pressure. If the solidus (crystal) and liquidus lines are known at atmospheric pressure and if Tg of the supercooled melt is known at atmospheric pressure (point G) then at any arbitrarily cooling rate:

a) the intersection of the crystalline lines ab and a’b’ with the melt lines ac and a’c’, (curves Vβ(T) and Vα(T) in dashed lines), gives the Vogel temperature T0, where the relaxation times diverge.

b) For the samples which have not crystallized, the glass lines Vβ(T) passing by the transition point G defining the glass transition temperature, are parallel to the crystal lines ab. Physical Aging of Glasses: The VFT Approach : The VFT Approach, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Properties of the Different Motions α  and β  in Glass Former Materials

15

c) the compressibility coefficient of the glass and crystalline samples are not very different (κg = κcrst) , of the order of κ l /2 for many glass formers.

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From these remarks one concludes that dTg/dP is greater than dT0/dP but lower than dT*/dP . Knowing Tm(P) (or T*(P) and T0(P), the VFT-WLF relation (rel.4b or 5b) permits to calculate the glass temperature Tg (t ~1/ν) at any time or frequency. The figure 3 shows that the Kohlrausch exponent and the volume at Tg is not constant when the pressure is changed. We will use these properties to estimate of the variations of Tg with pressure.

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Chapter 3

MODEL OF AGING: THE VFT RELAXATION FUNCTION

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3.1. THE EQUIVALENT TEMPERATURE In figure 2 we compare the variations of frequencies of the α and β motions and specific volume (and enthalpy) of a glass former materials as function of temperature. The samples have been quenched from the melt state to the glass state at temperature T (point A). Between Tg and T the contraction of the sample, due to the fast β motions, occurs in a time tF the Fourier time to reach the thermal equilibrium (tF = e2/a , e sample thickness, a thermal conductivity). At point A densification and change of the physical properties occur during the aging time (line ABBm). During aging at constant T the relaxation time of the α motions is given by rel.3b, indicating that the α and β motions are coupled, we used this relation but we note that the Kohlrausch exponent is no longer n(T) but n(T’). We call T’ the equivalent temperature of the glass. At point A the structure of the glass, just formed, is represented by point A0 , The volume VA is the volume of the melt in equilibrium at temperature T’. The equivalent temperature is different from the Tool temperature, corresponding to the intersection of the linear V(T) curves (at constant ageing time) of the melt and glass states. We call τ0i the VFT relaxation time corresponding to Ti' . For the times t < τ0i = 1/ v0 there is no aging due to the α motions. For aging time t > τ0i at constant T (point B) densification due to the α motions occurs; at point B the equivalent temperature is T’, the Kohlrausch exponent, n(T’) decreases continuously from n( Ti' ) at t = τ0i to n(T) at tm = τm . The characteristic times τ0i , (Figure2) is called the incubation time and τm the final time or the equilibrium VFT time. In the aging time domain time τ0i < t < τm the relation between T and the equivalent temperature T’ is:

y (t) [V (t,T )−V∝ (T )] ; y(t)= T' = T + V∝ (T ) αl

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Jacques Rault

y(t) is the relative difference of volumes at time τm > t > τ0i. V∝ (T )

is the equilibrium

volume obtained for long aging time, t > τm , deduced from the thermal expansion coefficient of the liquid measured above Tg. As the variations of volume are always small ΔV/V~10-3, V∝ (T ) is considered as a constant. The frequency of the α motions during aging is then :

τα = τ *0 exp

Eβ T ' − T0 ; n(T' ) = n(T' ) RT T * −T0

(9a)

In the following for more clarity and simplicity we will use often the modified VFT law :

(T − Tg + y / α l ) (T' − Tg) τ (T ,T' ) = − C1 = −C1 (9b) log α (T' − Tg + C2 ) τg (T − Tg + C2 + y / α l ) Rel.9 will be called the generalized VFT law. In most of the experimental results analyzed hereafter the aging temperature (and therefore the equivalent temperature) is not very far from Tg (measured by DSC at 10°C/min), Tg > T > Tg+15°C. Generally in the aging experiments T’ - Tg 0 σ0 C2 αl

(47b)

If a stress σ is applied, the β and then the α motions are activated, we have seen that the WLF constant C1 is proportional to the activation energy, Εβ , of the β motions. For more clarity in the last relation we have assumed that the effect of the hydrostatic pressure is negligible and that the activation volume v does not depend on T and σ :

E β (σ ) = E β − σ v = E β (1 −

σ σ ) ; C1 (σ ) = C1 (1 − ) σ0 σ0

(48)

σ0= Eβ/v is proportional the cohesive energy[34],[84]. In the above relations we have assumed, for more clarity, that the hydrostatic component of the stress tensor has no influence on the mobility of the individual and collective motions, this is not true for compression and tension tests (see references in ref.84). At the yield stress the Deborah criteria gives :

σy C αg log t a + log ε' = 1 (1− ) (Ta − T ) C2 α l σ0

C αg log t a + log ε' = 1 (Ta − T) for σ y σ 0 < 1 C2 α l

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when the temperature is changed from T1 to T2, the σ (log ta) curves can be put into coincidence by a translation log a parallel to the time axis (Figure23c):

C αg log a = log t a1 / t a2 ≈ 1 (T1 − T2 ) C2 α l

(49b)

The constants kij of rel.44 are then :

kε ' ta = kTta

dσ y C α = σ 0 2 l (log t a + log ε' ) ; dT C1 α g

dσ y dσ y σ 0 C2 α l =k = = = ' Tε' d log t a d log ε Ta − T C1 α g

(49c)

In a similar way one can calculate the yield stress for the thermal treatment of figure 23b. From these relations we conclude that :

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a) The characteristic times ta and 1/ε’ have exactly the same influence on σy. This property has been verified by Chow [85]. In Figure23c we report schematically the data of this author, the yield stress of PS at Tg-20 °C as function of log ε’ for three aging times. σy is constant (dotted line) if ta and 1/ε’ increase of the same factor (in the figure these parameters increase of 2 and 4 orders of magnitude). b) The slope kε' ta is independent on the test temperature (below Tg-5 °C). This property has been observed by several authors, see for example Figure2 of ref.85 or Figure6 of the paper of Aboulfaraj et al [86] c) The slope kTta decreases with T. Bowden reported this property; see Figure7 of Chap. 5 of the Haward book [11].

Final Relaxation Time Chow found that the “effect of physical aging on the yield stress (and modulus) diminish after a critical aging time, which is function of the annealing temperature”. Similar conclusion was given by Aboulfaraj et al. In Figure23d the VFT relaxation time at Tg-5 °C is given by the VFT law (linear approximation) : log τ/τg=(C1/C2)(Tg-T) =1.1 h (C1=15, C2=50 °C) , this is the value tm1 reported by these authors in Figure6 of their reference. With the same approximation we verified that at Tg-10 °C the characteristic time tm2 is of the order 100 h. Near Tg the VFT law log tm1/ tm2 ~ (C1/C2)( T1-T2) is observed (linear approximation). Such agreement between the final and VFT times can be verified in other published works. In conclusion when the temperature is changed the σy(log ta) curves can be put in coincidence by a translation log a parallel to the time axis (Figure23c). This translation is α g α l ~1/3 smaller than the translation predicted by the VFT law (rel.48b). This property was observed by Struik and latter on by Aboulfarad et al. [86].

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5.2. CREEP Creep properties at short and long times have a great importance for industrial applications. In his book Struik [9] has postulated that the creep compliance of glasses can be written in the following form : ⎛ t ⎞n J = J g exp ⎜ ⎟ (50a)

⎝τ ⎠

Jg been the creep compliance of the glass at short time (or at low temperature ). Many authors have used this relation for describing the short and long time creep compliance of polymers (crosslinked or not) and composites (see for examples ref. [87,88,89]). This relation, called hereafter the Struik law, similar to that used for metal (Andrade law with n ~ 0.3) is in fact not relevant here for polymer and composite; entanglements and cross-links impede continuous flow. At long times, J given by the above relation diverges, experimentally this is not observed. To avoid this divergence the authors have postulated that the time t must be replaced by a effective time λ, first introduced by Morland and Lee [90] in 1960. Then the effective time theory has been developed by Struik and by other authors (see for example references in the paper of Brinson et al [87]). This phenomenological theory will not be discussed here. It is important to remark that the compliance of the glasses (of polymers) must tend to the rubber compliance JR at long times t > τ. The glass compliance for polymers (cross-linked or not) must be put on the form:

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⎡ ⎛ t ⎞n ⎤ J = J g + ΔJ ⎢1 − exp − ⎜ ⎟ ⎥ ; ΔJ = (J R − J g ) ⎝ τ ⎠ ⎥⎦ ⎢⎣

(50b)

Only a few authors have proposed this form (the KWW function) to fit the creep properties of polymers glasses [85,91,92,93,94,95] and the relaxation of the magnetic susceptibility of metallic glasses (see for example references in the paper of Rivas et al [96]). A comparison of the fit of these two relaxation functions with experimental data on PMMA and PVC is given in the review paper of Read [93]. In most of the published works, creep is studied in the domain of time t < ta . At short times these two relations are equivalent, the reported values of the experimental compliance varies from Jg to 3 Jg for polymers and composites and from Jg to 10 Jg for amorphous materials such as bitumen, shellac, sugar and cheese (see Figure10 to 13 of the Struik book9), the curves J(log t) for t > ta and for t > ta are called respectively the momentary (or short term) creep curves and the long term creep curves. Typical for polymers Jg = 10-9 Pa-1 and JR = 10--6 Pa-1 the two relaxation times τ(stuik) and τ(KWW) in rel.50 verify the relation : n log (τ(Struik) / τ(KWW)) = n log (τ1 / τ2) ~ 3

(50c)

In the following we call τ1 and τ2 the relaxation times deduced from the experimental fit with relation 50a,b. This relation is obviously verified experimentally at short times. The Struik and KWW relations 50a,b describe respectively the short and long term compliances (see for example Figure5 of ref.91).

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Figure24a gives the form of the compliance of a polymer near Tg , in a large domain of time (or frequency), the whole curve have been built using the so-called time temperature superposition principle (see for example the Ferry book10). Near Tg the sigmoidal creep curve extends over 3 decades of time, the domain of the α transition. The β transition extends in a time domain of 1.18 decade, we have assumed that the β motions are elementary motions (Debye process).

Figure 24. Creep of glasses.a) Creep compliance, J, of a polymer glass as function of the time. The material has been aged below Tg during a time ta and 10 ta. The curves J(t) (β process) are superimposable by the horizontal translation log a (parallel to the log t axis) and a small vertical translation b (not shown).The whole curves in the α region are obtained near Tg or by applying the principle of superposition. At low temperature the β transition is observed.(schematic after Read91) b) Short term compliance t ta > t m

(51)

where t0 is the incubation time (τ0 i in rel.13) and kV the constant given in rel.20a, of the order of 10-3, is calculated via the VFT relaxation equation. We recall that this approximation of the VFT relaxation equation (rel.15a) is valid for aging time greater than the incubation time t0 but smaller than the final (VFT) time tm. Typically the relative variation of Jg is 2 10-2 per decade of aging time [9,87] (see Figure24 of the Struik book). Variations, 1 10-2 per decade of

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81

time, have been found by Read et al for PMMA, PVC, PC and PBT, see references in ref.93. This is one order of magnitude greater than the value predicted by the above relation. The vertical shift factor depends also on the temperature; for example changing the temperature of a copolymer (case of SAN, fig 31 of the Struik book) from 90 to 40 °C, at constant aging time, produces a relative variation of Jg of 0.07. This value must be compared to the relative volume variation 0.015 due to the thermal expansion (in polymer glasses, αg ∼ 2 to 3.5 10-4 cm3/g K). Noting the accuracy of the fit and the fact that the values of the compliance cannot be obtained for very short times, t 0. For low stress and in a domain of temperature, depending on the nature of the polymer glass, the shift rate is equal to 1. Struik has clearly demonstrated that the origin of the variation of log a with the aging time is due to the densification of the material. We schematize in Figure25a,b the variations of volume (just before the creep test) and creep shift factor of PS and PVC with the aging time ta (see Figure82 and 86 of ref.14). log a is the shift of a creep curve relative to the one measured at 25 minutes after the simple quench from above Tg to the temperature Tfinal. The Struik procedure for analyzing the memory effects in these polymers, aged at different times and different temperatures T1, T2 .. and then tested at Tfinal, has been recalled in section 4.1.3 (see Figure14). In Figure 25c report the schematic aspect of the variation of log a with V deduced from the schematic Struik curves in Figurea,b. This author gave these two important conclusions:

a) V and log a present same type of non monotonic variations with ta. Maximum (peak) values are obtained for the same aging time tmax. Same tmax values are obtained from the enthalpy recovery curves (Adachi et al56 , see fig 14), and from enthalpy relaxation [31]

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b) The curve log a versus V is linear (monotone), the inverse of the slope is: ka = dV / d

log a ~ 1 10-3 cm3/g per decade of time, independent on the exact aging and thermal history of the sample. Using rel.52a, we obtain the relation:

ka =

dV dV 1 = = V kV ≈ kV d log a μ d log t a μ

(53)

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the constant kV ~ C2 α l / C1 is predicted by the VFT relaxation equation and by the VFT approximation (rel.12b and 20a).

Figure 25. a,b) Memory effect in glasses. Variations of the volume V and creep shift factor log a with the logarithm of the time at the temperature Tfinal. log a is measured arbitrarily by shifting the creep curve with a reference curve of a sample quenched from above Tg to Tfinal. The glasses have been aged at different temperatures below Tfinal, during the same aging time ta (see Figure14). Experiments of Struik, see Figure82, 85 of ref.9 on the memory effects. c) Creep shift factor log a as function of the volume V at constant temperature deduced from Figurea,b. The samples have been submitted to different aging and thermal treatments (after Struik, see Figure85 of ref.9). The shift factor does not depend on the thermal history of the sample but only on the volume of the sample just before the creep test. d) Relaxation time τ1 of PVC and SAN as function of the supercooling Tg-T. The aging time at T is constant. τ1 is deduced from the fit of the data of Struik on PVC and SAN (Figure30,31 of ref.9) with rel.49a.

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The identity ka = kV ~ 1 to 1.8 10-3 cm3/g per decade of time (for small stress, μ =1) confirms the conclusion of Struik. The creep properties, at a fixed temperature, low stress and for t0 < t < tm, depend only on the specific volume measured just before the creep test. The similarity between the J and V curves versus log t would suggest that the incubation time t0 (of V or H) and the relaxation time τ1 (of J) are equal. The main question is : why these two times are not stricto-sensus equal, although varying in the same way ?

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Origin of the Time T* The origin of rel.52b has never been discussed in the literature. In our model the constant, log t*, is dependent on the initial state of the glass before aging; in other words t* is an equivalent aging time or incubation time due to the preparation of the glass before the isothermal aging process and the creep measurements. The effect of cooling rate on the relaxation curves V(log t) and on the incubation times are schematized in Figure26. The starting glass formers Q and SC has been quenched (Q) or slow cooled (SC) from the melt to T. The starting volumes Vstart of the samples can be calculated via the VFT relaxation equation if the exact thermal history is known. Then from the equivalent temperatures T1’ and T2’, one calculates the incubation times t01(Q) and t02(SC). Typically from Figure18 (case of PS) we note that a change of the cooling rate q of two orders of magnitude (20 to 0.2 °C/s) produces a relative change of volume ΔVstart =2.8 10-3. At Tg-T =10°C the change of equivalent temperature T’ between the two glasses before aging is of the order of 9 °C then there is about 3 orders of magnitude between the incubation times of the two samples (linear approximation of the VFT law gives t01=2 102 s and t02=2 105 s case of PS , see Figure18). In general creep measurement implies two steps

a) The sample quenched from above Tg to T1 is aged during a time ta at T1, the variation

of volume during aging is ΔVstart = kV log ta / t01 for t01< tTg the dielectric relaxation function verifies the KWW relation. The relaxation time is given by the VFT time and the Kohlrausch exponent by the linear relation n~ T-T0 (rel.1d or 1e). Below Tg if the aging time is equal or greater than tm (VFT time) then the glass must be considered as a liquid at equilibrium and the same relation applied, only τ= tm and n have changed. For short and long times the creep of a glass aged for a long time ta > tm is then given by the KWW relation. For a glass just formed at temperature T the creep compliance would be given by the KWW function but now the parameters τ and n are dependent on the aging time (and obviously on σ). For simplicity reasons we assume that at low stress the compliance is given by a distribution of KWW functions:

⎡ ⎛ t ⎞n j ⎤ ⎢ J = Jg + 1− exp − ⎜ ⎟ ⎥ ⎢ N ⎝ τ j ⎠ ⎥⎦ j=1 ⎣

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ΔJ

J =N

∑

(58a)

At equilibrium (melt or solid, T’=T, t0= tm, ni=n(T) ), the KWW rel.1a is obtained. In the following we assume that the change of Jg with aging and test times is negligible. At short time the parameters τ1 and n1 are deduced from the initial state, at long time τN and nN are deduced from the final state at equilibrium. We apply this relation to a polymer quenched to different temperatures below Tg, supercooling ΔT from 3 to 50 °C. We assume that the ratio of the expansion coefficients is α g / α l = 0.5 therefore the equivalent T’ is known. We assume that the WLF parameters are C1=15 and C2=50 °C, then the incubation and final relaxation times,τ1 and τN, are calculated via the VFT-WLF relation. During creep the volume is approximated by a linear function of log t (rel.20, Figure 10a), then the coupled parameters τj and nj verify the relations :

τi = τ1 +

i (τ N − τ 1 N

)

;

ni = n1 +

i (n N − n1 N

)

(58b)

The initial and final values τ1 τN and n1 nN are given in the following table for the different supercooling. Tg-T (°C) 3,00 6,00 12,0 18,0 24,0 30,0 50,0

Tg-T' (°C) 1,50 3,00 6,00 9,00 12,0 15,0 25,0

log τi/τg 0,46 0,96 2,0 3,3 4,7 6,4 15

log τm/τg 0,96 2,0 4,7 8,4 14 22 infinite

ni 0,320 0,310 0,290 0,270 0,250 0,230 0,163

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nm 0,310 0,290 0,250 0,210 0,170 0,130 0,00

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In figure 29 we report the calculated creep curves with N=10, a larger number will not change the shape of the curves. The main interest of this phenomenological model is that at large supercooling the creep curves cannot be described by a single KWW function. For high stress we must take into account the dependence of the relaxation time with σ via rel.55. We think that the above crude model can explain the observations of Struik. In Figure29a we define, for the compliance curve at Tg-24°C, the relaxation times t0 and tm,. These times are reported in Figure29b as function of the supercooling, we verify the relations t0 ~τi and tm~τm. Same relations were found from the VFT RF curves and the KWW function giving the volume after a down T-jump (Figure10). In PVC at a supercooling, ΔT=30 and 60 °C, the creep curve can be approximated by a logarithm law J ~ log t over a time domain of 3 decades (see Figure103 and 121 of ref.9), similar linear creep curves have been observed by Gates et al (fig9 of ref.[103]) on composites in a domain of 4 decades of time. In the above model we note that, for supercooling higher than ΔT=18 °C, the compliance varies linearly with log t in large domain of time (more than 4 decades).

Figure 29. Long term creep of a polymer glass, calculated from rel.58 (distribution of KWW function). The WLF constants and the expansion coefficients are given in the text, the samples have been cooled rapidly from above Tg to T. a) Compliance versus the logarithm of the test time for different supercooling ΔT= Tg-T. b) Logarithm of the apparent incubation t0 and final times tm deduced from the creep curves as function of the supercooling.

We recall that in case of down T-jump experiments, the volume recovery can be approximated by a series of exponentials (a distribution of relaxation). The fit is good for long times and not for short times (see Figure10a). We arrive at the same conclusion for the creep curves.

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5.3. STRESS RELAXATION In non equilibrium glasses the structure (volume) is continuously changing then the relaxation time is continually evolving. Stress relaxation (and creep) cannot be described by a simple KWW function (with a single relaxation time). As for creep the stress relaxation is generally measured in a small domain of time; the relaxation modulus during test time varies generally from Eg to 0.5 Eg. The modulus of the melt (relaxed modulus), ER ~10-3 Eg, is never approached, excepted when the test temperature is near or above Tg and the time long enough (about the final VFT time tm). Also it must be recall that glassy polymer are non linear materials; the response to a given input (stress or strain) depends on the duration and the magnitude of the previously applied load or strain. References can be found in the paper of Yee et al [104].

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The “Classical” Stress Relaxation Method To study the non linear effects, an initial strain in the form of a ramp is applied and then held constant (Figure30). The modulus is measured and from the fit with the KWW function one deduces the relaxation time τ and the Kohlrausch exponent n. We give here an example of stress relaxation of a polymer glass. In the experiments of Yee et al the samples, PC, have been prepared by slow cooling from melt to Tstart =125 °C. Then the samples of same specific volume are cooled to different temperatures below Tstart and the stress modulus is measured after an initial deformation, ε=0.1% . From the stress relaxation curve these authors deduce the relaxation time and the Kohlrausch exponent n (1-n, according there definition).

a) Kohlraush exponent: n varies linearly with T (see Figure7 of ref.[104]), the slope is ∂n ∂T = 4 10−3 K-1. In the generalized VFT model the variation of temperature dT produced a variation of the equivalent temperature dT ′ = (α g α l ) dT , then rel.1d gives:

αg dn ∂ T ′ − T0 1 = ( )= ≈ 3 10 −3 K −1 α l T * −T0 dT dT T * −T0

(59)

b) In PC67 α g / α l = 0.5 and T*-T0 ~ 150 °C. There is a good agreement between the experimental variations of n with T and those predicted by the Generalized VFT model. c) Relaxation time τ :In a large domain of temperature log τ varies linearly with T. As noted by these authors the extrapolated values for Tg = 145 °C is about the characteristic time at Tg , tg=100 s . In our model the characteristic time τ=104 s, measured at the initial temperature Tstart =125 °C, is the incubation time t* of the initial material and depends on the thermal history of the sample (slow cooling). The relaxation time, called induction time in our model, verifies the generalized VFT law, which states that n and τ vary inversely. As expected the product n log τ plotted in Figure30b as function of T verifies rel.57b.

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Figure 30. Stress relaxation as function of the initial strain of a polymer glass: a) The tickle method of Yee et al [104] b,c) Kohlrausch exponent n multiplied by the logarithm of relaxation time as function of temperature at constant strain (b) and as function of strain at constant temperature (c). The parameter n and logt are deduced from the whole relaxation curve (b) and by the tickle relaxation curve (c) results of Yee et al taken from Figure7-8 (b) and from Figure15-16 (c) of ref.104.

The “Tickle” Stress Relaxation Method To study the relaxation spectrum at various strain amplitudes a perturbation technique was developed by these authors, see Figure30a. A “tickle”, a strain of amplitude εt=0.1%, was superposed to the first strain ε at a time tt =105 , at fixed temperature T=35 °C . After this time the stress relaxation curve due to the first strain levels of, the stress relaxation curve due to the “tickle” is then measured as function of the initial strain ε (from 0.1 to 5 %). From these relaxation curves the relaxation time τt and the Kohlrausch exponent nt are deduced. The important conclusions of these authors are:

a) Kohlrausch exponent: nt increases linearly from 0.1 (ε=0.1%) to 0.35 (ε=5%) and

 6 , the yield strain. The limit value, nt (εy), is about the extrapolates to 0.4 for ε∼ε y= value of the Kohlrausch exponent at Tg. To explain these coincidence it is necessary to relate these measurements of nt and Tg to the times of measurement; a Tg value or a yield strain value without specifying the time of measurement is meaningless.

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b) Relaxation time: τt decreases with the initial strain. The extrapolated value at εy= 6%

is τt (εy) =104 s and not 102 s the characteristic time to measure (or to define) the “classical “ Tg.

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In Figure30c we have plotted the product nt log τt as function of the initial stain. For low values of ε the accuracy on the parameters nt and τt are obviously bad, then we conclude that nt log τt is constant as predicted by the correlation law rel.57b. The opposite variations of nt and τt with the strain are very similar to the variations of n and τ with the temperature, this is obvious : plastic flow (more exactly deformation) is accelerated by increasing temperature and/or stress. The constant nt log τt = n log τ is a characteristic of the glass former material depending on T and T’ and on Eβ , T* , T0 (C1, C2). The equivalent temperature T’ depends on T , the thermal expansion coefficients and on the previous thermal history of the glass as we have discussed thoroughly in the previous sections. It is important to remark the similarity of this stress relaxation experiment at different strains (therefore under different initial stress) with short time creep experiments under different stress reported in Figure28.

Remark on Modulus at Fixed Strain In general the modulus is measured at fixed strain (and fixed time), but the state of the glass depend on the strain (stress) and the time of application. A large strain (compression or torsion) produces a softening of the material, a partial erasure of the physical aging. This has been studied by various authors, see for example reference [105]. If the strain is very low, for example not exceeding 10-5, then the modulus is independent of the strain (during the time experiment). If the glass former materials are aged at a temperature near Tg and then quenched to low temperature then the glasses do not evolve during the time of experiment. Long experiments can be done on these “stable” materials. Etienne et al[106] have measured by mechanical spectroscopy the modulus of such materials as function of frequency and aging time (PC annealed at 403°K during an aging time ta and quenched to room temperature). They fitted the experimental results with the KWW function. The relaxation time and the Kohlrausch exponent of these glasses as function of the log ta are given in Figure4 and 5 of there reference. The exponent n decreases linearly with log ta the slope dn / d log ta is about 0.02 per decade, similar values are obtained by short time creep measurements rel.57c. The relaxation time verifies rel.52b with t*=103 s. The product n log τ is found constant of the order of 0.6 ± 0.05 (τ in s), as expected from rel.57b. In conclusion from the fit of modulus, stress relaxation and short term creep with the WKK and Struik relations one deduces a relaxation (incubation) time τta,σ and an KWW exponent n(ta,σ) which vary in an opposite manner with the incubation time t* of the initial state, the aging time ta, the stress σand with temperature. In these three types of experiments on polymer glasses the observed correlation law, n(ta,σ) log τta,σ ~ constant, is a consequence of the generalized VFT law.

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Chapter 6

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CONCLUSION The VFT law is one of the most general law observed in the nature. It is observed in all non crystalline molten materials (metallic, mineral, organic, polymer) in an extremely wide frequency domain (12 to 16 decades) and in a wide domain of temperature of the liquid state. We have extended this law to glass materials. Above Tg the relaxation time of the α motions is dependent on the β motions via the Kohlrausch exponent n (the inverse of the number of individual β motions implied in a cooperative motion). n varies linearly with the volume therefore with the temperature. Below Tg we have assumed that n has the same volume dependence and therefore a different temperature dependence. This permits to define the equivalent temperature T’ of the glass which depends on T, Δα and aging time. The relaxation time τα of the α motions is then dependent on T and T’, and therefore is time dependent. The evolution of T’, n and τ during aging is schematized in Figure2,3. The fact that τα depend on T’ on V) is suggested by the Grüneinsen relation. In non equilibrium systems (T < Tg), the generalized VFT law (rel.9a) explains: the kinetics of aging during up and down T-jump and during complex thermal treatment (memory effects). The (blurred) concepts of free volume, configurational heat capacity and entropy, dynamical and spacial heterogeneities, energy landscape, phase transition with one or several order parameters, etc... have been ignored. The glass is a liquid which evolves towards equilibrium and the glass transition is a kinetic transition; its arbitrarily definition (onset of aging, onset of the transition, end of the transition, maximum in the derivative of volume and enthalpy etc.. ) depends on the time measurement (and on previous history of the material). The generalized differential VFT-relaxation equation (rel.15) explains, without adjustable parameters, the various experimental forms (monotone and non monotone) of the relaxation curves (V and H) during isothermal and non isothermal aging. The macroscopic properties, V(T) (or H(T)), of the glass aged below Tg are described by the physical parameters of the liquid (the WLF constants C1, C2 and the characteristic temperatures T0 and T* deduced from viscoelasticity measurements) and the thermal expansion coefficients (and heat capacities) of the solid and liquid. The parallel variations of V and H with T and with the aging time is a consequence of the Grüneisen relation. We have given the correspondence between the calculated solutions of the VFT-RE and the stressed (and compressed) exponentials. We emphasize that the VFT-RE solutions fit the experimental results in a better way than the stressed exponentials. The kinetics approach given here explains very simply the non

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exponential and non linear behavior of the relaxation of glasses, via the non linearity parameter b = C1/C2 α l . The characteristic incubation and final times (t0 , tm) observed only by a few authors in simple T-jumps are explained by the VFT relation. During contraction only the α process is observed, during expansion both α and β processes are observed. In a two T-jumps experiment the memory time tmax is given by rel.21c and 23, this confirms the importance of the β process during structural relaxation. We have given the VFT relation for materials under pressure. The empirical relations of Flytas and Johari et al are deduced from this relation. The main effect of pressure is to increase the characteristic temperatures T0 and T* and the activation energy of the β motions (rel.36). The observed differences between the Constant Temperature Formation and Constant Pressure Formation of glasses have been explained in terms of aging (rel.41,42). In this paper we have introduced the concept of Deborah glass temperatures during cooling and heating, during pressurization and depressurization. These temperatures (rel.42) give an estimate of the onset of the glass transition (calculated by solving the VFT-RE). Finally we have analyzed the mechanical properties of polymer glasses in the framework of this generalized VFT model. This model explains the variations of the yield stress with strain rate, temperature and aging time (rel.44-49). The parallel variations of volume and creep compliance observed by Struik has been recalled and extended (rel.53,56). We conclude that in aged glasses the incubation times deduced from volume recovery (and enthalpy) and from the short term creep (at low stress) are equal (Figure26). The horizontal shift factor, log a, its dependence on aging time, stress and temperature are deduced from the generalized VFT law (rel.55). We have emphasized the importance of the role of the glass formation on the initial incubation time t* before aging and creep measurement (rel.52b,54b). Creep, modulus and stress relaxation can be described at short time by the ansatz KWW function, the inverse variations of the KWW parameters n and log τ with T, V, ta and σ are explained by the generalized VFT law. We think that in the light of the macroscopic properties described and interpreted here we should answer in the future to several questions, some are given hereafter:

a) As the liquid and amorphous solid obey to the same generalized VFT relation, what is the physical origin of the characteristic temperatures T* and T0, of there dependence on P ? From the intra and inter molecular potentials of a glass former is it possible to predict these temperatures ? b) During aging, in particular during up T-jump (expansion), why the relaxation time tmax due to the β process (same activation energy) is several order greater than τβ ? whereas the β process during down T-jump (contraction) is faster or equal to the time to reach the thermal equilibrium. c) Does the microstructure, size of the fluctuation of density (heterogeneities) dependent on the exact thermal treatment and not only on the temperature? Are these heterogeneities different in the solid and in the liquid of same specific volume? What is the real process of contraction and expansion, by migration of interstitials and vacancies, by nucleation and propagation of domains? d) We have seen in Annex B that the aging processes of atactic polymer like PMMA present some difference with the glass formers (PS, PC, PVAc, B2O3 ), We think that

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Conclusion

e)

f) g)

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h)

i)

j)

k)

l)

97

the aging process depends on the exact configuration of the chains (on the local tacticity of the chains). Are theses materials more heterogeneous than the others materials? In pure isotactic and syndiotactic PMMA (obtained in the glassy state) are the aging processes different from that observed in atactic PMMA ? Are the aging properties of atactic PMMA after T-jump experiments similar to those described here ? We have described the “fan structure” of isotherms and isobars observed on 3 glass formers. Is it a general behavior of glass former materials? What are the origins of the characteristic parameters T**, P** and V** ? Why V tends to level off for high pressure, P > P**/3 ? Are the compressibilities of the glass and crystalline states (of crystallizable materials) really equal, whatever are P and T ? Are they independent on the thermal treatment, on the process (CTF, CPF) of glass formation, on aging ? Are the physical aging effects similar in glass former materials whatever is the pressure? The application of a pressure on a same material change continuously the characteristic temperatures T* and T0 and the expansion coefficients αg and al α l of glass formers (and the melting temperature Tm of crystallizable glasses), we think that the study of the kinetics of aging of a same glass (same Mie potential) under different pressures would bring new insights on the dynamic of glassy systems and give some responses to the above questions e,f. Is there a similarity between creep and strain relaxation? In the sequential method of Struik for analyzing the creep behavior, each creep sequence is followed by a strain relaxation. The glass returns to its original dimension under the internal stress. Is this strain relaxation process (inverse to the creep process) can be described by the KWW function? In that case what are the relations between the parameters τ and n characterizing the two sequential processes? What is the origin of the Struik-Andrade law describing the short term creep of all types of glass former materials? In entangled polymers and crosslinked polymers the deformation after creep can be recovered, with time or by annealing. In other glass forming materials the deformation due to the flow is irreversible. Is there any plastic deformation near or far below the yield stress? Various authors (see for examples ref.57,[106]) claim that anelastic and plastic deformation occur at low stress even far below Tg . Is this distinction only due to a difference in the relaxation times? In that case it is difficult to define a cross-over between the elastic and anelastic (or plastic) behavior. The activation volumes Vβ and v rule the dependence of the Tg with the hydrostatic pressure and the flow (more exactly deformation) under a shear stress (yield and creep). Are these activation energies of the same order of magnitude? What is the exact dependence of the critical stress, σ0 =Eβ/v, (rel.48,49,55) with the nature of the glass former material and with the yield stress. Are σ0 and σy vary in the same way with the temperature? Isotropic and oriented polymer glasses age in the same way, they have same Tg, same relaxation enthalpy and equivalent creep properties. Is this similarity verified for long times of aging and/or near Tg? Dimensional relaxation (shrinkage) and creep

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along the draw direction are opposite. What is the relative importance of these two effects as function of the creep stress? m) Arrhenian and VFT glasses (Strong and fragile glasses) present some similarities, although the relaxation time does not follow the same law (Arrhenius and VFT laws). For strong glasses, are the dielectric relaxation, stress relaxation and compliance described by a pure exponential, a distribution of exponentials [107] or by the KWW function? What is the order of magnitude of the width of the relaxation function, and therefore of the Kohlrausch exponent at Tg? As shown by Struik the creep behavior below Tg in these two types of glasses is similar, in strong glasses it would be of the most importance to study the relaxation time determined by (short term) creep measurements as function of aging time and temperature. The similar behavior of the relaxation time τ1 with the aging time would suggest that the state of the glass can be characterized by the real and equivalent temperatures. To verify this property, the thermal expansion coefficients of the liquid and solid states must be known with accuracy. n) As noted earlier by Kovacs the physical aging is not dependent on the geometry of the samples. In the last decade several authors provided evidence for an accelerated physical aging in thin films, see references in the paper of Cangialosi et al [118]. In films the thickness dependence of physical aging disappears for thickness above several microns. The main question is then to know if there are really two different mechanisms for volume contraction and expansion in thick and thin films. Is there an intenal length scale for diffusion which would rule the redistribution of local inhomogeneities (free volume ?) during aging time and/or during a temperature jump?

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ANNEXE A. THE GRÜNEISEN PARAMETER The anharmonicity of potential in a crystal is usually expressed in terms of microscopic Grüneinsen parameters γi defined by [33,40]

γi =- d log νi / d log V = d log θi/ d log V

(A1)

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where νi is the frequency of the ith vibration mode and V the volume. By spectroscopic measurements on PE under pressure and volume Wu et al [108] show that these parameters are dependent on the lattice frequency; for optical modes 3000 cm-1 >q >1000 cm-1 γi varies from 0 to 0.1, for low frequency modes 72 cm-1 and 11 cm-1 γi is 1.53 and 0.47. In crystalline solids, using the Einstein model for optical bands and Debye model for acoustical bands, the Einstein or Debye temperature θi can be calculated in the quasi harmonic approximation. If one assumes that: a) All bands can be characterized by the sametemperature θ b) b ) The Debye temperature is proportional to the sound velocity u and dependent on the mass density ρ; θ ∼u ρ3Different approximations have been proposed to estimate the Grüneisen functions γ when the temperature variation of θ is small a good estimate[35] of γ is:

γm = -

⎛ d log u ⎞ 1 d log θ = − +B ⎜ ⎟ ; B=1/χ ⎝ dP ⎠T 3 d logV

(A2)

Other relations based on different assumptions have been proposed, references can be found in the book of Barron and White[35] and in the paper of Wu et al[107]. The anharmonic properties of crystalline and amorphous polymers have been studied by Brillouin spectroscopy. The sound frequency ν L (q) of a longitudinal polarized sound mode (LPM of a given wave-vector q) varies with the mass density ρof the material: In amorphous materials the LPM Grüneisen parameter γL:

γL (q) =

d log ν L (q) d log ρ

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(A3)

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is found to be fairly independent of temperature but presents a discontinuity at Tg. One will find in ref.[109,110,111] values of this parameter for different polymer glass. These microscopic Grüneisen parameters which concern one or a few vibration modes are different from the macroscopic Grüneisen parameter derived from thermodynamic considerations:

⎡

dP ⎤ α V ⎥ = κ CV ⎣ d(U /V ) ⎦V

γ = γ (T , P) = ⎢

(A4)

γ is often called the macroscopic or thermal Grüneisen parameter. Common features of all these parameters are: a) they are equal to zero for harmonic approximation (Δα = 0). b) they are very weakly dependent on the temperature (Δγ/ΔΤ=- 0.5 to -2 10-2 K-1 for glassy and semi-crystalline polymers,Δγ/ΔΤ=- 0 to 2 10-3 K-1 for mineral glasses (sodium silicate above 100°C see Figure5.21 of ref.[35]).

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Only a few authors have compared the different Grüneisen parameters. For various glass polymers Wada et al[108] and Warfield [109] found that the LPM parameter γL is about 4 times the thermal Grüneisen γ. Krüger found that γL of PVAc present a small discontinuity at Tg. In the literature there is however no comparison of these parameters in the melt and amorphous states in glass formers. To understand the physics of glasses it would be necessary to answer the two following questions: a) is the Grüneisen parameter γ (γi or γL ) really different across the glass transition ? b) in the glass state is this parameter dependent on the thermal treatment (aging in the glass state) ? No detailed studies permit to reply two these questions. In the table A we compare the values of the thermal Grüneisen parameters γ l and γ a (rel.A4) for different glass formers.

The values of the thermal expansion coefficient, compressibility and heat capacity in the liquid and amorphous states are found in the book of Donth15 and in the Kovacs7 review. The exact conditions of measurement of Cv, α andκ are not reported in these references, the fact that α and κ depend on T and P in the melt state would lead to an uncertainty on the γ value. In figure A we have plotted the ratio γ l / γ a as function of Tg for the two sets of data. In

rel.A4, Cp has been replaced by Cv which is generally easily measured. From this figure, taking into account the accuracy of measurements and the above assumption (Cp ~Cv), we can conclude that the dimensionless Grüneinsen parameters are of the order of 1 and are not very different across the transition ( γ l ~ γ a ), this is relevant with the fact that the parameter γL is

found to be temperature independent. Relation A4, called the Grüneisen relation, links the the macroscopic parameters of the solid and melt states (α κ and CV ). An equivalent empirical relation linking the microscopipc parameters is recalled hereafter.

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Annexe A. The Grüneisen Parameter

101

Table A . Values of the Grüneisen parameters γ l and γ a of several glass formers above and below Tg deduced from rel.A4 (CP has been replaced by CV)

Donth, from table 2.11 of Réf. 15

γl

γa

B2O3

CKN

PVAc

Glycerol

n-propanol

Selenium

Poly isobutene

PVC

0.518

1.81

0.789

1.43

5.71

2.75

1.01

0.887

0.321

2.11

0.743

2.69

9.26

2.15

0.442

0.718

183

95

304

198

350

Tg 550 340 303 (K) Kovacs, from table 1 of réf. 7

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γl

PS

PIB

Poly isoprene

PMMA

Glucose

Glycerol

0.608

1.52

1.54

0.58

1.13

1.39

1.83

0.405

1.13

2.68

200

373

305

190

γa

0.970

Tg (K)

375

305

Figure A. Ratio of the thermal Grüneisen parameters, γ l and γ a , of the melt and glass states as function of the glass temperature: glass former materials of table A.

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102

Jacques Rault Relation between the compensation and the Grüneisen empirical laws

In many conduction and thermomechanical processes which imply high activation energies ( > kT), large compared to typical excitations of the system, the measured property X (for example the reaction rate) can be put on the form:

X = X 0 exp − ( ΔH − TΔS ) / kT = X 0 exp −

ΔH 1 1 ( − ) k T Tc

A5

Where X0 and Tc are constants, Tc is called the compensation temperature. In these processes the entropy term TΔS is proportional to the enthalpy ΔΗ. 70 years ago Meyer and Neldel found for some oxide semiconductors prepared or annealed under different conditions that the DC conductivity follows the above empirical rule (often called the Meyer-Neldel law, see references in the review paper of Yelon et al [116] ). Several authors have shown that diffusion in solids obeys the same law. In that case the activation energy ΔH and the activation volume ΔV, measured by varying the temperature and pressure, verify the relation : ΔV = k κ ΔH A6 κ is the isothermal compressibility, the compensation constant k is found of the order of 4. If

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* the α process is considered ( see §2.3, rel.7) then ΔV = Vα* and ΔH = Eα Relation A6 first given by Keyes [117] have been discussed by Lawson [118]; the ratio ΔS/ΔH ~ΔV/ΔH is found characteristic of the material (metal, semiconductor, polymer) and not on the nature of the diffusing impurities. The compensation law has been observed in various polymer glasses by thermally stimulated creep, depolarized current and by yield studies, see references in [119]. In the following we assume that: a) κ is weakly dependent on T. b) The variations of volume and enthalpy of the material are proportional to the activation volume and energies. In that case differentiation of rel.A6 gives:

Cp 1 dV = kκ V dT V

A7

This is the Grüneisen relation, rel.A4. The Grüneisen and compensation parameters are of the same order of magnitude, for polymers γ = k ~ 4. We conclude that the change of volume and enthalpy during aging time ta due to the α motions are then given by :

* V* ΔV (t a ) Vα β = = =γκ ΔH (t a ) Eα Eβ

A8

We have assumed that the activation parameters of the α and β processes verify rel.7d as it has been observed in some cases (see §2.3.). It is important to verify the application of the above relation for Arhenian (strong) and VFT (fragile) glasses submitted to various thermal history.

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ANNEXE B. THE MULTIPLE GLASS TRANSITIONS Different types of collective motions (α transition) can occur in Glasses due to structural and/or chemical inhomogeneities. We give two such examples.

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CHAIN STEREOREGULARITY The glass temperature of stereoregular polymers depend on the tacticity [21]; for example, the Tg(1 Hz) of syndiotactic and isotactic PMMA are 403 K and 336 K . Tg of atactic PMMA (388 K) is intermediary between these two extreme values as expected from the plastification rule10. We think that in these last disordered materials there is a (discrete or uniform) distribution of the tacticity arrangements along the chain. This heterogeneous arrangement would lead to a distribution of Tg. This fact is suggested by the presence in several atactic polymers of a larger a peak or/and the presence of a secondary peak called α’ located between the β and α peaks. The individual β motions (Tβ=285-300 K in PS) are weakly affected by the tacticity [21]. The effect of aging of these materials show some differences with the classical materials described in this review. The kinetics of aging of atactic PMMA during simple and double T-jump is briefly reported hereafter as an example.

Simple T-jump The samples have been aged for the same time (3000 min) at different q temperatures below 115 °C. Then the DSC thermograms are recorded at several heating rate. The temperature Tm corresponding to the enthalpy relaxation peak is given in FigureBa. As for PS the relation log q ~ 1/Tm is verified, but the activation energy deduced from this figure is not constant. It varies continuously from Eα to Eβ when the aging temperature decreases from Tα to Τβ . Similar results have been reported by Bershtein et al (see Figure2.14 of ref.20). The relaxation enthalpy (FigureBb) varies linearly with the temperature and with log t, as for the glass former materials previously studied (PS, PVAc, etc..) but the slope kH decreases with decreasing temperature and the incubation time (1 to 5 min) does not vary with temperature. As reported in Figure 11-12 the slope in most glass formers is constant (faint dependence on the supercooling) and the incubation time varies of several order of magnitude for Tg > T >Tg-30 °C.

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Figure B.a ) Temperature (T=Tm) of the endotherm peaks of atactic PMMA versus the logarithm of the heating rate. The sample filled symbols have been aged 3000 min at different temperatures (heavy lines). The samples (cross and circle) have been submitted to the double down T-jump, at 105 °C then at 60 °C (dashed lines). Lines are guides for the eyes. b ) Relaxation enthalpy of samples annealed at different temperatures versus the aging time. The incubation time t0 is constant between 1 and 5 min. Lines are guides for the eyes.c ) DSC Thermograms J =dH/dt of PMMA quenched to room temperature (curve 1) and PMMA submitted to the double down T-jump 200 min at 105 °C and 3000 Min at 60 °C (curve 2), heating rate q=10 °C/min. Curve 3 is the substracted curve J’=J(aged) –J(non aged) d ) Temperature Tm2 of the low temperature endotherm peak of PMMA as function of the aging time at 70 °C. Samples are submitted to the double down T-jump experiments (140°C to 100°C then to 70°C) and to a simple down T-jump (140°C to 70 °C). Heavy line: linear fit with the data of the simple quenching experiment (at 70 °C).

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Annexe B. The Multiple Glass Transitions

105

These properties suggest that the nature of the motions in PMMA change continuously when T decreases. The main question is: are the motions at two different temperatures independent ?

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Double Down T-Jump The samples have been aged at two temperatures; at T1=105 °C for 200 min and then at T2 =60 °C for 3000 min. The DSC thermograms are shown in FigureBc. The double aging produces two enthalpy peaks of enthalpy ΔH1 and ΔH2 we note that the enthalpy relaxation and the temperatures Tm1 and Tm2 of the two peaks are equal to the corresponding peaks of the samples which has been submitted to the corresponding simple annealing at T1 and T2. This is illustrated in FigureBd which gives Tm2 as function of the aging time at 70 °C . The samples have been quenched directly at 70 °C or submitted to the double down T-jump (T1=100°C T2=70 °C). A previous aging at 100 °C , 20 °C below Tg(100s), does not change the kinetics of aging at 70 °C (Tm2 peak). We conclude that in these materials there is a distribution of independent cooperative motions that are likely due to the stereo-regularity of the polymer chains and which are operative in different domains of temperature. In atactic PS some authors found the presence of pre-peak below the glass transition, we have not observed this effect in our PS materials [31]. Certainly the aging properties of atactic polymer glasses depend on the exact structure of the chains, therefore on the conditions of synthesis. Remark on Physical Aging Struik has noted that the creep shift rate μ at low stress is equal to 1 in a very small domain (10 to 20 °C) of temperature for PVC and PMMA and a wide domain (100 to 150 °C) for PC and Polysulphone, PSU (see Figure15 of ref.[9]). We think that this difference between these two series of polymer glasses would be due to the presence of different local stereo-regularities along the atactic chains (PVC and PSU).

STRUCTURAL HETEROGENEITY Isotactic PP is a semi crystalline polymer. As reported by many authors the amorphous phase near the crystallites is submitted to constrains due to the presence of the crystalline surface and/or the tie molecules which connect the crystallites. This constrained amorphous phase has a higher Tg than the free amorphous phase (no crystallinity). Hutchinson et al [112], and Botev [113] have shown that two different domains of aging exist ; the classical domain of aging for T