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PHYSICAL AGING OF GLASSES
K. Ngai, I. Manning
To cite this version:
K. Ngai, I. Manning. PHYSICAL AGING OF GLASSES. Journal de Physique Colloques, 1982, 43 (C9), pp.C9-607-C9-610. �10.1051/jphyscol:19829121�. �jpa-00222427�
JOURNAL DE PHYSIQUE
Colloque C9, supplement au n°12, Tome 43, decembre 1982 page C9-607
PHYSICAL AGING OF GLASSES
K.L. Ngai and I . Manning
U.S. Naval Research Laboratory, Washington D.C. 20375. U.S.A.
Résumé.- On représente un modèle qui donne une description unifiée de la dis- persion des temps de relaxation dans les fonctions viscoêlastiques et le vieil- lissement des propriétés viscoêlastiques des verres, des métaux amorphes et des polymères. Notre calcul conduit à un module de relaxation de la forme :
G(t)=G0 exp[-|t/T ) 1 _ n] , avec T =[(l-n)enYEn T Q ]1/1 - 1 1. Dans notre modèle les effets de vieillissement se traduisent surtout par une augmentation de n au fur et à mesure du vieillissement ce qui produit un décalage non uniforme de tout le spectre des temps de relaxation. Ce modèle est en accord avec les résultats expérimentaux récents de Chai et Me Crum qui soulignent la nécessité impérieuse de tenir compte des effets de vieillissement. La classe des fonctions de complaisance correspondantes au module de relaxation ci-dessus est évaluée et discutée.
Abstract.- A model is presented which gives a unified description of the dis- persion of relaxation times in the viscoelastic functions and the aging of viscoelastic properties of glasses and amorphous metals and polymers. Our treatment leads to a relaxation modulus of the form G(t)=G exp[-(t/T ) ] , with X = [(l-n)en^En T ] . In our model aging effects are dominated by an increase of n as aging proceeds, resulting in a non-uniform shift of the entire spectrum of relaxation times. The model is consistent with recent experimental results of Chai and McCrum which impose a stringent demand on any treatment of viscoelastic aging. The class of compliance functions correspond- ing to the above relaxation modulus are evaluated and discussed.
Amorphous solids are not in equilibrium at temperatures below their glass transition temperature T . The material continues to approach the equilibrium state in the process calle% physical aging [1,2 J with all of the material properties which changed drastically during glass formation continuing to gradually change, and in the same direction. The material becomes stiffer and more brittle, its damping increases as does its creep and stress relaxation rates, dielectric constant, die- lectric loss, etc. Physical aging is a basic feature of the glassy state; it is found in all glasses and proceeds in a very similar way in all glasses whether polymeric, monomeric, organic, or inorganic. In this work, we are concerned with the viscoelastic compliance function J(t) which is measured at an elapsed time t after the material was quenched from the molten state to a temperature T between TR
and T , where TRis the temperature of the highest secondary transition. The com"
plianft is thus 5 function of all three parameters, J = J(t; t ,T), with
t « t . (1) e
Based on the concept of free volume, Struik has formulated a model for physical aging [1]. A key feature of this model is that every retardation time X. is altered by the same shift factor a(T,t ) to become T•= a T.. This time-temperature corre- spondence is familiar and widely accepted for temperatures well above the glass transition [3] , but remains to be established quantitatively for temperatures below the glass transition. According to this model the curve J(t), plotted as a function of log t with t and T fixed, has a shape which is characteristic in the following sense: For a given material at a given temperature T, any two such curves corre- sponding to two different aging times t can be brought into complete coincidence by making a (major) shift horizontally, parallel to the log t axis, combined with a (minor) shift vertically, parallel to the J axis. Providing the condition of eq. (1) is satisfied, this time-temperature correspondence is thought to hold for all aging times t stretching out to infinity.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19829121
C9-608 JOURNAL DE PHYSIQUE
Chai and McCrum have recently subjected the Struik model to an experimental test which imposes a stringent demand on any model of viscoelasticity in physical aging [4]. Their test is based on high-precision experimental determinations of
. d
J = - J(t). Let the temperature T be fixed and let to be a reference aging time.
dt
They point out that, for all aging times t satisfying eq. (11, it follows from linear viscoelastic theory that Struik's the-temperature correspondence implies
where a(te) is the shift factor of Struik's time-temperature correspondence and b(t ) and c(t ) are functions related to the compliance functions at t equal to zero andeinf hity. Correspondingly,
. t
log J (---- ; ):t = log J (t ; te) + log a(te) - log b(t,) ,
a(te) which implies that
d lo J
= constant independent of t .
For isotactic polypropylene which was quenched from 80°C to T=40°C, they found that the derivative (4) varied smoothly from -0.749 to -0.858 as t varied from 0.18 ks to 191 ks. Thus, at least for this material, it appears thate the Struik model is not consistent with the experimental findings of Chai and McCrum.
Our aim in this work is to point out, on the other hand, that a recently advanced unified theory of low-frequency dynamical responses in condensed matter in general leads, in a very natural way, to a model of viscoelastic aging which appears to be capable of organizing a vast range of aging data, as does the Struik model, and which seems to be fully compatible with the experimental demands of Chai and McCrum.
The unified theory involves an application of Wigner's statistical theory of energy levels, orginally developed for nuclear excitations, to the low-frequency dynamical responses of condensed matter [ S ] . A new class of excitations, termed correlated states, was found whose coupling to relaxation processes often dominates the long- time behavior of relaxation, fluctuation and dissipation properties of condensed matter. The coupling mechanism has been discussed in detail for the specific case of dielectric polarization [6] and, to a much lesser extent, for volume and enthalpy ion recovery of amorphous polymers [Ref. 7 and Ngai K.L. and Bendler J.T., unpub- lished].
In the latter case the theory considers a primary species of polymer molecular con- figuration with a configurational degree of freedom whose equation of motion is partly governed by a potential surface with multiple minima separated by activation barriers. For temperatures near T the important configurations are expected to be modes involving backbone monomer uEits with varying dihedral angles, possibly mixed with bond-bending and side-group torsions. In addition to the force due to the above potential barrier, the primary species experiences time-dependent influences from neighboring polymer chains as well as from non-bonded groups in the same chain.
The application of Wigner's random matrices to the low-lying excitations of the primary species leads to the appearance of correlated states, whose density of states is linear in energy E providing the energy is less than a characteristic energy E :
N(E) = kE , for E < Ec (5
The physlcs behind this is level repulsion. For glasses the frequency w corre- sponding to E is generally of the order of 10 GHz. The theory then appfies the density of $tates, together with the perturbation-theory result for the quantum-mechanical transition probability for a correlated-state excitation or de-excitation of a given energy, to obtain a macroscopic relaxation function of the form
with
where y = 0.5777 and 0 < n < 1.
The density of states (5) together with the above transition probability imply a divergence in the way that the correlated states which couple to the macroscopic relaxation cluster in energy about the singular point E=O; the divergence is analogous to the infrared divergence in the theory of Cerenkov radiation of a fast charged particle and in the theory of x-ray absorption edges of metals.
The above expressions (6) and (7) constitute a universal form for the description of the effects of correlated states in the fluctuation, dissipation, and relaxation properties of condensed matter. For dielectric polarizability the function G of eq. (6) turns out to be the polarization correlation function, while for visco- elastic relaxation it is the relaxation modulus. The present model of aging 1s one application among many others of this unified theory. In addition to the very wide range of phenomena cited in reference [5], the unified theory has been applied to molecular relaxations near the glass transition [8], volume recovery of poly- mers [7], dynamics and rheology of polymer melts [ 9 ] , and plasticity [lo].
References 181 and [7] discuss the dependence of n on t and T: As aging time t proceeds, the number of correlated states and also the ienteraction strength betweeg the primary species and the correlated states both increase because of the volume decrease. Correspondingly, n(t ) monotonically increases with t . In our model, aging effects are dominated byethis monotonic increase which hag two major mani- festations: (1) The dispersion shape of the spectrum of relaxation times broadens, changing according to eq. (6). This constitutes a breakdown of the analogue of thermorheological simplicity discussed on page 14 of Ref. [l] and represented by eq. (2), and found to be violated by Chai and McCrum. The other major effect is:
(2) The time scale shifts according to eq. (7). For w r >>I, which is usually the case, r of eq. (7) is monotonic increasing in n and theFeOfore in t .
P
The compliance function J(t) corresponding to the modulus G(t) satisfies the rela- tionship
The method of Hopkins and Hamming has been used to obtain the compliance functions corresponding to the modulus (6) of our model [Ill. For t<<t-, the numerical
P
results are in good agreement with the empirical law J = exp[(t/t which goes back to Kohlrausch in 1866 [12]. It is useful to note the limiting form
with
JOURNAL DE PHYSIQUE
In the above, Qk is defined as
which works out to be
The behavior of the compliance function is illustrated in Fig. 1. In that figure, one can consider n as replacing t as the parameter denoting aging. That is, J is given as a function of t for repreEentative values of n/t ) .
Figure 1. A representative family of compliance functions J(t) corresponding to the relaxa- tion modulus of eqs. (6) and (7).
The correspondence is given by eq. (8). The parameter n is a function of aging time and tempera- ture, and increases as aging proceeds. Thus, n can be consid- ered as replacing t in denoting the evolution of agigg.
Log t
For t/r <<1 our model predicts that the derivative (4) will be independent of t, as found b5 Chai and McCrum. In fact, that derivative in the small time limit is just -n. For isotactic propylene with aging times of 0.18, 0.72, 2.9, 11.5, 46.1, and 191 ks, Chai and McCrum found derivative values corresponding to n values of 0.749, 0.774, 0.788, 0.820, 0.842, and 0.858.
Thus, our model for physical aging appears to correctly predict not only the time dependence of J in Fig. 1 for fixed t and T, but also the increase in n for increasing t which, in our model, is theedominant effect of physical aging.
References
STRUIK L.C.E., Physical Aging in Amorphous Polymers and Other Materials (Elsevier Pub. Co.. Amsterdam) 1978.
STERNSTEIN S.,
rea at.
Matl. ~ c i . Tech. (1977) 541.But see McCRUM N.G., Bull. Am. Phys. Soc. (1982) 392.
CHAI C.K. and McCRUMM N.G., Polymer 21 (1980) 706.
NGAI K.L., Comments Solid State Phys. 9 (1979) 127 and 9 (1980) 141.
NGAI K.L. and WHITE C.T. , Phys. Rev. (1979) 2475.
BENDLER J. and NGAI K., Polymer Preprints 22 (1981) 287.
NGAI K.L., Polymer Preprints 2 (1981) 289.
NGAI K.L. and Rendell R.W., Polymer Preprints, to be published.
NGAI K.L., Rendell R.W., and MANNING I., Bull. Am. Phys. Soc. 27 (1982) 261 MANNING I., NGAI K.L., RENDELL R.W., Bull. Am. Phys. Soc. 27i (1982) 261.
STRUIK L.C.E., Europhys. Conf. Absts. 5 (1980) 135.
