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RELAXATION PHENOMENA AND THE TIME-TEMPERATURE SUPERPOSITION
PRINCIPLE
F. Povolo, M. Fontelos, E. Hermida
To cite this version:
F. Povolo, M. Fontelos, E. Hermida. RELAXATION PHENOMENA AND THE TIME-
TEMPERATURE SUPERPOSITION PRINCIPLE. Journal de Physique Colloques, 1987, 48 (C8),
pp.C8-353-C8-357. �10.1051/jphyscol:1987852�. �jpa-00227156�
JOURNAL DE PHYSIQUE
Colloque C8, supplbment au n012, Tome 48, dbcembre 1987
RELAXATION PHENOMENA AND THE TIME-TEMPERATURE SUPERPOSITION PRINCIPLE
F. POVOLO(
,
M. FONTELOS*and E .
HERMIDA* *CEN de Grenoble, DBpartement de Recherche Fondamentale, Service de Physique/MP., BP 85 X , F-38041 Grenoble Cedex, France
" ~ o n s i g l i o Nazionale delle Ricerche, Istituto di Ricerche su Tecnologia dei Polimeri e Reologia, Via Toiano 6. I-80072 Arc0 Felice, Napoli, Italy
1
* Universidad de BUenOS Aires, Facultad de Ciencias Exactas y Naturales, Depto. de ~l'sica, pabell& 1, Ciudad Universitaria, RA-1428 Buenos Aires, Argentina
Abstract. 'The time-temperature superposition principle. frequently used to describe relaxation phenomena in solids is discussed with~n the framework of general functions leading to scaling with translation parallel to the horizontal axis. Finally, the concepts developed are applied to particular functions used in the literature to describe relaxation phenomena in polymers and In crystalline solids.
introduction
The so called time-temperature superposition principle is w~dely used to extend the range of the measurements when transient or dynamic viscoelastic parameters are studied In polymers. as a function of time ( o r frequency) and at different temperatures ( 1 - 8 ) . According to this principle, trme and temperature are equivalent. that is, a given property measured for short times at a given temperature is identical with one measured for longer times at a lower temperature. except that the curves are shifted on a logarlthmlc time (or logarithmic frequency) axis. They can be superimposed once more by proper scale changes on this axis. Similarly. portions of the response curves can be observed at different temperatures and these curve segments can then be shifted along the log(time) axis to construct a composite curve or master curve. applicable for a given temperature. extending many decades of time. The shift factor for a curve segment is designated by aT. log aT being the horizontal displacement to allow it to join smoothly into the master curve. This Is the factor by which the time scale is altered due to the difference in temperature. and Is. a function of temperature. Furthermore, for all linear viscoelastic materials over a limited temperature range the horizontal shift factors are given by the empirical Willlams-Landel- Ferry (WLF) equation (6)
log a~ = -C1 (Ts) (T-Ts) / CC2(Ts) +(T-Ts) I ( 1 )
where T is the temperature, T, is a reference temperature and C 1 . Cg depend on Ts.
The tcme-temperature superposition principle is also used to describe the anelastic behaviour of crystalline solids ( 9 ) . particularly for the Standard Anelastic Solld Model (SAS). In fact. the dynamic response of the SAS model leads to Debye type equations for the description of the dynamic modulus and the internal friction.
it is the purpose of this paper to show that the time-temperature superposition principle can be considered as a particular case of general functlons leading to scaling. with a translation path parallel to the abscissa. Furhtermore. Eq. ( 1 ) will be obtained from the general formalism presented and Its properties will be discussed. Finally, the concepts developed will be applied to expresslons commonly used to describe the viscoelastic behavtour of polymers and the anelastic behavlour of crystall~ne solids.
("on leave from : ~omisi6n Nacional de Energia Atomica. Depto. de Materiales, Av. del Libertador 8250.
RA-1429 Buenos Aires. Argentina, and Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales, Depto. de ~isica. pabell& 1, Ciudad Universitaria. RA-1429 Buenas Aires, Argentina
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987852
JOURNAL
DE
PHYSIQUETheory
A form for the family of curves in the (x.y)-plane. at different z levels. which superimpose under a translation path along a glven direction. is glven by (10, 11).
where g is a real function. continuous. slngle-valued and differentiable. A, 13. C. a. b. c.
d are real 'constants and h(z) is a real function of z. In this case. the translation path in the tx.y)-plane 1s given by (10.11)
with the additional scaling conditions
Ay/Ah(z) =(Ac-Ca) / ( Ba-Ab) (4) and ah t z)
/ax=(
Ba-Ab) / ( Cb-Bc) ( 5) where A indicates finite increments of the corresponding variable.Scaling with a translation path parallel to the abscissa
I f the translat~on path is parallel to the x-axis. the scaling conditions for Eq. ( 2 ) are
reduced to
Ay/Ax=(Ac-Ca) / ( Cb-Bc) =O (6) : Ay/Ah(z) =(Ac-Ca) / ( Ba-Ab) =O ( 7)
Since Ay=O. Eqs. (6) to ( 8 ) imply that
(Ac-Ca)=O (9) ; (Ab-Ba) =O (10) and (Cb-Bc)=O
Eqs. ( 9 ) to ( 1 1 ) impose some restrictions on the form of Eq. ( 2 ) . compatible wlth a translation path parallel to the abscissa. These particular forms of Eq. (2). together with the correspondfng relationships between the increments of x and h ( z ) . are given In Table 1.
Table 1
--
Function Relationship between increments
Applications
The superposition of the response curves. by a shifting along the log-(time) or log- (frequency) axis. to construct a composite or master curve according with the time- lemperature superposition principle. implies, within the context of the scallng property, that the different curves must be related by scaling with a translation path parallel to the horizontal axis.. For example. the relaxation behaviour of dilute polymer solutions IS
frequently descrlbed by expressions of the type
where
R is the gas constant. p is the density of the material. G' and G" are the storage and loss shear moduli. respectively, N is the number of Gaussian subchains in which the polymer molecule is divided. o=2rrv where v is the applied frequency and M IS the molecular weight.
The longest relaxation time TI is only a function of T. With the change of variables x=lnv.
y=G'/Tp or GC'/Tp. h(z) = l n q ( T ) it is easily seen that Eq. ( 12) can be written as
where
A similar expression can be obtained for Eq. ( 13). Then. Eqs. (12) and (13) lead to a function of the type given by the second row of Table 1. with A=]. B=0. C = l . b = l . d=-RIM for Eq. ( 12) and d=O for Eq. ( 131
.
For both equationsFurthermore. on taking the logarithm of both sides of Eqs. (12) and (13) it is easily seen that a plot of iog(G'/Tp) or log (G"/Tp) versus logv also leads to the same type of g function. Then. a representation of both functions in this way leads to scaling with a translation path parallel to the logv axis, or, which is equivalent, to the applicability of the time-temperature superposition principle. In fact. these are the plots generally used i n the literature.
Equation ( 1) can be obtained from the scaling conditions since log aT is equivalent to Ax and the scaling conditions for all the functions given in Table 1 imply that
and. i f
where p, q and r are constants, independent of z. it is easy to show that Eq. ( 1 ) can be obtained from Eqs. ( 18) and ( 19). with the change of variables Ax=log a ~ . z=T,. A F T - T s and
Furthermore. if
where ro Is the pre-exponentlal factor, U IS the activation energy and k is Boltmann's constant. then
which does not lead to Eq. ( 1 ) . If TI
.
however. is given by ( 12)then. h(z) can be written In the form of Eq. ( 1 9 ) . leading to Eq. ( 1 ) for a ~ . Moreover. if the glass transttlon temperature is selected as a reference to construct the master curve.
that is. if Ts=Tg in Eq. ( 1 ) . then C1 and C2 are constant for ail polymers and Eq. (1) reduces to
log a>-=-GI (T-Tg) / (C2iT-Tg) (24)
C8-356 JOURNAL DE PHYSIQUE
With the change of variables x=T. y=log aT and z=Tg. Eq. (24) can be written as
which is a function of the type of that given by the second row of Table 1. wlth A=I/C2. R=O.
C=-1/C2. b=-1/C1, d=O and htz)=z. Then
showing that Eq. (24) has a scaling relationship in the log a~ versus T diagram. In addition.
since ATg=fl the magnitude of the translation depends only on the difference between the glass transit~on temperatures of the different systems. Then if a system is selected as reference. a master log aT versus T curve can be constructed for all the systems by translations parallel to the T-axis. Moreover, by using the formalism given In the Theory it can be shown that if several log aT versus T curves are plotted for different systems and at different reference temperatures. they can be superimposed to form a master curve along the translation path. p. given by ( 13)
where the subscripts 1 and 2 indicate the corresponding quantities for each system. The translation path is different for different pair of curves since f i depends on Ts and Tg. If Tsl-Tg1=Ts2-T 2-6, where 6 is constant for ail systems. the translation path is parallel to the T-ax~s (p=8) -and a master curve can be constructed by translations parallel to the abscissa. This explains the fact that Williams, tandei and Ferry ( 6 ) found a master log aT versus T curves for all systems and for 6=50 K . since the reference system was selected at Ts=Tg+50 K.
The dynamical response of the SAS wlll be considered as a last example. The Internal frlction, tan@ is given by (9)
where the relaxatlon time T depends on T and it will be assumed that a and w are independent from T. With the change of varfables x=lnw, y=tan@, z=T and h(z) =ln7(T)
.
Eq.(28) can be written as
Eq. (29) has the form of the function given by the second row of Table 1, with A=1. 0=0.
C=l and b = l / a and. consequently.
Then, the different tan@ versus Inw curves. obtained at different temperatures. are related by scaling along the abscissa. Furthermore. if T is given by Eq. (21) and tan@ is measured as a function of temperature. at d~fferent w , Eq. (30) leads to
whlch is the expression generally used to obtain U (9). A similar analysis can be made for the dynamic modulus. which IS given by an expression similar to Eq. (12)
Conclusions. A general formal~sm was glven that can be used to study the translation properties of the functions commonly used for tne descrlptlon of the relaxation behaviour O f
sol~ds. Furthermore. tt has been shown that the t~me-temperature superposltlon prlnclple means that the response functtons used to descrfbe the relaxat~on behavlour must belong to the famlly of general functions with scallng along the absclssa
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Acknowledgements
One of the authors ( F . P . ) would like to thank the CENG. where part of this work was performed during a sabbathical leave. for hospitality and financial support. Another author (M. F . ) would like to thank the International Centre for Theoretical Physics, Trieste. Italy, for the permanence at the ITPR through the ICTP Programme for Training and Research i n Italian Laboratories.
This work was supported i n part by the Consejo Nacional de lnvestigaciones Gientificas y Tecnicas (CONICET) and the "Proyecto Multlnacional de Tecnologia d e Materiales" (OAS- CNEA)
