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α-relaxation and molecular dynamics in glass-forming polymeric systems
860
Journal of Non-Crystalline Solids 131-133 (1991) 860-869 North-Holland
a-relaxation and molecular dynamics in glass-forming polymeric systems J. Colmenero Departamento de Fisica de Materiales, Unioersidad del Pals Vasco, PO Box 1072, 20080 San Sebastidn, Spain
During recent years there has been a great deal of experimental effort in the study of the relaxation behaviour above the glass-liquid transition of glass-forming systems. This behaviour is directly related to the dynamics of such systems in the supercooled liquid-like equilibrium state. Therefore it is related to the glass-liquid transition phenomena. In the special case of glass-forming polymeric systems, the situation is much more complex due to the fact that the glass-liquid transition is spread-out into the glass-rubber-liquid transition. Then, the relaxation behaviour in this temperature range shows two main steps, the a-relaxation and the terminal zone. The main results obtained to date can be summarized as follows. The relaxation response usually shows a non-exponential behaviour (time or frequency domain). Moreover, the corresponding timescale displays a non-Arrbenius temperature dependence. However, in spite of these results, several questions still remain unsolved or partially solved. Some of them are: the lack of thermorheological simplicity of time-temperature superposition of viscoelastic data in the glass-rubber-liquid transition range; the relationship among the dynamic behaviours obtained from different probes; can the dynamics of the a-relaxation be described in a wide time range (1-10 -1° s) by using the same temperature dependence of the characteristic timescale and the same shape parameters?. Another interesting question is how the a-relaxation behaviour, corresponding to a polymer in the rubber state, changes when such a system gradually becomes a glass by cooling through the rubber-glass transition. The aim of this paper is to review the work recently made in this line, not only by the present group but also by other groups working in this area.
1. Introduction D u r i n g recent years a great d e a l o f effort has b e e n m a d e in the s t u d y of the d y n a m i c s of g l a s s f o r m i n g systems a b o v e their g l a s s - t r a n s i t i o n temp e r a t u r e range. N o w a d a y s it is well k n o w n that the d y n a m i c a l b e h a v i o u r of such systems is directly related to the g l a s s - t r a n s i t i o n p h e n o m e n a , F r o m an e x p e r i m e n t a l p o i n t of view, the m a i n results o b t a i n e d can b e s u m m a r i z e d as follows, (1) T h e timescales characteristic of the dyn a m i c s (e.g. r e l a x a t i o n times) follow a n o n arrhenius t e m p e r a t u r e b e h a v i o u r which is u n u s u a l in physics. I n general, this b e h a v i o u r can b e well p a r a m e t r i z e d b y m e a n s of the V o g e l - F u l c h e r law, at least in the e x p e r i m e n t a l range usually covered b y the r e l a x a t i o n techniques ( t y p i c a l l y 1 0 - 6 - 1 0 0 s).
(2) A given r e l a x a t i o n function, ~ ( t ) , usually shows a p e c u l i a r n o n - e x p o n e n t i a l b e h a v i o u r that c a n b e e m p i r i c a l l y d e s c r i b e d b y the K o h l r a u c h W i l l i a m s - W a t t s ( s t r e t c h e d e x p o n e n t i a l function) law. (3) T h e c o m p l e x susceptibilities, which det e r m i n e the r e s p o n s e o f a system to a given excitation, show a clear n o n - D e b y e f r e q u e n c y behaviour. This can b e d e s c r i b e d b y m e a n s of several e m p i r i c a l d i s t r i b u t i o n f u n c t i o n s o f D e b y e processes, the H a v r i l i a k - N e g a m i one b e i n g the m o s t general. (4) T h e scattering curves, S(q, to), o b t a i n e d b y quasi-elastic s c a t t e r i n g e x p e r i m e n t s , such as inc o h e r e n t quasielastic n e u t r o n scattering, d i s p l a y non-lorentzian behaviour. H o w e v e r , in spite of these results, several exp e r i m e n t a l q u e s t i o n s still r e m a i n u n s o l v e d o r par-
0022-3093/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
J. Colmenero / a-relaxation and molecular dynamics
tially solved. Some of them are as follows. (i) The relationship among the dynamical behaviours obtained from different probes. In particular, how to compare relaxation techniques with scattering ones; what should be the q-dependence of the magnitudes characteristic of the dynamics?, (ii) The current results reported about the dynamics of glass-forming systems correspond to the temperature behaviour above the glass-transition range. However, almost nothing is known about the change of behaviour (if any) of the a-process across the experimental glass-transition temperature range. Is the a-process completely arrested at the glass-transition temperature range? If not, how would it be possible to measure the dynamics of the a-process just below the glasstransition? (iii) Another interesting question is whether the dynamics of the a-process can be described in a wide-time range (macroscopical/microscopical) (1-10 -l° s) by using the same temperature dependence and the same spectral shape, All the results and open questions mentioned above apply to glass-forming liquids in general, However, in the special case of glass-forming polymeric systems, the situation becomes much more complex due to the fact that the glass-liquid transition is spread-out into the glass-rubberliquid transition. Then, the relaxation/dynamical behaviour in this temperature range shows two main steps, the a-process and the 'terminal zone', The lack of the thermorheological simplicity of time-temperature superposition of viscoelastic data has recently been reported by several authors and from different techniques, The aim of this paper is to present some recent experimental results corresponding to these open questions. As far as I know, there are no definitive theoretical explanations for the results which will be below described. However, in some cases I would like to mention some theoretical approaches which in my opinion could account for the observed behaviour at least qualitatively, The results described here fall into the following topics: (a) relaxation behaviour of glass-forming polymeric systems around Tg; (b) dynamics of the a-process above Tg, microscopical and macroscopical scales; (c) lack of thermorheological simplicity in polymers,
861
2. Relaxation behaviour of glass-forming polymeric systems around Tg One of the more powerful experimental techniques for studying the dynamics of glass-forming polar polymer systems is the well known dielectric relaxation spectroscopy. The standard frequency range available is from 10 Hz to 10 5 or 10 6 nz, although with adequate experimental arrangements it is possible to extend this range. Then, this technique allows us to investigate the dynamics in a timescale which ranges from 10 -1 to 10 -6 s, i.e. in the supercooled liquid-like state above the experimental glass-transition. However, in order to investigate the dielectric response by this technique, at the temperature range just below Tg, the frequency range we have to use ( - 10 -3 Hz) implies some practical problems. In addition, at this range, the glassy polymeric system evolves towards the liquid-like equilibrium state during experimental times. As a consequence, the dielectric relaxation obtained in such conditions by these classical methods should correspond to the liquidlike zone instead of the actual glassy state. There is, however, a thermally stimulated technique known as thermally stimulated depolarization current (TSDC), which could be successfully used in order to monitor the temperature behaviour of the mean timescale of the dipolar motions just below Tg. This technique is a particular time-domain technique with an equivalent frequency range of 10-2-10 -5 Hz. Over the past few years, we have used this technique in order to investigate the molecular motions just below the glass-transition of different glassy polymers like Poly(carbonate), Polyarilate, Phenoxy etc [1-5]. However, the standard TSDC experimental procedure does not easily allow information to be obtained about the non-exponential behaviour of the dielectric relaxation function, ~(t). Moreover, the standard TSDC experimental procedure is a non-isothermalprocedure and this complicates the analysis and the interpretation of the experimental data corresponding to the temperature range below Tg. For this reason, we have recently developed [6] a new TSDC isothermal experimental procedure that allows us to obtain not only the temperature behaviour of the mean dielectric relaxation time just below the glass-transition, but also the tempera-
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J. Colmenero / a-relaxation and molecular dynamics
ture dependence of the non-exponential behaviour. In this experimental procedure, the sample is polarized above Tg and, without removing the polarization field, cooled down at constant rate (10 K/min) until the measurement temperature is reached. Then after a waiting time, tw, the electric field is removed and the depolarization current is measured as a function of time, the value of t w determines the 'state' of the glassy sample. In order to study the change of the dielectric relaxation function through the glass-transition we usually take t w as short as possible ( - 10 s). We will call this technique isothermal depolarization current (IDC). More details about this technique can be found in [6,7]. Then, by combining standard dielectric spectroscopy with the time-domain technique described above (IDC), we can investigate the change of the dielectric relaxation function through the glass-transition. To date we have studied three different polymers Poly (sulfone of bisphenol-A) (PSF), Poly (vinyl acetate) (PVAc) and Poly (vinyl chloride) (PVC). The results obtained are summarized in the following, The dielectric relaxation corresponding to the a-relaxation is not completely arrested just below the glass-transition. The experimental dielectric relaxation function shows a clear non-exponential behaviour both, above and below Tg. This be°
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.
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60 40 20 . . . . . 0 390
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Fig, 2. Percentage rate (%) of the enthalpy recovered, AH(tm)/AHoo(%), during the measuringtime, tm, of the IDC technique at different temperatures below the glass-transition of PSF.
haviour can be well described by assuming a stretched exponential functional form for the time-domain dielectric relaxation function ~(t) ep(t)a exp[[ - ]~[ t / T]B] /p , where Tp is a characteristic timescale and fl is a non-exponential parameter. The temperature dependence of zp displays a crossover from a Vogel-Fulcher (Tp = TOexp[A/(T - To)]) behaviour (liquid-like zone above Tg) to an Arrhenius-like one (glassy zone just below Tg). The value of the fl-parameter also changes belowthr°ugh v Tg of towards about a l 0.35. a more u or e lessT constant g
10' 102 Xp(s)00 1°-2
AH(tm) 100 (%)
I
2"~21000/T(l/K)214
I
2.6
Fig. 1. Temperature behaviour of the characteristic dielectric relaxation parameters B and zp around the glass-transition of Poly (sulfone) (PSF). D, dielectricspectroscopymeasurements; o, time domain measurements (IDC technique, see the text),
Both the apparent activation energy and the B-value below the glass-transition are similar for all the polymers investigated. The apparent activation energy, Ea, ranges from 0.8 to 1 eV (from 18 to 23 Kcal/mol) and the fl-value is close to 0.35 as has been mentioned above. The results corresponding to PSF can be seen in fig. 1 as an example. This figure shows the crossover of Tp(T) from Vogel-Fulcher behaviour towards an Arrhenius-like one across Tg as well as the fl(T) behaviour. The values of the VogelFulcher parameters and the apparent activation energy are also shown in the figure. However, one of the main objections o n e c a n raise to the results obtained by the IDC procedure is if the measurements actually correspond to a
J. Colmenero / a-relaxation and molecular dynamics
clear 'glassy state', i.e. if during the measuring time the system evolves towards the equilibrium, state. To clarify this point, we have plotted in fig. 2 the percentage of the recovered enthalphy, A H ( t m)/A H~(%), during the IDC measuring time, t m, at different temperatures below Tg. The data shown in fig. 2 correspond to PSF. The enthalpy recovery measurements are described in ref. [8]. As can be seen in the figure, AH(tm)/AH~(% ) is almost zero in the temperature range T _< Tg - 30 ° Then, both the ~'p(T) and the fl(T) in this range can be considered as representatives of the 'glassy state'as formed during the cooling procedure, In relation to the crossover shown by ~p(T) across the glass-transition, it is interesting to point out that molecular transport characteristic magnitudes, other than dielectric relaxation times (viscosity and diffusion coefficient), also show this temperature dependence around the glass-transition of both, polymeric [9] and non-polymeric [10] glass-forming systems. This 'universal' behaviour can be understood in the framework of a kinetic free-volume model approach which describes how a supercooled liquid loses its equilibrium state when it gradually becomes a glass through the glass-transition temperature range. A general description of this phenomenological approach can be found in our previous papers [10-12]. In relation to the non-exponential behaviour, as far as I know there are no theoretical predictions about the temperature dependence of fl either above or below Tg. If we interpreted the r-value as a measure of the distribution of the relaxation times, a monotonous decrease, such as the one found by us below Tg, should indicate a widening of the distribution. See as an example fig. 3, where the distribution functions of relaxation times corresponding to a stretched exponential function with fl = 0.5 (above Tg) and fl = 0.35 (below Tg) have been plotted. They are obtained by means of the inverse laplace transformation (CONTIN program) This kind of behaviour also could qualitatively be interpreted in the frame of the freevolume approach abovementioned [12,13]. On the other hand, in the frame of the coupling model (see for examples refs. [14,15], fl can be interpreted as a measure of the coupling strength between the relaxing unit and the complex system. Then, a
863
0.3 ,.--, ~ ~"
0.25 0.2
o 0.5 •
I
o
.35
o
o
o 0.15
••• o•
0.1
• o• ,g 0.05 o• o•o • gO•© 0 ..... " • ~ g ~ Q ................... o • le-07 le-06 le-05 le-04 le-03 log t Fig. 3. Distribution functions of relaxation times, L(ln t),
obtainedby inverse Laplace transformation from stretched exponential functions with fl = 0.35 and fl = 0.5.
decrease of fl should indicate an increase of the coupling. This would be in agreement with a transition from a liquid-like state to a solid-like one as the liquid ~ glass transition. Of course these are only qualitative explanations. Therefore, the the•retical interpretation of the fl(T) behaviour is in general still completely open. Finally, in order to give some physical meaning to the a-dielectric dynamics found below Tg, we can try to apply the coupling model. In the framework of this model, the apparent activation energy measured, Ea, is related to the energy corresponding to the elementary process, Era, through the coupling parameter r , i.e. E m = Earl. In our case, Ea is in the range 18-23 k c a l / m o l and fl is close to 0.35. Therefore, E m should range from - 5 to 9 kcal/mol. This is a reasonable range for the energy barrier to local conformational changes. These primitive motions of the a-relaxation below Tg could be interpreted as some kind of precursor of the glass-transition. We called these precursors a-relaxation in a previous paper [16]. In this respect, it is important to point out that Ngai and Yee found [17] a value of 7.5 k c a l / m o l for the primitive activation energy of the a-relaxation of PVC starting from sub-Tg stress-relaxation measurements. This is close to the value we found in PVC by dielectric measurements. Ngai and Yee interpret this elementary process as internal rotation isomerism in completely agreement with our interpretation.
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J. Colmenero / a-relaxation and molecular dynamics
3. Dynamics of the a-process above the glass-transition. Microscopical and macroscopical scales
The experimental results that will be described here are concerned with two open questions about the dynamics of glass-forming systems mentioned in the introduction. They are: how can the results obtained by different probes be related? Can the results corresponding to the time or frequency range characteristic of the standard relaxation techniques be extended to a more microscopical scale, or not? To go into these questions in more depth, we have investigated the dynamics of the a-process of a glass-forming polymeric system, Poly (vinyl methyl ether) (PVME), by means of standard dielectric spectroscopy (10 to 106 Hz), nuclear magnetic resonance (NMR) and incoherent quasielastic neutron scattering. First of all we measured the complex dielectric susceptibility in the frequency range 10-105 Hz under isothermal conditions. The temperature range covered was from Tg + 10 to Tg + 50. The value of Tg of PVME as is measured by standard calorimetric measurements is about 250 K. The experimental system and the procedure used by us have already been described in previous papers (see, for example, ref.[1]). The shape of the dielectric relaxation function does not appear to change
E;"
H-N fit ~
ix= .65 ~ c ~ a
le-01
le+01
PVME Tr = 270 K "~ ~~
~
.......... le+03 le+05 f (Hz)
le+07
Fig. 4. Master curve of the imaginary part of the complex dielectric constant of PVME built at a reference temperature of 270 K. o experimental points; - - , Havriliak-Negami fit withT=0.68 and a=0.65 (seethe text).
~ = .........................
(~( t )o.a 0.6
........
44
o4 o.2 0le-07
........... le-06 le-05 le-04 le-03 le-02 le-01 le+00 log t
Fig. 5. Time domain Fourier transform of the HavriliakNegami relaxation function with y = 0.68 and ct = 0.65. Continuous line is a stretched exponential fit with a r-value of
0.44. 'with the temperature. Then, a master plot can be built by the horizontal scaling of the log-log plots of the imaginary part of the dielectric susceptibility. This master curve can be perfectly described by means of the well-known Havriliak-Negami relaxation function [18] X,-X*(~0) 1 q)* (w) ., (1) X , - Xf [1 + ( i t o ' r H N ) v] where X, and Xr are the unrelaxed and completely relaxed susceptibility, respectively. The values of the Havriliak-Negami parameters deduced from the fit were a--0.65 and 3' = 0.68 The experimental master curve and the H - N fit are shown in fig. 4. It is interesting to point out that the time domain Fourier transform of this H - N function with the values of the above-mentioned Y parameters can be perfectly fit by stretched exponential curve ~(t)~t e x p [ - ( t / T ) ~] with a fl value of 0.44. This is shown in fig. 5. From both the temperature dependence of the shift factor used for building the master curve and the H - N fit of the master curve, we can deduce the "rHN(T ) behaviour in the measuring temperature range. This behaviour will be commented below. The dielectric results obtained by us are similar to those obtained by Kremer et al. [19]. Then, after these results, we try to fit the relaxation data corresponding t o a m o r e microscopiCal time scale (RMN), as well as the neutron
J. Colmenero / a-relaxation and molecular dynamics
scattering ones, by using the same spectral 'shape', i.e. the same H - N parameter values, In the case of nuclear magnetic resonance we have measured the 13C spin-lattice relaxation time, T1 and Tip, corresponding to the CH group in the main chain of PVME. These two magnitudes are related to the spectral density, J(¢0), by equations
suc as
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2
.
.
.
.
.
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[3
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o o
le-04
o
le-06
j y
le-10 6
+ J(~0 H - ~0c)]
(2)
(see ref. [20] for a complete description of the different parameters), Therefore, in order to describe the T1 and Tip experimental behaviour in the above-mentioned framework, we have built a Havriliak-Negami ( H - N ) spectral function: -lim (
o~
.
2
n "ICY, [3j(toc) + 6j(~0H + ¢0C) T1-1-- 10r6"
J("~) =
.
le-02 x (T)(s)
le-08
~2
.
865
1
8
10
12
14
16
1000/(V-190)(l/K)
Fig. 7. Temperature behaviour of the Havriliak-Negami relaxation time obtained from dielectric measurements: O, our measurements;D, from ref. [19]. +, NMR measurements; m, neutron scattering data. The continuous line is a Vogel-Fulcher fit with a To value of 190 K. The temperature behaviour of the zero-shear viscosity is also shown for comparison.
)
[1 + ( i o ~ . N ( T ) ) ~ ] "
"
(3)
dielectric measurements. It is important to point
More details about the experimental procedures can be found in refs. [20,21]. As can be seen in the fig. 6, the temperature behaviour of T1 and Tip can be well described by using an H - N functional form for J(to) and the values of the H - N parameters deduced from the
out that the T1 behaviour reported by Monnerie et al. [22] and corresponding to two different frequencies can also be well described by the same parameters (see fig. 6). From the fitting procedure, we obtain the temperature dependence of "rHN(T ) in the N M R experimental range. Now we can compare the relaxation time values obtained from both techniques, i.e. dielectric spectroscopy and NMR. As can be seen in fig. 7, the temperature behaviours of ZHN obtained from both techniques follow the same Vogel-Fulcher law. Then, it appears that the dynamics of PVME above the glass-transition in a wide time range (10-a°-10 -1 S) Can be described by using the same temperature behaviour of the main characteristic time scale and the same spectral shape. It is also interesting to point out that the temperature behaviour of the viscosity is different from the one corresponding to the a-relaxation time (see fig. 7). This point will be discussed in the next section. On the other hand, the dynamics of the a-proc e s s on a microscopical timescale can also be measured by m e a n s o f quasielastic N e u t r o n
1.0e+01
TI'Tlp(S)
75.4MHz~ o ) . ~ , - j •& tx A
1.0e+00 ~ - - ~ - 1.0e-Ol ~-_ ~ "
~ 1.0e-02
[] ~ 2 5 M n z
2 MHz 5 "
~ 313 ~ z
2
~
1.0e-03 1.0e-04 2.3
..................... 2.5
2.7
2.9
3.1
3.3
1000/T (K -1) Fig. 6. Temperature behaviour of 13C spin-lattice relaxation times, T1 (@) and Txp (It) measured by us. (zx) and ([3) con'espond to T1 data reported in ref. [22] and corresponding to two different frequencies. Continuous lines are the temperature behaviour of T1 and 1"1oobtained through eqs. (2) and (3) with the parameter values mentioned in the text.
scattering. We have performed incoherent quasielastic neutron scattering o n P V M E by m e a n s o f I N 1 0 and IN13 spectrometers a t t h e I L L - I n s t i t u t e
866
J. Colmenero / a-relaxation and molecular dynamics
in Grenoble. The incident wavelengths used by us were ~ = 6.28 A (IN10) and ~ = 2.23 ,~ (IN13), giving an energy resolution of B E - 1 /zeV and 8 E - 8/~eV, respectively. The Q-range covered by us was roughly between 0.2-2 A-1. The measurements were performed at two temperatures 324 K and 350 K. More experimental details can be found in ref. [21]. The main results obtained can be summarized as follows. The quasi-elastic scattering curves, S(q, to), obtained cannot be described well by a single Lorentzian function, Therefore, in order to fit these curves, we have built a scattering function, S(q, to), starting from the Havriliak-Negami frequency relaxation function as in the case of the NMR-measurements. S(q, to) can be expressed as ( ) 1 . Im S(Q, to) [1 + (ito'rHy(Q, T ) ) v ] ~ (4) Here the characteristic timescale, *HN, not only depends on T but also on Q. The values of a and "t chosen for fitting the quasielastic curves were the values used for fitting both the dielectric relaxation data and NMR-data, i.e., a = 0.65 and ~ = 0.68. Then the values of *rtN(Q, T ) were determined from the fit of the quasielastic curves (see ref. [21] for more details about the fitting procedure followed by us). The Q-variation of ~HN can be described within the experimental error by a law ~'(Q) ~ Q-4. Then, this Q-dependence allows us to compare the relaxation time values obtained from the N M R measurements to the corresponding values of the neutron measurements. The relaxation times obtained from N M R measurements at 325 and 350 K are similar to the extrapolated (by ~'(Q) tx Q - 4 ) neutron relaxation time values corresponding to a Q-value of about 0.7 .~-1. This allows us to define a spatial scale for the motions detected by N M R of about ~ - 1/0.7, i.e. ~ - 1.4 .~. However, the main result I would like to point out here is that the THN (Q = 0.7 .~-1) values obtained from neutron scattering data also fit the common temperaturf behaviour of the relaxation times obtained for dielectric spectroscopy and N M R data (see fig. 7). Therefore, the main conclusion of this section
of the paper is that, at least in the case of PVME, the dynamics above the glass-transition over a wide timescale (10-1-10 -~° s) can be well described by using the same 'spectral shape', i.e. the same Havriliak-Negami parameters or the same equivalent fl-value of the stretched exponential function. Moreover, the characteristic timescale, THN, deduced from these fits can also be described by using only one Vogel-Fulcher functional form. Then, this implies a self-consistent description of the dynamics obtained by very different probes. The major implications of these results concerns to the predictions of the mode-coupling theory of the dynamics of glass-forming liquids [23]. This theory predicts the existence of a dynamic instability at some critical temperature Tc above the experimental glass'transiti°n (T~ - Tg + 40 ° C)" The main magnitude of this theory is the density correlation function. Above Tc, this magnitude should perform a two-step decay. A fast local motion (called fl-process although with a different meaning from that which is usually referred to as fl-process in polymer science) is followed by the primary a-process. With decreasing temperature the a-process is slowed down dramatically until it does not occur below Tc. These predictions should imply a change in the temperature behaviour of the experimental characteristic timescale of the dynamics close to Tc. Moreover, a different 'spectral shape' of the relaxation function should also be expected above and below T¢. Although the theory was developed for simple classical fluids, during past few years several tests of the theory have been tried by using different glass-forming systems including some polymers [24-26]. However, in our opinion the tests made are not definitive in such complicated systems and several points remain cloudy. The results here presented indicate that at least in PVME we have not found any evidence of a critical temperature from the temperature behaviour of the characteristic timescale of the a-process as it is measured by different probes. Moreover, we can describe the dynamics in a wide time range (10-1-10 -l° s;) by using the same spectral shape parameters. Of course, this is only one example and many carefully controlled new experiments in this line would be necessary in order to clarify these points.
J. Colmenero / a-relaxation and molecular dynamics
4. Lack of thermorheological simplicity in polymers The last part of this paper refers to the question of the lack of thermorheological simplicity in polymers, i.e., the failure of the time-temperature superposition of viscoelastic data from the glassrubber transition to the terminal zone. This behaviour was first reported by Plazek and coworkers [27,28] and afterwards by several authors [29-32]. This behaviour, which clearly distinguishes glass-forming polymeric systems from other more simpler glass-forming systems, can be interpreted as a consequence of the fact that, in polymers, the glass-liquid transition is spread out into the glass-rubber-liquid transition. Due to this fact, the dynamical behaviour in this temperature range shows two main steps known as the a-relaxation (associated with the glass-rubber transition) and the terminal relaxation (associated with the rubber-liquid transition). These two processes display both different temperature behaviour and different time/frequency behaviour, Recently, several experimental studies including photon correlation [33,34] and dielectric investigation [35] together with the original viscoelastic data have shown that the shift factor of the a-relaxation varies more rapidly than the shift factor of the terminal zone as temperature is lowered towards Tg. Moreover, the terminal relaxation of polymer melts is also well described in the time domain by stretched exponential functions with r-values close to 0.6. However, this value appears to be affected by the polydispersity of the considered polymer. This value of /3 is also different from that usually found for the a-relaxation ( / 3 -
867
a-relaxation and the terminal relaxation, respectively. Here I would like to present some new results corresponding to different polymers. They are: Bisphenol-A-polycarbonate (PC); Poly(aryl ether sulfone) (PSF); Poly(hydroxyether of bisphenol-A) (Phenoxy, PH)) and a co-polymer of Bisphenol-A and the Isophthalic and Terephthalic acids (Polyarilate, PAr). In order to test the question of the lack of thermorheological simplicity as well as the coupiing-model predictions, we have performed a careful study of the mechanical relaxation behaviour (frequency range 10-x to 102 Hz) in the glass-rubber-liquid transition region of the polymers mentioned above. We have also made zeroshear viscosity measurements in a wide range (Tg + 10) to (Tg + 80) by means of a parallel plate squeezing flow technique [36,37]. The experimental details of this work can be found in ref [38]. The main results can be summarized as follows. For the four polymers investigated, the two dispersion regions corresponding to a-relaxation and terminal relaxation can be detected by mechanical relaxation measurements. The master curves corresponding to each of one of these two processes can be well described by means of stretched exponential functions. The values of the r-parameter found are shown in table 1. The temperature behaviour of the shift factors corresponding to both processes can be perfectly described by VogelFulcher equations (a. r oc e x p { B / [ T - T o ] } ) . The values of the Vogel-Fulcher parameters B and TO for the four polymers are also shown in table 1. As can be seen it is possible to assume only one TO value for the temperature behaviour of the two processes corresponding to each polymer. This
0.4). At this point, it is interesting to point out that the different values of fl as well as the different temperature behaviour of the a-relaxation and the terminal relaxation appear to be correlated in the framework of coupling model [15]. This model
Table 1 Values of the parameters characteristic of both a-relaxation and terminal-relaxation for different polymers investigated. Values corresponding to PIP are from ref. [35]
B~
B~
To
B~
predicts that
PAr PSF
0.43 0.44
0.49 0.63
415 422
1408 1250
PC
0.44 0.64 385
d In ate fl'~ d 1 / T
d In aTn fin d l I T '
(5)
where aT~ and avn are the shift factors of the
PH PIP
0.44 0.40
0.60 0.54
342 180
1149 607 504
B~ B./B.
B~/B~
1205 833
1.1 1.4
1.2 1.5
833
1.5
1.4
526 427
1.4 1.3
1.3 1.2
868
J. Colmenero / a-relaxation and molecular dynamics
fact allows the c o m p a r i s o n of the two processes within the f r a m e w o r k of the c o u p l i n g model. T h e t e m p e r a t u r e b e h a v i o u r of the t e r m i n a l r e l a x a t i o n a n d the zero-shear viscosity is the s a m e within the e x p e r i m e n t a l error. T h e values of the different p a r a m e t e r s shown in t a b l e 1 allow us to test the c o u p l i n g p r e d i c t i o n s m e n t i o n e d above, i.e. eq. (5). I n the f r a m e w o r k of the c o u p l i n g model, this e q u a t i o n c a n b e d e d u c e d
by assuming that the elementary microscopical process for the a - r e l a x a t i o n a n d the t e r m i n a l rel a x a t i o n is the s a m e (i.e. the s a m e m i c r o s c o p i c a l friction factor) b u t the c o u p l i n g of this e l e m e n t a r y process to the surrounding media is different. If the t e m p e r a t u r e b e h a v i o u r of the shift factors of the two processes c a n b e d e s c r i b e d b y V o g e l F u l c h e r equations, then eq. (5) reduces to fl~/fla = B J B , r
(7)
T h e c o r r e s p o n d i n g values of this e q u a t i o n are shown in table 1. This t a b l e also shows for c o m -
parison the values of this relationship corresponding to dielectric m e a s u r e m e n t s in C i s - P o l y i s o p r e n e ( s a m p l e PIP-38) d e d u c e d f r o m ref. [35]. I n this
particular polymer the two processes discussed here are dielectrically active. A s can b e seen in table 1, the d a t a c o r r e s p o n d ing to the four p o l y m e r s i n v e s t i g a t e d for us as well as the d a t a of C i s - P o l y s o p r e n e verify the p r e d i c t -
ions of the coupling model very well. Therefore, it appears that the lack of thermorheological simplicity is n o w clearly e s t a b l i s h e d
and that the coupling model gives a qualitative plausible explanation to this question.
5. Conclusion I n o r d e r to u n d e r s t a n d the d y n a m i c s of g l a s s -
forming polymeric systems and the proper nature of the glass-transition, it w o u l d b e necessary n o t o n l y to d e v e l o p e new theoretical a p p r o a c h e s b u t also to carefully perform new series of m e a s u r e m e n t s b y different techniques i n c l u d i n g r e l a x a t i o n
and scattering. This w o r k was s u p p o r t e d in p a r t b y U P V / E H U Project: UPV89 C O D . 206.215-0029/89 a n d b y
CICYT-Project: MEC MAT89-0186. The authors also t h a n k G i p u z k o a k o F o r u A l d u n d i a for p a r t i a l financial s u p p o r t .
References [11 A. Alegria, J. Colmenero, J.J. del Val and J.M. Barandiarhn, Polymer 26 (1985) 913. [2] J.J. del Val, A. Alegria, J. Colmenero and C. Lacabanne, J. Appl. Phys. 59 (1986) 3829. [3] J.J. del Val, A. Alegria, J. Colmenero and J.M. Barandiarhn, Polymer 27 (1986)1771. [4] J.J. del Val and J. Colmenero, Polym. Bull. 17 (1987) 489. [51 J.J. del Val, J. Colmenero, C. Mijangos, G. Martinez and J.L. Millhn, Makromol. Chem. 190 (1989) 3257. [6] A. Alegria, L. Goitiandia, I. Telleria and J. Colmenero, in: Basic Features of the Glassy State, eds. J. Coimenero and A. Alegria (World Scientific, Singapore, 1990) p. 246. [7] A. Alegria, L. Goitiandia, I. Telleda and J. Colmenero, these Proceedings, p. 457. [81 H.S. Chen, J. Non-Cryst. Solids 46 (1982) 289. [9] D. Ehlich and H. Sillescu, Macromolecules 23 (1990) 1600. [10] J. Colmenero, A. Arnau and A. Alegria, Current Topics on Non-Crystalline Solids, eds. M.D. Bar6 and N. Clavagera (World Scientific, Singapore, 1986)p. 183. [11] A. Alegria, J.M. Barandiarhn and J. Colmenero, Phys. Status Solidi (b)120 (1983) 349. [12] J. Colmenero, A. Alegria, J.J. del Val and J.M. Alberdi, Makromol. Chem. Macromol. Symp. 20/21 (1988) 397. [13] A. Alegria, J.M. Barandiarhn and J. Colmenero, Phys. Status Solidi (b)124 (1984) 409. [14] K.L. Ngai and T. White, Phys. Rev. B20 (1979) 2475. [151 K.L. Ngai, in: Basic Features of the Glassy State, eds. J. Colmenero and A. Alegria (World Scientific, Singapore, 1990) p. 265. [16] J. Colmenero, A. Alegria, J.M. Alberdi, J.J. del Val and G. Ucar, Phys. Rev. B35 (1987) 3995. [17] K.L. Ngai and A.F. Yee, J. Polym. Sci. B (1991), in press. [18] S. Havriliak and S. Negammi, J. Polym. Sci. C14 (1966) 99. [19] A. Zetsche, F. Kremer, W. Jung and H. Schulze, Polymer 31 (1990) 1833. [201 J.M. Alberdi, F. Alvarez, A. Alegria and J. Colmenero, in: Basic Features of the Glassy State, eds. J. Colmenero and A. Alegria (World Scientific, Singapore, 1990) p. 252. [21] J. Colmenero, A. Alegria, J.M. Alberdi and B. Frick, these Proceedings, p. 229. [22] R. Dejean de la Batie, F. Laupr&re and L. Monnerie, Macromolecules 21 (1988) 2045. [23] W. G~Stze,in: Liquids, Freezing and the Glass Transition, Proceedings of the Les Houches Summer School, Course LI, July 1989, eds. D. Levesque, J.P. Hansen and J. Zinn-Justin (North Holland, Amsterdam, 1991), to be published.
J. Colmenero / a-relaxation and molecular dynamics
[24] D. Richter, B. Frick and B. Farago, Phys. Rev. Lett. 61 (1988) 2465. [25] B. Frick, Colloid Polym. Sci. 80 (1989) 164. [261 B. Frick, D. Richter and L. Fetters, in: Basic Features of the Glassy State, eds. J. Colmenero and A. Alegria (World Scientific, Singapore, 1990) p. 204. [27] D.J. Plazek, J. Polym. Sci. 6 (1968)) 621; Polym J. 12 (1980) 43. [28] D.J. Plazek and V.M. O'Rourke, J. Polym. Sci. 9 (1971) 209. [29] R.W. Gray, G. Harrison and J. Lamb, Proc. R. Soc. London 356 (1977) 77. [30] D.J. Plazek, M.J. Rosner and D.L. Plazek, J. Polym. Sci. B26 (1988) 473. [31] J.Y. Cavaille, J. Perez, C. Jourdan and G.P. Johari, J. Polym, Sci. Polym, Phys. 25 (1987) 1847.
869
[32] J.M. Alberdi, A. Alegrta, E. Macho and J. Colmenero, Polym. Bull. 18 (1987) 39. [33] K.L. Ngai and G. Fytas, J. Polym. Sci. Polym. Phys. 24 (1986) 1683. [34] G. Fytas, G. Floudas and K.L. Ngai, Macromolecules 23 (1990) 1104. [35] D. Boese and F. Kremer, Macromolecules 23 (1990) 829; Macromolecules 23 (1990) 1826. [36] E. Macho, A. Alegria and J. Colmenero, Polym. Eng. Sci. 27 (1987) 810. [37] E. Macho, A. Alegria and J. Cotmenero, J. Appl. Phys. 64 (1988) 642. [38] E. Macho, A. Alegria and J. Colmenero, submitted to Macromolecules.
Journal of Non-Crystalline Solids 131-133 (1991) 860-869 North-Holland
a-relaxation and molecular dynamics in glass-forming polymeric systems J. Colmenero Departamento de Fisica de Materiales, Unioersidad del Pals Vasco, PO Box 1072, 20080 San Sebastidn, Spain
During recent years there has been a great deal of experimental effort in the study of the relaxation behaviour above the glass-liquid transition of glass-forming systems. This behaviour is directly related to the dynamics of such systems in the supercooled liquid-like equilibrium state. Therefore it is related to the glass-liquid transition phenomena. In the special case of glass-forming polymeric systems, the situation is much more complex due to the fact that the glass-liquid transition is spread-out into the glass-rubber-liquid transition. Then, the relaxation behaviour in this temperature range shows two main steps, the a-relaxation and the terminal zone. The main results obtained to date can be summarized as follows. The relaxation response usually shows a non-exponential behaviour (time or frequency domain). Moreover, the corresponding timescale displays a non-Arrbenius temperature dependence. However, in spite of these results, several questions still remain unsolved or partially solved. Some of them are: the lack of thermorheological simplicity of time-temperature superposition of viscoelastic data in the glass-rubber-liquid transition range; the relationship among the dynamic behaviours obtained from different probes; can the dynamics of the a-relaxation be described in a wide time range (1-10 -1° s) by using the same temperature dependence of the characteristic timescale and the same shape parameters?. Another interesting question is how the a-relaxation behaviour, corresponding to a polymer in the rubber state, changes when such a system gradually becomes a glass by cooling through the rubber-glass transition. The aim of this paper is to review the work recently made in this line, not only by the present group but also by other groups working in this area.
1. Introduction D u r i n g recent years a great d e a l o f effort has b e e n m a d e in the s t u d y of the d y n a m i c s of g l a s s f o r m i n g systems a b o v e their g l a s s - t r a n s i t i o n temp e r a t u r e range. N o w a d a y s it is well k n o w n that the d y n a m i c a l b e h a v i o u r of such systems is directly related to the g l a s s - t r a n s i t i o n p h e n o m e n a , F r o m an e x p e r i m e n t a l p o i n t of view, the m a i n results o b t a i n e d can b e s u m m a r i z e d as follows, (1) T h e timescales characteristic of the dyn a m i c s (e.g. r e l a x a t i o n times) follow a n o n arrhenius t e m p e r a t u r e b e h a v i o u r which is u n u s u a l in physics. I n general, this b e h a v i o u r can b e well p a r a m e t r i z e d b y m e a n s of the V o g e l - F u l c h e r law, at least in the e x p e r i m e n t a l range usually covered b y the r e l a x a t i o n techniques ( t y p i c a l l y 1 0 - 6 - 1 0 0 s).
(2) A given r e l a x a t i o n function, ~ ( t ) , usually shows a p e c u l i a r n o n - e x p o n e n t i a l b e h a v i o u r that c a n b e e m p i r i c a l l y d e s c r i b e d b y the K o h l r a u c h W i l l i a m s - W a t t s ( s t r e t c h e d e x p o n e n t i a l function) law. (3) T h e c o m p l e x susceptibilities, which det e r m i n e the r e s p o n s e o f a system to a given excitation, show a clear n o n - D e b y e f r e q u e n c y behaviour. This can b e d e s c r i b e d b y m e a n s of several e m p i r i c a l d i s t r i b u t i o n f u n c t i o n s o f D e b y e processes, the H a v r i l i a k - N e g a m i one b e i n g the m o s t general. (4) T h e scattering curves, S(q, to), o b t a i n e d b y quasi-elastic s c a t t e r i n g e x p e r i m e n t s , such as inc o h e r e n t quasielastic n e u t r o n scattering, d i s p l a y non-lorentzian behaviour. H o w e v e r , in spite of these results, several exp e r i m e n t a l q u e s t i o n s still r e m a i n u n s o l v e d o r par-
0022-3093/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
J. Colmenero / a-relaxation and molecular dynamics
tially solved. Some of them are as follows. (i) The relationship among the dynamical behaviours obtained from different probes. In particular, how to compare relaxation techniques with scattering ones; what should be the q-dependence of the magnitudes characteristic of the dynamics?, (ii) The current results reported about the dynamics of glass-forming systems correspond to the temperature behaviour above the glass-transition range. However, almost nothing is known about the change of behaviour (if any) of the a-process across the experimental glass-transition temperature range. Is the a-process completely arrested at the glass-transition temperature range? If not, how would it be possible to measure the dynamics of the a-process just below the glasstransition? (iii) Another interesting question is whether the dynamics of the a-process can be described in a wide-time range (macroscopical/microscopical) (1-10 -l° s) by using the same temperature dependence and the same spectral shape, All the results and open questions mentioned above apply to glass-forming liquids in general, However, in the special case of glass-forming polymeric systems, the situation becomes much more complex due to the fact that the glass-liquid transition is spread-out into the glass-rubberliquid transition. Then, the relaxation/dynamical behaviour in this temperature range shows two main steps, the a-process and the 'terminal zone', The lack of the thermorheological simplicity of time-temperature superposition of viscoelastic data has recently been reported by several authors and from different techniques, The aim of this paper is to present some recent experimental results corresponding to these open questions. As far as I know, there are no definitive theoretical explanations for the results which will be below described. However, in some cases I would like to mention some theoretical approaches which in my opinion could account for the observed behaviour at least qualitatively, The results described here fall into the following topics: (a) relaxation behaviour of glass-forming polymeric systems around Tg; (b) dynamics of the a-process above Tg, microscopical and macroscopical scales; (c) lack of thermorheological simplicity in polymers,
861
2. Relaxation behaviour of glass-forming polymeric systems around Tg One of the more powerful experimental techniques for studying the dynamics of glass-forming polar polymer systems is the well known dielectric relaxation spectroscopy. The standard frequency range available is from 10 Hz to 10 5 or 10 6 nz, although with adequate experimental arrangements it is possible to extend this range. Then, this technique allows us to investigate the dynamics in a timescale which ranges from 10 -1 to 10 -6 s, i.e. in the supercooled liquid-like state above the experimental glass-transition. However, in order to investigate the dielectric response by this technique, at the temperature range just below Tg, the frequency range we have to use ( - 10 -3 Hz) implies some practical problems. In addition, at this range, the glassy polymeric system evolves towards the liquid-like equilibrium state during experimental times. As a consequence, the dielectric relaxation obtained in such conditions by these classical methods should correspond to the liquidlike zone instead of the actual glassy state. There is, however, a thermally stimulated technique known as thermally stimulated depolarization current (TSDC), which could be successfully used in order to monitor the temperature behaviour of the mean timescale of the dipolar motions just below Tg. This technique is a particular time-domain technique with an equivalent frequency range of 10-2-10 -5 Hz. Over the past few years, we have used this technique in order to investigate the molecular motions just below the glass-transition of different glassy polymers like Poly(carbonate), Polyarilate, Phenoxy etc [1-5]. However, the standard TSDC experimental procedure does not easily allow information to be obtained about the non-exponential behaviour of the dielectric relaxation function, ~(t). Moreover, the standard TSDC experimental procedure is a non-isothermalprocedure and this complicates the analysis and the interpretation of the experimental data corresponding to the temperature range below Tg. For this reason, we have recently developed [6] a new TSDC isothermal experimental procedure that allows us to obtain not only the temperature behaviour of the mean dielectric relaxation time just below the glass-transition, but also the tempera-
862
J. Colmenero / a-relaxation and molecular dynamics
ture dependence of the non-exponential behaviour. In this experimental procedure, the sample is polarized above Tg and, without removing the polarization field, cooled down at constant rate (10 K/min) until the measurement temperature is reached. Then after a waiting time, tw, the electric field is removed and the depolarization current is measured as a function of time, the value of t w determines the 'state' of the glassy sample. In order to study the change of the dielectric relaxation function through the glass-transition we usually take t w as short as possible ( - 10 s). We will call this technique isothermal depolarization current (IDC). More details about this technique can be found in [6,7]. Then, by combining standard dielectric spectroscopy with the time-domain technique described above (IDC), we can investigate the change of the dielectric relaxation function through the glass-transition. To date we have studied three different polymers Poly (sulfone of bisphenol-A) (PSF), Poly (vinyl acetate) (PVAc) and Poly (vinyl chloride) (PVC). The results obtained are summarized in the following, The dielectric relaxation corresponding to the a-relaxation is not completely arrested just below the glass-transition. The experimental dielectric relaxation function shows a clear non-exponential behaviour both, above and below Tg. This be°
'
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.
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/
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.
.
.
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I
~
AH
.
.
.
.
.
.
.
.
.
.
.
80
60 40 20 . . . . . 0 390
g
410
430 T(K)
450
470
Fig, 2. Percentage rate (%) of the enthalpy recovered, AH(tm)/AHoo(%), during the measuringtime, tm, of the IDC technique at different temperatures below the glass-transition of PSF.
haviour can be well described by assuming a stretched exponential functional form for the time-domain dielectric relaxation function ~(t) ep(t)a exp[[ - ]~[ t / T]B] /p , where Tp is a characteristic timescale and fl is a non-exponential parameter. The temperature dependence of zp displays a crossover from a Vogel-Fulcher (Tp = TOexp[A/(T - To)]) behaviour (liquid-like zone above Tg) to an Arrhenius-like one (glassy zone just below Tg). The value of the fl-parameter also changes belowthr°ugh v Tg of towards about a l 0.35. a more u or e lessT constant g
10' 102 Xp(s)00 1°-2
AH(tm) 100 (%)
I
2"~21000/T(l/K)214
I
2.6
Fig. 1. Temperature behaviour of the characteristic dielectric relaxation parameters B and zp around the glass-transition of Poly (sulfone) (PSF). D, dielectricspectroscopymeasurements; o, time domain measurements (IDC technique, see the text),
Both the apparent activation energy and the B-value below the glass-transition are similar for all the polymers investigated. The apparent activation energy, Ea, ranges from 0.8 to 1 eV (from 18 to 23 Kcal/mol) and the fl-value is close to 0.35 as has been mentioned above. The results corresponding to PSF can be seen in fig. 1 as an example. This figure shows the crossover of Tp(T) from Vogel-Fulcher behaviour towards an Arrhenius-like one across Tg as well as the fl(T) behaviour. The values of the VogelFulcher parameters and the apparent activation energy are also shown in the figure. However, one of the main objections o n e c a n raise to the results obtained by the IDC procedure is if the measurements actually correspond to a
J. Colmenero / a-relaxation and molecular dynamics
clear 'glassy state', i.e. if during the measuring time the system evolves towards the equilibrium, state. To clarify this point, we have plotted in fig. 2 the percentage of the recovered enthalphy, A H ( t m)/A H~(%), during the IDC measuring time, t m, at different temperatures below Tg. The data shown in fig. 2 correspond to PSF. The enthalpy recovery measurements are described in ref. [8]. As can be seen in the figure, AH(tm)/AH~(% ) is almost zero in the temperature range T _< Tg - 30 ° Then, both the ~'p(T) and the fl(T) in this range can be considered as representatives of the 'glassy state'as formed during the cooling procedure, In relation to the crossover shown by ~p(T) across the glass-transition, it is interesting to point out that molecular transport characteristic magnitudes, other than dielectric relaxation times (viscosity and diffusion coefficient), also show this temperature dependence around the glass-transition of both, polymeric [9] and non-polymeric [10] glass-forming systems. This 'universal' behaviour can be understood in the framework of a kinetic free-volume model approach which describes how a supercooled liquid loses its equilibrium state when it gradually becomes a glass through the glass-transition temperature range. A general description of this phenomenological approach can be found in our previous papers [10-12]. In relation to the non-exponential behaviour, as far as I know there are no theoretical predictions about the temperature dependence of fl either above or below Tg. If we interpreted the r-value as a measure of the distribution of the relaxation times, a monotonous decrease, such as the one found by us below Tg, should indicate a widening of the distribution. See as an example fig. 3, where the distribution functions of relaxation times corresponding to a stretched exponential function with fl = 0.5 (above Tg) and fl = 0.35 (below Tg) have been plotted. They are obtained by means of the inverse laplace transformation (CONTIN program) This kind of behaviour also could qualitatively be interpreted in the frame of the freevolume approach abovementioned [12,13]. On the other hand, in the frame of the coupling model (see for examples refs. [14,15], fl can be interpreted as a measure of the coupling strength between the relaxing unit and the complex system. Then, a
863
0.3 ,.--, ~ ~"
0.25 0.2
o 0.5 •
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o
.35
o
o
o 0.15
••• o•
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• o• ,g 0.05 o• o•o • gO•© 0 ..... " • ~ g ~ Q ................... o • le-07 le-06 le-05 le-04 le-03 log t Fig. 3. Distribution functions of relaxation times, L(ln t),
obtainedby inverse Laplace transformation from stretched exponential functions with fl = 0.35 and fl = 0.5.
decrease of fl should indicate an increase of the coupling. This would be in agreement with a transition from a liquid-like state to a solid-like one as the liquid ~ glass transition. Of course these are only qualitative explanations. Therefore, the the•retical interpretation of the fl(T) behaviour is in general still completely open. Finally, in order to give some physical meaning to the a-dielectric dynamics found below Tg, we can try to apply the coupling model. In the framework of this model, the apparent activation energy measured, Ea, is related to the energy corresponding to the elementary process, Era, through the coupling parameter r , i.e. E m = Earl. In our case, Ea is in the range 18-23 k c a l / m o l and fl is close to 0.35. Therefore, E m should range from - 5 to 9 kcal/mol. This is a reasonable range for the energy barrier to local conformational changes. These primitive motions of the a-relaxation below Tg could be interpreted as some kind of precursor of the glass-transition. We called these precursors a-relaxation in a previous paper [16]. In this respect, it is important to point out that Ngai and Yee found [17] a value of 7.5 k c a l / m o l for the primitive activation energy of the a-relaxation of PVC starting from sub-Tg stress-relaxation measurements. This is close to the value we found in PVC by dielectric measurements. Ngai and Yee interpret this elementary process as internal rotation isomerism in completely agreement with our interpretation.
864
J. Colmenero / a-relaxation and molecular dynamics
3. Dynamics of the a-process above the glass-transition. Microscopical and macroscopical scales
The experimental results that will be described here are concerned with two open questions about the dynamics of glass-forming systems mentioned in the introduction. They are: how can the results obtained by different probes be related? Can the results corresponding to the time or frequency range characteristic of the standard relaxation techniques be extended to a more microscopical scale, or not? To go into these questions in more depth, we have investigated the dynamics of the a-process of a glass-forming polymeric system, Poly (vinyl methyl ether) (PVME), by means of standard dielectric spectroscopy (10 to 106 Hz), nuclear magnetic resonance (NMR) and incoherent quasielastic neutron scattering. First of all we measured the complex dielectric susceptibility in the frequency range 10-105 Hz under isothermal conditions. The temperature range covered was from Tg + 10 to Tg + 50. The value of Tg of PVME as is measured by standard calorimetric measurements is about 250 K. The experimental system and the procedure used by us have already been described in previous papers (see, for example, ref.[1]). The shape of the dielectric relaxation function does not appear to change
E;"
H-N fit ~
ix= .65 ~ c ~ a
le-01
le+01
PVME Tr = 270 K "~ ~~
~
.......... le+03 le+05 f (Hz)
le+07
Fig. 4. Master curve of the imaginary part of the complex dielectric constant of PVME built at a reference temperature of 270 K. o experimental points; - - , Havriliak-Negami fit withT=0.68 and a=0.65 (seethe text).
~ = .........................
(~( t )o.a 0.6
........
44
o4 o.2 0le-07
........... le-06 le-05 le-04 le-03 le-02 le-01 le+00 log t
Fig. 5. Time domain Fourier transform of the HavriliakNegami relaxation function with y = 0.68 and ct = 0.65. Continuous line is a stretched exponential fit with a r-value of
0.44. 'with the temperature. Then, a master plot can be built by the horizontal scaling of the log-log plots of the imaginary part of the dielectric susceptibility. This master curve can be perfectly described by means of the well-known Havriliak-Negami relaxation function [18] X,-X*(~0) 1 q)* (w) ., (1) X , - Xf [1 + ( i t o ' r H N ) v] where X, and Xr are the unrelaxed and completely relaxed susceptibility, respectively. The values of the Havriliak-Negami parameters deduced from the fit were a--0.65 and 3' = 0.68 The experimental master curve and the H - N fit are shown in fig. 4. It is interesting to point out that the time domain Fourier transform of this H - N function with the values of the above-mentioned Y parameters can be perfectly fit by stretched exponential curve ~(t)~t e x p [ - ( t / T ) ~] with a fl value of 0.44. This is shown in fig. 5. From both the temperature dependence of the shift factor used for building the master curve and the H - N fit of the master curve, we can deduce the "rHN(T ) behaviour in the measuring temperature range. This behaviour will be commented below. The dielectric results obtained by us are similar to those obtained by Kremer et al. [19]. Then, after these results, we try to fit the relaxation data corresponding t o a m o r e microscopiCal time scale (RMN), as well as the neutron
J. Colmenero / a-relaxation and molecular dynamics
scattering ones, by using the same spectral 'shape', i.e. the same H - N parameter values, In the case of nuclear magnetic resonance we have measured the 13C spin-lattice relaxation time, T1 and Tip, corresponding to the CH group in the main chain of PVME. These two magnitudes are related to the spectral density, J(¢0), by equations
suc as
.
2
.
.
.
.
.
.
[3
'
o o
le-04
o
le-06
j y
le-10 6
+ J(~0 H - ~0c)]
(2)
(see ref. [20] for a complete description of the different parameters), Therefore, in order to describe the T1 and Tip experimental behaviour in the above-mentioned framework, we have built a Havriliak-Negami ( H - N ) spectral function: -lim (
o~
.
2
n "ICY, [3j(toc) + 6j(~0H + ¢0C) T1-1-- 10r6"
J("~) =
.
le-02 x (T)(s)
le-08
~2
.
865
1
8
10
12
14
16
1000/(V-190)(l/K)
Fig. 7. Temperature behaviour of the Havriliak-Negami relaxation time obtained from dielectric measurements: O, our measurements;D, from ref. [19]. +, NMR measurements; m, neutron scattering data. The continuous line is a Vogel-Fulcher fit with a To value of 190 K. The temperature behaviour of the zero-shear viscosity is also shown for comparison.
)
[1 + ( i o ~ . N ( T ) ) ~ ] "
"
(3)
dielectric measurements. It is important to point
More details about the experimental procedures can be found in refs. [20,21]. As can be seen in the fig. 6, the temperature behaviour of T1 and Tip can be well described by using an H - N functional form for J(to) and the values of the H - N parameters deduced from the
out that the T1 behaviour reported by Monnerie et al. [22] and corresponding to two different frequencies can also be well described by the same parameters (see fig. 6). From the fitting procedure, we obtain the temperature dependence of "rHN(T ) in the N M R experimental range. Now we can compare the relaxation time values obtained from both techniques, i.e. dielectric spectroscopy and NMR. As can be seen in fig. 7, the temperature behaviours of ZHN obtained from both techniques follow the same Vogel-Fulcher law. Then, it appears that the dynamics of PVME above the glass-transition in a wide time range (10-a°-10 -1 S) Can be described by using the same temperature behaviour of the main characteristic time scale and the same spectral shape. It is also interesting to point out that the temperature behaviour of the viscosity is different from the one corresponding to the a-relaxation time (see fig. 7). This point will be discussed in the next section. On the other hand, the dynamics of the a-proc e s s on a microscopical timescale can also be measured by m e a n s o f quasielastic N e u t r o n
1.0e+01
TI'Tlp(S)
75.4MHz~ o ) . ~ , - j •& tx A
1.0e+00 ~ - - ~ - 1.0e-Ol ~-_ ~ "
~ 1.0e-02
[] ~ 2 5 M n z
2 MHz 5 "
~ 313 ~ z
2
~
1.0e-03 1.0e-04 2.3
..................... 2.5
2.7
2.9
3.1
3.3
1000/T (K -1) Fig. 6. Temperature behaviour of 13C spin-lattice relaxation times, T1 (@) and Txp (It) measured by us. (zx) and ([3) con'espond to T1 data reported in ref. [22] and corresponding to two different frequencies. Continuous lines are the temperature behaviour of T1 and 1"1oobtained through eqs. (2) and (3) with the parameter values mentioned in the text.
scattering. We have performed incoherent quasielastic neutron scattering o n P V M E by m e a n s o f I N 1 0 and IN13 spectrometers a t t h e I L L - I n s t i t u t e
866
J. Colmenero / a-relaxation and molecular dynamics
in Grenoble. The incident wavelengths used by us were ~ = 6.28 A (IN10) and ~ = 2.23 ,~ (IN13), giving an energy resolution of B E - 1 /zeV and 8 E - 8/~eV, respectively. The Q-range covered by us was roughly between 0.2-2 A-1. The measurements were performed at two temperatures 324 K and 350 K. More experimental details can be found in ref. [21]. The main results obtained can be summarized as follows. The quasi-elastic scattering curves, S(q, to), obtained cannot be described well by a single Lorentzian function, Therefore, in order to fit these curves, we have built a scattering function, S(q, to), starting from the Havriliak-Negami frequency relaxation function as in the case of the NMR-measurements. S(q, to) can be expressed as ( ) 1 . Im S(Q, to) [1 + (ito'rHy(Q, T ) ) v ] ~ (4) Here the characteristic timescale, *HN, not only depends on T but also on Q. The values of a and "t chosen for fitting the quasielastic curves were the values used for fitting both the dielectric relaxation data and NMR-data, i.e., a = 0.65 and ~ = 0.68. Then the values of *rtN(Q, T ) were determined from the fit of the quasielastic curves (see ref. [21] for more details about the fitting procedure followed by us). The Q-variation of ~HN can be described within the experimental error by a law ~'(Q) ~ Q-4. Then, this Q-dependence allows us to compare the relaxation time values obtained from the N M R measurements to the corresponding values of the neutron measurements. The relaxation times obtained from N M R measurements at 325 and 350 K are similar to the extrapolated (by ~'(Q) tx Q - 4 ) neutron relaxation time values corresponding to a Q-value of about 0.7 .~-1. This allows us to define a spatial scale for the motions detected by N M R of about ~ - 1/0.7, i.e. ~ - 1.4 .~. However, the main result I would like to point out here is that the THN (Q = 0.7 .~-1) values obtained from neutron scattering data also fit the common temperaturf behaviour of the relaxation times obtained for dielectric spectroscopy and N M R data (see fig. 7). Therefore, the main conclusion of this section
of the paper is that, at least in the case of PVME, the dynamics above the glass-transition over a wide timescale (10-1-10 -~° s) can be well described by using the same 'spectral shape', i.e. the same Havriliak-Negami parameters or the same equivalent fl-value of the stretched exponential function. Moreover, the characteristic timescale, THN, deduced from these fits can also be described by using only one Vogel-Fulcher functional form. Then, this implies a self-consistent description of the dynamics obtained by very different probes. The major implications of these results concerns to the predictions of the mode-coupling theory of the dynamics of glass-forming liquids [23]. This theory predicts the existence of a dynamic instability at some critical temperature Tc above the experimental glass'transiti°n (T~ - Tg + 40 ° C)" The main magnitude of this theory is the density correlation function. Above Tc, this magnitude should perform a two-step decay. A fast local motion (called fl-process although with a different meaning from that which is usually referred to as fl-process in polymer science) is followed by the primary a-process. With decreasing temperature the a-process is slowed down dramatically until it does not occur below Tc. These predictions should imply a change in the temperature behaviour of the experimental characteristic timescale of the dynamics close to Tc. Moreover, a different 'spectral shape' of the relaxation function should also be expected above and below T¢. Although the theory was developed for simple classical fluids, during past few years several tests of the theory have been tried by using different glass-forming systems including some polymers [24-26]. However, in our opinion the tests made are not definitive in such complicated systems and several points remain cloudy. The results here presented indicate that at least in PVME we have not found any evidence of a critical temperature from the temperature behaviour of the characteristic timescale of the a-process as it is measured by different probes. Moreover, we can describe the dynamics in a wide time range (10-1-10 -l° s;) by using the same spectral shape parameters. Of course, this is only one example and many carefully controlled new experiments in this line would be necessary in order to clarify these points.
J. Colmenero / a-relaxation and molecular dynamics
4. Lack of thermorheological simplicity in polymers The last part of this paper refers to the question of the lack of thermorheological simplicity in polymers, i.e., the failure of the time-temperature superposition of viscoelastic data from the glassrubber transition to the terminal zone. This behaviour was first reported by Plazek and coworkers [27,28] and afterwards by several authors [29-32]. This behaviour, which clearly distinguishes glass-forming polymeric systems from other more simpler glass-forming systems, can be interpreted as a consequence of the fact that, in polymers, the glass-liquid transition is spread out into the glass-rubber-liquid transition. Due to this fact, the dynamical behaviour in this temperature range shows two main steps known as the a-relaxation (associated with the glass-rubber transition) and the terminal relaxation (associated with the rubber-liquid transition). These two processes display both different temperature behaviour and different time/frequency behaviour, Recently, several experimental studies including photon correlation [33,34] and dielectric investigation [35] together with the original viscoelastic data have shown that the shift factor of the a-relaxation varies more rapidly than the shift factor of the terminal zone as temperature is lowered towards Tg. Moreover, the terminal relaxation of polymer melts is also well described in the time domain by stretched exponential functions with r-values close to 0.6. However, this value appears to be affected by the polydispersity of the considered polymer. This value of /3 is also different from that usually found for the a-relaxation ( / 3 -
867
a-relaxation and the terminal relaxation, respectively. Here I would like to present some new results corresponding to different polymers. They are: Bisphenol-A-polycarbonate (PC); Poly(aryl ether sulfone) (PSF); Poly(hydroxyether of bisphenol-A) (Phenoxy, PH)) and a co-polymer of Bisphenol-A and the Isophthalic and Terephthalic acids (Polyarilate, PAr). In order to test the question of the lack of thermorheological simplicity as well as the coupiing-model predictions, we have performed a careful study of the mechanical relaxation behaviour (frequency range 10-x to 102 Hz) in the glass-rubber-liquid transition region of the polymers mentioned above. We have also made zeroshear viscosity measurements in a wide range (Tg + 10) to (Tg + 80) by means of a parallel plate squeezing flow technique [36,37]. The experimental details of this work can be found in ref [38]. The main results can be summarized as follows. For the four polymers investigated, the two dispersion regions corresponding to a-relaxation and terminal relaxation can be detected by mechanical relaxation measurements. The master curves corresponding to each of one of these two processes can be well described by means of stretched exponential functions. The values of the r-parameter found are shown in table 1. The temperature behaviour of the shift factors corresponding to both processes can be perfectly described by VogelFulcher equations (a. r oc e x p { B / [ T - T o ] } ) . The values of the Vogel-Fulcher parameters B and TO for the four polymers are also shown in table 1. As can be seen it is possible to assume only one TO value for the temperature behaviour of the two processes corresponding to each polymer. This
0.4). At this point, it is interesting to point out that the different values of fl as well as the different temperature behaviour of the a-relaxation and the terminal relaxation appear to be correlated in the framework of coupling model [15]. This model
Table 1 Values of the parameters characteristic of both a-relaxation and terminal-relaxation for different polymers investigated. Values corresponding to PIP are from ref. [35]
B~
B~
To
B~
predicts that
PAr PSF
0.43 0.44
0.49 0.63
415 422
1408 1250
PC
0.44 0.64 385
d In ate fl'~ d 1 / T
d In aTn fin d l I T '
(5)
where aT~ and avn are the shift factors of the
PH PIP
0.44 0.40
0.60 0.54
342 180
1149 607 504
B~ B./B.
B~/B~
1205 833
1.1 1.4
1.2 1.5
833
1.5
1.4
526 427
1.4 1.3
1.3 1.2
868
J. Colmenero / a-relaxation and molecular dynamics
fact allows the c o m p a r i s o n of the two processes within the f r a m e w o r k of the c o u p l i n g model. T h e t e m p e r a t u r e b e h a v i o u r of the t e r m i n a l r e l a x a t i o n a n d the zero-shear viscosity is the s a m e within the e x p e r i m e n t a l error. T h e values of the different p a r a m e t e r s shown in t a b l e 1 allow us to test the c o u p l i n g p r e d i c t i o n s m e n t i o n e d above, i.e. eq. (5). I n the f r a m e w o r k of the c o u p l i n g model, this e q u a t i o n c a n b e d e d u c e d
by assuming that the elementary microscopical process for the a - r e l a x a t i o n a n d the t e r m i n a l rel a x a t i o n is the s a m e (i.e. the s a m e m i c r o s c o p i c a l friction factor) b u t the c o u p l i n g of this e l e m e n t a r y process to the surrounding media is different. If the t e m p e r a t u r e b e h a v i o u r of the shift factors of the two processes c a n b e d e s c r i b e d b y V o g e l F u l c h e r equations, then eq. (5) reduces to fl~/fla = B J B , r
(7)
T h e c o r r e s p o n d i n g values of this e q u a t i o n are shown in table 1. This t a b l e also shows for c o m -
parison the values of this relationship corresponding to dielectric m e a s u r e m e n t s in C i s - P o l y i s o p r e n e ( s a m p l e PIP-38) d e d u c e d f r o m ref. [35]. I n this
particular polymer the two processes discussed here are dielectrically active. A s can b e seen in table 1, the d a t a c o r r e s p o n d ing to the four p o l y m e r s i n v e s t i g a t e d for us as well as the d a t a of C i s - P o l y s o p r e n e verify the p r e d i c t -
ions of the coupling model very well. Therefore, it appears that the lack of thermorheological simplicity is n o w clearly e s t a b l i s h e d
and that the coupling model gives a qualitative plausible explanation to this question.
5. Conclusion I n o r d e r to u n d e r s t a n d the d y n a m i c s of g l a s s -
forming polymeric systems and the proper nature of the glass-transition, it w o u l d b e necessary n o t o n l y to d e v e l o p e new theoretical a p p r o a c h e s b u t also to carefully perform new series of m e a s u r e m e n t s b y different techniques i n c l u d i n g r e l a x a t i o n
and scattering. This w o r k was s u p p o r t e d in p a r t b y U P V / E H U Project: UPV89 C O D . 206.215-0029/89 a n d b y
CICYT-Project: MEC MAT89-0186. The authors also t h a n k G i p u z k o a k o F o r u A l d u n d i a for p a r t i a l financial s u p p o r t .
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