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The phenomenology and models of the kinetics of volume and enthalpy in the glass transition range
528
Journal of Non-Crystalline Solids 131-133 (1991) 528-536 North-Holland
Discussion Session
The phenomenology and models of the kinetics of volume and enthalpy in the glass transition range D i s c u s s i o n leaders: G.B. M c K e n n a a n d C.A. AngeU
G.B. McKenna *: This session was precipitated after Ron Rendell gave his talk on the coupling model and a rather heated debate was started on the constitutive relations and the physics problems in the volume recovery experiments. I thought what I'd do first is put up this outline viewgraph so that we have some points that we can talk about. There may be some other things that we want to bring up as the discussion goes along but let's try and go through this first so we are all talking about the same thing. Let's first look at the points that we want to make and address and see where we have agreements and disagreements. First I'd like to go through and ask what are the experiments that are of interest (and there may be other ones that some other people will want to point out) in describing the phenomenology of the kinetics of volume or enthalpy recovery near the glass-transition range. Austen and I have put together this list of the experiments that we think are important. I don't want to have to describe these unless people really don't understand what they are because we have to assume we're talking some common language here. First, there are the symmetry of approach experiments. There is the %fr paradox which comes out of these asymmetry of approach experiments. Then there is the memory effect. These are basically T-jump experiments in which you do the measurement in an isothermal state, but there are non-isothermal experiments such as DSC where you do temperature scans. I don't know if there are any others that people think are of interest to describe.
* Transcribed from recording.
I think that as we go through the phenomenology, there are some essential ingredients that are required to describe these experiments at least qualitatively. These are things that have been seen since... Andr6 Kovacs would tell me since the days of S i m o n . . . but at least since the days of Tool. The retardation times depend on glassy structure or the fictive temperature if you like. Out of the memory effect comes the observation that we need to have some sort of non-exponential retardation or relaxation function, some sort of non-exponential response function to describe this. A single exponential won't do it. In addition to those two things, because we're describing this phenomenologically and the behavior looks like a viscoelasticity problem, we need some sort of constitutive law. And that's what originally precipitated this session. Even if we put all the correct physics in but use the wrong constitutive law, it's not going to come out. There are the N a r a y a n a s w a m y - M o y n i h a n type of models. I think they're more or less equivalent at least as constitutive laws although not necessarily in the details of the physics that go into them. There is the K o v a c s - A k l o n i s - H u t c h i n s o n - R a m o s model or K A H R model as it is known in the polymer community. There is also now the Scherer-Hodge model which puts in the additional A d a m - G i b b s assumption. Then there's the N g a i - R e n d e l l model which is another form of constitutive model. If there are any others that people want to talk about, we'd be glad to do that. Austen, do you have any comments? C.A. Angeli *: I would like to welcome everybody here. I think everybody has come because there was a smell of blood on the floor the other day
0022-3093/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
Discussion Session / Kinetics of volume and enthalpy
and people like that sort of thing. I think we should start off with a brief re-statement from both Ron Rendell and Connie Moynihan to emphasize the initial points of contention and then let it roll on from there. Perhaps we can have some additional information. I know that at least one member of the audience, Oguni, has isothermal relaxation measurements in this area, and he may be able to make a contribution. R. IV. Rendell §: In the presentation, I had focussed on one feature of the structural relaxation models and that is the form of the relaxation function in the non-linear regime. Although the relaxation is found to be approximately K W W in structural equilibrium from linear response experiments, K W W is actually not used in any of the structural relaxation models. Rather some generalization of K W W is used. The K W W function from linear response has traditionally been generalized to the non-linear regime using the concept of 'reduced time' where t / r in K W W is replaced by f d t / ~ . Reduced time model: /3 Non-linear
Linear
The reduced time method is an assumption and must be tested by experiment. The reduced time expression is sometimes still referred to as K W W and this can cause confusion. Alternative generalizations can be considered which are not equivalent to reduced time. I have used one here based on the cot~ling model approach. The coupling model instead uses a time-dependent rate. Coupling model: e x p - f01 d t ' ( ~ c t ' ) -- n ~%IT, Tf] Non-linear exp - ( t /'r )1 - , , Linear where ¢ =
[(1
-
n),o'2"ro]1/{1-').
§ F r o m written version submitted afterwards.
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Although the reduced time and coupling model expressions both reduce to the K W W function in linear response when T is constant, they have different properties in non-linear structural relaxation. Even in linear response, the coupling model reduces to K W W plus an additional relation which relates the K W W ~- to r0: n
11/(l-n)
~"= rztL1 _ n) O~cr0]
We have verified this relation between ,r and ~'0 using linear response data from m a n y different processes and materials. This relation is also not compatible with reduced time. C.T. Moynihan *: I had a number of comments during the discussion but what I wanted to do to start out with was to show you something that I didn't get a chance to show you the other day because I ran out of time. These are some data we've recently obtained on a lead-silicate glass where in the non-linear regime we weren't able to get a fit. We tried just about everything we could think of in modifying the assumptions that are made to try to generate a fit and so far failed. I'll just go through this very quickly. We have a lot of experiments but I'll just illustrate it by one where we took the sample well above the glass-transition temperature, cooled it very quickly to some temperature Ta, an annealing temperature, and then held it for various amounts of time. This was all done on the DSC. After annealing isothermally for a certain amount of time on the DSC, we would reheat the material through the glass-transition region allowing us to measure the enthalpy that had relaxed out or measure its fictive temperature. The Tg for this glass is about 710 K. We did this at three temperatures and these are the best fits we could generate using the A d a m - G i b b s model. We also used the N a r a y a n a s w a m y model which assumes an Arrhenius temperature dependence and produced fit curves which were absolutely identical. The thing to note is that if we're not too far from equilibrium, say for annealing at 700 K, we get a rather good fit and in fact the sample does come to equilibrium. If you go a little lower in temperature to 680 K, it's still not too bad. But when you try to anneal out at very very low temperatures, we don't even come close.
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Discussion Session / Kinetics Of volume and enthalpy
Scherer has made this observation. I mentioned it earlier with regard to the glycerol data. The problem is that the non-linearity corrections we've made don't work very well far from equilibrium. There's some indication from the comparison of our glycerol data with the Birge and Nagel data that in fact they don't even work very close to equilibrium and all we're seeing is a sort of a compromise fit that doesn't match the fls, for example, that you get from the linear data. We tried to fix up these fits in a variety of ways. Note the formalism we used to handle the KWW function. This is simply because it is more economical of computer time to break it up into an equivalent relaxation time distribution. Whether you handle this directly in this form using the superposition principle or you handle it in this other form using fl really doesn't make a bit of difference. They're totally equivalent. One of the key assumptions in handling the non-linearity is that the various relaxation times depend upon the temperature and the average fictive temperature of the system. We tried violating that assumption by letting them depend partly on the individual fictive temperatures for the order parameters using an additional adjustable parameter, y. As we started going away from the original Tool or Narayanaswamy approach, the fits simply got worse. So this is not where the problem lies. We tried modifying the shape of the KWW function, thinking that perhaps we haven't got it quite right. Since trouble seems to come in at short times, we added a tail to it. So we modified our weighting factors, varying this constant C (renormalizing of course before we did the calculation). A search through parameter space found an effective distribution of relaxation times which was identical to the original best fit KWW. So this is not where the problem lies. We tried, like Scherer did, to abandon thermo-rheological simplicity to allow for fl to be temperature dependent. The way you do this is to assume that you have a distribution of activation energies. No improvement. Finally we tried changing the functional form of the non-linearity correction, modifying it using an odd power of Tf - T to speed it up relatively when the sample was above equilibrium and slow it down when it was below No improvement.
I don't know where the difficulty is. I also have a s u s p i c i o n l a n d I think we've got to crank this through in more d e t a i l - - t h a t the proposal by Ngai and Rendell is, in the end, somewhat contained in at least some of these things. That we've done equivalent things, tried testing the data, and don't find much improvement, so there's a real problem here. The final thing I'd like to note with regard to fitting this classical data of Kovacs is that, when trying to deal with non-linearity, you're worried about the start of the relaxation. That things are either slower or faster than they ought to be if you had assumed linearity. What happens way down at the tail when you're close to equilibrium usually doesn't get included in the typical fits. You're almost at equilibrium and if you miss that by half a percent, nobody worries about it. I think fitting the tail end of that %ff is not addressing what the real problem is. C.A. Angell *: I was wondering whether Dr Oguni from Tokyo Institute of Technology might be able to throw some light on this. The reason I'm directing it to him is that I'm aware that he and also the people in Osaka do very fine isothermal relaxation and are in a position to add information to this situation on conditions far from equilibrium. A.J. Kovacs *: Mr chairman, can I just reply to Moynihan before going on further because he, in a certain way, misquoted me I think. I would like to ask him a couple of questions and then we can go further. How do you know where the equilibrium is - was it determined accurately or by some extrapolation? It is not difficult to extrapolate but this is not necessarily the real equilibrium. The theories contain the real equilibrium. The driving force depends on the departure of your system from the real physical equilibrium and not on an extrapolation. C.T. Moynihan *: In many cases, we've annealed for such a long time that the properties are no longer changing with time within our experimental precision. The values we measure at that point agree very well in the cases where we extrapolate using the liquid and glass-like heat capacities to estimate what the enthalpy should be at that low annealing temperature.
Discussion Session / Kinetics of volume and enthalpy
A.J. Kovacs *: Was this the case for this last curve
you showed us? C.T. Moynihan *: Yes, it was. That was part of our results but we annealed as low as 650 K, which was our lowest temperature of experiment, outside the DSC in this case, until things did not change any more. So there's no doubt where equilibrium lies, at least in so far as we can determine it in a few months. A.J. Kovacs *: I don't think that in the DSC you annealed for 1000 hours. C.T. Moynihan *: No, these were separate furnace anneals. I didn't show you all the data, just the particular example that happened to be done in the DSC. There are other samples annealed in outside furnaces for some months' time. A.J. Kovacs *: I would like to make the general comment that for experiments where the approach to equilibrium is by contraction, it is much better and more precise to make the measurement by volume than by DSC because then for each point you have to recycle. C.T. Moynihan *: The more complicated your response experimentally, the more accurately you're able to pin down the parameters. There's very little that's more complicated than a DSC curve with a shoulder on it or an undershoot and an overshoot. We found that if you look strictly at approach curves, sometimes the fits are not too bad. But you really have problems when you try to fit the derivatives which we measured directly. The predictions there were based on fitting the reheating curves as well as where the sample annealed isothermally. A.J. Kovacs §: If I understood correctly, there is some suspicion about the accuracy of the CeTf~ ( = -- (V -- V ~ ) - 1 d v / d t) values as derived from the volume, v, expansion isotherms of PVAc, and thus about the reliability of the %ff 'gap' reported in my review paper in 1963 [[1] A.J. Kovacs, Adv. Polym. Sci. 3 (1963) 394] and shown by Dr R. Rendell in his talk. If this suspicion were founded, the discrepancy should also appear in the %ff-values derived from the volume contraction isotherms, obtained at the same temperature and using the same procedure as for expansion. Nevertheless, the contraction isotherms do converge asymptotically towards the equilibrium volume at
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the same value of the experimental time, thus displaying no '/'eft gaps. As a consequence, the latter cannot originate from experimental inaccuracy. Rather they result in a genuine difference between the approach of the volume toward its equilibrium value either from above or from below. The difference between the contraction and expansion isotherms is in fact clearly apparent on the approach of v towards its equilibrium values, v~, depicted by fig. 17 in ref. [1] from which the relevant %ef isotherms shown by Dr Rendell have been derived. Note that the experimental error in this figure is of the order of the size of the datapoints. Finally, I would like to mention that these experiments on PVAc have been repeated many times without showing discrepancies other than those inherent to the experimental error ( A v / v = + 1 0 - 5 ) . Furthermore, similar %ff gaps have also been obtained with a glucose and a polystyrene glass [J.M. Hutchinson and A.J. Kovacs, unpublished]. Consequently, the %ff gap is a quite general and highly significant phenomenon which reveals some major discrepancies of retardation models, at least when applied to volume (and presumably to enthalpy) recovery of glass-forming systems. The %ff paradox was identified as early as 1963. Subsequently, the multi-parameter K A H R model was developed in 1979 to describe many other features of volume recovery, but it was clearly recognized at that time that it would not explain the %ff paradox. This problem is complex and I have a better idea now than I had in 1963 about its origin. Perhaps the Ngai-Rendell coupling model will be an answer, but I am not sure. G.B. M c K e n n a *: This is a question of which experiments should be described. The %ff paradox is something that's happening when you get very close to equilibrium. And the models aren't fitting when you're very close to equilibrium in that case, except perhaps for Ron Rendell and Kia Ngai's model which seems to be able to fit that part of it. However, Moynihan also finds a breakdown of the basic formalisms when you're far from equilibrium.
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Discussion Session / Kinetics of volume and enthalpy
J.M. Hutchinson §: Moynihan's contention is that the equation [xAh
-- rio exp l---h-r-- + (l-y)(1-x) -~
y(1-x)
Ah
RTflave
Ah]
RTfi
(1)
gives a worse fit than does Ti
=
'rio exp ~
+
~-T-~v2
.
(2)
This cannot be so. The added adjustable parameter cannot possibly make the fit worse. Equation (1) is identical to eq. (2) when y = 1, and must therefore give, at worst, a fit as good as eq. (2). C.T. Moynihan *: Nature seems to like y = 1. When you go to the other extreme and set y = 0, you predict approach curves that are totally nonphysical and with enormous undershoots which are in no way observed experimentally. The fact that you use the average structure to modify the relaxation times for non-linearity is a built-in assumption that started with Tool and continued with Narayanaswamy and so forth. But it is in fact an assumption. We've tested this experimentally and it turns out that it's the best assumption that could be made. Rather than letting things depend on individual fictive temperatures, it just depends on the overall structure. J.M. Hutchinson *: But when you have an adjustable parameter, it cannot be worse. C.T. Moynihan *: It is worse. The two extremes are that the relaxation time depends only on the average fictive temperature and the other extreme is that each individual order parameter relaxation time depends upon its own fictive temperature. Simon Rekhson tried a few calculations of this sort too, I believe. What happens is that the short relaxation times, if you set y = 0, get hyper-accelerated. The long relaxation times get hyper-retarded. And the data just don't behave that way. It's like using a log-Gaussian instead of a KWW. It just doesn't fit. S. Matsuoka §: The spectrum for the non-equilibrium glass depends on the degrees of departure from the equilibrium. In my lecture on Wednes-
day [these Proceedings, p. 293], I presented the T-dependence of the stress relaxation data and proposed a distribution of sizes for configurational cooperative domains as the origin of the distribution of activation energies. If the distribution is assumed independent of T in the temperature range of interest in equilibrium then, when the temperature is suddenly shifted, fl will decrease if cooled and increase if suddenly heated. During the subsequent recovery process, fl will tend towards the equilibrium value. The change of fl with time (one is not at liberty to change it in any arbitrary manner here) will definitely improve the curve fitting for the very well-known - l o g r vs. 8 by Kovacs. R.W. Rendell §: With reference to Moynihan's earlier comments, I think a basic point of the structural relaxation models is still being missed. Different generalizations of KWW to the non-linear regime are not necessarily equivalent. The coupling model version cannot be transformed into the reduced time version. And the results can be quite different if each of them is used to calculate structural relaxation. And this is true even if we use the same value of fl = 1 - n and the same functional form for ¢[T, Tf] in both relaxation functions. The reduced time function has an integral over 1 / r and then the non-exponentiality is introduced by raising the whole integral to a power of ft. By contrast, the coupling model has an extra factor of (o~ct) -n inside the integral and that is what introduces the non-exponentiality. The reduced time deals first with the non-linearity through the integral over 1 / ¢ and then afterwards puts in the non-exponentiality. The coupling model deals with the non-linearity and the non-exponentiality at the same time within the integral. In particular, Moynihan had tried several new forms for ~'[T, Tf] in the reduced-time model for the lead-silicate glass but none of these modifications is equivalent to the coupling model. And there may be yet other generalizations of KWW, besides the coupling model and reduced-time approaches, which could be considered. The functional form to be used for z[T, Tf] in either model should also come from experiment. The only attempt I am aware of to obtain experimental information on ,r[T, Tf] was by Kovacs,
Discussion Session / Kinetics of volume and enthalpy Stratton and Ferry on PVAc. They measured the shear moduli G ' ( ~ ) , G"(~0) for similar samples and sample histories and constructed shift factors as a function of T and time after quench or equivalently, T and Tf. Additional experiments that give information about T[T, Tf] are very much needed. M. Oguni §: Physical aging has been studied in the time-domain and frequency-domain. But the two methods have to be considered separately. When it is studied by the frequency-domain method, the sample is located almost in the equilibrium state. On the other hand, when studied by the time-domain method, it should be located in the state apart from equilibrium. We have studied isothermally the enthalpy relaxation by using an adiabatic calorimeter, and showed already the result of vapor-deposited amorphous butyronitrile which is located in the state quite far from the equilibrium. When the enthalpy relaxation was fitted with the KWW equation, we obtained the small value of fl of 0.1 or so. Usually the fl value is around 0.6 or 0.5. Next we followed the enthalpy relaxation of liquid quenched propylene glycol which was studied by Professor Nagel's group by the method of frequency domain. Nagel's group obtained the fl value of 0.61 _+ 0.04. Then a r-value in agreement with the data of Nagel's group was obtained only when we took the temperature jump of zero. C.A. Angell *: So he's saying that when he's watching the relaxation far from equilibrium, the r-value is very much smaller in the case of butyronitrile and certainly in the same direction for propylene glycol. That would apparently need to be incorporated into any of these fits. C. 7". Moynihan *: No attempt has been made to account for the non-linearity via any one of these techniques. It depends upon which way you're approaching equilibrium - - if you're coming down it's going to look faster and faster whereas if you're going up it's going to look slower and slower compared with what it would be if you were very very close to equilibrium. I think that's why fl is changing. As he does his experiment very close to equilibrium, he'll converge on the value you would get from a linear experiment. So it sounds all very consistent, but it's not to be inter-
533
preted that the KWW function isn't working or something like that. J. O'Reilly §: The fls obtained from calorimetric (enthalpy) relaxation do not account for non-linearity and therefore cannot be compared directly with those by DSC which do. C.A. Angell *: When we were talking about experiments close to equilibrium, Dr Kovacs was pointing out that DSC is not as accurate a method as dilatometry. But I think the quality of the data that Oguni has obtained probably are very comparable. G.B. McKenna §: I submit that the difference in volumetric measurements and DSC experiments can be compared by an analogy with mechanical measurements. First, consider that the stimulus is temperature and the volume or enthalpy changes are the responses. The T-jump dilatometric experiments (if we had a linear system) are equivalent to a step-strain experiment. So at constant temperature (or strain) after imposition of the stimulus, we measure directly the response (volume or stress). In the typical DSC experiments, the measurement (or at least the scanning part) is equivalent to a constant rate of strain experiment. In these cases one measures some integral of the response function. On the other hand, in volume experiments we measure directly the extensive thermodynamic variable while in DSC we measure a derivative of the enthalpy, i.e. Cp. Each type of measurement will have advantages and disadvantages, although I prefer the T-jump experiment and expect that those who do DSC will prefer those types of measurements. L. Struik §: Comparing dilatometry and DSC, dilatometry has the advantage that the thermal equilibrium time is greater than that of DSC sample. In cooling cycles, this is a great disadvantage because of the high non-linearity of the process. C.T. Moynihan *: Sometimes in the DSC we try to fit the heat capacity curves but, as in the experiments I just described, the samples were annealed isothermally and we subsequently use the DSC to measure the total amount of enthalpy that the sample had lost. It's like measuring the area under a melting peak effectively. In this case it doesn't matter if your data are a little smeared out because your sample is not exactly isothermal. You
534
Discussion Session / Kinetics of volume and enthalpy
still get the right energy out. So, the DSC data aren't always measuring derivatives, sometimes they are measuring total quantities, and this was what I showed up there. J. O'Reilly §: DSC and adiabatic calorimetry are different by a factor of 10: 1% by DSC and 0.1% by adiabatic calorimetry. The sensitivity of the adiabatic calorimeter requires longer equilibration times but measures smaller changes in energy. The DSC measures larger changes in shorter times. Dilatometry measures AT to 10 -5 cma/g but the accuracy in Aa and ATf is less. ATfdil is Av _ 10- 5 cm3/g = 0.025 ° C. Aetr8 4 × 1 0 - 4 / K The precision of the DSC is _+1% in Cp and A H -- 0.01 J / g , yet the sensitivity of AH 0.01 J / g AT~o~c= ~ - 0.20 J / g K ----0.05 o C. Dilatometry is two times more precise in ATf measurements. G.B. McKenna §: I think that we should return to one of the focuses of this discussion and that is the combination of points 2 and 3 in our points for discussion vis h vis the Ngai-Rendell model. Here I don't know if the apparent resolution of the zefr paradox by this model is due to the fact that the constitutive model proposed is responsible or is it that the underlying physics (essential ingredients, points) of the Ngai-Rendell model are responsible? If we look at the constitutive law, it is not the same as the K A H R or Narayanaswam y - M o y n i h a n type models, since it allows the spectrum of relaxation times to change in shape as a function of the entire thermal history in a different way. On the other hand, there is also some 'new' physics in the formulation of the Ngai-Rendell coupling model and we need to ask if the new and better fits to Teff require the constitutive law or the new physics or both? I.M. Hodge §: I have performed preliminary calculations of d T f / d T that indicate that, using the same values of x, Ah* and r , the effective-time expression in which fl lies outside integral gives very different results than the Ngai-Rendell expression, where fl is part of the integrand. The
heat capacity overshoot calculated using the Ngai-Rendell expression was broader and lower. Comparisons of the two expressions in fitting experimental heat capacity data for a range of histories have not yet been attempted. C.A. Angell *: Ron, do you have any feeling for which way your model would change the results after having seen the direction in which, using isothermally determined parameters, Moynihan failed to predict the overshoot for glycerol? R. W. Rendell §: I haven't yet looked at the glycerol DSC data using Birge and Nagel's linear response parameters with the coupling model. However, for the same set of parameters, the DSC peak from the coupling model is found to be broader than the corresponding peak from the reduced time model. For a given experimental DSC curve I expect the coupling model to require a larger value of fl = 1 - n than the reduced-time method. Since Moynihan had to use a fl which was smaller than Birge and Nagel's value, the change in using the coupling model is expected to be in the right direction to agree with the experiments. However, a quantitative comparison remains to be done. J.-C. Bauwens §: The KWW relation is not the result of a differential equation. I prefer a hyperbolic law:
where ~ is the phenomenon and To the time constant, because it is the result of the differential equation dq~/~ 0 = d t / z , where r = ToC. For instance the Rouse model gives a = 1, then:
fl = 1 / ( a + 1). E. Oleynick §: I would like to call your attention to some experimental data using calorimetry and volume measurements in one experiment which probably is not known to most of those here. We measured simultaneously for one sample of polystyrene, using the technique of temperature jumps, the relaxation of volume and the relaxation of
Discussion Session / Kinetics of oolumeand enthalpy
enthalpy. Unfortunately I didn't bring my slides with me but I have a paper here. We put the dilatometer inside the differential calorimeter and measured the change of the volume. We equilibrated the sample at T = 120 ° C for a long time. It was then put into the second calorimeter which also contained the dilatometer, at temperatures between T = 90 ° C and approximately T = 70 ° C. This was for contraction of the volume. In other cases, we kept the sample at T = 120 ° C for some time and then put it at the lower temperature for something like 1000 min. Then we made the relaxation at a higher temperature so that the sample becomes expanded. At the same time we measured the volume and the rate of enthalpy relaxation. So we have at any moment the absolute relaxation time, without any scanning calorimetry, of volume and enthalpy. If anyone would like to look at the results, I have them here. We have the data for the rate of volume relaxation and the rate of enthalpy relaxation for both contraction and expansion. I would like to emphasize two points here. First, we have found, for all histories and all measurements that we did, a one-to-one correspondence between enthalpy relaxation and volume relaxation. N o difference, as you can see. Second, there is an important point which I cannot explain but would hke to put to you. The problem is that the amount of energy required for changing the volume differs from ACp/Act, where a is the coefficient of thermal expansion, by 30-40%. This is at least one order of magnitude larger than our error for the one-toone correspondence between volume and enthalpy. I don't know why it is, but it's observed experimentally. A.J. Kovacs §: Your data seem to confirm the %ff gap. S. Rekhson §: I agree that your (Rendell et al.) computations provide a better account for the Teff paradox than those using the T o o l - N a r a y a n a s w a my (TN) model. However, in your computations the shift function is adjustable, whereas in T N computations it is fixed by an Arrhenius or A d a m s - G i b b s relationship. The shift function resulting from your computations for T = 35 ° C has twice the activation energy of the shift function that you needed to account for the behavior at
535
40 o C. This is unrealistic. I understand that you now have this difficulty resolved. Could you elaborate on this? R.W. Rendell §: In collaboration with J.J. Aklonis, we have previously examined Kovacs' volume recovery data for PVAc using the coupling model relaxation function. To test the model, we need n and T[T, Tf]. Kovacs, Stratton and Ferry (KSF) had measured the shear moduli G'(~0) and G"(~0) for similar samples and sample histories in order to obtain a shift factor for T. However, the G " curves changed shape during volume recovery and the curves could not be reduced. This indicates that n changes during the structural relaxation in PVAc. Values of n could not be obtained from the G " curves since only 1.5 decades in frequency were available. The G ' curves could be approximately superimposed and K S F constructed a shift factor which could be represented as a modified Doolittle equation for T[T, T~]. Information about n from different PVAc samples is available from Plazek's shear creep compliance data and Mashimo's dielectric data. Our fit of Kovacs' %ff data at T = 3 5 ° C reproduced closely the shapes of the curves and the size of the 'expansion gap' using the K S F shift factor. The shapes and expansion gaps at T = 4 0 ° C could also be closely fit using the K S F shift factor but the position of - log O'efr relative to the T = 35 o C curve is slightly too large. The best fit of the T = 4 0 ° C data required deviations from the K S F shift factor as we showed in our paper. To examine these comparisons more carefully, we should obtain values of n from the same Kovacs samples and the dependence of T on Tr which does not depend on shifting the shear moduli in a situation where they are changing shape. This could be done if G " were available over several decades of frequency. An indication that these effects may improve the comparison with the T = 40 ° C data is seen in an exercise where we assumed that TO has a modified Doolittle form rather than T. F r o m the coupling model, TO should not be affected by changes in n. Obtaining parameters from the equilibrium KSF shift factor, we could reproduce the shapes, expansion gaps, and relative positions of the 35 and 40 ° C data. G. Williams §: The essential phenomena of iso-
536
Discussion Session / Kinetics of volume and enthalpy
thermal and non-isothermal relaxations of the thermodynamic properties of non-polymeric and polymeric glass-forming systems have been rationalized, at least semi-quantitatively, using the double-box model (KAHR model) or the stretched exponential function (KWW function). It has become apparent that, although semi-quantitative agreement may be obtained between experimental results and the phenomenological models, e.g. for specific heat data, for the memory effect, and for the non-linear isothermal volume relaxation data, the fits may not be made quantitative for sensitive variables (e.g. Cp, which is ( ~ H / ~ T ) p ) . Much discussion has surrounded the question of the use of particular forms for T(T, t). I wish to add the comment that the stretched exponential function, ¢p(t) = exp - (t/T o )~, is likely to be inadequate for t/T o << 1, as we found in our original work on the dielectric a-relaxation for amorphous polymers measured as e(~o) in the frequency domain. Also, the theory of molecular time-correlation functions requires d ~ p ( t ) / d t ~ O as t ~ 0 , whereas the KWW function gives infinite slope as t ~ 0. Therefore, it seems likely that the stretched exponential function will not give a good description of thermodynamic relaxation data for t/T o << 1, and special attention should be given to this aspect of the analysis of such data. S. Matsuoka §: The importance of a 'good' con-
stitutive equation is not just to fit data well, but also that the 'constants' have some physically tractable values. I show you first the log T vs. 8 fit, using the equation we published in Macromolecules, i.e. In T= ~A# - T ( Z + Z* ) + C for thermodynamic recovery, vs. A~ l n T = -k-T Z + C for dielectric relaxation in which A/~, Z(Tf) are identical. This leads to a hypothesis that the density fluctuations are larger and longer than the rotational (dielectric) fluctuations. L. Struik §: Part of the problem with the theory for volume relaxation may be due to the fact that there are also considerable contributions from fland a-processes, which follow different laws and have a different sensitivity to fictive temperature or free volume. In a recent paper (Polymer) I showed that the secondary peaks in volume relaxation may be as high as the peak at Tg. So, a description by a simple formula such as KWW, may lead to problems. I suggest applying the method shown by Dr Read [these Proceedings, p. 408] to separate such processes.
Journal of Non-Crystalline Solids 131-133 (1991) 528-536 North-Holland
Discussion Session
The phenomenology and models of the kinetics of volume and enthalpy in the glass transition range D i s c u s s i o n leaders: G.B. M c K e n n a a n d C.A. AngeU
G.B. McKenna *: This session was precipitated after Ron Rendell gave his talk on the coupling model and a rather heated debate was started on the constitutive relations and the physics problems in the volume recovery experiments. I thought what I'd do first is put up this outline viewgraph so that we have some points that we can talk about. There may be some other things that we want to bring up as the discussion goes along but let's try and go through this first so we are all talking about the same thing. Let's first look at the points that we want to make and address and see where we have agreements and disagreements. First I'd like to go through and ask what are the experiments that are of interest (and there may be other ones that some other people will want to point out) in describing the phenomenology of the kinetics of volume or enthalpy recovery near the glass-transition range. Austen and I have put together this list of the experiments that we think are important. I don't want to have to describe these unless people really don't understand what they are because we have to assume we're talking some common language here. First, there are the symmetry of approach experiments. There is the %fr paradox which comes out of these asymmetry of approach experiments. Then there is the memory effect. These are basically T-jump experiments in which you do the measurement in an isothermal state, but there are non-isothermal experiments such as DSC where you do temperature scans. I don't know if there are any others that people think are of interest to describe.
* Transcribed from recording.
I think that as we go through the phenomenology, there are some essential ingredients that are required to describe these experiments at least qualitatively. These are things that have been seen since... Andr6 Kovacs would tell me since the days of S i m o n . . . but at least since the days of Tool. The retardation times depend on glassy structure or the fictive temperature if you like. Out of the memory effect comes the observation that we need to have some sort of non-exponential retardation or relaxation function, some sort of non-exponential response function to describe this. A single exponential won't do it. In addition to those two things, because we're describing this phenomenologically and the behavior looks like a viscoelasticity problem, we need some sort of constitutive law. And that's what originally precipitated this session. Even if we put all the correct physics in but use the wrong constitutive law, it's not going to come out. There are the N a r a y a n a s w a m y - M o y n i h a n type of models. I think they're more or less equivalent at least as constitutive laws although not necessarily in the details of the physics that go into them. There is the K o v a c s - A k l o n i s - H u t c h i n s o n - R a m o s model or K A H R model as it is known in the polymer community. There is also now the Scherer-Hodge model which puts in the additional A d a m - G i b b s assumption. Then there's the N g a i - R e n d e l l model which is another form of constitutive model. If there are any others that people want to talk about, we'd be glad to do that. Austen, do you have any comments? C.A. Angeli *: I would like to welcome everybody here. I think everybody has come because there was a smell of blood on the floor the other day
0022-3093/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
Discussion Session / Kinetics of volume and enthalpy
and people like that sort of thing. I think we should start off with a brief re-statement from both Ron Rendell and Connie Moynihan to emphasize the initial points of contention and then let it roll on from there. Perhaps we can have some additional information. I know that at least one member of the audience, Oguni, has isothermal relaxation measurements in this area, and he may be able to make a contribution. R. IV. Rendell §: In the presentation, I had focussed on one feature of the structural relaxation models and that is the form of the relaxation function in the non-linear regime. Although the relaxation is found to be approximately K W W in structural equilibrium from linear response experiments, K W W is actually not used in any of the structural relaxation models. Rather some generalization of K W W is used. The K W W function from linear response has traditionally been generalized to the non-linear regime using the concept of 'reduced time' where t / r in K W W is replaced by f d t / ~ . Reduced time model: /3 Non-linear
Linear
The reduced time method is an assumption and must be tested by experiment. The reduced time expression is sometimes still referred to as K W W and this can cause confusion. Alternative generalizations can be considered which are not equivalent to reduced time. I have used one here based on the cot~ling model approach. The coupling model instead uses a time-dependent rate. Coupling model: e x p - f01 d t ' ( ~ c t ' ) -- n ~%IT, Tf] Non-linear exp - ( t /'r )1 - , , Linear where ¢ =
[(1
-
n),o'2"ro]1/{1-').
§ F r o m written version submitted afterwards.
529
Although the reduced time and coupling model expressions both reduce to the K W W function in linear response when T is constant, they have different properties in non-linear structural relaxation. Even in linear response, the coupling model reduces to K W W plus an additional relation which relates the K W W ~- to r0: n
11/(l-n)
~"= rztL1 _ n) O~cr0]
We have verified this relation between ,r and ~'0 using linear response data from m a n y different processes and materials. This relation is also not compatible with reduced time. C.T. Moynihan *: I had a number of comments during the discussion but what I wanted to do to start out with was to show you something that I didn't get a chance to show you the other day because I ran out of time. These are some data we've recently obtained on a lead-silicate glass where in the non-linear regime we weren't able to get a fit. We tried just about everything we could think of in modifying the assumptions that are made to try to generate a fit and so far failed. I'll just go through this very quickly. We have a lot of experiments but I'll just illustrate it by one where we took the sample well above the glass-transition temperature, cooled it very quickly to some temperature Ta, an annealing temperature, and then held it for various amounts of time. This was all done on the DSC. After annealing isothermally for a certain amount of time on the DSC, we would reheat the material through the glass-transition region allowing us to measure the enthalpy that had relaxed out or measure its fictive temperature. The Tg for this glass is about 710 K. We did this at three temperatures and these are the best fits we could generate using the A d a m - G i b b s model. We also used the N a r a y a n a s w a m y model which assumes an Arrhenius temperature dependence and produced fit curves which were absolutely identical. The thing to note is that if we're not too far from equilibrium, say for annealing at 700 K, we get a rather good fit and in fact the sample does come to equilibrium. If you go a little lower in temperature to 680 K, it's still not too bad. But when you try to anneal out at very very low temperatures, we don't even come close.
530
Discussion Session / Kinetics Of volume and enthalpy
Scherer has made this observation. I mentioned it earlier with regard to the glycerol data. The problem is that the non-linearity corrections we've made don't work very well far from equilibrium. There's some indication from the comparison of our glycerol data with the Birge and Nagel data that in fact they don't even work very close to equilibrium and all we're seeing is a sort of a compromise fit that doesn't match the fls, for example, that you get from the linear data. We tried to fix up these fits in a variety of ways. Note the formalism we used to handle the KWW function. This is simply because it is more economical of computer time to break it up into an equivalent relaxation time distribution. Whether you handle this directly in this form using the superposition principle or you handle it in this other form using fl really doesn't make a bit of difference. They're totally equivalent. One of the key assumptions in handling the non-linearity is that the various relaxation times depend upon the temperature and the average fictive temperature of the system. We tried violating that assumption by letting them depend partly on the individual fictive temperatures for the order parameters using an additional adjustable parameter, y. As we started going away from the original Tool or Narayanaswamy approach, the fits simply got worse. So this is not where the problem lies. We tried modifying the shape of the KWW function, thinking that perhaps we haven't got it quite right. Since trouble seems to come in at short times, we added a tail to it. So we modified our weighting factors, varying this constant C (renormalizing of course before we did the calculation). A search through parameter space found an effective distribution of relaxation times which was identical to the original best fit KWW. So this is not where the problem lies. We tried, like Scherer did, to abandon thermo-rheological simplicity to allow for fl to be temperature dependent. The way you do this is to assume that you have a distribution of activation energies. No improvement. Finally we tried changing the functional form of the non-linearity correction, modifying it using an odd power of Tf - T to speed it up relatively when the sample was above equilibrium and slow it down when it was below No improvement.
I don't know where the difficulty is. I also have a s u s p i c i o n l a n d I think we've got to crank this through in more d e t a i l - - t h a t the proposal by Ngai and Rendell is, in the end, somewhat contained in at least some of these things. That we've done equivalent things, tried testing the data, and don't find much improvement, so there's a real problem here. The final thing I'd like to note with regard to fitting this classical data of Kovacs is that, when trying to deal with non-linearity, you're worried about the start of the relaxation. That things are either slower or faster than they ought to be if you had assumed linearity. What happens way down at the tail when you're close to equilibrium usually doesn't get included in the typical fits. You're almost at equilibrium and if you miss that by half a percent, nobody worries about it. I think fitting the tail end of that %ff is not addressing what the real problem is. C.A. Angell *: I was wondering whether Dr Oguni from Tokyo Institute of Technology might be able to throw some light on this. The reason I'm directing it to him is that I'm aware that he and also the people in Osaka do very fine isothermal relaxation and are in a position to add information to this situation on conditions far from equilibrium. A.J. Kovacs *: Mr chairman, can I just reply to Moynihan before going on further because he, in a certain way, misquoted me I think. I would like to ask him a couple of questions and then we can go further. How do you know where the equilibrium is - was it determined accurately or by some extrapolation? It is not difficult to extrapolate but this is not necessarily the real equilibrium. The theories contain the real equilibrium. The driving force depends on the departure of your system from the real physical equilibrium and not on an extrapolation. C.T. Moynihan *: In many cases, we've annealed for such a long time that the properties are no longer changing with time within our experimental precision. The values we measure at that point agree very well in the cases where we extrapolate using the liquid and glass-like heat capacities to estimate what the enthalpy should be at that low annealing temperature.
Discussion Session / Kinetics of volume and enthalpy
A.J. Kovacs *: Was this the case for this last curve
you showed us? C.T. Moynihan *: Yes, it was. That was part of our results but we annealed as low as 650 K, which was our lowest temperature of experiment, outside the DSC in this case, until things did not change any more. So there's no doubt where equilibrium lies, at least in so far as we can determine it in a few months. A.J. Kovacs *: I don't think that in the DSC you annealed for 1000 hours. C.T. Moynihan *: No, these were separate furnace anneals. I didn't show you all the data, just the particular example that happened to be done in the DSC. There are other samples annealed in outside furnaces for some months' time. A.J. Kovacs *: I would like to make the general comment that for experiments where the approach to equilibrium is by contraction, it is much better and more precise to make the measurement by volume than by DSC because then for each point you have to recycle. C.T. Moynihan *: The more complicated your response experimentally, the more accurately you're able to pin down the parameters. There's very little that's more complicated than a DSC curve with a shoulder on it or an undershoot and an overshoot. We found that if you look strictly at approach curves, sometimes the fits are not too bad. But you really have problems when you try to fit the derivatives which we measured directly. The predictions there were based on fitting the reheating curves as well as where the sample annealed isothermally. A.J. Kovacs §: If I understood correctly, there is some suspicion about the accuracy of the CeTf~ ( = -- (V -- V ~ ) - 1 d v / d t) values as derived from the volume, v, expansion isotherms of PVAc, and thus about the reliability of the %ff 'gap' reported in my review paper in 1963 [[1] A.J. Kovacs, Adv. Polym. Sci. 3 (1963) 394] and shown by Dr R. Rendell in his talk. If this suspicion were founded, the discrepancy should also appear in the %ff-values derived from the volume contraction isotherms, obtained at the same temperature and using the same procedure as for expansion. Nevertheless, the contraction isotherms do converge asymptotically towards the equilibrium volume at
531
the same value of the experimental time, thus displaying no '/'eft gaps. As a consequence, the latter cannot originate from experimental inaccuracy. Rather they result in a genuine difference between the approach of the volume toward its equilibrium value either from above or from below. The difference between the contraction and expansion isotherms is in fact clearly apparent on the approach of v towards its equilibrium values, v~, depicted by fig. 17 in ref. [1] from which the relevant %ef isotherms shown by Dr Rendell have been derived. Note that the experimental error in this figure is of the order of the size of the datapoints. Finally, I would like to mention that these experiments on PVAc have been repeated many times without showing discrepancies other than those inherent to the experimental error ( A v / v = + 1 0 - 5 ) . Furthermore, similar %ff gaps have also been obtained with a glucose and a polystyrene glass [J.M. Hutchinson and A.J. Kovacs, unpublished]. Consequently, the %ff gap is a quite general and highly significant phenomenon which reveals some major discrepancies of retardation models, at least when applied to volume (and presumably to enthalpy) recovery of glass-forming systems. The %ff paradox was identified as early as 1963. Subsequently, the multi-parameter K A H R model was developed in 1979 to describe many other features of volume recovery, but it was clearly recognized at that time that it would not explain the %ff paradox. This problem is complex and I have a better idea now than I had in 1963 about its origin. Perhaps the Ngai-Rendell coupling model will be an answer, but I am not sure. G.B. M c K e n n a *: This is a question of which experiments should be described. The %ff paradox is something that's happening when you get very close to equilibrium. And the models aren't fitting when you're very close to equilibrium in that case, except perhaps for Ron Rendell and Kia Ngai's model which seems to be able to fit that part of it. However, Moynihan also finds a breakdown of the basic formalisms when you're far from equilibrium.
532
Discussion Session / Kinetics of volume and enthalpy
J.M. Hutchinson §: Moynihan's contention is that the equation [xAh
-- rio exp l---h-r-- + (l-y)(1-x) -~
y(1-x)
Ah
RTflave
Ah]
RTfi
(1)
gives a worse fit than does Ti
=
'rio exp ~
+
~-T-~v2
.
(2)
This cannot be so. The added adjustable parameter cannot possibly make the fit worse. Equation (1) is identical to eq. (2) when y = 1, and must therefore give, at worst, a fit as good as eq. (2). C.T. Moynihan *: Nature seems to like y = 1. When you go to the other extreme and set y = 0, you predict approach curves that are totally nonphysical and with enormous undershoots which are in no way observed experimentally. The fact that you use the average structure to modify the relaxation times for non-linearity is a built-in assumption that started with Tool and continued with Narayanaswamy and so forth. But it is in fact an assumption. We've tested this experimentally and it turns out that it's the best assumption that could be made. Rather than letting things depend on individual fictive temperatures, it just depends on the overall structure. J.M. Hutchinson *: But when you have an adjustable parameter, it cannot be worse. C.T. Moynihan *: It is worse. The two extremes are that the relaxation time depends only on the average fictive temperature and the other extreme is that each individual order parameter relaxation time depends upon its own fictive temperature. Simon Rekhson tried a few calculations of this sort too, I believe. What happens is that the short relaxation times, if you set y = 0, get hyper-accelerated. The long relaxation times get hyper-retarded. And the data just don't behave that way. It's like using a log-Gaussian instead of a KWW. It just doesn't fit. S. Matsuoka §: The spectrum for the non-equilibrium glass depends on the degrees of departure from the equilibrium. In my lecture on Wednes-
day [these Proceedings, p. 293], I presented the T-dependence of the stress relaxation data and proposed a distribution of sizes for configurational cooperative domains as the origin of the distribution of activation energies. If the distribution is assumed independent of T in the temperature range of interest in equilibrium then, when the temperature is suddenly shifted, fl will decrease if cooled and increase if suddenly heated. During the subsequent recovery process, fl will tend towards the equilibrium value. The change of fl with time (one is not at liberty to change it in any arbitrary manner here) will definitely improve the curve fitting for the very well-known - l o g r vs. 8 by Kovacs. R.W. Rendell §: With reference to Moynihan's earlier comments, I think a basic point of the structural relaxation models is still being missed. Different generalizations of KWW to the non-linear regime are not necessarily equivalent. The coupling model version cannot be transformed into the reduced time version. And the results can be quite different if each of them is used to calculate structural relaxation. And this is true even if we use the same value of fl = 1 - n and the same functional form for ¢[T, Tf] in both relaxation functions. The reduced time function has an integral over 1 / r and then the non-exponentiality is introduced by raising the whole integral to a power of ft. By contrast, the coupling model has an extra factor of (o~ct) -n inside the integral and that is what introduces the non-exponentiality. The reduced time deals first with the non-linearity through the integral over 1 / ¢ and then afterwards puts in the non-exponentiality. The coupling model deals with the non-linearity and the non-exponentiality at the same time within the integral. In particular, Moynihan had tried several new forms for ~'[T, Tf] in the reduced-time model for the lead-silicate glass but none of these modifications is equivalent to the coupling model. And there may be yet other generalizations of KWW, besides the coupling model and reduced-time approaches, which could be considered. The functional form to be used for z[T, Tf] in either model should also come from experiment. The only attempt I am aware of to obtain experimental information on ,r[T, Tf] was by Kovacs,
Discussion Session / Kinetics of volume and enthalpy Stratton and Ferry on PVAc. They measured the shear moduli G ' ( ~ ) , G"(~0) for similar samples and sample histories and constructed shift factors as a function of T and time after quench or equivalently, T and Tf. Additional experiments that give information about T[T, Tf] are very much needed. M. Oguni §: Physical aging has been studied in the time-domain and frequency-domain. But the two methods have to be considered separately. When it is studied by the frequency-domain method, the sample is located almost in the equilibrium state. On the other hand, when studied by the time-domain method, it should be located in the state apart from equilibrium. We have studied isothermally the enthalpy relaxation by using an adiabatic calorimeter, and showed already the result of vapor-deposited amorphous butyronitrile which is located in the state quite far from the equilibrium. When the enthalpy relaxation was fitted with the KWW equation, we obtained the small value of fl of 0.1 or so. Usually the fl value is around 0.6 or 0.5. Next we followed the enthalpy relaxation of liquid quenched propylene glycol which was studied by Professor Nagel's group by the method of frequency domain. Nagel's group obtained the fl value of 0.61 _+ 0.04. Then a r-value in agreement with the data of Nagel's group was obtained only when we took the temperature jump of zero. C.A. Angell *: So he's saying that when he's watching the relaxation far from equilibrium, the r-value is very much smaller in the case of butyronitrile and certainly in the same direction for propylene glycol. That would apparently need to be incorporated into any of these fits. C. 7". Moynihan *: No attempt has been made to account for the non-linearity via any one of these techniques. It depends upon which way you're approaching equilibrium - - if you're coming down it's going to look faster and faster whereas if you're going up it's going to look slower and slower compared with what it would be if you were very very close to equilibrium. I think that's why fl is changing. As he does his experiment very close to equilibrium, he'll converge on the value you would get from a linear experiment. So it sounds all very consistent, but it's not to be inter-
533
preted that the KWW function isn't working or something like that. J. O'Reilly §: The fls obtained from calorimetric (enthalpy) relaxation do not account for non-linearity and therefore cannot be compared directly with those by DSC which do. C.A. Angell *: When we were talking about experiments close to equilibrium, Dr Kovacs was pointing out that DSC is not as accurate a method as dilatometry. But I think the quality of the data that Oguni has obtained probably are very comparable. G.B. McKenna §: I submit that the difference in volumetric measurements and DSC experiments can be compared by an analogy with mechanical measurements. First, consider that the stimulus is temperature and the volume or enthalpy changes are the responses. The T-jump dilatometric experiments (if we had a linear system) are equivalent to a step-strain experiment. So at constant temperature (or strain) after imposition of the stimulus, we measure directly the response (volume or stress). In the typical DSC experiments, the measurement (or at least the scanning part) is equivalent to a constant rate of strain experiment. In these cases one measures some integral of the response function. On the other hand, in volume experiments we measure directly the extensive thermodynamic variable while in DSC we measure a derivative of the enthalpy, i.e. Cp. Each type of measurement will have advantages and disadvantages, although I prefer the T-jump experiment and expect that those who do DSC will prefer those types of measurements. L. Struik §: Comparing dilatometry and DSC, dilatometry has the advantage that the thermal equilibrium time is greater than that of DSC sample. In cooling cycles, this is a great disadvantage because of the high non-linearity of the process. C.T. Moynihan *: Sometimes in the DSC we try to fit the heat capacity curves but, as in the experiments I just described, the samples were annealed isothermally and we subsequently use the DSC to measure the total amount of enthalpy that the sample had lost. It's like measuring the area under a melting peak effectively. In this case it doesn't matter if your data are a little smeared out because your sample is not exactly isothermal. You
534
Discussion Session / Kinetics of volume and enthalpy
still get the right energy out. So, the DSC data aren't always measuring derivatives, sometimes they are measuring total quantities, and this was what I showed up there. J. O'Reilly §: DSC and adiabatic calorimetry are different by a factor of 10: 1% by DSC and 0.1% by adiabatic calorimetry. The sensitivity of the adiabatic calorimeter requires longer equilibration times but measures smaller changes in energy. The DSC measures larger changes in shorter times. Dilatometry measures AT to 10 -5 cma/g but the accuracy in Aa and ATf is less. ATfdil is Av _ 10- 5 cm3/g = 0.025 ° C. Aetr8 4 × 1 0 - 4 / K The precision of the DSC is _+1% in Cp and A H -- 0.01 J / g , yet the sensitivity of AH 0.01 J / g AT~o~c= ~ - 0.20 J / g K ----0.05 o C. Dilatometry is two times more precise in ATf measurements. G.B. McKenna §: I think that we should return to one of the focuses of this discussion and that is the combination of points 2 and 3 in our points for discussion vis h vis the Ngai-Rendell model. Here I don't know if the apparent resolution of the zefr paradox by this model is due to the fact that the constitutive model proposed is responsible or is it that the underlying physics (essential ingredients, points) of the Ngai-Rendell model are responsible? If we look at the constitutive law, it is not the same as the K A H R or Narayanaswam y - M o y n i h a n type models, since it allows the spectrum of relaxation times to change in shape as a function of the entire thermal history in a different way. On the other hand, there is also some 'new' physics in the formulation of the Ngai-Rendell coupling model and we need to ask if the new and better fits to Teff require the constitutive law or the new physics or both? I.M. Hodge §: I have performed preliminary calculations of d T f / d T that indicate that, using the same values of x, Ah* and r , the effective-time expression in which fl lies outside integral gives very different results than the Ngai-Rendell expression, where fl is part of the integrand. The
heat capacity overshoot calculated using the Ngai-Rendell expression was broader and lower. Comparisons of the two expressions in fitting experimental heat capacity data for a range of histories have not yet been attempted. C.A. Angell *: Ron, do you have any feeling for which way your model would change the results after having seen the direction in which, using isothermally determined parameters, Moynihan failed to predict the overshoot for glycerol? R. W. Rendell §: I haven't yet looked at the glycerol DSC data using Birge and Nagel's linear response parameters with the coupling model. However, for the same set of parameters, the DSC peak from the coupling model is found to be broader than the corresponding peak from the reduced time model. For a given experimental DSC curve I expect the coupling model to require a larger value of fl = 1 - n than the reduced-time method. Since Moynihan had to use a fl which was smaller than Birge and Nagel's value, the change in using the coupling model is expected to be in the right direction to agree with the experiments. However, a quantitative comparison remains to be done. J.-C. Bauwens §: The KWW relation is not the result of a differential equation. I prefer a hyperbolic law:
where ~ is the phenomenon and To the time constant, because it is the result of the differential equation dq~/~ 0 = d t / z , where r = ToC. For instance the Rouse model gives a = 1, then:
fl = 1 / ( a + 1). E. Oleynick §: I would like to call your attention to some experimental data using calorimetry and volume measurements in one experiment which probably is not known to most of those here. We measured simultaneously for one sample of polystyrene, using the technique of temperature jumps, the relaxation of volume and the relaxation of
Discussion Session / Kinetics of oolumeand enthalpy
enthalpy. Unfortunately I didn't bring my slides with me but I have a paper here. We put the dilatometer inside the differential calorimeter and measured the change of the volume. We equilibrated the sample at T = 120 ° C for a long time. It was then put into the second calorimeter which also contained the dilatometer, at temperatures between T = 90 ° C and approximately T = 70 ° C. This was for contraction of the volume. In other cases, we kept the sample at T = 120 ° C for some time and then put it at the lower temperature for something like 1000 min. Then we made the relaxation at a higher temperature so that the sample becomes expanded. At the same time we measured the volume and the rate of enthalpy relaxation. So we have at any moment the absolute relaxation time, without any scanning calorimetry, of volume and enthalpy. If anyone would like to look at the results, I have them here. We have the data for the rate of volume relaxation and the rate of enthalpy relaxation for both contraction and expansion. I would like to emphasize two points here. First, we have found, for all histories and all measurements that we did, a one-to-one correspondence between enthalpy relaxation and volume relaxation. N o difference, as you can see. Second, there is an important point which I cannot explain but would hke to put to you. The problem is that the amount of energy required for changing the volume differs from ACp/Act, where a is the coefficient of thermal expansion, by 30-40%. This is at least one order of magnitude larger than our error for the one-toone correspondence between volume and enthalpy. I don't know why it is, but it's observed experimentally. A.J. Kovacs §: Your data seem to confirm the %ff gap. S. Rekhson §: I agree that your (Rendell et al.) computations provide a better account for the Teff paradox than those using the T o o l - N a r a y a n a s w a my (TN) model. However, in your computations the shift function is adjustable, whereas in T N computations it is fixed by an Arrhenius or A d a m s - G i b b s relationship. The shift function resulting from your computations for T = 35 ° C has twice the activation energy of the shift function that you needed to account for the behavior at
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40 o C. This is unrealistic. I understand that you now have this difficulty resolved. Could you elaborate on this? R.W. Rendell §: In collaboration with J.J. Aklonis, we have previously examined Kovacs' volume recovery data for PVAc using the coupling model relaxation function. To test the model, we need n and T[T, Tf]. Kovacs, Stratton and Ferry (KSF) had measured the shear moduli G'(~0) and G"(~0) for similar samples and sample histories in order to obtain a shift factor for T. However, the G " curves changed shape during volume recovery and the curves could not be reduced. This indicates that n changes during the structural relaxation in PVAc. Values of n could not be obtained from the G " curves since only 1.5 decades in frequency were available. The G ' curves could be approximately superimposed and K S F constructed a shift factor which could be represented as a modified Doolittle equation for T[T, T~]. Information about n from different PVAc samples is available from Plazek's shear creep compliance data and Mashimo's dielectric data. Our fit of Kovacs' %ff data at T = 3 5 ° C reproduced closely the shapes of the curves and the size of the 'expansion gap' using the K S F shift factor. The shapes and expansion gaps at T = 4 0 ° C could also be closely fit using the K S F shift factor but the position of - log O'efr relative to the T = 35 o C curve is slightly too large. The best fit of the T = 4 0 ° C data required deviations from the K S F shift factor as we showed in our paper. To examine these comparisons more carefully, we should obtain values of n from the same Kovacs samples and the dependence of T on Tr which does not depend on shifting the shear moduli in a situation where they are changing shape. This could be done if G " were available over several decades of frequency. An indication that these effects may improve the comparison with the T = 40 ° C data is seen in an exercise where we assumed that TO has a modified Doolittle form rather than T. F r o m the coupling model, TO should not be affected by changes in n. Obtaining parameters from the equilibrium KSF shift factor, we could reproduce the shapes, expansion gaps, and relative positions of the 35 and 40 ° C data. G. Williams §: The essential phenomena of iso-
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Discussion Session / Kinetics of volume and enthalpy
thermal and non-isothermal relaxations of the thermodynamic properties of non-polymeric and polymeric glass-forming systems have been rationalized, at least semi-quantitatively, using the double-box model (KAHR model) or the stretched exponential function (KWW function). It has become apparent that, although semi-quantitative agreement may be obtained between experimental results and the phenomenological models, e.g. for specific heat data, for the memory effect, and for the non-linear isothermal volume relaxation data, the fits may not be made quantitative for sensitive variables (e.g. Cp, which is ( ~ H / ~ T ) p ) . Much discussion has surrounded the question of the use of particular forms for T(T, t). I wish to add the comment that the stretched exponential function, ¢p(t) = exp - (t/T o )~, is likely to be inadequate for t/T o << 1, as we found in our original work on the dielectric a-relaxation for amorphous polymers measured as e(~o) in the frequency domain. Also, the theory of molecular time-correlation functions requires d ~ p ( t ) / d t ~ O as t ~ 0 , whereas the KWW function gives infinite slope as t ~ 0. Therefore, it seems likely that the stretched exponential function will not give a good description of thermodynamic relaxation data for t/T o << 1, and special attention should be given to this aspect of the analysis of such data. S. Matsuoka §: The importance of a 'good' con-
stitutive equation is not just to fit data well, but also that the 'constants' have some physically tractable values. I show you first the log T vs. 8 fit, using the equation we published in Macromolecules, i.e. In T= ~A# - T ( Z + Z* ) + C for thermodynamic recovery, vs. A~ l n T = -k-T Z + C for dielectric relaxation in which A/~, Z(Tf) are identical. This leads to a hypothesis that the density fluctuations are larger and longer than the rotational (dielectric) fluctuations. L. Struik §: Part of the problem with the theory for volume relaxation may be due to the fact that there are also considerable contributions from fland a-processes, which follow different laws and have a different sensitivity to fictive temperature or free volume. In a recent paper (Polymer) I showed that the secondary peaks in volume relaxation may be as high as the peak at Tg. So, a description by a simple formula such as KWW, may lead to problems. I suggest applying the method shown by Dr Read [these Proceedings, p. 408] to separate such processes.