Mark G. Vangel,
Statistical Engineering Division, ITL
Gregory B. McKenna
Polymers Division, MSEL
When polymer glasses are equillibrated at an initial temperature (Ti) and then placed in a water bath at a second temperature (Tf), the volume of the polymer will change smoothly until the material attains the new equilibrium temperature. A measure of how `far' the volume (v(t)) of a specimen is from equilibrium (vf) at time t is . The negative of the derivative of the logarithm of , a measure of the rate of approach to equilibrium, was defined as the `effective' recovery time, , by A.J. Kovacs in a very influential 1964 article summarizing an extensive experimental program on volume recovery of poly(vinyl acetate).
Kovacs claimed that when approaching the same equibrium temperature Tf from different initial temperatures, say Ti and , the values differed for values as small as could be reliably measured. This is somewhat paradoxical, since it suggests that the specimen `remembers' Tiwhen it is close to a very different Tf. A recent publication by a prominent researcher has called this assertion into question by arguing that Kovacs' experimental uncertainty is much larger than he realized.
It is of considerable theoretical importance to establish to what extent Kovacs was correct, and we have addressed this question by means of a thorough statistical analysis of 96 experiments done by Kovacs and his students, for poly(vinyl acetate) at many initial and final temperatures. We conclude that Kovacs was essentially correct in his assessment of experimental uncertainty, and that the recent article critical of Kovacs experimental work overstates this uncertainty, primarily by ignoring the positive correlation among measurements of made close together in time.
We used a propagation-of-errors argument to express the uncertainty in divided-difference estimates of in terms of the uncertainty in measurements of , and the correlation among these measurements. The uncertainty in was bounded above using physical considerations, and the correlation among measurements of was estimated from Kovacs' many replicated experiments. Together, these results enabled us to calculate approximate confidence intervals on , for curves having different Tis, but the same Tf. From these confidence bands one can quantify how small needs to be in order to conclude that the values are not statistically distinguishable. This analysis is illustrated in the figure for a pair of experimental curves.
These quantitative results were corroborated by a second statistical analysis, using a repeated measures model, and by a qualitative graphical analysis.
Figure 24: Estimated with 95% individual confidence bands, as functions of t, for initial temperatures and , and final temperature . The curves were estimated from A.J. Kovacs' experimental data on volume recovery of poly(vinyl acetate).
Date created: 7/20/2001