3.3.12 Volume Recovery of Poly(Vinyl Acetate)

Mark G. Vangel,
Andrew L. Rukhin,
Stefan D. Leigh

Statistical Engineering Division, ITL

Gregory B. McKenna

Polymers Division, MSEL

B. Lotz,
C. Straupe
Institute Charles Sadron, Strasborg, France

When polymer glasses are equillibrated at an initial temperature (T_i) and then placed in a water bath at a second temperature (T_f), 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 (v<sub>f</sub>) at time t is $\delta(t) = (v(t)-v_f)/v_f$. The negative of the derivative of the logarithm of $\vert\delta(t)\vert$, a measure of the rate of approach to equilibrium, was defined as the `effective' recovery time, $\tau_{\rm eff}$, 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 T_f from different initial temperatures, say T_i and $T_i^{\prime}$, the $\tau_{\rm eff}$ values differed for $\delta$values as small as could be reliably measured. This is somewhat paradoxical, since it suggests that the specimen `remembers' T_iwhen it is close to a very different T_f. 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 $\delta(t)$ made close together in time.

We used a propagation-of-errors argument to express the uncertainty in divided-difference estimates of $\tau_{\rm eff}(t)$ in terms of the uncertainty in measurements of $\delta(t)$, and the correlation among these measurements. The uncertainty in $\delta(t)$ was bounded above using physical considerations, and the correlation among measurements of $\delta(t)$was estimated from Kovacs' many replicated experiments. Together, these results enabled us to calculate approximate confidence intervals on $\tau_{\rm eff}(t)$, for curves having different T_is, but the same T_f. From these confidence bands one can quantify how small $\delta(t)$ needs to be in order to conclude that the $\tau_{\rm eff}$ 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 $\log(\tau_{\rm eff})$ with 95% individual confidence bands, as functions of t, for initial temperatures $T_i=25^{\circ}{\rm C}$and $T_i^{\prime}=35^{\circ}{\rm C}$, and final temperature $T_f=40^{\circ}{\rm C}$. The curves were estimated from A.J. Kovacs' experimental data on volume recovery of poly(vinyl acetate).

       

Date created: 7/20/2001
Last updated: 7/20/2001
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