The τ-effective paradox revisited: An extended analysis of Kovacs' volume recovery data on poly(vinyl acetate)

G. B. McKenna, M. G. Vangel, A. L. Rukhin, S. D. Leigh, B. Lotz, C. Straupe

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In 1964 Kovacs (Kovacs, AJ, Transition vitreuse dans les polymeres amorphes. Etude phenomenologique. Fortschr Hochpolym-Forsch 1964;3:394-507) published a paper in which he analyzed structural (volume) recovery data in asymmetry of approach experiments. Kovacs used a parameter referred to as τ- effective (τ(eff)) which is defined in terms of the volume departure from equilibrium δ as τ(eff)/-1 = - 1/δ dδ/dt. In plots of the log(l/τ(eff)) vs. δ Kovacs observed an apparent paradox in that the values of τ(eff) did not converge to the same point as δ approached zero (i.e. equilibrium). Hence the equilibrium mobility of the structural recovery seemed path dependent. Also, the apparent paradox was accompanied by a spreading of the curves for τ(eff) in the up-jump experiments which has come to be known as the expansion gap. While it is currently accepted that the paradox itself does not exist because the curves will converge if the measurements are made closer to δ = 0 (Kovacs' estimates of τ(eff) were made for values as small as δ ≃ 1.6 x 10-4), the existence of the expansion gap is still a subject of dispute. This is particularly relevant today because recent models of structural recovery have claimed 'success' specifically because the expansion gap was predicted. Here we take the data Kovacs published in 1964, unpublished data from his notebooks taken at the same time, as well as more recent data obtained at the Institut Charles Sadron under his tutelage in the late 1960s and early 1980s. We then examine them using several different statistical analyses to test the following hypothesis: the value of τ(eff) as |δ| → 1.6 x 10-4 for a temperature jump from τ(i) to T(o) is significantly different from the value obtained for the temperature jump from T(j) to T(o). The temperatures T(i) or T(j) can be either greater or less than T(o). If the hypothesis is rejected, the τ(eff)-paradox and expansion gap need to be rethought. If the hypothesis is accepted, then the argument that reproduction of the expansion gap is an important test of structural recovery models is strengthened. Our analysis leads to the conclusion that the extensive set of data obtained at 40°C support the existence of an expansion gap, hence an apparently paradoxical value of τ(eff), for values of |δ| ≥ 1.6 x 10-4. However, at smaller values of |δ| it appears that the values τ(eff) are no longer statistically different and, in fact, the data suggest that as |δ| → 0 all of the τ(eff) values converge. In addition, data for experiments at 35°C do not have sufficient accuracy to support the expansion gap for such small values of |δ| because the duration of the experiments is significantly longer than those at 40°C. Consequently the data readings taken at 35°C were made at longer time intervals and this leads to dramatically reduced error correlations.

Original languageEnglish
Pages (from-to)5183-5205
Number of pages23
Issue number18
StatePublished - Aug 1999


  • Expansion gap
  • Structural recovery
  • Volume recovery


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