Pressure Dependence of Relaxation Times

Equation 31 can be generalized to give the ratio ai2 of a particular relaxation time T/ in state 2 to that in state 1, in terms of the free volumes f and in these states  

If states 1 and 2 refer to atmospheric pressure Pq and a higher pressure P respectively, we have (at constant temperature) 

Within the limited pressure range where jS/ can be considered constant, the 

Compressibility in atm plotted against pressure in atm for polystyrene at several different temperatures as indicated. (Hellwege, Knappe, and Lehmann.)  

Analysis of data following procedures analogous to the use of equations 22,23,37, and 38 can provide values for/o and j8/ (if B is set equal to unity). This has been done successfully for the pressure dependence of viscosities of some ordinary liquids and of bulk relaxation times of polyvinyl acetate. For the latter, the value of /o at a particular reference temperature found from pressure dependence is consistent with that found from temperature dependence. In all cases, j8/ is less than half of jS, so the major portion of the volume change in compression is due to reduction of the occupied volume—in contrast to thermal expansion, where the major portion (often about two-thirds) is due to increase in the free volume. The compressibility jS/ is presumably closely similar to the difference Aj3 between the total compressibilities just above and just below Tg. This identification is subject to an important reservation, however, as described in Section 2 below. 

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The derivation of equation 59 corresponds to the statement that the temperature (or pressure) dependence of relaxation times at constant free volume is zero i.e.. 

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