Cooperative dipolar relaxation in polymers

We may attribute 7 to a major change in the segmental mobility of the polymer chains. Above Ts there is sufficient mobility, a sort of micro-Brownian 

If we assume that the excess rate of expansion of the rubber over that of the glass is entirely attributable to an increasing free volume, we may write 

This equation has the same form as the well-known WLF equation (Williams, Landel and Ferry, 1955) that correlates the mechanical behaviour of all polymers near their Tg provided we set Tg = Tx (Tz measured by the same method for each polymer). From experimental results one finds that 

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