Volume Continuity
(Temperature-Pressure Differential at Tg)

At the glass transition we can assume two-phase equilibrium, that is, the pure-component molar volumes in the liquid (rubbery) and glassy state should be identical:

dVglass(T,p) = dVliquid(T,p)


Vglass(T,p) = Vliquid(T,p)

At equilibrium, the pressure and the volume are the same in both phases. Hence

(∂Vglass / ∂T)p dT + (∂Vglass / ∂P)T dP = (∂Vliquid / ∂T)p dT + (∂Vliquid / ∂P)T dP

The temperature and pressure coefficient of the molar volume can be written in terms of volumetric thermal expansion and isothermal compressibility, respectively:

(∂V / ∂T)P = V α

(∂V / ∂P)T = -V β


Vglass αglass dT + Vglass βglass dP = Vliquid αliquid dT + Vliquid βliquid dP

A two-phase equilibrium of a pure material has only one degree of freedom. Therefore, the temperature and pressure changes are not independent on the glass transition phase boundary. Since Vglass = Vliquid, the differential (∂V / ∂P)T along the glass-liquid boundary is


  • Summary

    Thermal Transition

    A second-order transition occurs as a discontinuity in the thermal expansion coefficient (constant pressure) or as a discontinuity in the compressibility coefficient (constant temperature).

  • A discontinuity in a plot of the thermal expansion coefficient versus temperature marks the occurence of a second-order transition.

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