Kohlrausch–Williams–Watts equation

synonym: KWW Equation
https://doi.org/10.1351/goldbook.12778
Equation describing the enthalpy relaxation, \(\Delta H\) which is observed during the annealing of polymers in the glassy state at a temperature, \(T_{\rm{a}}\), lower than \(T_{\rm{g}}\) \[\Delta H(t_{\rm{a}}) = \Delta c_{p}(T_{\rm{g}} - T_{\rm{a}}) \exp\!\left[- \left (\frac{t_{\rm{a}}}{\tau} \right)^{\beta} \right]\] \(\Delta H\) = difference of the enthalpy in the initial, glassy state, and the enthalpy of the glassy state after annealing; \(t_{\rm{a}}\) = time of annealing; \(\Delta c_{p}\) = difference in heat capacity at the glass-transition temperature, \(T_{\rm{g}}\), and the starting temperature of the annealing process, \(T_{\rm{a}}\); \(\tau\) = relaxation time at \(T_{\rm{a}}\); \(\beta\) = exponential factor.
Notes:
  1. The exponential factor \(\beta\) has a value between 0 and 1, and characterises the asymmetry of the distribution of relaxation times.
  2. Moduli, stress-strain behaviour, and molecular motions are affected by enthalpy relaxation, which reflects physical aging.
Source:
PAC, 2013, 85, 1017. (Glossary of terms relating to thermal and thermomechanical properties of polymers (IUPAC Recommendations 2013)) on page 1031 [Terms] [Paper]