Frequency Dependent Heat Capacity and the Glass Transition

4 FREQUENCY DEPENDENT HEAT CAPACITY AND THE GLASS TRANSITION 

Heat content stored in molecular motions, such as vibration, is assumed to be rapid when compared with the modulation of the temperature kinetic effects influence heat flow, but only through reaction rates. This contrasts with cases where the heat in molecular vibrations itself is not rapid and some sort of kinetics plays a role in the heat capacity, or at least in some of the heat taken up and released. A relaxation time appears in the rate of change of enthalpy. 

Perhaps the simplest example of how a time-scale can be involved in heat flow to and from a sample, and thereby give an (apparent) dependence of heat capacity upon frequency, is where the thermal conductivity of the sample is in some sense poor (or, equivalently, the sample s specific heat is very large). For simplicity, we may look at the case of a homogeneous spherical sample inside a locally uniform part of the calorimeter. More realistic cases are less easy to analyse, but the qualitative effects are much the same. 

Using complex notation, the cyclic part of the temperature can be written as 

This gives rise to a complex heat capacity of 

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