Fictive Temperature of a Glassy System in Terms of Frequency Dependent Specific Heat : A Memory Function Approach

Abstract

When a glass forming liquid is cooled at a fast rate in its supercooled state, the liquid can fall out of equilibrium and the relaxation becomes progressively slower with time. The slowly evolving nonequilibrium state of the glassy system can be described by introducing a fictive temperature which describes the instantaneous state of the nonequilibrium system. The fictive temperature T f (t) is time dependent. During cooling, the fictive temperature is higher than the actual temperature. After the cooling or quenching has ceased, the fictive temperature approaches the final temperature at a rate that depends on the relaxation properties of the liquid. In this work we use linear response theory to connect the time dependence of the fictive temperature to memory function which is shown to be related to the frequency dependent specific heat which itself depends on the fictive temperature T f (t). Thus, one requires a self-consistent calculation to capture the interdependence of relaxation rate and structural response function. We present a numerical calculation where we apply our relations to silica where the relaxation function that describes the frequency dependent specific heat and is modeled as a stretched exponential William-Watts (WW) function, while the relaxation time is modeled as a Vogel-Fulcher-Tammann (VFT). We calculate the fictive temperature self-consistently. T f (t) exhibits the fall out from actual temperature as time (t) progresses.