From Frequency Dependent Specific Heat to Fictive Temperature of a Glassy Liquid

Abstract

Upon rapid quenching of temperature of a glass forming liquid, the system falls out of equilibrium due its finite relaxation time. Additionally, the relaxation becomes progressively slower with time. The created nonequilibrium state of the glassy system is conveniently described by introducing a fictive temperature which provides 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 { \it 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.