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

Biman Bagchi

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.

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 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 . 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. exhibits the fall out from actual temperature as time (t) progresses.

Paper Structure

This paper contains 25 sections, 38 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Kinetic models, fictive temperature, and connection to Ising--Glauber dynamics
  3. Frequency-dependent specific heat and enthalpy fluctuations
  4. Theoretical analysis : Linear response theory
  5. Linear Response Based Analysis
  6. Relation to microscopic fluctuations.
  7. Connection with the structural relaxation function.
  8. Reduced time : The internal clock
  9. Relation to frequency dependent specific heat, $C_p"(\omega)$.
  10. Use of VFT for temperature dependence of viscosity & the Maxwell viscoelastic relation
  11. Determination of $\Phi$
  12. (A) Frequency domain determination
  13. (B) Time domain measurements.
  14. Commonly employed forms, and fits
  15. Requirement of self‐consistency
  16. ...and 10 more sections

Figures (1)

  • Figure 1: Fictive temperature trajectories during linear cooling of silica. The figure shows the Tool-type plots of the fictive temperature $T_f(t)$ versus bath temperature $T(t)$ for two different linear cooling rates: 1 K/s (blue) and 2 K/s (red), starting from 2000 K and ending at 750 K. The dashed black line denotes the equilibrium condition ($T_f = T$). At high temperatures, $T_f$ follows $T$ closely because structural relaxation is rapid and the system remains in equilibrium. As the temperature decreases through the glass-transition range (approximately 1300--1100 K), the structural relaxation time $\tau(T)$ increases sharply, causing $T_f$ to fall out of equilibrium and freeze at a constant value corresponding to the final structural state. The slower cooling rate (1 K/s) allows more time for structural relaxation, yielding a lower frozen-in fictive temperature compared to the faster 2 K/s quench. The curves were generated by numerically integrating the Tool equation $\frac{dT_f}{dt} = \frac{T(t) - T_f(t)}{\tau(T)}$, using a logistic form for the structural relaxation time: $\tau(T) = \tau_{\infty} + \frac{\tau_0}{1 + \exp\!\left[\frac{T - T_g}{\Delta}\right]}$, where $\tau_{\infty}=10^{-9}\,\mathrm{s}$, $\tau_0=10^{4}\,\mathrm{s}$, $T_g = 1200\,\mathrm{K}$, and $\Delta = 100\,\mathrm{K}$. This choice ensures a smooth and physically realistic increase in $\tau(T)$from nanoseconds at high $T$ to hours near the glass transition.