Integro-Differential Equation for the Non-Equilibrium Thermal Response of Glass-Forming Materials: Analytical Solutions

Abstract

An integro-differential equation describes the non-equilibrium thermal response of glass-forming substances with a dynamic (time-dependent) heat capacity to fast thermal perturbations. We found that this heat transfer problem could be solved analytically for a heat source with an arbitrary time dependence and different geometries. The method can be used to analyze the response to local thermal perturbations in glass-forming materials, as well as temperature fluctuations during subcritical crystal nucleation and decay. The results obtained can be useful for applications and a better understanding of the thermal properties of glass-forming materials, polymers, and nanocomposites.

Integral Equations: Theories, Approximations, and Applications

“…"Integro-Differential Equation for the Non-Equilibrium Thermal Response of Glass-Forming Materials: Analytical Solutions" has been published by A.A. Minakov and C. Schick [2]. In this study, they have studied the non-equilibrium thermal response of glass-forming substances with a dynamic (time-dependent) heat capacity to fast thermal perturbations based on an integro-differential equation.…”

Temperature Relaxation in Glass-Forming Materials under Local Fast Laser Excitations during Laser-Induced Microstructuring

“…Thus, the thermal response of glass-forming materials to a thermal perturbation at time t depends on the temperature at earlier times. The effect of the long-term relaxation of dynamic heat capacity significantly affects the dynamics of local temperature changes T(t, r) [68,69]. In turn, the rate of change in local temperature T(t, r) can significantly affect the microstructuring of glass-forming materials [19,37,43,44,70].…”

“…In this paper, we will focus on modeling such thermal processes, taking into account the relaxation effect of the dynamic heat capacity c dyn (t). The dynamic behavior of glass-forming materials under fast local thermal perturbations can be described using the integrodifferential heat equation with "memory" [68,69]. This equation has an analytical solution, at least in spherical, cylindrical, and planar geometries [68,69,71].…”

“…The dynamic behavior of glass-forming materials under fast local thermal perturbations can be described using the integrodifferential heat equation with "memory" [68,69]. This equation has an analytical solution, at least in spherical, cylindrical, and planar geometries [68,69,71]. This work aims to determine the dynamics of local temperature changes, especially the local rate of the temperature change R(t, r) associated with laser-induced thermal excitations during laser processing of glasses.…”

A Procedure for Factoring and Solving Nonlocal Boundary Value Problems for a Type of Linear Integro-Differential Equations

“…Integro-differential equations arise in many areas in sciences and engineering. For example, Fredholm and Volterra integro-differential equations are used in [1][2][3] for modeling neural networks; Fredholm integro-differential equations are utilized in [4,5] to model the response of plates and shells in the theory of elasticity; Volterra integro-differential equations are employed in [6][7][8] to simulate thermal problems in the glass-forming process, drug concentrations in pharmacokinetics, and Hepatitis B Virus infection in medicine, respectively.…”