|M.Sc Thesis||Department of Mechanical Engineering|
|Supervisor:||Assoc. Prof. Gendelman Oleg|
|Full Thesis text|
The objective of this work is to investigate the process of non-stationary heat conduction in models of one-dimensional nano-scale dielectric crystals.
The most commonly used macroscopic equation of heat conduction is based on the Fourier law of heat transfer which admits the non-physical propagation of the temperature field with infinite velocity due to its parabolic nature. As more appropriate alternatives, equations of a higher order - such as the hyperbolic second order Cattaneo-Vernotte (CV) equation - have been proposed. The CV equation predicts the possibility of oscillatory behavior of the temperature profile, in contrast to the purely diffusive solutions of Fourier’s equation of heat conduction.
In this work, the non-stationary conduction problem in one dimension is investigated from the microscopic point of view. Particularly, we focus on comparing three features of the CV equation to the numerical behavior of our microscopic models: 1) The CV equation suggests that there exists a specific wavelength of the initial temperature distribution which divides between two types of decays (oscillatory and diffusive) of the respective modes. 2) It is a linear equation. 3) It has the second order with respect to time.
The models chosen to be investigated are the Frenkel-Kontorova, φ4 and φ4- models. These models are described by finite chains of linearly coupled particles, with an external (model-specific) on-site potential. The method of the investigation is to conduct molecular dynamics simulations of the models: using the Langevin thermostat, the chains are brought to desired initial harmonic and periodic temperature distributions. Then the thermostat is “turned off”, and relaxation towards an equilibrium steady state is obtained. Finally, the relaxation profiles are analyzed. Due to the systems' comparatively large size, the results were obtained using parallel computing.
The results show that a critical wavelength can indeed be observed in all our three models and indeed three regimes of relaxation to a steady temperature do exist: oscillatory, critical and diffusive. In addition, linearity was observed almost completely in all the different regimes, in each of the models. However, in contrast to the linear and second-order nature of the CV equation, an investigation of the frequency content of the resulting relaxation profiles reveals the existence of more than one possible frequency which is/are time-variant. The latter result is observed in relaxation profiles of systems with a sufficiently large wavelength of temperature distribution, whereas systems having a profound oscillatory nature show a behavior similar to the CV equation.