Ballistic heat transport in semiconductors within the extended irreversible thermodynamics theory

The transport in miniaturised electronic devices requires to go beyond simple hydrodynamic descriptions. So, carrier transports in nano-length devices is no longer dominated by collisions among the particles but the ballistic transport governs at this level. This type of transport occurs when the perturbation characteristic time is of the order of the relaxation time and the mean free path is of the order of the device's dimension. The aim of the present work is to calculate the ballistic velocity of heat transport by using a continued-fraction technique in the framework of extended irreversible thermodynamics. This ballistic speed must be less or equal to the maximum value c0 corresponding to phonon speed. Note that the classical Fourier equation is only able to describe the diffusion regime and the Maxwell-Cattaneo equation predicts a second sound propagation at speed c0/Ö3 but it is silent about the ballistic behaviour .The transport in miniaturised electronic devices requires to go beyond simple hydrodynamic descriptions. So, carrier transports in nano-length devices is no longer dominated by collisions among the particles but the ballistic transport governs at this level. This type of transport occurs when the perturbation characteristic time is of the order of the relaxation time and the mean free path is of the order of the device's dimension. The aim of the present work is to calculate the ballistic velocity of heat transport by using a continued-fraction technique in the framework of extended irreversible thermodynamics. This ballistic speed must be less or equal to the maximum value c0 corresponding to phonon speed. Note that the classical Fourier equation is only able to describe the diffusion regime and the Maxwell-Cattaneo equation predicts a second sound propagation at speed c0/Ö3 but it is silent about the ballistic behaviour

The Hodograph Equation for Slow and Fast Anisotropic Interface Propagation

Using the model of fast phase transitions and previously reported equation of the Gibbs-Thomson-type, we develop an equation for the anisotropic interface motion of the Herring-Gibbs-Thomson-type. The derived equation takes the form of a hodograph equation and in its particular case describes motion by mean interface curvature, the relationship 'velocity - Gibbs free energy', Klein-Gordon and Born-Infeld equations related to the anisotropic propagation of various interfaces. Comparison of the present model predictions with the molecular-dynamics simulation data on nickel crystal growth (obtained by Jeffrey J. Hoyt et al. and published in Acta Mater. 47 (1999) 3181) confirms the validity of the derived hodograph equation as applicable to the slow and fast modes of interface propagation. This article is part of the theme issue 'Transport phenomena in complex systems (part 1)'. © 2021 The Authors.Data accessibility. Electronic supplementary material on asymptotic analysis of the hyperbolic phase field model are attached to the main text of the manuscript. Authors’ contributions. Both authors contributed to the derivation, analytical treatments and analysis of results. A.S. carried out calculations for comparison of the analytical equation with data of molecular dynamics simulations. Competing interests. We declare we have no competing interests. Funding. The funding has been made for P.K.G. by German Science Foundation (DFG-Deutsche Forschungsgemeinschaft) under the Project GA 1142/11-1. Acknowledgements. Authors thank Jeffrey J. Hoyt for his valuable explanations about molecular dynamic simulation data of Ni. P.K.G. acknowledges financial support of German Science Foundation (DFG-Deutsche Forschungsgemeinschaft). A.S. thanks M. Bennai for hosting the present work in the research activities of LPMC

Thermodynamics of rapid solidification and crystal growth kinetics in glass-forming alloys

Thermodynamic driving forces and growth rates in rapid solidification are analysed. Taking into account the relaxation time of the solute diffusion flux in the model equations, the present theory uses, in a first case, the deviation from local chemical equilibrium, and ergodicity breaking. The second case of ergodicity breaking may exist in crystal growth kinetics of rapidly solidifying glass-forming metals and alloys. In this case, a theoretical analysis of dendritic solidification is given for congruently melting alloys in which chemical segregation does not occur. Within this theory, a deviation from thermodynamic equilibrium is introduced for high undercoolings via gradient flow relaxation of the phase field. A comparison of the present derivations with previously verified theoretical predictions and experimental data is given. This article is part of the theme issue 'Heterogeneous materials: Metastable and nonergodic internal structures'. ©2019 The Author(s) Published by the Royal Society

Solute trapping phenomenon in binary systems and hodograph-equation within effective mobility approach

The phase field model is developed by the effective mobility approach to slow and rapid solidification. The phase field model equations are reduced to the hodograph equation for solid-liquid interface movement which is applied to the problem of solute trapping in a binary alloy. A specific method based on the one-point Cauchy problem is developed for solution of the hodograph equation with the solute diffusion equation. The method is tested in comparison with the rapid solidification of Si–0.1 at.% As alloy previously analyzed experimentally and using phase field modelling

Kinetics of rapid crystal growth: Phase field theory versus atomistic simulations

Kinetics of crystal growth in undercooled melts is analyzed by methods of theoretical modeling. Special attention is paid to rapid growth regimes occurring at deep undercoolings at which non-linearity in crystal velocity appears. A traveling wave solution of the phase field model (PFM) derived from the fast transitions theory is used for a quantitative description of the crystal growth kinetics. The "velocity - undercooling" relationship predicted by the traveling wave solution is compared with the data of molecular dynamics simulation (MDS) which were obtained for the crystal-liquid interfaces growing in the 100-direction in the Ni50Al50 alloy melt. © Published under licence by IOP Publishing Ltd

Hodograph-equation for rapid solidification of Si-0.1 at.% As alloy melt

Hyperbolic-type equations of both phase field and concentration arising from a phase-field model for fast phase transformations in binary dilute systems yield in the one-dimension moving frame of reference to the concentration-and phase field governing equations, respectively. These equations have been solved numerically and applied to the case of Si-0.1 at.% As binary alloy [P.K. Galenko et al., Phys. Rev. E 84, 041143 (2011)]. In this paper, the coupling of the hodograph equation for the interface with the solute diffusion equation leads to an exact analytical solution of the one-point Cauchy problem of an ordinary differential equation in a parametric form. Application of this solution to the case of Si-0.1 at.% As gives (i) the same tendency of concentration variation along dimensionless spatial coordinate (ii) the same values of interface velocity with a very slight difference in the value of concentration for a given undercooling at the interface. Based on the results obtained, the established hodograph-equation confirms again its usefulness to predict, for instance, certain aspects of rapid solidification processes for binary alloys

A brief glimpse of extended irreversible thermodynamics theory and some of its applications

In the last decade, it has been shown that the so-called extended irreversible thermodynamics (EIT) has a wider range of applications and a deep insight on open physical problems than classical irreversible thermodynamics in the research on processes out of equilibrium. The purpose of the present paper is to exhibit the basis and some current applications of EIT, as for instance, the generalized absolute temperature, the behavior of entropy in hyperbolic heat conduction, the flux saturation behavior and an approximate value of ballistic phase velocity.In the last decade, it has been shown that the so-called extended irreversible thermodynamics (EIT) has a wider range of applications and a deep insight on open physical problems than classical irreversible thermodynamics in the research on processes out of equilibrium. The purpose of the present paper is to exhibit the basis and some current applications of EIT, as for instance, the generalized absolute temperature, the behavior of entropy in hyperbolic heat conduction, the flux saturation behavior and an approximate value of ballistic phase velocity