Second-Order Dual Phase Lag Equation. Modeling of Melting and Resolidification of Thin Metal Film Subjected to a Laser Pulse

Ewa Majchrzak,Bohdan Mochnacki

doi:10.3390/math8060999

Abstract

The process of partial melting and resolidification of a thin metal film subjected to a high-power laser beam is considered. The mathematical model of the process is based on the second-order dual phase lag equation (DPLE). Until now, this equation has not been used for the modeling of phase changes associated with heating and cooling of thin metal films and the considerations regarding this issue are the most important part of the article. In the basic energy equation, the internal heat sources associated with the laser action and the evolution of phase change latent heat are taken into account. Thermal processes in the domain of pure metal (chromium) are analyzed and it is assumed that the evolution of latent heat occurs at a certain interval of temperature to which the solidification point was conventionally extended. This approach allows one to introduce the continuous function corresponding to the volumetric fraction of solid or liquid state at the neighborhood of the point considered, which significantly simplifies the phase changes modeling. At the stage of numerical computations, the authorial program based on the implicit scheme of the finite difference method (FDM) was used. In the final part of the paper, the examples of numerical computations (including the results of simulations for different laser intensities and different characteristic times of laser pulse) are presented and the conclusions are formulated.

Highlights

Heat transfer through thin films subjected to an ultrafast laser pulse is of vital importance in microtechnology applications and is a reason that the problems related to the fast heating of solids have become a very active research area
The mathematical model of macroscale heat conduction is based on the parabolic Fourier equation
The computations were performed for the thin metal film of the thickness G = 100 nm made of chromium

Summary

Introduction

Heat transfer through thin films subjected to an ultrafast laser pulse is of vital importance in microtechnology applications and is a reason that the problems related to the fast heating of solids have become a very active research area. One can already find books devoted to this type of non-Fourier heat conduction model, for example, [10,11,12,13] In this brief literature review, the selected papers on analytical and numerical solutions of first-order DPLE will be listed. The main subject of these papers (except [37]) is related to the construction of algorithms for numerical modeling of problems described by second-order DPLE (the different variants of FDM are used). At the beginning of the part of the article devoted to own research, the assumed form of the dualphase lag equation and the mathematical formulas determining the laser action and the evolution of phase change latent heat are presented Both phenomena are taken into account by an introduction to the energy equation of the functions determining the efficiency of internal heat sources. The conclusions resulting from the performed research are formulated

Governing Equations

Numerical Model Based on FDM f

Results of Computations

Conclusions

Methods