Modelling of transient heat transport in metal films using the interval lattice Boltzmann method

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In the paper a description of heat transfer in one-dimensional crystalline solids is presented. The lattice Boltzmann method based on Boltzmann transport equation is used to simulate the nanoscale heat transport in thin metal films. The coupled lattice Boltzmann equations for electrons and phonons are applied to analyze the heating process of thin metal films via laser pulse. Such approach in which the parameters appearing in the problem analyzed are treated as constant values is widely used, but in the paper the interval values of relaxation times and electron-phonon coupling factor are taken into account. The problem formulated has been solved by means of the interval lattice Boltzmann method using the rules of directed interval arithmetic. In the final part of the paper the results of numerical computations are shown.


  • [1] M. Eshraghi, S.D. Felicelli, “An implicit lattice Boltzmann model for heat conduction with chase chandler”, Int. J. of Heat and Mass Transfer 55, 2420–2428 (2012).

  • [2] R.A. Escobar, C.H. Amon, “Thin film phonon heat conduction by the dispersion lattice Boltzmann method”, J. Heat Transfer 130(9), 092402 1–8 (2008).

  • [3] G.R. McNamara, G. Zanetti, “Use of the Boltzmann equation to simulate lattice-gas automata”, Phys. Rev. Lett. 61 (20), 2332–2335 (1988).

  • [4] F. Higuera, S. Succi, R. Benzi, “Lattice gas dynamics with enhanced collisions”, Europhys. Let. 9 (4), 345–349 (1989).

  • [5] R. Benzi, S. Succi, M. Vergassola, “The lattice Boltzmann equation: theory and applications”, Phys. Rep. 222 (3), 145–197 (1992).

  • [6] E. Majchrzak, G. Kałuża, “Heat flux formulation for 1D dual-phase lag equation”, J. of Applied Mathematics and Computational Mechanics 14(1), 71–78 (2015).

  • [7] E. Majchrzak, B. Mochnacki, “Sensitivity analysis of transient temperature field in microdomains with respect to the dual phase lag model parameters”, Int. J. for Multiscale Computational Engineering 12(1), 65–77 (2014).

  • [8] E. Majchrzak, J. Dziatkiewicz, “Analysis of ultashort laser pulse interactions with metal films using a two-temperature model”, J. of Applied Mathematics and Computational Mechanics 14(2), 31–39 (2015).

  • [9] A. Maroufi, C. Aghanajafi, “Analysis of conduction–radiation heat transfer during phase change process of semitransparent materials using lattice Boltzmann method”, J. of Quantitative Spectroscopy & Radiative Transfer 116, 145–155 (2013).

  • [10] S.C. Mishra, B. Mondal, T. Kush, B.S.R. Krishna, “Solving transient heat conduction problems on uniform and non-uniform lattices using the lattice Boltzmann method”, Int. Communications in Heat and Mass Transfer 36, 322–328 (2009).

  • [11] D. Wang, Z. Qu, Y. Ma, “An enhanced Gray model for nondiffusive heat conduction solved by implicit lattice Boltzmann method”, J. of Heat and Mass Transfer 94, 411–418 (2016).

  • [12] P.E. Hopkins, P.M. Norris, “Contribution of ballistic electron transport to energy transfer during electron-phonon nonequi in thin metal films”, J. Heat Transfer, 131(4), 043208 1–8 (2009).

  • [13] J.K. Chen, D.Y. Tzou, J.E. Beraun, “A semiclassical two-temperature model for ultrafast laser heating”, Int. J. of Heat and Mass Transfer 49, 307–316 (2006).

  • [14] A.A. Joshi, A. Majumdar, “Transient ballistic and diffusive phonon heat transport in thin films”, J. of Appl. Physics 74(1), 31–39 (1993).

  • [15] A. Piasecka Belkhayat, A. Korczak, Modelling of Transient Heat Transport in One-Dimensional Crystalline Solids Using the Interval Lattice Boltzmann Method, 363–368 Taylor & Francis Group, London (2014).

  • [16] A. Piasecka Belkhayat, A. Korczak, “Numerical modelling of the transient heat transport in 2d silicon thin film using the interval lattice Boltzmann method”, J. of Appl. Mathematics and Comp. Mechanics 13 (2), 95–103 (2014).

  • [17] A. Piasecka Belkhayat, “The interval lattice Boltzmann method for transient heat transport”, Scientific Research of the Institute of Mathematics and Computer Science 1(8), 155–160 (2009).

  • [18] M. Kaviany, A.J.H. McGaughey, J.D. Chung, “Integration of molecular dynamics simulations and Boltzmann transport equation in phonon thermal conductivity analysis”, Heat Transfer 4, 277–287 (2003).

  • [19] J.B. Lee, K. Kang, S.H. Lee, “Comparison of theoretical models of electron-phonon coupling in thin gold films irradiated by femtosecond pulse lasers”, Materials Transactions 52 (3), 547–553 (2011).

  • [20] S.M. Markov, “On directed interval arithmetic and its applications”, J. of Universal Comp. Science 1, 514–526 (1995).

  • [21] A. Piasecka Belkhayat, A. Korczak, “Numerical modelling of the transient heat transport in a thin gold film using the fuzzy lattice Boltzmann method with α-cuts”, J. of Applied Mathematics and Computational Mechanics 15(1), 99–111 (2016).

  • [22] B. Mochnacki, A. Piasecka Belkhayat, “Numerical modeling of skin tissue heating using the interval finite difference method”, Molecular & Cellular Biomechanics 10 (3), 233–244 (2013).

  • [23] A. Piasecka Belkhayat, “Interval boundary element method for 2D transient diffusion problem”, Engineering Analysis with Boundary Elements 32 (5), 424–430 (2008).

  • [24] A. Piasecka Belkhayat, Interval Boundary Element Method for Imprecisely Defined Unsteady Heat Transfer Problems, Monograph (321), Silesian University of Technology, Gliwice, 2011, (in Polish).

  • [25] A. Neumaier, Interval Methods for System of Equations, Cambridge University Press, Cambridge, New York, Port Chester, Melbourne, Sydney, 1990.

  • [26] S.S. Ghai, W.T. Kim, R.A. Escobar, et al., “A novel heat transfer model and its application to information storage systems”, J. Appl. Physics 97, 10P703 1–3 (2005).

  • [27] S. Pisipati, J. Geer, B. Sammakia, B.T. Murray, “A novel alternate approach for multiscale thermal transport using diffusion in the Boltzmann Transport Equation”, Int. J. of Heat and Mass Transfer 54, 3406–3419 (2011).

  • [28] R.A. Escobar, S.S. Ghai, M.S. Jhon, C.H. Amon, “Multi-length and time scale thermal transport using the lattice Boltzmann method with application to electronics cooling”, J. of Heat and Mass Transfer 49, 97–107 (2006).

  • [29] K. Venkatakrishnan, B. Tan, B.K.A. Ngoi, “Femtosecond pulsed laser ablation of thin gold film”, Optics & Laser Technology 34, 199 – 202 (2002).

  • [30] X. Zhang, S.S. Chu, J.R. Ho, Grigoropoulos C.P., “Excimer laser ablation of thin gold films on a quartz crystal microbalance at various argon background pressures”, Appl. Physics A 64, 545–552 (1997).

  • [31] B.S. Yilbas, A.Y. Al-Dweik, Bin Mansour S., “Analytical solution of hyperbolic heat conduction equation in relation to laser short-pulse heating”, Physica B 406, 1550–1556 (2011).

  • [32] R. Escobar, B. Smith, C. Amon, “Lattice Boltzmann modeling of subcontinuum energy transport in crystalline and amorphous microelectronic devices”, J. of Electronic Packaging 128 (2), 115–124 (2006).

  • [33] J.K. Chen, J.E. Beraun, “Numerical study of ultrashort laser pulse interactions with metal films”, Numerical Heat Transfer, Part A, 40, pp. 1–20, 2001.

  • [34] E. Majchrzak, B. Mochnacki, A.L. Greer, J.S. Suchy, “Numerical modeling of short pulse laser interactions with multi-layered thin metal films”, CMES: Computer Modeling in Engineering and Sciences, 41 (2), 131–146 (2009).

  • [35] E. Majchrzak, B. Mochnacki, J.S. Suchy, “Numerical simulation of thermal processes proceeding in a multi-layered film subjected to ultrafast laser heating”, J. of Theor. and Appl. Mechanics 47 (2), 383–396 (2009).

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