International Journal of Thermal Sciences 48 (2009) 34–49 www.elsevier.com/locate/ijts A hyperbolic two-step model based ﬁnite difference scheme for studying thermal deformation in a double-layered thin ﬁlm exposed to ultrashort-pulsed lasers Tianchan Niu, Weizhong Dai ∗ Mathematics and Statistics, College of Engineering and Science, Louisiana Tech University, Ruston, LA 71272, USA Received 1 August 2007; received in revised form 14 January 2008; accepted 4 February 2008 Available online 4 March 2008 Abstract Hyperbolic two-step micro heat transport equations have attracted attention in thermal analysis of thin metal ﬁlms exposed to ultrashort-pulsed lasers. In this article, we develop a ﬁnite difference scheme for studying thermal deformation in a 2D double-layered micro thin ﬁlm exposed to ultrashort-pulsed lasers. This scheme is obtained based on the hyperbolic two-step model with temperature-dependent thermal properties coupled with the dynamic equations of motion. The method is illustrated by investigating thermal deformation in a gold layer on a chromium padding layer.  2008 Elsevier Masson SAS. All rights reserved. Keywords: Hyperbolic two-step model; Finite difference scheme; Ultrashort-pulsed laser; Thermal deformation 1. Introduction Ultrashort-pulsed lasers with pulse durations of the order of sub-picosecond to femtosecond domain have the ability of lim- iting the undesirable spread of the thermal process zone in the heated sample [1]. They have been widely applied in structural monitoring of thin metal ﬁlms [2], laser micromachining [3] and patterning [4], structural tailoring of microﬁlms [5], and laser synthesis and processing in thin-ﬁlm deposition [6]. For an ultrashort-pulsed laser, the heating involves high-rate heat ﬂow from electrons to lattices in the picosecond domains. The energy equations describing the continuous energy ﬂow from hot electrons to lattices during non-equilibrium heating can be written as [7–9]: C e (T e ) ∂T e ∂t =∇·  k e (T e ,T l )∇ T e  − G(T e − T l ) + S (1) C l ∂T l ∂t = G(T e − T l ) (2) * Corresponding author. Tel.: +1 318 257 3301; fax: +1 318 257 2562. E-mail address: dai@coes.latech.edu (W. Dai). where C e (T e ) = A e T e ,k e (T e ,T l ) = k 0 (T e /T l ), T e is electron temperature, T l lattice temperature, k 0 thermal conductivity in thermal equilibrium, C e and C l volumetric heat capacity, G electron-lattice coupling factor, S laser heating source, and ∇ the gradient operator. Here, A e ,C l , k 0 and G are all positive constants. The above coupled Eqs. (1) and (2), often referred to as parabolic two-step micro heat transport equations, have been widely applied in analysis of microscale heat transfer [7–17]. In particular, Wang et al. [18–20] developed a ﬁnite difference scheme based on the above model for studying thermal defor- mation in a two-dimensional thin ﬁlm exposed to ultrashort- pulsed lasers. However, when the characteristic heating time (which is either the laser pulse during or the time needed to heat a material to a certain temperature) is much shorter than the electron relaxation time of free electrons (the mean time for electrons to change their states) in a metal, the parabolic two- step model may be inadequate to describe the continuous en- ergy ﬂow from hot electrons to lattices during non-equilibrium heating (see Fig. 1 in [11]). As Qiu and Tien [11] pointed out, the relaxation time increases dramatically, as the temperature decreases, from 0.04 ps at room temperature to about 10 ps at 10 K. They [9] developed the hyperbolic two-step heat trans- port equations based on the macroscopic averages of the electric 1290-0729/$ – see front matter  2008 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ijthermalsci.2008.02.001