Discrete space-time model for heat conduction: Application to size- dependent thermal conductivity in nano-films S.L. Sobolev Institute of Problems of Chemical Physics, Academy of Sciences of Russia, Chernogolovka, Moscow Region 142432, Russia article info Article history: Received 29 October 2016 Received in revised form 6 December 2016 Accepted 16 December 2016 Keywords: Nano films Ballistic-diffusive heat conduction Boundary temperature jump Effective thermal conductivity Discrete model abstract An analytical discrete-variable model has been developed to describe heat conduction in nano-sized sys- tems. The model assumes that the system consists of a homogeneous array of cells with characteristic size h; each cell interacts with the nearest neighbors in discrete time step s and all the cells compute their new state simultaneously. In the continuum limit h ! 0 and s ! 0, the model reduces to classical heat diffusion equation of parabolic type or heat conduction equation of hyperbolic type, depending on the choice of scaling invariant. The model is applied to heat conduction in nano-films with emphasis on the transition from the diffusive to ballistic heat transport, which occurs with decreasing film thickness. This model provides a simple method for predicting in a self consistent manner the effective cross-plane thermal conductivities, the temperature jump at the boundaries, the heat flux across the film, and the temperature gradient within the film as functions of the film thickness. The results are in good agreement with molecular dynamic and Monte Carlo simulations. Ó 2016 Elsevier Ltd. All rights reserved. 1. Introduction The classical heat conduction theory, which is based on the local equilibrium assumption, leads to the linear relation q ¼krT between the heat flux q and the temperature gradient rT , where k is the bulk thermal conductivity [1]. This relation known as Four- ier law suggests that the heat flux q(x, t) at a space point x and at a time moment t depends on the temperature gradient at the same space-time point, i.e. rT ðx; tÞ: In other words, Fourier law is local both in time and space. Strictly speaking heat transport is an inher- ently nonlocal phenomenon [1–12]. The heat flux at a point depends on the history of the heat carriers reaching the point at time t and the carriers arrive at the point in space having brought the energy from other points. Thus, there are essentially two important non-Fourier effects: (i) the one is related to the time lag between the heat flux and the corresponding temperature gra- dient - it can be called as a time non-local effect, which describes relaxation to local equilibrium, (ii) the other is a space non-local effect, which takes into account that the carriers come to a point from another distant point. Several theoretical methods have been proposed to study the local nonequilibrium effects [1–20]. Primar- ily, more attention has been paid to study the time-nonlocal (or relaxation) effects (see [1–6] and references therein), which, in particular, have been observed in metals under ultra-short laser irradiation [8] or during ultrafast phase transformations [6,17,18,20]. Recently, the trend towards miniaturization of elec- tronic devices has increased the interest in space nonlocal effects during nano-scale heat conduction [1,2,7–47]. One of the most important characteristics of nano-scale heat conduction is that the thermal conductivity of nanostructures such as thin films, superlattices, nanowires and nanotubes is reduced significantly from that of the corresponding bulk materials depending on the sizes of nanostructures [1,2,9,11,13–16,19–43]. This effect has been observed, for example, in silicon and germanium films [14,2 3–25,27,29,31,32,36,38,41], two-dimensional black phosphorus [43], polycrystalline aluminum nitride [35], graphene and ultrathin graphite [22,28,40,42]. Molecular dynamic (MD) [24–26] and Monte Carlo (MC) [27] or simulations also demonstrate that the effective thermal conductivity of nano-systems is significantly lower than the bulk value and decreases with the system size. The size-dependent thermal conductivity implies the breakdown of the Fourier law in nanoscale heat transfer where the mean- free-path (MFP) of phonons h is comparable or even much larger than the material length scale L. As a consequence, the heat trans- port is no longer diffusive (i.e. dominated by collisions amongst the particles of the system) but becomes ballistic (i.e. dominated by collisions with the walls). The size-dependent thermal conductiv- ity has been considered on the bases of Boltzmann’s equation [9,11], EIT [1,14–16], phonon hydrodynamic equation [29], Lan- dauer approach [23]. Usually, the non-Fourier heat conduction http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.12.051 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved. E-mail address: sobolev@icp.ac.ru International Journal of Heat and Mass Transfer 108 (2017) 933–939 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt