Contents lists available at ScienceDirect International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts Effective interface thermal resistance and thermal conductivity of dielectric nanolayers Kamal Alaili * , Jose Ordonez-Miranda, Younès Ezzahri Institut Pprime, CNRS, Université de Poitiers, ISAE-ENSMA, F-86962 Futuroscope Chasseneuil, France ARTICLE INFO Keywords: Dielectric nanolayers Boltzmann transport equation Effective interface thermal resistance Effective thermal conductivity Ballistic-diffusive heat conduction ABSTRACT Analytical expressions for the effective interface thermal resistance (ITR) and thermal conductivity k of dielectric nanolayers are derived and analyzed, based on the analytical solution of the phonon Boltzmann transport equation under the gray relaxation time approximation. This is achieved by using accurate expressions for the temperature and one-dimensional heat flux propagating across nanolayers supporting a diffusive phonon scat- tering at their interfaces. It is shown that the effective ITR between two layers can be symmetric on their thermal properties, such that its asymptotic value in the ballistic regime is higher than that in the diffusive one. In the ballistic-diffusive regime, the effective ITR depends strongly on the ratio = λ L l / , between the layer thickness L and mean free path l of phonons. Our predictions for the effective ITR in the ballistic regime are in good agreement with those of the diffuse mismatch model, while they differ by about 16% in the diffusive regime. On the other hand, k increases with λ until reaching saturation for bulk layers and agrees rather well with previous predictions reported in the literature. The obtained results could be useful for analytically describing the heat transport in dielectric nanothin films and superlattices, in which the gray approximation is valid. 1. Introduction In recent years, the thermal transport in thin solid films with thickness ranging from tens of nanometers to micrometers has become an important research topic in fundamental science [1–4], due to its importance and potential applications in electronics and photonics [5]. As is well known, when the thickness of a thin film becomes compar- able to the mean free paths (MFP) of its heat carriers (phonons and/or electrons) [6], heat conduction deviates from the predictions of the diffusive Fourier's law, due to their ballistic dynamics described by the Boltzmann transport equation (BTE) [7,8]. Significant efforts have been devoted to analytically and numerically solve the BTE, whose solution is rather difficult to determine, owing to its high dimensionality. Fuchs and Sondheimer derived analytical solutions of the BTE for electrons undergoing partially specular and partially diffuse boundary scattering [9,10]. Mazumder and Majumdar used a Monte-Carlo method to study the phonon transport along a silicon thin film including phonon dis- persion and polarizations [11]. Based on the discrete-ordinates method and the gray relaxation time approximation, Majumdar numerically solved the BTE for the temperature and heat flux in a dielectric thin film, considering that its two surfaces are black phonon emitters [7]. More recently, Chen [12] obtained an implicit solution of the BTE to study the ballistic phonon transport in the cross-plane direction of superlattices and analyzed the inconsistent definitions of temperature at the interfaces. Chengyun [13] derived a semi-analytical solution of the frequency-dependent transient BTE using the method of degenerate kernels to study phonon transport in both the diffusive to ballistic re- gimes. More recently, a recent paper by Ordonez-Miranda et al. [14] presented explicit analytical solutions of the phonon BTE under the gray relaxation time approximation for the steady-state and modulated components of the temperature of dielectric layered systems. In this latter work, the effects of the interface mismatch are taken into account through the reflectivity and transmissivity of phonons, without invol- ving the effective interface thermal resistance (ITR). Two common models for estimating the effective ITR are the acoustic mismatch model (AMM) and the diffusive mismatch model (DMM) [15,16]. In the DMM, the incident phonon loses all memory of its direction and polarization, after being reflected or transmitted at the interface, while in the AMM, its direction of propagation is controlled by laws similar to those of Snell-Descartes in geometric optics. The AMM can explain experimental data for specular interfaces mainly, but fails to describe the heat transport across diffusive interfaces [15,17], which are better described by the DMM [18]. As the thickness of a dielectric thin film reduces to values comparable or smaller than the phonon MFP, size effects are expected to appear on the effective ITR, as is the case of its effective thermal conductivity [19–21]. This is the case https://doi.org/10.1016/j.ijthermalsci.2018.05.024 Received 31 October 2017; Received in revised form 12 April 2018; Accepted 14 May 2018 * Corresponding author. E-mail addresses: kamal.alaili@univ-poitiers.fr (K. Alaili), jose.ordonez@cnrs.pprime.fr (J. Ordonez-Miranda), younes.ezzahri@univ-poitiers.fr (Y. Ezzahri). International Journal of Thermal Sciences 131 (2018) 40–47 1290-0729/ © 2018 Elsevier Masson SAS. All rights reserved. T