The Thermal Discrete Dipole Approximation (T-DDA) for near-field radiative heat transfer simulations in three-dimensional arbitrary geometries

Highlights

• The T-DDA is proposed for near-field thermal radiation simulations.

• The T-DDA is verified against analytical results for two spheres.

• The T-DDA is employed for simulations involving two cubes of silica.

Abstract

A novel numerical method called the Thermal Discrete Dipole Approximation (T-DDA) is proposed for modeling near-field radiative heat transfer in three-dimensional arbitrary geometries. The T-DDA is conceptually similar to the Discrete Dipole Approximation, except that the incident field originates from thermal oscillations of dipoles. The T-DDA is described in details in the paper, and the method is tested against exact results of radiative conductance between two spheres separated by a sub-wavelength vacuum gap. For all cases considered, the results calculated from the T-DDA are in good agreement with those from the analytical solution. When considering frequency-independent dielectric functions, it is observed that the number of sub-volumes required for convergence increases as the sphere permittivity increases. Additionally, simulations performed for two silica spheres of 0.5 μm-diameter show that the resonant modes are predicted accurately via the T-DDA. For separation gaps of 0.5 μm and 0.2 μm, the relative differences between the T-DDA and the exact results are 0.35% and 6.4%, respectively, when 552 sub-volumes are used to discretize a sphere. Finally, simulations are performed for two cubes of silica separated by a sub-wavelength gap. The results revealed that faster convergence is obtained when considering cubical objects rather than curved geometries. This work suggests that the T-DDA is a robust numerical approach that can be employed for solving a wide variety of near-field thermal radiation problems in three-dimensional geometries.

Introduction

Radiation heat transfer between bodies separated by distances greater than the dominant thermal wavelength is limited by Planck's blackbody distribution. In this far-field regime, radiative heat exchange predictions in three-dimensional (3D) complex geometries are tractable using well-established numerical techniques such as the discrete ordinates method and Monte Carlo approaches [1], [2]. In the near-field regime of thermal radiation, which refers to the case where bodies are separated by sub-wavelength gaps, heat transfer can exceed by several orders of magnitude the blackbody limit [1], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]. The enhancement beyond Planck's distribution is due to the extraneous contribution to energy transport by waves evanescently confined within a distance of about a wavelength normal to the surface of a thermal source. These modes include evanescent waves generated by total internal reflection of a propagating wave at the material–gap interface as well as resonant surface waves such as surface phonon-polaritons and surface plasmon-polaritons.

To account for tunneling of evanescent modes and wave interference, near-field heat transfer problems are modeled using fluctuational electrodynamics [1], [3], [15]. In this formalism, thermal emission is modeled in Maxwell's equations by stochastic currents that are related to the local temperature of the source via the fluctuation-dissipation theorem. So far, the vast majority of near-field radiative heat transfer predictions have been restricted to simple canonical geometries. This is due to the fact that near-field thermal radiation problems have been mainly solved by deriving analytical expressions for dyadic Green's functions (DGFs); this approach is referred to as the DGF method. The DGF method provides exact results, but becomes intractable when dealing with 3D arbitrarily-shaped objects. Over the past years, the DGF approach has been applied to various cases: two bulks [4], [5], [6], [15], [16], [17], [18], [19], two films [20], [21], [22], [23], two structured surfaces [24], two nanoporous materials [25], one-dimensional layered media [26], [27], [28], cylindrical cavity [29], two dipoles [30], [31], [32], two large spheres [33], [34], dipole-surface [35], dipole-structured surface [36], sphere-surface [34], [37], two long cylinders [38], two nanorods [39], [40], two gratings [41] and N small objects (compared to the wavelength) modeled as electric point dipoles [42].

With the rapid advances in nanofabrication, near-field thermal radiation is becoming an important part of heat transfer engineering. Indeed, near-field thermal radiation may find application in imaging [43], thermophotovoltaic power generation [44], [45], [46], [47], [48], nanomanufacturing [49], [50], thermal management of electronic devices [51], thermal rectification through a vacuum gap [52], [53] and radiative property control [54], [55], [56] to name only a few. Due to these numerous potential applications, there is a need for predicting near-field heat exchange in 3D complex geometries. Numerical procedures, namely the finite-difference time-domain (FDTD) method [57], [58], [59], the finite-difference frequency-domain (FDFD) method [60] and the boundary element method (BEM) [61] have been applied recently to near-field thermal radiation calculations. Both FDTD and FDFD approaches suffer from large computational time, while the BEM is difficult to apply when dealing with heterogeneous materials. In this work, the Thermal Discrete Dipole Approximation (T-DDA) is proposed for simulating near-field heat transfer between 3D arbitrarily-shaped objects. The Discrete Dipole Approximation (DDA), extensively used for predicting electromagnetic wave scattering by particles, is based on discretizing objects into cubical sub-volumes behaving as electric point dipoles [62], [63], [64], [65], [66]. The T-DDA follows the same general procedure as the DDA, except that the incident field is induced by thermal fluctuations of dipoles instead of being produced by an external illumination.

The objective of this paper is to formulate the T-DDA and to test the method against exact results obtained from the DGF approach. In Section 2, the physical and mathematical formulation of the problem is provided. Next, the T-DDA is derived starting from the stochastic Maxwell equations and the associated solution procedure is detailed. In the fourth section, the T-DDA is verified against exact results for two spheres separated by a sub-wavelength vacuum gap; a problem involving two cubes is presented afterwards. Concluding remarks are finally provided.