Temperature and electric potential fields of an interface crack in a layered thermoelectric or metal/thermoelectric material

Highlights

• The interface crack problem in a layered thermoelectric material is studied.

• Electric impermeable and heat semi-permeable crack model is adopt.

• Closed-form solutions of thermoelectric fields at crack tip are obtained.

Abstract

The interface crack problem in a layered thermoelectric or metal/thermoelectric material subjected to thermoelectric loadings is studied based on the nonlinear governing equations and complex variable method. The electric impermeable and heat semi-permeable crack boundary conditions are adopted. Explicit and closed-form solutions of temperature and electric potential on the interface crack surfaces, and crack tip thermoelectric fields are obtained. The influence of crack thickness to length ratio and thermal conductivity of air inside the crack on electric current density and heat flux intensity factors at crack tip is discussed. It is found that electric current density and heat flux intensity factors at the interface crack tip between metal interconnector and thermoelectric material are much bigger than those in homogenous or layered thermoelectric material. This explains why the failure (or microcrack initiation) of a thermoelectric generation/cooler element always appears at the interface between the metal electrode and thermoelectric material.

Introduction

The research and development of thermoelectric materials have attracted significant interests recently owing to their capability in converting heat energy into electric energy directly utilizing the Seebeck effect and vice versa [1], [2]. Being all solid state and without moving parts, thermoelectric devices can be applied to waste heat recovery [3], refrigeration [4], carbon reduction [5] and solar energy harvesting [6]. Design of thermoelectric intelligent devices is calling for a better understanding of the responses of these materials subjected to applied temperature and electric potential loadings. The influence of the Thomson effect on the performance of a thermoelectric cooler was studied by Huang et al. [7]. Finite element analysis of nonlinear fully coupled thermoelectric materials based on continuum mechanics and the Galerkin method was carried out by Pérez-Aparicio et al. [8]. A general, three-dimensional thermoelectric material model by taking the Joule heating, Fourier's heat conduction, the Thomson effect, the Peltier effect, the convection and radiation heat transfer, and temperature dependent properties of semiconductor materials into account was presented by Wang et al. [9]. A continuum theory for thermoelectric medium following the general framework of continuum mechanics and conforming to basic thermodynamic laws was developed by Liu [10]. Dynamic response characteristics of thermoelectric generator predicted by a 3D heat-electricity coupled model was investigated by Meng et al. [11].

Massive efforts on thermoelectric materials have been directed towards improving their figure of merit and energy conversion efficiency [12], [13], [14]. The pioneering works of [15], [16] have shown that the energy conversion efficiency of a layered thermoelectric composite can be higher than all its constituents due to the nonlinear distribution of temperature and electric potential inherent in thermoelectric transport. This conclusion may point to a new route for high figure of merit thermoelectric materials. The formulas for thermoelectric devices that operate in parallel were derived by Mahan [17]. A rigorous nonlinear asymptotic homogenization theory to analyze the coupled transport of electricity and heat in thermoelectric composite materials was developed by Yang et al. [18]. On the other hand, it is well known that thermoelectric materials are typical brittle solids. Defects/imperfections produced during the manufacturing and/or in-service condition, such as cracks, can cause geometric, temperature and electric potential discontinuities across the crack faces, and thermoelectric fields concentrations at the crack tip, thus contribute to critical crack growth. The crack problem in a thermoelectric medium under thermal-electric loads was considered by Zhang and Wang [19]. The effective material properties of thermoelectric composites with elliptical fibers were studied by Wang [20]. The general solutions for the two-dimensional problem in a cracked thermoelectric materials were derived by Song et al. [21]. Further, an overwhelming majority of thermoelectric devices fracture failure begins at the interface between metal electrode and thermoelectric material, or two different thermoelectric materials. Cui et al. [22] observed that cracks occurred near the interface between Sn₉₅Ag₅ and Bi₂Te₃ after thermal stability testing at 192 °C annealing for 200 h. The interface characterization of nickel contacts to bulk bismuth tellurium selenide was reported by Lyore et al. [23]. CoSb₃-based thermoelectric element with insertion of Ti layer by means of spark plasma sintering was obtained by Zhao et al. [24], and they found that the cracks appeared near the interface due to the different thermal expansion coefficient of CoSb₃ and Ti. Therefore, the investigation of interface crack problem in layered thermoelectric or metal/thermoelectric materials is very important.

In view of the above literature analysis, the purpose of this paper is to present the solutions of temperature and electric potential distribution for an interface crack problem in a layered thermoelectric or metal/thermoelectric heterogeneous material. The influence of the thermal conductivity of air inside the crack on the crack tip fields is also studied based on the finite thickness crack model. The paper is organized as follows. Firstly, the governing equations of thermoelectric material are outlined. Next, solutions of electric and temperature fields of a crack-free and a cracked layered thermoelectric material are obtained in closed-form in Sections 2 Governing equations, 3 Temperature and electric potential fields of a crack-free layered thermoelectric material, respectively. Some numerical results are given in Section 4. Finally, concluding remarks on interface crack problem are made.