Room temperature mechanical properties of natural-mineral-based thermoelectrics

Abstract

Low cost, highly efficient thermoelectric materials for waste heat recovery applications can be made by combining the naturally occurring thermoelectric mineral tetrahedrite (Cu₁₀Zn₂As₄S₁₃) and the synthetic compound Cu₁₂Sb₄S₁₃. To better utilize this material in waste heat harvesting applications, it is essential to characterize the material’s mechanical properties including elastic modulus, hardness, and fracture toughness. In this study, powders of Cu₁₀Zn₂As₄S₁₃ were mixed with varying amounts of Cu₁₂Sb₄S₁₃ and then densified by hot pressing. The room temperature mechanical properties were investigated as a function of (i) composition and (ii) ball milling time. Elastic moduli were measured using resonant ultrasound spectroscopy. Hardness and fracture toughness were determined by Vickers indentation technique.

Appendices

Appendix 1: Crack length as a function of load and relationship to equation 3

As reviewed by Ponton and Rawlings [39, 40], a number of expressions exist in the literature for calculating fracture toughness from Vickers indentation cracks. In particular, dependence of crack length, c, on load F can depend on the indentation crack profile. For example, Palmquist cracks (which have separate, semi-elliptical cracks on either side of the indentation impression [39]) have a different c versus F dependence than fully developed radial cracks [16, 39, 40]. To highlight the c versus F relationship, Eq. 3 can be rewritten as

$$ F = Dc^{ 1. 5} $$

(6)

where $D = K{}_{c}/\xi (E/H)^{0.5}$. In order to experimentally determine the c versus F dependence of the Vickers indentation cracks in this work, a separate crack length versus indentation load study was performed in which a 0.50 Syn specimen was indented at loads of 0.98, 1.96, 2.94, and 4.90 N with 10 indentations at each load. The resulting average radial crack length versus load data was least-squares fit to Eq. 6, with a coefficient of determination, R ², of 0.983 (Fig. 9), indicating that the indentation data are consistent with the fully developed radial crack system assumed by Eq. 3 [16].

Fig. 9

A plot of c ^1.5 versus indentation load, where c is the radial crack length induced by Vickers indentation on a 0.50 Syn composition specimen. The straight line represents a least-squares fit to Eq. 6. The crack length versus load behavior is consistent with that expected from Eq. 3

Appendix 2: Estimation of theoretical densities for intermediate compounds

In order to determine the porosity of the specimens included in this study, the theoretical densities of the intermediate compounds must be known. The natural mineral Cu_12−x (Zn, Fe) _x As₄S₁₃ and the intermediate compounds included in this study are solid solutions of tetrahedrite (Cu₁₂Sb₄S₁₃) and the synthetic compound tennantite (Cu₁₂As₄S₁₃) [5, 6]. In the natural mineral, Zn and Fe are substituted on the Cu site [5, 6]. However, the ratio of Zn and Fe on the Cu site can vary within a bulk specimen of the natural mineral, making precise theoretical density calculations difficult. In order to estimate the theoretical density of the intermediate compounds, ρ _{theo est}, the following interpolation was employed,

$$ \rho_{{{\text{theo}}\;{\text{est}}}} = \frac{{Mf_{\text{tet}} + Mf_{\text{ten}} }}{{\frac{{Mf_{\text{tet}} }}{{\rho_{\text{tet}} }} + \frac{{Mf_{\text{ten}} }}{{\rho_{\text{ten}} }}}} = \frac{{f{}_{\text{tet}} + f_{\text{ten}} }}{{\frac{{f_{\text{tet}} }}{{\rho_{\text{tet}} }} + \frac{{f_{\text{ten}} }}{{\rho_{\text{ten}} }}}} = \frac{1}{{\frac{{f_{\text{tet}} }}{{\rho_{\text{tet}} }} + \frac{{f_{\text{ten}} }}{{\rho_{\text{ten}} }}}} $$

(7)

In Eq. 7, the theoretical densities of the tetrahedrite (ρ _tet) and tennantite (ρ _ten) were calculated from the mass per unit cell and the lattice parameter data from Wuensch [41] and Pfitzner et al. [42], respectively, which gave theoretical densities ρ _tet = 5.05 g cm⁻³ ρ _ten = 4.67 g cm⁻³. Also, M is the total specimen mass and f _tet and f _ten are the mass fractions of the tetrahedrite and tennantite, respectively. In this calculation we assume that since (i) the mass differences among the Cu, Fe, and Zn ions are small, (ii) the Zn and Fe impurity concentration in the natural mineral is low, (iii) the crystal structure is the same for Cu₁₂Sb₄S₁₃, Cu₁₂As₄S₁₃, the natural mineral Cu_12−x (Zn, Fe) _x As₄S₁₃ and each of the intermediate compounds, then the variations in the Zn and Fe impurity levels and the details of how they are incorporated into the structure of each composition will not substantially affect the theoretical density of the intermediate compounds.

Given these assumptions, most of the difference among the theoretical densities of the intermediate compounds will be due to the changes in the Sb/As ratio thus the interpolation between the theoretical densities of Cu₁₂Sb₄S₁₃ and the natural mineral (Eq. 7) should give reasonable estimates of the theoretical densities of the intermediate compounds. The theoretical densities for all of the compositions in this study are given in Table 1, along with the calculated porosities of the sintered specimens.