Abstract
Multi‐phase thermoelectric materials are mainly investigated these days due to their potential of lattice thermal conductivity reduction by scattering of phonons at interfaces of the involved phases, leading to the enhancement of expected thermoelectric efficiency. On the other hand, electronic effects of the involved phases on thermoelectric performance are not always being considered, while developing new multi‐phase thermoelectric materials. In this chapter, electronic effects resulting from controlling the phase distribution and morphology alignment in multi‐phase composite materials is carefully described using the general effective media (GEM) method and analytic approaches. It is shown that taking into account the specific thermoelectric properties of the involved phases might be utilized for estimating expected effective thermoelectric properties of such composite materials for any distribution and relative amount of the phases. An implementation of GEM method for the IV–VI (including SnTe and GeTe), bismuth telluride (Bi2Te3), higher manganese silicides (HMS) and half‐Heusler classes of thermoelectric materials is described in details.
Keywords
- thermoelectric
- GEM
- multi‐phase
1. Thermoelectrics
Climate changes, due to fossil fuels combustion and greenhouse gases emission, cause deep concern about environmental conservation. Another pressing issue is sustainable energy production that is coupled with depletion of conventional energy resources. This concern might be tackled by converting the waste heat generated in internal‐combustion vehicles, factories, computers, etc. into electrical energy. Converting this waste heat into electricity will reduce fossil fuel consumption and emission of pollutants. This can be achieved by direct thermoelectric (TE) converters, as was successfully demonstrated by development of various highly efficient TE material classes, including Bi2Te3 [1–3] for temperatures,
2. Multi‐phase thermoelectric materials
In the last few decades, major trend is to move from pristine single‐crystal TE compositions towards polycrystalline multi‐phase materials. One of the reasons for that is improved shear mechanical strength of polycrystalline materials compared to single crystals, exhibiting high compression, but very low transverse strengths, required to withstand high thermal and mechanical gradients applied in practical applications. Another reason is the possibility of phonon scattering by the involved interfaces as mentioned above. Most of the TE materials investigated these days are being synthesized by powder metallurgy approach under high uniaxial mechanical pressures, deforming involved grains and phases into anisotropic geometrical morphologies, which affect the electronic transport properties. Besides, a certain amount of porosity (as a second phase) is in many cases unavoidable, adversely affecting TE transport properties. Furthermore, many of currently employed TE materials (e.g. Bi2Te3 and HMS) are crystallographic anisotropic with optimal TE transport properties along preferred orientations. Some researches of such materials for TE applications do not consider crystallographic anisotropy, while assuming, that randomly oriented grains of different crystallographic planes cancel each other in polycrystalline samples. Yet, some anisotropy can exist also in such materials in case of highly anisotropic specific properties (e.g. mechanical properties), leading to textured polycrystal. For example, texture development of non‐cubic polycrystalline alloys was attributed to multiple deformation modes applied in each grain, twinning resulting in grain reorientation and strong directional grain interactions [12]. Specifically, in Bi2Te3, for example, exhibiting highly anisotropic layered crystal structure consists of 15 parallel layers stacked along crystallographic
Besides of metallurgical phases, individual transport properties of two species (e.g. light and heavy holes in
In this chapter, effective TE properties (Seebeck coefficient,
We consider in this chapter a simple formulation for modelling of multi‐phase TE materials, originated from materials science aspects, such as inter‐diffusion, alloying, dissolution, phase transitions, phase separation, phase segregation, precipitation, recrystallization and other phenomena, that can take place in operation conditions of TE modules, especially TE power generation modules exposed to high thermo‐mechanical stresses.
3. TE GEM effective equations for two‐phase materials
Effective TE properties of two‐phase composites can be accurately predicted by GEM method, Eqs. (1)–(3) [4, 33–35]:
These three GEM equations, Eqs. (1)–(3), are usually employed for calculating effective Seebeck coefficient (
Similarly, substituting
Please note that although for the case of parallel alignment, effective electrical and thermal conductivity, Eq. (4), follow a simple rule of mixture, a more complicated dependency is apparent for series alignment, Eq. (6). Yet, as shown in Eq. (7), for this latter case, effective electrical resistivity,
Substituting of Eq. (6) in Eq. (1), leads in this case to Eq. (8):
While investigating Eqs. (5) and (8) , for the cases of parallel and series alignment, respectively, it can be easily seen, that for both cases, effective Seebeck coefficient depends not only on individual Seebeck coefficients of the two phases, but also on other electronic transport properties, electrical conductivity of the involved phases for the case of parallel alignment, Eq. (5), and thermal conductivity of the involved phases for the case of series alignment, Eq. (8). An explanation for this observation is given in the next section.
4. Analytical effective equations for multi‐phase materials
In order to extend GEM, Eqs. (1)–(3) listed above for two‐phase composite materials, into higher‐ordered composites with three or more coexisting phases, a simple analytical model for calculating effective TE properties of several conductors, subjected to external electrical and thermal gradients, can be applied. For this purpose, two boundary conditions explained above, can be examined; one for conductors connected in parallel to both thermal and electrical applied gradients and the other for conductors connected in series.
4.1. Thermoelectric phases in parallel
In the case of three distributed conductors oriented in parallel to external temperature, Δ
Total electrical current
Considering definition of Seebeck coefficient as derivative of applied voltage with respect to temperature for non‐current flowing condition, Eq. (11), a simple manipulation of Eq. (10) gives Eq. (12), which describes effective Seebeck coefficient,
Using specific parameters (resistivity
where,
While considering, volume fractions,
From electrical analogue shown in Figure 1(c), effective electrical and thermal conductivities can also be easily derived, as expressed in Eqs. (16) and (17), respectively:
It is noteworthy that applying the same approach for higher
4.2. Thermoelectric phases in series
Equivalent description for the case of three distributed conductors oriented in series to external temperature and electrical potentials gradients is shown in Figure 2(a).
For this case, a similar analysis is presented, taking into account individual thermal gradients applied on each of the phases. Taking into account that the first, second and third phases are subjected to temperature differences of (
Under adiabatic heat conduction conditions, where no lateral heat losses are apparent, the heat flow,
where
Using expression (19), the numerator terms of Eq. (18) can be easily described in terms of expressions (20):
In the rightmost equation of expression (19),
Combining Eqs. (21) and (22) leads to Eq. (23):
Substitution of the expression of
Substitution of temperature differences derived in Eqs. (20) and (24) into Eq. (18) results in the expression of
Applying the same considerations described above, effective electrical and thermal conductivities can also be derived, as expressed in Eqs. (26) and (27), respectively:
Similarly to the previous case of parallel‐connected phases,
5. Practical examples and applications
Prior to describing the full potential of the GEM concept on optimizing performance of multi‐phase TE materials, two general examples highlighting the potential of the method for monitoring the microstructure and phase morphology are described.
While analysing measured electrical and thermal conductivities of Cu following different spark plasma sintering (SPS) conditions, resulting in porosity levels in the range of 0–30%, a good agreement to GEM equations, Eqs. (2) and (3), was observed while assuming homogeneous dispersion (
Specifically, for TE materials, it was recently shown that upon introduction of MoSe2 phase into layered
A similar approach was recently applied for investigation of the morphological effects on TE properties of Ti0.3Zr0.35Hf0.35Ni1+δSn alloys following phase separation into half‐Heulser Ti0.3Zr0.35Hf0.35NiSn and Heusler Ti0.3Zr0.35Hf0.35Ni2Sn phases [15]. In this research, it was found that although phases’ orientation was aligned in intermediate level (
Another very interesting implementation of GEM approach was recently applied to estimate effective room temperature Seebeck coefficient and electrical resistivity values of a randomly morphological oriented homogeneous mixture of (001) and (hk0) grains in anisotropic polycrystalline HMS TE samples [12]. Applying GEM analysis to homogeneous distribution of (001) and (hk0) oriented grains (
Implementation of GEM concept in three‐phase TE materials, based on Eqs. (15)–(17) and (25)–(27), was recently shown for quasi‐ternary GeTe‐PbTe‐SnTe system [37, 38]. Specifically, it was shown that phase separation of solution‐treated (ST) Pb0.25Sn0.25Ge0.5Te composition (phase B in Figure 3c) into Pb‐rich, Pb0.33Sn0.3Ge0.37Te (phase A), and Ge‐rich, Pb0.1Sn0.17Ge0.73Te (phase C) phases is apparent in the system. In this system, prolonged thermal treatments at each temperature resulted, at the first stages, in three phases, parent B phase and two decomposed A and C phases. This stage is terminated by full decomposition into A and C, where only these phases are apparent. Furthermore, a lamellar alignment of the phases was observed at the first 24 h of thermal treatment, while prolonged treatments were resulted in spheroidization, due to reduced surface area free energy at this configuration. It was also observed that
6. Concluding remarks
In this chapter, the potential of GEM approach to optimize electronic properties of multi‐phase thermoelectric materials in terms of compositional or morphological considerations is shown in details. This approach already proved itself in monitoring of the densification rate of powder metallurgy processed materials, as well as in the determination of compositional modifications in binary systems just by measuring one of the transport properties. It is just beginning to approach the true potential to optimize thermoelectric transport properties of multi‐phase materials, such as those containing embedded nano‐features for reduction of the lattice thermal conductivity, where electronic contribution of the involved phase is usually neglected. It was shown that method does not just explain unexpected electronic trends in such materials, but might be employed for prediction of synthesis routes for optimizing thermoelectric figure of merit based on different compositions or alignment morphologies.
Based on the pointed above examples, it is obvious that for TE power generators operating at low (<300°C), intermediate (300–500°C) and high (>500°C) temperature ranges, Bi2Te3, PbTe/GeTe and HMS/half‐Heusler‐based compositions might be employed. In such systems, identifying compositions enabling phase separation or precipitation into multi‐phases, according to specific phase diagram, has a potential to reduce lattice thermal conductivity. Yet, for maximizing TE potential, optimal geometrical alignment of the phases should be identified. Using the proposed approach, based on individual TE transport properties of the involved phases, optimal geometrical alignment direction might be identified, leading to enhanced TE performance, enabling a real contribution to the society by reducing our dependence on fossil fuels and by minimizing emission of greenhouse gases.
Acknowledgments
The work was supported by the Ministry of National Infrastructures, Energy and Water Resources grant (3/15), No. 215‐11‐050.
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