*2.1. Phonon Thermal Conductivity*

An important quantity for the description of phonon scatter processes is the materialdependent Debye temperature *θ*D. Several publications address the identification of this variable. Different values are identified in dependency of the used methodology as discussed in [9]. The average value of *θ*<sup>D</sup> = 418 K has developed as the state of the art [8,10,11] and is used in this study. The thermal resistance of the phonon-phonon scatter *W*p−<sup>p</sup> is defined by the following:

$$\mathcal{W}\_{\text{P}^{-}\text{P}}(\theta) = \frac{A \cdot \theta}{\theta\_{\text{D}}}.\tag{5}$$

The constant *A* is calculated by Julian's modification of the Liebfried–Schlömann equation [8,10,12]. The authors in [10] conclude that the alloy components have a minor influence on the thermal resistance of the phonon–phonon scatter. The values of *A* = 0.412 m K/W and *θ*<sup>D</sup> = 418 K for pure *α*-iron can be used for the calculation. The authors in [11] suggest a correction term for *ϑ* > *θ*<sup>D</sup> to consider the thermal expansion. The temperature range in this study is limited to values of 498 K. The correction term is neglected because the maximum influence is 0.2% in the considered temperature range. The phonon–phonon scatter *W*p−<sup>p</sup> can be simplified using the introduced simplifications as follows:

$$\mathcal{W}\_{\text{P}^{-}\text{P}}(\theta) = 9.86 \times 10^{-5} \,\text{m}/\text{W} \cdot \theta \tag{6}$$

The formulation of the thermal resistance of the phonon–electron scatter *W*p−<sup>e</sup> shows a temperature dependency. A formulation of *W*p−<sup>e</sup> at a temperature that is equal to the Debye temperature *θ*<sup>D</sup> is given with the following:

$$W\_{\text{p-e}} = 2.69 \times 10^{-2} \,\text{mK/W} \tag{7}$$

Additional therms need to be considered for temperatures below the Debye temperature *θ*<sup>D</sup> [10,11]. In this study, only small differences from the Debye temperature are considered, and the additional therms are neglected. A similar simplification is used in [8] for alloys. The thermal resistance of the phonon–impurity scatter *W*p−<sup>i</sup> describes the interaction between impurities such as the alloys and the Fe-lattice. The process can be modeled as follows [8]:

$$\mathcal{W}\_{\text{P}^{-\text{i}}} = B \cdot \sum\_{i} c\_{\text{a,i}} \cdot \Gamma\_{\text{i} \prime} \tag{8}$$

with an experimental constant *B*, the impurity parameter Γ<sup>i</sup> and the atomic content *c*a,i of each alloy *<sup>i</sup>*. The value of *<sup>B</sup>* is given in [8] with *<sup>B</sup>* = 1.36 × <sup>10</sup>−<sup>2</sup> m K/W. The impurity factor Γ can be calculated based on weighted ratios of the molar masses and the molar volumes between the alloy contents *i* and the Fe-lattice. The values for nine different alloys are given in [8]. The value for silicon, aluminum, and manganese is equal to 0.59, 0.05, and zero, respectively. This means that the influence of aluminum is almost negligible, and the influence of manganese is not present. The phonon thermal conductivity *k*<sup>p</sup> can be calculated as the inverse of a sum of the three influences *W*p−p(*ϑ*), *W*p−e, and *W*p−i:

$$k\_{\mathbb{P}}(\theta) = \left(9.86 \times 10^{-5}/\text{K} \cdot \theta + 2.69 \times 10^{-2} + 1.36 \times 10^{-6} \cdot \sum\_{i} \text{c}\_{\text{a,i}} \cdot \Gamma\_{i}\right)^{-1} \text{W}/(\text{m}\,\text{K}). \tag{9}$$
