**1. Introduction**

#### *1.1. Controversial Points of View*

Entropy is a central quantity in thermoelectrics, but seldom has it been addressed as such. The basic physical quantity that is known today as entropy is widely considered to be a derived quantity according to the approaches by Clausius [1–3] and Boltzmann [4–6] to quantify its value in certain situations. Both the perception of entropy as a derived quantity and the underestimation of its role in thermal processes are seen as residual outcomes of the Ostwald-Boltzmann battle, which is worth recalling and constitutes another chapter in the tragicomical history of thermodynamics [7]. In the frame of this work, entropy is considered to be a basic quantity. The benefits of this controversial point of view are made obvious on the example of thermoelectric materials.

#### *1.2. Implications of Natural Philosophy*

Clausius intended to borrow terms for important quantities from the ancient languages, so that they may be adopted unchanged in all modern languages. He proposed to call the quantity *S*, which had been introduced by him, the entropy of the body, from the Greek word τρoπη (tropy), transformation [1–3]. Intentionally, he formed the word entropy to be as similar as possible to the word energy. In his opinion, the two quantities to be denoted by these words are so nearly allied in their physical meanings that a certain similarity in designation is desirable [1–3].

The importance of entropy was underlined by Gibbs in the very first words of his treatise on thermodynamics: "The comprehension of the laws which govern any material system is greatly facilitated by considering the energy and entropy of the system in the various states of which it is capable" [8,9]. However, the "Energeticist" [10] school in Germany, which rejected atomism and other matter theories, postulated energy as the primary substance in nature, and considered entropy as a superfluous derived concept [11–13]. The protagonist was Ostwald, cofounder of physical chemistry and its Nestor in Germany, and behind it was the natural philosophy of Mach [6,14,15]. Soon, the "Energeticist" school attracted much critical attention not only by the British pioneers [16] but also from a younger generation of German physicists [11]. The young Sommerfeld witnessed a memorable debate at the 1895 Assembly of the German Society of Scientists and Physicians in Lübeck, in which Boltzmann "like a bull defeated the torero [Helm as substitute to Ostwald] despite all his art of fencing [14]." In a follow-up critique, Boltzmann [17,18] condemned Ostwald's "Energetics" not only for perceived mathematical and physical error, but also for its false promise of easy rewards [11]. However, Ostwald never admitted that he had been defeated, and the object of the dispute has been kept alive to the present day [19,20]. Even though the personalities have changed over time, the battle has been newly inflamed in the controversy regarding the Karlsruhe Physics Course [21], which resulted in removing the entropy-treating educational course from German schools [22].

Today, the dissipation or "degradation" of energy is often treated without clear reference to entropy [19,20]. Preference is given to thermal energy ("heat") or enthalpy. Textbooks on classical thermodynamics take the approach of Clausius to quantify entropy in equilibrium conditions as the definition of entropy, which then is perceived as an energy-derived quantity. The success of Boltzmann's principle (called so by Einstein [6]) to quantify entropy in partitioned systems in equilibrium [23] renders it often to be a statistics-derived quantity [24]. However, the special cases considered herein do show only certain aspects of entropy, which should be considered in a wider context. By not considering entropy as a central basic quantity, clearness is lost, and uncertainty even creeps over authors who endeavor for accuracy and clarity when it comes to the description of thermal phenomena.

#### *1.3. Evolution of Thermodynamics*

The field of thermodynamics has evolved from the aim of understanding the thermodynamical engine (i.e., the steam engine) [11], which by principle operates under non-equilibrium conditions. However, for several reasons, thermodynamics has been limited to equilibrium conditions for a long time. For its suggestion to use entropy under non-equilibrium conditions, Planck's PhD thesis [25] was heavily criticized [19,20]. Planck was likely then intimidated and did not deepen this approach to entropy [19,20]. Alternately, the elegance and success of Gibbs' treatise on using equilibrium conditions did pave the way for thermodynamics under equilibrium conditions.

It took several decades until Callen [26,27] and de Groot [28] independently formulated a theory to describe thermodynamic systems in non-equilibrium conditions. This theory was helpful for quantitatively describing thermoelectric phenomena. However, the primary focus was the entropy production in irreversible processes and, thus, the excess entropy. No attention was given to entropy itself and its ability, which in older terms could be mentioned as the motive power of entropy, to drive a steam engine [29–31] or thermoelectric generator [32–34].

#### *1.4. Modern Thermodynamics*

Consistent with Falk [35], Fuchs [32], and Strunk [23,31], the author holds the view that entropy should be considered as a fundamental quantity. The characteristics of a fundamental quantity unfold from its relations with other fundamental quantities. Concise theories have been developed by Fuchs [32], Job & Rüffler [36,37], and Strunk [23,31,38].

In context of the development of physical concepts, it is worth noting that the basic physical quantity that is known today as entropy, was named quantity of heat by Joseph Black (1728–1799) [39–41] and

calorique by Sadi Carnot (1796–1832) [29,30,40]. Indeed, calorique is the French word for quantity of heat. In his 1911 Presidential address to the Physical Society of London, Hugh Longbourne Callendar [29] outlined Carnot's calorique (i.e., entropy) as a quantity, that "any schoolboy could understand". Moreover, Callendar underlined that Carnot's calorique reappeared as a triple integral in Kelvin's 1852 paper, as the thermodynamic function of Rankine and as equivalence-value of a transformation in the 1854 paper of Clausius, and as entropy in the 1865 paper of Clausius [2] along with an abstract redefinition. No one at that time appears to have realized that entropy was merely calorique under another name. Callendar closed his remarks with the advice to distinguish a quantity of heat from a quantity of thermal energy.

Traditionally, thermal energy is called "heat". Concordant with Callendar [29] and Fuchs [32], in the author's opinion, heat is not energy, and entropy is the true measure of a quantity of heat as opposed to a quantity of thermal energy. Thus, the use this term for thermal energy should be avoided [42]. For clarity, the traditional term "heat" is put into quotation marks when it addresses the thermal energy. In this approach, entropy is a basic quantity. Thermoelectrics is an example par excellence to show the benefits of this philosophical perspective.

#### *1.5. Entropy in Thermoelectrics*

In the context of thermoelectrics, according to Boltzmann's principle, entropy is considered as a statistics-derived quantity when it is used to quantify the effect of spin and orbital degrees of freedom on the Seebeck coefficient in strongly correlated electron systems [43,44]. This, however, is a minor aspect. The approach by Clausius, to consider entropy as an energy-derived quantity does not play a significant role either.

In the so-called theory of thermodynamics of irreversible processes, as developed by Callen [26,27] and de Groot [28], it is rather the case that the thermal energy is derived from the entropy. Entropy is a fundamental quantity that is central to thermoelectrics. These texts can be read with grea<sup>t</sup> earning if entropy is considered as an indestructible substance-like quantity that is able to flow through the thermoelectric material and carries the thermal energy. The concept of energy carriers was developed by Falk et al. [45] and Herrmann [21].

However, the theory of thermodynamics of irreversible processes has the tendency to focus on the irreversibly produced excess entropy, but not on the entropy itself. Instead, energetic quantities are preferred. In §60 of his textbook, de Groot [28] presents an alternative presentation of thermoelectricity by the use of entropies of transfer, for which he has stated that the theory becomes somewhat more elegant compared to using energies of transfer. Unfortunately, he has not deepened this approach.

In a preceding paper [34], the author has shown that the rehabilitation of entropy into the theory by Callen [26,27] and de Groot [28] leads to a vivid description of thermoelectric devices. Like electrical charge carries the electrical energy, entropy carries the thermal energy. Thermal induction of an electrical current and electrical induction of a thermal current become understandable.

#### *1.6. Aim of This Work*

Like the preceding paper by the author [34], the present work aims to contribute to a better understanding of thermoelectrics by reconsidering it by treating entropy and electric charge as basic quantities of equal rank. This is semantically considered by naming the part of energy that flows together with entropy the thermal energy and part of energy flowing together with electrical charge the electrical energy. The energy flux through the thermoelectric material can thus be divided into thermal power and electrical power. Power conversion, which is in the focus of this article, implies that the system under consideration is not in equilibrium, but instead flown through by substance-like quantities. For the case of thermoelectric materials, these are entropy, electric charge, and energy.

By recalling the historical development of the perception of entropy, obstacles are identified, which have hindered the recognition of its important role in the field of thermoelectrics. The confused traditional approach and the use of model devices are avoided. Both power conversion and the

efficiency of power conversion are accessed quantitatively for a thermoelectric material apart from a device. New physical insight into thermoelectrics is gained on the level of the thermoelectric material rather than on the device level. On the material's voltage–electrical current curve, distinct working points are identified (see Table 1), which not only allow for quantification of the material's properties and performance under specific operational conditions, but also relate generator mode (thermal-to-electrical power conversion) and entropy pump mode (electrical-to-thermal power conversion) of the same material to each other.

**Table 1.** Working points on the voltage–electrical current curve of a thermoelectric material in both operational modes, as addressed in this work.


The results are worked out in detail, and the outcome from the formalism is graphically illustrated and explained. The simplicity of thermoelectrics is clarified. The findings are linked to the outcome of the traditional approach to thermoelectrics and state-of-the-art thermoelectric materials.
