Thermogalvanic cells convert waste heat directly to electric work. There is an abundance of waste heat in the world and thermogalvanic cells may be underused. We discuss theoretical tools that can help us understand and therefore improve on cell performance. One theory is able to describe all aspects of the energy conversion: nonequilibrium thermodynamics. We recommend to use the theory with operationally defined, independent variables, as others have done before. These describe well-defined experiments. Three invariance criteria serve as a basis for any description: of local electroneutrality, entropy production invariance, and *emf*’s independence of the frame of reference. Alternative formalisms, using different sets of variables, start with ionic or neutral components. We show that the heat flux is not the same in the two formalisms and derive a new relationship between the heat fluxes. The heat flux enters the definition of the Peltier coefficient and is essential for the understanding of the Peltier heat at the electrode interfaces and of the Seebeck coefficient of the cell. The Soret effect can occur independently of any Seebeck effect, but the Seebeck effect will be affected by the presence of a Soret effect. Common misunderstandings are pointed out. Peltier coefficients are needed for the interpretation and design of experiments.

## I. INTRODUCTION

The conversion of chemical to electrochemical energy in batteries, fuel cells, and electrolysis cells has always been a central subject in electrochemical science and technology. For recent overviews, see Refs. 1 and 2. The conversion of heat to electrochemical energy has been much less studied, in theory as well as in practice. Predominant converters of this type have been composed of semiconductors,^{3} and experimental results have often been reported without much theory (see, e.g., Refs. 4 and 5).

The conversion from thermal to electrochemical energy has, however, obtained increased attention over the last few years.^{6–10} Recent studies have included cells with ionic liquids,^{9} aqueous solutions,^{6,7} solid electrolytes,^{11} and non-aqueous electrolytes of molten salts.^{12–16} The use of gas electrodes in such cells^{8,14} is also interesting. The increased activity could be linked to current scarcity of electric energy or an abundance of low-grade waste heat.^{17} Few thermogalvanic cells are found in operation, however.

These facts provide a strong motivation to add to the development of this field. It becomes important to ask: How can this particular conversion be better described, understood, and used to design new experiments and cells that eventually can enhance the development? In this work, we shall examine theories available to describe energy conversion in thermogalvanic cells, with the purpose of being able to recommend in the end routes for further developments and tools suited for this development.

The relevant theory, nonequilibrium thermodynamics (NET), with a multitude of alternative expressions, may presently be difficult to access, and without a detailed theoretical description, experimental results cannot be well understood. A better description will provide guidelines for interpretation, but will also help design new experiments. The present perspective is written with this in mind; we want to provide the user with more accessible tools that can facilitate new practical research and development.

NET has its basis in the last century.^{18–24} But the detailed expressions from that time have barely been applied in this century. The Seebeck coefficient, which is the ratio of the thermoelectric potential and the temperature difference between two heat reservoirs, has, for instance, erroneously been taken as the temperature coefficient of the isothermal cell *emf*.^{25–27} A good theory may help the user to avoid errors and extract more information from the cell. But central electrochemistry textbooks^{1,28,29} are neither concerned with nonisothermal cells nor with the cell’s entropy production. We shall see here that there is an advantage to include the entropy balance among the governing equations. This offers new relations that are central for the conversion of thermal to electric energy.

Nonequilibrium thermodynamic theory dates back to Onsager, while descriptions of thermogalvanic cells originate from Agar.^{21} In Agar’s terminology, a nonisothermal electrochemical cell with electrolyte and electrodes is called a thermogalvanic cell. We shall adopt this name, as the simpler name “thermoelectric cell” is used for converters composed of semiconductors.^{30,31} Thermoelectric converters do not have electrodes.

The present work aims to make NET more accessible for thermogalvanic cells by pointing at variables that are essential to understand the energy conversion in these cells. The aim is to help users navigate in the jungle of possibilities and give advice on suitable tools. We shall use the theory of discrete systems^{19} as it now appears to deal with the layered structure of electrochemical cells.^{24,32}

In the application of NET to a thermogalvanic cell, we first have to select which variables to choose. The common set of variables contain all ions and the solvent molecule; see, e.g., de Groot and Mazur,^{20} Haase,^{22} and followers.^{33,34} Another set, pioneered by Katchalsky and Curran,^{19} Kedem and Leaf,^{35} and the school of Førland *et al.*,^{12,13,23} proposed to use electro-neutral component variables. To have two sets, that both describe the same electrolyte and the electrodes, is in many ways an advantage. It allows for consistency checks in the description. It can put the treatment of molten salt mixtures or ionic liquids on the same basis as aqueous solutions. Nevertheless, one may ask if there is a favored set under certain conditions. We shall see that the two sets differ in the way they describe the cell and even in the interpretation of the same phenomena. While this is interesting and useful; we will still argue throughout the paper that the set of neutral variables has an important advantage in its closeness to the measurements. Since this set has less attention per today, we shall recommend that this attention increases for the benefit of a better total understanding of this type of energy conversion.

We shall here choose a system with an aqueous electrolyte to demonstrate principles.^{21,22} The type of relations that we derive will apply to all types of cells, however.

We start with the two ways to choose thermodynamic variables and discuss how they can be used to describe the same cell performance. We shall see, for instance, that the expression for the measurable heat flux in the system is crucial for interpretation of Peltier heat effects. This flux is central in the conversion from thermal to electrochemical energy, and we find that it depends on the set of components that are used. This calls for care in the choice of description. In order to keep things brief, we shall *not* give complete derivations of equations but skip intermediate steps when they are documented in the literature, e.g., by Agar,^{20–22} or Katchalsky and Curran.^{19}

The outline of the paper is as follows: We present the system in Sec. II and discuss three conditions that need be obeyed in any cell description in Sec. III. These are the local electroneutrality condition, the requirement for entropy production invariance with respect to choice of variables, and the independence of the *emf* on the choice of frame of reference. We show in Sec. IV how the conditions are obeyed for two sets of variables, the ionic and the electroneutral component variables. Relationships between the variable sets can then be extracted. Transport coefficients are next defined and related to each other (see Sec. V). The detailed thermal contributions to the electric potential of a thermogalvanic cell follow. In particular, we address the origin of the thermoelectric potential in terms of measurable variables (see Sec. VI). This gives a background for several recommendations: for interpretation of literature in the field, for how to decompose data from experiments, and for proposals for new experiments. Some perspectives on thermoelectric energy conversion follow (see Sec. VII).

## II. THE SYSTEM AND ITS SUBSYSTEMS

We choose a system that is complicated enough to contain all issues of interest, but not unduly complicated. A relatively simple and well-studied thermogalvanic cell has an aqueous alkali chloride electrolyte^{7,21,22,36,37} and two identical electrodes reversible to chloride ion. Data are available for cells with different uniform salt concentrations, as well as for cells where a concentration gradient develops over time, due to an applied temperature gradient and/or an electric field. This has motivated our choice of example cell,

Here, *T*_{a} and *T*_{c} are the temperatures of the anode and cathode, respectively. Leads are connected to the potentiometer at *T*_{0}. The salt concentration in the electrolyte may vary across the cell, denoted by *c*(*x*). Subscripts a and c refer to the anode and cathode, respectively. On the anode surface on the left-hand side, the reaction produces electrons in the connecting lead (a), and chloride ions in the adjacent electrolyte (e),

The reverse reaction takes place on the cathode, on the right-hand side. The system is illustrated in Fig. 1. We divide the system into five subsystems following Kjelstrup and Bedeaux:^{24} two electronic leads, two electrode surfaces, and the bulk of the electrolyte. The surfaces are two-dimensional autonomous area elements containing the electrochemical reaction.

When a positive electric current density, *j*, is passing the cell from left to right, there is heat taken from the reservoir at the a, e junction, reversibly connected with the surface reaction and the fluxes into and out of the surface at this location. This determines the Peltier heat of the electrode, here indicated by a red (left) arrow.^{38} The Peltier heat has contributions from all the changes that take place at the surface. A quantification of these contributions is the target of this work. The Peltier heat is Π^{a,e} on the left-hand side and Π^{e,c} on the right-hand side (blue arrow). Superscript combination a, e (e, c) means the interface between a and e (e and c).

Under stationary state and reversible conditions, we have from the first law of thermodynamics that net heat supplied can be transformed into work done on the surroundings,

Here, Δ*φ* is the measured electric potential difference and *F* is Faraday’s constant. The Δ-symbol means a difference between the right-hand side value (c) and the left-hand side one (a).

At uniform composition, and identical electrodes, the Seebeck coefficient is defined by

In the present example, the Seebeck coefficient is positive and equal to 0.63 mV/K (at 298 K and 0.01M NaCl) according to Kjelstrup Ratkje *et al.* (see Ref. 36). When Δ*T* > 0, the cell does work (Δ*φ* > 0). The coefficient is thus a measure of the ability of the thermogalvanic cell to transform heat into work.

Work is done when heat is transferred from high to low temperature. Equation (2) must not be confused with the temperature coefficient of the isothermal cell, dΔ*φ*/d*T*.^{6,26,27,39} This temperature coefficient is derived from the plot of *emf* vs the (uniform) temperature at which the *emf* is measured. The derivative of the curve is equal to *T*Δ*S*/*F*, where Δ*S* is the entropy change of the cell. The cell entropy change is further related to both Peltier heats, cf. Eq. (72).

By changing the conditions of the measurement, by introducing Δ*μ*_{NaCl} ≠ 0, we change the value of the ratio Δ*φ*/Δ*T*. A gradient in chemical potential can arise in the thermal field to produce Δ*μ*_{NaCl} ≠ 0. A well-defined measurement situation arises when the thermal driving force is balanced by a chemical driving force, and there is no mass flux. We have then achieved the state of Soret equilibrium, and the corresponding Seebeck coefficient is defined by

The two expressions (2) and (3) represent different measurements and are thus distinguishable. The first neglects any Soret effect; the other contains Soret equilibrium.

The name *Peltier heat* dates back to the early literature in the field (see Refs. 18, 19, and 40). It is defined as the heat needed to keep the interface temperature constant when one faraday of positive charge is passing from the left to right in the external circuit.^{23} The electrode interface, say on the left-hand side, receives the heat Π^{a,e} from the surrounding reservoir in order to compensate for reversible heat changes due to the electrode reaction and transport into and out of the electrode chamber (see, e.g., Refs. 12, 13, 16, and 23).

It is common to describe the electrolyte by its ionic components.^{28,29} Single ion properties are not measurable, however,^{41,42} so Kedem and Leaf^{35} argued that a formulation in terms of variables of neutral components might also be useful, the main reason being that a closer handle can be obtained on measured results. In this work, we present expressions for both sets of components and discuss their advantages, disadvantages, and differences. Doing this, we hope to be useful to a broad range of users. With equivalent sets of descriptions (formalisms) available, we also have the possibility of consistency control of assumptions. A common set of conditions, which both formulations must obey, will first be presented.

## III. THREE CONDITIONS FOR ALL CELLS

Any description of a thermogalvanic cell must obey (1) local electroneutrality, (2) entropy production invariance, and (3) independence of the *emf* on the frame of reference. The meaning of these conditions will be formulated for the example cell in Fig. 1.

### A. The condition of electroneutrality

The standard assumption of electroneutrality means that the concentrations of cations and anions are the same; here,

The condition applies to any *control volume* in the bulk phases of the cell and to the electrode surface. For a discussion of the validity of this assumption, see Jackson.^{43} The control volume of the electrode surface may be polarized^{44} but is still electroneutral. It follows that any thermodynamic property of an electrolyte, *Y*_{i}, say the entropy, *S*_{NaCl}, the enthalpy *H*_{NaCl}, or the chemical potential, *μ*_{NaCl}, is the sum of the ionic contributions in the control volume,

Everywhere across the cell, the electric field is constant, independent of whether there is a charge separation or not.^{44}

### B. The condition of entropy production invariance

According to the second law of thermodynamics, the entropy production in any control volume (control area) is always positive, *σ* ≥ 0. The entropy production is absolute and is therefore also invariant to a change in the set of variables. The entropy balance of the control volume in the electrolyte is

Here, the change in the entropy density with time *t*, *∂s*/*∂t*, is equal to minus the net change in the entropy flux across the volume, −*∂J*_{s}/*∂x*, plus the entropy production, *σ*. The entropy flux, *J*_{s}, is similarly invariant to changes in variables. *We shall see that this will make the heat flux particular for the choice of variables.* This observation has an effect on the contributions to the Peltier heats in Eq. (1), which so far has gone unnoticed.

Entropy production invariance means furthermore that *σ* = *∑*_{j}*J*_{j}*X*_{j} is the same for any choice of frame of reference for the mass flux *J*_{j}. Here, *X*_{j} is the driving force conjugate to the flux *J*_{j}. Each flux is a linear combination of all driving forces, *J*_{j} = *∑*_{k}*L*_{jk}*X*_{k}. Using the invariance of *σ*, and these linear relations, relations can be obtained between sets of coefficients in the flux equations. Such relations have been used to great advantage for diffusion in multicomponent mixtures,^{45} but they are often missing in the electrochemical literature.^{2,4,46}

### C. *emf*–independence of choice of frame of reference

The electric current density does not depend on the frame of Ref. 24, but the mass fluxes will depend on the frame of reference. The value of the *emf* of any electrochemical cell, as measured with well-defined electrodes, cannot depend on the frame of reference chosen for the fluxes.^{47} This applies also to the thermoelectric power, as well as to the Seebeck coefficient. This condition has been much used in electrochemistry to develop relations between transport coefficients.^{22,37,47}

## IV. IONIC- AND ELECTRONEUTRAL FORMALISMS

The invariance conditions of Sec. III leads to relations between fluxes in the ionic- and the electroneutral formalisms. We review each formalism separately before we compare.

### A. Ions as components

In the formalism using ionic components, we follow Agar,^{21} de Groot and Mazur,^{20} and Haase.^{22} The variables of the electrolyte are $cNa+$ and $cCl\u2212$. The dissociation of NaCl in water is assumed to be complete, and there is negligible dissolution of AgCl in the electrolyte. The independent mass fluxes in the solvent water frame of reference are

The electric current density is not an independent flux in this scenario. The entropy production *σ* of the electrolyte in the nonisothermal cell is^{20–22}

The driving force conjugate to the entropy flux is −∇ ln *T*, where *T* is the absolute temperature. The other driving forces are the negative gradients of the electrochemical potential of each ion divided by the temperature, i.e., $\u2212\u2207\mu \u0303Na+/T$ and $\u2212\u2207\mu \u0303Cl\u2212/T$, respectively. The tilde symbol refers to the electrochemical potentials that were defined by Guggenheim,^{41,42}

Here, $\mu Na+$ is the chemical potential of the sodium ion, $\mu Cl\u2212$ is the chemical potential of the chloride ion, and *ψ* is the Maxwell potential, or the scalar potential in electrodynamics.

In terms of ionic properties, the entropy flux is given by^{20,22}

Superscript *C* is added to make clear that the heat flux $Jq\u2032C$ belongs to a description using ionic components. This flux is called the reduced heat flux by de Groot and Mazur.^{20}

The entropy production of a thermostatted electrode surface has two terms: the electric driving force and the reaction Gibbs energy.^{44} The reaction rate *r* is proportional to the electric current density, *r* = *j*/*F*. With ions as components, the entropy production of the anode surface is

Subscript a, e means that the difference is taken between the electrolyte and the electronic lead. The reaction Gibbs energy on the anode, Δ_{r}*G*^{s,a}, is the sum of chemical potentials of the products minus the chemical potentials of the reactants.^{24} On the anode side, we obtain

A similar equation can be written for the cathode side. We observe that the Nernst equation appears at reversible conditions (*σ*^{s,a} = 0). It takes the form *F*Δ_{a,e}*ψ* = −Δ_{r}*G*^{s,a}.

### B. Neutral components

When we describe the cell using neutral components, we follow Kedem and Leaf,^{35} Katchalsky and Curran,^{19} and Førland *et al.*^{23} The use of neutral components to describe electrochemical cells is more common with electrolytes of molten salts.^{12,13,16} Here, the independent fluxes that describe mass transport in the electrolyte are one salt flux and the electric current density,

The entropy production of the nonisothermal bulk electrolyte is^{19,23,35}

The entropy flux is again conjugate to the driving force −∇ ln *T*. The driving force, which is conjugate to the salt flux, *J*_{NaCl}, is given by minus the gradient in the chemical potential of the salt times the inverse temperature. Electric power, −*j*∇*φ*, is added to the system’s control volume via the electrodes that define the boundary conditions. Here, ∇*φ* is the gradient in the electric potential, as measured with Ag|AgCl-electrodes.^{23} Each product on the right-hand side is independent of the other terms.

In the neutral component formalism, the entropy flux in the electrolyte is^{19,44}

The reduced heat flux, $Jq\u2032N$, is measurable, as the other quantities in the expression are absolute. Superscript *N* has been added to denote that the flux applies to a description using neutral components. We examine the relationship to $Jq\u2032C$ in Sec. IV C.

By choosing neutral components only, the entropy production of anode surface becomes

The reaction Gibbs energy, Δ_{n}*G*^{s,a}, is now the difference in Gibbs energy of the *neutral* components taking part in the electrode reaction. For the anode, we have

A similar expression applies to the cathode. All terms on the right-hand side of this equation can be measured. This was not so for variables in Eq. (12) that contain the chemical potential of the chloride ion. For reversible conditions (*σ*^{s,a} = 0), we obtain the Nernst equation in the neutral component formalism, *F*Δ_{a,e}*φ* = −Δ_{n}*G*^{s,a}.

### C. Relations between sets of fluxes and forces

Both sets of variables presented above have been used to describe energy conversion in thermogalvanic cells. From entropy production invariance, we obtain links between the variables. The term containing the thermal driving force in the electrolyte is the same in both descriptions, cf. Eqs. (8) and (14). This term disappears when the temperature is constant. We therefore compare the expressions by comparing the two other terms on the right-hand side under these equations, cf. Refs. 23 and 44.

The electric current density is the difference between the ionic fluxes,

With the present electrodes, the salt flux can everywhere be identified by the flux of the cation,^{19,23}

With these identifications, it follows from entropy production invariance that

For the electrolyte, this means that

These relations, which apply for a system with chloride reversible electrodes, have been established since long.^{19,23} With cation-reversible electrodes, the difference in the electrochemical potential refers to the anion. Their connection to the measured electric potential (*emf*) in the external circuit is further discussed in Secs. V and VI. As was pointed out by the inventor,^{41,42} there is no operational definition of the chemical potential difference of a single ion, but the electrochemical potential difference has an operational definition and can be measured, as evident from Eq. (20).

Entropy production invariance means also that the entropy flux of the control volume in the electrolyte keeps its value (see Sec. III). This gives from Eqs. (10) and (15)

The difference between the heat fluxes in the ionic and the neutral component description is equal to the temperature times the entropy of the species that carry charge at the boundaries (the chloride-reversible electrodes). Neither heat flux depends on a frame of reference. The relation is new, and we shall see that it plays a central role in the understanding of local heat effects in the thermogalvanic cell, cf. Sec. V, because the Peltier heat depends on the heat flux. In all systems without charge transfer (with interdiffusion and thermal diffusion alone), the two heat fluxes are the same.

### D. The measurable heat flux and the entropy production

The aim of this work is to provide tools that enable the user to interpret and design experiments on thermogalvanic cells. As evident from the name, energy conversion in thermogalvanic cell involves heat transport. It is therefore of interest to replace the entropy flux by the proper heat flux in Eqs. (8) and (14).

For the ionic component formalism, we obtain

and in the neutral component formalism, we have

The gradients in the electrochemical potential in Eq. (24) and the gradient in chemical potential in Eq. (25) are here evaluated at constant temperature,^{20,24} meaning that

The first flux–force product in the two expressions for entropy production, Eqs. (24) and (25), are no longer the same due to Eq. (23).

So far, we have put an emphasis on the equivalence of the two descriptions: the ionic and the neutral component formalisms. Both descriptions are correct. But some differences can be noted: While the neutral component formalism is using variables that all can be measured, this is not so in the ionic formalism. The neutral component formalism may therefore have an advantage, cf. the Discussion section.

## V. FLUX–FORCE EQUATIONS FOR THE ELECTROLYTE

In order to be able to use data available in the literature, we provide coefficient relations that follow from the invariance criteria of Sec. III. The choice we make for the entropy production, Eq. (24) or (25), has an impact on the flux equations and therefore on the transport coefficients. We give the coefficients that derive from the ionic set of flux–force relations, the set that derive from the neutral component formulation and the links between them.

The flux equations in the ionic formalism follow from Eq. (24). They are

The electrochemical potential gradients and ∇*ψ* are related by Eq. (9), which provides the link to the *emf*.

According to the entropy production (25), the flux–force set for neutral components are

Both coefficient matrices are symmetric. The diagonal coefficients in Eq. (28) are directly related to the coefficients of Fourier, Fick, and Ohm.^{24} They are often tabulated. Coupling coefficients are discussed below. All coefficients are readily available by molecular dynamics simulations using the Green–Kubo formulas; see Refs. 45 and 48 and references therein.

### A. Coefficients for coupled transport of mass and charge

The coefficients in the bottom right corner of the flux equations, in both Eqs. (27) and (28), describe coupled transport of mass and charge. The entropy production invariance (cf. Sec. III B), the operational definition of the electrochemical potential [Eq. (20)], and the expression for the electric current in terms of ion fluxes [Eq. (18)] give the following relations between the coefficients (see also Refs. 19, 22, 24, and 29):

Four measurements characterize the system:^{19} of electric conductivity, salt diffusion, salt transference, or the *emf*. The electric conductivity is operationally defined by

where the measured potential difference Δ*φ* refers to the particular set of electrodes in use. The conditions ∇*μ*_{NaCl} = 0 and ∇*T* = 0 mean that *κ* is a property of the salt solution. The right-hand side equality relates the measurement to Onsager coefficient *L*_{φφ}.

The *emf*-method gives one route to the transference coefficient of NaCl.^{49} The Hittorf experiment gives another operational definition of this property,^{23}

Onsager symmetry provides a well-known possibility to check for consistency between the *emf*- and the Hittorf method. A relation between the transference coefficient and the transport number follows from Eq. (19),

The salt diffusion coefficient is measured in the absence of electric current. The left-hand side equality gives the operational definition, while the right-hand side equality relates Onsager coefficients to this definition,

The mobility model is often used for strong electrolytes.^{22,29} In this model, it is assumed that cations move independently of anions. As the coupling coefficients are proportional to the concentration squared, they are approximately zero in dilute solutions, *L*_{+−} = *L*_{−+} ≈ 0. The approximation gives the mobility model

The ionic conductivities are here related to ionic mobilities, *u*_{i}, i.e., the ionic velocity of *i* in a unit electric field. The electric conductivity of the solution becomes

The transport number of the cation, Na^{+}, defined as the fraction of the electric current carried by this ion, obtains the simple form

A similar expression is found for the anion. The corresponding diffusion coefficient for an ideal solution is^{29}

The coefficients of the neutral-component description can be written in terms of the ionic mobilities as

The diffusional mobility is equal to the electric mobility in this scenario. These equations were, for instance, given by Newman.^{29} The model of Gering,^{50} typically used in lithium battery models,^{51} contains many of these expressions. Haase^{22} gave the more general expressions valid for *L*_{+−} ≠ 0. All transport coefficients in the matrices can be can be simulated using the fluctuation dissipation theorem.^{45,48}

### B. Coefficients for coupled transport of heat and mass

We return to the matrix of coefficients of Eq. (28). When ∇*φ* = 0, the upper left corner describes a short-circuited system. We are instead interested in the case *j* = 0, which is the condition under which we define the thermal conductivity, diffusion coefficient, thermal diffusion coefficient, or heat of transfer. Under uniform composition and *j* = 0, we measure the thermal conductivity *λ* by recording the transfer of measurable heat across the system due to a temperature gradient

The coefficient *ℓ*_{qq} is found from the general relation

which is obtained by eliminating the electric driving force in Eq. (28). The diffusion coefficient has the same operational definition as before [see Eq. (33)].

The heat of transfer has the operational definition^{19}

There is access to the coupling coefficient *ℓ*_{qμ} also via the Soret coefficient^{19}

The coefficient *D*_{T,NaCl} is the thermal diffusion coefficient, defined by

Thermal diffusion has often been neglected in electrochemical texts.^{29,51} In their treatment of transport in colloidal solutions, Würger^{52} gave a model for the thermal diffusion coefficient in terms of enthalpy differences; see also Haase.^{22}

The thermal conductivity is the same in the ionic as well as in the neutral component formalism because $Jq\u2032N$ = $Jq\u2032C$ when *j* = 0. Heats of transfer have been defined for ions using Eq. (27). For the cation, we have

A similar definition is written for the anion. The definition requires a system with non-zero electric current, unlike the Definition 41. Entropy production invariance leads to the relation

where $qNaCl*$ is defined by Eq. (41). Furthermore,

### C. Coefficients for coupled transport of heat, mass, and charge

Seebeck and Peltier coefficients can be defined for each of the five layers of the cell. Consider first the contribution to the Seebeck coefficient from the homogeneous electrolyte labeled e,

It follows from the Onsager symmetry, *L*_{φq} = *L*_{qφ}, that

where *π*^{N,e} is the Peltier coefficient of the electrolyte, and *N* refers to the use of neutral components. Equation (49) applies to the electrolyte (the e-phase). Similar equations apply to the other phases, but also to the overall system.^{44} They can be used to find contributions to the overall thermoelectric potential.

The Peltier *coefficient* is defined in the neutral component formalism as the ratio of the reduced heat flux in this formalism, $Jq\u2032N$, and the electric current density, *j*/*F*, at isothermal and uniform cell conditions. For the electrolyte, this gives

Unlike the heat of transfer of Eq. (41), this coefficient is a direct consequence of the electric current. A similar expression applies to the a-phase on the left-hand side and to the c-phase and on the right-hand side.

The Peltier *heat* of Eq. (1) is a boundary effect, composed of two Peltier *coefficients*.^{44} The entropy balance for reversible conditions at the interface a,e gives in the neutral component formalism^{44}

The same expression can be written with the measurable heat flux in the ionic formalism. The heat flux has a discontinuity at the interfaces. This discontinuity is differently described in the two formalisms, cf. Eq. (23). But the net amount of heat exchanged with the surroundings must be the same. Further comments are given in Sec. VI.

## VI. *EMF*-CONTRIBUTIONS FROM THE ELECTRODES AND THE ELECTROLYTE

The total cell *emf* can be found by adding subsystem contributions. We proceed to formulate the cell *emf* in the two formalisms. Premises of the two sets were presented in Sec. III, and the sets were related in Sec. IV.

The electrode regions are thermostatted, and there is no salt concentration gradient in the outset of the case study (*T*^{a}, *T*^{c} and *c*^{a} = *c*^{b} are constant). The boundary conditions apply to the initial experimental state when Δ*μ*_{NaCl,T} = 0. We assume that the variation in the entropy with temperature is negligible in the interval Δ*T* = *T*^{c} − *T*^{a}.

### A. The neutral component formalism

The total thermoelectric potential consists of contributions from two electronic leads, the two electrode surfaces, and the electrolyte. For the two electronic leads, the net contribution is^{44}

where $Se\u2212*$ is the transported entropy of the electron in the electronic leads.

For the electrode surfaces and the electrolyte, we use the Peltier heats of the anode and cathode interfaces. The Peltier coefficient of the electrolyte is given by heat flux in the electrolyte. In terms of the entropy flux and the component flux of Eq. (15), we first obtain^{14}

The entropy connected to charge transport follows from Agar,^{53}

This equation defines the entropy transported by the ions. Transport numbers were defined in Sec. V A. The Peltier coefficients of the e- and the a-phases at the electrode surfaces become

The Peltier heat is the difference of the Peltier coefficients

The expression for Π^{e,c} has opposite sign and temperature *T*^{c} = *T*^{a} + Δ*T*. The overall *emf* becomes^{23,24}

The Seebeck coefficient *η*_{S,0} at uniform composition, cf. Eq. (2), is accordingly

The Seebeck coefficient varies with *S*_{NaCl}.^{7,22,36} The entropy is equal to the standard entropy $SNaCl0$ minus 2*R* ln *a*_{NaCl}. The standard entropy, 115.5 J/mol K, refers to a 1M solution at 298 K.^{36} Activity data are given by Ref. 54. A proportionality with ln *a*_{NaCl} has been verified for the electrolyte concentration range of 0.03–0.6M for NaCl.

A chemical potential gradient will develop in the thermal field after some time, eventually reaching the Soret equilibrium state. The expression for Δ*φ* then obtains a contribution from the chemical potential gradient.^{23} This contribution, coming from the second equation in Eq. (28), is proportional to Δ*T* at *J*_{NaCl} = 0. It will bring the heat of transfer into the expression for the Seebeck coefficient. We compute from the two last equations in Eq. (28),

The measured difference in the Seebeck coefficients of Eqs. (2) and (3) can therefore be used to find the heat of transfer (or the Soret coefficient). The difference between the two measurements provides us with the heat of transfer, $qNaCl*$, when the transport number is known. The heat of transfer can also be found from the Soret coefficient [Eq. (41)] in the absence of *j*.

The time-evolution of the thermogalvanic potential from one state to the other, which reflects the impact of thermal diffusion (Soret effect) on the thermogalvanic potential, can be used to find the diffusion coefficient. The potential evolves in time as a simple exponential function. After some time *t* > *θ*/3, we have^{9,12,53,55,56}

The diffusion coefficient, *D*_{NaCl}, can be found from this equation by fitting experimental data to the theoretical curve using the characteristic time *θ* = *h*^{2}/(*D*_{NaCl}*π*^{2}), where *h* is the cell length.

Neither the expression for the thermoelectric potential nor the Seebeck coefficient can depend on the frame of reference for the mass flux—here, on the definition of $tNa+$. Using the cation frame of reference at stationary state, see Ref. 23 for details, we obtain

This in combination with Eq. (59) leads to the relation

The transported entropy of chloride can be computed from Eq. (61) once the transported entropy of electrons (a small term) and the thermodynamic entropies are known. For a discussion of the transported entropy of electrons and ions and the nature of these properties, see Kjelstrup.^{57}

The entropy that accompanies mass diffusion (the heat of transfer divided by temperature) has in the literature been divided into Eastman (hat-) entropies^{23,40}

where the transported entropy of the ion is written as the sum of the ionic entropy and the Eastman entropy,

The identification is consistent with Eq. (47). The Eastman entropy has frequently been neglected in the sums on the right-hand side of Eq. (64). This assumption is only good if the heat of transfer is negligible in Eq. (63).

The Seebeck coefficient at the stationary state gives direct information on the transported entropy of the chloride ion. Knowing this and the heat of transfer, we can find the transported entropy for the sodium ion from Eq. (62). Numerous experiments have been done in the past to find the transported entropies in this manner.^{21,22}

### B. The ion component formalism

In the ion component formalism, the measured cell potential Δ*φ* does not appear explicitly but is identified by the electrochemical potential difference of the charge carrier at the electrode surfaces, cf. Eq. (9). The Soret and Peltier coefficients in the ionic formalism can be obtained from the flux-force relations (27) (see, e.g., Ref. 58). We adopt for simplicity the mobility model with *L*_{+−} = 0 and obtain for the sodium ion

The similar procedure for chloride ion gives

The Peltier coefficient is therefore

The Peltier coefficient of the electrolyte becomes

The expression can be shown to be true also if *L*_{+−} ≠ 0.^{22} This relation has been taken as evidence for charge separation and used to explain the Seebeck coefficient measurement.^{59–64} This expression is not sufficient to understand the *emf*, however, as electrode contributions are lacking. The Soret effect exists also in electroneutral systems without electrodes, while the Peltier effect does not, as is clear from the neutral component formalism.

The relationship between the Peltier coefficient of the neutral component description, and the description in terms of ions, as obtained from Eq. (23), can be used to find equivalent expressions for the Peltier heat,

The entropy of chloride ion, $SCl\u2212$, originates from the electrode reaction at this surface; see the Introduction. This term does not appear in the neutral formulation. The difference between $Jq\u2032N$ and $Jq\u2032C$ follows. It is the combination of the contributions from the bulk electrolyte and the electrode surfaces that both formalisms give the same expression for the electrode Peltier heat of the thermogalvanic cell. Equations (55) apply to the neutral formalism. By adding and subtracting the entropy of the chloride ion in Eq. (59), we obtain the equivalent equations of the ionic formalism.

These expressions are furthermore consistent with the following relation for the total cell *emf*:

or

The Maxwell potential difference across the whole cell is Δ*ψ*. This is not measurable, unlike what has been stated, e.g., by Garrido.^{65} The difference between the measured Δ*φ* and the difference in the Maxwell potential, Δ*ψ*, is equal to the difference in the electron chemical potential between the terminals of the potentiometer. Only for isothermal conditions and constant composition, $\Delta \mu e\u2212=0$ and Δ*φ* = Δ*ψ*.

## VII. DISCUSSION AND PERSPECTIVES

### A. Two equivalent, but different descriptions

We have presented two equivalent ways to describe the thermoelectric potential of thermogalvanic cells, called the ionic and the neutral component formalisms, respectively. Both formalisms obey three conditions: the conditions of electroneutrality, entropy production invariance, and reference-free *emf*-results. In this sense, they are equivalent.

To have two equivalent ways to describe the same cell is useful. When the descriptions agree, the end result can be expected to be correct. Interesting is that it also gives two interpretations. In the ionic formalism, the electric potential difference is viewed as composed of electrostatic potential differences across the cell (differences in Maxwell potentials). In the neutral component formalism, the focus is on electric or chemical work, created by movement of heat down the temperature gradient, directly connected with the conversion process.

The Maxwell potential difference is related to the *emf* [Eqs. (20) and (21)] but cannot be measured. In his original work, Guggenheim^{41} stated that the splitting of the electrochemical potential $\mu \u0303i$ into a chemical part, *μ*_{i}, and an electrical part, *z*_{i}*Fψ*, cf. Eqs. (9) and (70), is without physical significance. The electrostatic Maxwell potential, *ψ*, is not defined in terms of physical realities but is rather a mathematical construct that serves as the potential for the hypothetical fluid electricity that is introduced in electrodynamics. Since an electric charge does not flow independent of the particles that carry it, a difference in the quantity *ψ* between two points does neither constitute a single driving force for charge transfer nor is it the quantity that is measured by a voltmeter. There is no doubt that the net driving force is given by the gradient in the electrochemical potential of the charge carrier relevant to the electrode, cf. Eq. (20). The conceptual difference between *φ* and *ψ* have, however, implications for the understanding of their origin and of the origin of their local contributions. Lack of a separation of the two has led to misconceptions in the literature (see Subsection VII B).

In this work, we have pointed out differences between the two formalisms. The local heat fluxes are not the same in the two formalisms. The heat flux of the ionic description is composed of terms that cannot be measured. The entropy of the chloride ion enters in this formalism; compare Eqs. (54) and (69). This quantity is absent in the entropy flux in the neutral component formalism. The Peltier coefficients differ accordingly, but the Peltier *heats* remain the same in both formalisms, as they must do. Measurements cannot be used to distinguish between the two descriptions.

We have furthermore seen that the ionic formalism does not separate between electroneutral processes and charge transferring processes. The Seebeck effect may appear as a type of Soret effect in the ionic description.^{58,66} We know, however, that thermal diffusion can take place without transfer of electric charge at electrodes, cf. Eq. (28). A description using neutral components will enable us to make a clear-cut separation of the two phenomena. This can be useful when we want to understand the origin of energy conversion.

The ionic formalism dominates the field of electrochemistry.^{21,22} But the neutral component formalism^{35} is also in use, mostly with strong electrolytes, ionic liquids, or molten salts.^{12,13,16,23,32,67–69} In this situation, several questions arise. Why has one formulation played a lesser role? Given certain premises, which description should be preferred? Measurable quantities are central for progress in science. Such quantities are required for theory validation and check of approximations. We advise below that more emphasis be put on the neutral component description in future works on thermogalvanic cells.

### B. Misinterpretations

The existence of two descriptions can be helpful in the discussion of experimental results. However, the existence of two formulations may also have added confusion about thermoelectric phenomena in the past. Some examples of this follow.

Recent literature has incorrectly identified Δ*S* of a symmetrical cell with the Seebeck coefficient of the cell.^{6,10,26,27,39,70} The temperature variation of the *emf* of an *isothermal* cell is from basic thermodynamics equal to the entropy change of the cell reaction, dΔ*φ*/d*T* = Δ*S*/*F*.^{23} But the Seebeck coefficient of Eq. (2) is not the temperature variation of the *emf* of an *isothermal* cell.

The *difference* in Peltier heats of the two electrodes can, however, be related to the cell entropy change at *T*. By subtracting the Peltier heat of Eq. (56) from the corresponding expression for the cathode,^{23} we obtain

There are several wrong statements in the literature related to this expression. The authors have concluded that the Peltier heats in Eq. (72) are small if Δ*S* is small. A small Δ*S* can, however, have two large contributions with opposite signs.^{71} Another statement is that the electrodes do not contribute to *η*_{S} of Eq. (2). This depends very much on the type of electrodes.

The measured Seebeck coefficient has also been associated with the Maxwell potential difference of the cell.^{4,67} Doing this, one neglects the chemical potential difference of the electrons in Eq. (70).

The frame of reference of fluxes has also caused confusion.^{72,73} Chikina *et al.*^{58} used the barycentric frame of reference for their mass fluxes. The driving force conjugate to the mass flux depends on the choice, but this has been overlooked. To shift the frame of reference means to shift the flux as well as the driving force, while the entropy production is the same.

The Seebeck coefficients can be measured with high precision with a high impedance potentiometer. But the system may need a long time to achieve Soret equilibrium, so as to obtain *η*_{S,S}. Thermal diffusion is slower than ordinary diffusion, and it may take days to come to a stationary state. Such a state could therefore have been missed, e.g., by Wang *et al.*^{5} and Buckingham and Aldous.^{25} The balance of forces in Soret equilibrium was not recognized by Sehnem and Janssen^{60} and Bonetti *et al.*^{67} Sehnem and Janssen^{60} reported time-variations in huge thermogalvanic potentials, cf. Eq. (60). They related the measured potential to Δ*ψ*. and not to diffusion. A thermal field might well lead to polarization, for instance, of pure water,^{74} but it does not necessarily produce electrons in an outer circuit. Electrode contributions have not always been recognized.^{4}

In the neutral component formalism, the Soret coefficient concerns diffusion of an electroneutral component, not charge separation. In this formalism, the Soret effect takes place in the absence of a net electric current, so thermal diffusion is electroneutral. The definition Eq. (42) relates the heat of transfer, and the Soret coefficient, in the neutral component formalism. A division into ionic components is possible, however [see Eqs. (62)–(64)].

### C. Large Seebeck coefficients: Systematic studies

Lack of available theory may hamper experiments and applications. From theory, we conclude using Onsager relations that huge Seebeck effects can be expected for materials with large Peltier heats. Cells with gas electrodes are then of interest, as the electrode reaction will involve conversion of a gas, which is known to have a large entropy.^{13,14,16} Large entropy changes at electrodes could also be promoted by the use of complexing agents. Relevant results from aqueous electrolytes and mercury reversible electrodes were provided by Ikeshoji and Ratkje.^{75} A steep increase in the Seebeck coefficient of 0.35 mV/K could be related to the onset of complex formation of Hg^{2+} with EDTA. Bonetti *et al.*^{67} reported huge Seebeck coefficients, 7 mV/K, for tetrabutylammonium nitrate, 0.1M in 1-dodecanol, and electrodes made using platinum wires. The results were related to structure-making effects of tetra-alkylammonium ion, clearly an entropic effect. Electrode reactions may involve water oxidation or reduction, but this was not discussed. Ionic liquids or hydrogels may also be expected to produce locally large entropy changes. Indeed, Horike *et al.*^{61} observed Seebeck coefficients varying up to 10 mV/K with ionic hydrogels as electrolytes and silver electrodes (which are not reversible in the system used). Large values and variations with complexing agents were also observed by Wang *et al.*^{5}

More studies, such as these, would be interesting for work aimed to enhance thermoelectric energy conversion. Few systematic studies exist on how the Seebeck coefficient varies with the type of materials, materials entropy, in particular. There are few studies on how to optimize the Seebeck coefficient.

## VIII. CONCLUSION AND PERSPECTIVES

Progress in the field of thermoelectric energy conversion may help us make use of the abundance of low-grade heat in the world. Thermogalvanic cells have scarcely been described in the literature, even in classical books on nonequilibrium thermodynamics;^{19,20} see, however, Haase.^{22} Agar’s article^{21} provided an early milestone. We have shown that a larger effort in this field may enhance progress.

Using past and recent studies^{32,44,69} as a stepping stone, we have shown how thermogalvanic cells can be described in terms of ionic or electroneutral component variable sets. The two formalisms have been connected by three invariance criteria. The relations that followed may facilitate consistency checks of the theory. Taking a simple example for the sake of illustration, we have used the invariance criteria to help clarify several misconceptions in contemporary literature, e.g., incorrect interpretations of the Seebeck coefficient have been given in terms of temperature coefficient of one half cell potential or of the isothermal cell. Progress in the field can be hampered when a wrong theory is used. We have introduced several tools in terms of transforms between variables in order to avoid this situation.

Explicit formulas for thermoelectric effects may help guide new experimental designs. The formulas for thermogalvanic potentials suggest that we need to look for systems with large entropy changes at the electrodes.^{44} We have pointed out the lack of systematic studies on reactions with complex formation, gas electrodes: variables that promote large Seebeck coefficients. To measure at stationary state conditions gives information on the Soret equilibrium state and can be used to find heats of transfer. The heat of transfer is often neglected but is non-negligible in ionic systems. Systematic investigations of heats of transfer in relevant materials seem crucial. It is essential to understand all contributions to the heat flux. In this study, we have shown that the heat flux in the ionic formalism differs from that in the neutral component formalism.

Among the descriptions examined, we propose that the one recommended by Kedem and Leaf,^{35} which uses a set of neutral components, be used more extensively. With this description, one can systematically add new independent variables, always with the possibility of experimental control of approximations. The approach belongs to a general description that also applies, say to electroacoustics,^{76} sedimentation and electrophoresis,^{77} and avoids the use of the Maxwell potential gradient as a driving force.^{41,42}

A particular example cell was used to illustrate the analysis in the present work. The arguments presented are of a general nature, however. One may envision that several types of reference cells can be established, i.e., for different types of electrolytes. This may help establish more systematic analyses. In the neutral component formalism, the theoretical expressions are set up following the procedure outlined above and in Refs. 24 and 78. The entropy production determines the flux equations. For example, Eq. (28) follows directly from Eq. (14). Several of the comments presented here are also not limited to thermogalvanic cells, but also apply to all electrochemical cells. The discussion on choice of variable sets, for instance, also has a bearing on those.^{60}

## ACKNOWLEDGMENTS

The authors are grateful for the financial support from the Norwegian Research Council, Center of Excellence Funding Scheme for Project No. 262644 Porelab. Øivind Wilhelmsen is thanked for a critical review of the manuscript in its last stages.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Signe Kjelstrup**: Conceptualization (lead); Data curation (equal); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Methodology (lead); Validation (lead); Writing – original draft (lead); Writing – review & editing (equal). **Kim R. Kristiansen**: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal). **Astrid F. Gunnarshaug**: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). **Dick Bedeaux**: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal).

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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