On the thermopower of ionic conductors and ionic capacitors

Thermoelectric materials are extensively studied for energy applications such as the conversion of low-grade waste heat into electrical power. As an important performance parameter, the thermopower (or Seebeck coefficient) $S$ describes the electric current generated by a temperature gradient, or the voltage difference arising between hot and cold boundaries Bell2008 .

Onsager’s reciprocal relations link the Seebeck coefficient $S=Q/qT$ to the Peltier heat of transport $Q$, which is the enthalpy carried by a charge $q$ moving in an electric field. Thermoelectric effects were first observed for metals, with $S$ much smaller than the natural unit $k_{B}/e=86\mathrm{\mu V/K}$. Stronger effects occur in semiconductors, where the thermopower ranges from 1 to 20 $k_{B}/e$ Bubnova2011 ; Russ2016 ; Gregory2018 ; Xiao2017 . These numbers are rationalized in terms of Mott’s formula, accounting for electronic band structure effects, doping, Anderson localization well below the Fermi surface, and an energy-dependent mobility $\mu(E)$ Cutler1969 ; Fritsche1971 ; Kang2017 .

Ionic conductors differ from electronic devices in two fundamental aspects: First, ions cannot be transferred to electrodes, and thus cannot directly generate thermoelectric currents. Second, in general there are several carrier species and, at least in the absence of redox reactions, the number of each of them is conserved. Accordingly, $S$ consists of the sum of ion-specific contributions, which are independent of concentrations as long as correlation effects are negligible Onsager1927 ; Demery2016 . In this paper we consider a binary electrolyte, where the Seebeck coefficient reads as

with weight factors $w_{\pm}$ and the heats of transport $Q_{\pm}$ of cations and anions. In aqueous solution, the solvation enthalpies of common salt ions are of the order $Q_{\pm}\sim k_{B}T$ Podszus1908 ; Takeyama1983 ; Agar1989 ; Salez2017 ; Sehnem2020 . The resulting Seebeck coefficent, $S\sim k_{B}/e$, was shown to drive colloidal thermophoresis Putnam2005 ; Wuerger2008 ; Vigolo2010 ; Eslahian2014 ; Lin2018 . In recent years, much higher values up to 300 $k_{B}/e$ were reported for polymer-based electrolytes Zhao2016 ; Wang2017 ; Li2019 , small mobile ions in gels or solid matrices Kim2016 ; Han2020 , or ionic liquids Zhao2019 .

There seems to be no general agreement regarding the weight factors $w_{\pm}$ for the carrier specific contributions to the Seebeck coefficient. If all previous works agree on the fact that they are proportional to concentration $w_{\pm}\propto n_{\pm}$, discrepancies arise with respect to the dependencies on ion valency and mobility; the latter appears when identifying $w_{\pm}$ with Hittorf transport numbers $t_{\pm}$ that account for the relative conductivity of each ion species. At present it is not clear which description is correct for the Seebeck coefficient of ionic conductors. This question is of practical interest: Because of the large mobility contrast of polymer electrolytes, different weight factors may even result in opposite signs of the Seebeck coefficient $S$.

In the present note we study the thermoelectric properties of ionic conductors as open or closed systems, where the former exchange charges with the environment and the latter don’t. Starting from two experimental situations with well-defined boundary conditions, we find that the corresponding Seebeck coefficients may significantly differ from each other. We discuss our results in view of recent experiments Zhao2016 ; Wang2017 ; Li2019 ; Kim2016 ; Han2020 ; Zhao2019 , and compare with what is known for electronic materials.

We consider an electrolyte solution of positive and negative charge carriers with concentrations $n_{\pm}$ and mobilities $\mu_{\pm}$. In simple monovalent electrolytes, overall charge neutrality imposes $n_{+}=n_{-}$. In complex systems, the concentrations of positive and negative carriers need not to be identical. Then the charge density reads as

where $\varrho_{f}$ are fixed charges. As examples we note polyelectrolyte complexes where the number difference of mobile ions $n_{+}\neq n_{-}$ is compensated by the charge $\varrho_{f}$ of the solid matrix De2017 ; Ostendorf2019 , or ionic liquids where the ions form immobile charged aggregates, thus leaving unlike numbers $n_{\pm}$ of mobile ions Agostino2018 . In the bulk one always has $\varrho=0$, yet there may be net surface charges at the sample boundaries.

Many non-equilibrium situations in physical chemistry are well described by Onsager’s linear relations between fluxes and forces deGroot1962 . In the present case, this means that the ion currents are linear in the thermodynamic and electric forces,

where the three contributions account for electrophoresis in an electric field $E$, thermodiffusion in a temperature gradient due to the heat of transport $Q_{\pm}$, and gradient diffusion with Einstein coefficient $D_{\pm}=k_{B}T\mu_{\pm}$.

As a consequence of the linear-response approximation, the non-equilibrium properties arise only from the generalized forces $E$, $\nabla T$, and $\nabla n_{\pm}$, whereas the coefficients are evaluated at thermal equilibrium, and thus do not depend on position. Thus throughout this paper, all bulk quantities such as $T$, $n_{\pm}$, $Q_{\pm}$,… are taken as constants. The only exception occurs in Section IV.3 where we evaluate surface effects, which are irrelevant for the thermoelectric coefficients but merely complete the physical picture. In the case of a 1D geometry, even the generalized forces are constants, and none of the bulk properties depend on position.

Thermoelectric effects are usually defined in terms of the heat flow driven by an electric field and the charge current due to a temperature gradient, with the Peltier and Seebeck linear response coefficients. In electronic materials, this is realized by connecting the hot and cold boundaries to electrodes. A similar situation occurs for ionic conductors which are coupled to reservoirs at different potential or temperature Nernst1889 , as illustrated in Fig. 1a.

The electric conductivity $\sigma$ is defined through Ohm’s law for the current, $I_{E}=\sigma E$. Collecting the contributions of positive and negative carriers in $I_{E}=e(J_{+}-J_{-})$, we have

as illustrated in Fig. 1c. The relative contributions of cations and anions are expressed through Hittorf transport numbers,

Besides the charge current $I_{E}$, the electric field induces a heat flow: Because of their electrostatic self-energy and the interaction potential with the surrounding material, the ions carry a heat of transport $Q_{\pm}$, resulting in the heat current

where the second identity defines the Peltier coefficient $\Pi$, which is readily expressed as

On the other hand, a temperature gradient $\nabla T$ gives rise to thermodiffusion of the mobile ions deGroot1962 : According to the second law, the excess enthalpy $Q_{\pm}$ flows towards the cold and drags the ions at velocities $-\mu_{\pm}Q_{\pm}\nabla T/T$, as illustrated in Fig. 1b. Thus the temperature gradient drives an electric current

where the last identity defines the Seebeck coefficient $S$. With the above expressions for $\sigma$ and the transport numbers $t_{\pm}$ one finds

As expected, the thermoelectric coefficients verify Onsager’s reciprocal relation $S=\Pi/T$ deGroot1962 .

In Fig. 2 we plot the Seebeck coefficient (9) as a function of the heat of transport ratio $Q_{+}/Q_{-}$, for different values of the mobility ratio $\mu_{+}/\mu_{-}$. Not surprisingly, a large cation mobility results in $S>0$, whereas highly mobile anions favor a negative Seebeck coefficient.

The above thermoelectric coefficients describe an ionic conductor sandwiched between two reservoirs at different temperature or potential. Most experiments on ionic systems, however, are done on closed systems where the charge carriers cannot enter or leave. Then the linear response for heat and electric currents is valid only for the transient behavior after switching on the fields, or if the applied electric and temperature fields oscillate in time. For oscillatory fields the validity is restricted to sufficiently high frequency, $\omega\tau\gg 1$, where $t\ll\tau=\lambda^{2}/D$ is the diffusion time of ions over one Debye screening length. Similarly, $S$ describes the behavior after switching on the temperature gradient, for times shorter than $\tau$, which is much shorter than the time scale for the emergence of the bulk ion gradients, $D/L^{2}$, over the system size $L$. The transient behavior requires to solve the continuity equation $\partial_{t}n_{\pm}+\mathrm{div}J_{\pm}=0$ Bonetti2015 ; Ly2018 ; Janssen2019 ; Sehnem2020 .

Now we turn to a closed system which exchanges heat with the surrounding but which does not transfer charges. After switching on the temperature gradient, the electric current $I_{T}$ accumulates charges at the hot and cold boundaries, which in turn give rise to an electric field. After a transient time, the system attains a steady state with constant surface charge and zero ion currents (Fig. 1d), which is characterized by the thermoelectric field

or by the corresponding thermopotential $-\hat{S}(T_{H}-T_{C})$ between hot and cold boundaries. Previous works on ionic thermoelectrics often assume, more or less explicitly, that the steady state is characterized by the Seebeck coefficient (9), that is, $\hat{S}=S$.

For the sake of simplicity we have so far considered monovalent binary electrolyte solutions. In order to account for more complex systems, containing mixtures of different salts or acids Putnam2005 ; Vigolo2010 ; Wuerger2008 ; Eslahian2014 ; Sehnem2020 , or multivalent ions Sehnem2020 , we give the general expression, with several cations and anions species $i$, of concentration $n_{i}$ and valency $z_{i}$.

The Seebeck coefficient of an open system is readily generalized in terms of the partial conductivity $\sigma_{i}=z_{i}^{2}e^{2}n_{i}\mu_{i}$, resulting in

and Hittorf transport numbers $t_{i}=\sigma_{i}/\sigma$.

Now we turn to the steady-state of a closed system and generalize the coefficient $\hat{S}$. Inserting the zero-current and zero-charge conditions,

in the charge current $I=e\sum_{i}z_{i}J_{i}$, eliminates the ion gradients. Solving for the electric field $E$ one readily obtains

This form generalizes the expressions used previously for the mixed electrolyte NaCl_xOH_1-x where $z_{i}=\pm 1$ Wuerger2008 ; Vigolo2010 ; Eslahian2014 . A more complex form occurs if one of the species is a macroion, for example a charged polymer or a colloidal particle, where the thermally driven velocity cannot be expressed by the product of Peltier heat $Q$ and mobility $\mu$, and where both the numerator and the denominator in (24) depend on the ratio of phoretic and diffusive mobilities Majee2012 .

As the main result of this paper, we found that the thermoelectric properties of ionic conductors depend crucially on the boundary conditions: For an open system, the Seebeck coefficient is defined through the thermally induced current $I_{T}$, where the heat of transport of positive and negative carriers is weighted with Hittorf’s transport numbers $t_{\pm}$. The resulting coefficient (9) depends on the mobilities, similar to what is known for electronic systems Fritsche1971 .

For a closed system, on the contrary, the coefficient $\hat{S}$ is defined through the macroscopic thermoelectric field, or the potential difference between the hot and cold boundaries,

The thermopower $\hat{S}$, given in Eq. (18), differs significantly from the Seebeck coefficient $S$ and in particular does not depend on the ion mobilities, as is obvious when comparing the weight factors $\hat{t}_{\pm}$ and $t_{\pm}$. Recent experiments on various ionic thermoelectric materials Zhao2016 ; Kim2016 ; Wang2017 ; Li2019 ; Zhao2019 ; Han2020 do not generate currents but a thermopotential, and thus are described by the coefficient $\hat{S}$.