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. 2025 Jan 10;17(3):4984–4995. doi: 10.1021/acsami.4c19795

Contributions of Both the Eastman Entropy of Transfer and Electric Double Layer to the Electromotive Force of Ionic Thermoelectric Supercapacitors

Shohei Horike †,‡,§,*, Vikas Nandal ∥, Qingshuo Wei ⊥,*, Kazuhiko Seki ∥,*
PMCID: PMC11917421  PMID: 39789981

Abstract

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Recently, ionic thermoelectric supercapacitors have gained attention because of their high open circuit voltages, even for ions that are redox inactive. As a source of open circuit voltage (electromotive force), an asymmetry in electric double layers developed by the adsorption of ions at the electrode surfaces kept at different temperatures has previously been proposed. As another source, the Eastman entropy of transfer, which is related to the Soret coefficient, has been considered. Herein, we theoretically estimated the open circuit voltages generated in the Stern layer, the diffuse layer and by the Eastman entropy of transfer. The Grahame equation has been generalized to consider the temperature gradient in the diffuse layer. The ion coverage difference between the hot and cold electrodes and the open circuit voltage are obtained by solving self-consistent equations using the adsorption isotherm. The results are compared with experimental results using a metal electrode and a conductive polymer-based electrode. We show the possible origin of the high ionic Seebeck effect caused by the asymmetry in the coverages of adsorbed ions in terms of the various types of interface capacitance factor at the hot and cold electrodes.

Keywords: thermo-electrochemical cell, supercapacitors, Seebeck effect, electric double layer, Stern layer, Grahame equation

1. Introduction

Recently, the electromotive force (EMF) of ionic thermoelectric supercapacitors has been increased using the ionic Seebeck effect, where ions are inactive regarding redox reactions.1−8 Unlike thermogalvanic cells, where the open circuit voltage is essentially governed by the Gibbs free energy difference associated with redox reactions, the origin of thermoelectrics using redox-inactive ions has not been fully elucidated. In thermogalvanic cells, configurational entropy at different temperatures has been reported to enhance the Gibbs free energy difference and thus the open circuit voltage.9−26 In ionic thermoelectric supercapacitors, the EMF is obtained under a temperature difference in an open circuit condition, even though the ions are redox inactive. The EMF is linearly related to the temperature difference and the open circuit voltage per temperature difference as high as 10 mV/K has been achieved using redox-inactive ions.1−3,6,7 Using the same electrolyte, the sign and the magnitude of the EMF can be altered by changing the electrodes, which indicates the importance of interactions between the electrolyte and the electrodes, even though the ions are redox inactive.2−4,6,7,27

As a source of EMF, an asymmetry in electric double layers developed by the adsorption of ions at the electrode surfaces kept at different temperatures has been proposed,2,3 but theoretical estimations have not been performed while fully accounting for the temperature gradient. As another source of EMF, the effect of the Eastman entropy of transfer, which is related to the Soret coefficient, has been proposed.23 The analytical expression of ionic thermopower expressed in terms of the Eastman entropy of transfer is known under charge neutrality condition.23,28,29 However, the charge neutrality is not satisfied inside electric double layer.

Herein, we adopt a transport equation inside electric double layer under inhomogeneous temperatures to calculate the open circuit voltage of ionic thermoelectrics.30−32 We also consider a phenomenological parameter expressing the Eastman entropy of transfer.6,33−35 Our model consists of an electrolyte between parallelly placed electrodes with different temperatures. The magnitude of electric double layers depends on the boundary condition representing the interaction between ions and electrodes set in the transport equation. In this way, the EMF generated in the bulk of the electrolyte by the Eastman entropy of transfer and that generated by the electric double layer can simultaneously be considered. We obtain an analytical expression of the EMF in terms of ion concentrations and electrode temperatures.

On one-hand, the EMF expressed in terms of ion concentrations can be rewritten in terms of the coverages of adsorbed ions on the hot and cold electrode surfaces using the adsorption isotherm. On the other hand, the EMF can be expressed using the Grahame equation generalized to take into account the temperature gradient in the diffuse layer; the generalized Grahame equation is expressed in terms of the surface charge density which depends on the coverage of adsorbed ions. These two expressions of the EMF are equated to evaluate numerically the coverage of adsorbed ions. We also derive the EMF originating from the Stern layer using the difference in the coverage. These theoretical results are used to interpret the experimental results obtained using the Ag and PEDOT:PSS electrodes.

2. Materials and Experimental Methods

2.1. Chemicals

1-Ethyl-3-methylimidazolium chloride (EMIM+Cl–) was purchased from Tokyo Chemical industry Co., Ltd. Poly(3,4-ethylenedioxythiophene)/poly(4-styrenesulfonate) (PEDOT:PSS) (PH1000, Clevios) was purchased from H.C. Starck. Ethylene glycol (EG) (>99.5%) and 1,5,7-triazabicyclo[4.4.0]dec-5-ene (TBD) were purchased from TCI Chemicals. Silver foil (50 μm, 99%) was purchased from Nilaco Co., Japan.

2.2. Electrode Preparation

A solution of PEDOT:PSS (3 mL) with 5 wt % EG was drop-cast onto a slide glass (25 mm × 75 mm) and then heated on a hot plate at 70 °C for 2 h followed by 150 °C for 30 min. Once all the solvent had evaporated, the resulting film exhibited a thickness of approximately 6–8 μm, with a typical conductivity of around 800 S cm–1. For the silver electrode device, we first cut the silver foil to a size of 25 mm × 75 mm. The surface was then polished with sandpaper to remove any silver oxide layer that might be present.

2.3. Characterization

The thermovoltage was evaluated using the custom setup detailed in prior research.12 Two vertical Peltier cooler/heaters (Ampere UTC-200A) were used to control the temperature of the system. The ionic thermopower measurement cell consists of a polyether ether ketone (PEEK) container with a thickness of 1 cm and a hole with a diameter of 1.5 cm. A small hole in the wall of the container allowed the injection of the electrolyte. The PEEK container was sandwiched by the two electrodes (e.g., PEDOT on glass substrates), which were sealed with silicone grease.

After injecting the electrolyte to fill the container, the system was completely sealed with silicone grease and the cell was placed between the two Peltier elements. The top of the cell was kept at a constant temperature of 25 °C, while the bottom temperature was varied from 25 to 35 °C. This configuration achieved a temperature difference of about 8 °C. The temperature at the electrodes was measured using type K thermocouples. The temperature data on two electrodes were collected, while the voltage difference between the two electrodes was measured by Voltage/Temp unit (HIOKI U8550).

3. Experimental Results

Previous studies on electromotive force measurement have typically been conducted under open conditions and sometimes involved the use of polymers and additives. However, recent research has demonstrated that complex effects can influence the voltage generation in ionic thermoelectric supercapacitors.36 Therefore, in order to streamline the experimental setup, we opted to conduct our measurements under closed conditions, as illustrated in Figure S1 of Supporting Information Part S1. In this setup, a liquid electrolyte was introduced into a cell formed by two electrodes. The device was then placed between two Peltier devices to establish a temperature difference across the electrodes. The temperature difference and output voltage were subsequently measured using attached thermocouples.

We selected 1-ethyl-3-methylimidazolium chloride (EMIM+Cl–) as the electrolyte in water, maintaining a fixed concentration of 10 g/L. For electrodes, we opted for two distinct materials based on insights gained from our previous investigations.2,3 One is silver (Ag) electrode and the other is the conducting polymer electrode poly(3,4-ethylenedioxythiophene)/poly(4-styrenesulfonate) (PEDOT:PSS). While the potential reaction between Cl– and Ag is a concern, the system demonstrates the highest power output, as shown in our previous work.8 This makes it a valuable example to include and discuss its behavior within the scope of this study. According to prior studies, the temperature dependence of the Ag/AgCl electrode potential is only 0.22 mV/K.37 Thus, even if AgCl forms on the Ag paste electrodes, its contribution to the observed Seebeck coefficient should be minimal or negligible. To address this concern and expand the discussion, we have also investigated PEDOT:PSS electrodes in this work. Compared to metal electrodes, PEDOT:PSS electrodes offer more tunable parameters, such as carrier density, enabling better control over the system’s properties.

As shown in Figure 1A,B, the output voltages show a noticeable increase corresponding to the applied temperature difference. Consequently, it becomes possible to extract the ionic Seebeck coefficients from the slopes observed in the ΔV–ΔT plots, as shown in Figure 1C,D. Specifically, for the silver electrode, the calculated ionic Seebeck coefficient is +0.94 mV/K, while the PEDOT:PSS electrode yields a relatively smaller value of 0.57 mV/K.

Figure 1.

Figure 1

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(A) The temporary change of the potential difference measured by Ag electrodes and (B) PEDOT:PSS electrodes in response to the supplied temperature difference. (C) Plot of the electromotive force as a function of temperature difference for Ag electrodes and (D) PEDOT:PSS electrodes. In (A) and (B), the line color corresponds to the axis color.

It is noteworthy that these results show a significant disparity compared to measurements made under open conditions.2,3 In the previous setup, the silver electrode showed an ionic Seebeck coefficient of about 10 mV/K, while the PEDOT:PSS electrodes showed a negative ionic Seebeck coefficient. In an open system, both the electrolyte and the electrode are exposed to air, which can introduce various external effects. In contrast, a closed system is fully sealed, allowing only interactions at the electrode/electrolyte interface. Although further investigation of these contributing parameters is ongoing, it is beyond the scope of our current study, which is primarily focused on theoretical analysis. Therefore, in this discussion, we focus only on the ionic Seebeck coefficient determined under closed conditions.

4. Results of Continuity Equation

We consider a half-cell with the hot electrode facing toward the electrolyte and another half-cell with the cold electrode facing toward the electrolyte. Using the results of the half-cells, we calculate the potential difference between the hot and cold electrodes with an electrolyte between them.

As shown in Figure 2, we introduce one-dimensional coordinates to express the positions of mobile ions in the electrolyte. The x axis is normal to the electrode, and the value of the x-coordinate increases with the increasing distance from the electrode. The mobile cation and anion concentrations are denoted by n+(x) and n–(x), respectively. The location of the mobile ions in the electrolyte nearest to the electrode is x = 0, and the concentration of one of the mobile ion species at x = 0 is fixed as ni by assuming fast equilibration between the surface adsorbed ions and the mobile ions at x = 0 (Figure 2). Accordingly, if the cations are adsorbed to the electrodes, the boundary condition for the mobile cations equilibrated with the adsorbed cations is given by n+(0) = ni, where n+(x) is the concentration of mobile cations at x. A reflecting boundary condition is imposed at x = 0 for the other ions (anions if the cations are adsorbed to the electrodes). We first consider the half-cell where the temperature of the electrolyte at x = L is denoted by T(L).

Figure 2.

Figure 2

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Schematic of interface between the electrode and electrolyte. We account for the adsorption layer and diffusion region. ni indicates the mobile ion concentration at the closest distance from the electrode in the diffusion region. ni is a constant and can be regarded as the boundary condition used for obtaining a profile of mobile ion concentration. θ indicates the coverage of adsorbed ions on the electrode surface. ni and θ are related through adsorption–desorption equilibrium. If θ decreases by increasing the temperature, ni increases accordingly. The one-dimensional coordinate is introduced only in the diffusion region. Mobile cations and anions are present between x = 0 and x = L. The temperatures at x = 0 and x = L are denoted by T0 and T(L), respectively. One of the mobile ion species is assumed to be adsorbed at the electrode, and the other ions are reflected at x = 0.

We introduce the diffusion constants of the cation and anion, denoted by D+ and D–, respectively. Using the Einstein relation, the diffusion constant is related to the mobility of the respective species as D±(x) = μ±kBT(x), where kB is the Boltzmann constant, and T(x) is the temperature at x. Herein, + and – indicate the properties of cations and anions, respectively. The valency of the ions is denoted by z±, and the elementary charge is denoted by q. For example, cations and anions with the valency of 1 are denoted by z+ = 1 and z– = −1, respectively.

In free space without boundaries, the temperature gradient affects the mass current density, where the proportionality constant is termed the Eastman entropy of transfer. We consider the case where the Eastman entropy of transfer Inline graphic is expressed as Inline graphic.33,34 Considering a point ionic particle in a heterogeneously distributed temperature gradient, α± = 1 has been derived using the Fokker–Planck equation.30 The Eastman entropy of transfer is proportional to kB using the classical expression of the Boltzmann equation.38 The probability current density (j±) can be expressed as6,26,34,35,39−41

4. 1

where E indicates the electric field. We introduce an electrostatic potential denoted by ψ(x), which satisfies

4. 2

By considering the open circuit condition given by j± = 0, we can make the following approximation

4. 3

when the density gradient can be ignored. Inline graphic can be interpreted as the ionic Seebeck coefficient expressed by the differential form. However, if the potential gradient (negative of the electric field) is absent, j± = 0 leads to

4. 4

Inline graphic is regarded as the Soret coefficient in the differential form. Below, we account for the effects of both the potential gradient and density gradient in the electric double layer.

We consider the open circuit condition and set j± = 0

4. 5

Since both n± and T depend on x, we divide eq 5 by n±kBT and obtain

4. 6

By integrating from 0 to L, we find

4. 7

When the Debye length is smaller than L, eq 7 simplifies to (see Supporting Information Part S2)

4. 8

As shown in Figure 3, we consider that the electrolyte is sandwiched between the hot electrode with the temperature Th and the cold electrode with the temperature Tc. The electrostatic potential at the hot electrode is denoted by ψh and that at the cold electrode is denoted by ψc. We study the potential difference when the concentration of one of the mobile ion species at the hot electrode is a constant denoted by nh and the concentration of the same mobile species at the cold electrode is a constant denoted by nc. First, we consider the case where this mobile ion species is a cation, and the measurement procedure can be described as follows. In equilibrium, both electrodes are cold. Then, the temperature of one of the electrodes is raised to Th, and the surface fraction of adsorbed cations in the adsorption layer is changed from that at Tc. Accordingly, the mobile cation concentration at the hot electrode is changed from nc to nh. If the surface fraction of adsorbed cations in the adsorption layer is decreased by raising the temperature and changing the adsorption–desorption equilibrium, nh tends to increase from nc. In the subsequent section, nh/nc in the steady state is discussed based on the Langmuir isotherm and its extension.

Figure 3.

Figure 3

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Schematic of the electrolyte sandwiched between the hot electrode with the temperature Th and the cold electrode with the temperature Tc. The cold electrode is grounded. Coverage of the hot and cold electrodes by cations in the adsorption layer are denoted by θh and θc, respectively. We study the potential difference when the concentration of one of the mobile cations at the hot electrode and the cold electrode are constants denoted by nh and nc, respectively. In equilibrium, both electrodes are cold. Then, the temperature of one of the electrodes is raised to Th, and the surface fraction of adsorbed cations in the adsorption layer is decreased from that at Tc. Accordingly, the mobile cation concentration at the hot electrode is increased from nc to nh. Δn/nc, where Δn = nh – nc, is obtained in Section 6.

Considering the interface between the hot electrode and the electrolyte, T(0), ψ(0), and ni can be expressed as Th, ψh, and nh. For this arrangement, we rewrite eq 8 as

4. 9

Similarly, we can write an equation for the electrolyte at the cold electrode surface

4. 10

We can eliminate ψ(L) and n+(L) from eqs 9 and 10

4. 11

When the temperature of the cold electrode is kept constant and the temperature of the hot electrode is varied, it is convenient to express eq 11 as

4. 12

We define the voltage difference as

4. 13

As shown in Figure 3, the cold electrode is grounded and we set ψc = 0. Using eq 12

4. 14

where α+ is related to the Eastman entropy of transfer Inline graphic by Inline graphic.

On an analogy with semiconductors, the ionic Seebeck coefficient defined by −ΔV/(Th – Tc) is positive (ΔV < 0) for p-type and negative (ΔV > 0) for n-type polarities; the definition of the ionic Seebeck coefficient is corroborate by the last term on the right-hand side of eq 14, i.e., ΔV ≈ −[kBTh/(z+q)]α+ ln(Th/Tc), where kB/q = 8.6 × 10–5 V/K; α+ > 0 corresponds to the positive ionic Seebeck coefficient, and the p-type polarity here means that cations are assumed to be adsorbed to the electrodes, whereas anions are assumed to be reflected by the electrodes. Using eq 1, we note that α+ is positive when cations tend to migrate from the hot side to the cold side, and α+ is negative when cations tend to migrate from the cold side to the hot side. Equation 14 indicates that ΔV increases toward negative values if α+ becomes more positive. ΔV increases toward positive values if α+ becomes more negative. ΔV expressed by α+ alone in eq 14 originates from the polarity in the electric double layer. When cations are adsorbed to the electrodes, ΔV is expressed by α+ alone as shown in eq 14. When anions are adsorbed to the electrodes, ΔV is expressed by α– alone. The appearance of α+ or α– in the expression of ΔV by the polarity is in sharp contrast to the cases of semiconductors, where the choice of α+ or α– in the expression of ΔV is naturally selected by the dark carriers of semiconductors (α+ for p-type semiconductors and α– for n-type semiconductors).

When nh/nc > 1, we have, ln(nh/nc) > 0 and ln(Th/Tc) > 0. Both conditions correspond to p-type behaviors with the positive ionic Seebeck coefficient. Therefore, if the mobile cation concentration at the hot electrode is larger than that at the cold electrode and α+ > 0, the effect of the concentration gradient induces an EMF corresponding to p-type polarity, and the effect of the positive Eastman entropy of transfer also induces an EMF corresponding to p-type polarity. Conversely, when nc/nh > 1, we have, ln(nh/nc) < 0 while ln(Th/Tc) > 0. According to eq 14, ln(nh/nc) < 0 corresponds to n-type behavior with the negative ionic Seebeck coefficient if ln(Th/Tc) is ignored. Therefore, if the mobile cation concentration at the hot electrode is smaller than that at the cold electrode and α+ > 0, the effect of the concentration gradient induces an EMF corresponding to n-type polarity, whereas the effect of the Eastman entropy of transfer induces an EMF corresponding to p-type polarity.

Overall, for nh/nc > 1, both the concentration gradient and the positive Eastman entropy of transfer increase the thermoelectric voltage toward negative values and promote the p-type character. For nc/nh > 1, the concentration gradient and the positive Eastman entropy of transfer are competitive. The results are summarized in Table 1.

Table 1. Cation Adsorption Responding to ΔT.

  α+ > 0 Inline graphic α+ < 0 Inline graphic
nh > nc p-type [−ΔV/(Th – Tc) > 0] competition
nh < nc competition n-type [−ΔV/(Th – Tc) < 0]
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Equation 14 is derived by assuming that the mobile cation concentration responds to the temperature change of an electrode, where mobile anions are reflected by electrodes. The concentration of mobile anions nearest to the hot electrode and that nearest to the cold electrode can be denoted by n–(h) and n–(c), respectively (n–(h) and n–(c) are the short notations for n–(0) when the electrode is hot and cold, respectively). These are not given, but n–(h)/n–(c) can be derived once nh/nc and Th/Tc are given, as shown in Supporting Information Part S3.

In the opposite case where the mobile anion concentration responds to the temperature change of an electrode, where cations are reflected by electrodes, the EMF can be expressed using anion concentrations at the electrodes and the Eastman entropy of transfer of anions. Here, we only show the summary in Table 2. The derivation is shown in Supporting Information Part S4.

Table 2. Anion Adsorption Responding to ΔT.

  α– > 0 Inline graphic α– < 0 Inline graphic
n–h > n–c n-type [−ΔV/(Th – Tc) < 0] competition
n–h < n–c competition p-type [−ΔV/(Th – Tc) > 0]
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Here, in eq 14, ΔV is expressed by the Eastman entropy of transfer for the ion species responding directly to the temperature change of an electrode through the change in adsorption equilibrium. We termed p-type polarity when cations respond to the temperature change and n-type polarity when anions respond to the temperature change. The first term on the right-hand side of eq 14 can be interpreted as the Nernst equation by assuming local equilibrium in the vicinity of each electrode.26,40 Here, we consider the change in adsorption equilibrium by introducing the temperature change of an electrode. In the Nernst equation of eq 14, the temperature is given by that of hot electrode because the cold electrode is grounded. If the cold electrode is not grounded, the Nernst equation leads to eq 11, where α+ = 0. The Nernst equation has been considered for proton conductors with negatively charged fixed backbones,24,26,40,42 where protons migrate from the hot electrode to the cold electrode, resulting in a higher proton density at the cold side.24,26,40,42 In these previous works on proton conductors, the last term in eq 14 has not been considered.

In addition to the Nernst equation, ΔV is expressed by the Eastman entropy of transfer for the ion species obeying adsorption equilibrium. Previously, the ionic Seebeck coefficient has been proposed for binary electrolytes as Inline graphic or Inline graphic by assuming charge neutrality (details of derivation are given in Supporting Information Part S5).11,23,28,29,35,43−45 As shown in Supporting Information Part S6, Inline graphic is more appropriate for ionic thermoelectric supercapacitors in a steady state, where ions are inactive regarding redox reactions. The charge neutrality is violated inside electric double layer. We show that the open circuit voltage can be expressed using the concentration and the Eastman entropy of transfer of adsorbed species when charge neutrality is violated. Notice that charge neutrality is violated in the vicinity of electrodes by the adsorption equilibrium responding to temperature difference.

Direct comparison of our results to the ionic Seebeck coefficient obtained under the charge neutrality condition is possible, if we assume Δn/nc ≪ 1 and ΔT/Tc ≪ 1, where Δn = nh – nc and ΔT = Th – Tc. By applying the Taylor expansion to eq 14, we obtain

4. 15

where Inline graphic is introduced. If the first term on the right-hand side of eq 15 can be ignored, the ionic Seebeck coefficient defined by −ΔV/ΔT can be approximately given by Inline graphic. For z+ = 1, we find Inline graphic, which can be larger in magnitude than Inline graphic obtained by assuming the charge neutrality condition, if Inline graphic and the ionic Seebeck coefficient is positive. This is one of the important results of the paper. Below, we study in details the influence of the first term on the right-hand side of eq 15 on the ionic Seebeck coefficient.

The EMF generated in the electrolyte has been considered either by using the Eastman entropy of transfer obtained under the charge neutrality condition or by the difference in the electric double layer of cold and hot sides without taking into account the temperature gradient in the electric double layer.2−4,6,7,23,27 However, the EMF owing to the electric double layer under the temperature gradient should naturally include the Eastman entropy of transfer in the double layer, where the charge neutrality condition is broken. Equation 15 expresses the EMF originating from the electric double layer, where the temperature gradient is taken into account. Equation 15 is expressed using the Eastman entropy of transfer for the ion species responding directly to the temperature change of an electrode instead of the Eastman entropy of transfer obtained under the charge neutrality condition.

The evaluation of eq 15 requires the value of Δn = nh – nc. Δn can be expressed in terms of the ion coverages on the hot and cold electrode surfaces using the adsorption isotherm. Therefore, ΔV of eq 15 can be expressed in terms of the ion coverages. On the other hand, ΔV can be expressed by the Grahame equation generalized to take into account the temperature gradient in the diffuse layer; ΔV is expressed in terms of the surface charge density which depends on the ion coverage. The asymmetry in the ion coverages on the hot electrode and the cold electrode surfaces can be determined by equating these two expressions of ΔV; the resulting equation is the self-consistent equation that determines the asymmetry in the ion coverages. The construction of the self-consistent equation and the EMF obtained from the numerically exact solution of the self-consistent equation together with an analytical approximate expression of the EMF are presented in the subsequent sections. We also calculate the EMF originating directly from the adsorbed ions by using the Stern model; the difference between the coverage of adsorbed ions on the hot and cold electrode surfaces is required for calculation of the EMF using the Stern model.

5. Influence of Adsorption Isotherm

We have shown that the sign of the open circuit voltage may depend on the relative magnitudes of nh (or n–h) and nc (or n–c) when cations (anions) are adsorbed and anions (cations) are reflected at the electrode. Here, we study the relative magnitudes of nh (or n–h) and nc (or n–c) using the Langmuir or Frumkin adsorption model (see Supporting Information Part S7 for details).

We consider that the coverage of adsorbed ions in the adsorption layer denoted by θ is equilibrated with ni at temperature Ti of the electrode, where ni is the concentration of mobile cations at the shortest distance from an electrode (see Figure 2). ni represents either nh (or n–h) or nc (or n–c). We also introduce θh (θ–h) and θc (θ–c) to represent the coverage of the hot and cold electrodes by cations (anions) in the adsorption layer, respectively.

According to the Langmuir or Frumkin adsorption model (see Supporting Information Part S7 for details), θ can be expressed as

5. 16

with the adsorption equilibrium constant given by

5. 17

where ΔEeff(θ) indicates effective adsorption energy expressed as

5. 18

If ΔEads is negative, ions are easily adsorbed; we have ΔEads(θ) < 0 if the adsorbed state is more stable compared with the desorbed state of the ion. Interactions between adsorbed ions are taken into account effectively by h0 using the Frumkin isotherm. If the interaction between adsorbed ions is repulsive, the Frumkin interaction factor is positive (h0 > 0). If the interaction between adsorbed ions is attractive, the Frumkin interaction factor is negative (h0 < 0). In Supporting Information Part S6, we show that θ-dependence originates from interactions between adsorbed ions taken into account effectively by the Frumkin interaction factor. For simplicity, we ignore the θ-dependence for drawing figures.

It is convenient to rewrite eq 16 as

5. 19

By identifying ni as nh and nc, Ti as Th and Tc, we obtain

5. 20

By substituting eq 20, eq 14 can be expressed as

5. 21

We also present the case where the adsorption of anions, instead of cations, responds to a temperature change in Supporting Information Part S6. If we assume Δθ/θc ≪ 1 and ΔT/Tc ≪ 1, where Δθ = θh – θc, eq 21 simplifies to

5. 22

where we ignored the Frumkin interaction factor [h0 = 0 in eq 18]. If the first term on the right-hand side of eq 22 can be ignored, the ionic Seebeck coefficient defined by −ΔV/ΔT can be approximately given by Inline graphic. The ionic Seebeck coefficient is increased from Inline graphic by the amount of −ΔEads/(z+qTc), where we have ΔEads < 0 when cations tend to be adsorbed to the electrodes. This is another important result of the paper. In the subsequent section, we study the influence of the first term on the right-hand side of eq 22 on the ionic Seebeck coefficient.

In reality, θh/θc should be influenced by the value of ΔEads. However, the determination of θh/θc requires precise knowledge of the change in electrostatics associated with the change in adsorption–desorption equilibrium with the varying temperature of an electrode. So far, θh and θc are regarded as free parameters and we restricted ourselves to the general trends of thermoelectric capacitors, rather than elucidating quantitative details. In principle, θh and θc are related to the electric charges on the electrodes and can be determined by solving an electrostatics model. The theoretical model may be specific to a particular combination of electrode and electrolyte. Below, we introduce an electrostatic model of the PEDOT:PSS electrode.

6. Electrostatic Theoretical Results

Here, we focus on changes in the adsorption–desorption equilibrium of cations with the varying temperature of PEDOT:PSS electrodes. We assume that cations can be adsorbed to PEDOT:PSS electrodes by considering that PEDOT:PSS is negatively charged because of SO–3-groups on PSS.46,47 It has been experimentally demonstrated that −ΔV/ΔT increases by increasing PSS contents in the electrode.4 Anions are reflected by the PEDOT:PSS electrodes, which is consistent with the results of recent molecular dynamics simulations.47 The coverage of adsorbed cations in the adsorption layer is denoted by θ, and the temperature of the electrode is denoted by Ti. The electrostatic potential applied to mobile ions at the closest distance from the electrode is denoted by Ψi. When the temperature gradient in the diffuse layer cannot be ignored, the Grahame equation can be generalized for α = α+ = α– as (see Supporting Information Part S8)

6. 23

where Inline graphic for x ≥ 0,48Tm = (Th + Tc)/2 indicates the average temperature, nb is the bulk concentration of mobile cations, ϵr is the relative dielectric constant of electrolyte, and σ(θ) is the surface charge density for a combined medium composed of an electrode and an adsorption layer with the coverage of adsorbed ions given by θ. The generalized Grahame equation is one of the important results of the paper. We assume σ(θ) = z–qρs(1 – θ), where ρs indicates the surface density of the negative charge on the electrode. If the electrode is completely covered with adsorbed cations, the electric field from the electrode is completely diminished. By considering that the combined medium composed of an electrode and an adsorption layer is negatively charged (σ(θ) < 0), because z– is negative, z+qΨi/(kBTi) is negative; accordingly z+qΨi/(kBTi) given by eq S47 in Supporting Information Part S8 is negative.

When electrolyte is present between the cold and hot electrodes, we obtain the following from eq 23

6. 24

where Tm = (Th + Tc)/2. By applying the same approximation to derive eq 14 from eq 11, we obtain

6. 25

As shown in Supporting Information Part S9, θh/θc can be determined by equating eqs 21 and 25 using σ(θ) = z–qρs(1 – θ) for a given θc. We express Δθ analytically up to the linear order of ΔT and expand ΔV up to the linear order of ΔT. (See Supporting Information Part S10.) Under the condition of

6. 26

and h0 = 0 (h0 is the Frumkin interaction factor in eq 18), we obtain (see Supporting Information Part S10 for the derivation)

6. 27

where we have Inline graphic. The ionic Seebeck coefficient (−ΔV/ΔT) depends on ΔEads as indicated in eq 22. The factor 2θc/(1 + 2θc) arises by taking into account the first term on the right-hand side of eq 22. Equation 27 is the main result of the paper. By considering ΔEads < 0, the ionic Seebeck coefficient is increased from Inline graphic owing to the last term in eq 27. By considering that 2θc/(1 + 2θc) is a monotonically increasing function of θc (the coverage of adsorbed cations on the cold electrode), the larger ionic Seebeck coefficient can be expected from eq 27 by increasing θc. However, Inline graphic on the left-hand side of eq 26 decreases by increasing θc and eq 26 provides an upper bound for θc; the linear relation given by eq 27 holds under the condition of eq 26.

Equation 27 is derived by assuming a simple screening function of electrode charge density by adsorbed cations expressed by σ(θ) = z–qρs(1 – θ). The resultant ionic Seebeck coefficient given by eq 27 is independent of surface charge density of electrodes; the electrode charge density influences the ionic Seebeck coefficient indirectly through the change in the coverage θc. Equation 27 indicates that the ionic Seebeck coefficient is governed by both the Eastman entropy of transfer for the ion species obeying adsorption equilibrium and the adsorption energy, if the surface charge density of electrodes is sufficiently high (eq 26). The ionic Seebeck coefficient is not expressed directly in terms of ρs in eq 27; eq 27 might hold independently from the details of the electrode charge density and may hold for metal electrodes, where negative image charges of cations projected inside the electrodes are also screened by adsorbed cations.

In Figure 4, we analyze the data shown by Figure 1D, where the electrolyte is 10 g/L of EMIM+Cl– in water and the electrodes are PEDOT:PSS; the measured ionic Seebeck coefficient was 0.57 mV/K. As described in Supporting Information Part S9, we determine θh/θc by equating eqs 21 and 25 using σ(θ) = z–qρs(1 – θ) for a given θc. By assuming ρs = 5.0 × 1017 m–2 (negative charge density of PEDOT is one on the surface area of 2 × 10–18 m2), the linear dependence of EMF on the temperature difference is reproduced within the measured range of temperature difference. The slope increases by increasing α, θc and by decreasing ΔEads (by increasing |ΔEads|), as can be understood from eq 27. (See Figure S3 in Supporting Information Part S9 for details.) The dashed line obtained using eq 27 reasonably approximates the numerical results of ΔV as shown in Figure 4A.

Figure 4.

Figure 4

Open in a new tab

(A) (Red) Dots indicate the electromotive force (EMF) as a function of temperature difference measured using EMIM+Cl– as the electrolyte in water (10 g/L) and PEDOT:PSS electrode (Figure 1D). The upper solid (black) line and the upper dashed (black) line represent theoretical EMF originating from the diffuse layer obtained using eq 21 with eqs 25 and 27, respectively. The lower solid (red) line and the lower dashed (red) line represent the numerical results of EMF originating from both the Stern layer and diffuse layer; the EMF of the Stern layer is obtained using eq 32 with Cs = 20 μF cm–2, where θh/θc is determined by equating eqs 21 and 25. Parameters for theoretical calculation are θc = 0.5, α = 4, z+ = 1, z– = −1, ϵr = 80 (water), nb = 4.1 × 1025 m–3 (0.068 mol/L), Tc = 298 K, ΔEads = −0.2 eV, h0 = 0, and ρs = 5.0 × 1017 m–2 unless otherwise stated. (B) nh/nc (red) and θh/θc (black) as a function of temperature difference. Red circles and black dots (red long dashed line and black short dashed line) indicate nh/nc and θh/θc obtained using eq 21 with eq 25 for α = 4 (α = 1), respectively. Red thick solid line and black thick solid line represent the results of eqs 31 and 29, respectively. (C) θc = 0.03. (Red) Dots (measured EMF) are the same as in (A). The upper solid (black) line and the lower solid (red) line represent the numerical results of EMF originating from the diffuse layer and those of EMF originating from both the Stern layer and diffuse layer, respectively.

We show calculated nh/nc and θh/θc as a function of temperature difference in Figure 4B. nh/nc increases with increasing the temperature difference, while θh/θc decreases with increasing the temperature difference, both of which support the original assumptions regarding the anisotropy in double layers caused by temperature difference. These theoretical results originate from adsorption isotherm and electrostatic interaction between ions and electrodes. Indeed, by applying the same approximation to derive eq 27, we obtain

6. 28
6. 29
6. 30
6. 31

where ΔEads < 0 and 0 ≤ θc ≤ 1. (See Supporting Information Part S10.) If −2θcΔEads/(kBTc) > α is satisfied, Δθ (Δθ = θh – θc) decreases and Δn (Δn = nh – nc) increases by increasing ΔT as originally considered in Section 4 and shown in Figure 3. In Figure 4B, nh/nc and θh/θc for α = 4 are close to those for α = 1; the results are consistent with eqs 29 and 31 for the parameters used to draw Figure 4. Δθ/θc and Δn/nc are governed by the adsorption equilibrium and are virtually independent of the Eastman entropy of transfer for these cases. By ignoring the influence of the Eastman entropy of transfer, eqs 28 and 30 can be further simplified as eqs 29 and 31, respectively.

Since θh/θc is determined by equating eqs 21 and 25 using σ(θ) = z–qρs(1 – θ) for a given θc, we can estimate ΔV originating from the Stern layer for a given θc by considering that the area of the Stern layer is proportional to the coverage; ΔV is proportional to the difference between the coverage of adsorbed ions on the hot and cold electrode surfaces. We denote ΔV of the Stern layer by ΔVS, which can be expressed as49,50

6. 32

where Cs indicates the Stern layer capacitance per unit area; the typical values of Cs are 20–25 μF cm–2.49−53 Using the expression, Cs = ϵrsϵ0/H, where H ≈ 5 Å is the typical thickness of the Stern layer and ϵrs = 11.3 is the typical value of the relative dielectric constant of the Stern layer,53 we obtain Cs = 20 μF cm–2, which is used in Figure 4.

In Figure 4A, |ΔV| of diffuse layer is consistent with the experimental data shown by dots, but ΔV + ΔVS shown in the lower (red) solid line deviates significantly from the dots because |ΔVS| is approximately three times larger than |ΔV| of diffuse layer, if θc = 0.5 is assumed. In Figure 4C, θc = 0.03 is assumed. ΔV + ΔVS correlates with the dots; |ΔVS| is close to |ΔV|. By reducing θc from 0.5 to 0.03, the coverage difference between the hot and cold electrodes is also reduced, which reduces |ΔVS|. Figure 4 indicates that the significant amount of EMF is generated in the Stern layer compared to that in the diffuse layer. In particular, the large fraction of the ionic Seebeck coefficient of about 2–3 mV/K for θc = 0.5 might originate from the Stern layer.

In Figure 4A, we show ΔV + ΔVS, where the simple expression of ΔVS obtained by substituting eq 29 into eq 32 is used

6. 33

According to Figure 4A, the numerical exact results of ΔVS originating from the Stern layer and those of ΔV originating from the diffuse layer can be approximated by eqs 33 and 27, respectively. Therefore, these approximate expressions can be used to estimate the EMF originating from the Stern layer and the diffuse layer. The factor θc(1 – θc) in eq 33 indicates that the larger ΔVS might be expected for the intermediate coverage at the cold electrode (θc). Equation 33 is an important result to estimate the EMF origination from the Stern layer.

In general, EMF originating from the Stern layer is inversely proportional to the Stern layer capacitance; the smaller capacitance results in the larger EMF. By assuming flat electrodes, we considered the typical values of the Stern layer capacitance per unit area as 20–25 μF cm–249−53 and estimated the ionic Seebeck coefficient of about 2–3 mV/K by considering EMF originating from both the Stern layer and the diffuse layer. If the electrodes are porous, we need to consider the Stern layer capacitance per unit effective area, which would be reduced from the Stern layer capacitance per area of the flat surface.54,55 If the Stern layer capacitance per unit effective area is reduced by surface roughness, the ionic Seebeck coefficient can be increased if the other parameters are kept constant.

If the oxide layer of the thickness dox is formed on the electrode, we need to consider the oxide capacitance in addition to the Stern layer capacitance. The EMF originating from both the Stern layer and oxide layer might be considered using ΔVS = z+qρs/Ceff, where 1/Ceff = 1/Cs + 1/Cox and Cox = ϵoxϵ0/dox is the oxide capacitance per unit area. Using ϵox = 5 and dox = 10 nm, we find Cox ≈ 443 nF cm–2, which is orders of magnitude smaller than Cs. Therefore, we estimate Ceff ≈ Cox. The above estimation illustrates the important contribution from the capacitance from the electrodes, which might be relevant if the space charge layer is developed inside the semiconductor electrodes. Recently, the high apparent ionic Seebeck coefficient of 13.6 mV K–1 has been reported using the stainless-steel electrodes under corrosion.56 The high apparent ionic Seebeck coefficient might originate from the difference in the capacitance developed between the high temperature electrode and the low temperature electrode with different degree of corrosion.

For simplicity, we ignored the influence of temperature dependence in the capacitance of the Stern layer; the evaluation of the temperature dependence in the capacitance of the Stern layer requires the temperature dependence in the dielectric constant of the Stern layer.50 We also assume that the difference between the Fermi level of the hot electrode and that of the cold electrode originates mainly from the potential difference in the electrolyte; we ignore possible Fermi level pinning by surface defects or impurity states.57,58

Before closing this section, we summarize the relation among factors controlling the ionic Seebeck effect, i.e., the Eastman entropy of transfer for the ion species obeying adsorption equilibrium, the adsorption energy of the ion species, and the Stern layer capacitance per unit area. According to eq 27, the Eastman entropy of transfer can be ignored if α+ < |ΔEads/(kBTc)| holds. By comparing eq 27 with eq 33, we note that the dominant contribution to the ionic Seebeck effect is caused by the Stern layer capacitance per unit area when Inline graphic is satisfied apart from the factors related to θc. By using ρs = 5.0 × 1017 m–2 and Tc = 298 K, the condition can be explicitly given by Cs < 310 μF/cm2; the Stern layer with the lower relative dielectric constant would increase the ionic Seebeck coefficient. Similarly, the interface layer on the electrode, such as an oxide layer, might also increase the ionic Seebeck coefficient. On one hand, θc and ΔEads resulting from the ion–electrode interaction, together with the surface properties such as the Stern layer capacitance, would significantly influence the ionic Seebeck coefficient. On the other hand, the Eastman entropy of transfer should be larger than |ΔEads/(kBTc)| to influence the ionic Seebeck coefficient. Such quantitative comparison of the controlling factors might be useful for the development of rational approaches to increase the ionic Seebeck coefficient with an appropriate sign.

7. Conclusion

We have estimated the EMF generated in the diffuse layer under the temperature gradient by both the Eastman entropy of transfer and the polarity in the diffuse layer. We have also estimated the EMF generated in the Stern layer by considering the asymmetry in the coverages of adsorbed ions on the hot and cold electrode surfaces.

Herein, one of the mobile ion species is assumed to be adsorbed at the electrode while the other ions are assumed to be reflected. We show that the EMF corresponding to p-type polarities can be expected if cations are adsorbed with sufficiently large adsorption energy and the Eastman entropy of transfer for the cations is positive. The EMF corresponding to n-type polarities can be expected if anions are adsorbed with sufficiently large adsorption energy and the Eastman entropy of transfer for the anions is positive. The EMF is expressed by the Eastman entropy of transfer for the ion species responding directly to the temperature change of an electrode, through the change in adsorption equilibrium. The result is in sharp contrast to the EMF of the Eastman entropy of transfer estimated under the charge neutrality condition, which is expressed using the Eastman entropy of transfer of both cations and anions.23,28,29

By calculating the electrostatic potential from the adsorbed ions under adsorption–desorption equilibrium, the EMF has been evaluated semiquantitatively when the adsorbed ion concentration is given. For this purpose, the Grahame equation is generalized to take into account the temperature gradient (eq 23). According to the generalized Grahame equation, the EMF originating from the diffuse layer is expressed by the coverage of adsorbed ions on the hot electrode and that on the cold electrode. By taking into account the difference in the coverage of adsorbed ions between the hot and cold electrode, while ignoring the temperature dependence in the surface charge density of the pure electrode in the absence of adsorbed ions, the EMF originating from the diffuse layer has been derived as a function of the Eastman entropy of transfer and the adsorption energy of the ion species responding directly to the temperature change (eq 27).

The theoretical results are further supported by modulating the doping state of PEDOT using two methods: H2SO4 doping and treatment with 1,5,7-triazabicyclo[4.4.0]dec-5-ene (TBD), as demonstrated in our previous studies.59,60 For the H2SO4-doped samples, the conductivity of PEDOT increased to approximately 2000 S/cm, resulting in an ionic Seebeck coefficient of 3.7 mV/K for EMIM+Cl– (see Figure S4A of Supporting Information Part S10). In contrast, the TBD-treated films exhibited reduced conductivity (approximately 100 S/cm) and a corresponding ionic Seebeck coefficient of 0.39 mV/K for EMIM+Cl– (see Figure S4B of Supporting Information Part S10). The untreated PEDOT/PSS films showed an intermediate value, with an ionic Seebeck coefficient of 0.57 mV/K.

These results partially support our model. In our model, the higher ionic Seebeck coefficient can be achieved by increasing the difference in cation coverage between the hot and cold electrodes, as cations can adsorb onto PEDOT:PSS electrodes due to the negatively charged nature of PSS. Following H2SO4 treatment, the electrostatic interaction between PEDOT and PSS is weakened, leading to conformational changes in the PEDOT:PSS structure, phase segregation between the PSS and PEDOT chains, and alterations in doping concentration. The increase in electrical conductivity of PEDOT:PSS resulting from H2SO4 treatment is attributed to these combined effects.59−62 Additionally, the morphological changes influence the disparity in cation coverage between the hot and cold electrodes, contributing to an increase in the ionic Seebeck coefficient. Conversely, under dedoping with TBD treatment, we anticipate an opposite effect. Therefore, the ion coverage induced by doping affects the ionic Seebeck coefficient in a manner consistent with changes in the electrical conductivity of PEDOT:PSS. However, it is important to note that both H2SO4 doping and TBD dedoping introduce additional ions into the film, which may influence the measured ionic Seebeck coefficients.

The theoretical result might apply for conducting polymer electrodes. However, careful consideration might be required to apply for metal electrodes; in principle, the image charges in the electrodes should be considered for metal electrodes, where the surface charge density of the metal electrode is equal to the negative of spatial integration of ion charge density on the solution side, which depends on the temperature difference.63 The surface charge density in the pure metal electrode, therefore, depends on the temperature difference through the temperature difference in the ion distribution. As long as the image charge effect can be ignored in the EMF, the theoretical result (eq 27) applies for electrodes other than conducting polymer electrodes.

As mentioned above, the difference between the coverage of adsorbed ions on the hot and cold electrode surfaces is determined using the generalized Grahame equation. Using the Stern model and the difference between the coverage of adsorbed ions, the EMF origination directly from the adsorbed ions is calculated (eq 33). It turned out that the EMF of the Stern layer is larger than the EMF of the diffuse layer when the typical values of the Stern layer capacitance are used.

The theoretical results are compared with experimental results using a metal electrode and a PEDOT:PSS electrode in a closed system, where the electrolyte is fully enclosed. The measured p-type behaviors, as well as the values of EMF, are consistent with the theoretical results, where the Seebeck coefficient of up to 2–3 mV/K can be obtained using the generalized Grahame equation and the Stern model.

Previously, the silver electrode showed an ionic Seebeck coefficient of about 10 mV/K, while the PEDOT:PSS electrodes showed a negative ionic Seebeck coefficient for open systems, where electrolytes are exposed to the atmosphere.2,3 The origin of the discrepancy under open-air conditions should be clarified. In general, the oxide capacitance and/or the space charge capacitance might give rise to the dominant EMF owing to their low capacitance compared to both the diffuse layer capacitance and the Stern layer capacitance; the large Seebeck coefficient exceeding 2–3 mV/K might be caused by the oxide capacitance and/or the space charge capacitance. If the oxide capacitance is different between the closed and the open systems, EMF might depend on the closed and open conditions. The other reasons may be solvent evaporation leading to concentration changes, electrode surface changes, and environmental variations, which only occurs in open systems. Moreover, we have not considered the effect of latent heat, which may provide a higher temperature gradient. Although the present study is limited to closed systems, our results offer baseline information that can be used to further enhance the EMF in ionic thermoelectric cells.

Acknowledgments

S.H. and Q.W. acknowledge the support from NEDO Uncharted Territory Challenge 2050, Japan.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsami.4c19795.

  • Schematic of experimental setup, derivation of eq 8, concentration of mobile anions reflected by an electrode, the mobile anion concentration responding to the temperature change of an electrode, ionic Seebeck coefficient under charge neutrality, adsorption isotherm, ΔV as a function of the coverage of adsorbed anions, generalized Grahame equation, coverage of ions on an electrode, and ΔV–ΔT plots for H2SO4 and TBD treated PEDOT:PSS (PDF)

The authors declare no competing financial interest.

Supplementary Material

am4c19795_si_001.pdf (1.9MB, pdf)

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