Thermal efficiency of a thermocell made of Prussian blue analogues

Recently, it was reported that a thermocell can convert temperature into electric energy by using the difference in the thermal coefficient (α = dV/dT) of the redox potential (V) between the cathode and anode materials. Among battery materials, Prussian blue analogues (PBAs) are promising materials for thermocell, because α changes from approximately −0.3 mV/K in NaxMn[Fe(CN)6]0.83 3.5 H2O (NMF83) to approximately 1.3 mV/K in NaxCo[Fe(CN)6]0.92,9H2O (NCF90). In this work, we systematically investigated the thermal efficiency (η) of the NMF83/NCF90 thermocell relative to the difference (ΔT) between low (TL = 282 K) and high (TH = 292–338 K) temperatures. We found that the thermal efficiency (η) increased proportionally with ΔT. The linear increase in η is ascribed to the linear increase in the cell voltage (Vcell) and the charge (QNCF90) extracted from NCF90. Moreover, η reached 3.19% at ΔT = 56 K, which corresponds to 19% of the Carnot efficiency (ηcarnot = 17.0%). We further confirmed that the magnitude of QNCF90 is quantitatively reproduced by the slopes of the discharge curves of NMF83 and NCF90.

(10-30%). The discharge curves of Co-and Mn-PBAs show characteristic plateaus, the redox reaction of which are well assigned using X-ray absorption spectroscopy 18,19 . PBAs are also promising materials for thermocells because the magnitude and sign of α can be controlled by the chemical composition 20  In our previous work 5 , we demonstrated that the NCF71/NCF90 thermocell can convert temperature into electric energy with 1% thermal efficiency between T L = 295 K and T H = 323 K. In this work, we systematically investigated the thermal efficiency (η) of the NMF83/NCF90 thermocell against the temperature difference (ΔT) between T L and T H . The η of the NMF83/NCF90 thermocell is expected to be higher than that of the NCF71/ NCF90 thermocell because Δα (=α cathode − α anode ~ 1.7 mV/K) is much larger in the former cell. We found that η increases in proportion to ΔT and reaches 3.19% at ΔT = 56 K. The linear increase in η is ascribed to the linear increase in the cell voltage (V cell ) and charge (Q NCF90 ) extracted from NCF90. We further confirmed that the magnitude of Q NCF90 is quantitatively reproduced by the slopes of the discharge curves in NMF83 and NCF90. Figure 2 shows the discharge curves of the (a) NMF83 and (b) NCF90 films. The curve of NMF83 [(a)] shows two plateaus (plateaus I and II) near 1.1 and 0.6 V vs. Ag/AgCl. Plateau I (<20 mAh/g) near 1.1 V is ascribed to the reaction 19 6 ] 0.9 . Plateau VI (>50 mAh/g) near 0.5 V is ascribed to the reaction: Na 0.6 Co 3+ [Fe 2+ (CN) 6 ] 0.9 + Na + + e − → Na 1.6 Co 2+ [Fe 2+ (CN) 6 ] 0.9 .

Discharge curve of a half-cell
The difference (Δα = α cathode − α anode ) in α is a crucial parameter for a thermocell, because V cell increases in proportion to Δα. Fukuzumi et al. 20 systematically investigated the α values of prototypical PBAs against the Na concentration (x). In NMF83, α gradually decreases from 1.4 to − 0.4 mV/K with increasing in x. At plateau II, α becomes negative (approxmately −0.3 mV/K). In NCF90, α is ~0.4 mV/K and ~1.3 mV/K in plateau III and IV, respectively. Therefore, we can maximize Δα if we chose NMF83 (plateau IV) and NCF90 (plateau II) as the anode and cathode, respectively. As discussed later, the slope of the plateau is another crucial parameter. In this work, we define the discharge curves of NMF83 [ Fig. 1(a)] and NCF90 [ Fig. 1(b)] as V NMF83 (Q) and V NCF90 (Q), respectively. The slopes, i.e., dV NMF83 /dQ and dV NCF90 /dQ, are roughly evaluated as −3.5 and −0.5 mV/(mAh/g) at plateaus II and IV, respectively.

Thermal Cycle Measurement of Thermocell
We fabricated a two-pole beaker-type thermocell; for this thermocell, the anode, cathode and electrolyte are an as-grown NMF83 film, pre-oxidized NCF90 film, and an aqueous solutions containing 17 mol/kg NaClO 4 , respectively. The as-prepared thermocell was slowly cooled down to T L (=282 K). At T L , the thermocell shows a finite V cell (~0.1 V), which was discharged to 0 V under the constant current condition. If T cell is slowly increased by ΔT(=T H − T L ) in the open circuit condition, the redox potentials of the anode and cathode change by α anode ΔT and α cathode ΔT, respectively, in the warming process. We expect a thermally induced change in V cell as larger as ΔT(α cathode − α anode ). In other words, the electric energy is thermally stored in the thermocell. The amount of stored electric energy can be evaluated by the discharge process to 0 V under a constant current condition. The sweep area on the voltage-charge diagram corresponds to the stored electric energy. The behaviour of V cell in the cooling process is opposite to that in the warming process. In this cooling process, the redox potentials of the anode and cathode change by −α anode ΔT and −α cathode ΔT, respectively. We expect a thermally induced change in V cell as larger as −ΔT(α cathode − α anode ). The stored electric energy can be extracted by the discharge process to 0 V under constant current condition. Therefore, our thermocell can convert temperature to electric energy in the thermal cycle between T L and T H . Figure 3 shows a prototypical example of the thermal cycle of the NMF83/NCF90 thermocell at ΔT = 40 K. Black and red colours represent the data obtained in the first and second cycles, respectively. In the (a) warming  In the (c) cooling process, V cell linearly decreases with the decrease in T cell at a rate of 1.3 mV/K. At T L , the cell shows a finite voltage (V cell = −44 mV). In the (d) discharge process, Q NCF90 was 8.4 mAh/g. The electric work (W L ) at T L is 2.82 meV/NCF90. The data in the second cycle are highly similar to those in the first cycle. Figure 4 shows another example of the thermal cycle at ΔT = 56 K. In the (a) warming process, V cell linearly increases with the increase in T cell at a rate of 1.1 mV/K. At T H (=338 K), the cell shows a finite voltage (V cell = 60 mV). In the (b) discharge process at T H , V cell linearly decreases with the extracted charge, where Q NCF90 is 28.0 mAh/g, which is 26.7% of the discharge curve of NCF90 [ Fig. 1(b)], and W H is 12.80 meV/NCF90. In the (c) cooling process, V cell linearly decreases with the decrease in T cell at a rate of 1.2 mV/K. At T L (=282 K), the cell shows a finite voltage (V cell = −65 mV). In the (d) discharge process, Q NCF90 is 20.0 mAh/g and W L is 9.70 meV/ NCF90. Figure S1 shows the discharge curves at T H and T L against ΔT.

Thermal Efficiency Against ΔT
The thermal efficiency (η) is defined by W H + W L /Q H , where Q H is the inflow of heat. Table 1 shows W H , W L , Q H , and η against ΔT. Furthermore, W H (W L ) was roughly evaluated as qV cell /2 at T H (T L ), where q is the final extracted charge per NCF90 in the discharge process at T H (T L ) and Q H was evaluated as (C anode + C cathode )ΔT, where C anode (C cathode ) is the heat capacity of anode (cathode) material. We neglected the heat capacity of the electrolyte because the amount of electrolyte is minimized in the current thermocell made of redox-capable solids. Using the specific heat (=4.16 meV/K per formula unit) of the ideal Na 2 Co[Fe(CN) 6 ] in the Dulong-Petit law, Q H is expressed as 4.16(1 + n)ΔT per NCF90, where n [=n NMF83 /n NCF90 . where n NMF83 (n NCF90 ) is the number of the the NMF83 (NCF90) units] is the molar ratio. Additionally, W H , W L , Q H , and η monotonously increases with ΔT, and η reaches 3.19% at ΔT = 56 K, which corresponds to19% of the Carnot efficiency (η carnot = 17.0%).

Comparison with Previous Works
In Table 2, we compare the cell parameters and η in the current thermocell with those in previously reported thermocells. In the thermocells reported in ref. 2 and ref. 3 , ehe electrolyte is different between the anode and cathode and is separated by a membrane. However, the thermocells made of two types of PBA solids (ref. 5 and this work) use the same electrolyte in the anode and cathode. As discussed in the following section, η increases linearly with ΔT. We should compare η/ΔT, rather than η, among the thermocells, and the η/ΔT values of the thermocells made of PBA solids are comparable to those of the thermocells reported in ref. 2 and ref. 3 .

Discussion
We turn the discussion to the ΔT dependence of V cell , Q NCF90 , and η. Figure 5(a) shows the ΔT dependence of V cell . The open and closed circles represent the data at T H and T L , respectively, and V cell increases linearly with ΔT, as indicated by the solid line. The coefficient (=1.15 mV/K) is comparable to α NCF90 − α NMF83 (~ 1.6 mV/K). Thus, we experimentally confirm the relationship of V cell = ΔT(α NCF90 − α NMF83 ). Figure 5(b) shows the ΔT dependence of Q NCF90 . The open and closed circles represent the data at T H and T L , respectively. We quantitatively evaluate Q NCF90 at T H from the discharge curves, V NMF83 (Q) and V NMF83 (Q), of NMF83 and NCF90. By applying the Tayloer expansion, V cell at T H is expressed as Q NCF90 dV NCF90 /dQ + (−Q NNF83 dV NMF83 / dQ), where Q NCF90 (Q NMF83 ) is the final extracted charges from NCF90 (NMF83). The first and second terms are negative, because dV NCF90 /dQ < 0, dV NMF83 /dQ < 0, and Q NMF83 < 0. The charge neutrality between the anode and cathode imposes a constraint condition of m NMF83 Q NMF83 = −m NCF90 Q NCF90 , where m NMF83 (m NCF90 ) is the mass of NMF83 (NCF90). We obtaine Q NCF90 = −V cell /(dV NCF90 /dQ + m −1 dV NMF83 /dQ), where m = m NMF83 /m NCF90 . We note that m = n (M NMF83 /M NCF90 ), where M NMF83 (=338.5) and M NCF90 (=324.0) are the molecular weights of NMF83 and NCF90, respectively, and dV NMF83 /dQ and dV NCF90 /dQ are roughly evaluated as −3.5 and −0.5 mV/ (mAh/g). In Fig. 5(b), these calculated Q NCF90 values at T H are plotted with crosses. The calculated Q NCF90 satisfactorily reproduces the experimentally-obtained Q NCF90 . We further confirmed the strong correlation between the calculated and experimentally-obtained Q NCF90 data at T L (Fig. S2), thus, we demonstrating the empirical relationship of Q NCF90 = −V cell /(dV NCF90 /dQ + m −1 dV NMF83 /dQ). These arguments clearly indicate that the slope of the discharge curve is a determining parameter of Q NCF90 . In other words, the flatter the discharge curve becomes, the higher η becomes.

Summary
We systematically investigated V cell , Q NCF90 , and η of the NMF83/NCF90 thermocell against ΔT. These three quantities increase linearly with ΔT, and η reaches 3.19% at ΔT = 56 K, which corresponds to 19% of the Carnot efficiency (η carnot = 17.0%). We further confirmed that the magnitude of Q NCF90 is quantitatively reproduced by the slopes of the discharge curves of NMF83 and NCF90. These observations unambiguously illustrate us the strategy for enhancing η of the thermocell, that is, to explore and/or develop materials with a higher |α| and a flatter discharge curve. The film area was 1.0 cm 2 . The NMF83 film consists of crystalline particles of a few hundred nm and the colour is white. The NCF90 film also consists of crystalline particles of a few hundred nm and the colour is light green.  The film thicknesses were determined by a profilometer (BRUKER Dektak3030). The chemical composition of the film was determined by the inductively coupled plasma (ICP) method and CHN organic elemental analysis. The synchrotron radiation X-ray powder diffraction (XRD) measurements were performed at BL02B2 beamline 22 at the SPring-8. The films were removed from the ITO glass and were filled in 300 μm glass capillaries. The capillary was placed at the Debye Scherrer camera. The XRD patterns were monitored with a one-dimensional semiconductor detector (MYTHEN, Dectries Ltd.). The exposure time was 5 min. The wavelength of the X-rays (=0.69963 Å) was calibrated by the cell parameter of a standard CeO 2 powder. Figure S3 shows the magnified diffraction pattern of NMF83 and NCF90. All reflections in NMF83 are assigned to a face-centrered cubic (fcc) structure (Fm 3 m; Z = 4), whereas those in NCF90 are assigned to a trigonal (hexagonal setting) structure (R 3 m; Z = 3). The cell parameters were refined using the Rietan-PF program 23 , and a = 10.5210(2) Å in NMF83, a H = 7.4353(4) Å and c H = 17.4758(11) Å in NCF90.

Method
The charge/discharge curves of the NMF83 and NCF90 films were measured with a potentiostat (HokutoDENKO HJ1001SD8) using a three-pole beaker-type cell. The working, referential, and counter electrodes were the PBA film, a standard Ag/AgCl electrode, and Pt, respectively. The electrolytes consisted of aqueous solutions containing 17 mol/kg NaClO 4 . The charge/discharge rate was 0.5 C. The cut-off voltage was ranged from 0.20 to 1.15 V vs. Ag/AgCl. The mass of each film was evaluated using thickness, area, and density. We confirmed that the actual densities of the NMF83 and NCF90 films were 0.71 and 0.58 of the ideal density, respectively.
Thermal cycle measurement of thermocell. The thermocell is a two-pole beaker-type cell (Fig. S4). The anode, cathode and electrolyte are the as-grown NMF83 film, pre-oxidized NCF90 film, and aqueous solutions containing 17 mol/kg NaClO 4 , respectively. Pre-oxidization of the NCF90 film was performed at V upper = 0.65 V against Ag/AgCl in aqueous solutions containing 17 mol/kg NaClO 4 . The as-prepared thermocell was slowly cooled to T L (=282 K). At T L , the thermocell show a finite V cell (~0.1 V), which was discharged to 0 V under a constant current condition (0.1 C). The discharge rate was defined by the inverse of the charging time [hour] of the NCF90 film.
The thermal cycle measurement consists of four processes: (a) warming process from T L to T H , (b) discharge process at T H , (c) cooling process from T H to T L , and (d) discharge process at T L . In the (a) warming process, T cell was slowly increased from T L to T H in the open circuit condition, and T cell was monitored by a platinum resistance thermometer in the electrolyte. At (b) T H , the thermally-charged cell was discharged at 0.1 C. In the (c) cooling process, T cell was slowly decreased from T H to T L in the open circuit condition. At (d) T L , the thermally-charged cell was discharged at 0.1 C. T L was fixed at 282 K and T H was changed from 292 K to 338 K.