Vibrational entropy as an indicator of temperature coefficient of redox potential in conjugated polymers

To attain a "smart" society, it is crucial to develop novel energy-harvesting technologies that produce electric energy efficiently and inexpensively from human body heat or waste heat near room temperature. Recently, it was demonstrated that environmental heat can put a battery on the charged state by using the difference Δα in the thermal coefficient α (= dV/dT) of the redox potential (V) between the cathode and anode materials.^1–6) Hereafter, we call such a battery as "tertiary battery", because it is charged by the environmental heat, not by the electric energy. The tertiary battery generates electric energy in the thermal cycle between low (T_L) and high (T_H) temperatures. In the warming process, one can obtain the cell voltage (V_cell) of ΔαΔT (ΔT = T_H − T_L) and can extract the accumulated electric energy by discharging the cell at T_H. Similarly, one obtains V_cell of −ΔαΔT in the cooling process. Shibata et al.⁴⁾ demonstrated that high thermal efficiency η of 1.0% was attained between T_L (= 295 K) and T_H (= 323 K) in a tertiary battery consisting of two kinds of Prussian blue analogs with different α. Thus, α^5,7,8) is the key parameter that determines the performance of the tertiary battery. From a thermodynamic point of view, α is nothing but $\tfrac{{\rm{\Delta }}S}{e}$ with the elementary charge e (≥0) and difference ΔS in entropies (S) of the system between the reduced and oxidized states. Importantly, the redox process of a battery influences not only the solid electrode but also the electrolyte. Therefore, ΔS consists of the solid (ΔS_solid) and electrolyte (ΔS_electrolyte) components.⁹⁾

Among the battery materials, conjugated polymers have advantages, like light weight, flexibility, environmental frendness.^10–13) For example, polyacetylene,¹⁴⁾ poly(p-phenylene) (PPP),^14,15) and polythiophene,¹⁶⁾ show reversible redox process between the reduced, neutral and oxidized states. Polythiophene and its derivatives have been extensively investigated in applications, since they are highly stable in their neutral states, easy to structural modification, and processable in solution.^17–19) The poly(3-alkylthiophenes), a prototype of this class of compounds, exhibit the best device characteristics.^20–26) On the other hand, fluorene-based polymers show quite high fluorescence quantum yields of 0.6–0.8 in solid and have been applied to light-emitting diodes due to their chemical and thermal stabilities.^27–30) Figure 1 shows prototypical thiophene and fluorene-based polymers, that is, poly(3-hexylthiophene-2,5-diyl) (P3HT), poly(9,9-dioctylfluorene-co-dithienyl-benzothiadiazole) (F8TBT), poly(9,9-dioctyl-fluorene-co-bithiophene) (F8T2), poly(9,9-dioctylfluorene-co-N-(4-butylphenyl)diphenylamine) (TFB), poly(9,9-dioctylfluorene-co-N,N'-bis(4-butylphenyl)-N,N'-diphenyl-1,4-benzenediamine) (PFB), together with the energy levels (E_HOMO) of the highest occupied molecular orbital (HOMO) and that (E_LUMO) of the lowest unoccupied molecular orbital (LUMO). The main chains of the fluorene-based polymers consist of fluorene, triphenylamine, benzothiadiazole, and thiophene molecules. The former three molecules are classified as aryl groups. In this sense, we can control the number ratio r (= $\tfrac{{n}_{\mathrm{thio}}}{{n}_{\mathrm{thio}}+{n}_{\mathrm{aryl}}};$ where n_thio and n_aryl are the number of the thiophenes and aryl groups, respectively) of thiophene in the main chain from r = 0.0 in TFB and PFB, r = 0.4 in F8TBT, r = 0.5 in F8T2, to r = 1.0 in P3HT.

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In this work, we systematically investigated α of P3HT, F8TBT, F8T2, PFB, and TFB with controlling r. We found that α steeply increases with r from 0.19 mV K⁻¹ (TFB) at r = 0.0 to 1.08 mV K⁻¹ (P3HT) at r = 1.0. This observation suggests that α in thiophene is higher than that in benzene. This trend is well explained by variation of the vibrational entropy S_vib between the neutral and oxidized states of the molecules.

2.1. Sample preparation and characterization

2.2. Electrochemical properties

2.3. Determination of α

The temperature coefficient (α) of the cathode material can be expressed as α_cell − α₀, where α_cell and α₀ are the coefficient of the cell and anode material, respectively. We determined α_cell of the beaker-type cell fabricated with the electrodes and electrolyte materials mentioned above. The temperature coefficient (α_Li) of Li in the electrolyte (EC/DEC containing 1 mol l⁻¹ LiClO₄) is 0.76 mV K⁻¹.⁹⁾ Therefore, α is obtained using the equation α = α_cell + 0.76 mV K⁻¹. To avoid the potential fluctuation that is prominently seen in the neutral (as-grown) state, the polymers are partially oxidized (arrows in Fig. 2) in the beaker-type cell. V_cell was carefully measured as a function of the cell temperature (T), which was monitored with a Pt resistance thermometer in the electrolyte. Increase or decrease of T was slowly controlled at a rate of 0.3 K min⁻¹. In the low-V plateau, the redox potential was too unstable to determine reliable α.

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3.1. Electrochemical properties

3.2. Determination of α

Figure 3 shows V_cell of the Li/polymer cell against T: (a) P3HT, (b) F8TBT, (c) F8T2, (d) TFB, and (e) PFB. Red and blue marks represent that the data observed in the heating and cooling runs, respectively. No thermal hysteresis was observed in V_cell, which indicates that temperature gradient as well as sample deterioration are excluded in our experiment. α_cell of the Li/polymer cell was evaluated by least-squares fittings (see straight lines in Fig. 3). In P3HT [(a)], whose main chain contains only thiophene, α_cell (= 0.32 mV K⁻¹) is positive. In fluorene-based polymers [(b)–(e)], in which part of or all thiophenes are substituted by aryl groups, α_cell is negative.

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Figure 4 shows interrelation between α (= α_cell + 0.76 mV K⁻¹) of polymer and r. We found that α steeply increases with r from 0.19 mV K⁻¹ (TFB) and 0.27 mV K⁻¹ (PFB) at r = 0.0, 0.42 mV K⁻¹ (F8TBT) at r = 0.4, 0.38 mV K⁻¹ (F8T2) at r = 0.6 to 1.08 mV K⁻¹ (P3HT) at r = 1.0. This strong correlation between α and r suggests that α in polythiophene is much higher than those in aryl groups. The correlation further implies that α strongly depends on the molecules in the main chain, and hence, can be controlled by the design of the main chain.

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Now, let us consider the reason why α in polythiphens is higher than those in aryl groups. Here, recall that ΔS consists of the solid (ΔS_solid) and electrolyte (ΔS_electrolyte) components. In the present case, ΔS_electrolyte is originated from the desolvation and intercalation of ClO${}_{4}^{-}$, and hence, is difficult to evaluate. Fortunately, ΔS_electrolyte can be regarded as a constant, because the solvent (EC/DEC containing 1 mol l⁻¹ LiClO₄) and ion (ClO${}_{4}^{-}$) are the same for the five Li/polymer cells. Therefore, we concentrate our attention to the difference in ΔS_solid between polythiophene and aryl groups.

Now, let us consider the contribution of the vibrational entropy (S_vib) to α, which is expressed as $\tfrac{{S}_{\mathrm{vib}}^{\mathrm{neu}}-{S}_{\mathrm{vib}}^{\mathrm{oxi}}}{e}$, where ${S}_{\mathrm{vib}}^{\mathrm{neu}}$ and ${S}_{\mathrm{vib}}^{\mathrm{oxi}}$ are the vibrational entropies in the neutral and oxidized states of the molecules, respectively. In the oxidization process, an electron is removed from the HOMO level with bonding character. The removal weakens the force constant between the neighboring atoms and lowers the frequencies of the vibrational modes.^31–34) Such red-shifts of the modes increase ${S}_{\mathrm{vib}}^{\mathrm{oxi}}$, because S_vib is express as ${k}_{{\rm{B}}}{{\rm{\Sigma }}}_{i=1}^{3N-6}[\tfrac{{x}_{i}{e}^{-{x}_{i}}}{1-{e}^{-{x}_{i}}}-\mathrm{ln}(1-{e}^{-{x}_{i}})]$, where N is the number of atoms in the molecule. x_i (= $\tfrac{{\hslash }{\omega }_{i}}{{k}_{{\rm{B}}}T}$, where ${\hslash }{\omega }_{i}$ and k_B are the vibrational energy of the ith mode and the Boltzmann constant, respectively) is the vibrational energy in the unit of the thermal energy.

As shown in Fig. 4, α correlates with r in spite of different kinds of the aryl groups (fluorene, triphenylamine, and benzothiadiazole). Thus, we considered the phenyl ring (R-C₆H₅) as the simplest representative of such aryl groups. The oxidization process removes an electron from the π-bond region, and hence, has negligible effect on the vibrational energies (x_i) in the side groups. In this sense, substitution of hydrogens for the side groups of thiophene and phenyl ring has no serious effect on evaluation of α (= $\tfrac{{S}_{\mathrm{vib}}^{\mathrm{neu}}-{S}_{\mathrm{vib}}^{\mathrm{oxi}}}{e}$). Thus, α of polythiophene and aryl groups can be approximated to α of thiophene (C₄H₄S) and benzene (C₆H₆) molecules, respectively.

To make a quantitative discussion, we performed a density functional theory (DFT) calculation of C₄H₄S, C₄H₄S⁺, C₆H₆ and C₆H${}_{6}^{+}$. The calculations were performed at the B3LYP/6-311G(d, p) level.³⁵⁾ After the structural optimization, frequencies of the vibrational modes were evaluated in the C_2v (C₄H₄S, C₄H₄S⁺), ${D}_{6{\rm{h}}}$ (C₆H₆) and ${D}_{2{\rm{h}}}$ (C₆H${}_{6}^{+}$) symmetries. The symmetry change observed in C₆H₆ is ascribed to the Jahn-Teller deformation into a quinoid structure. Figure 5(a) shows the frequencies of the vibrational modes in C₄H₄S (open circles) and C₄H₄S⁺ (closed circles). The removal of an electron slightly reduces the frequencies. Figure 5(b) shows the frequencies of the vibrational modes in C₆H₆ (open circles) and C₆H${}_{6}^{+}$ (closed circles). For convenience of explanation, irreducible representations in the ${D}_{2{\rm{h}}}$ symmetry is used for C₆H₆. The removal of an electron significantly reduces the frequencies, especially, in the lowest-lying ${B}_{2{\rm{g}}}$ and ${B}_{3{\rm{g}}}$ modes.

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In Table II, we listed ${S}_{\mathrm{vib}}^{\mathrm{neu}}$ and ${S}_{\mathrm{vib}}^{\mathrm{oxi}}$ of C₄H₄S and C₆H₆, together with their contribution to α. The contribution is larger in C₄H₄S (−0.038 mV K⁻¹) than in C₆H₆ (−0.129 mV K⁻¹). Thus, the DFT calculation qualitatively explains why α in polythiophene is higher than those in aryl groups. Here, recall that ΔS consists of ΔS_solid and ΔS_electrolyte. ΔS_electrolyte, which can be regarded as a constant, is unknown. Therefore, it is impossible to compare the absolute values (or signs) of the calculated α and those of the experimentally obtained α.

Table II.  Vibrational entropies in the neutral (${S}_{\mathrm{vib}}^{\mathrm{neu}}$) and oxidized (${S}_{\mathrm{vib}}^{\mathrm{oxi}}$) states of C₄H₄S and C₆H₆ at 298.15 K. α is expressed as $\tfrac{{S}_{\mathrm{vib}}^{\mathrm{neu}}-{S}_{\mathrm{vib}}^{\mathrm{oxi}}}{e}$.

Molecule	${S}_{\mathrm{vib}}^{\mathrm{neu}}$ (meV K−1)	${S}_{\mathrm{vib}}^{\mathrm{oxi}}$ (meV K−1)	α (mV K−1)
C4H4S	0.150	0.188	−0.038
C6H6	0.191	0.320	−0.129

In conclusion, we investigated α of P3HT, F8TBT, F8T2, PFB, and TFB with controlling the number ratio (r) of the thiophene in the main chain. We found a strong correlation between α and r, and interpreted it in terms of variation of S_vib. Our analysis strongly suggests that S_vib is a useful indicator for evaluating the magnitude of α in conjugated polymers.

This work was supported by JSPS KAKENHI (Grant Number JP17H01137). This work was supported by a joint research with Focus Systems Corporation.