Electrochemical implications of modulating the solvation shell around redox active organic species in aqueous organic redox flow batteries

Significance

The development of cost-effective batteries for long-duration grid scale energy storage will be accelerated using frameworks to rapidly screen and select battery components. Herein, we show that the solvent reorganization energy calculated from the Born equation (with reference to an electrolyte’s composition) is predictive of the electrolytes’ device level performance. This descriptor was found to correlate with key transport and kinetic properties over a range of electrolyte compositions and pH values, succinctly capturing the multicomponent interactions between the electrolyte salts and solvent. This enables the initial high-throughput screening of electrolyte candidates with minimal experimentation. Applied to aqueous redox flow batteries employing organic redox active species, we predict high-performance electrolyte compositions, enabling significantly enhanced device performance.

Abstract

Organic and organometallic reactants in aqueous electrolytes, being composed of earth-abundant elements, are promising redox active candidates for cost-effective organic redox flow batteries (ORFBs). Various compounds of ferrocene and methyl viologen have been examined as promising redox actives for this application. Herein, we examined the influence of the electrolyte pH and the salt anion on model redox active organic cations, bis((3-trimethylammonio) propyl)- ferrocene dichloride (BTMAP-Fc) and bis(3-trimethylammonio) propyl viologen tetrachloride (BTMAP-Vi), which have exhibited excellent cycling stability and capacity retention at ≥1.00 M concentration [E. S. Beh, et al. ACS Energy Lett. 2, 639–644 (2017)]. We examined the solvation shell around BTMAP-Fc and BTMAP-Vi at acidic and neutral pH with SO₄^2-, Cl⁻, and CH₃SO₃⁻ counterions and elucidated their impact on cation diffusion coefficient, first electron transfer rate constant, and thereby the electrochemical Thiele modulus. The electrochemical Thiele modulus was found to be exponentially correlated with the solvent reorganizational energy (λ) in both neutral and acidic pH. Thus, λ is proposed as a universal descriptor and selection criteria for organic redox flow battery electrolyte compositions. In the specific case of the BTMAP-Fc/BTMAP-Vi ORFB, low pH electrolytes with methanesulfonate or chloride counterions were identified as offering the best balance of transport and kinetic requirements.

The market-driven adoption of renewable power sources such as solar and wind due to the decrease in the unsubsidized levelized cost of electricity (currently cheaper than every source except natural gas) necessitates the adoption of grid scale energy storage to ensure grid reliability and resiliency (1). The stringent requirements for the economic viability of these sources, such as a cost of <$100/KWh (2), levelized cost of storage of <$0.05/kWh-cycle (3), and 10 to 100 h of discharge (4) at rated power, imposes significant constraints on the possible technologies that meet these targets. Redox flow batteries (RFBs), which decouple the power and energy output of a battery by storing the redox active material outside the battery itself, are a promising grid scale energy storage technology (5). Various RFBs employing elemental, metallic actives such as Fe-Cr (6), all-V (7, 8), Zn-Ce (9), and V-Ce (10, 11) have been the subject of active research over the past half century. The capacity of these systems is determined by the actives’ solubility (typically ∼1 M) and the number of electrons transferred per redox active ion. The typical 1 M concentration and the single electron transfer possible with a vast majority of these systems caps their theoretical specific capacity at 26.8 Ah ⋅ L⁻¹ with the energy density being restricted to a maximum of ∼35 Wh ⋅ L⁻¹ (assuming a generous ∼1.5 V operation). The cost-conscious decision to utilize the cheapest solvent, water, focuses developmental efforts on improving the solubility and number of electrons transferred within the aqueous electrochemical window.

Recent developments in utilizing water-soluble organic/organometallic actives, some with multiple redox centers, has led to the demonstration of organic RFBs (ORFBs) with reasonably long cycle life and all-V RFB-like capacity. The key incentive for moving to these organic redox active species is the potential for multielectron redox processes at each electrode and the >1 M solubility limit of these organic species. For example, a two-electron transfer redox process at each electrode and a 1.5 M solubility limit for a hypothetical organic active species yields a theoretical specific capacity of 80.4 Ah ⋅ L⁻¹. To realize this potential, a variety of RFBs based on methyl viologen (12–16), (2,2,6,6-Tetramethylpiperidin-1-yl)oxyl (12–14), ferrocene (15, 16), various quinones (17–20), ferricyanide (17), and alloxazine (21) actives in acidic, neutral, and alkaline pH conditions have been developed. The performance of some of these RFBs have resulted in plans for large-scale deployment of such systems. In other systems, extraordinarily long cycles lives in the order of centuries (200 y to 50% capacity loss) have been projected (22). These organic actives are invariably composed of chemicals that are low cost at scale (e.g., the viologen paraquat is an extensively used herbicide). Thus, after initial capital investments in scaled-up production, the cost of these materials would be lower than typical inorganic RFB actives (23). This is due to the fact that these organic actives can result in RFBs with higher specific energy, thereby lowering the cost on a capacity normalized basis (24).

Given the stringent cost consideration for grid scale deployment, any ORFB needs to maximize its energy density and power density. Engineering solutions exist to improve most factors that influence energy density and power density, like the reactant concentration, current density, and cell/stack component electrical resistance, but a key parameter, the open circuit voltage (OCV), is inherently a function of the redox potentials of the solvated actives. A selected chemistry with low discharge polarization and high OCV would exhibit superior energy and power densities. Thus, we examined the effect of the supporting electrolyte components on the transport and kinetic properties of these organic actives, as it is the solvated ions that take part in the interactions of interest.

We selected a configuration with bis(3-trimethylammonio) propyl)ferrocene dichloride (BTMAP-Fc) catholyte coupled with bis(3-trimethylammonio)propyl viologen tetrachloride (BTMAP-Vi) anolyte in an aqueous ORFB as our model system. At neutral pH, this system has shown high capacity, high stability, and low crossover (16). The present study examined the effect of pH and counter ion selection on the transport and kinetics of the model organic actives BTMAP-Fc and BTMAP-Vi. Given concerns with alkaline stability, only acidic (H₂SO₄, HCl, and CH₃SO₃H) and neutral (Na₂SO₄, NaCl, and NaCH₃SO₃) pH aqueous electrolytes were considered (25). The schematic of the ORFB and the various electrolyte compositions considered are depicted in Fig. 1. The effect of the pH and counter anion on the diffusion coefficient, hydrodynamic radius, first electron transfer rate constant, and solvent reorganization energy ($\lambda$) was examined using standard electrochemical methods. The electrochemical Thiele modulus ($h_T$), combining the diffusion coefficient and first electron transfer rate constant, was found to be strongly correlated with the solvent reorganization energy ($\lambda$). Detailed discussions of $\lambda$ in the context of Marcus–Hush (M-H) kinetics and its advantages over the widely used Butler–Volmer (B-V) kinetics formulation have been presented in our prior publications (26–29). We have previously shown that similar strong predictive correlations exist between $h_T$ and $\lambda$ in nonaqueous electrolytes (29). Having established that such correlations hold for aqueous and nonaqueous electrolytes at up to 1 M dissolved salt concentrations, extensions of this framework to highly concentrated water-in-salt, ionic liquid, and deep eutectic solvent-based electrolytes could be fruitful. The optimal range of electrolyte $\lambda$ that resulted in high OCVs was determined, and recommendations on both optimal electrolyte pH and choice of counterion are provided. Although Marcus theory predicts an inverse relationship between the heterogeneous electron transfer kinetics and the solvent reorganization energy, here, we conclusively demonstrate the practical predictive applicability of M-H theory to electrochemical device engineering by showing how Marcus theory parameters are related to (and, hence, predictive of) device level metrics like the Thiele modulus.

Fig. 1.

Schematic of an ORFB assembled with BTMAP-Fc catholyte and BTMAP-Vi anolyte incorporating an anion exchange membrane separator. The pH and anion combinations in the supporting electrolyte are tabulated below the schematic. The electron and ion flow during charging is depicted using a solid line, and the equivalent flow during the discharge process is depicted using dashed lines.

Materials and Methods

Chemicals.

The electrochemical measurements reported herein were carried out in 1 M aqueous solutions of BTMAP-Fc and BTMAP-Vi with six different supporting electrolytes. The supporting electrolytes used were sulfuric acid (H₂SO₄) (Sigma-Aldrich, ≥99%), HCl (Sigma-Aldrich, American Chemical Society [ACS] reagent 37%), CH₃SO₃H (Sigma-Aldrich, ≥99%), Na₂SO₄ (Sigma-Aldrich, ACS reagent ≥99%), NaCl (Sigma-Aldrich, ACS reagent ≥99%), and NaCH₃SO₃ (Sigma-Aldrich, ≥98%). All the supporting electrolytes used in the study were at 1 M concentration.

Synthesis of Model Organic Redox Active Species.

The organic redox active species BTMAP-Fc and BTAP-Vi, detailed in the work of Beh et al. (16), were chosen as the model compounds (their respective structures are depicted in Fig. 1). The compounds were synthesized as detailed in the supplementary information of that paper, and successful synthesis was confirmed by H¹ NMR. All air and moisture sensitive steps in the synthesis were carried out in a MBraun Ar-filled glove box with <0.5 ppm of O₂ and H₂O. The salts were prepared and stored in their Cl⁻ forms and suitably ion exchanged using amberlite ion exchange resins into the desired form before electrochemical measurements. An additional supporting electrolyte was added following ion exchange.

Viscosity Measurements.

The viscosity of all the six electrolytes was measured using Ubbelohde viscometer. The concentration of all the solutions were 1 M. All the measurements were done at 25 °C. The viscometer was vertically inserted into a constant temperature bath and charged by pouring sample in lower meniscus. Suction was applied to the tube, and efflux time in the capillary was noted. Kinematic viscosity was calculated by multiplying efflux time to viscometer constant. From kinematic viscosity, dynamic viscosity was calculated by dividing the kinematic viscosity by density of the solution. The viscosities are listed in SI Appendix, Table S3.

Electrochemical Measurements.

The synthesized BTMAP-Fc-Cl and BTMAP-Vi-Cl salts were stored in a MBraun argon-filled glove box with H₂O and O₂ levels <0.5 ppm to prevent moisture absorption. The salts were weighed and removed from the glove box prior to each experimental run, and the electrolytes were prepared in a fume hood.

The electrochemical measurements were carried out in a setup that consisted of a small-volume electrochemical cell (Pine Instruments, RRPG223) with polytetrafluoroethylene (PTFE) stoppers and openings for working, counter, and reference electrodes and gas purge inlet and exit. A 3-mm-diameter glassy carbon (GC) disk was used as the working electrode; the counter electrode consisted of a Pt mesh attached to a Pt wire, and the reference electrode was Ag/AgCl. The electrochemical measurements were performed using a Gamry Instruments Series-G single channel potentiostat. The GC was chosen as a working electrode because it is made up of a noncatalytic material in the potential window of interest and will equally reduce or promote surface adsorption of electroactive species in all the six electrolytes such that kinetic parameters can be compared.

The electrolyte’s conductivity measurements were carried out with a two-electrode cell using electrochemical impedance spectroscopy with a Solartron analytical potentiostat (1470 Cell test system) connected to a 1252A Frequency Response Analyzer. A total of 30 mL of solution was placed in a beaker with two titanium fiber felt electrodes (Fuel Cell Store) separated by a distance of 1.6 cm with an immersed area of 1.2 cm². The potentiostat was used to measure the impedance response in the frequency range 300,000 to 1,000 Hz. The high-frequency resistance was estimated from Bode plots corresponding to a phase angle of zero (SI Appendix, Fig. S5). Conductivity was calculated using the following equation:

$$\mathit{\kappa }=\frac{\mathit{l}}{\mathit{A}*\mathit{R}},$$ [1]

where κ is conductivity in Siemens per centimeter (S/cm), l is distance between electrodes, A is cross-sectional area of sample, and R is resistance.

Results and Discussion

Electrochemical Characterization.

The redox characteristics of the BTMAP-Fc and BTMAP-Vi actives were initially evaluated using cyclic voltammetry (CV) as depicted in Fig. 2. The CV characteristics (after iR correction), such as the cathodic and anodic peak potentials (E_pc and E_pa, respectively), peak separation ($\mathrm{\Delta }{E}_p$), half-wave potential ($E_{\frac{1}{2}}$), and the transfer coefficient ($\alpha$), are tabulated for both the catholyte and anolyte compositions in Tables 1 and 2, respectively. The transfer coefficient, $\alpha$, was calculated using the following equation (30):

$$\mathit{\alpha }=\frac{1.86\mathit{RT}}{\mathit{F}\left({\mathit{E}}_{\mathit{p}}-{\mathit{E}}_{\frac{\mathit{p}}{2}}\right)},$$ [2]

where R (universal gas constant) = 8.314 J ⋅ mol^-1 ⋅ K⁻¹, T is the temperature in K, F (Faraday’s constant) = 96,485 C ⋅ mol⁻¹ , and $E_{\frac{p}{2}}$ is the half-peak potential. The pH, ionic strength, and counterions were found to significantly influence the half-wave potentials (calculated as $E_{\frac{1}{2}}=(E_{pc}+E_{pa})/2$), which were taken to closely approximate the formal potentials of the redox species (10, 30). $E_{\frac{1}{2}}$ is approximated as the formal potential following Bard and coworkers (30) because at a low scan rate, the peak current ratio is approximately equal to 1 (SI Appendix, Table S6). Low pH electrolytes exhibited a greater difference between the half-wave potentials of BTMAP-Fc and BTMAP-Vi (Table 3), suggesting that ORFBs employing low pH electrolytes would exhibit higher OCV and, hence, greater peak power densities (all other overpotential contributions being equal). The magnitudes of the anodic and cathodic currents were invariably found to be higher in low-pH electrolytes, except in the case of BTMAP-Fc with methanesulfonate counterion, where the pH = 7 electrolyte exhibited higher currents. All redox species were found to exhibit >59 mV overpotential for both anodic and cathodic reactions, indicating that the reactions are not fully reversible (30). This indicated significant charge and discharge overpotentials. Thus, any increases in OCV conferred by the use of low-pH electrolytes were deemed to be advantageous because, all overpotentials being equal, the higher OCV cell will exhibit higher operating voltage. Furthermore, low-pH electrolytes were invariably found to exhibit higher conductivity compared to pH = 7 electrolytes (SI Appendix, Table S1). H₂SO₄ was found to exhibit the highest conductivity. Since the other overpotential contributions from ohmic losses and concentration polarization would be the same between systems (given the same separator and pumping rate), the choice of low-pH electrolytes is advantageous from the perspective of lowering ohmic losses.

Fig. 2.

CVs of BTMAP-Fc and BTMAP-Vi aqueous solutions with the following supporting electrolytes: H₂SO₄ (A), Na₂SO₄ (B), HCl (C), NaCl (D), CH₃SO₃H (E), and NaCH₃SO₃ (F). All CVs were measured at 225 mV ⋅ s⁻¹ and represent the average of three scans. WE: GC; CE: Pt; RE: Ag/AgCl.

Table 1.

Voltammetric properties for BTMAP-Fc containing electrolytes obtained from CVs in Fig. 2

Electrolyte	Epc (V)	Epa (V)	ΔEp (V)	E1/2 (V)	α
H2SO4	0.43	0.38	0.06	0.41	0.71
HCl	0.45	0.37	0.08	0.41	0.74
CH3SO3H	0.48	0.40	0.08	0.44	0.67
Na2SO4	0.49	0.41	0.08	0.45	0.77
NaCl	0.47	0.40	0.08	0.44	0.67
NaCH3SO3	0.51	0.41	0.10	0.46	0.38

Table 2.

Voltammetric properties for BTMAP-Vi containing electrolytes obtained from CVs in Fig. 2

Electrolyte	Epc (V)	Epa (V)	ΔEp (V)	E1/2 (V)	α
H2SO4	−0.46	−0.35	−0.11	−0.41	0.53
HCl	−0.43	−0.35	−0.07	−0.39	0.59
CH3SO3H	−0.43	−0.32	−0.11	−0.38	0.54
Na2SO4	−0.43	−0.28	−0.15	−0.36	0.31
NaCl	−0.37	−0.30	−0.07	−0.33	0.52
NaCH3SO3	−0.39	−0.27	−0.12	−0.33	0.38

Table 3.

Supporting electrolyte-dependent OCV values for BTMAP-Fc/BTMAP-Vi ORFBs

Electrolyte (1 M)	Ionic strength	E1/2 (V) (BTMAP-Fc [1 M])	E1/2 (V) (BTMAP-Vi [1 M])	OCV (V)
H2SO4	3.00	0.41	−0.41	0.82 ± 0.01
HCl	1.00	0.41	−0.39	0.80 ± 0.02
CH3SO3H	1.00	0.44	−0.38	0.82 ± 0.01
Na2SO4	3.00	0.45	−0.36	0.80 ± 0.01
NaCl	1.00	0.44	−0.33	0.77 ± 0.00
NaCH3SO3	1.00	0.46	−0.33	0.79 ± 0.01

The transfer coefficient, $\alpha$, indicates the tendency of a given redox species in its transition state to go to its oxidized or reduced form for the one-electron transfer reactions under consideration here (30). A value of $\alpha$ = 0.5 indicates that oxidation and reduction (i.e., the anodic and cathodic reactions) were equally facile. The $\alpha$ values listed in Table 1 are for the oxidation (anodic process) of BTMAP-Fc, while values in Table 2 are for the reduction (cathodic process) of BTMAP-Vi since the organic actives were synthesized in their “discharged” state. Tafel analysis using these $\alpha$ values provide an indication of the overpotentials required to drive the anodic and cathodic reactions. The Tafel equation is as follows:

$$\mathit{\eta }=\mathit{a}+\mathit{b}\hspace{0.17em}\mathbf{l}\mathbf{o}\mathbf{g}\left({\mathit{i}}_{\mathit{k}}\right),$$ [3]

where

$$\mathit{a}=\frac{2.3\mathit{RT}}{\mathit{\alpha F}}\log {\mathit{i}}_0\hspace{0.17em}\text{and}$$ [4]

$$\mathit{b}=-\left(\frac{2.3\mathit{RT}}{\mathit{\alpha F}}\right).$$ [5]

Here, i₀ is the exchange current density in milliampere per square centimetre (mA cm⁻²). The value of the Tafel slope for a one-electron transfer reaction with the $\alpha =0.5$ is 118 mV ⋅ dec⁻¹. In this ideal case, the ratio of anodic and cathodic Tafel slopes is 1 and their difference is 0. Since the overall reactions for both the catholyte and anolyte are a one-step and one-electron transfer, the sum of cathodic and anodic transfer coefficients ${\alpha }_c+{\alpha }_a=1$ (31). A higher Tafel slope value indicates that greater overpotential is required to drive a given increase in current (and hence rate of reaction). The absolute values of the anodic and cathodic Tafel slopes for all compositions of the anolyte and catholyte are tabulated in Table 4. The NaCl supported electrolytes came closest to the ideal Tafel slope ratio of one and exhibited the lowest average difference in Tafel slopes for both the anolyte and catholyte. Another promising composition (based on the same analysis) was the CH₃SO₃H-supported electrolyte. Comparing the Tafel slopes with the OCVs in Table 3, high OCV electrolytes with low Tafel slopes (hence, low overpotential) will result in high operating potentials. Thus, we predict that low-pH electrolytes with CH₃SO₃⁻ counterions will result in the lowest average charge and discharge overpotentials and the highest ORFB OCVs, thereby resulting in the highest peak power values. Having established this experimental baseline, the fundamental cation transport and heterogenous electron kinetics were examined to determine possible correlation with the solvent reorganization energy. Similar to our previous studies in nonaqueous systems, if $\lambda$ can serve as a universal nonheuristic descriptor for selecting high-performance electrolyte compositions, this would significantly reduce the amount of trial-and-error experimentation with various electrolyte compositions required to arrive at the optimum.

Table 4.

Calculated anodic and cathodic Tafel slopes for BTMAP-Fc and BTMAP-Vi electrolytes

Electrolyte	BTMAP-Fc		BTMAP-Vi
	Anodic	Cathodic	Anodic	Cathodic
	(mV ⋅ dec−1)	(mV ⋅ dec−1)	(mV ⋅ dec−1)	(mV ⋅ dec−1)
H2SO4	80.58 ± 1.09	222.35 ± 7.93	126.65 ± 1.98	110.80 ± 1.48
HCl	80.20 ± 0.79	224.77 ± 6.04	144.28 ± 3.32	100.18 ± 1.60
CH3SO3H	87.74 ± 0.93	181.06 ± 4.05	126.61 ± 1.46	110.77 ± 1.14
Na2SO4	75.10 ± 0.69	277.77 ± 9.19	87.74 ± 0.93	181.06 ± 4.05
NaCl	88.16 ± 0.62	179.08 ± 2.56	119.76 ± 1.34	116.61 ± 1.24
NaCH3SO3	162.92 ± 4.51	92.83 ± 1.41	93.76 ± 0.70	159.70 ± 2.04

Diffusion Coefficient and Hydrodynamic Radii of the Solvated Model Organic Cations.

CVs at different potential scan rates were obtained as shown in SI Appendix, Figs. S1 and S2. These voltammograms displayed the characteristic 30/α mV shift in the reduction peak position of the CV for a 10-fold increase in scan rate. Based on the peak separations tabulated in Tables 1 and 2, the Nicholson–Shain (N-S) equation (32, 33) was used to relate the peak currents to the scan rate. The relationship is as follows:

$${\mathit{i}}_{\mathit{p}}=2.99\hspace{0.17em}\mathbf{X}\hspace{0.17em}{10}^5{\mathit{n}}^{\frac{3}{2}}{\mathit{\alpha }}^{\frac{1}{2}}\mathit{AC}{\mathit{D}}^{\frac{1}{2}}{\mathit{\nu }}^{\frac{1}{2}}\hspace{0.17em},$$ [6]

where i_p (A) is the peak current, n is the number of electrons involved in the electrode reaction (n = 1), α is the charge transfer coefficient, A (squared centimeters) is the electrode area, C in mole per cubic centimeters (mol ⋅ cm⁻³) is the concentration of electroactive organic compounds, T (K) is the temperature, D (squared centimeters per second) is the diffusion coefficient of electroactive organic compounds, and $\nu$ (V ⋅ s⁻¹) is the scan rate at which the CV was recorded. The N-S plots are depicted in SI Appendix, Figs. S3 and S4. The plots were linear with square root of scan rate as expected, and their slopes were used to calculate the diffusion coefficient of electroactive organic compound. The redox reactions considered in this work do not involve any bond formation or breakage. Under such conditions, the variation in the radius of the reduced and oxidized species can be quite small (30) and results in $\frac{i_{pa}}{i_{pc}}\sim 1$ (which is the case at small scan rates for the system under consideration). Thus, the diffusion coefficient of the oxidized and reduced state is taken to be the same [following Bond and coworkers (34)], and $E_{\frac{1}{2}}=E_o+\frac{RT}{nF}\left(\frac{D_R}{D_o}\right)$ and $E_{\frac{1}{2}}\hspace{0.17em}\sim \hspace{0.17em}{E}_o$. The diffusion coefficients of the catholyte and anolyte in the various supporting electrolytes considered are depicted in Figs. 3A and 4A, respectively. The hydrodynamic radius of the solvated ions in solution is directly correlated with the mean free path and typically exhibits an inverse relationship with diffusion coefficient as per the Stokes–Einsten relation (35). According to the Stokes–Einstein equation,

$$\mathit{D}=\frac{{\mathit{k}}_{\mathit{B}}\mathit{T}}{6\mathit{\pi \eta }{\mathit{r}}_{\mathit{hyd}}},$$ [7]

where D (cm² ⋅ s⁻¹) is diffusion coefficient, k_B (1.38064852 × 10⁻²³ m² ⋅ kg ⋅ s⁻² ⋅ K⁻¹) is Boltzmann’s constant, T (298 K) is temperature, η (Pa s) is the dynamic viscosity, and r_hyd (meters) is the hydrodynamic radius.

Fig. 3.

Transport and kinetic properties of BTMAP-Fc aqueous solution in acidic electrolytes (H₂SO₄, HCl, and CH₃SO₃H) and neutral electrolytes (Na₂SO₄, NaCl, and NaCH₃SO₃). (A) Diffusion coefficient, (B) hydrodynamic radius, (C) rate constant, and (D) solvation reorganization overpotential. The error bars represent SE from at least three independent measurements. The corresponding numerical values are tabulated in SI Appendix, Table S3.

Fig. 4.

Transport and kinetic properties of BTMAP-Vi aqueous solution in acidic electrolytes (H₂SO₄, HCl, and CH₃SO₃H) and neutral electrolytes (Na₂SO₄, NaCl, and NaCH₃SO₃). (A) Diffusion coefficient, (B) hydrodynamic radius, (C) rate constant, and (D) solvation reorganization overpotential. The error bars represent SE from at least three independent measurements. The corresponding numerical values are tabulated in SI Appendix, Table S3.

The diffusion coefficient is also influenced by electrolyte composition–dependent intermolecular forces, which are macroscopically measured as the electrolyte’s viscosity. The viscosities of all the six electrolytes were measured using an Ubbelohde viscometer, listed in SI Appendix, Table S2. The viscosity increased with change in counterion in the order SO₄²⁻ > CH₃SO₃⁻ > Cl⁻. The hydrodynamic radius in different supporting electrolytes is depicted in Figs. 3B and 4B. The expectation that the hydrodynamic radius would increase in the order Cl⁻ < HSO₄⁻ < CH₃SO₃⁻ with increasing counterion radii was not borne out for either cation. The diffusion coefficient was found to exhibit an inverse trend with the hydrodynamic radii for both cations as expected from the Stokes–Einstein equation. Nevertheless, the ratios of cation D between electrolytes were not the same as ratios of r_hyd, indicating the influence of the viscosity. Also, while the D in acidic and neutral media were of the same order of magnitude for BTMAP-Fc, D decreased by an order of magnitude for BTMAP-Vi when transitioning from acidic to neutral media. The decrease in D resulted in a concomitant increase in r_hyd. This is attributed to greater solvation by polar water molecules of the BTMAP-Vi (with four N⁺ charge centers) in neutral media. We attribute the lack of a similar effect in case of BTMAP-Fc to the $\delta$⁺ charges from the two N⁺ charge centers being restricted to one end of the cation as opposed to the more uniform distribution in BTMAP-Vi. Given this complex interplay of factors, the organic cation diffusion coefficient cannot be readily increased by just choosing counterions of smaller ionic radii. There is therefore a need to experimentally determine D for new organic active–based electrolyte compositions at present.

Electrochemical Kinetics and the Solvation Overpotential.

The rate constant was calculated using the Kochi and Klingler method (36). A transient method (CV) was employed to study this system given fears of possible side reactions. BTMAP-Vi electrolyte undergoes dimerization leading to the formation of a charged transfer complex (37–39), and the ferrocene or ferrocenium may adsorb on the electrode and poison it (40, 41). We have previously screened electrolytes by applying M-H theory to steady-state voltammetry data when such side-reaction considerations were not warranted (29). Kochi and Klingler evaluated the standard rate constant [k (cm ⋅ s⁻¹)] from CV curves by means of the relationship:

$$\mathit{k}=2.18{\left(\frac{\mathit{\alpha vnFD}}{\mathit{RT}}\right)}^{0.5}\exp \left(-\frac{{\mathit{\alpha }}^2\mathit{nF}\mathrm{\Delta }{\mathit{E}}_{\mathit{P}}}{\mathit{RT}}\right),$$ [8]

where α is the transfer coefficient for the redox reaction (listed in Tables 1 and 2), and ΔE_p (V) is the peak-to-peak separation of the anodic and cathodic waves. CV measurements on both reactants gave an oxidation rate constant of order 10⁻¹ cm ⋅ s⁻¹ for BTMAP-Fc and a reduction rate constant of order 10⁻² cm ⋅ s⁻¹ for BTMAP-Vi in various supporting electrolytes. Thus, BTMAP-Vi was identified as the rate-limiting side. These reactions are much faster than those of common inorganic species and are also faster than those of most other organic or organometallic reactants that have been used in RFBs, as tabulated in SI Appendix, Table S4. These measurements utilized a noncatalytic GC electrode to obtain a baseline rate constant value. Based on the rate-limiting electrode reaction, recommendations can then be made for the use of catalysts. We deliberately avoided the use of a catalytic electrode, as we have previously found that catalytic electrodes may actually be detrimental to outer-sphere electron transfer processes and lead to incorrect identification of the rate-limiting electrode reaction (28). Given the high-rate constant values for BTMAP-Fc and BTMAP-Vi redox reactions compared to typical RFB electrodes, the use of catalysts may not be warranted at either electrode. BTMAP-Vi being rate limiting suggests (at first glance) that the HCl-supporting electrolyte, which results in the highest rate constant for BTMAP-Vi, should be chosen for the overall system. However, the overall combination of bulky cations and fast first electron transfer kinetics results in BTMAP-FC/BTMAP-Vi ORFBs being mass transport limited under some conditions. Thus, a judicious optimum of D and k is needed. Given the counterintuitive trends in D and fast first electron transfer to the solvated cations, we examined the solvation energy for further insights.

The solvent reorganization energy ($\lambda$), which is the amount of energy required to change the solvation shell of the reactants to that of the product during an electrochemical reaction, plays a prominent part in the M-H kinetic formulation (42). There are two contributions to λ, one from the changes in the intramolecular vibrations (the inner-sphere reorganization energy, λ_i) and the second from the changes in solvent orientation coordinates (the outer sphere reorganization energy, l_o) (30, 43–45):

$$\mathit{\lambda }=\hspace{0.17em}{\mathit{\lambda }}_{\mathit{i}}+\hspace{0.17em}{\mathit{\lambda }}_{\mathit{o}}.$$ [9]

The inner component is described by summing over the normal vibrational modes of the species using molecular theory:

$${\mathit{\lambda }}_{\mathit{i}}=\frac{1}{2}{\displaystyle \sum }_{\mathit{j}}{\mathit{k}}_{\mathit{j}}{\left({\mathit{q}}_{\mathit{R}}-{\mathit{q}}_{\mathit{O}}\right)}_{\mathit{j}}^2,$$ [10]

where k_j is the force constant for each oscillator, and q is the displacement in the normal mode coordinates.

The outer sphere reorganization energy is usually approximated by modeling the reactants and products as spheres and the solvent as a dielectric continuum (Born theory), as seen in the work of Bard and Faulkner (30), Compton et al. (43, 44, 46), Savéant and Costentin (47), Marcus (43, 46, 48), and Chidsey (43, 48). The Born equation (49) can be used for estimating the electrostatic component of Gibbs free energy of solvation of an ion and was used to estimate ${\lambda }_o$. It is an electrostatic model that treats the solvent as a continuous dielectric medium and is as follows:

$${\mathit{\lambda }}_{\mathit{o}}=\frac{{\mathit{e}}^2}{8\mathit{\pi }{\mathit{\epsilon }}_{\mathit{o}}}\left(\frac{1}{{\mathit{a}}_{\mathit{o}}}-\frac{1}{2\mathit{d}}\right)\left(\frac{1}{{\mathit{\epsilon }}_{\mathit{op}}}-\frac{1}{{\mathit{\epsilon }}_{\mathit{r}}}\right),$$ [11]

where e is the elementary charge, ε_o is the permittivity of free space, a_o is the radius of the reactant, ε_op is optical permittivity equal to the square of refractive index of solvent, ε_r is relative permittivity, and d is the distance from the reactant surface to the metal surface.

As, d > a_o,

$${\mathit{\lambda }}_{\mathit{o}}=\frac{{\mathit{e}}^2}{8\mathit{\pi }{\mathit{a}}_{\mathit{o}}{\mathit{\epsilon }}_{\mathit{o}}}\left(\frac{1}{{\mathit{\epsilon }}_{\mathit{op}}}-\frac{1}{{\mathit{\epsilon }}_{\mathit{r}}}\right).$$ [12]

When the oxidizing and reducing species are similar in structure, internal reorganizational energy is smaller than λ_o. Therefore, we can use λ_o as an estimate for reorganizational energy.

$$\mathit{\lambda }={\mathit{\lambda }}_{\mathit{o}}=\hspace{0.17em}\mathrm{\Delta }\mathit{G}=\hspace{0.17em}-\frac{N_A{z}^2{e}^2}{8\pi r_{hyd}{\epsilon }_o}\left(\frac{1}{{\epsilon }_{op}}-\frac{1}{{\epsilon }_r}\right),$$ [13]

where N_A (6.02214076 × 10²³ mol⁻¹) is Avogadro’s constant, z is charge of the ion, e (1.6022 × 10⁻¹⁹ C) is elementary charge, ε₀ (8.8541878128 × 10⁻¹² F ⋅ m⁻¹) is permittivity of free space, r_hyd is effective radius of ion (i.e., hydrodynamic radius), ε_op (1.775) is optical permittivity, and ε_r (78.4) is relative permittivity of the solvent (i.e., water). λ₀ encompasses both the ease of solvation of a given species and the ease of oxidizing or reducing it by calculating |λ_o/F|, the solvation reorganization overpotential required to rearrange or break the solvation shell of a given reactant. The |λ_o/F| values for BTMAP-Fc and BTMAP-Vi are provided in Figs. 3D and4D, respectively. Assuming all other overpotential contributions are equal for a given cation in various supporting electrolytes, |λ_o/F| will then determine the overall overpotential and, hence, the voltage efficiency of a full ORFB cell. The effective radii of the cationic complexes were understood to be a function of both the radii of the counterions and also their charge density. Thus, a nominally larger counterion with a high charge density results in an overall smaller complex by reducing the distance between the charge centers. |λ₀/F| can also be interpreted to indicate the occurrence of an inner-sphere versus outer-sphere electron transfer process as we have detailed elsewhere (28). Given the large |λ_o/F| values (greater than even the $\mathrm{\Delta }{E}_p$) for all electrolyte compositions, outer-sphere electron transfer was indicated in all cases. Given the complex interplay between D, k, and |λ_o/F|, we elected to simultaneously optimize for both D and k by examining the impact of |λ_o/F| on the electrochemical Thiele modulus (29).

Electrochemical Thiele Modulus and Ion Solvation.

The transport and kinetic parameters of the various electrolytes considered (depicted in Figs. 3 and 4) were considered in totality by using the electrochemical Thiele modulus (h_t) with hydrodynamic radius as the characteristic length (l) (29). The derivation of h_t can be found in Sankarasubramanian et al. (29) The value of k (as calculated using the Kochi–Klingler equation) was not mass transport limited and hence satisfied the key boundary condition [from the derivation of the h_t relation in our previous work (29)] that the reactant concentration at the surface is finite. The electrochemical Thiele modulus involves a potential dependent k (unlike the constant k in the typical isothermal reaction engineering case) and is related to ${\lambda }_o$ as follows:

$${\left({\mathit{h}}_{\mathit{t}}\right)}^2=\frac{\mathit{k}{\mathit{r}}_{\mathit{hyd}}}{\mathit{D}}=\frac{{\mathit{i}}_{\mathit{o}}{\mathit{r}}_{\mathit{hyd}}{\mathit{e}}^{-4\mathit{RT}\mathbf{\eta }\left(\frac{{\mathbf{\lambda }}_{\mathbf{o}}}{\mathbf{F}}\right)}}{\mathbf{n}\mathbf{F}{\mathbf{C}}_{\mathbf{i}\mathit{bulk}}\mathit{D}},$$ [14]

where $\eta$ (V) is the applied overpotential and C_ibulk in mole per cubic meters (mol ⋅ m⁻³) is the concentration of the electroactive species in the electrolyte bulk. The transport dependencies and the overpotential drive force are captured in the groupings A and B which are as follows:

$$\mathit{A}=\hspace{0.17em}\frac{{\mathit{i}}_{\mathit{o}}{\mathit{r}}_{\mathit{hyd}}}{\mathbf{n}\mathbf{F}{\mathbf{C}}_{\mathbf{i}\mathit{bulk}}{\mathit{D}}_{\mathit{i}}},$$ [15]

$$\mathit{B}=\hspace{0.17em}-4\mathit{RT}\mathbf{\eta }.$$ [16]

The A and B values are listed in SI Appendix, Table S5. A increased with cation size as expected based on the correlation between D and r_hyd. B also increased with cation size as expected based on the correlation between |λ_o/F| and r_hyd. This served as validation of the applicability of this model to the present case. Plots of ln(h_T²) versus |λ_o/F| are depicted in Fig. 5 and exhibited a linear relationship (within experimental error), as expected. To utilize the Thiele modulus to comment on the relative importance of transport versus kinetics in a given system, it is useful to consider the effectiveness factor (${\eta }_{eff}$; not to be confused with the overpotential, $\eta$), which is defined as the ratio of the rate of a reaction when reactant availability is diffusion limited to the rate of reaction at maximum (bulk electrolyte) concentration. The one-dimensional model (29) applied herein allows us to calculate ${\eta }_{eff}$ using the relationship for a slab geometry and, hence, ${\eta }_{eff}=\tanh \left(h_t\right)/h_t$ (50). A limiting case when h_t is small is that η_eff → 1. All the Thiele moduli in our system are in the order of 10⁻² (SI Appendix, Table S3), which would result in η_eff → 1, indicating that the redox reactions of both BTMAP-Fc and BTMAP-Vi are occurring at close to their maximum possible reaction rates (i.e., at rates we would expect if the near electrode reactant concentration was close to the bulk concentration). Thus, we predict that increases in the electrolyte flowrates will have minimal influence on the overall power output from the ORFB. The |λ_o/F| values were found to numerically correlate with the OCV values listed in Table 3. Based on this correlation and Eq. 14, we recommend that (in the absence of experimental data) new electrolyte compositions be chosen based on higher absolute solvation energy calculated from the Born equation (Eq. 13). This will result in higher h_T, which, in turn, will result in higher currents at the cell level. For the vast majority of electrochemical reactions, given that the r_hyd will be in the order of 10⁻⁹ m, h_T << 1, resulting in ${\text{η}}_{\text{eff}}\to 1$.

Fig. 5.

Correlation between solvated reorganization overpotential (|λ_ο/F|) and the electrochemical Thiele Modulus (incorporating the diffusion coefficient, electron rate constant, and hydrodynamic radius) for (A) BTMAP-Fc and (B) BTMAP-Vi with various counterions in acidic and neutral electrolytes.

Conclusions

The choice of supporting electrolyte greatly influences the transport and kinetic properties in the anolyte and catholyte of an RFB. In this work, we develop a nonheuristic method to select aqueous supporting electrolyte compositions (pH and counterion) for organic redox actives using the solvent reorganization energy (${\lambda }_{\text{o}}$) from M-H kinetic theory as a universal descriptor. Overall, solvent reorganizational energy and hence the electrolyte contribution to the overpotential (i.e., solvation overpotential |λ_o/F|) are found to be inversely correlated with the natural logarithm of the square of the electrochemical Thiele modulus [${\left(h_t\right)}^2=\frac{k{r}_{hyd}}{D}$)]. Based on our analysis, we conclude the following:

• 1)Effectiveness factor ${\eta }_{eff}\to 1$ for both the BTMAP-Fc and BTMAP-Vi redox couples indicates that electrolyte flowrate or optimization for better mass transport would result in minimal ORFB performance improvements.
• 2)Given the exceptionally high first electron transfer rate constants, even on GC, the introduction of catalysts at the ORFB electrodes is not necessary.
• 3)Both redox couples exhibit outer-sphere electron transfer mechanisms.
• 4)Barring possible side reactions, we recommend the use of low-pH or CH₃SO₃H- or HCl-supported electrolytes to improve the energy efficiency of the BTMAP-Fc/BTMAP-Vi ORFB.

The key finding is that the solvation energies calculated from the Born equation fit into our electrochemical Thiele modulus framework. This has the great advantage of allowing for the initial high-throughput screening of electrolyte candidates with minimal experimentation.

Data Availability

All study data are included in the article and/or SI Appendix.