We report viscous flow properties of a redox-active organic molecule, *N*-(2-(2-methoxyethoxy)ethyl)phenothiazine (MEEPT), a candidate for non-aqueous redox flow batteries, and two of its radical cation salts. A microfluidic viscometer enabled the use of small sample volumes in determining viscosity as a function of shear rate and concentration in the non-aqueous solvent, acetonitrile, both with and without supporting salts. All solutions tested show Newtonian behavior over shear rates of up to 30 000 s^{−1}, which was rationalized by scaling arguments for the diffusion-based relaxation time of a single MEEPT molecule without aggregation. Neat MEEPT is flowable but with a large viscosity (412 $mPa\u22c5s$ at room temperature), which is ∼1000 times larger than that of acetonitrile. MEEPT solutions in acetonitrile have low viscosities; at concentrations up to 0.5 M, the viscosity increases by less than a factor of two. From concentration-dependent viscosity measurements, molecular information was inferred from intrinsic viscosity (hydrodynamic diameter) and the Huggins coefficient (interactions). Model fit credibility was assessed using the Bayesian Information Criterion. It is found that the MEEPT and its charged cations are “flowable” and do not flocculate at concentrations up to 0.5 M. MEEPT has a hydrodynamic diameter of around 8.5 Å, which is almost insensitive to supporting salt and state of charge. This size is comparable to molecular dimensions of single molecules obtained from optimized structures using density functional theory calculations. The results suggest that MEEPT is a promising candidate for redox flow batteries in terms of its viscous flow properties.

## I. INTRODUCTION

Non-aqueous redox flow batteries (NAqRFBs) utilizing solutions of redox-active organic molecules (ROMs) are competitive for reliable electrochemical energy storage systems due to their scalable energy capacity, large electrochemical stability windows, and potentially long operating lifetimes.^{1–6} In NAqRFBs, charges are stored in ROMs and transported by ionic supporting salts, both dissolved in an organic solvent. From a techno-economic study,^{2,7} the concentration of active materials must be larger than 1 M for NAqRFBs to be competitive, with a target concentration of 5 M. Previous research has shown that highly concentrated electrolytes can result in dramatically increased viscosities.^{8} The performance of RFBs is heavily dependent on the viscosity of electrolytes, with higher viscosities having a negative influence on key transport properties,^{9} such as ionic conductivity^{10} and diffusivity.^{11,12} Viscosity is also directly related to pumping costs.^{13} Bindner *et al.* reported 8% to 11% of total power loss from pumping for a vanadium RFB.^{14} Understanding the origin of viscosity differences in these complex fluids is therefore of interest in the development of solutions with favorable properties.

Studies evaluating the viscous flow properties of electrolytes for RFBs show that the concentration-dependent viscosities of electrolytes can be affected by ROM size^{15,16} and its state of charge,^{17,18} solution temperature,^{19} inclusion of additives,^{20} and ionic strength of supporting salts.^{21} Great efforts have been made in predicting electrolyte viscosity, e.g., models based on Eyring’s absolute theory,^{22} transition-state theory,^{23–25} and the Advanced Electrolyte Model (AEM).^{9,26} However, no universal method accounts for all factors that affect the viscosity, and experimental calibration is needed for some model parameters. Furthermore, few publications discuss non-Newtonian behavior of ROM electrolytes, which might arise from flow-induced conformation of ROMs^{27} and the break-up of interactions between them.^{28} Non-Newtonian analysis matters in non-equilibrium molecular dynamics (NEMD) simulations^{29–33} as the NEMD simulation may study flows in the large Weissenberg number (Wi) regime (very high shear rates), where viscosities show non-Newtonian behavior. For instance, Zhang *et al.*^{34} used NEMD to calculate the shear viscosity of imidazolium-based ionic liquids at shear rates in the range 10^{7}–10^{9} s^{−1}, where shear-thinning was observed. Therefore, to optimize the operating condition of RFBs, experimental viscous flow properties of ROM solutions must be considered.

In this study, the viscous flow properties of *N*-(2-(2-methoxyethoxy)ethyl)phenothiazine (MEEPT) were studied using a microfluidic viscometer [Fig. 1(a)]. This enables viscosity measurement using small sample volumes and a wide range of shear rates around two orders of magnitude. MEEPT is a ROM that shows promise for meeting grid-scale energy storage requirements, as evidenced by its high current density and long duration cycling.^{35} However, its viscosity and other transport properties, such as diffusivity and ionic conductivity, have not been reported. Here, we measured the concentration-dependent viscosities of neutral MEEPT, its tetrafluoroborate radical-cation salt (MEEPT-BF_{4}), and its bis(trifluoromethanesulfonyl)imide salt (MEEPT-TFSI) at different concentrations in acetonitrile (ACN) and in ACN-based electrolytes containing different supporting salts, namely, tetraethylammonium tetrafluoroborate (TEABF_{4}) and lithium bis(trifluoromethanesulfonyl)imide (LiTFSI). For concentration-dependent viscosities, the Einstein,^{36} Huggins,^{37} and extended Jones–Dole equations^{38} were used to fit and understand the measured viscosities. From fit results, molecular information, such as the intrinsic viscosity, hydrodynamic diameter, Huggins coefficient, and presence of ion interactions can be determined. The residual sums of squares (RSS) was used and the Bayesian Information Criteria (BIC) of the fits were compared to determine the most credible fit. A simple estimate of the relaxation time scale of MEEPT was used to understand the observed Newtonian behavior of MEEPT solutions.

The hydrodynamic diameter of MEEPT remains essentially unchanged with the addition of supporting salts and is comparable to the molecular structure dimensions, which suggests minimal flocculation and small solvation shells for MEEPT in supporting salt environments. The Huggins coefficients are of the same order of magnitude but larger than that of Brownian hard spheres in shear flow as derived by Batchelor,^{39} suggesting that interactions other than hydrodynamic interactions and steric effects contribute to viscous dissipation. Our results ultimately show that (i) both neutral and charged MEEPT are “flowable” ROMs, and their solution viscosities increase with concentration as a result of excluded volume and interactions between ROMs; (ii) addition of supporting salts does not have a dramatic influence on the hydrodynamic diameter of MEEPT, but it does affect the interactions between MEEPT molecules; (iii) the viscosities increase in the charged form of MEEPT, which arises from the volume excluded by corresponding anions; and (iv) there is minimal flocculation within the concentrations and shear rates tested, and the critical shear rate, the rate at which the solution begins to show shear-thinning, is larger than limits experimentally accessed.

## II. MATERIALS AND METHODS

### A. Materials

The redox-active molecule MEEPT ($>$ 98%, 301.40 g/mol) was obtained from Tokyo Chemicals, Inc., and its radical cation salts, MEEPT-BF_{4}^{35} and MEEPT-TFSI,^{40} were synthesized as described in the supplementary material. Two supporting salts, LiTFSI (Aldrich, 99.95%) and TEABF_{4} (Aldrich, 99%), were chosen because these are both commonly used in RFBs and they affect the solubility of charged MEEPT differently, as shown in Table I. All ROMs and salts were dissolved in acetonitrile (ACN, Aldrich, 99.8%) to reduce ion associations as ACN is one of the most polar organic solvent with a wide electrochemical stability window^{41,42} and, therefore, commonly employed in NAqRFBs.

. | . | . | Maximum concentration . |
---|---|---|---|

ROM . | Electrolyte . | Solubility (M) . | tested in this study (M) . |

MEEPT | ACN | Miscible | 3.78 |

0.5 M TEABF_{4}/ACN | Miscible | 1.00 | |

0.5 M LiTFSI/ACN | Miscible | 1.00 | |

MEEPT-BF_{4} | ACN | 0.55 | 0.45 |

0.5 M TEABF_{4}/ACN | 0.45 | 0.36 | |

0.5 M LiTFSI/ACN | 0.81 | 0.43 | |

MEEPT-TFSI | ACN | 1.54 | 1.00 |

0.5 M TEABF_{4}/ACN | 1.11 | 1.00 | |

0.5 M LiTFSI/ACN | 1.18 | 1.00 |

. | . | . | Maximum concentration . |
---|---|---|---|

ROM . | Electrolyte . | Solubility (M) . | tested in this study (M) . |

MEEPT | ACN | Miscible | 3.78 |

0.5 M TEABF_{4}/ACN | Miscible | 1.00 | |

0.5 M LiTFSI/ACN | Miscible | 1.00 | |

MEEPT-BF_{4} | ACN | 0.55 | 0.45 |

0.5 M TEABF_{4}/ACN | 0.45 | 0.36 | |

0.5 M LiTFSI/ACN | 0.81 | 0.43 | |

MEEPT-TFSI | ACN | 1.54 | 1.00 |

0.5 M TEABF_{4}/ACN | 1.11 | 1.00 | |

0.5 M LiTFSI/ACN | 1.18 | 1.00 |

### B. Viscometry

The dynamic viscosity of solutions was measured using a microfluidic viscometer m-VROC (RheoSense, Inc.). Only microliter sample volumes are required by this viscometer, and free surface effects,^{43} evaporation, and contamination can be avoided during the measurement due to internal flow. The measuring chip is made from borosilicate glass containing a rectangular slit flow channel with a uniform cross section, as shown in Fig. 1(a).^{44} The fluid sample is pushed by a syringe pump at a constant volume flow rate, *Q*, which is related to the apparent shear rate, $\gamma \u0307app$, by^{27}

where *w* and *h* are the width and the height of the channel, respectively (2 mm × 50 *µ*m). The actual shear rate at the wall, $\gamma \u0307$, is related to $\gamma \u0307app$ as^{27}

For Newtonian fluids and laminar flow, the shear stress, *τ*, is linearly proportional to $\gamma \u0307app$, so $\gamma \u0307=\gamma \u0307app$. For non-Newtonian fluids, the shear rate needs to be modified using Eq. (2) to calculate the true viscosity. Four pressure sensors are mounted at the boundary wall in the channel to detect the pressure drop from the inlet to the outlet as a function of position along the channel, and *τ* is calculated from the pressure drop as^{27}

where *∆p* is the measured pressure drop and *L* is the channel length over which *∆p* is measured (15 mm). Two different microchips were used, each with the same dimensions but different pressure measurement ranges, having maximum measurable pressure drops, *∆p*_{max}, of 10 000 and 40 000 Pa, respectively, and with a minimum measurable pressure drop of 1% of the maximum. These limits, together with the maximum flow rate, bound the range of measurable viscosity vs shear rate. The dynamic viscosity is defined as^{45}

Experimental limits constrain the range of $\gamma \u0307$ available to explore non-Newtonian behavior, which are converted to experimental windows,^{43} as shown in the viscosity vs shear rate plots. For each solution, viscosities were measured within a wide shear rate range (5000 s^{−1}–30 000 s^{−1}) for most samples, which varied depending on the viscosity of each solution. This corresponds to a relevant range for RFBs, which can be estimated from conditions in the literature. For instance, laboratory-scale RFBs^{46,47} may have channel diameters, 2*R*, as small as 1 mm –3 mm with a flow rate, *Q*, of 10 mL/min–100 mL/min. From the Poiseuille flow of a Newtonian fluid in a circular tube, the wall shear rate is $\gamma \u0307=4\pi QR3$, giving an upper-bound estimate of the shear rate around 100 s^{−1}–20 000 s^{−1}. The temperature was maintained at 25 °C using a ThermoCube circulator. For each sample, repeat measurements were done in triplicate with the same solution including separate syringe loading (except when the sample volume was limited for which only one or two measurements were possible), and the uncertainty was calculated as the standard deviation.

### C. Viscosity models

To describe the concentration-dependent viscosity, the equations used are listed here. For solutions containing uncharged particles, Einstein formulated an equation to theoretically calculate the viscosity of dilute molecular solutions of rigid, non-attracting spheres at different concentrations^{36} as

where *η*(*ϕ*) and *η*_{s} are the viscosities of the solution and solvent, respectively, and *ϕ* is the volume fraction of solute particles. For spherical particles, *ϕ* can be written as

where *d*_{H} is the hydrodynamic diameter of an individual particle, *c* is the concentration (moles per volume), and *N*_{A} is Avogadro’s constant. Thus, the Einstein equation can be written in terms of concentration as

where [*η*] is the intrinsic viscosity, which is a function of only *d*_{H}, such that

Experimentally, Huggins expanded this equation to a more concentrated regime, with a higher-order term representing the particle interactions,^{37}

which is called the Huggins equation, where *k*_{H} is the Huggins coefficient that quantifies the interactions between particles. For Brownian hard spheres with only hydrodynamic interactions and steric effects, Batchelor numerically derived the second-order coefficient^{39} to be

which corresponds to *k*_{H} = 0.992.

For salt solutions, the viscosity is most commonly described by the Jones–Dole equation,^{38} which is

This can be extended to higher concentrations as

where *A*, *B*, and *D* are fit coefficients and $Ac$ represents the interactions and mobility of ions, which can be calculated using the Falkenhagen theory.^{48,49} Here, the term *Bc* represents ion–solvent interactions, and *Dc*^{2} represents ion–dipole interactions and long-range Coulombic ion–ion interactions.

Note that according to these models, as the concentration of the solute increases, the viscosity of the solution increases, first linearly and then much more dramatically. The high viscosity can result in high pumping costs and reduced conductivity and therefore lower the efficiency of RFBs. For our purposes, we use the term “flowable” to mean that the fluid viscosity is sufficiently low to be within the range of viscosity of known RFB working fluids. For example, viscosities up to 1 $Pa\u22c5s$ have been used in working RFBs.^{50} Although there is no strict upper limit to the maximum possible viscosity in a RFB, since the system-level architecture could possibly be changed to accommodate higher viscosities, for practical purposes, we treat as “flowable” any RFB solution that has viscosity lower than 1 $Pa\u22c5s$.

In this study, the Einstein, Huggins, and extended Jones–Dole equations were used to describe the measured viscosities. Molecular information, such as the intrinsic viscosity, hydrodynamic diameter, Huggins coefficient, and presence of ion interactions, were determined from fit parameters. For comparing the hydrodynamic diameters, the representative geometries were optimized using Density Functional Theory (DFT) (see details in the supplementary material). As the parameters are highly sensitive to small uncertainties in measurement, great attention was paid to the fit procedure and the most credible fit was determined from the Bayesian Information Criterion (BIC), which is introduced in Sec. II D.

### D. Fit method and credibility

The fit and model selection were done using the “fitnlm” function in MATLAB,^{51} a commonly employed nonlinear regression model that uses an iterative generalized least squares algorithm, which in its most general form minimizes the residual sum of squares (RSS).^{52} The weighting function was specified as the experimental uncertainty (standard deviation) of each measured data point, which propagates the uncertainty in measurement to the fit. That is,

is minimized to determine the fit parameters. The data distribution and data density are important but subtle challenges to any data fit for the purpose of inference. In our case, we have higher data density at lower concentrations. Fitting to this uneven data distribution may emphasize the low concentration measurements, but we note that this is also the region most applicable to the theories described in Sec. II C.

To evaluate the hydrodynamic diameter and quantify the interaction of molecules in solution, the intrinsic viscosity and Huggins coefficient can be fit to the measured viscosities at given concentrations. However, the intrinsic viscosity is defined in the limit of *c* → 0,^{53} while all the measurements are done at finite concentrations. This is a known issue, and various approaches have been used to resolve this conflict, such as the single point method and extrapolation.^{54} Here, the extrapolation method was employed by rearranging the Huggins equation as

where *η*_{red} is called the reduced viscosity. The reduced viscosity vs concentration is fit to a linear function, and the intrinsic viscosity is evaluated as the intercept at zero concentration. Since the viscosity contributed by the solvent must be subtracted before extrapolation, small experimental errors can be amplified,^{55} and this is addressed by including a weighting function in the fit based on propagated uncertainty of the data so that the uncertainty in concentration is accounted for in the reduced viscosity, which is used in model fit and selection. The most credible fit is found by comparing the Bayesian Information Criterion (BIC) for each fit.

BIC is a criterion for model selection, which takes both the goodness of fit and the penalty term for the number of parameters in the model into account. It is an approximation of the full Bayes factors used to assess model credibility. While the full Bayesian calculation has been applied previously to complex fluid rheology,^{56} the BIC approximation is much less computationally involved. It is calculated as

where ln *L* is the log-likelihood, which represents the goodness of fit; *n* is the number of data points used for fit; and *k* is the number of parameters in the model. For linear regression, the log-likelihood function can be written in terms of the RSS as^{57}

The BIC is then calculated by

The second term on the right-hand side is the penalty term accounting for over-parameterizing. Lower BIC corresponds to better credibility. The BIC increases in magnitude with *n*, which only improves the credibility for sufficiently negative values of the square-bracket term, i.e., for sufficiently small RSS/*n*.

The viscosities of solutions of MEEPT and its cations were measured and analyzed using the methods mentioned above, and the results are discussed next.

## III. RESULTS AND DISCUSSION

### A. MEEPT: A flowable ROM

A wide range of concentrations of MEEPT/ACN solutions and neat MEEPT were studied. Neat MEEPT is a yellow, oily liquid, as shown in the photograph inset of Fig. 2(a). It is miscible with ACN in any proportion. The viscosities of neat MEEPT and MEEPT/ACN solutions were measured over a wide range of shear rates, and Newtonian viscosities were observed for all samples within 5% variation, as shown in Fig. 2(a). The error bars come from the standard deviation for repeat measurements.

The average Newtonian viscosity at each concentration was calculated by taking an average over viscosities at different shear rates; the error bars come from both the standard deviation and the uncertainty propagation, as shown in Fig. 2(b). The viscosity of neat MEEPT is 412 $mPa\u2009\u22c5\u2009s$, 1000 times larger than that of ACN. Therefore, as MEEPT concentration increases, the viscosity of MEEPT/ACN solutions increases, first linearly in the dilute regime and then more dramatically when the concentration becomes larger than 1 M. Note that the viscosity is plotted on a log scale, and it increases over three orders of magnitude. The trend follows a viscosity mixing law with a single coupling term,^{58} as detailed in the supplementary material.

### B. Viscosity of two supporting salt electrolytes

In RFB electrolytes, supporting salts are added to raise ionic conductivity and to provide counterions for charged ROMs. In this paper, the viscosities of two different supporting salt electrolytes, TEABF_{4}/ACN and LiTFSI/ACN, were measured at different concentrations at shear rates of 5000 s^{−1}–35 000 s^{−1}, as shown in Fig. 3. Both the TEABF_{4}/ACN and LiTFSI/ACN solutions show Newtonian behavior, within 5% variation.

The average Newtonian viscosities for both supporting salts were obtained by taking an average of viscosities, with error bars coming from standard deviation and uncertainty propagation, as shown in Fig. 4(a). The viscosity of the salts in ACN solutions increases faster for LiTFSI than for TEABF_{4}. From Fig. 4(b), it can be seen that unlike MEEPT, the reduced viscosities of TEABF_{4} and LiTFSI decrease at first and then increase linearly, possibly suggesting that the hydrodynamic diameters decrease at first, which warrants including a square root term in the viscosity model. In the Jones–Dole equation, the $Ac$ term represents the interaction and mobility of ions and this term becomes less influential as concentration increases. Therefore, for salt solutions, the Jones–Dole and the extended Jones–Dole equations are fitted, where the BICs of the extended Jones–Dole equation fits are found to be lower for both salts (see details in the supplementary material). The solid lines in Fig. 4 are the extended Jones–Dole equation fits, and the dashed lines in Fig. 4(b) are the extended Jones–Dole equation fit without the $Ac$ term. It can be seen that the two lines differ in the low concentration regime and then converge as the concentration increases. The *B* coefficient is interpreted as an intrinsic viscosity, and the hydrodynamic diameters and the Huggins coefficients are evaluated from the *B* and *D* coefficients, respectively, as shown in Table II. The hydrodynamic diameters of TEABF_{4} and LiTFSI are calculated to be 8.06 ± 0.77 and 9.48 ± 0.73 Å, respectively.

. | c_{max} (M)
. | A (L^{1/2}/mol^{1/2})
. | B (L/mol)
. | D (L^{2}/mol^{2})
. | d_{H} (Å)
. | k_{H}
. |
---|---|---|---|---|---|---|

TEABF_{4} | 0.53 | 0.08 ± 0.04 | 0.41 ± 0.12 | 0.25 ± 0.14 | 8.06 ± 0.77 | 1.49 ± 0.94 |

LiTFSI | 0.48 | 0.06 ± 0.05 | 0.67 ± 0.15 | 0.33 ± 0.22 | 9.48 ± 0.73 | 0.72 ± 0.52 |

. | c_{max} (M)
. | A (L^{1/2}/mol^{1/2})
. | B (L/mol)
. | D (L^{2}/mol^{2})
. | d_{H} (Å)
. | k_{H}
. |
---|---|---|---|---|---|---|

TEABF_{4} | 0.53 | 0.08 ± 0.04 | 0.41 ± 0.12 | 0.25 ± 0.14 | 8.06 ± 0.77 | 1.49 ± 0.94 |

LiTFSI | 0.48 | 0.06 ± 0.05 | 0.67 ± 0.15 | 0.33 ± 0.22 | 9.48 ± 0.73 | 0.72 ± 0.52 |

To study the viscosities of MEEPT and charged MEEPT cations in different supporting salt environments, the ROMs are dissolved in electrolytes containing 0.5 M supporting salts. The viscosity of the electrolytes of 0.5 M TEABF_{4} is the measured viscosity of TEABF_{4} in ACN; the same holds for 0.5 M LiTFSI. At a concentration of 0.5 M, the viscosity of the TEABF_{4}/ACN solution is 0.45 $mPa\u22c5s$, while that of the LiTFSI/ACN solution is 0.50 $mPa\u22c5s$.

### C. MEEPT in different supporting salt environments

For MEEPT in ACN electrolytes containing supporting salts, the conformation and interactions between MEEPT molecules might change, which affects the viscosities of MEEPT solutions. Thus, viscosities of MEEPT in different supporting salt environments are studied. The measurement results, which show Newtonian behavior within 3% variation, are shown in Fig. 5.

Similarly, the concentration-dependent average Newtonian viscosities for both solutions were calculated and plotted in comparison with the MEEPT/ACN solution, as shown in Fig. 6(a). At zero MEEPT concentration, the viscosities of the two electrolytes and the solvent are different, with 0.5 M LiTFSI/ACN > 0.5 M TEABF_{4}/ACN > ACN without supporting salt, but the increasing slopes are similar. To quantify the effect of supporting salts on the increase in viscosity vs MEEPT concentration, the reduced viscosity is plotted as a function of concentration (0.1 M–1.0 M), as shown by the data points in Fig. 6(b), and the solid lines are fits to the rearranged Huggins equation [Eq. (14)].

In the Huggins equation [Eq. (9)], only two factors are taken into account, namely, the excluded volume described by the linear term, [*η*]*c*, and the interactions between particles, represented by the quadratic term, *k*_{H}([*η*]*c*)^{2}. If higher order terms can be neglected, the reduced viscosity, *η*_{red}, as shown in the rearranged Huggins equation [Eq. (14)], is a linear function of *c*. First, to determine the critical concentration at which the higher order terms become important and the Huggins equation no longer holds, the most credible fit in terms of concentration range for the Huggins equation was found. This was done by fitting the reduced viscosity vs concentration data to the rearranged Huggins equation [Eq. (14)] with different numbers of data points. This was done by changing the maximum concentration used to fit, that is, three points from 0.1 M to 0.3 M were fit to the equation, then four points from 0.1 M to 0.4 M, and so on, until the maximum concentration tested was used. The BIC of each fit were compared, and the one with the minimum BIC is the most credible fit. The corresponding concentration range is the most appropriate range for the Huggins equation, beyond which the equation is no longer suitable. The fit results at each sample size are shown in the supplementary material, and the fit lines over different concentration ranges are plotted in Fig. 7. From fit results for MEEPT/ACN with no salt (red), the deviation of fit when using data at sufficiently high concentrations can be seen, as at such high concentrations, the viscosity increase is more dramatic and the Huggins equation no longer holds. From the fit of MEEPT/ACN solutions with 0.5 M TEABF_{4} (orange) and with 0.5 M LiTFSI (green), the fit lines using fewer data points at low concentrations are observed to deviate from the data due to the large uncertainty of the measurements in the low concentration regime. The BIC of each fit is plotted vs the number of data points used for fit. It can be seen that for MEEPT/ACN with no salt (red), the BIC decreases with the number of data points, reaching its minimum at *n* = 7. This corresponds to a concentration range of 0.1 M to 0.8 M, and the BIC then increases as *n* becomes larger. This means that until 0.8 M, the higher order term in the Huggins equation can be neglected and it is appropriate to use the equation to describe the concentration-dependent viscosity. For solutions with 0.5 M TEABF_{4} (orange) and with 0.5 M LiTFSI (green), the BIC decreases with *n*, reaching its minimum at *n* = 8, which corresponds to a maximum concentration, $cmax$, of 1.0 M. For solutions with 0.5 M LiTFSI, the minimum BIC at three data points was neglected as it gives unreasonable fit parameters, which comes from the large fluctuation at small concentrations. The most credible fit results are shown as solid lines in Fig. 6, where in the reduced viscosity plot, the intercept represents [*η*], and *k*_{H} is determined from the slope. Therefore, *d*_{H} of MEEPT can be determined from [*η*] using Eq. (8), which are 8.52 ± 0.07, 7.72 ± 0.12, and 8.56 ± 0.16 Å for MEEPT/ACN solutions without supporting salt, with 0.5 M TEABF_{4}, and with 0.5 M LiTFSI, respectively. Table III summarizes these results.

. | c_{max} (M)
. | η_{s} ($mPa\u22c5s$)
. | [η] (L/mol)
. | d_{H} (Å)
. | k_{H}
. |
---|---|---|---|---|---|

No salt | 0.80 | 0.34 ± 0.01 | 0.49 ± 0.01 | 8.52 ± 0.07 | 1.95 ± 0.10 |

TEABF_{4} (0.5 M) | 1.00 | 0.46 ± 0.01 | 0.36 ± 0.02 | 7.72 ± 0.12 | 4.94 ± 0.29 |

LiTFSI (0.5 M) | 1.00 | 0.50 ± 0.01 | 0.49 ± 0.03 | 8.56 ± 0.16 | 2.28 ± 0.21 |

. | c_{max} (M)
. | η_{s} ($mPa\u22c5s$)
. | [η] (L/mol)
. | d_{H} (Å)
. | k_{H}
. |
---|---|---|---|---|---|

No salt | 0.80 | 0.34 ± 0.01 | 0.49 ± 0.01 | 8.52 ± 0.07 | 1.95 ± 0.10 |

TEABF_{4} (0.5 M) | 1.00 | 0.46 ± 0.01 | 0.36 ± 0.02 | 7.72 ± 0.12 | 4.94 ± 0.29 |

LiTFSI (0.5 M) | 1.00 | 0.50 ± 0.01 | 0.49 ± 0.03 | 8.56 ± 0.16 | 2.28 ± 0.21 |

The optimized structures of the neutral MEEPT molecule and their relative energies are shown in Fig. 8, calculated using Density Functional Theory (DFT). The energy differences of the four optimized MEEPT molecules are within 0.2 eV. The dimensions of MEEPT molecules are specified. The molecular structure dimensions, which are about 7 Å–11 Å, are comparable to the hydrodynamic diameters determined from viscosity measurements and are almost equal within the experimental error in different supporting salts. This means that the addition of supporting salts does not have a significant effect on the hydrodynamic diameter of MEEPT.

The Huggins coefficients, which quantify the molecular interactions, are 1.95 ± 0.10, 4.94 ± 0.29, and 2.28 ± 0.21, calculated from the most credible fits to the rearranged Huggins equation. Compared with 0.992, which is the Huggins coefficient derived by Batchelor for Brownian hard spheres with only hydrodynamic interactions and steric effects, those for MEEPT in different supporting salts are larger. This suggests that there are extra interactions between MEEPT molecules that contribute to the viscous dissipation at finite concentrations.

MEEPT solutions containing different supporting salts represent the uncharged state of RFBs; when charged, solution viscosities are unknown because the excluded volume of the anion and the effects of solubility limits of its charged form have not been quantified so far. The impact of supporting salts on the viscosities of the MEEPT radical cation salt solutions is also unknown. The viscous flow properties of MEEPT radical cation salt solutions are addressed next.

### D. Charged MEEPT cations

The two radical cation salts of MEEPT are investigated here: MEEPT-BF_{4} and MEEPT-TFSI. The solubilities of the species are around 0.5 M and 1 M, respectively, varying within 60% in different supporting salt environments, as shown in Table I. MEEPT-TFSI is more soluble than MEEPT-BF_{4} in all three conditions. For each radical cation, the addition of the supporting salt changes the solubility, and different supporting salts have different effects; TEABF_{4} reduces solubility more dramatically.

The viscosity measurements for MEEPT-BF_{4} in three different environments (ACN, 0.5 M TEABF_{4}/ACN, and 0.5 M LiTFSI/ACN) are shown in Fig. 9. The solutions are Newtonian within 3% variation. The average Newtonian viscosities at different concentrations were calculated and plotted in Fig. 10, with error bars coming from standard deviation and uncertainty propagation. The slopes of MEEPT-BF_{4} viscosities are almost the same for all three solutions.

For charged cation solutions, the measured viscosities are fit to the Einstein [Eq. (5)], Huggins [Eq. (9)], Jones–Dole [Eq. (11)], and extended Jones–Dole equations [Eq. (12)]. The details are shown in the supplementary material. The extended Jones–Dole equation fits have the lowest BIC for all three solutions, and the results are shown as solid lines in Fig. 10.

From Fig. 10(b), it can be seen that unlike MEEPT, the reduced viscosities of MEEPT-BF_{4} decrease at first and then increase linearly, similar to that of supporting salts. This also suggests that the extended Jones–Dole equation should be used for fit. The dashed lines in Fig. 10(b) are the extended Jones–Dole equation fits without the $Ac$ term. Thus, the hydrodynamic diameter and the Huggins coefficient of MEEPT-BF_{4} are calculated from the *B* and *D* coefficients, as shown in Table IV. The hydrodynamic diameters of MEEPT-BF_{4} are 10.00 ± 0.59, 9.38 ± 0.52, and 8.58 ± 1.15 Å, respectively, in the three electrolytes, with Huggins coefficients of 1.33 ± 0.40, 2.54 ± 0.50, and 5.27 ± 2.56, respectively. Therefore, we can conclude that the addition of supporting salts decreases the hydrodynamic diameter of MEEPT-BF_{4} while promoting the interactions between MEEPT-BF_{4} species.

. | c_{max} (M)
. | A (L^{1/2}/mol^{1/2})
. | B (L/mol) . | D (L^{2}/mol^{2})
. | d_{H} (Å)
. | k_{H}
. |
---|---|---|---|---|---|---|

No salt | 0.45 | 0.08 ± 0.04 | 0.79 ± 0.14 | 0.82 ± 0.20 | 10.00 ± 0.59 | 1.33 ± 0.40 |

LiTFSI (0.5 M) | 0.43 | 0.10 ± 0.03 | 0.65 ± 0.10 | 1.08 ± 0.17 | 9.38 ± 0.52 | 2.54 ± 0.57 |

TEABF_{4} (0.5 M) | 0.36 | 0.16 ± 0.05 | 0.50 ± 0.20 | 1.30 ± 0.36 | 8.58 ± 1.15 | 5.27 ± 2.56 |

. | c_{max} (M)
. | A (L^{1/2}/mol^{1/2})
. | B (L/mol) . | D (L^{2}/mol^{2})
. | d_{H} (Å)
. | k_{H}
. |
---|---|---|---|---|---|---|

No salt | 0.45 | 0.08 ± 0.04 | 0.79 ± 0.14 | 0.82 ± 0.20 | 10.00 ± 0.59 | 1.33 ± 0.40 |

LiTFSI (0.5 M) | 0.43 | 0.10 ± 0.03 | 0.65 ± 0.10 | 1.08 ± 0.17 | 9.38 ± 0.52 | 2.54 ± 0.57 |

TEABF_{4} (0.5 M) | 0.36 | 0.16 ± 0.05 | 0.50 ± 0.20 | 1.30 ± 0.36 | 8.58 ± 1.15 | 5.27 ± 2.56 |

In addition to MEEPT-BF_{4}, we also analyzed MEEPT-TFSI, which is more soluble than MEEPT-BF_{4}, as shown in Table I. The results of its viscosity measurements are shown in Fig. 11 in the same electrolyte environments as before. The viscosities of MEEPT-TFSI in different solutions show Newtonian behavior even at 1 M, within 4% variation. Therefore, average Newtonian viscosities were calculated and plotted for the concentration-dependence analysis.

Figure 12 shows the average Newtonian viscosities and reduced viscosities of MEEPT-TFSI in the three electrolytes. Similar to MEEPT-BF_{4}, at concentrations up to 0.5 M, the viscosities of radical cation solutions increase slowly. However, as the concentration further increases, the higher order effects become significant and the viscosities increase dramatically as MEEPT-TFSI approaches its maximum solubility limits.

From Fig. 12(b), we can see that the reduced viscosities show increasing trend in the low concentration regime, which suggests an increasing apparent hydrodynamic diameter and a negative *A* coefficient in the Jones–Dole equation, which contradicts Falkenhagen theory.^{48,49} Currently, we are unable to describe such concentration-dependent viscosities using any equation available. Therefore, the apparent hydrodynamic diameters, $dapp=12B5\pi NA1/3$, were evaluated at a single concentration of 0.2 M, where the $Ac$ terms become insignificant and the reduced viscosities begin to increase linearly with concentration. The results are shown in Table V.

From the hydrodynamic diameter calculations, we see that the diameters for both MEEPT-BF_{4} and MEEPT-TFSI increase compared to the neutral MEEPT and remain almost unchanged with the addition of supporting salts. The increase in hydrodynamic diameters is due to the excluded volume of anions, which is shown in Fig. 13. The DFT-optimized models show that the molecular structure dimensions of MEEPT-BF_{4} and MEEPT-TFSI are larger compared with that of neutral MEEPT.

When the concentration of MEEPT-TFSI becomes larger than 0.5 M, the viscosities of solutions increase dramatically, which means that the interactions between MEEPT-TFSI molecules and between MEEPT-TFSI and supporting salts become more significant. In our work with other ROMs and supporting salt electrolytes, we have shown both experimentally and theoretically that the concentration of maximum ionic conductivity is associated with the concentration at which the viscosity begins to increase non-linearly,^{59} which is about 0.5 M in this case. It can be seen from Fig. 12(a) that the viscosity of MEEPT-TFSI in the 0.5 M TEABF_{4}/ACN solution increases faster and exceeds that of other solutions at 0.5 M, suggesting that the interactions in 0.5 M TEABF_{4}/ACN are stronger compared to those in others.

For MEEPT/ACN solutions (Fig. 6), we observed that the viscosity increases linearly with concentration up to 0.5 M, no matter what supporting salts are used, and the increase is by less than a factor of two, up to 1.0 M. However, for MEEPT-TFSI/ACN solutions (Fig. 12), the viscosity increases dramatically at concentrations exceeding 0.5 M. This means that in the operation of NAqRFBs using MEEPT as the ROM, when MEEPT solutions are charged, the reconstitution of fluids at such high concentrations can greatly affect the viscous flow properties of the electrolyte, including viscosity, diffusion, and ionic conductivity.

Table VI summarizes the results for all solutions tested.

. | . | Hydrodynamic . | . | . | . |
---|---|---|---|---|---|

ROM . | Electrolyte . | diameter (Å) . | Viscosity ($mPa\u22c5s$) . | Concentration (M) . | Analyzed by . |

MEEPT | ACN | 8.52 ± 0.07 | 0.58 ± 0.01 | 0.80 | Huggins equation |

0.5 M TEABF_{4}/ACN | 7.72 ± 0.12 | 0.91 ± 0.01 | 1.00 | ||

0.5 M LiTFSI/ACN | 8.56 ± 0.16 | 1.03 ± 0.01 | 1.00 | ||

MEEPT-BF_{4} | ACN | 10.00 ± 0.59 | 0.45 ± 0.01 | 0.45 | Extended Jones–Dole equation |

0.5 M TEABF_{4}/ACN | 9.38 ± 0.52 | 0.67 ± 0.01 | 0.42 | ||

0.5 M LiTFSI/ACN | 8.58 ± 1.15 | 0.78 ± 0.01 | 0.36 | ||

MEEPT-TFSI | ACN | 11.68 ± 0.22 | 0.43 ± 0.01 | 0.20 | Einstein equation one-point method |

0.5 M TEABF_{4}/ACN | 11.31 ± 0.26 | 0.54 ± 0.01 | 0.20 | ||

0.5 M LiTFSI/ACN | 10.08 ± 0.37 | 0.62 ± 0.01 | 0.20 |

. | . | Hydrodynamic . | . | . | . |
---|---|---|---|---|---|

ROM . | Electrolyte . | diameter (Å) . | Viscosity ($mPa\u22c5s$) . | Concentration (M) . | Analyzed by . |

MEEPT | ACN | 8.52 ± 0.07 | 0.58 ± 0.01 | 0.80 | Huggins equation |

0.5 M TEABF_{4}/ACN | 7.72 ± 0.12 | 0.91 ± 0.01 | 1.00 | ||

0.5 M LiTFSI/ACN | 8.56 ± 0.16 | 1.03 ± 0.01 | 1.00 | ||

MEEPT-BF_{4} | ACN | 10.00 ± 0.59 | 0.45 ± 0.01 | 0.45 | Extended Jones–Dole equation |

0.5 M TEABF_{4}/ACN | 9.38 ± 0.52 | 0.67 ± 0.01 | 0.42 | ||

0.5 M LiTFSI/ACN | 8.58 ± 1.15 | 0.78 ± 0.01 | 0.36 | ||

MEEPT-TFSI | ACN | 11.68 ± 0.22 | 0.43 ± 0.01 | 0.20 | Einstein equation one-point method |

0.5 M TEABF_{4}/ACN | 11.31 ± 0.26 | 0.54 ± 0.01 | 0.20 | ||

0.5 M LiTFSI/ACN | 10.08 ± 0.37 | 0.62 ± 0.01 | 0.20 |

### E. Discussion to rationalize Newtonian behavior

With the inference of the hydrodynamic diameter in all the tested compositions, we can *a posteriori* rationalize the observation of Newtonian behavior up to a shear rate of 30 000 s^{−1}. We do this by considering an estimate of a diffusion-based relaxation time^{60,61} for a single MEEPT molecule,

where *k*_{B}*T* is the thermal energy, and postulating that non-Newtonian effects in steady shear become significant only when the Weissenberg number, defined as

becomes order unity. This number expresses the ratio between the rate at which the structure of the particle distribution is deformed by shear flow and the rate of Brownian diffusion that helps recover the equilibrium conformation. The Weissenberg number is a general concept for fluid nonlinearity,^{62} and here, it relates specifically to the concept of a Péclet number, a ratio of the advection rate to diffusion rate. We note that in the expression in Eq. (18), the time scale *λ* is a lower-bound estimate since *λ* would increase sensitively with size (e.g., association of multiple molecules), increase linearly with background viscosity, and increase significantly at higher concentrations when intermolecular interactions become important.

For ACN solutions, with *η*_{s} = 0.34 $mPa\u22c5s$, containing MEEPT with known hydrodynamic dimeter *d*_{H} = 8.5 Å, with negligible Coulombic or van der Waals forces, flowing at a shear rate $\gamma \u0307$ ranging from 5000–30 000 s^{−1}, and at temperature *T* = 298 K, the relaxation time scale *λ* is 0.12 ns, and the corresponding Weissenberg (Péclet) number ranges from 6 × 10^{−7} to 4 × 10^{−6}, much less than unity. From this, the solutions are expected to be Newtonian, as observed in our experiments. To show shear-thinning behavior, the Weissenberg (Péclet) number must be order unity or larger. We can define a critical shear rate for the condition Wi = 1, yielding for our specific case the critical shear rate

which is much larger than the experimental limits. Thus, in our study, no shear-thinning was observed for any solution. Larger molecules, or aggregates of molecules, would change this estimate. For example, for a critical shear rate of 10 000 s^{−1} (within the range of the instrument used here), the critical diameter is around 80 nm, which is about one hundred times larger than a single MEEPT molecule. From this analysis, the lack of shear-thinning implies that no structures of such size are contained within the complex fluid compositions tested here.

## IV. CONCLUSIONS

In summary, the viscous flow properties of MEEPT and its radical cation form were studied in different supporting salt environments, and it is shown that the solution viscosity of both neutral and charged MEEPT increases as a result of excluded volume and species interactions; the excluded volume is comparable to the molecular size, while the interactions are greater than the hydrodynamic interactions. The hydrodynamic diameter of neutral MEEPT remains unchanged with the addition of supporting salts, while for charged MEEPT-BF_{4}, supporting salts decrease the hydrodynamic diameter and enhance interactions. For MEEPT-TFSI, which has a larger solubility compared with MEPPT-BF_{4}, the effects of interactions on the dramatic increase in viscosity at sufficiently high concentration were observed, and the apparent hydrodynamic diameters were calculated in this study, whose changes are relatively small with the addition of supporting salt. All the solutions are Newtonian, which is explained by comparing the time scale of the shear rate to the diffusion-based relaxation time estimate.

It should be noted that the molecular information analysis is based on a continuum assumption, while the sizes of MEEPT and ACN are different by only one order of magnitude. Nonetheless, the hydrodynamic diameters determined from viscometric analysis are comparable to molecular structure dimensions, providing a reasonable estimate. The effect of supporting salts on bulk viscosity was studied, but more complex interactions and conformations of the molecule, such as ion association and liquid solvation, will be the subject of subsequent publications. Other transport properties, such as diffusivity and ionic conductivity vs concentration, remain to be studied and will enable better prediction of the performance of MEEPT RFBs as well as gaining a better understanding of species association and solvation.

Importantly, the results presented here suggest that MEEPT is a promising ROM candidate with respect to transport properties as it and its charged cation are “flowable” up to 0.5 M. The result also confirms that the MEEPT/ACN solutions behave like Newtonian liquids over a wide range of shear rates. For molecules with similar size, Newtonian behavior is expected as a result of the relaxation time scale. The model fit and selection method used here also provides a template to analyze the measured solution viscosity, from which molecular information can be inferred.

## SUPPLEMENTARY MATERIAL

The viscosity of LiTFSI and TEABF_{4} and the fit details of MEEPT and MEEPT-BF_{4} are given in the supplementary material.

## DATA AVAILABILITY

The data that support the findings of this study are available within the article and its supplementary material.

## ACKNOWLEDGMENTS

This work was supported by the Joint Center for Energy Storage Research (JCESR), an Energy Innovation Hub funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences. Zhou Yu and Lei Cheng gratefully acknowledge the computing resources provided on Bebop, a high-performance computing cluster operated by the Laboratory Computing Resource Center at Argonne National Laboratory.

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_{3}SO

_{3}H + VOSO

_{4}+ H

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_{4}+ H

_{2}SO

_{4}+ H

_{2}O) based on Eyring’s theory