From 5b21ae3baae342eb283b9d6fb85aba11e649e9c6 Mon Sep 17 00:00:00 2001 From: Pierre Beaujean Date: Tue, 20 Aug 2024 11:17:09 +0200 Subject: [PATCH] grammar + abstract --- nitroxides.tex | 88 +++++++++++++++++++++++++------------------------- 1 file changed, 44 insertions(+), 44 deletions(-) diff --git a/nitroxides.tex b/nitroxides.tex index a97adff..01332f8 100644 --- a/nitroxides.tex +++ b/nitroxides.tex @@ -59,7 +59,7 @@ country={Belgium}} \begin{abstract} - This paper investigates the impact of solute-solvent effects on the redox potentials of nitroxides, with a focus on ionic interactions caused by the presence of electrolytes found in different environment such as batteries. The analysis of various nitroxide families shows that ion-substituent interactions, especially in aromatic systems, significantly influence complex stability. In particular, in acetonitrile, the hydroxylamine anion and its cation exhibit strong interactions near the nitroxyl moiety, but only if the nitroxyl is well positioned. The study also confirm that an electrostatic interaction model can predict the effects of substituents, aromaticity, and ring size on redox potentials of nitroxides. Concerning the impact of the environment, solute-ion interactions \replaced{and ion-pairs formation play a crucial role in modulating the redox potential.}{play a crucial role. This study reveals that moderate electrolyte concentrations stabilize charged compounds as described by the the Debye-Hückel (DH) model, and higher concentrations lead to ion-pair formation, both affecting redox properties. } + This paper investigates the impact of solute-solvent effects on the redox potentials of nitroxides, with a focus on ionic interactions caused by the presence of electrolytes found in different environment such as batteries. The analysis of various nitroxide families shows that ion-substituent interactions, especially in aromatic systems, significantly influence complex stability. \replaced{Indeed, the counterion typically prefers to position near the substituent. However, in acetonitrile, the hydroxylamine anion and its cation show strong interactions when the cation is near the nitroxyl moiety}{In particular, in acetonitrile, the hydroxylamine anion and its cation exhibit strong interactions near the nitroxyl moiety, but only if the nitroxyl is well positioned}. The study also confirm that an electrostatic interaction model can predict the effects of substituents, aromaticity, and ring size on redox potentials of nitroxides. Concerning the impact of the environment, solute-ion interactions \replaced{and ion-pairs formation play a crucial role in modulating the redox potential.}{play a crucial role. This study reveals that moderate electrolyte concentrations stabilize charged compounds as described by the the Debye-Hückel (DH) model, and higher concentrations lead to ion-pair formation, both affecting redox properties. } \end{abstract} @@ -101,20 +101,20 @@ \section{Introduction} -The quest for efficient and sustainable energy storage solutions has intensified research into various battery chemistries. Among these, nitroxide-based batteries have attracted, since their first application in 2002 \cite{nakaharaRechargeableBatteriesOrganic2002}, significant attention due to their high theoretical capacity \cite{friebeSustainableEnergyStorage2019,ernouldNitroxidesBatteryrelatedApplications2021,keDesigningStrategiesAdvanced2023}. They are generally used, like other radical polymers, as cathode materials \cite{okaRadicalPolymersRechargeable2020a,assummaNewConductingCopolymer2020}. Beyond their application in batteries, nitroxides have been extensively studied and utilized in various fields. They serve as spin labels in electron paramagnetic resonance spectroscopy \cite{torricellaNitroxideSpinLabels2021}, as mediators for organic radical reactions and polymerizations \cite{tebbenNitroxidesApplicationsSynthesis2011,leifertOrganicSynthesisUsing2023}, in organic electronic devices \cite{jiAirStableOrganicRadicals2020,xieNitroxideRadicalPolymers2021}, and as antioxidants in biological systems \cite{souleChemistryBiologyNitroxide2007,lewandowskiNitroxidesAntioxidantsAnticancer2017,prescottBiologicalRelevanceFree2017}. +The quest for efficient and sustainable energy storage solutions has intensified research into various battery chemistries. Among these, nitroxide-based batteries have attracted, since their first application in 2002 \cite{nakaharaRechargeableBatteriesOrganic2002}, significant attention due to their high theoretical capacity \cite{friebeSustainableEnergyStorage2019,ernouldNitroxidesBatteryrelatedApplications2021,keDesigningStrategiesAdvanced2023}. They are generally used, like other radical polymers, as cathode materials \cite{okaRadicalPolymersRechargeable2020a,assummaNewConductingCopolymer2020}. Beyond their application in batteries, nitroxides have been extensively studied and utilized in various fields. They serve as spin labels in electron paramagnetic resonance spectroscopy \cite{torricellaNitroxideSpinLabels2021}, as mediators for organic radical reactions and polymerization \cite{tebbenNitroxidesApplicationsSynthesis2011,leifertOrganicSynthesisUsing2023}, in organic electronic devices \cite{jiAirStableOrganicRadicals2020,xieNitroxideRadicalPolymers2021}, and as antioxidants in biological systems \cite{souleChemistryBiologyNitroxide2007,lewandowskiNitroxidesAntioxidantsAnticancer2017,prescottBiologicalRelevanceFree2017}. Nitroxides are organic compounds containing the nitroxyl/aminoxyl radical functional group \cite{berlinerHistoryUseNitroxides2012}. This group can exist in three redox states: the nitroxide radical ($>$\ce{N-O^.}, also shortened as \ce{N^.} in this article), which can be oxidized to the oxoammonium cation ($>$\ce{N+=O}, abbreviated \ce{N+}) or reduced to the hydroxylamine anion ($>$\ce{N-O-}, abbreviated \ce{N-}), as depicted in Fig.~\ref{fig:states}. The remarkable stability of the nitroxide radical arises from the delocalization of the unpaired electron over the nitrogen and oxygen atoms, combined with the steric protection provided by the substituents, and environmental effects due to the interaction with solvent and ions \cite{grynovaOriginScopeLongRange2013,grynovaSwitchingRadicalStability2013}. \begin{figure}[!h] \centering \includegraphics[width=.7\linewidth]{Figure1} - \caption{Oxidized (left) and reduced (right) forms of the the nitroxide radical (center).} + \caption{Oxidized (left) and reduced (right) forms of nitroxide radical (center).} \label{fig:states} \end{figure} -Experimental studies have demonstrated that the performance of nitroxide-based batteries is influenced by substituents \cite{sugaCathodeAnodeActivePoly2007}, solvation, and the nature of the electrolytes used \cite{armandIonicliquidMaterialsElectrochemical2009,strehmelRadicalsIonicLiquids2012,wylieIncreasedStabilityNitroxide2019b}. Understanding the interplay between the nitroxides, the solvents, and the electrolytes is therefore crucial for the rational design of high-performance batteries. Computational studies using quantum chemical methods provide valuable insights into the crucial solvation effects, enabling the prediction and tuning of redox properties. +Experimental studies have demonstrated that the performance of nitroxide-based batteries is influenced by substituents \cite{sugaCathodeAnodeActivePoly2007}, solvation, and the nature of the electrolytes used \cite{armandIonicliquidMaterialsElectrochemical2009,strehmelRadicalsIonicLiquids2012,wylieIncreasedStabilityNitroxide2019b}. Understanding the interplay between the nitroxides, the solvents, and the electrolytes is therefore crucial to the rational design of high-performance batteries. Computational studies using quantum chemical methods provide valuable insights into the crucial solvation effects, enabling the prediction and tuning of redox properties. -From a phenomenological perspective, two approaches can be used: at low concentrations in electrolytes, the Debye-Hückel (DH) theory \cite{kontogeorgisDebyeHuckelTheoryIts2018,silvaDerivationsDebyeHuckel2022,silvaImprovingBornEquation2024} provides an initial estimate for interactions within an ionic liquid. While improvements have been proposed over the years to better account for ion-solvent interactions, particularly by including dipole-ion \cite{silvaImprovingBornEquation2024} and quadrupole-ion interactions \cite{slavchovQuadrupoleTermsMaxwell2014,slavchovQuadrupoleTermsMaxwell2014a,coxQuadrupolemediatedDielectricResponse2021}, there have been only a few attempts \cite{matsuiDensityFunctionalTheory2013,xiaoReorganizationEnergyElectron2013,xiaoMolecularDebyeHuckelApproach2014} to incorporate DH theory into the prediction of redox potentials. There is also a limited implementation of DH theory in the polarizable continuum model \cite{cossiInitioStudyIonic1998}. +From a phenomenological perspective, two approaches can be used: at low concentrations in electrolytes, the Debye-Hückel (DH) theory \cite{kontogeorgisDebyeHuckelTheoryIts2018,silvaDerivationsDebyeHuckel2022,silvaImprovingBornEquation2024} provides an initial estimate for interactions within an ionic liquid. While improvements have been proposed over the years to better account for ion-solvent interactions, particularly by including dipole-ion \cite{silvaImprovingBornEquation2024} and quadrupole-ion interactions \cite{slavchovQuadrupoleTermsMaxwell2014,slavchovQuadrupoleTermsMaxwell2014a,coxQuadrupolemediatedDielectricResponse2021}, there have been only a few attempts \cite{matsuiDensityFunctionalTheory2013,xiaoReorganizationEnergyElectron2013,xiaoMolecularDebyeHuckelApproach2014} to incorporate DH theory into the prediction of redox potential. There is also a limited implementation of DH theory in the polarizable continuum model \cite{cossiInitioStudyIonic1998}. At high concentrations (such as in ionic liquids), ion-pair formation can be expected \cite{marcusIonPairing2006}. Alongside electrostatic models \cite{krishtalikElectrostaticIonSolvent1991,lundDielectricInterpretationSpecificity2010}, various theoretical calculations of redox potentials have been performed \cite{mehtaTheoreticalInvestigationRedox2007,quAccurateModelingEffect2016,taherkhaniInvestigationIonPairs2022}, indicating that such interactions can be significant. This has also been recently investigated experimentally by Mugisa et al. \cite{mugisaEffectIonparingKinetics2024}, who assessed the impact of complexation on the thermodynamics and kinetics of the reduction of charged metal complexes. @@ -197,7 +197,7 @@ \subsection{Debye-Hückel theory} G^\star_{SCRF}(X) &= \Braket{\Psi(X)|{\hat{H}+\frac{1}{2}\hat{R}}|\Psi(X)} + G_{thermo}[\Psi(X)] + G_{nonelst}(X) \nonumber\\ &= E[\Psi(X)] + G_{thermo}[\Psi(X)] + \underbrace{G_{elst}[\Psi(X)] + G_{nonelst}(X)}_{\Delta G^\star_{S,SCRF}(X)}, \label{eq:scrf} \end{align} -where $\Psi(X)$ is the wavefunction of $X$ (minimized under the application of $\hat R$, so not equal to the gas phase wavefunction), $\hat H$ is the electronic Hamiltonian, leading to the electronic energy, $E[\Psi(x)]$, $\hat R$ is the reaction field operator (generally recognized to give rise to the electrostatic contribution to the solvation energy, $G_{elst}$), $G_{thermo}$ are the thermal contributions to the Gibbs free energy derived from thermostatistic analysis, and $G_{nonelst}$ collects the non-electrostatic contributions (cavitation, dispersion, etc) to the solvation energy. Therefore, using the notation of Figure \ref{fig:th} (and assuming no change in the geometry of $X^z$), $ \Delta G^\star_{S,SCRF}(X^z) = \Delta G_d + \Delta G_s + \Delta G_{c}$. $ \Delta G^\star_{S,SCRF}(X^z)$ is, therefore, an approximation to $\Delta G^\star_S(X^z)$, since $\Delta G^\star_{DH}$ is missing. +where $\Psi(X)$ is the wavefunction of $X$ (minimized under the application of $\hat R$, so not equal to the gas phase wavefunction), $\hat H$ is the electronic Hamiltonian, leading to the electronic energy, $E[\Psi(x)]$, $\hat R$ is the reaction field operator (generally recognized to give rise to the electrostatic contribution to the solvation energy, $G_{elst}$), $G_{thermo}$ are the thermal contributions to the Gibbs free energy derived from thermostatistic analysis, and $G_{nonelst}$ collects the non-electrostatic contributions (cavitation, dispersion, etc.) to the solvation energy. Therefore, using the notation of Figure \ref{fig:th} (and assuming no change in the geometry of $X^z$), $ \Delta G^\star_{S,SCRF}(X^z) = \Delta G_d + \Delta G_s + \Delta G_{c}$. $ \Delta G^\star_{S,SCRF}(X^z)$ is, therefore, an approximation to $\Delta G^\star_S(X^z)$, since $\Delta G^\star_{DH}$ is missing. On the other hand, the Debye-Hückel (DH) theory provides another estimate of $\Delta G_{S}^\star$ \cite{bockrisModernElectrochemistryIonics1998}. Indeed, assuming that an ion $X^z$, bearing a charge $q = z\,e$ ($e$ is the elementary charge), can be approximated by a sphere of radius $a$ and that the ions in the solution are distributed according to Maxwell-Boltzmann statistics, one obtains the corresponding solvation energy as \cite{kontogeorgisDebyeHuckelTheoryIts2018,silvaTrueHuckelEquation2022,silvaImprovingBornEquation2024}:\begin{align} \Delta G^\star_{S,DH}(X^z) @@ -214,9 +214,9 @@ \subsection{Debye-Hückel theory} \begin{equation} \Delta G^\star_{Born}(X^z) =\frac{q^2}{8\pi\varepsilon_0\,a}\,\left[\frac{1}{\varepsilon_r}-1\right], \label{eq:born}\\ \end{equation} - The Born part [Eq.~\eqref{eq:born}] is generally dominant in solvation energies predicted by this model (Fig.~S1). - - While the SCRF approach neglects $\Delta G^\star_{DH}$, the DH approach approximates $\Delta G_{d}$ and $\Delta G_{c}$ (with the Born model) and neglects $\Delta G_{s}$. Therefore, by combining Eqs.~\eqref{eq:scrf} and \eqref{eq:adh}, one defines:\begin{equation} +The Born part [Eq.~\eqref{eq:born}] is generally dominant in solvation energies predicted by this model (Fig.~S1). + +While the SCRF approach neglects $\Delta G^\star_{DH}$, the DH approach approximates $\Delta G_{d}$ and $\Delta G_{c}$ (with the Born model) and neglects $\Delta G_{s}$. Therefore, by combining Eqs.~\eqref{eq:scrf} and \eqref{eq:adh}, one defines:\begin{equation} G^\star(X^z) = G^\star_{SCRF}(X^z) + \Delta G^\star_{DH}(X^z), \label{eq:gtot} \end{equation} to be used in Eq.~\eqref{eq:nernst}. @@ -237,7 +237,7 @@ \subsection{Modeling the effect of the substituents on the nitroxides and its re \begin{equation} U_q(r) =\frac{q}{4\pi\varepsilon_0} \left[\frac{\mu_x}{r^2} + \frac{Q_{xx}}{r^3}\right], \label{eq:Er} \end{equation} -where $q \in \{-1, 1\}$ is the charge of the redox center, assuming a non-charged substituent. The different quantities (dipole moment, $\mu_x$, and traceless quadrupole moment, $Q_{xx}$) are evaluated through a single-point calculation on a simplified structure, using the geometry of the radical where the \ce{N-O^.} moiety is substituted by \ce{CH2}. Since the alignment of the dipole with the charge needs to be accounted for, this geometry is oriented such that the $x$-axis passes through the the carbon bearing the substituent (set as the origin) and the nitrogen. This definition differs from the original model, as Zhang and co-workers \cite{zhangEffectHeteroatomFunctionality2018} did not consider multiple positions for a given substituent. They also only focused on the oxidation, while both redox processes are considered here. +where $q \in \{-1, 1\}$ is the charge of the redox center, assuming a non-charged substituent. The different quantities (dipole moment, $\mu_x$, and traceless quadrupole moment, $Q_{xx}$) are evaluated through a single-point calculation on a simplified structure, using the geometry of the radical where the \ce{N-O^.} moiety is substituted by \ce{CH2}. Since the alignment of the dipole with the charge needs to be accounted for, this geometry is oriented such that the $x$-axis passes through the carbon bearing the substituent (set as the origin) and the nitrogen. This definition differs from the original model, as Zhang and co-workers \cite{zhangEffectHeteroatomFunctionality2018} did not consider multiple positions for a given substituent. They also only focused on the oxidation, while both redox processes are considered here. \subsection{Impact of ion-pair formation on redox potentials} @@ -254,11 +254,11 @@ \subsection{Impact of ion-pair formation on redox potentials} \newcommand{\arrwy}[3]{ \draw [transform canvas={yshift=0.3ex},arrows = {-Stealth[harpoon]}] (#1) -- (#2) node[midway,above]{#3}; \draw [transform canvas={yshift=-0.3ex},arrows = {-Stealth[harpoon]}] (#2) -- (#1); - } - \newcommand{\arrwx}[3]{ + } + \newcommand{\arrwx}[3]{ \draw [transform canvas={xshift=0.3ex},arrows = {-Stealth[harpoon]}] (#1) -- (#2) node[midway,left]{#3}; \draw [transform canvas={xshift=-0.3ex},arrows = {-Stealth[harpoon]}] (#2) -- (#1); - } + } \begin{tikzpicture}[node distance=2cm] \node (N1) {\ce{N+ + A- + C+ + 2e-}}; \node[right=1cm of N1] (N1c) {\ce{N+A^- + C+ + 2e-}}; @@ -291,7 +291,7 @@ \subsection{Impact of ion-pair formation on redox potentials} \item at the equilibrium of redox reactions, $c_{ox} = c_{rad}$ (for $K_1$) and $c_{red} = c_{rad}$ (for $K_2$), and \item the redox potentials of the ion-pair complexes are smaller than the one of the free species. \end{inparaenum} -Within these assumption, the following electrolyte concentration-dependent (formal) redox potentials are obtained:\begin{align} +Within these assumptions, the following electrolyte concentration-dependent (formal) redox potentials are obtained:\begin{align} E^f_{abs}(\ce{N+|N^.}) &= E^0_ {abs}(\ce{N+|N^.})+\frac{RT}{F}\,\ln\left[\frac{1+K_{11}\,[X]+K_{12}\,[X]^2}{1+K_{01}\,[X]+K_{02}\,[X]^2}\right],\label{eq:ef1}\\ E^f_{abs}(\ce{N^.|N-}) &= E^0_ {abs}(\ce{N^.|N-})+\frac{RT}{F}\,\ln\left[\frac{1+K_{21}\,[X]+K_{22}\,[X]^2}{1+K_{11}\,[X]+K_{12}\,[X]^2}\right],\label{eq:ef2} \end{align} @@ -396,13 +396,13 @@ \section{Compounds and computational chemistry aspects} \label{sec:methodo} In this study, the set of nitroxides considered by Hodgson \textit{et al.} (compounds \textbf{1}-\textbf{54}) is examined, supplemented with a few additional compounds to increase the number of experimental values (\textbf{55}-\textbf{61}). All structures are depicted in Fig.~\ref{fig:nitroxides}. The \ce{AC} pair, consisting of \ce{BF4^-} (\ce{A^-}) and \ce{NMe4^+} (\ce{C^+}), is used as electrolyte. These ions were chosen because they are good models of electrolytes used in the experimental measurements discussed in Section \ref{sec:exp}. While other pairs have been shown to have a larger effect on the redox potential \cite{wylieImprovedPerformanceAllOrganic2019a}, the trends among the families and substituents should remain consistent. \begin{figure}[!p] -\centering -\includegraphics[width=\linewidth]{Figure7} -\caption{Selected nitroxides, sorted by families. Compounds \textbf{1}-\textbf{54} are from Ref.~\citenum{hodgsonOneElectronOxidationReduction2007}. If the compound number is written in red, experimental (oxidation) potentials are available in water, while in acetonitrile if the number is underlined.} -\label{fig:nitroxides} + \centering + \includegraphics[width=\linewidth]{Figure7} + \caption{Selected nitroxides, sorted by families. Compounds \textbf{1}-\textbf{54} are from Ref.~\citenum{hodgsonOneElectronOxidationReduction2007}. If the compound number is written in red, experimental (oxidation) potentials are available in water, while in acetonitrile if the number is underlined.} + \label{fig:nitroxides} \end{figure} -Geometry optimizations and subsequent vibrational frequency calculations were performed at the density functional theory (DFT) level with the $\omega$B97X-D exchange-correlation functional (XCF), and with the 6-311+G(d) basis set. The solvent effects are included using the SMD approach \cite{marenichUniversalSolvationModel2009} approach. All calculations were performed with Gaussian 16 C02 \cite{g16}. With other possible candidates, this XCF has been demonstrated to provide reliable geometries (Ref.~\citenum{wylieImprovedPerformanceAllOrganic2019a}) and redox potentials \cite{flores-leonarFurtherInsightsDFT2017,maierG4AccuracyDFT2020} (see also Fig.~S4). For compound \textbf{1}-\textbf{54}, the geometries obtained by Hodgson et al.~\cite{hodgsonOneElectronOxidationReduction2007} have been used as starting point, taking advantage of their extensive conformational search. All radical forms are considered to have a doublet ground state [$\braket{S^2}=\frac{3}{4}$]. Then, the same calculations were performed in acetonitrile for the subset of compounds for which experimental redox potentials are available (Fig.~\ref{fig:nitroxides}). The geometries of the different complexes (Fig.~\ref{fig:cip}) were then optimized at the same level of approximation, for which different positions of the counterions have been assessed (\textit{vide infra}). Finally, to study the influence of the substituent on the redox potential with the model presented in Section \ref{sec:eleczhang}, single point calculations are performed at the $\omega$B97X-D/6-311+G(d) level in gas phase, using the optimized geometries of the radical states of each nitroxides (in water) in which the $>$\ce{N-O^.} moiety is substituted by \ce{CH_2} (the rest of the geometry is kept fixed). +Geometry optimizations and subsequent vibrational frequency calculations were performed at the density functional theory (DFT) level with the $\omega$B97X-D exchange-correlation functional (XCF), and with the 6-311+G(d) basis set. The solvent effects are included using the SMD approach \cite{marenichUniversalSolvationModel2009} approach. All calculations were performed with Gaussian 16 C02 \cite{g16}. With other possible candidates, this XCF has been demonstrated to provide reliable geometries (Ref.~\citenum{wylieImprovedPerformanceAllOrganic2019a}) and redox potentials \cite{flores-leonarFurtherInsightsDFT2017,maierG4AccuracyDFT2020} (see also Fig.~S4). For compound \textbf{1}-\textbf{54}, the geometries obtained by Hodgson et al.~\cite{hodgsonOneElectronOxidationReduction2007} have been used as starting points, taking advantage of their extensive conformational search. All radical forms are considered to have a doublet ground state [$\braket{S^2}=\frac{3}{4}$]. Then, the same calculations were performed in acetonitrile for the subset of compounds for which experimental redox potentials are available (Fig.~\ref{fig:nitroxides}). The geometries of the different complexes (Fig.~\ref{fig:cip}) were then optimized at the same level of approximation, for which different positions of the counterions have been assessed (\textit{vide infra}). Finally, to study the influence of the substituent on the redox potential with the model presented in Section \ref{sec:eleczhang}, single point calculations are performed at the $\omega$B97X-D/6-311+G(d) level in gas phase, using the optimized geometries of the radical states of each nitroxide (in water) in which the $>$\ce{N-O^.} moiety is substituted by \ce{CH_2} (the rest of the geometry is kept fixed). Since all thermochemical quantities are $\kappa$-dependent, analyses were performed using custom Python scripts. When required (e.g., in Eq.~\eqref{eq:dh}), the value of $a$ (the radius of the solute cavity) is taken as half the largest distance between any pair of two atoms in the molecule. Although this is an approximation, it provides a consistent method to treat all molecules proportionally to their size and this approach is consistent with other publications \cite{matsuiDensityFunctionalTheory2013}. Furthermore, a value of $\varepsilon_{r,water}=80$ for water and $\varepsilon_{r,acetonitrile}=35$ for acetonitrile is used. These relative permittivities correspond to those of the pure solvents and are known to be lower for the respective electrolyte solutions \cite{silvaTrueHuckelEquation2022}. These variations can be substantial; for example, $\varepsilon_r \approx 70$ for a solution containing \SI{1}{\mol\per\kilo\gram} of \ce{NaCl} in water \cite{kontogeorgisDebyeHuckelTheoryIts2018, silvaTrueHuckelEquation2022}, but they depend on the nature of the electrolyte, so it was not considered here. @@ -413,7 +413,7 @@ \section{Results and discussion} \label{sec:results} \subsection{Structure-activity relationships} \label{sec:sar} -Oxidation and reduction potentials of the nitroxide radicals in water, grouped by family, are plotted in Fig.~\ref{fig:family} (see also Table S1). In comparison to the most simple compound, \textbf{1}, modifying the molecular structure and/or adding substituents generally increases both the oxidation and reduction potentials. Regarding structural impacts, six-membered ring compounds (P6O and APO) exhibit higher reduction potentials than their five-membered ring counterparts (P5O and IIO). Additionally, the incorporation of one or two aromatic rings (IIO and APO) enhances both the oxidation and reduction potentials. +Oxidation and reduction potentials of the nitroxide radicals in water, grouped by family, are plotted in Fig.~\ref{fig:family} (see also Table S1). In comparison to the simplest compound, \textbf{1}, modifying the molecular structure and/or adding substituents generally increases both the oxidation and reduction potentials. Regarding structural impacts, six-membered ring compounds (P6O and APO) exhibit higher reduction potentials than their five-membered ring counterparts (P5O and IIO). Additionally, the incorporation of one or two aromatic rings (IIO and APO) enhances both the oxidation and reduction potentials. \begin{figure}[!h] \centering @@ -423,7 +423,7 @@ \subsection{Structure-activity relationships} \label{sec:sar} \end{figure} Regarding the impact of substituents, it is noteworthy that non-substituted nitroxides within each family (\textit{i.e.}, \textbf{2}, \textbf{14}, \textbf{23}, and \textbf{36}) have generally one of the lowest oxidation and reduction potentials within their respective groups. Several trends emerge based on the nature of the substituent: \begin{inparaenum}[(i)] - \item shielding the radical center with ethyl groups instead of methyl groups (\textbf{7}, \textbf{19}, and \textbf{28}) results in a decrease in potentials (particularly the reduction potential), likely due to changes in inductive effects, + \item shielding the radical center with ethyl groups instead of methyl groups (\textbf{7}, \textbf{19}, and \textbf{28}) results in a decrease in potential (particularly the reduction potential), likely due to changes in inductive effects, \item protonation of \ce{-NH2} (\textit{i.e.}, \textbf{4} vs \textbf{11}, \textbf{16} vs \textbf{21}, and \textbf{25} vs \textbf{35}) increases the potentials, especially in P5O, \item multiple substitutions by \ce{-COOH} (\textit{e.g.}, \textbf{8} vs. \textbf{9} and \textbf{10}) also increase the potentials, though the effect is less pronounced in IIO (\textbf{30}-\textbf{33}) and APO (\textbf{41}-\textbf{48}), \item consistently with the model described in Section \ref{sec:eleczhang}, compounds with mesomeric donor substituents (\ce{-NH2}, \ce{-OH}, \ce{-OMe}) have lower potentials than those with acceptor substituents (\ce{-COOH}, \ce{-NO2}), particularly in aromatic systems (\textit{e.g.}, \textbf{49} vs \textbf{52}), and @@ -435,26 +435,26 @@ \subsection{Structure-activity relationships} \label{sec:sar} The electrostatic interaction model [Eq.~\eqref{eq:Er}] provides more insights. Results are presented in Fig.~\ref{fig:corr} (see also Table S4). It should be noted that this model fails to account for the effect of substituting methyl groups with ethyl groups. Moreover, including the disubstituted compounds (e.g., \textbf{9}, \textbf{10}, \textbf{20}, ...) worsens the correlation ($R^2 \sim 0.5$ and 0.3 for oxidation and reduction, respectively). Compounds \textbf{56} and \textbf{58} remain outliers for reduction. Therefore, all these sets of compounds are treated as outliers in the following discussion. -Though the correlation is poorer for reduction than for oxidation (probably because the electron delocalization means that the nitrogen atom is not the atom that should be used to define the origin in that case), this model helps explaining the general trends. For instance, the increase in oxidation (and reduction) potential for aromatic compounds correlates with an increase in quadrupole moment ($Q_{xx} > \SI{5}{\elementarycharge\bohr\squared}$ for most members of IIO or APO). Additionally, modifications due to donor/acceptor substituents are linked to changes in the dipole moment. For example, aromatic compounds with \ce{-NH2} as a substituent (\textit{e.g.}, \textbf{51}) are characterized by $\mu_{x} < 0$, which increases for compounds with \ce{-COOH} (\textit{e.g.}, \textbf{39}) or \ce{-NO2} (\textit{e.g.}, \textbf{54}). Furthermore, members of P5O generally present a smaller value of $U_q$ than P6O (\textit{e.g.}, \textbf{17} versus \textbf{5}), which correlates with the increase in oxidation potential observed between these two families. The same trend is observed between APO and IIO. +Though the correlation is poorer for reduction than for oxidation (probably because the electron delocalization means that the nitrogen atom is not the atom that should be used to define the origin in that case), this model helps explain the general trends. For instance, the increase in oxidation (and reduction) potential for aromatic compounds correlates with an increase in quadrupole moment ($Q_{xx} > \SI{5}{\elementarycharge\bohr\squared}$ for most members of IIO or APO). Additionally, modifications due to donor/acceptor substituents are linked to changes in the dipole moment. For example, aromatic compounds with \ce{-NH2} as a substituent (\textit{e.g.}, \textbf{51}) are characterized by $\mu_{x} < 0$, which increases for compounds with \ce{-COOH} (\textit{e.g.}, \textbf{39}) or \ce{-NO2} (\textit{e.g.}, \textbf{54}). Furthermore, members of P5O generally present a smaller value of $U_q$ than P6O (\textit{e.g.}, \textbf{17} versus \textbf{5}), which correlates with the increase in oxidation potential observed between these two families. The same trend is observed between APO and IIO. This model also accounts for some effects due to the position of the substituent (see, e.g., \textbf{49}-\textbf{51}), which was not the case with the original model by Zhang and co-workers (resulting in weak correlations, $R^2 \leq 0.3$). Finally, although this model is not directly applicable to positively charged substituents (\textbf{11}, \textbf{21}, and \textbf{35}), for which the dipole and higher multipole moments are ill-defined, the only term of Eq.~\eqref{eq:Er} would be $(q\,q')/r$ (where $q'$ is the charge of the substituent), resulting in a destabilizing interaction with \ce{N+} and \ce{N^.}, while stabilizing \ce{N-} (Fig.~\ref{fig:dipole}), which correlates well with the increase in oxidation and reduction potentials for these compounds. \begin{figure}[!h] -\centering -\includegraphics[width=\linewidth]{Figure9} -\caption{Relationship between absolute oxidation (top) and reduction (bottom) potentials of nitroxides and the electrostatic potential between the redox center ($>$\ce{N-O^.}) and the substituent, as computed using Eq.~\eqref{eq:Er} at the $\omega$B97X-D/6-311+G(d) level in water (SMD), in the limit of $[\ce{X}]=\SI{0}{\mole\per\liter}$. Triangular marker ($\blacktriangle$) indicates results that are excluded from the correlation (see text).} -\label{fig:corr} + \centering + \includegraphics[width=\linewidth]{Figure9} + \caption{Relationship between absolute oxidation (top) and reduction (bottom) potentials of nitroxides and the electrostatic potential between the redox center ($>$\ce{N-O^.}) and the substituent, as computed using Eq.~\eqref{eq:Er} at the $\omega$B97X-D/6-311+G(d) level in water (SMD), in the limit of $[\ce{X}]=\SI{0}{\mole\per\liter}$. Triangular markers ($\blacktriangle$) indicates results that are excluded from the correlation (see text).} + \label{fig:corr} \end{figure} \clearpage \subsection{Impact of the solvent} \label{sec:solv} -The solvent exerts a significant stabilizing effect on the charge. In the gas phase (Table S2), $E^0_{abs}(\ce{N+}|\ce{N^.})$ values are around \SI{7}{\volt} (and up to \SI{10}{\volt} for \textbf{11}, \textbf{21}, and \textbf{35}), while $E^0_{abs}(\ce{N^.}|\ce{N-})$ values are approximately \SI{0.3}{\volt} (around \SI{3}{\volt} for \textbf{11}, \textbf{21}, and \textbf{35}). The modifications due to the solvent, primarily resulting from the stabilization of the charges (as indicated by the Born model), but also including moderate geometry changes, amount to about \SI{2}{\volt} (which corresponds to an free Gibbs energy change of \SI{200}{\kilo\joule\per\mole}). +The solvent exerts a significant stabilizing effect on the charge. In the gas phase (Table S2), $E^0_{abs}(\ce{N+}|\ce{N^.})$ values are around \SI{7}{\volt} (and up to \SI{10}{\volt} for \textbf{11}, \textbf{21}, and \textbf{35}), while $E^0_{abs}(\ce{N^.}|\ce{N-})$ values are approximately \SI{0.3}{\volt} (around \SI{3}{\volt} for \textbf{11}, \textbf{21}, and \textbf{35}). The modifications due to the solvent, primarily resulting from the stabilization of the charges (as indicated by the Born model), but also including moderate geometry changes, amount to about \SI{2}{\volt} (which corresponds to a free Gibbs energy change of \SI{200}{\kilo\joule\per\mole}). -The difference between redox potentials computed in water and acetonitrile is reported in Fig.~\ref{fig:watvsac} (see also Table S3). The oxidation potential is minimally affected, while the reduction potentials show a disparity greater than \SI{0.5}{\volt}. Indeed, while the linear regressions are similar in both case (in particular, the slope is of 1.2), the lower values for the reduction potential results in large differences. +The difference between redox potentials computed in water and acetonitrile is reported in Fig.~\ref{fig:watvsac} (see also Table S3). The oxidation potential is minimally affected, while the reduction potentials show a disparity greater than \SI{0.5}{\volt}. Indeed, while the linear regressions are similar in both cases (in particular, the slope is of 1.2), the lower values for the reduction potential results in large differences. In a first approximation, the Born model [Eq.~\eqref{eq:born}] can explain the electrostatic contribution to these changes. For oxidation, the change in potentials between the two solvents, $E^0_{\text{acetonitrile}} - E^0_{\text{water}}$, is proportional to $\varepsilon_{r,\text{acetonitrile}}^{-1} - \varepsilon_{r,\text{water}}^{-1}$ (about \SI{0.1}{\volt} for $a = \SI{3}{\angstrom}$ using Eq.~\eqref{eq:born}), which is positive (assuming \ce{N^.} is neutral, which is valid for the subset of compounds considered). For reduction, it is proportional to $\varepsilon_{r,\text{water}}^{-1} - \varepsilon_{r,\text{acetonitrile}}^{-1}$, having the same magnitude but opposite sign. However, the non-electrostatic contribution to the solvation energies must also be considered. The cavitation term is expected to be larger in water due to hydrogen bonding than in acetonitrile \cite{marenichUniversalSolvationModel2009}. In this case, both effects are of the same order of magnitude and they counterbalance each other in the case of oxidation, while they add together in the case of reduction. @@ -473,7 +473,7 @@ \subsection{Impact of the solvent} \label{sec:solv} \clearpage \subsection{Impact of the electrolytes} \label{sec:elect} -So far, the concentration of electrolyte, $[X]$, has been maintained at zero. To evaluate its impact on the redox potentials, the DH correction itself, Eq.~\eqref{eq:dh}, is initially examined in Fig.~\ref{fig:DH}. As anticipated due to the amplitude of the charges involved ($z=\pm 1$), it remains small within the concentration range considered here (a few tenths of millivolts for $[X] \leq \SI{1}{\mole\per\liter}$ and larger in acetonitrile), increasing with $[X]$. Its sign differs between oxidation and reduction potentials, since it only affects the charged species, and while \ce{N+} is a reactant, \ce{N-} a product. Additionally, it decreases for compounds belonging to the IIO and APO families (as they are larger molecules with larger $a$), while it is amplified for species with a net positive charge (\textbf{11}, \textbf{21}, \textbf{35}), for which the correction for oxidation and reduction potentials is negative. +So far, the concentration of electrolyte, $[X]$, has been maintained at zero. To evaluate its impact on the redox potentials, the DH correction itself, Eq.~\eqref{eq:dh}, is initially examined in Fig.~\ref{fig:DH}. As anticipated due to the amplitude of the charges involved ($z=\pm 1$), it remains small within the concentration range considered here (a few tenths of millivolts for $[X] \leq \SI{1}{\mole\per\liter}$ and larger in acetonitrile), increasing with $[X]$. Its sign differs between oxidation and reduction potentials, since it only affects the charged species, and while \ce{N+} is a reactant, \ce{N-} is a product. Additionally, it decreases for compounds belonging to the IIO and APO families (as they are larger molecules with larger $a$), while it is amplified for species with a net positive charge (\textbf{11}, \textbf{21}, \textbf{35}), for which the correction for oxidation and reduction potentials is negative. \begin{figure}[!b] @@ -492,10 +492,10 @@ \subsection{Impact of the electrolytes} \label{sec:elect} \end{inparaenum} \begin{figure}[!h] -\centering -\includegraphics[width=.8\linewidth]{Figure12} -\caption{$\Delta G^\star_{cplx}$ of compound \textbf{4} as a function of counterion position. The distance between the redox center (the nitrogen in the oxidized form, $>$\ce{N+=O}, or the oxygen in the reduced form, $>$\ce{N-O-}) and the counterion are also given. Calculations were performed at the $\omega$B97X-D/6-311+G(d) level in water (black) and acetonitrile (blue) using SMD, in the limit of $[X]=\SI{0}{\mole\per\liter}$.} -\label{fig:pos-anion} + \centering + \includegraphics[width=.8\linewidth]{Figure12} + \caption{$\Delta G^\star_{cplx}$ of compound \textbf{4} as a function of counterion position. The distance between the redox center (the nitrogen in the oxidized form, $>$\ce{N+=O}, or the oxygen in the reduced form, $>$\ce{N-O-}) and the counterion are also given. Calculations were performed at the $\omega$B97X-D/6-311+G(d) level in water (black) and acetonitrile (blue) using SMD, in the limit of $[X]=\SI{0}{\mole\per\liter}$.} + \label{fig:pos-anion} \end{figure} In both solvents, the radical (\ce{N^.}) interacts with \ce{C+} (\textit{i.e.}, \ce{NMe4+}) in the first position, near $>$\ce{N-O^.} \cite{zhangEffectHeteroatomFunctionality2018}. Then, in water, the complexation energies for the \ce{N+A-} and \ce{N^-C+} pairs are positive and significant. The difference in complexation energies between the two positions is generally small (a few \si{\kilo\joule\per\mole}), but the second position is favored in most of the cases. Interestingly, this type of interaction corresponds to a larger nitroxide-to-counterion distance, indicating a smaller electrostatic interaction between the redox center and the counterion and suggesting that the substituent also plays a role in lowering the complexation energy. Another contributing factor is the quadrupole-ion interaction due to the aromatic moieties present in compounds from the IIO and APO families, which is particularly important in the \ce{N^-C+} pair. Here, the difference in energy between the two positions is more pronounced. @@ -503,7 +503,7 @@ \subsection{Impact of the electrolytes} \label{sec:elect} In acetonitrile, however, the lower dielectric constant leads to a reduced charge screening. A significant decrease of the complexation energy (10 to \SI{20}{\kilo\joule\per\mole}) is observed for the \ce{N^-C+} pair when \ce{C+} is near the nitroxyl group (first position). This aligns with the model for ion pair formation [Eq.~\eqref{eq:pair}], particularly when considering the impact of the ratio between the radii of the cavities. Since \ce{NMe4+} has a larger radius (\SI{2.1}{\angstrom}) than \ce{BF4-} (\SI{1.5}{\angstrom}), the former is closer in size to nitroxides ($>$\SI{3}{\angstrom}). The second position remains generally favored in the \ce{N+A-} pair, though only by a few \si{\kilo\joule\per\mole}. Using the most stable positions for each complex, the corresponding equilibrium constants are reported in Fig.~\ref{fig:Kx1} (see also Tables S7-S8). The average value for each family in water is presented in Table \ref{tab:Kx1}. In general, the order $K_{11} < K_{01} < K_{21} < 1$ indicates that the interaction between the hydroxylamine anion and the cation is the least destabilizing. While these equilibrium constants are of the same order of magnitude ($K_{x1} \sim \num{e-4}$) for \textbf{1}, significant differences appear in the other families. These trends are primarily driven by enthalpy (Fig.~S6): while $\Delta H^\star_{cplx}$ varies significantly among complexes (and is generally $<0$), the entropic contribution remains relatively constant ($-T\Delta S^\star_{cplx} \sim \SI{40}{\kilo\joule\per\mole}$). -In particular, in water, the equilibrium constants $K_{01}$ are one order of magnitude larger and $K_{21}$ are two order of magnitude larger for aromatic compounds (IIO and APO), due to ion-quadrupole interactions. The latter family contains compounds with some of the lowest complexation energies. However, correlating the nature of the substituent with the complexation energy remains challenging. In the literature, perpendicular $\pi$-cation interactions are stabilized by electron-rich aromatic rings (and thus donor substituents), while $\pi$-anion interactions are stabilized by electron-poor rings (acceptor substituents) \cite{pappFourFacesInteraction2017}. Despite this, compounds with either donor (\textit{e.g.}, \textbf{42}) or acceptor (\textit{e.g.}, \textbf{54}) substituents exhibit small complexation energies, likely due to variations in ion positioning across different systems. Regarding acetonitrile, while $K_{21}$ increases for the reasons discussed above, but this is not necessarily the case for the other constants. +In particular, in water, the equilibrium constants $K_{01}$ are one order of magnitude larger and $K_{21}$ are two orders of magnitude larger for aromatic compounds (IIO and APO), due to ion-quadrupole interactions. The latter family contains compounds with some of the lowest complexation energies. However, correlating the nature of the substituent with the complexation energy remains challenging. In the literature, perpendicular $\pi$-cation interactions are stabilized by electron-rich aromatic rings (and thus donor substituents), while $\pi$-anion interactions are stabilized by electron-poor rings (acceptor substituents) \cite{pappFourFacesInteraction2017}. Despite this, compounds with either donor (\textit{e.g.}, \textbf{42}) or acceptor (\textit{e.g.}, \textbf{54}) substituents exhibit small complexation energies, likely due to variations in ion positioning across different systems. Regarding acetonitrile, while $K_{21}$ increases for the reasons discussed above, but this is not necessarily the case for the other constants. \begin{figure}[!h] \centering @@ -519,10 +519,10 @@ \subsection{Impact of the electrolytes} \label{sec:elect} & $pK_{01}$ & $pK_{11}$ & $pK_{21}$ \\ \hline AMO & 4.13 & 3.37 & 3.88 \\ -P6O & 3.23 $\pm$ 0.67 & 4.25 $\pm$ 0.71 & 6.24 $\pm$ 1.36 \\ -P5O & 4.37 $\pm$ 1.82 & 4.32 $\pm$ 1.65 & 2.33 $\pm$ 1.84 \\ -IIO & 3.24 $\pm$ 0.54 & 3.36 $\pm$ 0.38 & 1.69 $\pm$ 0.52 \\ -APO & 3.12 $\pm$ 0.42 & 3.48 $\pm$ 1.01 & 1.12 $\pm$ 1.18 \\ + P6O & 3.23 $\pm$ 0.67 & 4.25 $\pm$ 0.71 & 6.24 $\pm$ 1.36 \\ + P5O & 4.37 $\pm$ 1.82 & 4.32 $\pm$ 1.65 & 2.33 $\pm$ 1.84 \\ + IIO & 3.24 $\pm$ 0.54 & 3.36 $\pm$ 0.38 & 1.69 $\pm$ 0.52 \\ + APO & 3.12 $\pm$ 0.42 & 3.48 $\pm$ 1.01 & 1.12 $\pm$ 1.18 \\ \hline \end{tblr} \caption{Mean value of the cologarithm ($pK = -\log_{10}K$) of the complexation equilibrium constants \added{for ion-pair formation in} each family (reported as mean $\pm$ standard deviation), as computed at the $\omega$B97X-D/6-311+G(d) level in water using SMD and $[X]=\SI{1}{\mole\per\liter}$.} @@ -537,10 +537,10 @@ \subsection{Impact of the electrolytes} \label{sec:elect} \begin{figure}[!h] -\centering -\includegraphics[width=\linewidth]{Figure14} -\caption{Value of the cologarithm ($pK = -\log_{10}K$) of the complexation equilibrium constants: $pK_{02}$ (round markers, $\bullet$), $pK_{12}$ (triangular markers, $\blacktriangle$) and $pK_{22}$ (square markers, $\blacksquare$) for the 3 oxidation state of nitroxides, as computed at the $\omega$B97X-D/6-311+G(d) level in water (top) and acetonitrile (bottom) using SMD and $[X]=\SI{1}{\mole\per\liter}$. The dashed line is used to help visualization. } -\label{fig:Kx2} + \centering + \includegraphics[width=\linewidth]{Figure14} + \caption{Value of the cologarithm ($pK = -\log_{10}K$) of the complexation equilibrium constants: $pK_{02}$ (round markers, $\bullet$), $pK_{12}$ (triangular markers, $\blacktriangle$) and $pK_{22}$ (square markers, $\blacksquare$) for the 3 oxidation state of nitroxides, as computed at the $\omega$B97X-D/6-311+G(d) level in water (top) and acetonitrile (bottom) using SMD and $[X]=\SI{1}{\mole\per\liter}$. The dashed line is used to help visualization. } + \label{fig:Kx2} \end{figure} As expected, the equilibrium constants are smaller by about four orders of magnitude ($\Delta G^\star_{cplx} \sim \SI{40}{\kilo\joule\per\mole}$) than those previously discussed. In water \added{(Table }\ref{tab:Kx2}\added{)}, the general order is $K_{22}\added{<} K_{02} < K_{12} \added{ < 1}$. However, for many compounds in the IIO and APO families, $K_{02}$ is larger than $K_{22}$, which is attributed to the interaction between the \ce{NMe4+} cation and the aromatic moiety present in these compounds. In acetonitrile, the \ce{N^-AC} complexes are again more stable than the others, consistently with previous observations \cite{wylieImprovedPerformanceAllOrganic2019a}. Thus, the dielectric constant significantly impacts the equilibrium constants of these ion-triplets. This is further confirmed by the observation that the stabilization of \ce{N^.AC} is less pronounced in this study than in Ref.~\citenum{wylieImprovedPerformanceAllOrganic2019a}, which employed a solvent with an even lower dielectric constant. @@ -595,7 +595,7 @@ \section{Conclusions and outlooks} \label{sec:conclusion} In this paper, the impact of different solute-solvent effects on the redox potentials of nitroxides has been assessed using quantum chemistry approaches, with a particular emphasis on ionic interactions. The calculation have been performed at the DFT level ($\omega$B97X-D/6-311+G*), with implicit solvation (SMD) to account for the impact of the solvent. -Different families of nitroxides have been considered: 5- (P6O, APO) and 6-membered rings (P5O, IIO) containing the nitroxyl moiety, and with (IIO, APO) or without an aromatic system (P5O, P6O) in close vicinity. The impact of such structural changes as well as of substituent, on the redox potentials is largely explained by the electrostatic interaction model (Fig.~\ref{fig:dipole}) developed by Zhang and co-workers \cite{zhangEffectHeteroatomFunctionality2018}: thought the dipole interaction between the substituent and the redox center can explain, in first approximation, the variation that are observed, the inclusion of the quadrupole moment is necessary to explain the increases of both the oxidation and reduction potentials of aromatic nitroxides. Furthermore, acceptor substituents (such as \ce{NO2}) further increase both potentials. While the impact remains moderate (+\SI{0.4}{\volt}), it is hoped that this will provide design rules for future investigations. +Different families of nitroxides have been considered: 5- (P6O, APO) and 6-membered rings (P5O, IIO) containing the nitroxyl moiety, and with (IIO, APO) or without an aromatic system (P5O, P6O) in close vicinity. The impact of such structural changes as well as of substituents, on the redox potentials is largely explained by the electrostatic interaction model (Fig.~\ref{fig:dipole}) developed by Zhang and co-workers \cite{zhangEffectHeteroatomFunctionality2018}: thought the dipole interaction between the substituent and the redox center can explain, in first approximation, the variation that are observed, the inclusion of the quadrupole moment is necessary to explain the increases of both the oxidation and reduction potentials of aromatic nitroxides. Furthermore, acceptor substituents (such as \ce{NO2}) further increase both potentials. While the impact remains moderate (+\SI{0.4}{\volt}), it is hoped that this will provide design rules for future investigations. While the Born model [Eq.~\eqref{eq:born}] shows that the solvent tends to stabilize charges due to changes in the dielectric constant (especially in polar solvents), other, more subtle effects arise from solute-ion interactions caused by the presence of electrolytes. These electrolytes are found in moderate concentrations (\textit{i.e.,} \SI{0.1}{\mole\per\liter}) during the experimental measurement of redox potentials and in higher concentrations (>$\SI{1}{\mole\per\liter}$) in ionic liquids used for batteries. Their impact is twofold: @@ -606,7 +606,7 @@ \section{Conclusions and outlooks} \label{sec:conclusion} Both effects have been examined: when the charge of the compound and of the electrolyte constituents is moderate the correction proposed by the Debye-Hückel model is sufficient. However, the formation of pairs depends on the redox state of the nitroxide and the nature of the intermolecular interactions, which goes beyond a simple pair formation model (such as the one found in Fig.~\ref{fig:ionpair}). Indeed, two positions are possible for the ion: near the redox center of the nitroxide, and closer to its substituent, if any. The ion-substituent interaction (in the second position) generally leads to more favorable complexes (especially when the molecule contains aromatic moieties). However, in acetonitrile, the interaction between the reduced form (hydroxylamine anion) and its cation, positioned near the $>$\ce{N-O-} moiety, is the strongest. This seems to be the case in other low-dielectric environments, as noted by others in an even less polar solvent (using methanol, $\varepsilon_r$ = 25, in Ref.~\citenum{wylieImprovedPerformanceAllOrganic2019a}). It was, however, not possible to correlate the impact of the substitution on the formation of ion-pairs, but it was noticed that the favorable interactions between \ce{N-} and \ce{C+} was systematically hampered by the nitroxyl in an axial position in P6O. This is another important design rule for future applications. -Finally, a comparison with experiment has been performed. It results in an excellent correlation, but the impact of the corrections presented above is small in the solvents considered here (water and acetonitrile) and with the concentrations of electrolytes used experimentally. As a matter of fact, it would be interesting to compare redox potentials measured under different conditions (such as in ionic liquids). Another factor that should be investigated is the temperature, which would affect both the DH correction (through $\kappa$, Eq.~\eqref{eq:kappa2}) and the complexation equilibrium constant (though the entropic contribution). For example, conventional lithium-ion batteries can operate up to \SI{60}{\degreeCelsius} \cite{maTemperatureEffectThermal2018}, and ionic liquids are stable over extended temperature ranges. The modification of the dielectric constant of the solution with increasing electrolyte concentration \cite{kontogeorgisDebyeHuckelTheoryIts2018, silvaTrueHuckelEquation2022}, is another point that should be considered in future studies. +Finally, a comparison with the experiment has been performed. It results in an excellent correlation, but the impact of the corrections presented above is small in the solvents considered here (water and acetonitrile) and with the concentrations of electrolytes used experimentally. As a matter of fact, it would be interesting to compare redox potentials measured under different conditions (such as in ionic liquids). Another factor that should be investigated is the temperature, which would affect both the DH correction (through $\kappa$, Eq.~\eqref{eq:kappa2}) and the complexation equilibrium constant (though the entropic contribution). For example, conventional lithium-ion batteries can operate up to \SI{60}{\degreeCelsius} \cite{maTemperatureEffectThermal2018}, and ionic liquids are stable over extended temperature ranges. The modification of the dielectric constant of the solution with increasing electrolyte concentration \cite{kontogeorgisDebyeHuckelTheoryIts2018, silvaTrueHuckelEquation2022}, is another point that should be considered in future studies. \section*{Notes} The author declare no competing financial interest.