a few last adjustment before discussion

Author: pierre-24

TODO.md

@@ -15,3 +15,4 @@
 - [ ] mark (and address) missing values in Tables
 - [ ] tries to move labels as much as possible in graphs
 - [ ] PACS code, MSC code?
+- [ ] Fig. or Figure?

nitroxides.tex

@@ -114,7 +114,8 @@ \section{Introduction}
 Therefore, while seminal studies by Coote and co-workers \cite{hodgsonOneElectronOxidationReduction2007,blincoExperimentalTheoreticalStudies2008} have focused on the impact of substituents on the redox potential of nitroxides, later investigations by other groups have also considered the impact of electrolytes and the various interactions between them, including electrostatic interactions. For example, in 2019, Wylie and co-workers \cite{wylieImprovedPerformanceAllOrganic2019a,wylieIncreasedStabilityNitroxide2019b} demonstrated that interactions between the ionic liquid and nitroxide can increase the redox potential by more than \SI{3}{\volt} with a well-chosen pair of electrolytes. Prior to this, Zhang \textit{et al.} focused on stabilizing radical-ion interactions \cite{zhangInteractionsImidazoliumBasedIonic2016,zhangEffectHeteroatomFunctionality2018}, showing that interactions between a cation and the nitroxyl group are driven by both electrostatic and dispersion effects.
 
 
-In this study, the impact of solvation at low and high concentrations in electrolytes on the redox potential of various nitroxides is investigated. Since the impact of different electrolytes has already been addressed, the focus is on the effects of the skeleton bearing the nitroxyl group, categorized into five families (Fig.~\ref{fig:families}), and the substituents. To facilitate comparison with experimental data, two solvents are considered: water and acetonitrile, for which experimental results are available \cite{morrisChemicalElectrochemicalReduction1991,goldsteinStructureActivityRelationship2006,blincoExperimentalTheoreticalStudies2008,zhangEffectHeteroatomFunctionality2018}. Different (semi-)quantitative models are employed at each step to aid in the interpretation of the results.
+In this study, the impact of solvation at low and high concentrations in electrolytes on the redox potential of various nitroxides is investigated. Since the impact of different electrolytes has already been addressed, the focus is on the effects of the skeleton bearing the nitroxyl group, categorized into five families (Fig.~\ref{fig:families}), and the substituents. To facilitate comparison with experimental data, two solvents are considered: water and acetonitrile, for which experimental results are available \cite{morrisChemicalElectrochemicalReduction1991,goldsteinStructureActivityRelationship2006,blincoExperimentalTheoreticalStudies2008,zhangEffectHeteroatomFunctionality2018}. 
+Different (semi-)quantitative models are employed at each step to aid in the interpretation of the results. It should be noted that the reduced form (hydroxylamine anion) is generally not found in solution \cite{israeliKineticsMechanismComproportionation2005}, as further proton additions (depending on the pH) are involved to form the hydroxylamine or the hydroxylamonium cation. These species were not considered in this article, and thus only experimental oxidation potentials (first reaction in  Fig.~\ref{fig:states}) are compared to the theoretical predictions.
 
 This paper is organized as follows: Section \ref{sec:theory} introduces key concepts and models. The methodology used in this study is detailed in Section \ref{sec:methodo}. The results are then presented in four parts: the impact of substituents on the redox properties is discussed in Section \ref{sec:sar}, followed by an analysis of the effects of solvents in Section \ref{sec:solv}, and the influence of electrolytes in Section \ref{sec:elect}. Finally, a comparison between theoretical predictions and experimental results is provided in Section \ref{sec:exp}. Conclusions and future outlooks are presented in Section \ref{sec:conclusion}.
 
@@ -180,12 +181,12 @@ \subsection{Debye-Huckel theory}
 		\draw[blue,thick,-latex] (4.5,0) -- +(1.5,0) node[midway,above]{$\Delta G^\star_{DH}$};
 		\draw[blue,thick,-latex] (7,1) -- ++(0,.75) -- node[midway,above]{$-\Delta G_S^\star(X^z)$} ++(-7,0) -- ++(0,-.75);
 	\end{tikzpicture}
-	\caption{Thermodynamic cycle to compute the energy of solvatation of an ion, $X^z$, in a electrolyte (solvent characterized by a $\varepsilon = \varepsilon_0\,\varepsilon_r$ dielectric constant and by a ``cloud'' of other ions). $\Delta G_d$ is the discharge of $X^z$ in gas phase, $\Delta G_s$ is the solvatation of $X$, $\Delta G_c$ is the charging of $X$ in $\varepsilon$, and $\Delta G^\star_{DH}$ is the addition of the other ions.}
+	\caption{Thermodynamic cycle to compute the energy of solvation of an ion, $X^z$, in a electrolyte (solvent characterized by a $\varepsilon = \varepsilon_0\,\varepsilon_r$ dielectric constant and by a ``cloud'' of other ions). $\Delta G_d$ is the discharge of $X^z$ in gas phase, $\Delta G_s$ is the solvation of $X$, $\Delta G_c$ is the charging of $X$ in $\varepsilon$, and $\Delta G^\star_{DH}$ is the addition of the other ions.}
 	\label{fig:th}
 \end{figure}
 
 
-On the one hand, at the quantum chemistry (QC) level, the solvatation energy is generally treated implicitly, thanks to a self-consistent reaction field approach (SCRF) \cite{herbertDielectricContinuumMethods2021}: \begin{align}
+On the one hand, at the quantum chemistry (QC) level, the solvation energy is generally treated implicitly, thanks to a self-consistent reaction field approach (SCRF) \cite{herbertDielectricContinuumMethods2021}: \begin{align}
 	G^\star_{SCRF}(X) &= \Braket{\Psi|{\hat{H}+\frac{1}{2}\hat{R}}|\Psi} + G_{th}[\Psi] + G_{nonelst}(X) \nonumber\\
 	&= E[\Psi] + G_{th}[\Psi] + \underbrace{G_{elst}[\Psi] + G_{nonelst}(X)}_{\Delta G^\star_{S,SCRF}(X)}, \label{eq:scrf}
 \end{align}
@@ -205,7 +206,7 @@ \subsection{Debye-Huckel theory}
 in which $\kappa$ is the inverse of the Debye screening length, defined from:\begin{equation}
 	\kappa^2 = \sum_i \frac{n_i\,q_i^2}{\varepsilon_0\varepsilon_r\,k_B\,T}, \label{eq:kappa2}
 \end{equation}
-where $n_i$ is the number density ($n_i = N_i / V = c_i\,\mathcal{N}_a$ where $\mathcal{N}_a$ is the Avogadro number and $c_i$ is the concentration in ion $i$) of ion of type $i$, $k_B$ is the Boltzmann constant (\SI{1.380649e-23}{\joule\per\kelvin}), and $T$ is the temperature (assumed to be \SI{298.15}{\kelvin}).  $\kappa$ is  proportional to the ionic strength of the solution, $I = \frac{1}{2}\sum_i c_i\,z_i^2$.  The Born part is generally dominant in solvatation energies predicted by this model (Fig.~S1).
+where $n_i$ is the number density ($n_i = N_i / V = c_i\,\mathcal{N}_a$ where $\mathcal{N}_a$ is the Avogadro number and $c_i$ is the concentration in ion $i$) of ion of type $i$, $k_B$ is the Boltzmann constant (\SI{1.380649e-23}{\joule\per\kelvin}), and $T$ is the temperature (assumed to be \SI{298.15}{\kelvin}).  $\kappa$ is  proportional to the ionic strength of the solution, $I = \frac{1}{2}\sum_i c_i\,z_i^2$.  The Born part [Eq.~\eqref{eq:born}] is generally dominant in solvation energies predicted by this model (Fig.~S1).
 
 In the limit of $\kappa\to 0$,  $\Delta G^\star_{DH} = 0$ and thus $\Delta G^\star_S \approx \Delta G^\star_{born} = \Delta G_d + \Delta G_c$.  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}