domain: spi chim

text:
Olivier Alévêque

Eric Levillain

Olivier Aleveque
email: olivier.aleveque@univ-angers.fr

Use of phase angle in alternating voltammetry on redox self-assembled monolayers with intermolecular interactions

Keywords:

We present a methodology based on the study of the frequency dependence of the phase angle in Alternative Voltammetry to estimate the standard electrochemical rate constant for heterogeneous electron transfer reactions in the case of strong intermolecular interactions occur between the redox forms.

This work is a follow-up to our previous study which was dedicated only to the study of peak currents.

Introduction

Heterogeneous electrochemical processes on redox-responsive materials (i.e., conductive polymers, multilayer, thin films and self-assembled monolayers) are known to play a key role in fields such as electrochemical sensors, electrocatalysis-based devices, electrochemical synthesis, molecular electronics and the reliable estimation of the electrochemical rate constant has become, among other parameters, a challenge to understand the interfacial electrochemical mechanisms involved.

In this context, since their discovery [START_REF] Nuzzo | Adsorption of bifunctional organic disulfides on gold surfaces[END_REF], self-assembled monolayers (SAMs) have become an ideal system for studying interfacial phenomena as closely as possible to proposed theories, thanks to their simplicity of preparation, their relatively good stability, and the possibility of modulating all the parts of the grafted molecule (anchoring group, spacer, functional group, etc.) in order to modify the properties studied [START_REF] Love | Self-Assembled Monolayers of Thiolates on Metals as a Form of Nanotechnology[END_REF].

To provide relevant information on electrochemical processes involved, many models and theories have been developed [START_REF] Eckermann | Electrochemistry of redox-active self-assembled monolayers[END_REF][START_REF] Bard | Electrochemical Methods: Fundamentals and Applications[END_REF] and studied specifically by more or less complex electrochemical techniques (cyclic voltammetry [START_REF] Laviron | Adsorption, autoinhibition and autocatalysis in polarography and in linear potential sweep voltammetry[END_REF][START_REF] Gonzalez | Electrochemical determination of kinetic parameters of surface confined redox probes in presence of intermolecular interactions by means of Cyclic Voltammetry. Application to TEMPO monolayers in gold and platinum electrodes[END_REF][START_REF] Waelder | Beyond the Laviron Method: A New Mathematical Treatment for Analyzing the Faradaic Current in Reversible, Quasi-Reversible, and Irreversible Cyclic Voltammetry of Adsorbed Redox Species[END_REF],

impedance spectroscopy [START_REF] Nahir | The distribution of standard rate constants for electron transfer between thiolmodified gold electrodes and adsorbed cytochrome c[END_REF], square-wave voltammetry [START_REF] Reeves | Application of square wave voltammetry to strongly adsorbed quasireversible redox molecules[END_REF][START_REF] Mirčeski | Theoretical and experimental study of the surface redox reaction involving interactions between the adsorbed particles under conditions of square-wave voltammetry[END_REF], square-wave voltcoulometry [START_REF] Gonzalez | Square Wave Voltcoulommetry Analysis of the Influence of the Electrostatic Environment on the Electrochemical Functionality of Redox Monolayers[END_REF], chronoamperometry [START_REF] Ravenscroft | Kinetics of electron transfer to attached redox centers on gold electrodes in nonaqueous electrolytes[END_REF][START_REF] Gonzalez | Kinetic Implications of the Presence of Intermolecular Interactions in the Response of Binary Self-Assembled Electroactive Monolayers[END_REF], small-amplitude alternative voltammetry (AV) [START_REF] Creager | A New Way of Using ac Voltammetry To Study Redox Kinetics in Electroactive Monolayers[END_REF][START_REF] Laviron | polarography and faradaic impedance of strongly adsorbed electroactive species: Part I. Theoretical and experimental study of a quasi-reversible reaction in the case of a Langmuir isotherm[END_REF][START_REF] Laviron | polarography and faradaic impedance of strongly adsorbed electroactive species: Part II. Theoretical study of a quasi-reversible reaction in the case of a Frumkin isotherm[END_REF][START_REF] Laviron | Polarography and faradaic impedance of strongly adsorbed electroactive species: Part III. Theoretical complex plane analysis for a surface redox reaction[END_REF], largeamplitude alternative voltammetry [START_REF] Guo | Fourier Transform Large-Amplitude Alternating Current Cyclic Voltammetry of Surface-Bound Azurin[END_REF][START_REF] Bond | Changing the Look of Voltammetry[END_REF][START_REF] Bell | Theoretical treatment of highfrequency, large-amplitude ac voltammetry applied to ideal surface-confined redox systems[END_REF][START_REF] Laborda | Theoretical-experimental synergy towards better understanding of interfacial electron transfer kinetics[END_REF] …). Some of them have made it possible to model systems in which strong intermolecular interactions occur between the redox forms [START_REF] Calvente | Intermolecular interactions in electroactive thiol monolayers probed by linear scan voltammetry[END_REF].

Cyclic voltammetry (CV) is known to be the most frequently used electrochemical technique to achieve this objective because, based on Laviron's model, the standard electrochemical rate constant (k0) can be deduced from the shift of the peak potentials as a function of the scan rate [START_REF] Laviron | General expression of the linear potential sweep voltammogram in the case of diffusionless electrochemical systems[END_REF]. However, this method remains very random for high scan rates, which limits the determination of electrochemical rate constants to not too large values. In order to provide a new alternative approach, our previous paper [START_REF] Alévêque | Alternative voltammetry on self-assembled monolayers: An original approach to estimate the electrochemical electron-transfer rate constants when electroactive adsorbed species interact[END_REF] described two methods based on alternative cyclic voltammetry (ACV) technique to estimate the standard electrochemical rate constants (k0) of immobilized redox molecules presenting intermolecular interactions and described by the general lateral interaction model initiated by Laviron [START_REF] Creager | A New Way of Using ac Voltammetry To Study Redox Kinetics in Electroactive Monolayers[END_REF] and update over time [START_REF] Alévêque | Electroactive self-assembled monolayers: Laviron's interaction model extended to non-random distribution of redox centers[END_REF][START_REF] Alévêque | Electroactive mixed self-assembled monolayers: Lateral interactions model updated to interactions between redox and non-redox species[END_REF].

The theoretical part, in which the redox reaction is described by Butler-Volmer (BV) kinetic formalism under

Nernstian conditions for the non-alternative component of the perturbating signal e(t), extends the Creager and Wooster's method [START_REF] Creager | A New Way of Using ac Voltammetry To Study Redox Kinetics in Electroactive Monolayers[END_REF] and allowed to define the expression of the ratio ION/IOFF during an ACV experiment.

In this work, we deal with the same problematic using the second important parameter of an AC signal, namely the phase angle. Even if this approach is difficult to implement with current technologies, the first results obtained are quite encouraging.

Experimental conditions 2.1 Reagents, chemical and preparation of SAMs at gold electrodes

The relevance of our approaches was tested via TEMPO-based self-assembled monolayers (C15-TEMPO)

on Au substrate, known to provide very intense attractive interactions between its oxidized species in methylene chloride. The synthesis, characterisations and preparation of C15-TEMPO were described in Ref [START_REF] Aleveque | Nitroxyl Radical Self-Assembled Monolayers on Gold: Versatile Electroactive Centers in Aqueous and Organic Media[END_REF].

Electrochemical measurements

Electrochemical experiments were carried out with a Biologic SP-300 potentiostat driven by the EC-Lab software including ohmic drop compensation. All electrochemical experiments were performed in a threeelectrode cell controlled at a temperature of 293 K in a glove box containing dry, oxygen-free (< 1 ppm) argon.

Working electrodes were a gold modified surface. Counter electrodes were platinum wires. Reference electrodes were Ag wire or Ag/AgNO3 (0.01 M CH3CN) in order to have a very fine determination of the potential. Experiments were recorded in dry HPLC-grade methylene chloride with tetrabutylammonium hexafluorophosphate (Bu4NPF6, electrochemical grade, Fluka) as supporting electrolyte.

All the procedures described in the following sections must be performed in a defined order: studies of the SAMS by 1) cyclic voltammetry to reduce the number of unknows, follow by 2) alternative voltammetry. Table 1 lists symbols used hereinafter.

Cyclic voltammograms

Cyclic voltammograms (CV) were essentially performed under Nernstian conditions (scan rate v << k0) to allow the extraction of the full width at half maximum (FWHM), peak potential (Ep) and peak intensity (ip). Interactions which influence these parameters can thus be estimated. In addition, Cdl can be roughly estimated by dividing the capacitive current by the scan rate   cap Cdl i / v  . The charge (Q) involved during a CV experiment is deduced by integration of the voltammetric signal, and the surface coverage of electroactive species is calculated as Q / (nFA)  with A the area of the electrode, F the faraday constant and n the number of electrons involved during the process.

Alternatives voltammograms

Alternative voltammetry (AV) experiments on the same SAMs were performed with a small amplitude voltage of 1-5 mV (EAC), a scan rate (v) of 1-5 mV.s -1 for ACV, and an acquisition time adjusted to obtain at least 36 points per alternative cycle whatever the frequency (f) used. Generally, at least five frequencies per decade were used between 1 mHZ and 500 Hz (i.e., the maximum frequency of our instrument) to obtain the minimum number of points needed to perform the proposed adjustment procedures. Ohmic drop compensation should be used to compensate for the resistance of the electrolytic solution as accurately as possible.

Generation of simulated voltammograms

As previously described [START_REF] Alévêque | Alternative voltammetry on self-assembled monolayers: An original approach to estimate the electrochemical electron-transfer rate constants when electroactive adsorbed species interact[END_REF], all simulated cyclic voltammograms (CV), alternative voltammograms (AV) and alternative cyclic voltammograms (ACV) were performed using SigmaPlot 14.5 and/or MATLAB R2020b softwares run on a conventional desktop computer (Intel Core i7-10810U CPU @ 1.10GHz 6 cores). We used a simple numerical method approximating the continuous variation of the applied potential as a series of potential steps ΔE. With an increment ΔE small enough, it is possible to assume that the oxidation (kb) and reduction (kr) rate constants are quasi constant during the step. This procedure is detailed in previous papers [START_REF] Honeychurch | Numerical simulation of Fourier transform alternating current linear sweep voltammetry of surface bound molecules[END_REF] and adapted to our formalism.

In the case of a reversible n-electron process of adsorbed electroactive species following the general lateral interaction model (vide infra), CV, AV or ACV voltammograms can be simulated with this set of equations:

kf kb O ne R                                                       O max max b O f O bb O O b f b f b f b O O O O b f bf d i nFA nFA k k dt k i 1 k i 1 i 1 i exp k i 1 k i 1 t k i 1 k i 1 k i 1 k i 1 k i 1 i 1 i i 1 exp k i 1 k i 1 t k i 1 k i 1                                                                                                                                                              OO 0' b 0 RR R OR O 0' f 0 OO O OR R i AC i AC j the valueof at step j nF k i 1 k exp 1 E i 1 E exp 2a i 2a i RT with nF k i 1 k exp E i 1 E exp 2a i 2a i RT E(i 1) E v t E sin w t E i 1 t v E sin i 1 t w          

Theoretical prerequisites on the models and formalisms used

The general lateral interaction model

The generalized lateral interaction model (GLI model) enables current-voltage behaviours (CV, ACV, …) to be simulated in the case of a reversible n-electron process of adsorbed electroactive species. It was updated to include the effects of non-random distributions of electroactive species, interactions between redox and nonredox species and the possibility to use binary electrolyte mixture [START_REF] Alévêque | Nitroxyl radical self-assembled monolayers: Generalized lateral interactions model used with binary electrolyte mixture[END_REF].

For ease of reading, here is a summary of the main hypotheses if we only select the case of interactions between redox species:

-Strongly adsorbed electroactive species are only oxidized and reduced at the adsorbed state.

-The electroactive species are theoretically grafted on the electrode according to a unimodal statistical distribution of the electroactive neighbours, with a spatial regularity (random or not). The surface occupied by an oxidized molecule, or a reduced molecule is the same.

-

The segregation level of the electroactive species is quantified with a normalized parameter   -The standard electrochemical rate constant (k0) at the heterogenous standard potential (E 0 ') is considered as a constant.

 (   1      ), defined for a normalized surface coverage        2
-aOO, aRR and aOR are the potential-independent interaction constants between molecules of O, molecules of R and molecules of O and R, respectively, and are positive for an attraction and negative for a repulsion.

-The influence of lateral interactions in the electro-chemical response of the surface redox probes is visualized through potential-independent global interaction parameters G and S, defined as

  OO RR OR G a a 2a    ,   RR OO S a a  with G ≤ 2. Overall,

 

G    changes the shape of the voltametric peak, i.e., the FWHM and the peak intensity, whereas

 

S    controls the position of the peak potential.

During a CV experiment under Nernstian conditions (k0 >> v), these parameters are defined as:

            0' p 22 max p G1 RT E ( ) E S nF n F vA I ( ) 2RT 2 -G R T 3 2 FWHM ( ) 2ln 2 2 3 G nF 2                                  (1)
With n, F, A, R, T, E and v having their usual meanings and E 0' the apparent standard surface potential.

General Lateral Interaction Function (GLI function)

Based on the General Lateral Interaction model, the GLI function has initially been developed to fit cyclic voltammetric peaks of SAMs under Nernstian conditions (v << k0) to extract the main characteristic parameters as the peak current (ip), the full width at half maximum (FWHM) and the peak potential (Ep).

This model [START_REF] Alévêque | A generalized lateral interactions function to fit voltammetric peaks of selfassembled monolayers[END_REF] is defined according to an approximation of  O to work around a problem of mathematical recursion and leads to directly link the potential E to  O .

 

                                                 n O O p
                          AC i
A differentiation is made according to the amplitude of the sinusoidal part: if EAC is high enough the technique is called Large Amplitude AV and conversely when EAC is small in comparison to the global change in voltage the technique is called Small Amplitude AV.

Here, small amplitude Alternative Cyclic Voltammetry is used under two major conditions:

1. The variation of the surface concentration of redox species must be small at each cycle to obtain a sinusoidal resulting current [START_REF] Eckermann | Electrochemistry of redox-active self-assembled monolayers[END_REF]. In this case, mathematical treatments based on equivalent circuits can be used, taking advantage of the linearization of the kinetics of the electrodes, which initially have an exponential dependence on the applied voltage. Typically, a small amplitude of 5 mV or less is chosen depending on the interactions, especially when the interactions produce a fast surface concentration variation of the redox state of electroactive species close to apparent standard potential.

2. Scan rate (v) must be lower than k0 in order to respect the Nernst conditions (DC component only considered) and, also, to have an AC current higher than the DC current (i.e. conditions similar to a multitude of AV at different potential).

The formalism used for Alternative (Cyclic) Voltammetry

From the above experimental conditions of AV, redox SAMs can be modelled with the Randles equivalent circuit (figure 1).

Depending on the potential applied, the Randles equivalent circuit presents two boundary simplifications:

1. When the faradaic contribution is negligible, far away from the apparent standard potential, the resulting current is only due to the double layer capacitance contribution. This case is denominated as OFF state:

           Cdl Cdl Cdl OFF OFF OFF i I sin wt i I sin wt (4a) 2.
When the faradaic contribution is not negligible (named here "tot" for total state), the resulting current is the sum of faradaic and double layer capacitance contributions.

At a given frequency, two peculiars ON states co-exist at specified potentials (or  O ) when the current magnitude is maximum (ION) and when the phase angle passes through a minimum (  ON ). We emphasize on this coexistence because these two cases may or may not occur at the same potential.

                              ON ON
                  * * * ON ON ON ON* State at fixed potentials * tot Cdl f tot tot tot ON i i i I sin wt i i I sin w t (4c)
Note that, if the applied potential is constant, the ION* states do not exist, Itot increasing continuously with frequency.

From the formalism devoted to AC polarograms in the case of strongly adsorbed and randomly distributed electroactive species [START_REF] Bard | Electrochemical Methods: Fundamentals and Applications[END_REF][START_REF] Laviron | Adsorption, autoinhibition and autocatalysis in polarography and in linear potential sweep voltammetry[END_REF][START_REF] Gonzalez | Electrochemical determination of kinetic parameters of surface confined redox probes in presence of intermolecular interactions by means of Cyclic Voltammetry. Application to TEMPO monolayers in gold and platinum electrodes[END_REF], Rct and Cads can be explicit by equation 5:

22 0 max 22 max RT 1 Rct k n F A 1 ko 1 RT Rct Cads Cads n F A                  (5) with                                                       RR R OR O OO O OR R RR R OR O OO O OR R 2a 2a 2a 2a 1 RO 2a 2a 2a 2a 1 OR RR R OR OO O 1 e e 1 2 a a e 1 2 a a e (6)
At the equilibrium potential (i.e., in small amplitude ACV experiment when 0 vk ), η can be expressed following:

        0' O OR OO R RR OR O nF EE 2 a a 2 a a 2G S G OO RT RO e e e                    (7)
with O  and R  are the values in the absence of alternative potential at

i E E v t    .
With simplifying assumptions such as a charge transfer coefficient ( 0.5 

) and a randomly distributed SAM (i.e.       ), equation 6 becomes:

            O OO OR R RR OR O RR OR a a a a S a a O R O O OO OR O R O O ee 2G 2G                                             (8)
It must be emphasized that Rct and Cads are not constant but depend on the surface coverage of reduced and oxidized species (i.e., the potential applied) and on the interaction parameters (G, S, aij). The total resulting current defined according to the Randles equivalent circuit associated to the formalism above (equation 5) can be expressed as follows: Itot can expressed using two different mathematical approaches:

                                                                        tot Cdl f Cdl Cdl f f
I                                                            22
           2 2 0 tot OFF 2 0 w k II w 1 k with       22 max n F A 1 Cdl RT (10a) or (Appendix A),   2 2 2 tot f Cdl Cdl f f 1 Cdl I I I 2I I sin X I       (10b) with                                  2 22 max 2 1 AC 2 2 2 2 max n F A RT RT X 1 2Cdl E n F A 1 ko w
In previous work [START_REF] Alévêque | Alternative voltammetry on self-assembled monolayers: An original approach to estimate the electrochemical electron-transfer rate constants when electroactive adsorbed species interact[END_REF], k0 was estimated by plotting the ratio of the magnitude of the peak current (Itot) to the magnitude of the background current (IOFF) of ACV experiments as a function of the logarithm of the frequency (f) followed by the fitting the sigmoid curve obtained and describing a cut-off phenomenon. Creager and Wooster's performed the same with an approach that does not consider interactions between grafted molecules [START_REF] Creager | A New Way of Using ac Voltammetry To Study Redox Kinetics in Electroactive Monolayers[END_REF].

The following part aims at expressing tot  with  and  parameters.

Theoretical study of the phase angle and analysis

In order to express tot  vs. frequency, its dependence with O  and thus with  and  must be clearly defined.

Via the common trigonometric formulas, the resultant current expressed with equation 9 can be expanded to:

                                                                       tot tot Cdl Cdl f f tot tot Cdl Cdl Cdl f f f tot tot Cdl Cdl f f Cdl Cdl f f
                                   Cdl Cdl f f Cdl f f Cdl Cdl f f f f A I
            (12)
and with equation 11, tot  can be extracted according to:

    0 Cdl f Cdl f f tot ff 0 f k I I sin arctan I I sin w A tan( ) B I cos k I cos arctan w                      and with         2 2 x sin arctan x 1x 1 cos arctan x 1x           2 00 Cdl f tot f kk I 1 I ww tan( ) I       2 0 AC 0 tot AC 22 k w CdlE 1 w k tan( ) PE w 1 ko w                         with 22 max n F A P RT   (13)
And, finally, after some tricky rearrangements to:

2 00 tot 0 kk Cdl Cdl 1 w tan( ) w P w P k        (14)
Equation 14 shows that  tot depends on the ratio 0 k / w , Cdl, and  and  parameters in which interaction parameters (G, S, aij) are included.

From the possible values of the parameters of this equation, the values of  tot are located between π/2 (capacitive behaviour when w tends to 0 or infinity) and 0. In this case, the trends of  tot and  tot tan( ) are the same, so future interpretations of the results will be conducted on either of these parameters. At a given  O (i.e., potential), the  tot vs. frequency plots (figure 2C) exhibit inflexion points, the ON* states at from an infinite limit when  O tends to 0 or 1 (figure 4).

Based on these finding, four strategies can be considered to reach k0:

1. Recording  tot at different frequencies for a given  O (i.e., potential), with Multi-frequency AV experiments (same setup as EIS experiments). The curve obtained can be modelled using equation 14. To achieve good accuracy via a curve fitting process of k0, the variation of  tot must be maximized, which means that the experiment must be ideally performed at a special  O that produces the minimum  * ON i.e.  Min .

2. Same strategy as before but carried out at several  O In this case, if the expression of  * ON can be determined (vide infra), the curve fitting process must be performed on this parameter as a function of

 * ON f .
3. Recording  tot at several  O at a given frequency, with an ACV experiment (or several AV experiments). The curve obtained can be modelled using equation 14. To achieve good accuracy of k0 via a curve fitting process, the variation of  tot must be maximized, which means that the experiment must be performed at a peculiar frequency that produces the minimum value of  ON i.e.  Min .

4. Same approach as before but carried out at several frequencies. In this case, if the expression of  ON can be determined (vide infra), the curve fitting process can be carried out on this parameter as a function of

  ON O, .

Fitting procedures

At a given potential, as a function of frequency

If the phase angle of the current is recorded at a given potential (i.e. at a given O  ),  and  are constant.

The phase angle only depends on the ratio 0 k / w and Cdl and equation 14 becomes:

                   OO O OO 2 0 E, E, 0 tot E, 0 E, E, k k Cdl Cdl w tan( ) w P w P k ( 15
)
As O  can be evaluated using GLI function, taking advantage of the

    O tot E,
tan( ) vs w plot via a curve fitting procedure makes it possible to estimate k0 and Cdl in one shot (strategy 1). This strategy is optimal if the variation of

    O tot E, tan (
) is maximal during the experiment: this requires estimating the value of  O that produce  Min .

Firstly, we need to reach the frequency at

    O * ON E,
, after finding the zeros (Appendix B) of the derivative of equation 15 with respect to w:

                  O * O ON E, O O E, 0 E, E, P w k 1 (16) At     * ON E, O
w , equation 14 becomes:  } for determining k0 and Cdl in one shot via equation 15.

                      OO O OO E, E, * ON E, E, E, Cdl Cdl tan( ) 2 1 PP (17a)             * ON E, O O O * ON E, 0 E, w Cdl tan( ) 2

At a given frequency, as a function of potential

At a given frequency (w) during an AV, the phase angle variation only depends on the on  and  , i.e. O  and equation 14 becomes:

         2 00 tot w 0 kk Cdl Cdl w tan( ) w P w P k ( 18
)
As O  can be evaluated using GLI function, fitting the

   tot w tan(
) vs O  plot via equation 18 makes it possible to estimate k0 Cdl in one shot (strategy 3). As with strategies 1, this strategy is optimal if the variation of

    O tot E, tan (
) is maximal during the experiment: this requires estimating a frequency that produces  Min .

Firstly, finding the zeros of the derivative of equation 18 with respect to O  allows to reach O  at

   ON w (i.e.
a specific couple (  ;  )). Mathematically complex with the presence of non-linearities, solutions can only be found at boundary conditions (  w0 or  w

), i.e., plateaux observed in figure 3:

                    ON 0 0 22 tot w O, , k w 00 k S S 1 1 w Cdl 1 w lim tan( ) k P k 2S ( 19
)
whose step-by-step development is detailed in Appendix C.

                   ON 0 0 2 00 tot w O, , 0 k w 0 0 k kk w Cdl 0 lim tan( ) f ;G;S;Cdl k w P w
whose value very influenced by interactions and Cdl can be determined in some conditions (  Cdl 0 or ) by Ferrari's and Cardan's method and whose step-by-step development produces an oversize equation [START_REF] Serret | Cours d'algèbre supérieure : professé à la Faculté des sciences de Paris[END_REF].

Consequently, this approach (strategy 3), while usable, is not optimal because it does not allow access to the best frequency to perform the experiment and therefore to estimate k0 with the highest accuracy. An approximate frequency can nevertheless be estimated by first carrying out strategy 1 at

   Min O,

2

, which makes it possible to determine experimentally  Min f at  Min .

If we perform this experiment at several given frequencies, it is possible to determine k0 from the ON states (strategy 4). Keeping in mind that O  is determined via the GLI function, it is possible to proceed the fitting of

   ON w tan(
) with respect to

    ON w O,
and w using equation 18 for determining k0 and Cdl in one shot.

Figure 5 highlights the principle of these strategies and shows the results of the curve fitting procedure and the estimation of k0 and Cdl using simulated alternative voltammograms with aOO=1, k0=100 s -1 and Cdl=5 µF as input parameters.

These four fitting procedures makes it possible to determine the values of k0 and Cdl with good accuracies and precisions in the case of interacting electroactive species.

These strategies were also tested with a lower number of points and with an added Gaussian noise (SI-3), and showed that strategies 1 and 3 remain the most robust under these conditions, which is explained by the fact that the fit uses only one variable as input (frequency for strategy 1 and O  for strategy 3) and not both simultaneously for strategy 2 and 4.

Fitting procedures on experimental data

Measurement feasibility

Equations describing the monitored parameters of an AV or ACV experiment (Itot and  tot ) are based on Randles model assuming a sinusoidal response to a sinusoidal solicitation. As for the amplitude of the current (Itot), where a very low potential amplitude (EAC) and therefore a very low scan rate (v) are required, some parameters must also be finely controlled to obtain a reliable measurement of the phase angle  tot in agreement the assumptions of the Randle's model and to be able to use the strategies proposed previously.

The number of points per cycle is the first critical point. As the phase angle is directly related to a time difference between the applied potential and the resulting current, the minimum resolution of the phase angle measurement therefore depends on the time base used (Δt) to increment the potential, which is a function of the number of points per cycle (N) and the frequency of the AC component. Consequently, and whatever the numerical determination method chosen (search for the maximum, offset of the power component, correlation of the signals e(t) and i(t)…), the phase angle resolution is the measurement of two adjacent points by:

      2 R 2 f t N with   1 t Nf 20)
According to equation 20,  tot measurement resolution depends mainly on the number of points per cycle.

The accurate measurement of small changes in the phase angle can therefore only be achieved by increasing the number of points, which leads to digitization problems even for newer potentiostats. The phase angle resolution dependence with N is clearly observed in figure 6A, where for example with N=36 a resolution  R =0.175 is insufficient to finely measure  tot variations.

Determination of

  ON O,
and  * ON w are also strongly impacted by the number of points, especially when the variation of  tot is small around these points. For example, in figure 6A, for N=36,

  ON O,
is in a wide interval between 0.45 and 0.69.

The potential amplitude (EAC) is the second critical point. The lower this factor is, the closer the induced current will be close to a perfect sinusoid and therefore to the theoretical results based on a linear response of the system.  tot measurement accuracy dependence with EAC is clearly observed in figure 6B, a convergence taking place by progressively decreasing the value of EAC.

A low potential amplitude and therefore the use of a low scan rate (v) is required to avoid excessive distortion of the AC voltammograms (see 3.3). EAC must nevertheless be large enough to measure a resulting current, which requires some compromises.

In conclusion, access to  tot remains a challenge in an AV or ACV experiment, requiring the system to be placed in restricted conditions (low potential amplitude, near zero scan rate and large number of points). With these technical considerations, the most appropriate measurement strategies will be chosen according to the characteristics of the potentiostat used to obtain experimentally accurate measurements with an acceptable number of points.

Theory vs. experience

To validate the above approaches, the frequency dependence of the phase angle in Alternative Voltammetry was studied through TEMPO-based SAMs (C15-TEMPO) on Au substrate, a well-known redox-responsive material. Indeed, previous works [START_REF] Aleveque | Nitroxyl Radical Self-Assembled Monolayers on Gold: Versatile Electroactive Centers in Aqueous and Organic Media[END_REF][START_REF] Aleveque | Nitroxyl radical self-assembled monolayers on gold: Experimental data vs. Laviron's interaction model[END_REF] shown that the analysis of CVs at different surface coverages under reversible conditions allows with the extraction of characteristic parameters such as the full width at half maximum (FWHM), peak potential (Ep) and peak intensity (ip) to quantify (with equation 1) the interactions between the adsorbed electroactive species. Associated to the knowledge of the chemical system involved (electrochemical system, isotherms of immobilization in the neutral state ...), the interaction parameters G, S and aOO of TEMPO-based SAMs were estimated at 1, -1 and 1 respectively. Figure 7A shows an experimental cyclic voltammogram of a TEMPO SAM presenting a surface coverage of electroactive species of 2.8 10 -10 mol.cm -2 (i.e., 10 10 max 2.8 10 0.62 5 10

        
with a mean value of max  according to references [START_REF] Calvente | Intermolecular interactions in electroactive thiol monolayers probed by linear scan voltammetry[END_REF][START_REF] Alévêque | Alternative voltammetry on self-assembled monolayers: An original approach to estimate the electrochemical electron-transfer rate constants when electroactive adsorbed species interact[END_REF][START_REF] Alévêque | Electroactive mixed self-assembled monolayers: Lateral interactions model updated to interactions between redox and non-redox species[END_REF][START_REF] Alévêque | Nitroxyl radical self-assembled monolayers: Generalized lateral interactions model used with binary electrolyte mixture[END_REF]) and a peak potential Ep closed to 0.820 V vs Ag reference electrode, on which alternative experiment will be performed hereafter.

In order to comply with the feasibility criteria and avoid measurement bias caused by different techniques, the experiments were carried out under the same conditions constrained by the potentiostat used (BIOLOGIC SP300):

  Frequency: points from 1 Hz to 500 Hz with 3 points per decade regularly spaced logarithmically Note that strategies 2 and 4 cannot be tested on TEMPO SAMs because these approaches require long experimental times and then very stable redox-responsive materials. For all the experiments of strategies 1 and 3, the phase angles were determined by correlation between e(t) and i(t), this method being the most accurate in the presence of noisy signals. Figure 7C and 7D show a good agreement of the experimental plots with the theory proposed and allow to estimate the parameters k0 and Cdl via a fitting procedure with a good accuracy and precision (Table 2).

The two strategies give quite similar results, strategy 1 being the most robust, especially on the accuracy of Cdl. Table 2 also shows the estimations made without considering the interactions. In this case, the results are not representative given the accuracy of the estimated values (Sk0 and SCdl) and prove the usefulness of the proposed model which takes the interactions into account.

In addition, all the results obtained with these two fitting procedures are compared with those obtained with the procedures described in the previous article and based on the ratio of Itot/IOFF.

The results estimated for k0 are quite similar (value and accuracy), and those found for Cdl are less precise compared to previous method despite fairly close estimated values, the slight differences observed can be explained by the environment, the difference between the batches and the stability of the SAMs.

This proves the validity of this phase angle model, even if it remains more difficult to implement with current technologies and involves the use of a very powerful instrument.

Conclusion

Accurately estimating the standard electrochemical rate constant (k0) for heterogeneous electron transfer reactions in the case of strong intermolecular interactions occur between the redox forms is therefore possible by following the phase angle during of AV experiments.

The validity of our methodological approach has been demonstrated experimentally on TEMPO-based SAMs and compared with previous results, but it must be emphasized that it remains difficult to implement with current technologies and implies the use of a potentiostat with a reliable frequency response analyzer.

At last, this methodology also confirms our previous work based on the ION/IOFF ratio.

Appendices

Appendix A

A1

Via the common trigonometric formulas, the resultant current expressed with equation 9 can be expanded to:

                                                                       tot tot
                                              Cdl Cdl f f Cdl f f Cdl Cdl f f f f A I
            (A2)
Itot can be determined combining A1 and A2:

                            22 22 tot Cdl f f f f 2 2 2 2 tot f Cdl Cdl f f 1 Cdl 1 f Cdl f f I A B I I sin I cos I I I 2I I sin X I with X I 2I I sin (A3) A2                                     0 f
                     2 2 1 AC 22 P 2Cdl X 1 E P 1 ko w (A6) Appendix B B1                    OO O OO 2 0 E, E, 0 tot E, 0 E, E, k k Cdl Cdl w tan( ) w P w P k (B1)                       O OO OO 2 tot E, 0 E, E, 0 22 0 E, E, d tan( ) k k Cdl Cdl dw P P k ww (B2)           O O tot E, * ON E, d tan( ) 0 at dw (B3)                 OO * O ON E, O E, E, 2 2 2 2 00 E, P w k k Cdl (B4)                   O * O ON E, O O E, 0 E, E, P w k 1 Cdl (B5)
By combining (B1) and (B5),

                             * ON E, OO O O ** OO ON E, ON E, OO 2 0 E, E, * 0 ON E, 0 E, E, w k k Cdl Cdl tan( ) w P w P k (B6)                                                               O O O O O O O OO OO E, E, E, * ON E, E, E, E, E, E, E, E, 1 P 1 Cdl Cdl 1 tan( ) P P 1 Cdl Cdl P 1 P Cdl (B7)                                                                       
B2                                                                             O OO OR

  normalized surface coverage of oxidized and reduced species is equal to the normalized surface coverage:      OR .



  Figure2Band 2C provide evidence of the complexity of the phenomena involved: a loss of the proportionality of the variation of  tot at different  O or frequencies, boundary effects and a shift of local minima (ON and ON* -magenta and blue crosses respectively). For example, in the absence of interaction, whatever  O , local ON* minima are located close to the same frequency (not true for  O 0 or 1 -pink dot line), and inversely whatever the frequency, local ON minima are always located at the same  O 2, which is not the case when interactions exist. However, there is always a single remarkable point, which is both an ON state and an ON* state, corresponding to the smallest phase angle called  Min for a single pair {   Min O,;  Min f

  values of the interaction coefficients (aOO) and of Cdl,  * ON and  * ON f are also greatly influenced by the values of interaction parameters and Cdl, but no plateau or constant values of  * ON f , apart

  Alternative Voltammetry at a fixed potential (EDC) and fixed frequency  Temporal acquisition resolution Δtacq= 0.5 ms      acq R 2 f t , that induces a poor resolution at high frequency  10 measurements cycles after a stabilization at EDC during 15s  Amplitude of the sinusoidal part, EAC = 1 mV Here are the ranges used in  and in frequency:   : 20 points around Ep with a step of 5 mV, which are transformed into  using the GLI function (equation 2)

Figure

  Figure 7B displays, as example, an experimental alternative voltammogram of the same TEMPO-SAM performed at the peak potential (     O

  Some conditions are necessary to validate the zeros obtained in (B16).

	In this case,    00 k  P G0 4Cdl       0 tot w w Cdl 1 w lim tan( ) k P k			(C2)
	Solutions between 0 and  are obtained if:                2 2 0 4CdlG P 8 Cdl 4CdlG P                4CdlG P                 O RR OR S a a 2 tot OO 3/2 O0 OO dtan( ) Cdl w e S 1 S d P k 2	(C3)
	   P 0 ; Cdl 0 ;     tot O dtan( ) 0 at    0 ON w   d2              2 OO S 1 S		0	(C4)
	The upper condition is true. The lower condition is not true because: The only solution between 0 and  is
	            2 2 4CdlG P 8 Cdl 4CdlG P 0      4CdlG P 8Cdl 0        ON 0 22 w O, , k S S 1 1 2S		(C5)
	 Cdl 4G	      8 P	0
	Cdl	   P 4G 8	0	with	 P0         4G 8 0 with G	   2 ; 2 and 	 	  0 ; 1
	Cdl cannot be positive, that is impossible.
	Case 2: 	  4CdlG P 0
	R ee RR OR a a a a O O In this case,  O R P G 4Cdl	O	S	RR a	OR a
	OO Solutions between 0 and  are obtained if: OR 2G 2G	(B11)
	O R 2G  4CdlG P OO     2 2 8 Cdl 4CdlG P O O   OO           P 0 ; Cdl 0 ; 0		  0		   4CdlG P
	 4CdlG P          2 2 8 Cdl 4CdlG P 0        O * ON E, 0; 2 , which means that the trend of       4CdlG P 8Cdl 0         Cdl 4G 8 P 0	tan(	 ON E,  *	 O		) is like the trend of	 ON E, 2*  tan (	 O		)
	 ON E,  2* tan (     O ) 4   P Cdl 0 Cdl                   OO OO E, E, E, E, Cdl 1 P P 4G 8 with  P0         4G 8 0 with G  4   Cdl P  2 ; 2 and              O O O 2G                        O O O O O 2G Cdl 1 P   0 ; 1	(B12)
	 Cdl cannot be positive, that is impossible.                      O 2* ON E, 3 O 2 OO O dtan ( ) 32Cd d l 2 P       O 2* ON E, Min O d tan ( d 0 at ) At last,   O 2 (B15) is the unique solution	                        2 O O P P Cdl G Cdl G Cdl 4 4 2	(B13) (B14)
	Zero of the red part is Appendix C	  O 2			(B15)
	Zeros of the blue part are           O 4CdlG P      2 00 tot w 0 kk Cdl Cdl w tan( ) w P w P k		 2 4CdlG P        2 2 4CdlG P 8 Cdl 4CdlG P  		(B16)	(C1)
	Case 1: 	  4CdlG P 0

Acknowledgments

The authors thank Flavy ALEVEQUE for her critical reading of the manuscript, Dr. Christelle GAUTIER for the synthesis of TEMPO-based molecules and the plateau CARMA of the SFR MATRIX (University of ANGERS) for the spectroelectrochemical characterisations.