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An Estimate of a Frequency Characterizing the Electrochemical Stability of a Gold Electrode Modified by MHDA Thiol in Different Ways

A theoretical investigation aimed at estimating a characteristic frequency in the medium-low frequency domain in which the impedance response of a given interface measured by electrochemical impedance spectroscopy (EIS) is almost constant, constitutes the basic idea of this work. A theoretical model was subsequently applied to the data resulting from EIS measurements performed on gold electrodes modified by various ways of 16-mercaptohexadecanoic acid (MHDA) thiol functionalization. Analysis of these data revealed a direct relationship between the way the substrate was modified and this characteristic frequency.


INTRODUCTION
LECTROCHEMICAL biosensors are sensors based on affinity for certain entities. The detection principle is essentially based on the exploitation of the electrical data resulting from the variation of the signal on the modified electrode. Their design requires little energy resources involving relatively low cost, they also offer an ease to miniaturization. All this makes them very attractive as biosensors. [1,2,[4][5][6][7][8][9] The most used electrochemical techniques to evaluate the variation of the surface deposit can be summarized as follows: differential voltametric pulse, cyclic voltammetry and chronoamperometry. The last ones are both transient techniques aimed at the analysis of electrochemical reactions. They provide the characteristics related to transport as well as thermodynamic phenomena. [10][11][12][13][14][15][16][17][18] EIS is one of the most powerful technique used to study interfaces, [18] based on the analysis of the impedance of the equivalent electrical circuit, it allows the quantification of the interfaces obtained by modification of the electrodes either spontaneously when they are immersed in electrolytes, or by electrodeposition, depending on the nature and the purpose of the work performed. Some of our recent works show the quantification of gold electrodes modified by different types of thiols. [19][20][21] Changes at the double layer of the electrode surface are detectable at low frequencies, [18,22] this requires large stabilization time of the harmonic cycles; hence the choice of the frequencies defining the experimental low frequencies range is crucial. The spontanious adsorption of self-assembled monolayer's (SAM's) of alkanethiolates on gold electrode has been the subject of numerous reports. Such SAMs-modified electrodes have been widely used in fundamental science and biosensing as ligand immobilization. [4,10,[23][24][25][26][27][28][29][30][31][32][33] This is mainly due to the ability to form stable and suitable monolayers for different application domains simply by changing their end-groups. However, the performance of functional SAMs depends on many parameters such as concentration, incubation time, temperature, the nature of the solvent, and on the applied potential in the case of an electrochemical characterization. [12] Various works based on frequency studies have been reported in the literature. Generally, the frequency lowest limit reached in the EIS measurements is in the order of 10 mHz. [34][35][36][37][38][39][40] The principal purpose of this work is to estimate a frequency limit defining a domain in the low frequency range, characterized by an impedance response tending to a steady state, with the aim to reduce the experiencing time. In this context experimental investigations were performed on different functionalization types of MHDA thiol by varying the incubation time and/or temperature, or by changing the electro-deposition potential for EIS measurements. The conductivity analysis of the thiolmodified gold electrode of MHDA, normalized with respect to the uncovered gold substrate (σAu-MHDA / σAu), and of its real part (σAu-MHDA), has allowed to delimit the frequency domain in which the frequency variation has a small effect on the conductivity.

THEORETICAL APPROACH
The physical origin of the constant phase element (CPE) is a controversial subject to day. In practice, the CPE expresses a frequency dispersion of the electric double layer capacitance; its impedance is expressed as follows: where j is the complex number, ω is the angular frequency (ω = 2 π f), Q is the frequency independent parameter and n is the CPE power. When n = 1, Q represents the pure capacitance, while for n ≠ 1 the system shows behavior that has been attributed to the surface heterogeneity or to the continuously distributed time constants for charge transfer reactions. For a parallel R-CPE combination (Figure 1.) the impedance becomes: The measured impedance (Z) is the sum of a real part called resistance and an imaginary part named reactance, in the other terms: The real and the imaginary parts of Z are given by: The phase displacement is given by: On the other hand, the real part of the conductivity can be related to the impedance as follows: where d is the thickness of the deposit and S is the active area of the electrode. Its modulus is: Knowing that the conductivity of most disordered and ionic materials obeys to the Josher's universal power law: [18] = + ac dc σdc is the relaxation time conductivity, with: which is called the dielectric time. [18]  The frequency dispersion of the conductivity is then represented by Aω n in many solid electrolytes; it expresses the CPE behavior when a periodic amplitude signal AC is applied.
The total conductivity in presence of dipolar relaxation processes has the following expression: Therefore the conductivity for a material with no dipolar relaxation (parallel RC circuit) is: In the case of a non-dispersive conductivity its real part tends to ∼ (d / S.R) at low frequencies, this expresses the invariability of the real (Z) with the frequency.
The presence of the CPE in the circuit introduces an ohmic loss Aω n previously mentioned, which affects the conductivity in particular for the transport in disordered media far from a viscous media where the number of defects is insignificant.
The frequencies range in the laboratory measurements varies from 0.1Hz to 100 KHz. The model of the conduction in this domain is principally governed by the ionic conduction by hop of the charge carrier in the well system potential created by the particles in the medium, and by relaxations phenomena related to the polarization. [41] Two types of relaxations are considered; the relaxation by deformation and the interfacial relaxation. [42]

THEORETICAL RESULTS
According to the Josher's power law [18] the conductivity is said non dispersive when Z' tends to R. Referring to the model circuit made up by a resistance R connected in parallel to a capacitance C, the real part of the impedance is equals to 90 % R at f = 0.3 fτ, where fτ is the dielectric frequency defined at the maximum of the imaginary part of the impedance. It can be considered that from this frequency, the real part of the impedance is practically independent of the frequency.
In another hand at f = 0.3 fτ the [Eq. (4)] becomes: The plot of the real part of the impedance Z' against n for 0.66 < n < 1 which is corresponding to 0.5 < cos(n ᴫ / 2) < 1 is illustrated by the curve (Figure 2a). Knowing that cos(n ᴫ / 2) represents the fraction of the dissipation energy Diss (%) of each harmonic cycle. [43] This curve shows that as soon as α approaches 0.85, we have Z' tends to 0.9R with an accuracy of 6 %. Therefore, from this limit corresponding to f = 0.3 fτ, it can be considered that the modulus of the impedance is not affected by the variation of the frequency, and that the CPE identifies with a capacitance.
When the value of the real part of the impedance (Z') is 0.9 R, the relation (4) gives: where fmin is a characteristic frequency defined the low boundary bandwidth in which the faradic current is considered as constant. For simplicity we introduce a function: 16(cos ) 9 2 2 ( ) 9 n n π π n n g n (16) such that fmin can be written as: The nearby plot ( Figure 2b) shows a curve varying linearly with n and tending to a limit of 0.3 fτ (n = 1) corresponding to an RC parallel circuit.
On the other hand, for n greater than 0.95, thus g(n) tends to: with an accuracy of 7 %, it leads to: Then: According to the relation (20), the real part of the current tends to zero as soon as f is less than fmin, this corresponds in term of a dielectric time to a period T greater than (3 R Q) 1/n . The dielectric time t -the invert of fminevolves according to the value of n in (20). For n comprised between 0.66 and 0.95 this time is approached by: When n is close to unity (0.95 ˂ n ˂ 1), this time is estimated as: At the limit value of n equal to the unity, this time is given by: This show that the dielectric time for these particular values given by relations (21), (22) and (23) decreases when n evolves from 0.66 to 1 and that the R-CPE parallel circuit evolves towards an R-C parallel circuit, with a limit dielectric time value (3 R Q) for n equal to the unity. This is the ultimate case of an insulating defects-free substrate.   (Figures 3a and  3b). The EC-lab software modeling program was used to fit impedance data. The dotted lines represented in the figures tend to a final frequency corresponding to the frequency in which the impedances Z equal to Rct (Z'' = 0). They were evaluated by a suitable fitting from the imaginary part of the impedance curves versus the frequency. The real and imaginary parts of the impedance as a function of the frequency, of SAMs-modified electrodes subjected to variable potentials are shown above (Figure. 3). The graph shows increasing curves as a function of the potential. They are evolving similarly to those obtained on a naked gold electrode subjected to an increasing polarization up to a potential value, close to the zero charge potential (PZC) of the electrode bare gold (0.18 V). [3] The subsequent curve (Figure 4) depicts the variations of the real and the imaginary parts of the impedance versus frequencies of SAMs for different temperatures and incubation times. The highest value of the dielectric frequency corresponds to the SAM-modified electrode, functionalized at ϑ = 55 °C for 1 hour 15 min, while the smallest value corresponds to the functionalization by the succession of two incubations: one during 1 hour 15 minutes at ϑ = 55 °C, followed by another for 1 hour at ϑ = 4 °C. As the dielectric conductivity [Eq. (10)] and [Eq. (11)] predict, the two conductive and insulating (blocking) behaviors are to be considered for these two cases.

EXPERIMENTAL, RESULTS AND DISCUSSION
The normalized conductivity of SAMs-modified gold electrode plotted in Figure 5 (from the dielectric frequency) depicts decreasing curves exhibiting a tendency of SAMs toward an ordered and stable state.
This with the exception of the SAMs-modified electrode under E = -0.3 V polarization (Figure 5b), that shows decreasing curves in almost uniform way compared to that not subjected to any polarization (Figure 5a), where the normalized conductivity plot illustrates a curve almost flat, and evolving with a higher magnitude order. This result could indicate that SAMs are not stable, evolving slowly to an ordered state, and developing defects which favor the adsorption of the ferrocynate on the SAMs/gold interface. This is why the relative conductivity is in the same magnitude order that of the bare gold electrode at equivalent conditions (important charge transfer through the pinholes).
Note that during conductivity measurements it is clear that it is difficult to go to frequencies that are too low of the order of mHz, because this requires a considerable time for the acquisition of a single value, therefore the   noise can affect the measurements, so we extrapolated the graph obtained experimentally up to the calculated value deduced from the theoretical model. This allowed us to conclude that the normalized conductivity modulus remain almost constant, corresponding to Z modulus equal to the charge transfer resistance (Z = Rct).
As illustrated by the curves of figure 6, the normalized real part of the conductivity varies as a function of the frequency with an order of magnitude between 1 and 1.3 and obeys the universal power law of Josher (12). The plots belonging to the dotted area vary in an order of magnitude between 1 and 1.1. It represents the region in which the real part of the conductivity is quite equal to that of Z ' = Rct, and the graphs of MHDA-thiol types where the conductivity is almost frequency-independent if one considers that the small variations (curves: a, d and e) result from the dipole relaxation in this region as predicted by Josher's universal power law (9) and (13), where A ω α can be neglected. Regarding the values of fmin extracted from Figure 6 recorded with their corresponding theoretical values fmin (α) and summarized in Table 1, we can notice that they are more and less the same as predicted by the model. The plot of Figure 7 graphically represents the variation of the normalized real part of the conductivity as a function of the phase shift. Plots inside the dotted zone have a lower ratio than 1.1, and the phase shift in this region is decreasing until zero, which corresponds to the state at Z '= Rct (thus Z" is zero). The small variations in the conductivity ratio as a function of the phase shift shown by some of the plots (b, e and d) indicate that the plots deviate a little from Josher's universal power law (9), however, this remains insignificant, and the contribution of the capacitive current to the total current is small, with a general tendency towards a faradic current, which is in good agreement with the normalized conductivity plots shown in Figure 5.
In summary, as we have been interested in parameters such as a characteristic frequency at which the faradic current is considered constant, and a resistance Rct which express the transfer of charges through the interface of the electrolyte with the electrode modified with gold. The Nyquist curves corresponding to the results obtained by EIS and modeled by a loop R-CPE are drawn up in Figure 8.
And since the charge transfer occurs at medium and Table 1. Variation of the dielectric frequency measured at E = -0.3 V for various functionalization ways of MHDA, and the corresponding minimal frequencies: theoretical fmin(α) and fmin deduced from Figure 6.  high frequencies, we opted for an equivalent circuit consisting of an Rct-CPE loop in series with the electrolyte resistance Rel which is almost constant and very weak comparing to Rct. It turns out that this circuit describes our interface faithfully. For certain modes of elaboration (ϑ = 4 °C, ϑ = 55°C + ϑ = 4°C, E = -0.3 V), all the points obtained experimentally can be included in this EEC, however for others (ϑ = 55°C, Electrodeposition). The deviations of the experimental values from the ideal semicircles are visible between the symbols and solid lines. The last two points of the Nyquist representation go slightly out of the fit curve obtained with Zsim software. As these points are generally attributed to the phenomena of diffusions through the SAMs, caused by the local imperfections of the SAMs and / or due the roughness of the substrate itself, [21] and which are not the aim of this work, we have excluded them from the equivalent circuit so as not to bias the values of Rct and CPE elements. The results of the fitting of the data obtained by EIS are drawn up in the Table 2. Fit errors are not exceeding 3 %. Rel values can be deduced directly from the offset of the Nyquist semicircle from the Zim axis. The salient point in these results is that the value of the exponent n is very close to 1, which is the value for a perfect capacitor. It is around 0.93 for the different elaboration methods, except for the case of electrodeposition where it is 0.84. The high values of Rct and low of Q respectively reflect the quality of the resulting coating relative to the bare gold electrode.

CONCLUSION
The main problem with electrochemical sensors is to delineate the low frequency threshold where key parameters such as charge transfer resistance Rct, used to quantify any changes occur on the surface. Based on the analogy between the impedances of a loop in R-CPE and another in RC, we tried to delimit the low frequency threshold obtained with a circuit in R-CPE, beyond the electric current is almost stable. This analogy made it possible to correlate the low frequency limit with the dielectric frequency (fτ) of the ligand through a multiplicative factor g(n). This result is confirmed when applied to the analysis of the real part of the conductivity and / or to the normalized total conductivity   We also note that from the characteristic frequency of the ligand [g(n) fτ], the real part of the current function does not evolve too much. This is particularly interesting for detection applications when the sensor is exposed to different ligand concentrations and for which the formation time of each deposited layer is closely related to the low frequency domain. When this time is evaluated from [g(n) fτ] of the receiver, it should facilitate the exploitation of the data by the optimization of the response time of the experiments, particularly when fτ is of the order of some Hertz; knowing that in terms of time, an harmonic cycle of EIS is meanly spent in the last 10 mHz. This concept once applied, would allow considerable time savings in bio-detection, and consequently early diagnosis.