Frequency Dispersion of Double Layer Capacitance of Polyaniline-Coated Electrodes Under the Conducting State

The double layer capacitance of polyaniline (PANi)-coated electrode in acidic solution exhibited little frequency dispersion under the emeraldine (electrically conducting) state, while it showed large frequency dispersion under the insulating state. The former has a possibility of working as such an ideal capacitor that it may generate neither heat in iterative charge-discharge processes nor leakage of stored charge. The frequency dispersion is generally expressed by f for ac-frequency f and a constant λ. This power law is ascribed to orientation of solvent dipoles rather than non-uniform distribution of ions. The value of λ under the conducting state was close to zero, implying the orientation to be facilitated by less interaction of solvents. The less interaction was supported indirectly with strain-stress curves of PANi by atomic force microscopy. λ-Values close to zero were retained even for thick PANi films.


Introduction
Polyaniline (PANi) exhibits electric conductivity in acidic solution under the oxidized state called emeraldine (MacDiarmid, et al., 1989;McManus, et al., 1987).The electric features of PANi-coated films have often been examined by use of ac-impedance techniques.Interpretation of the impedance data inevitably needs equivalent circuits composed of electronic elements such as ideal resistors and capacitors, which should be translated to electrode reaction mechanisms.Since equivalent circuits do not correspond uniquely to electrochemical reaction mechanisms (Brug, et al., 1984), suggested equivalent circuits do not always lead to an interpretation of the mechanisms.There are some complications (Deslouis, et al., 1999) that avoid unique interpretation especially for PANi films.They include (a) redox reactions which convert the states between the insulator and the conductor, (b) a function of electrically conducting porous PANi films as an electrode (Ren, et al., 1995), (c) hysteresis responding to such a long applied voltage change (Heinze, et al., 1987;Feldberg, et al., 1988) that it can be expressed quantitatively by logarithmic time-dependence (Aoki, et al., 1994;Odin, et al., 1992), (d) undistinguished Faradaic currents from capacitive ones (Heinze, et al., 1988;Feldberg, 1984), which has been extensively discussed (MacDiarmid, et al., 1989), and (e) complicated contribution of ions (Deslouis, et al., 1999;Benyaich, et al., 1996).Item (a) is coupled with (b) to expand or shrink the conducting zone over the film, called the propagation of the conducting zone (Aoki, et al., 1992;Aoki, et al., 1994).Since the expansion of the conducting zone increases the capacitive current through (d), it is not easy to estimate reaction mechanisms from equivalent circuits.
The basic concept of the frequency dispersion is not due to a combination of ideal capacitances and resistances, but lies in the charge q stored in a time-dependent DL capacitance C and the voltage V, as represented by the current, I = dq/dt = d(CV)/dt, according to the definition of C. Letting the angular velocity of the ac-voltage V 0 in amplitude be ω , the time-derivative for the ac voltage V = V 0 e iωt becomes ( ) where i is the imaginary unit.If the time t in dC/dt is replaced by ω through t → t' = 1/ω, eq.1 can be rewritten through dC/dt' = -ω 2 dC/dω as ( ) Since the current in eq.2 is composed of a series combination of the real term and the imaginary one, the equivalent circuit should be a parallel combination of the capacitance C p and the resistance R p ≡ -1/(ω 2 dC p /dω), where C in eq.2 was represented as C p for stressing the parallel.The parallel resistance, R p , is caused from a delay of the capacitance associated with the phase shift of 90 o .Since the capacitance itself brings about the response with the 90 o shift, R p turns out to be shifted by 180 o , belonging to an in-phase component.Then the impedance at the electrode|solution (E) interface is given by where the subscript E means the electrode|solution interface.The imaginary component of Z E often has shown a linear relation with the real component, i.e. a line in the Nyquist plot, of which slope -C pE /(ωdC pE /dω) is a constant.Letting the slope be 1/λ E for a constant λ E (close to 0), we obtain (eq.4) as a result of solving the differential equation (Aoki, et al., 2018).Values of λ E are independent of area of an electrode, i.e. independent of choosing a projected area or an area on molecular scale.The logarithmic law in eq. 4 has been verified for various solvents and electrodes (Aoki, et al., 2013;Hou, et al., 2013).Inserting eq.4 into eq.3 yields Since PANi is electrically conducting material especially at the oxidized form, it works as an electrode.The interface between the electrode-like conducting film and the solution should also provide a double layer impedance, denoted by Z F (film|solution) Therefore, there is the other impedance, Z F , at the film|solution interface, which is predicted to have a form similar to Z E , as illustrated in Fig. 1, i.e. ( A film resistance between the electrode and the film|solution interface, R F , is involved in the observed impedance.Furthermore, the solution resistance, R s , takes part in the total impedance, Z. Since the impedance, Z -R s , is given by the parallel combination of Z E and Z F + R F , we have (eq.7)Our subject is to evaluate C pF, and R F by variations of Z with frequency in the PANi films.Dependence of these values on frequency and dc-potentials may provide electronic properties of PANi films.Especially, we pay attention to values of λ F from dC pF /dω, which imply time-dependent parameters of the capacitor.The time-dependence is significant for energetic reversibility of supercapacitors in charge-discharge processes.We will find very small values of λ F , which may demonstrate reversibility of the charge/discharge processes from a physicochemical viewpoint.This work does not include any kinetics for the redox reaction because we want to extract the DL properties accurately not to be disturbed by the faradaic reactions. )]

Experiment
Water was distilled and then ion-exchanged with an ultrapure water system, CPW-100 (Advantec, Tokyo).All the chemicals were of analytical grade.The working electrode was a platinum wire 0.5 mm in diameter, which was inserted into solution by a given length ca. 8 mm.The accurate length was measured by means of an optical microscope.The inserted wire electrode is useful for avoiding the stray capacitance, which is caused at disk-shielded electrodes by crevice of insulator|electrode boundaries.The reference electrode and the counter electrode were, respectively, Ag|AgCl in saturated KCl and a platinum coil.Cyclic voltammetry and ac-impedance measurements were carried out with a potentiostat, Compactstat (Ivium Tech., Netherlands).Ac-impedance was obtained by applying ac-voltage of the 10 mV amplitude superimposed on a given value of dc-potential (E dc ) ranging from -0.1 to 0.5 V vs. Ag|AgCl in the frequency domain from 1 Hz to 10 kHz.
PANi films were synthesized with electrochemical oxidation of 0.1 M (= mol dm -3 ) aniline in 1.0 M sulfuric acid at the platinum wire.The oxidation was made by cyclic voltammetry of which potential ranged from -0.15 to 0.80 V vs. Ag|AgCl at a given cycle number, ranging from 40 to 300, at the scan rate 30 mV s -1 .Atomic force microscope (AFM) was Nanocute (Hitachi).Film thickness was estimated from the assumptions of the one electron reaction of leucoemeraldine to emeraldine per four-aniline unite (MacDiarmid, A.G., & Epstein, A.J., 1989) and the density 1.35 g cm -3 of PANi films.Then the charge density 0.01 C cm -2 for the redox reaction corresponds to 0.3 μm thickness.The film thickness was 0.5 μm, unless otherwise stated.

Results and Discussion
Fig. 2 shows Nyquist plots of a PANi film at three dc-potentials, E dc .The plots at the sufficiently (a) reduced and (c) oxidized PANi films show slanted lines rather than a vertical line, like the CPE behavior (Brug, et al., 1984;Lasia, et al., 1999;Nyikos, et al., 1985;Zoltowski, et al., 1998).When Z E >> Z F + R F , eq.7 becomes Z = Z F + (R s + R F ), which is composed of a series combination of the impedance at the film|solution interface and the resistances of the PANi film and the solution.The real and the imaginary parts are represented respectively by (eq.8) or Then the Nyquist plot shows a line with a slope of 1/λ F and an intercept of R s + R F on the Z 1 axis.On the other hands, eq.7 for Z E << Z F + R F becomes Z = R s + Z E , of which Nyquist plot is given by Since the plots in Fig. 2 for the reduced (a) and the oxidized (c) forms take lines, either of eq.9 or eq.10 is possible, i.e.
(either λ F or λ E ) = 0.5 (a) and 0.02 (c).The impedance values at the oxidized PANi (at 82 Hz to be marked as a cross) are much smaller than those at the reduced one.Since the values of the impedance vary largely with the redox state of PANi films, the difference in the slopes of the Nyquist plots should be ascribed to λ F rather than λ E .As a result, we can infer Z E >> Z F + R F .The intercept represents R s + R F , according to eq.9.Values of the resistance (R S +R F ) for E dc < 0.1 V are six times larger than those for E dc > 0.2 V in Fig. 3.In contrast, the resistivity of the reduced PANi film has been reported to be larger by a few order in magnitude than that of the oxidized form (McManus, et al., 1987;Paul, et al., 1985;Ofer, et al., 1990;Csahók, et al., 2000).Our result in Fig. 3 is consistent with the large ratio of the resistivity, because the difference between the maximum resistance E dc < 0.1 V and the minimum one E dc > 0.2 V can be attributed to the film resistance (R F ) through the redox reaction, the net ratio being a few order in magnitude.(eq.11)According to eq.6, the admittance can be rewritten through the f-dependent capacitance: Values of the imaginary part, Y F2 , calculated from eq.11 were plotted against the frequency in the logarithmic scale in Fig. 4. The linearity supports the validity of eq.12, of which slopes and intercepts give λ F and C pF,1Hz , respectively.5 shows the variation of C pF,1Hz with the dc-potentials on the right axis.The oxidized PANi which lies in the potential domain E dc > 0.15 V exhibits values of C pF,1Hz much larger than of the reduced one (in E dc < 0.15 V).The large capacitance values can readily be understood in terms of the porous, conducting PANi film, which provides huge net area of the electric conducting material, as if the electrode area were large.They are 1000 times as large as those at Pt|aqueous solution interfaces.In contrast, the reduced PANi exhibits 400-700 μF cm -2 , which is more than 10 times larger than at Pt|aqueous solution interfaces.The 10 time larger capacitance supports the assumption of Z E >> Z F + R F for eq.8 and eq.9, or supports that the capacitance at the film|solution interface is much predominant over that at the electrode|solution interface.As a result, the capacitance in Fig. 5  Fig. 5 also shows the variation of λ F with the dc-potentials on the left axis.The oxidized PANi provides values of λ F (= 0.04) as small as values at the highly oriented pyrolytic graphite (λ = 0.06) and thin (< 0.1 μm) films of graphene flakes in aqueous solutions (Wang, et al.,2015) in comparison with at platinum electrodes (λ ≈ 0.1) (Aoki, et al., 2013;Hou, et al., 2013;Zhao, et al., 2014;Hou, et al., 2014).Smaller values of λ imply less contribution of frequency dispersion or of the associated resistance expressed by ω -2 (dC/dω) -1 in eq.2.They are ascribed to high feasibility of the orientation of solvent dipoles (Aoki, 2016).The thicker are electrically conducting materials, the longer is the orientation time of dipoles generally (Wang, et al., 2015).PANi films are often as thick as of μm order, and hence are predicted to yield large values of λ F .The evaluated small values of λ F can be attributed to the high feasibility of the orientation of water molecules around the oxidized form of PANi molecular wires.
Figure 6.Force-depth curves when a cantilever of AFM was pressed on the PANi film under (a) the reduced state and (b) the oxidized one The feasibility may be related with microscopic Young's modulus of PANi.The mechanical properties such as macroscopic moduli of PANi films have been discussed in the context of the redox states (Mohamoud, et al., 2007;Valentová, et al., 2010;Han, et al., 2004;Yu, et al., 2009).The Young moduli of the oxidized state are smaller than those of the reduced one (Valentová, et al., 2010).We regarded the modulus as a slope of stress-strain curves or force-depth curves by the force mode of atomic force microscope (Passeri, et al., 2011).Fig. 6 shows the force-depth curves of the reduced and the oxidized forms of PANi films.The curves varied with locations of touching a cantilever with the film, but they were always composed of two lines with a critical point.Two lines with a junction have been exhibited in most PANi films (Mohamoud, et al., 2007;Valentová, et al., 2010;Han, et al., 2004;Yu, et al., 2009;Passeri, et al., 2011) although complicated curves appeared in a single polymer chain (Yu, et al., 2009).The slope of force against the depth, proportional to the Young modulus, of the oxidized form (8.8±3.8 mV nm -1 ) is smaller than that of the reduced one (10.5±5.9 mV nm -1 ), where these values were averaged for 46 curves at different locations, and the errors mean the standard deviation.Therefore, the oxidized form is higher feasibility of the orientation for generating the capacitance than the reduced form.This fact supports the dependence of λ F on the dc-potential.
Supercapacitors are desired to have such physical properties as a) high density of capacitance, b) high voltage, c) reversibility without heat generation, and d) the absence of leakage of current through the DL layer.Item d) is more essential from the physicochemical viewpoints than the others which have resorted practically to hybridization of polyaniline with carbons (Cheng, et al., 2013;Tayel, et al., 2016) as pseudocapacitors (Liu, et al., 2014;Gup, et al., 2015;Leary, et al., 2016;Liu, D., et al., 2015;Sarangapani, et al., 1996) and morphological variations I , II , III (Grover, et al., 2016;Chen, et al., 2013;Singu, et al., 2012).It includes two causes: one is the leakage due to time-variation of voltage through R pF (= -1/(ω 2 dC pF /dω)) and the other is due to redox reactions.The former is related with λ F .Since an increase in the film thickness generally increases λ F , item a) is competitive with λ F in d).In order to know a degree of the competition, we examine the relation between values of λ F and the film thickness or the redox charge involved in PANi films.We determined the amount of the redox charge from the linear sweep voltammograms on the assumption that the observed current, I(E), was the sum of the current, I rd (E), for the surface wave of the redox reaction and the capacitive current of the oxidized PANi.The capacitive component can be expressed in terms of the integral of kI rd (E), as illustrated in the inset of Fig. 7, where k is a proportional constant.The solution of the integral equation yields (Tezuka, et al., 1989) (eq.13) or (eq.13')where E in is the initial potential.A value of k was determined so that I rd (E) = 0 for sufficiently positive potential, e.g.0.4 V vs. Ag|AgCl.We regard the integral of I rd as the redox charge Q.We examined the reliability of k within 6 % errors for different film thicknesses and scan rates, as consistent with the previous result (Tezuka, et al., 1989).Fig. 7 shows the variation of λ F for the oxidized state (at E dc = 0.4 V) with the redox charge density, σ = Q/S for the projected surface area S. Values of λ F did not vary with the film thickness within the standard deviation.In contrast, C pF,1Hz increased proportionally with the film thickness (on the right axis), as conventionally predicted without complications.Therefore, item a) is not competed with λ F .Even a thick PANi film can keep values of λ F to be close to zero, i.e. an ideal capacitor.

Conclusion
Ac-impedance of PANi films obeys the equivalent circuit in Fig. 1, in which most of the ac-current flows through the film|solution rather than electrode|solution, represented by Z E >> Z F + R F .The capacitance of the PANi film is followed by the power law of the frequency, similar to the power law at the electrode|solution interface.The values are quite different, i.e., C pF >> C pE and λ F < 0.04 <λ E .Since the conducting state of the PANi can be regarded as an assembly of conducting wires, the former inequality is obvious for the electrode area.The latter is significant for applying PANi films to supercapacitors in the viewpoint of suppressing heat generation at the film|solution interface in charge-discharge processes.

Figure 2 .
Figure 2. Nyquist plots at the PANi-coated Pt electrode in 1 M H 2 SO 4 solution at E dc = (a) 0.0, (b) 0.15 and (c) 0.40 V vs. Ag|AgCl.The cross marks are at f = 82 Hz Fig. 3 shows variation of the intercept of the Nyquist plots on the Z 1 axis with the dc-potentials, together with the voltammogram of the PANi film in the upper part.The intercept represents R s + R F , according to eq.9.Values of the resistance (R S +R F ) for E dc < 0.1 V are six times larger than those for E dc > 0.2 V in Fig.3.In contrast, the resistivity of the reduced PANi film has been reported to be larger by a few order in magnitude than that of the oxidized form(McManus, et al., 1987;Paul, et al., 1985;Ofer, et al., 1990;Csahók, et al., 2000).Our result in Fig.3is consistent with the large ratio of the resistivity, because the difference between the maximum resistance E dc < 0.1 V and the minimum one E dc > 0.2 V can be attributed to the film resistance (R F ) through the redox reaction, the net ratio being a few order in magnitude.

Figure 3 .
Figure 3. (A) Variation of the resistance with the dc-potential, and (B) voltammogram of the PANi film in 1 M H 2 SO 4 solution at the scan rate of 10 mV s -1We let the admittance of Z F be Y F , i.e.,

Figure 4 .
Figure 4. Logarithmic plots of Y F2 against frequency at E dc = (a) 0.0 and (b) 0.4 V vs. Ag|AgCl should be C pF rather than C pE .Although values of the net area of the interface are ambiguous, values of λ E and λ F have been determined independently of the area from the plot in Fig. (4).

Figure 7 .
Figure 7. Variations of λ F (circles) and C pF,1Hz (triangles) with the charge density of the PANi film The bar means the standard deviation of λ F .The inset illustrates a linear sweep voltammogram (solid curve) and a charging current (dotted curve)