On Selection of the Perturbation Amplitude Required to Avoid Nonlinear Effects in Impedance Measurements

Transcription

1 n Seletion of the Perturbation Amplitude Required to Avoid Nonlinear Effets in Impedane Measurements Bryan Hirshorn, a Bernard ribollet, b and Mark E. razem a, * a Department of Chemial Engineering, University of Florida, Gainesville, Florida 32611, USA b UPR15 du CNRS, Université P. et M. Curie,, Plae Jussieu, Paris 75252, Frane (Reeived April 28; aepted 6 July 28) Abstrat. Numerial simulations of eletrohemial systems were used to explore the influene of large-amplitude potential perturbations on the measured impedane response. he amplitude of the input potential perturbation used for impedane measurements, normally fixed at a value of 1 mv for all systems, should instead be adjusted for eah experimental system. Guidelines are developed for seletion of appropriate perturbation amplitudes. A harateristi transition frequeny is defined that an be used to tailor a frequeny-dependent input signal to optimize signal-to-noise levels while maintaining a linear response. 1. Introdution Eletrohemial impedane spetrosopy is a powerful tehnique that has been used to investigate a broad range of experimental systems with very different eletrohemial properties. While it an be onsidered a generalized transfer funtion approah, eletrohemial impedane spetrosopy usually involves a measured urrent response to a potential input. In its ommon appliation, the tehnique relies on use of a small input signal amplitude to ensure a linear response whih an be interpreted using theories of linear transfer funtions. Most experimentalists employ a 1-mV input signal amplitude, but there is reason to expet, given the wide range of eletrohemial properties investigated with this tehnique, that this amplitude may not be optimal for some experimental systems. razem and o-authors have investigated the error struture of impedane measurements, using a measurement model approah to quantify both stohasti and bias errors in repliated spetra. 1 Minimization of stohasti errors serves to improve the regression analysis for interpretation of spetra. A large input amplitude generally redues the stohasti errors, but an amplitude that is too large results in errors assoiated with the nonlinear response. he seletion of appropriate input amplitudes has drawn interest in the literature. Darowiki investigated the effet of the input amplitude on the error of polarization or harge-transfer resistane obtained from impedane measurements. 5 He showed that the impedane spetrum of a nonlinear eletrial system depends on both the frequeny and amplitude of the input signal. *Author to whom orrespondene should be addressed. meo@he.ufl.edu Israel Journal of Chemistry Vol pp DI 1.156/IJC

2 13 He also demonstrated that the polarization resistane unorrupted by nonlinear effets an be determined by extrapolating to the zero value of the amplitude of the input signal. Diard et al. studied the dependene of impedane measurement error on the eletrode potential and the sinusoidal voltage amplitude for a Nernstian redox system. 6 hey showed that for their given system the impedane measurement error was independent of frequeny in the low frequeny range. In a separate work, Darowiki showed that, for systems with a non-negligible hmi resistane, the interfaial potential differs from the applied potential signal. 7 He derived an expression for the interfaial potential using a series expansion approah that relates the interfaial potential to the amplitude of the input signal, the input frequeny, the eletrolyte resistane, the double layer apaitane, and the kineti parameters. Darowiki found that, for all input amplitudes, the effetive interfaial potential hanges with frequeny due to the frequeny dependene of the harging urrent, having a maximum amplitude at low frequeny and tending toward zero at high frequeny. As a result of this effet, the influene of a large input amplitude hanges with frequeny. Darowiki provided a method for determining the frequeny for whih impedane measurements will be linear in harater. 7 Darowiki s observations were supported by the modeling work of Popkirov and Shindler, who developed syntheti data for a hargetransfer resistor obeying Butler Volmer kinetis in parallel with a double layer apaitane. 8 heir results showed that the perturbation amplitude had no effet on the impedane values in the high-frequeny range where the harging urrent dominates. Alternatively, in the low-frequeny range, a derease of the impedane values was observed with inreasing input signal amplitude. here has been signifiant effort to determine the linear impedane values when nonlinear errors are not negligible. Diard et al. quantified the deviation of the measured polarization resistane due to nonlinearity using a suessive derivative approah Diard et al. developed expressions for the eletrohemial response of a two-step reation to a sinusoidal perturbation that results in nonlinear impedane. 12 hey used numerial methods to show that deviation from the linearized system depended on the kineti parameters, the eletrode potential, the input amplitude, and the frequeny. Miloo used a aylor series method to determine the linear impedane response when the perturbation aused a nonlinear response. 13 From an experimental perspetive, Van Gheem et al. 1 and Blajiev et al. 15 used multisine broadband signals to detet nonlinearities in eletrohemial systems. hese groups were able to distinguish measurement errors aused by stohasti noise and errors aused by nonlinear distortions. Urquidi-Madonald et al. used an experimental approah to show that the ramers ronig transforms are insensitive to the ondition of linearity. 16 A large input perturbation is sometimes used to explore the system response over a large potential range in a single impedane measurement. Wilson et al. used Non-Linear Eletrohemial Impedane Spetrosopy (NLEIS) to investigate harge-transfer mehanisms in solid oxide materials. 17,18 he large-amplitude perturbations generated harmonis in the response, whih were used to gain insight into nonlinear harge-transfer mehanisms. Engblom et al. showed that two-eletron transfer mehanisms generated larger amplitude harmonis than did the orresponding one-eletron transfer mehanisms. 19 ada et al. used harmoni analysis to identify the onset of rak initiation and orrosion fatigue indued from ylial stressing of metalli materials. 2 Darowiki used harmoni analysis of the Warburg impedane for the determination of diffusion oeffiients. 21 Gabrielli et al. derived the harmonis generated from afel behavior and disussed the utility of harmoni analysis in providing basi kineti information. 22 In other works, harmonis are measured and evaluated to obtain kineti information in orrosion systems. 23, he objet of this work is to develop a generalized approah for the interpretation of the impedane response of nonlinear systems subjeted to large input perturbations. he impedane response was generated by use of Fourier integrals following the methods employed by ommerial impedane instrumentation. his work was used to provide guidelines for the appropriate perturbation amplitude whih an be used by the experimentalist. 2. heoretial Approah he nonlinear response in eletrohemial systems typially results from the potential dependene of Faradai reations. For example, both afel and Butler Volmer reation kinetis display an exponential dependene on the interfaial potential. he total urrent passed through the eletrode ontributes to harging the interfae and to the Faradia reation. hese ontributions are presented in parallel in the iruit presented in Fig. 1a, where the use of a box for the Faradai reation is intended to emphasize the ompliated and nonlinear potential dependene. Addition of an hmi harater of the eletrolyte auses the interfaial potential V to differ from the applied potential U. his effet is illustrated in Fig. 1b. he applied potential U an be expressed as a sinu- Israel Journal of Chemistry 8 28

3 135 (a) (b) () Fig. 1. Ciruit representations for the systems onsidered in the present work: (a) non-hmi Faradai system. (b) hmi Faradai system. () hmi onstant harge-transfer resistane system. soidal perturbation about a steady value U as U = U + DU os( ~ t) (1) where DU is the input amplitude, w is the input angular frequeny, and t is time. In the absene of an hmi resistane, as shown in Fig. 1a, the applied ell potential U and the interfaial potential V are equal. In the presene of an hmi resistane R e the applied ell potential is related to the interfaial potential by U = V + (i f + i C )R e (2) he Faradai urrent density an be expressed as i f = i [exp(b a (V V )) exp( b (V V ))] (3) or equivalently, i f = a exp(b a V) exp( b V) () where b a and b are the anodi and athodi oeffiients with units of inverse potential and inludes the exhange urrent i and the equilibrium potential differene V as a = i exp( b a V ) and = i exp(b V ). When b a and b are related through the symmetry fator, the general form of eq for independent reations simplifies to that of Butler Volmer kinetis. he apaitive urrent is expressed as dv ic = Cdl dt (5) where C dl is the double layer apaitane. he total urrent passing through the ell is the sum of the Faradai and apaitive ontributions, i.e., i = i f + i C (6) In the absene of an hmi resistane, i.e., as shown in Fig. 1a, eqs 1 6 an yield an analyti expression for urrent density as a funtion of applied potential U = V; i =- ~ C sin] ~ tg+ exp_ b ^V + os] ~ tghi dl a a - exp_ - b ^V + os] ~ tghi (7) he urrent and potential terms annot be separated in the more general ase given in Fig. 1b, and a numerial method must be employed. 3. Numerial Method A numerial method was used to estimate the timedependent urrent response to a sinusoidal potential input using the eletrial iruit presented as Fig. 1b for whih the harge-transfer resistane R t is a nonlinear funtion of potential. he relationship between urrent and potential an be expressed in the form of a single differential equation, dv R dt C R V e dl e R t = U + DU os] ~ t t] g m g (8) in whih R t (t) is a funtion of potential and, therefore, a funtion of time. Equation 8 an be solved analytially for fixed R t using the integrating fator approah. he equivalent iruit of suh a system is shown in Fig. 1. he solution of eq 8 for fixed R t an be expressed as where CdlReRt V] t g ~ = Aos] ~ tg+ sin] ~ tgm + V * ] Rt + Reg (9) 2-1 D U] Rt + Reg ] Rt + Reg 2 A = 2 = 2 + ~ G (1) Rt ] CdlReg ] RtCdlReg and V* is a onstant of integration. A similar approah was taken by Xiao and Lalvani, who solved a linearized form of the afel equation to develop expressions for potential and urrent in a orrosion system. 25 he value of the harge-transfer resistane at a given potential V(t) an be alulated from the slope of the interfaial polarization urve, i.e., R t (t) = ( a b a exp(b a V(t))+ b exp( b V(t))) 1 (11) Under the assumption that, for short time periods, i.e., small movements on the polarization urve, the hargetransfer resistane is onstant, an iterative proedure using eqs 9 and 11 was used to alulate the development of V and i as funtions of time. his proedure allowed for the omplete determination of the system desribed Hirshorn et al. / Nonlinear Effets in Impedane Measurements

4 136 by a potential-dependent harge-transfer resistane. he analyti equations derived for a fixed harge-transfer resistane an be used to approximate the solution to Fig. 1b for whih the harge transfer resistane varies with interfaial potential. he rationale for this approximation is developed in Setion.. he impedane response was alulated diretly for eah frequeny using Fourier integral analysis. 26 he fundamental of the real and imaginary omponents of the urrent signal, for example, an be expressed as 1 Ir ] ~ g = # I] t gos] ~ tgdt (12) and 1 Ij] ~ g =- # I] t gsin] ~ tgdt (13) respetively, where I(t) is the urrent signal, w is the input frequeny, and is the period of osillation. Similar expressions an be found for real and imaginary omponents of the potential signal. he real and imaginary omponents of the impedane an be found from Ur + juj Zr ] ~ g = Re( 2 (1) Ir + jij and Ur + juj Z j] ~ g = Im( 2 (15) I r + ji j respetively, where j represents the imaginary number. he advantage of the numerial approah employed here was that it ould be applied to general forms of nonlinear behavior, inluding onsideration of a potential-dependent apaitane.. Results and Disussion he objetive of this presentation is to make the analysis of system nonlinearity useful to the experimentalist. o that end, guidelines are provided to assess appropriate perturbation amplitudes as funtions of kineti and hmi parameters, and experimental methods are disussed for assessing the ondition of linearity. he frequeny dependene of the interfaial potential an be exploited to tailor input signals..1 Errors in Assessment of Charge-ransfer Resistane In the limit that the perturbation amplitude tends toward zero, the polarization resistane an be expressed as 2U R lim p, d DU 2i n (16) = " f ] g, where U is the ell potential, DU is the amplitude of the input ell potential signal, i f is the Faradai urrent density, i () is the onentration of speies i evaluated at the eletrode surfae, and g k is the frational surfae overage of adsorbed speies k. Equation 16 an be expressed in terms of an effetive harge-transfer resistane as 2V R lim t, d DU 2i n (17) = " f ] g, where V is the interfaial potential. For the Butler Volmer kinetis desribed in the previous setion, the linear value of the harge-transfer resistane is given as -1 R t, = ^aba exp^ba V h + b exp^-b V hh (18) where V represents the potential at whih the impedane measurement is made. he alulated impedane response is given in Fig. 2 with applied perturbation amplitude as a parameter. he system parameters were R e = 1 Ωm 2, a = = 1 ma/m 2, b a = b = 19 V 1, C dl = 2 mf/m 2, and V = V, giving rise to a linear harge-transfer resistane R t, = 26 Ωm 2. he results presented in Fig. 2 are onsistent with the observation of Darowiki that the measured harge-transfer resistane dereases with inreased amplitude of the perturbation signal. 5 i i k k Fig. 2. Calulated impedane response with applied perturbation amplitude as a parameter. he system parameters were R e = 1 Ωm 2, a = = 1 ma/m 2, b a = b = 19 V 1, C dl = 2 µf/m 2, and V = V, giving rise to a linear harge-transfer resistane R t, =26 Ωm 2. Israel Journal of Chemistry 8 28

5 137 As suggested by eq 16, the derease in the measured harge-transfer resistane with inreased amplitude is not a general result and depends on the polarization behavior. 9.2 ptimal Perturbation Amplitude A guideline for seletion of the perturbation amplitude needed to maintain linearity under potentiostati regulation an be obtained by using a series expansion for the urrent density. Similar series-expansion approahes that express deviations from linearity in eletrohemial systems have been provided by ooyman et al., 27 Gabrielli et al., 22 and Diard et al. 9,1,12 For a system that follows a Faradai reation, the urrent density response to an interfaial potential perturbation is given by hus, or where and V] t g = V + os] ~ tg (19) i f (t) = a exp(b a V) exp( b V) (2) i f ] t g = a exp^ba] V + os ~ tgh - exp^- b ] V + os ~ tgh (21) i f ] t g = a exp] ba os ~ tg - exp] -b os ~ tg (22) A aylor series expansion yields a = a exp^ba V h (23) = exp^-b V h () J ba os ~ t ba os ~ t N 1 + ba os ~ t + 2! + 3! + i f ] t g = a n n n ba os ~ t f + n! + f L P J b os ~ t b os ~ t N 1 - b os ~ t + 2! - 3! + - n n n b os ~ t f + n! + f L P (25) he mean value of the urrent i f (t) is, for equal to an integer number of yles, By taking into aount the formula i f 1 = # i f ] t gdt (26) 1 n 1 os xdx = n os x sin x + - n os # # n n - 1 n - 2 xdx (27) and observing that sin =, n n 1 n 2 os xdx = - n os - xdx # # (28) If n is an even number, n n 1 n 3 1 os xdx = - - # n n - 2 g 2 (29) and if n is an odd number, the value of the integral is equal to zero. hus, the mean value of i f (t) is i f / / (3) 3 2 n n n ba V b V 1 n D D = a e + 1 n 2 - e + n 2 n = 1 ^2 n! h o n = 1 ^2 n! h o Evaluation of the harmonis of the nonlinear urrent response an be ahieved by introdution of the trigonometri expressions os 2x = 2os 2 x 1 (31) and os 3x = os 3 x 3osx (32) By onsidering only the three first terms of the aylor series, i f (t) beomes J ba 3ba N 1 + ba os] ~ tg i f ] t g = a ba ba + os] 2~ tg+ os] 3~ tg L P J b 3b N 1 - b os] ~ tg b ba + os] 2~ tg- os] 3~ tg L P (33) he limitation to the first three terms of the aylor series gives for the mean value only the first term of the series (see eq 3). Equation 33 an be written as i f ] t g = i f + i f, 1 os] ~ tg+ i f, 2 os] 2~ tg+ i f, 3 os] 3~ tgf (3) where the d urrent is given by ba b i f = a 1-1 (35) and the first harmoni or fundamental is given by ba 3b i f, 1 = a ba + b (36) For smaller than. 2 a -, the varia- 2 2 aba - b Hirshorn et al. / Nonlinear Effets in Impedane Measurements

6 138 tion of the d urrent is smaller than 1 perent. For aba + b smaller than , the variation of the fundamental is smaller than.5 perent. aba + b Appliation of a large-amplitude potential perturbation to a nonlinear system results in harmonis that appear at frequenies orresponding to multiples of the fundamental or applied frequeny. Appliation of a large-amplitude potential perturbation to a nonlinear system hanges both the steady-state urrent density and the fundamental urrent response. he impliation of this result is that the impedane response will also be distorted by appliation of a large-amplitude potential perturbation. In the presene of a signifiant hmi resistane, the guideline for the low-frequeny perturbation amplitude is aba + b U. 2 1 b b R R e D = 3 3 (37) a a + t,obs where R t,obs is the effetive harge-transfer resistane measured at the given perturbation amplitude. hus, a larger perturbation amplitude should be applied for systems where (1 + R e /R t,obs ) is muh larger than unity. he rationale for eq 37 is developed in Setion.. he perent error in the low-frequeny impedane asymptote assoiated with use of a large-amplitude potential perturbation is given in Fig. 3 under the assumption of afel kinetis with b as a parameter. At a value of b =.2, the error in the low-frequeny impedane asymptote is.5 perent. he orresponding perturbation amplitude is 1. mv for a afel slope of 12 mv/ deade, and 5.2 mv for a afel slope of 6 mv/deade..3 Experimental Assessment of Linearity As indiated by Urquidi-Madonald et al., 16 the ramers ronig relations do not provide a useful tool for identifying errors assoiated with nonlinear response to a large perturbation amplitude. Sequential impedane measurements onduted with different perturbation amplitudes an be used to find the optimal input perturbation, but this proess is time onsuming. A more rapid assessment of a nonlinear system response an be obtained by observing distortions in Lissajous plots at low frequeny. Lissajous plots are presented in Fig. with perturbation amplitude and frequeny as parameters. he system parameters were R e = Ωm 2, a = = 1 ma/m 2, b a = b = 19 V 1, C dl = 2 mf/m 2, and V = V, giving rise to a linear hargetransfer resistane R t, = 26 Ωm 2. At the low frequeny of.16 Hz, a straight line is observed for a perturbation amplitude of 1 mv; whereas, a sigmoidal shape is evident for a perturbation amplitude of 1 mv. he Fig. 3. he error in the low-frequeny impedane asymptote assoiated with use of a large amplitude potential perturbation. Fig.. Lissajous plots with perturbation amplitude and frequeny as parameters. he system parameters were R e = Ωm 2, a = = 1 ma/m 2, b a = b = 19 V 1, C dl = 2 µf/m 2, and V = V, giving rise to a linear harge-transfer resistane R t, =26 Ωm 2. sigmoidal shape is seen beause the alulations were performed at V = V. A deviation from a straight line will be seen for large amplitudes at larger or smaller applied potentials, but the shape will be altered. At the larger frequeny of 16 Hz, the differenes between the smaller and larger perturbation amplitudes beomes less apparent, and the two urves superimpose as a perfet irle at large frequenies due to the domination of the apaitive urrent. Similar results are seen for the ase where R e, with the exeption that the Lissajous plot appears as a straight line at both low and high frequenies. he influene of nonlinearities is seen at low frequeny. Israel Journal of Chemistry 8 28

7 139 Fig. 5. Maximum variation of the interfaial potential signal as a funtion of frequeny for parameters U = 1 mv, R e = 1 Ωm 2, C dl =2 µf/m 2, and R t, = 26 Ωm 2. Fig. 6. Maximum variation of the interfaial potential signal as a funtion of frequeny for parameters U = 1 mv, R e = 1 Ωm 2, C dl =2 µf/m 2, R t, =26 Ωm 2. he solid urve is V max resulting from the numerial simulation. he dashed urve is V predited from eq 9 using R t,obs =19 Ωm 2, whih dereases from the linear value, R t, =26 Ωm 2, due to the large input perturbation.. Frequeny Dependene of the Interfaial Potential In the absene of hmi resistane or when a linear approximation is suffiient, the interfaial potential is a sinusoidal quantity and represents the amplitude of the interfaial potential. For large perturbations the interfaial potential signal ontains nonlinear distortions, thus, in the following disussion max represents the maximum variation of the interfaial potential signal. he alulated max is presented in Fig. 5 as a funtion of frequeny for a system with parameters DU = 1 mv, R e =1 Ωm 2, C dl = 2 mf/m 2, and R t, =26 Ωm 2. At high frequenies max is damped and tends toward zero. Equation 9, although derived for a onstant hargetransfer resistane, an be used to approximate the interfaial potential of a nonlinear system if the hargetransfer observed at low frequeny, R t,obs, is used in the equation. As shown in Fig. 6, max resulting from the numerial alulation is ompared to alulated from eq 9. he agreement shown between the numerial solution and the solution obtained using eq 9 onfirms that eq 9 is useful for approximating the behavior of the interfaial potential, even though it is derived from a onstant harge-transfer resistane. It should be noted that the sinusoidal time-domain approximation will not ontain the nonlinear distortions and will have maximum error at low frequeny. Inspetion of the low-frequeny and high-frequeny limits of eq 9 provides insight into the onditions at whih ideal linearity is approahed, i.e., DU Rt, obs lim = (38) ~ " R + R and lim = ~ " 3 t, obs DU R ~Cdl e e (39) respetively, where R t,obs is the observed harge-transfer resistane at the given perturbation amplitude. Although eqs 38 and 39 are derived for the linear system, the results shown in Fig. 6 onfirm that these equations are useful for approximating max, as long as the harge-transfer resistane R t is replaed by the harge-transfer resistane influened by a nonlinear response R t,obs. As shown in eq 38, max dereases in the low-frequeny range with inreasing hmi resistane. As shown in eq 39, max dereases in the high-frequeny range with inreasing frequeny. Both the limits of high hmi resistane and high frequeny approah the ondition of ideal linearity. he frequeny dependene of max and the orresponding Lissajous plots are shown in Figs. 7a and 7b, respetively. As shown in Figs. 7a and 7b, a linear response is obtained for a 1 mv input amplitude when the hmi resistane is large; whereas, a nonlinear response is seen for the same perturbation amplitude when the hmi resistane is small. his result is onsistent with eq 37. he linearity of the system response is governed by the value of max. A harateristi frequeny for the transition from the low-frequeny behavior to the high-frequeny behavior was obtained as 1 1 ~ t = R,obsC + () t dl ReCdl where w t is the infletion point of max versus frequeny, as shown in Fig. 8. his frequeny marks the transition from low-frequeny nonlinear behavior to high-frequeny linear behavior. Hirshorn et al. / Nonlinear Effets in Impedane Measurements

8 1 (a) (b) Fig. 8. Infletion point of V max is loated at the transitional frequeny defined by eq ( U = 1 mv, R e = 1 Ωm 2, C dl = 2 µf/m 2, R t, = 26 Ω m 2 ). (a) Fig. 7. Calulated results for parameters U = 1 mv, C dl = 2 µf/m 2, and R t, = 26 Ωm 2 with hmi resistane as a parameter. (a) Maximum variation of the interfaial potential signal as a funtion of frequeny. (b) he orresponding Lissajous plots at a frequeny of.16 Hz. he low-frequeny limit given by eq 38 is equivalent to that derived by Darowiki (see, e.g., eq 16 in Darowiki 28 ). he advantage of using eq 9 is that it approximates the interfaial potential aross all frequenies while providing a muh simpler expression than those derived by the series expansion approah used by Darowiki. he harge-transfer resistane was alulated using eq 11 for eah time-dependent value of V generated during the development of syntheti data. At eah frequeny, the harge-transfer resistane was averaged over a omplete sinusoidal yle yielding the effetive hargetransfer resistane. he onsequene of the hange in interfaial potential with frequeny is illustrated in Figs. 9a and 9b for parameters R e = 1 Ωm 2, C dl = 2 mf/ m 2, and R t, = 26 Ωm 2 with applied perturbation amplitude as a parameter. At low frequenies, the effetive harge-transfer resistane dereases with inreased in- (b) Fig. 9. Calulated results for parameters R e = 1 Ωm 2, C dl = 2 µf/m 2, and R t, = 26 Ωm 2 with applied perturbation amplitude as a parameter. (a) Maximum variation of the interfaial potential signal as a funtion of frequeny. (b) he effetive harge-transfer resistane as a funtion of frequeny. Israel Journal of Chemistry 8 28

9 11 (a) (a) (b) (b) Fig. 11. System with R e = 2R t and baseline noise that is onstant at 2 perent of low frequeny urrent signal. (a) U = 1 mv for all ω. (b) U = 3 mv for ω < 1 ω t and U = 3 mv for ω >1 ω t. Fig. 1. he effetive harge-transfer resistane with hmi resistane and input amplitude as parameters: (a) in dimensional form as a funtion of frequeny, and (b) in dimensionless form as a funtion of dimensionless frequeny. put amplitude as expeted. At higher frequenies, however, the effetive harge-transfer resistane approahes the linear value. As desribed by eq, max hanges value at the transition frequeny. Correspondingly, the effetive harge-transfer resistane hanges value at this transitional frequeny. For the 1 mv perturbation the variation in the harge-transfer resistane is signifiant. For the 1 mv perturbation the variation is negligible. In the presene of an hmi resistane max is damped in the limit of high frequeny and the values for the harge-transfer resistane will be superimposed. he effetive harge-transfer resistane is given in Fig. 1a as a funtion of frequeny for different values of hmi resistane and input amplitudes. he validity of eq is onfirmed by the superposition of the urves presented in Fig. 1b where the normalized effetive harge-transfer resistane is presented as funtions of normalized frequeny..5 ptimization of the Input Signal he results presented in Fig. 1b suggest that an optimized protool an be established for systems with hmi resistane. A smaller perturbation amplitude an be employed at frequenies below the transition frequeny defined by eq, and a larger amplitude an be employed at frequenies above the transition frequeny. As shown in Fig. 1a, at moderate to large values of hmi resistane the transition frequeny defined by eq is well within the experimentally assessable range. Large amplitude inputs an be employed at frequenies above the transition frequeny due to the dampening of the interfaial potential. For large values of hmi resistane the seond term in eq 37 beomes signifiant and influenes seletion of the appropriate input potential amplitude. o illustrate the onept, an eletrohemial system was modeled for whih the eletrolyti resistane was twie the value of the harge-transfer resistane. A onstant baseline noise of 2 perent of the low-frequeny urrent signal was added to the urrent signal. he resulting impedane response to the 1 mv input perturbation employed in ommon pratie is presented in Fig. 11a. Substantial satter is observed at all frequenies. he input signal an be modified in two ways. In the low-frequeny limit (eq 38), when R e = 2R t, the experi- Hirshorn et al. / Nonlinear Effets in Impedane Measurements

10 12 mentalist an use three times the input amplitude signal and still ahieve an adequate linear response. In the high-frequeny limit (eq 39), max is damped to zero and, aordingly, a muh higher input amplitude signal an be used. he impedane response given in Fig. 11b was obtained when a 3 mv voltage perturbation was introdued into the system for frequenies less than ten times the transitional frequeny and a 3 mv perturbation was introdued for frequenies greater than 1 times the transitional frequeny. he dashed line shows the impedane response that would have resulted if the 3 mv perturbation amplitude was employed for all frequenies. he satter was signifiantly redued using the input signal employed for Fig. 11b. he variableamplitude method yields more aurate results and provides a higher onfidene for the extration of system parameters..6 Potential-Dependent Capaitane A onstant double layer apaitane was used for the purposes of this work. In general, the apaitane is a funtion of potential. In the presene of a signifiant hmi resistane, the dampening of the interfaial potential above the transition frequeny defined by eq allows for adequate linearization of apaitane when the apaitive urrent dominates. For suffiiently low hmi resistane, the interfaial potential will not be damped and extration of apaitane values may be ompromised due to nonlinear effets. 5. Conlusions he amplitude of the input potential perturbation used for impedane measurements, normally fixed at a value of 1 mv for all systems, should instead be adjusted for eah experimental system. If system parameters suh as afel slope, harge-transfer resistane, and hmi resistane are known, eq 37 provides a useful guide for seletion of perturbation amplitude at low frequenies. he transition frequeny defined by eq an be used to tailor a frequeny-dependent input signal. When these parameters are unknown, distortions of low-frequeny Lissajous plots are assoiated with perturbation amplitudes that are too large to ensure a linear response. Aknowledgment. he authors express their sinere appreiation for a thorough, thoughtful, and helpful anonymous review. REFERENCES AND NES (1) Agarwal, P.; razem, M.E.; Garía-Rubio, L.H. Journal of the Eletrohemial Soiety 1992, 139, (2) Agarwal, P.; Crisalle,.D.; razem, M.E.; Garía-Rubio, L.H. Journal of the Eletrohemial Soiety 1995, 12, (3) Agarwal, P.; razem, M.E.; Garía-Rubio, L.H. Journal of the Eletrohemial Soiety 1995, 12, () razem, M.E. Journal of Eletroanalytial Chemistry 2, 572, (5) Darowiki,. Eletrohimia Ata 1995,, (6) Diard, J.; LeGorre, B.; Montella, C. J. Chim. Phys. 1992, 89, (7) Darowiki,. Eletrohimia Ata 1997, 2, (8) Popkirov, G.; Shindler, R. Eletrohimia Ata 1995,, (9) Diard, J.; LeGorre, B.; Montella, C. Journal of Eletroanalytial Chemistry 1997, 32, (1) Diard, J.; LeGorre, B.; Montella, C. Journal of Eletroanalytial Chemistry 1997, 32, (11) Diard, J.; LeGorre, B.; Montella, C. Journal of Eletroanalytial Chemistry 1997, 32, (12) Diard, J.; LeGorre, B.; Montella, C. Eletohimia Ata 1997, 2, (13) Miloo, R. Eletohimia Ata 1999,, (1) Gheem, E.V.; Pintelon, R.; Vereeken, J.; Shoukens, J.; Hubin, A.; Verboven, P.; Blajiev,. Eletohimia Ata 2, 9, (15) Blajiev,.; Pintelon, R.; Hubin, A. Journal of Eletroanalytial Chemistry 25, 576, (16) Urquidi-Madonald, M.; Real, S.; Madonald, D. Eletrohimia Ata 199, 35, (17) Wilson, J.; Shwartz, D.; Adler, S. Eletrohimia Ata 26, 51, (18) Wilson, J.; Sase, M.; awada,.; Adler, S. Eletrohemial and Solid-State Letters 27, 1, B81 B86. (19) Engblom, S..; Myland, J.; ldham,.b. Journal of Eletroanalyti Chemistry 2, 8, (2) ada, E.; Noda,.; umai, S.; suru,. Corrosion Siene 2, 6, (21) Darowiki,. Eletohimia Ata 199, 39, (22) Gabrielli, C.; eddam, M.; akenouti, H. In Materials Siene Forum rans.; eh Publiations: Switzerland, 1986, pp (23) Meszaros, L.; Meszaros, G.; Lengyel, B. Journal of the Eletrohemial Soiety 199, 11, () Bosh, R.; Bogaerts, W. Journal of the Eletrohemial Soiety 1996, 13, (25) Xiao, H.; Lalvani, S. Journal of the Eletrohemial Soiety 28, 155, C69 C7. (26) Wylie, C.R. Advaned Engineering Mathematis, 3rd ed.; MGraw-Hill: New York, (27) ooyman, D.; Sluyters-Rehbah, M.; Sluyters, J. Eletohimia Ata 1966, 11, (28) Darowiki,. Corrosion Siene 1995, 37, Israel Journal of Chemistry 8 28