Extension of the measurement model approach for deconvolution of underlying distributions for impedance measurements

Transcription

1 Electrochimica Acta 47 (2002) 2027/ Extension of the measurement model approach for deconvolution of underlying distributions for impedance measurements Mark E. Orazem *, Pavan Shukla, Michael A. Membrino Department of Chemical Engineering, University of Florida, PO Box , Gainesville, FL , USA Received 10 August 2001; received in revised form 24 September 2001 Abstract Electrochemical impedance spectra frequently reveal the influence of distributions of activation or relaxation processes, but methods for extracting information concerning these distributions are not well developed. Complex non-linear weighted regression of Voigt elements was applied to synthetic impedance data to identify the influence of stochastic noise and incomplete frequency ranges on the ability to resolve the expected distribution. The method was also applied to analysis of impedance data for heatseparated human stratum corneum. Regression of a Voigt series was found to provide a convenient way to identify a distribution of relaxation time constants corresponding to a given experimental spectrum. While the presence of stochastic noise reduces the number of Voigt elements that can be resolved, the parameters trace the same profile for the distribution of time constants. The technique should be particularly useful for experimental systems for which no deterministic model is available. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: Measurement model; Regression; Iontophoresis; Constant-phase elements 1. Introduction The objective of the present work was to develop a method for identification, from impedance spectra, of the distribution of time constants associated with activation or relaxation processes. In contrast to direct and indirect methods presented to date, this approach accounts explicitly for the error structure of the measurement. In that sense, it represents an extension of the measurement model technique introduced by Agarwal et al. [1 /3] to electrochemistry. Impedance spectra are generally interpreted in terms of physical models expressed either as equivalent electrical circuits or as deterministic models derived from proposed kinetic and mass transfer relations. The heterogeneity of electrode surfaces and electrode processes complicates Manuscript submitted to the special issue of Electrochimica Acta for the Fifth International Symposium on Electrochemical Impedance Spectroscopy, held in Marilleva, Italy on June 17/22, * Corresponding author. Tel.: ; fax: address: meo@che.ufl.edu (M.E. Orazem). this interpretation by adding a frequency dispersion to the impedance response. The corresponding distribution of time constants may itself provide insight into the system under investigation, and significant effort has been expended on developing methods to extract such distributions from impedance data. As reviewed by Lasia, [4,5] these problems are described by the Fredholm integral equation of the first kind h(e) K(u; E)f (u) du (1) g 0 where h(e) is an experimentally observed function of parameter E, K(u, E) is the kernel which is a function of E and of parameter u which has a distribution f(u). The distribution function f(u) is obtained from experiment. Determination of the distribution f(u) from experimental values h(e) is a type of inverse problem, which is inherently ill-posed and extremely sensitive to errors in h(e) [6]. Eq. (1) can be expressed in terms of an arbitrary distribution of Debye relaxation phenomena as /02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S ( 0 2 )

2 2028 M.E. Orazem et al. / Electrochimica Acta 47 (2002) 2027/2034 g(t) o(v)o dt (2) g 1 jvt 0 where o (v) is the complex permittivity and g(t) represents a distribution of relaxation time constants. Eqs. (1) and (2) can also be written in terms of impedance. One method for deconvolution of such spectra is to assume an a priori distribution such as that associated with the so-called constant-phase element shown in Fig. 1, i.e. R Z CPE (v) (3) 1 (jvrc) f or R Z CPE (v) 1 (jv) f RC where the exponent f is typically assumed to have a value between 0.5 and 1, and the behavior of a resistor in parallel with an ideal capacitor is recovered when f/ 1. Development of such expressions for impedance response is usually done in the context of proposed mechanisms for the observed frequency dispersion. For example, Macdonald developed such distributions in terms of distributions of activation energies [7 /11]. Lukács presented interpretations in terms of surface inhomogeneities [12,13]. Nahir and Bowden addressed the issue of distributed rate constants [14]. They demonstrated that if g(t) has lognormal distribution, i.e. a Gaussian distribution of logarithms of time constants, the CPE behavior of impedance spectra can be simulated. Li et al. presented a generalized equivalent circuit model to account for a distribution of rate constants for redox active monolayer assemblies [15]. The circuit model was a simple Randle type circuit with multiple arms in parallel with double-layer capacitance. The impedance response was simulated from the circuit, and the results were presented in the form of I peak / I background as a function of frequency. Various distribution functions for rate constants were explored, but no (4) experimental results were presented to confirm the simulated results. It is important to note that the a priori assumption of a lognormal distribution of time constants, implicit in use of a CPE, may not be valid for a given experimental system. Insight into the physical origin of such impedance responses may be lost when the interpretation of the spectra is limited to use of CPE models. Several authors have described the deconvolution of impedance data without assumptions concerning the type or shape of the distributions. VanderNoot concluded that, for impedance data, use of complex non-linear least-squares regression techniques would be preferable to use of the maximum entropy deconvolution techniques [16]. Dion and Lasia used regularization techniques to facilitate deconvolution, and they showed that the deconvolution approach can be successful for simple cases [5]. They concluded that, although it is possible to determine the distribution function in simple cases, the best way is to carry out the complex non-linear least-squares approximation to a known model and then to obtain the model parameters [5]. Stoynov and co-workers describe a robust method in which calculation of the local derivatives of the impedance with respect to frequency allows calculation of model parameters and their effective timeconstants for each frequency [17,18]. The variation of parameters with frequency allows visualization of the distribution of time constants for a given spectrum without a priori assumption of a distribution function. For cases where no known model is available, Dion and Lasia [5] suggested that the Voigt circuit, developed by Agarwal et al. [1/3] as a measurement model for analysis of errors in impedance data, can be used to obtain the correct distribution function in the case of a CPE element in the system. The measurement model approach, as presented here, differs from the methods proposed by Dion and Lasia [5] and by Stoynov [17] in that it takes into account explicitly the stochastic error structure of the measurement and minimizes artifacts introduced by data that are inconsistent with the Kramers /Kronig relations. 2. Method Fig. 1. Schematic representation of a constant-phase electrical circuit. The objective of the present work was to extend the use of the measurement model concept, developed for identification of the error structure of frequency-domain measurements, [1 /3] to generalized identification of distributions of time constants. In order to obtain a deconvolution of impedance data by non-linear regression, the integral within Eq. (2) was replaced by a summation, i.e.

3 M.E. Orazem et al. / Electrochimica Acta 47 (2002) 2027/ Z(v)R XN R k (5) k 1 jvt k In this way, a continuous distribution of Debye relaxation time constants was approximated by a discrete series of Voigt elements, shown in Fig. 2. This model was shown by Agarwal et al. to provide a statistically significant representation of general classifications of impedance data, including spectra containing inductive and/or capacitive loops, spectra influenced by mass transfer, and spectra represented by constant phase elements [1]. In the present work, a weighted non-linear regression of the Voigt model was applied to two different sets of impedance data. The regression software was written inhouse, and the weighting strategy was chosen to be appropriate for the data set regressed. Comparison to the known lognormal distribution associated with the constant-phase element model (see Eq. (3)) was used to demonstrate the influence of stochastic noise and truncation errors on the robustness of the deconvolution approach. For this comparison, the measurement model was regressed to synthetic data, generated from Eq. (3). The method was then applied to the impedance response of human skin. 3. Constant-phase element The synthetic impedance data used for this work were obtained from R Z(v)R (6) 1 (jv) f t RC are presented in Fig. 3. The synthetic data were distributed logarithmically at a spacing of 20 points per decade. The calculations were performed in double precision. To explore the influence of stochastic errors, random errors were introduced with a standard deviation corresponding to 1% of the modulus. The errors added to real and imaginary parts of the impedance were calculated independently but with the same standard deviation, following Agarwal et al. Comparison of Fig. 3 to published data suggests that the errors added can be considered to be on the upper end of what might be expected for typical impedance measurements. Indeed, experimentally-determined standard deviations of impedance data have been reported that are significantly Fig. 2. Schematic representation of a Voigt series. Fig. 3. Synthetic impedance data calculated using Eq. (6) with R 10 V, R10 3 V, t RC 0.1 s, and with f as a parameter. Symbols represent synthetic data with random noise added (s0.01jzj), and the lines represent values calculated at machine precision. These values were used as the input to obtain the results presented in Figs. 6/10. less than 0.1% of the modulus [19]. The lines shown in Fig. 3 correspond to synthetic data with no errors added, and the symbols show the data resulting from addition of random errors. Extreme scatter in the data is evident at low frequencies where the magnitude of the impedance is large. Proportional weighting was used for regression to data that were limited by numerical roundoff errors, and modulus weighting was used for regression to data with errors added. The number of parameters was increased sequentially until the confidence interval for one or more parameters included zero. The confidence intervals were calculated under the assumption that the problem could be linearized about the trial solution, an assumption that is justified because the residual errors were randomly distributed. The number of Voigt elements was then decreased such that all calculated parameters were statistically significant. 4. Heat-separated excised human skin Electrochemical impedance spectroscopy has been applied extensively for the investigation of skin transport properties, motivated, in part, by technique s ease of application combined with the relatively short time required to collect a spectrum (see, e.g. Kalia and Guy [20]). Typical Nyquist representations of skin impedance data exhibit a depressed semicircle, characteristic of a distribution of relaxation processes, and constant phase element models are often fit to skin impedance data. It has been proposed that the constant phase element is representative of a size or charge distribution of the aqueous pores which provide the transport pathways through skin [21].

4 2030 M.E. Orazem et al. / Electrochimica Acta 47 (2002) 2027/2034 Application of steady current changes the impedance of skin [22]. In experiments performed using hairless mouse skin, new pathways were observed to form after a period of time [23 /25]. The formation of new pathways suggests that skin properties are affected by application of electrical current at the rates used here. Changes in skin properties were also observed by a variety of techniques including electrochemical impedance spectroscopy, both during iontophoresis [26 /28] and in the absence of applied currents [29 /32]. The experimental technique applied in the present work is described by Membrino [33]. Heat-separated human cadaver skin was placed between two chambers of a diffusion cell joined by compression (see Fig. 4). The cell was held at a constant temperature of 32 8C. The thickness of the skin sample was about 100 mm; therefore, the membrane used included the entire stratum corneum in addition to the portions of the underlying epidermis. The stratum corneum is reported to be the primary barrier to transport. Skin samples were collected from the back and abdominal areas, and had little or no hair. The skin was visually examined before use to ensure the integrity of the membrane. Experiments for which impedance scans found a polarization resistance approximately an order of magnitude below the normal range were discarded under the assumption that macroscopic shunt paths were present that were not detected during visual inspection. Ag/AgCl driving electrodes were used (In Vivo Metric). Ag/AgCl reference electrodes (Micro Electrodes Inc.) were placed on either side of the membrane, as close to the surfaces as possible, to yield the best estimates for trans-membrane potential and current measurements. The electrochemical instrumentation consisted of a Solartron 1286 electrochemical interface and Solartron 1250 frequency response analyzer. At open-circuit, variable-amplitude galvanostaticallymodulated impedance spectroscopy was employed in which the current amplitude was adjusted at each frequency to maintain the voltage response of the skin below a predetermined value [34,35]. This approach avoids inducing large potential gradients, which can alter skin properties. Previously measured impedance values were used to predict the impedance at the frequency of the measurement being conducted. In-house software written for LabView was used to control input parameters for the impedance experiments. Impedance data are presented in Fig. 5. The total duration of the experiment was 4 h, and each impedance scan took less than 5 min to perform. Current was turned on for a period of about 15 min, during which the impedance scans shown in Fig. 5 were taken. The skin was allowed to relax for a period of about 15/20 min after the current was applied. The impedance scans taken during the relaxation process also showed evidence of relaxation. The magnitude of the current was increased systematically during the course of the experiment. The use of weighting strategies based on experimentally-determined stochastic error structure and the use of the Kramers /Kronig relations to identify the portion of the spectra suitable for analysis represent important features of the work presented here. The methods of Agarwal et al. [1/3] were used to identify the stochastic error structure and to ensure that the spectra used for the present analysis were consistent with the Kramers / Kronig relations. Data that were found to be inconsistent with the Kramers /Kronig relations were rejected. The threshold for rejection was that the real part of the measured impedance should fall outside the 95.4% confidence interval of the prediction based on regression of the measurement model to the imaginary part (see Agarwal et al. for a detailed discussion of the procedure). The rejection of inconsistent data minimized the role that non-stationary effects would otherwise have on the calculated distributions of time constants. The experimentally determined standard deviation of the measurement was smaller that 0.2% of the modulus. Fig. 4. Schematic representation of the diffusion cell used for the present work. Fig. 5. Impedance data collected for heat-separated human skin: (a) 0; (b) 0.07; (c) 0.14; (d) 0.29; (e) 0.57; and (f) 0.86 ma cm 2. The lines connecting frequencies of 5.37 khz and 365 Hz are drawn in temporal sequence.

5 M.E. Orazem et al. / Electrochimica Acta 47 (2002) 2027/ Results The distributions identified by use of the Voigt measurement model are presented in this section Constant-phase element Interpretation of a known distribution was employed to assess the influence of stochastic errors and of incomplete frequency spectra on the distribution of time constants ascertained by regression of a measurement model Influence of stochastic errors The first objective of this work was to examine the utility of the measurement model approach as a tool for identifying the distribution of Debye time constants with data that contain stochastic noise. The results obtained from regression to data with f/0.5 are presented in Fig. 6. To eliminate the influence of the number of Voigt elements obtained by regression, resistor values R k were normalized to the maximum value. Fifteen Voigt elements were obtained by regression to the synthetic data with noise level governed by machine precision. The error bars, corresponding to calculated 9/ 1s confidence intervals for the parameters, were too small to be seen on the scale of Fig. 6. The distribution can be approximated by an equation of the form R A 1 A R max t a1 a1 2 t0 t a2 a2 (7) t0 t t t 0 where A 1 /0.445, A 2 /0.005, a 1 /0.5, and a 2 /1/6. The parameter t 0 represents the characteristic time constant for the distribution, which had a value of ms when f/0.5. The parameters for Eq. (7) were t 0 obtained by inspection. Eq. (7), represented by a solid line in Fig. 6, is symmetric about t 0. The nearly Gaussian distribution seen in Fig. 6 is consistent with the result of Nahir and Bowden that a lognormal distribution of rate constants yields CPE behavior [14]. The number of statistically-significant Voigt elements that could be resolved for the data with noise added was substantially smaller, yet the distribution approximated by the corresponding parameters was in agreement with that identified for the noise-free case. The agreement between the two distributions is seen more clearly on a logarithmic scale, as shown in Fig. 7. The agreement is excellent at high frequencies (small values of t), but the influence of added noise was to reduce the number of Voigt elements that could be obtained with large time constants, corresponding to low frequencies. Similar agreement was seen at all values of f. As expected, at f/1, the method yields only one value for t. For f/0.9, the distribution for both the clean and noisy data, seen in Fig. 8, was much sharper than seen in Fig. 7. The parameters for the approximate distribution function, Eq. (7), were A 1 /1.25, A 2 /0.15, a 1 /2.5, a 2 /5/6, and t 0 /94.65 ms. The influence of noise on the approximate distribution obtained by regression of the measurement model to synthetic data was to reduce the ability to resolve Voigt elements with resistor values significantly below the maximum value, i.e. R k /R max /1. This effect was most evident for large time constants, corresponding to low frequencies Influence of incomplete spectra It is often not possible to measure a complete impedance spectrum. The lower frequency range can be limited because data taken at low frequencies require a long time, and system non-stationarity can cause the Fig. 6. Distribution of Debye relaxation time constants for synthetic data corresponding to a CPE with f0.5. The error bars correspond to 91s, calculated using a linear approximation. Fig. 7. Distribution of Debye relaxation time constants for synthetic data corresponding to a CPE with f0.5. The error bars correspond to 91s, calculated using a linear approximation.

6 2032 M.E. Orazem et al. / Electrochimica Acta 47 (2002) 2027/2034 Fig. 8. Distribution of Debye relaxation time constants for synthetic data corresponding to a CPE with f0.9. Fig. 10. The distribution of time constants obtained for the truncated data sets shown in Fig. 9. resulting data to be inconsistent with the Kramers / Kronig relations. To assess the utility of the measurement model approach for identification of the distribution of relaxation time constants in incomplete spectra, the measurement model was regressed to truncated data sets. The data sets used were the synthetic data with addition of 1% noise. The truncated data sets are presented in Fig. 9, and the corresponding distributions are presented in Fig. 10. The results show that the distribution obtained by regression of a measurement model to impedance data is influenced by absence of data at low frequencies, but for reasonably complete spectra, the approach provides a good identification of the distribution of relaxation time constants. Good results were obtained when the data set used for regressed contained all data with real values up to 84% of the maximum value. The corresponding value of the imaginary part of the impedance included the peak value of /Z j and extended to 50% of the peak value (see Fig. 9) Heat-separated excised human skin The measurement model was regressed to the impedance data presented in Fig. 5. While some evolution of the system was observed during the hydration period, the regression results, presented in Fig. 11, show that the corresponding distributions of relaxation time constants were unchanged. The distributions were roughly, but not exactly, symmetric with respect to a principal time constant. The results shown in Fig. 11 are consistent with the success of regression of the CPE circuit to impedance data for skin, but suggest that additional features are present at low frequencies. The impedance data in Fig. 5 show that the magnitude of the impedance decreased slightly upon imposition of a steady current of 0.07 ma cm 2. A comparison of the regression results for the hydration step (i/0 macm 2 ) and during application of a small Fig. 9. Truncated data sets corresponding to a CPE with f0.5 used for evaluating the influence of incomplete data sets on the ability of the measurement model to resolve a distribution of time constants. Vertical dashed lines and changes in shading are used to designate the data used to obtain Fig. 10. Fig. 11. The distribution of time constants obtained during the hydration period shown in Fig. 5. The time indicated corresponds to the beginning of the impedance measurement referenced to the initial time of immersion of the skin.

7 M.E. Orazem et al. / Electrochimica Acta 47 (2002) 2027/ applied current (i/0.07 ma cm 2 ), presented in Fig. 12, show that, on a qualitative basis, the corresponding distributions of relaxation time constants were unchanged. A new feature is seen, however, at small time constants, and the weight applied to the time constant distribution near 5 ms is reduced. These subtle features would not be seen by regression of a CPE model to the data. The impedance data in Fig. 5 show that the amount of the decrease in the magnitude of the impedance depended on the magnitude of the applied current density. A comparison of the regression results for the hydration step (i/0 macm 2 ) and during application of a large applied current (i/0.86 ma cm 2 ), presented in Fig. 13, shows that the corresponding distributions of relaxation time constants changed significantly. At large applied current densities, the characteristic time constant is shifted to smaller values at high frequencies (small t), and the shape of the distribution is changed at low frequencies (large t). The results show that impedance spectroscopy may provide insight into the manner in which large potential differences alter the properties of skin. These results are consistent with the observation of Membrino et al. [36] from impedance results that skin properties are changed once a critical potential drop across the skin is exceeded. 6. Conclusions Fig. 12. The distribution of time constants obtained during the hydration period and during application of a small applied current (i0.07 ma cm 2 ) as shown in Fig. 5. The time indicated corresponds to the beginning of the impedance measurement referenced to the initial time of immersion of the skin. Regression of a Voigt series provides a convenient way to identify a distribution of relaxation time constants corresponding to a given experimental spectrum without a priori assumption of a lognormal or other distribution function. For the purposes of identification of distributions, the Voigt measurement model has, therefore, a significant advantage over the use of constant-phase-element models. While the presence of stochastic noise reduces the number of Voigt elements that can be resolved, the parameters trace the same profile for the distribution of time constants. The approach provides an incorrect distribution if the measured frequency range is incomplete. Application of the proposed method to experimental data collected for heat-separated human skin shows that the technique can provide significant insight for experimental systems for which no deterministic model is available. It is clear, from the work presented here, that the distribution of time constants is not uniform and that the CPE model cannot describe all features of the impedance response. The deterministic model for skin must account for a distribution of properties. It must account also for low frequencies features that are, by casual inspection of the impedance data, clearly visible at large applied currents. These features are shown by the deconvolution method proposed here to be evident as well under conditions of open-circuit and low-applied currents. The method described herein can be considered to be a natural extension of the measurement model approach proposed for impedance data by Agarwal et al. [1/3] because the weighting strategy for the regression of the Voigt series was based on the experimentally-determined stochastic error structure and because only the data found to be consistent with the Kramers /Kronig relations were employed. Fig. 13. The distribution of time constants obtained during the hydration period and during application of a large applied current (i0.86 ma cm 2 ) as shown in Fig. 5. The time indicated corresponds to the beginning of the impedance measurement referenced to the initial time of immersion of the skin. Acknowledgements The contributions of Jean-Michel Pernaut, Departmento Quimica, Universidade Federal de Minas Gerais, Brazil, funding from the National Science Foundation,

8 2034 M.E. Orazem et al. / Electrochimica Acta 47 (2002) 2027/2034 and unrestricted contributions from ALZA Corporation are gratefully acknowledged. References [1] P. Agarwal, M.E. Orazem, L.H. García-Rubio, J. Electrochem. Soc. 139 (1992) 1917/1927. [2] P. Agarwal, O.D. Crisalle, M.E. Orazem, L.H. García-Rubio, J. Electrochem. Soc. 142 (1995) 4149/4158. [3] P. Agarwal, M.E. Orazem, L.H. García-Rubio, J. Electrochem. Soc. 142 (1995) 4159/4168. [4] A. Lasia, in: R.E. White, B.E. Conway, J.O. Bockris (Eds.), Modern Aspects of Electrochemistry, vol. 32, Plenum Press, New York, NY, 1999, p [5] F. Dion, A. Lasia, J. Electroanal. Chem. 475 (1999) 28/37. [6] S. Marco, J. Palacín, J. Samitier, IEEE Trans. Instrument. Measure. 50 (2001) 774/780. [7] J.R. Macdonald, J. Appl. Phys. 58 (1985) 1971/1978. [8] J.R. Macdonald, J. Appl. Phys. 58 (1985) 1955/1970. [9] R.L. Hurt, J.R. Macdonald, Solid State Ionics 20 (1986) 111/124. [10] J.R. Macdonald, J. Appl. Phys. 62 (1987) R51/R62. [11] J.R. Macdonald, J. Electroanal. Chem. 378 (1994) 17/29. [12] Z. Lukacs, J. Electroanal. Chem. 432 (1997) 79/83. [13] Z. Lukacs, J. Electroanal. Chem. 464 (1999) 68/75. [14] T. Nahir, E. Bowden, J. Electroanal. Chem. 410 (1996) 9 /13. [15] J. Li, K. Schuler, S.E. Creager, J. Electrochem. Soc. 147 (2000) 4584/4588. [16] T.J. VanderNoot, J. Electroanal. Chem. 386 (1995) 57/63. [17] Z. Stoynov, Polish J. Chem. 71 (1997) 1204/1210. [18] D. Vladikova, Z. Stoynov, L. Ilkov, Polish J. Chem. 71 (1997) 1196/1203. [19] M.E. Orazem, T.E. Moustafid, C. Deslouis, B. Tribollet, J. Electrochem. Soc. 143 (1996) 3880/3890. [20] Y.N. Kalia, R.H. Guy, J. Control. Release 44 (1997) 33/42. [21] K. Kontturi, L. Murtomäki, J. Hirvonen, P. Paronen, A. Urtti, Pharm. Res. 10 (1993) 381/385. [22] M.R. Prausnitz, Adv. Drug Deliv. Rev. 18 (1996) 395/425. [23] E.R. Scott, H.S. White, Solid State Ionics 53/56 (1992) 176/183. [24] E.R. Scott, H.S. White, J.B. Phipps, Anal. Chem. 65 (1993) 1537/ [25] E.R. Scott, A.I. Laplaza, H.S. White, J.B. Phipps, Pharm. Res. 10 (1993) 1699/1709. [26] Y.N. Kalia, F. Pirot, R.H. Guy, Biophys. J. 71 (1996) 2692/2700. [27] S. Ollmar, A. Eek, F. Sundstróm, L. Emtestam, Med. Progr. Technol. 21 (1995) 29/37. [28] A.D. Woolfson, G.P. Moss, D.F. McCafferty, A. Lackermeier, E.T. McAdams, Pharm Res. 16 (1999) 459/462. [29] T. Yamamoto, Y. Yamamoto, Med. Biol. Engin. Comput. 19 (1981) 302/310. [30] A. Rawlings, A. Davies, M. Carlomusto, S. Pillai, K. Zhang, R. Kosturko, P. Verdejo, C. Feinberg, Arch. Dermatol. Res. 288 (1996) 383/390. [31] H. Tanojo, H.E. Junginger, H.E. Boddé, J. Control. Release 47 (1997) 31/39. [32] L. Norlén, I. Nicander, B. Rozell, S. Ollmar, B. Forslind, J. Invest. Dermatol. 112 (1999) 72/77. [33] M.A. Membrino, Transdermal Delivery of Therapeutic Drugs by Iontophoresis, Ph.D. dissertation, University of Florida, Gainesville, FL, [34] P.T. Wojcik, P. Agarwal, M.E. Orazem, Electrochim. Acta 41 (1996) 977/983. [35] P.T. Wojcik, M.E. Orazem, Corrosion 54 (1998) 289/298. [36] M.A. Membrino, M.E. Orazem, E. Scott, J.B. Phipps, Minutes: Transdermal Administration: A Case Study, Iontophoresis, Éditions de Santé, Paris, France, 1997, p. 313.