The removal of the polarization errors in low frequency dielectric spectroscopy
11 The removal of the polarization errors in low frequency dielectric spectroscopyCamelia Prodan and Corina Bot Physics Department, New Jersey Institute of Technology, Newark, NJ 07102
Electrode polarization error is the biggest problem when measuring the low frequency dielectricproperties of electrolytes or suspensions of particles, including cells, in electrolytes. We present asimple and robust method to remove the polarization error, which we demonstrate to work onweak and strong ionic electrolytes as well as on cell suspensions. The method assumes noparticular behavior of the electrode polarization impedance; it makes use of the fact that theeffect dies out with frequency. The method allows for direct measurement of the polarizationimpedance, whose behavior with the applied voltages, electrode distance and ionic concentrationis investigated.Low frequency impedance spectroscopy (or dielectric spectroscopy) is an experimentaltechnique that records the low frequency variation of the impedance (or the complex dielectricpermittivity) of a sample as function of frequency. Because is noninvasive, this method is widelyused in many areas among which biophysics, pharmacology and geophysics. For exampleimpedance spectroscopy have been used to monitor the living cell and plasma membraneproperties or protein activity.
Impedance measurements were used to characterize the woundhealing process of monolayer cell tissues. The single cell DNA content was reported to bedetermined using impedance monitoring. In Pharmacology, cellular dielectric spectroscopy hasbeen used as a label free technology for drug discovery. In geophysics low frequencyimpedance measurements are used to look at mineral pore size.
7, 8
However, the experimentalresearch is mostly qualitative (comparison between impedances as different processes takeplace), despite the fact that there are theoretical studies,
9, 10 that could be used for quantitativeanalysis. The quantitative interpretation of the measurements is prevented by contamination ofthe signal with the electrode polarization. To obtain the true dispersion curves and thus tophysically interpret the measurements, the polarization error has to be removed from themeasurements.The available methods to remove the polarization effect work in special conditions and/orat narrow frequency ranges.
Most of them assume a specific behavior of the polarizationimpedance. A detailed presentation of the existing electrode polarization removal methods canbe found in Refs. .The advantages of the technique presented in this paper are that (i) is easy to apply, (ii) noneed for calibrations at any step and (iii) no assumption on the behavior of the electrodepolarization impedance. This allows for applications in all conditions and frequency ranges fromzero to gigahertz.The following three paragraphs present a description of the experimental set up and itstesting.a)
Description of the experimental set up.
We use the same experimental setup as in Ref. . Thesolution to be measured is placed between two parallel gold-plated electrodes that are enclosed ina cylindrical glass tube. The distance between the capacitor’s plates is controlled with amicrometer. The results reported here were obtained with electrodes of 28.1 mm in radius. Thedistance between the electrode plates was varied from 1 mm to 10 mm. We measure the complextransfer function is equal to: ZRT / = , where Z is the complex impedance of our measuringcell and R the reference resistor. R = 100 ! in these experiments. If the impedance Z were notcontaminated by the polarization effects and other possible factors, then the complex dielectricfunction, " ! j / * += , of the solution could be calculated from: Z = d / Aj !" * or " * = d / Aj ! R T . (1)b) Simple tests.
Before discussing the signal contamination by the polarization effect, it isinstructive to see how the setup performs without any post-data processing. The measurements ofindividual or combined electronic components such as resistors and capacitors proved that thereis very little noise in the data and the measured values are within 1% of the nominal values, evenat the very low frequency range. Measurements of the value of the relative permittivity of air, asextracted from the measured impedance Z with measuring cell filled with air, gave valuesremarkably closed to 1, given that no calibration was used in this experiment.c)
Samples.
The measurements reported here are for Millipore water, saline water with lower andthen higher concentrations of KCl, buffer solutions and live cells suspended in water.The following three paragraphs describe the methodology for measuring Z p and removing it fromthe raw data.a) Polarization effect.
Although the setup is capable of highly accurate impedance measurementsat low frequencies, the measured impedance Z doesn’t necessarily reflect the true impedance ofthe solution. It is now well known that, especially at low frequencies, Z will be contaminated bythe so-called “polarization” impedance, Z P . This effect is due to ionic charges accumulated at theinterface between the fluid and metallic electrodes and was carefully studied in several works.Here we mention that, because the effect takes place at the interfaces, the polarization impedanceappears in series with the true impedance of the solution, Z s . In other words, the measuredimpedance is the sum of the two: Z=Z s +Z p . Thus, if one develops a method to measure Z p , theintrinsic impedance of the solution can be obtained by subtracting Z p from the measured Z .b) Measuring Zp.
We will use Millipore water to exemplify the method. The dielectric functionof pure water is known to be constant ( $ r =78) for a frequency range spanning from 0Hz toseveral GHz. However, if we compute the dielectric function using Eq. 1, we obtain the graphshown in Figure 2 k (red line). The graph shows that from approximately 1kHz and up, $ r agreesextremely well with the nominal value of 78, while from 1kHz down, $ r starts increasing quitestrongly. The anomalous behavior of $ r at these low frequencies is quite typical and it is preciselydue to the polarization effects. But the main observation is that above 1kHz the effect ispractically gone. If we plot the real and imaginary parts of the impedance Z (red lines in Figure 2a and f), and the ideal impedance of the Millipore water, computed as: AjdZ s * / !" = , where * ! is the nominal complex dielectric function (blue lines in Figure 2 a and f), we see that the twographs match quite well above 1kHz. In fact, if we fit the experimental values of Z , above 1kHz,with a frequency dependent function Z fit ( " ) = d / Aj " ! , (2)with $ and % as fitting parameters, we obtain $ =78 and % =0.000070 S/m, in very good agreementwith the nominal values. But since it is known that the dielectric function and conductivity ofionic solutions are constant all the way to zero frequency, we can extrapolate Z s to lowerfrequencies. The difference between the measured Z and Z s is precisely the polarizationimpedance Z p . A plot of Z p is show in the inset of Figure 2 f (green line). The following methodology emerges:
To measure the polarization impedance for anunknown ionic solution one can do the following:i) Fit the experimental data for Z with the function Z fit given in equation (2), by giving a largeweight to the high frequency data and a low, or almost zero weight to the low frequency data.ii) From the fit, determine the true dielectric constant $ and conductivity % of the ionic solution.iii) Extrapolate Z s all the way to 0 Hz.iv) Compute Z p as the difference between Z and the extrapolated Z s .c) Behavior of Zp.
We will use the mili-Q water to answer several important questions about thepolarization impedance. Z P depends, in general, on the distance d between the capacitor plates, onthe applied voltage and on frequency. Figure 1 maps the dependence of Zp for mili-Q water onall these tree parameters. The methodology outlined above was applied for three values of thedistance between capacitor plates, d =1, 3, 5mm. For each d , the experiments were repeated withfive values of applied voltages per c entimeter: 0.1, 0.3, 0.5, 0.7 and 0.9V/cm. The appliedelectric fields were kept below the linear regime limit of 1V/cm. The steps i)-iii) were applied toresulting 30 measurements. Following this procedure, it we found the following.In all our measurements, we find that Z P is mainly reactive, in line with several otherworks . Then, in the log-log plot of the insets in Figure 2 the curves appear linear for a widerange of frequencies, implying that Z P goes with the frequency as a power law. This is again inline with previous works. The power law can be easily computed from the graphs. It is notablethat, this very specific behavior of Z p was used in the past to remove the polarizationimpedance.
11, 17
We can confirm now that both, the amplitude and the exponent of the power laware weakly dependent on the applied voltage.Another important finding is that Z P saturates at large values of d . This fact justifies the useof an existing method called “electrode distance variation technique” to correct for thepolarization effect. We found that this simple technique can be applied for electrolytes withlow conductivity; for higher conductivity, the method isn’t reliable.In the following three paragraphs we demonstrate that the methodology is robust and that itcan be applied to various ionic solutions. In particular, we describe how to remove thepolarization effect for live cell suspensions.a)
Weak ionic solutions.
In Figure 2 we present an application of the methodology to electrolyteswith low conductivity, more precisely, water with low concentrations of KCl. From left to right,the different columns in Figure 2 refer to 0 uM KCl, 1um KCl, 5 uM KCl, 10 um KCl and 20 uMKCl. It is well known that, when increasing the KCl concentration, the dielectric function of thesolution should remain constant at 78, while large increases in the conductivity should beobserved. In all columns, we can see an almost perfect match between the real parts of Z and Z fit .The fit is also perfect for the imaginary parts of Z and Z fit , if we look above 1kHz. The fittingprovided the following values: ! r = ± and ! = % have been measuredusing a regular conductivity probe. This demonstrates that, by an automated implementation ofthe steps i)-iv) described above, we were able to remove the polarization effects and obtain thetrue dielectric function and conductivity of all these solutions.b) Strong ionic solutions.
Low frequency dielectric spectroscopy has applications in many fields,but we are primarily interested in biological applications, such as measuring the dielectricfunctions of live cell suspensions. Now, many physiological buffers are known to have hugeconductivities. Thus, in order to apply our method to live cells, we must demonstrate that itworks for strong ionic solutions.As the conductivity of the electrolyte increases so does the frequency limit where thepolarization effect highly contaminates the data. Since our method relies on the informationcontained in the frequency domain where the polarization effects are small, the experimental datamust contain a good part of this domain. Thus, for strong ionic solutions, we had to go to muchhigher frequency. This why, in these experiments we replaced the initial signal analyzer (SR795) with another one (Solartron 1260), which allowed us to record data in the frequencywindow from 0Hz to 10 Hz.We give an application of the methodology to water and KCl, at much higher concentrations thanbefore, namely 0.1, 0.5 mM, and to HEPES with 5 mM concentration. The dielectric functionsbefore and after polarization removal are shown in Figure 3 left. For KCl solutions, themethodology provides the following values: ! r = ± and ! = ! r = and ! = % have been measured using the conductivity probe. Weobserved that for large concentrations of KCl the relative dielectric permittivity tends todecrease, this effect has been reported before. This has to do with the fact that K + and Cl - ionsdon’t have an intrisec dipole moment (meaning a smaller polarizability than water), howeverthey occupy space in the solution thus lowering the overall dielectric permittivity of the solution.Now we can say something about the measured frequency behavior of Z p with the increase of theKCl concentration (see Figure 4). The most notable things we observe are that the ImZ p has apower law behavior with an average exponent of 0.8 for uM concentrations, 0.6 for mMconcentrations of KCl and 0.5 for HEPES. These are in good agreement with previous studieswhere, for mM concentrations of KCl, the average exponent was 0.7. Moreover, the amplitudeappears to depend on the concentration of KCl, i.e
ImZ p goes as inverse proportional with thesolution’s conductivity, as expected. Also the frequency where Z p becomes negligible increaseswith the ionic concentration. This last observation shows that polarization removal becomesmore difficult at higher ionic concentrations.c) Colloidal suspensions.
We should point out from the beginning that the dielectric functions ofcolloidal suspensions are not constant with frequency. In fact, one of the main goals of dielectricspectroscopy is to capture and study these variations of $ with frequency. Thus, the methodologyoutlined above cannot be directly applied to these systems. However, one way to apply themethodology to colloids (including live cells) suspended in electrolytes or to samples saturatedwith electrolytes is as follows. Initially, the sample is measured and the raw values of Z sample arerecorded. The data are, of course, contaminated by the polarization effect. Now the keyobservations are: 1) the polarization effect is due to the electrolyte and not the colloidal particlesand 2) the dielectric function of the electrolyte is constant with frequency. Thus if we separatethe electrolyte from the colloids, we can measure Zp as before, which then can be removed from Z sample . Thus, the following methodology emerges : " ) Record the brute values of Z sample . " ) Remove the colloids from the solution. ) Apply stepts i)-iv) to the supernatant and determine Z p . & ) Remove Z p from Z sample to obtain the intrinsic impedance of the sample.In the case of live cell suspension, the step ) can be done by a gentle centrifugation. Fast opticaldensity (O.D.) measurements can be used to make sure there are no cells left in supernatant. Thecentrifugation should be gentle so that no cell rupture occurs during the process, otherwise theconductivity of the supernatant will be highly increased. An application of this methodology isshown in the right panel of Figure 3, to a suspension of live E-coli cells. In conclusion , this paper presented a simple technique for the removal of polarizationerrors in dielectric measurements at low frequencies. In contrast to previous works, ourmethodology requires no assumption on the behavior of the polarization impedance, other thanthe widely accepted fact that the effect dies out at high frequencies. Critical to this method is thatthe frequency span of the measurements must incorporate a high frequency window where thepolarization error becomes negligible. There is also no need for calibrations at any step in theexperiment.The method has been demonstrated to be robust for a class of electrolytes, includingphysiological buffers. Our study confirmed that the polarization impedance is reactive and that itvaries as a power law with the frequency. Our study also showed that the amplitude and theexponent of the power law are weakly dependent on the applied voltage. However, Z p showed adependence on the distance between the electrode plates, which becomes weaker for largerdistances. For weak ionic solutions, we found that the exponent of the power law is independentwhile the amplitude decreases linearly with the increase of ionic concentration in line withprevious work. Similar conclusions apply for strong ionic solutions. At last, the method was0applied to live E.coli cell suspension. Obtaining uncontaminated dielectric spectroscopy data inalpha region opens the possibility of many interesting applications, the most notable beingmeasuring the membrane potential as predicted in Ref. . Acknowledgement:
This work was supported by a grant from NJIT-ADVANCE which isfunded by the National Science Foundation (grant G. Leung, H. R. Tang, et al., J. of the Assoc. for Lab. Autom., 258 (2005). D. Nawarathna, J. R. Claycomb, J. H. Miller, et al., APL 86, 023902 (2005). G. R. Facer, D. A. Notterman, and L. L. Sohn, APL 78, 996 (2001). D. Nawarathna, J. H. Miller, and J. Claycomb, Phys. Rev. Lett. 95, 158103 (2005). C. R. Keese, J. Wegener, S. R. Walker, et al., PNAS 101, 1554 (2004). L. L. Sohn, O. A. Saleh, G. R. Facer, et al., PNAS 97, 10687 (2000). L. Slater, D. Ntarlagiannis, Y. Personna, et al., Geophys. Res. Letters 34 (2007). L. Slater, D. Ntarlagiannis, and D. Wishart, Geophysics 71, A1 (2006). C. Prodan and E. Prodan, J. of Physics: D 32, 335 (1999). C. Grozze and V. Shilov, J. of Colloids and Interface Science 309, 283 (2007). F. Bordi, C. Cametti, and T. Gili, Bioelectrochem. 54, 53 (2001). C. L. Davey and D. Kell, Bioelectrochem. and Bioenergetics 46, 91 (1998). V. Raicu, T. Saibara, and A. Irimajiri, Bioelectrochem.. Bioenergetics 48, 325 (1998). H. P. Schwan and C. D. Ferris, Rev. Sci. Instrum. 39, 481 (1968). C. L. Davey, G. H. Markx, and D. B. Kell, Eur. Biophys. J. 18, 255 (1990). H. Sanabria and J. H. Miller, Phys. Rev. E 74 (2006). M. R. Stoneman, M. Kosempa, et al., Phys. in med. and bio. 52, 6589 (2007). U. Kaatze and Y. Feldman, Meas. Sci. and Technol. 17, R17 (2006). Y. Feldman, E. Polygalov, I. Ermolina, et al., Meas. Sci. Technol. 12, 1355 (2001). C. Prodan, F. Mayo, J. R. Claycomb, et al., JAP 95, 3754 (2004). H. P. Schwan, Ann. NY Acad of Sci 148, 191 (1968). H. P. Schwan, Biophysik 3, 181 (1966). E. McAdams and J. Jossinet, Med.&Bio. Eng & Comp. 32, 126 (1994). K. R. Foster and H. P. Schwan, Critical Rev. in Biomed. Eng. 17, 25 (1989).2Figure Captions:
Figure 1. Real (1st row) and imaginary (2nd row) parts of the sample impedance for d=1mm(1st column), d=3mm (2nd column) and d=5mm (3rd column), for applied electric fields of 0.1(red), 0.3 (blue), 0.5 (green), 0.7 (magenta) and 0.9 (cayan) V/cm. Black line marks Z ideal .Figure 2. Dielectric measurements for water and water with 1!M KCl, 5!M KCl, 10!M KCl,20!M KCl: a) – e) real part of impedance; f) – j) imaginary part of impedance; k) – o) dielectricpermittivity. Red lines marked with circles represent the dielectric values without compensatingfor the electrode polarization effects while blue, solid lines represent the dielectric values afterremoving the polarization effects. The green lines represent the imaginary part of the electrodepolarization impedance.Figure 3. Dielectric permittivity versus frequency for (left) water with 0.1 and 0.5 mM KCl and5mM HEPES and (right) E.coli cell suspension for an optical density (O.D.) of 0.051. The dottedlines represent the real relative dielectric permittivity after the removal of the polarization error.Figure 4. Imaginary part of the electrode polarization impedance for the samples from Figures 2and 3 as function of frequency. The curves show a power law behavior with an average exponentequal to 0.8 for µµ