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Introduction to Electrochemical Impedance Spectroscopy Gamry Instruments

Basics of Electrochemical Impedance Spectroscopy

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An introduction to Electrochemical Impedance Spectroscopy (EIS) theory and has been kept as free from mathematics and electrical theory as possible. If you still find the material presented here difficult to understand, don't stop reading. You will get useful information from this application note, even if you don't follow all of the discussions.

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  • Introduction to Electrochemical Impedance SpectroscopyGamry Instruments

  • ImpedanceThe term impedance refers to the frequency dependant resistance to current flow of a circuit element (resistor, capacitor, inductor,etc.)Impedance assumes an AC current of a specific frequency in Hertz (cycles/s).Impedance: Z = E/IE = Frequency-dependent potentialI = Frequency-dependent currentOhms Law: R = E/IR = impedance at the limit of zero frequency

  • Reasons To Run EISEIS is theoretically complex (and can be expensive) why bother?The information content of EIS is much higher than DC techniques or single frequency measurements.EIS may be able to distinguish between two or more electrochemical reactions taking place.EIS can identify diffusion-limited reactions, e.g., diffusion through a passive film.EIS provides information on the capacitive behavior of the system.EIS can test components within an assembled device using the devices own electrodes.

  • Making EIS MeasurementsApply a small sinusoidal perturbation (potential or current) of fixed frequency Measure the response and compute the impedance at each frequency.Z = E/IE = Frequency-dependent potentialI = Frequency-dependent currentRepeat for a wide range of frequenciesPlot and analyze

  • Excitation and Response in EIS

    SineOutofPhase

    0-0.4

    0.0174524064-0.3878476962

    0.0348994967-0.3755772502

    0.0523359562-0.3631923998

    0.0697564737-0.3506969174

    0.0871557427-0.3380946094

    0.1045284633-0.3253893145

    0.1218693434-0.3125849028

    0.139173101-0.2996852747

    0.156434465-0.2866943596

    0.1736481777-0.2736161147

    0.1908089954-0.2604545236

    0.2079116908-0.2472135955

    0.2249510543-0.2338973638

    0.2419218956-0.2205098847

    0.2588190451-0.2070552361

    0.2756373558-0.1935375165

    0.2923717047-0.1799608435

    0.3090169944-0.1663293527

    0.3255681545-0.1526471963

    0.3420201433-0.1389185421

    0.3583679495-0.125147572

    0.3746065934-0.1113384808

    0.3907311285-0.0974954747

    0.4067366431-0.0836227706

    0.4226182617-0.0697245942

    0.4383711468-0.055805179

    0.4539904997-0.041868765

    0.4694715628-0.0279195974

    0.4848096202-0.0139619251

    0.50

    0.51503807490.0139619251

    0.52991926420.0279195974

    0.5446390350.041868765

    0.55919290350.055805179

    0.57357643640.0697245942

    0.58778525230.0836227706

    0.60181502320.0974954747

    0.61566147530.1113384808

    0.6293203910.125147572

    0.64278760970.1389185421

    0.6560590290.1526471963

    0.66913060640.1663293527

    0.68199836010.1799608435

    0.69465837050.1935375165

    0.70710678120.2070552361

    0.71933980030.2205098847

    0.73135370160.2338973638

    0.74314482550.2472135955

    0.75470958020.2604545236

    0.76604444310.2736161147

    0.77714596150.2866943596

    0.78801075360.2996852747

    0.798635510.3125849028

    0.80901699440.3253893145

    0.81915204430.3380946094

    0.82903757260.3506969174

    0.83867056790.3631923998

    0.84804809620.3755772502

    0.85716730070.3878476962

    0.86602540380.4

    0.87461970710.4120304599

    0.88294759290.4239354114

    0.89100652420.435711228

    0.89879404630.4473543228

    0.9063077870.4588611491

    0.91354545760.4702282018

    0.92050485350.4814520185

    0.92718385460.4925291803

    0.93358042650.5034563128

    0.93969262080.5142300877

    0.94551857560.5248472232

    0.95105651630.5353044851

    0.9563047560.545598688

    0.96126169590.5557266964

    0.96592582630.5656854249

    0.97029572630.5754718403

    0.97437006480.5850829613

    0.97814760070.5945158604

    0.98162718340.6037676642

    0.9848077530.6128355545

    0.98768834060.6217167692

    0.99026806870.6304086029

    0.99254615160.638908408

    0.99452189540.6472135955

    0.99619469810.6553216354

    0.99756405030.663230058

    0.99862953480.6709364544

    0.9993908270.6784384769

    0.99984769520.6857338406

    10.692820323

    0.99984769520.6996957657

    0.9993908270.7063580743

    0.99862953480.7128052194

    0.99756405030.719035237

    0.99619469810.7250462296

    0.99452189540.7308363661

    0.99254615160.7364038828

    0.99026806870.7417470837

    0.98768834060.7468643412

    0.9848077530.7517540966

    0.98162718340.7564148605

    0.97814760070.760845213

    0.97437006480.7650438048

    0.97029572630.7690093568

    0.96592582630.772740661

    0.96126169590.776236581

    0.9563047560.7794960518

    0.95105651630.7825180806

    0.94551857560.7853017468

    0.93969262080.7878462024

    0.93358042650.7901506725

    0.92718385460.792214455

    0.92050485350.7940369213

    0.91354545760.7956175163

    0.9063077870.7969557585

    0.89879404630.7980512402

    0.89100652420.7989036278

    0.88294759290.7995126616

    0.87461970710.7998781561

    0.86602540380.8

    0.85716730070.7998781561

    0.84804809620.7995126616

    0.83867056790.7989036278

    0.82903757260.7980512402

    0.81915204430.7969557585

    0.80901699440.7956175163

    0.798635510.7940369213

    0.78801075360.792214455

    0.77714596150.7901506725

    0.76604444310.7878462024

    0.75470958020.7853017468

    0.74314482550.7825180806

    0.73135370160.7794960518

    0.71933980030.776236581

    0.70710678120.772740661

    0.69465837050.7690093568

    0.68199836010.7650438048

    0.66913060640.760845213

    0.6560590290.7564148605

    0.64278760970.7517540966

    0.6293203910.7468643412

    0.61566147530.7417470837

    0.60181502320.7364038828

    0.58778525230.7308363661

    0.57357643640.7250462296

    0.55919290350.719035237

    0.5446390350.7128052194

    0.52991926420.7063580743

    0.51503807490.6996957657

    0.50.692820323

    0.48480962020.6857338406

    0.46947156280.6784384769

    0.45399049970.6709364544

    0.43837114680.663230058

    0.42261826170.6553216354

    0.40673664310.6472135955

    0.39073112850.638908408

    0.37460659340.6304086029

    0.35836794950.6217167692

    0.34202014330.6128355545

    0.32556815450.6037676642

    0.30901699440.5945158604

    0.29237170470.5850829613

    0.27563735580.5754718403

    0.25881904510.5656854249

    0.24192189560.5557266964

    0.22495105430.545598688

    0.20791169080.5353044851

    0.19080899540.5248472232

    0.17364817770.5142300877

    0.1564344650.5034563128

    0.1391731010.4925291803

    0.12186934340.4814520185

    0.10452846330.4702282018

    0.08715574270.4588611491

    0.06975647370.4473543228

    0.05233595620.435711228

    0.03489949670.4239354114

    0.01745240640.4120304599

    00.4

    -0.01745240640.3878476962

    -0.03489949670.3755772502

    -0.05233595620.3631923998

    -0.06975647370.3506969174

    -0.08715574270.3380946094

    -0.10452846330.3253893145

    -0.12186934340.3125849028

    -0.1391731010.2996852747

    -0.1564344650.2866943596

    -0.17364817770.2736161147

    -0.19080899540.2604545236

    -0.20791169080.2472135955

    -0.22495105430.2338973638

    -0.24192189560.2205098847

    -0.25881904510.2070552361

    -0.27563735580.1935375165

    -0.29237170470.1799608435

    -0.30901699440.1663293527

    -0.32556815450.1526471963

    -0.34202014330.1389185421

    -0.35836794950.125147572

    -0.37460659340.1113384808

    -0.39073112850.0974954747

    -0.40673664310.0836227706

    -0.42261826170.0697245942

    -0.43837114680.055805179

    -0.45399049970.041868765

    -0.46947156280.0279195974

    -0.48480962020.0139619251

    -0.50

    -0.5150380749-0.0139619251

    -0.5299192642-0.0279195974

    -0.544639035-0.041868765

    -0.5591929035-0.055805179

    -0.5735764364-0.0697245942

    -0.5877852523-0.0836227706

    -0.6018150232-0.0974954747

    -0.6156614753-0.1113384808

    -0.629320391-0.125147572

    -0.6427876097-0.1389185421

    -0.656059029-0.1526471963

    -0.6691306064-0.1663293527

    -0.6819983601-0.1799608435

    -0.6946583705-0.1935375165

    -0.7071067812-0.2070552361

    -0.7193398003-0.2205098847

    -0.7313537016-0.2338973638

    -0.7431448255-0.2472135955

    -0.7547095802-0.2604545236

    -0.7660444431-0.2736161147

    -0.7771459615-0.2866943596

    -0.7880107536-0.2996852747

    -0.79863551-0.3125849028

    -0.8090169944-0.3253893145

    -0.8191520443-0.3380946094

    -0.8290375726-0.3506969174

    -0.8386705679-0.3631923998

    -0.8480480962-0.3755772502

    -0.8571673007-0.3878476962

    -0.8660254038-0.4

    -0.8746197071-0.4120304599

    -0.8829475929-0.4239354114

    -0.8910065242-0.435711228

    -0.8987940463-0.4473543228

    -0.906307787-0.4588611491

    -0.9135454576-0.4702282018

    -0.9205048535-0.4814520185

    -0.9271838546-0.4925291803

    -0.9335804265-0.5034563128

    -0.9396926208-0.5142300877

    -0.9455185756-0.5248472232

    -0.9510565163-0.5353044851

    -0.956304756-0.545598688

    -0.9612616959-0.5557266964

    -0.9659258263-0.5656854249

    -0.9702957263-0.5754718403

    -0.9743700648-0.5850829613

    -0.9781476007-0.5945158604

    -0.9816271834-0.6037676642

    -0.984807753-0.6128355545

    -0.9876883406-0.6217167692

    -0.9902680687-0.6304086029

    -0.9925461516-0.638908408

    -0.9945218954-0.6472135955

    -0.9961946981-0.6553216354

    -0.9975640503-0.663230058

    -0.9986295348-0.6709364544

    -0.999390827-0.6784384769

    -0.9998476952-0.6857338406

    -1-0.692820323

    -0.9998476952-0.6996957657

    -0.999390827-0.7063580743

    -0.9986295348-0.7128052194

    -0.9975640503-0.719035237

    -0.9961946981-0.7250462296

    -0.9945218954-0.7308363661

    -0.9925461516-0.7364038828

    -0.9902680687-0.7417470837

    -0.9876883406-0.7468643412

    -0.984807753-0.7517540966

    -0.9816271834-0.7564148605

    -0.9781476007-0.760845213

    -0.9743700648-0.7650438048

    -0.9702957263-0.7690093568

    -0.9659258263-0.772740661

    -0.9612616959-0.776236581

    -0.956304756-0.7794960518

    -0.9510565163-0.7825180806

    -0.9455185756-0.7853017468

    -0.9396926208-0.7878462024

    -0.9335804265-0.7901506725

    -0.9271838546-0.792214455

    -0.9205048535-0.7940369213

    -0.9135454576-0.7956175163

    -0.906307787-0.7969557585

    -0.8987940463-0.7980512402

    -0.8910065242-0.7989036278

    -0.8829475929-0.7995126616

    -0.8746197071-0.7998781561

    -0.8660254038-0.8

    -0.8571673007-0.7998781561

    -0.8480480962-0.7995126616

    -0.8386705679-0.7989036278

    -0.8290375726-0.7980512402

    -0.8191520443-0.7969557585

    -0.8090169944-0.7956175163

    -0.79863551-0.7940369213

    -0.7880107536-0.792214455

    -0.7771459615-0.7901506725

    -0.7660444431-0.7878462024

    -0.7547095802-0.7853017468

    -0.7431448255-0.7825180806

    -0.7313537016-0.7794960518

    -0.7193398003-0.776236581

    -0.7071067812-0.772740661

    -0.6946583705-0.7690093568

    -0.6819983601-0.7650438048

    -0.6691306064-0.760845213

    -0.656059029-0.7564148605

    -0.6427876097-0.7517540966

    -0.629320391-0.7468643412

    -0.6156614753-0.7417470837

    -0.6018150232-0.7364038828

    -0.5877852523-0.7308363661

    -0.5735764364-0.7250462296

    -0.5591929035-0.719035237

    -0.544639035-0.7128052194

    -0.5299192642-0.7063580743

    -0.5150380749-0.6996957657

    -0.5-0.692820323

    -0.4848096202-0.6857338406

    -0.4694715628-0.6784384769

    -0.4539904997-0.6709364544

    -0.4383711468-0.663230058

    -0.4226182617-0.6553216354

    -0.4067366431-0.6472135955

    -0.3907311285-0.638908408

    -0.3746065934-0.6304086029

    -0.3583679495-0.6217167692

    -0.3420201433-0.6128355545

    -0.3255681545-0.6037676642

    -0.3090169944-0.5945158604

    -0.2923717047-0.5850829613

    -0.2756373558-0.5754718403

    -0.2588190451-0.5656854249

    -0.2419218956-0.5557266964

    -0.2249510543-0.545598688

    -0.2079116908-0.5353044851

    -0.1908089954-0.5248472232

    -0.1736481777-0.5142300877

    -0.156434465-0.5034563128

    -0.139173101-0.4925291803

    -0.1218693434-0.4814520185

    -0.1045284633-0.4702282018

    -0.0871557427-0.4588611491

    -0.0697564737-0.4473543228

    -0.0523359562-0.435711228

    -0.0348994967-0.4239354114

    -0.0174524064-0.4120304599

    -0-0.4

    PhaseShift

    Voltage Magnitude

    Current Magnitude

    Applied Voltage

    Measured Current

    Time

    Magnitude

    Sheet1

    XRad(x)Sin(x)-30Rad(x+30)sin(x+30)0.8

    000-30-0.5235987756-0.4

    10.01745329250.0174524064-29-0.5061454831-0.3878476962

    20.0349065850.0348994967-28-0.4886921906-0.3755772502

    30.05235987760.0523359562-27-0.471238898-0.3631923998

    40.06981317010.0697564737-26-0.4537856055-0.3506969174

    50.08726646260.0871557427-25-0.436332313-0.3380946094

    60.10471975510.1045284633-24-0.4188790205-0.3253893145

    70.12217304760.1218693434-23-0.401425728-0.3125849028

    80.13962634020.139173101-22-0.3839724354-0.2996852747

    90.15707963270.156434465-21-0.3665191429-0.2866943596

    100.17453292520.1736481777-20-0.3490658504-0.2736161147

    110.19198621770.1908089954-19-0.3316125579-0.2604545236

    120.20943951020.2079116908-18-0.3141592654-0.2472135955

    130.22689280280.2249510543-17-0.2967059728-0.2338973638

    140.24434609530.2419218956-16-0.2792526803-0.2205098847

    150.26179938780.2588190451-15-0.2617993878-0.2070552361

    160.27925268030.2756373558-14-0.2443460953-0.1935375165

    170.29670597280.2923717047-13-0.2268928028-0.1799608435

    180.31415926540.3090169944-12-0.2094395102-0.1663293527

    190.33161255790.3255681545-11-0.1919862177-0.1526471963

    200.34906585040.3420201433-10-0.1745329252-0.1389185421

    210.36651914290.3583679495-9-0.1570796327-0.125147572

    220.38397243540.3746065934-8-0.1396263402-0.1113384808

    230.4014257280.3907311285-7-0.1221730476-0.0974954747

    240.41887902050.4067366431-6-0.1047197551-0.0836227706

    250.4363323130.4226182617-5-0.0872664626-0.0697245942

    260.45378560550.4383711468-4-0.0698131701-0.055805179

    270.4712388980.4539904997-3-0.0523598776-0.041868765

    280.48869219060.4694715628-2-0.034906585-0.0279195974

    290.50614548310.4848096202-1-0.0174532925-0.0139619251

    300.52359877560.5000

    310.54105206810.515038074910.01745329250.0139619251

    320.55850536060.529919264220.0349065850.0279195974

    330.57595865320.54463903530.05235987760.041868765

    340.59341194570.559192903540.06981317010.055805179

    350.61086523820.573576436450.08726646260.0697245942

    360.62831853070.587785252360.10471975510.0836227706

    370.64577182320.601815023270.12217304760.0974954747

    380.66322511580.615661475380.13962634020.1113384808

    390.68067840830.62932039190.15707963270.125147572

    400.69813170080.6427876097100.17453292520.1389185421

    410.71558499330.656059029110.19198621770.1526471963

    420.73303828580.6691306064120.20943951020.1663293527

    430.75049157840.6819983601130.22689280280.1799608435

    440.76794487090.6946583705140.24434609530.1935375165

    450.78539816340.7071067812150.26179938780.2070552361

    460.80285145590.7193398003160.27925268030.2205098847

    470.82030474840.7313537016170.29670597280.2338973638

    480.8377580410.7431448255180.31415926540.2472135955

    490.85521133350.7547095802190.33161255790.2604545236

    500.8726646260.7660444431200.34906585040.2736161147

    510.89011791850.7771459615210.36651914290.2866943596

    520.9075712110.7880107536220.38397243540.2996852747

    530.92502450360.79863551230.4014257280.3125849028

    540.94247779610.8090169944240.41887902050.3253893145

    550.95993108860.8191520443250.4363323130.3380946094

    560.97738438110.8290375726260.45378560550.3506969174

    570.99483767360.8386705679270.4712388980.3631923998

    581.01229096620.8480480962280.48869219060.3755772502

    591.02974425870.8571673007290.50614548310.3878476962

    601.04719755120.8660254038300.52359877560.4

    611.06465084370.8746197071310.54105206810.4120304599

    621.08210413620.8829475929320.55850536060.4239354114

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    Sheet2

    Sheet3

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    -0-0.4

    PhaseShift

    Voltage Magnitude

    Current Magnitude

    Applied Voltage

    Measured Current

    Time

    Magnitude

  • EIS Data PresentationEIS data may be displayed as either a vector or a complex quantity.A vector is defined by the impedance magnitude and the phase angle.As a complex quantity, Ztotal = Zreal + ZimagThe vector and the complex quantity are different representations of the impedance and are mathematically equivalent.

  • Vector and Complex Plane Representations of EIS VectorComplex PlaneMagnitudeReal Impedance, ZImaginary Impedance, Z = Phase Angle

  • EIS data may be presented as a Bode Plot or a Complex Plane (Nyquist) PlotBode PlotNyquist Plot

    Chart4

    2.29667929095.0000303995-2.37040472215.0000303995

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    Log Freq (Hz)

    Log Modulus (Ohm)

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    EISCURVE

    DataFileName'Copeland_EIS_Dummy.DTA'

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    TitlePotentiostatic EIS

    Date12/9/1999

    Time10:41:55

    ChartTitlePotentiostatic EIS 'Copeland_EIS_Dummy.DTA' 12/9/1999-10:41:55

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    Pstat: pc4Start Frequency: 100000 HzEnd Frequency: 0.001 HzEOC: 0.000048359 VDC Pot.: 0 VAC Pot.: 10 mV(rms)Area: 1 cm2Electrode: 7.87 gm/cm3, 27.92 g/equivConditioning: OFFDelay: ON, s

    Notes:

    Results:

    EISCURVE

    &F

    Page &P

    Potentiostatic EIS 'Copeland_EIS_Dummy.DTA' 12/9/1999-10:41:55

    &F

    Page &P

    #REF!

    &F

    Page &P

    #REF!

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    Log Freq (Hz)

    Real (Ohm)

    -Imag (Ohm)

    Potentiostatic EIS 'Copeland_EIS_Dummy.DTA' 12/9/1999-10:41:55

    Real (Ohm)

    -Imag (Ohm)

    Chart3

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    -Imag (Ohm)

    EISCURVE

    DataFileName'Copeland_EIS_Dummy.DTA'

    PointCount75

    TitlePotentiostatic EIS

    Date12/9/1999

    Time10:41:55

    ChartTitlePotentiostatic EIS 'Copeland_EIS_Dummy.DTA' 12/9/1999-10:41:55

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    Disp2.MatchToListRealImaginary

    Disp3.ChartNameChart 11

    Disp3.EnablePointsFlag

    Disp3.AxisRange03500-2002569.5099818512

    Disp3.FitAxisRange

    Disp3.LabelsReal (Ohm)-Imag (Ohm)

    Disp3.MatchToListNyquist

    Curve.FreqCurve.RealCurve.ImagCurve.ModCurve.PhzCurve.SigmaFitCurve.RealFitCurve.ImagFitCurve.ModFitCurve.PhzDisp.Freq.DataDisp.Real.DataDisp.Imag.DataDisp.Mod.DataDisp.Phz.DataDisp.Sigma.DataAnal.Real.DataAnal.Imag.DataAnal.Mod.DataAnal.Phz.DataDisplay1.PointStatusDisplay2.PointStatusDisplay3.PointStatusPtTog

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    63105.5198.313-7.05642198.4385018897-2.037852209414.801.98E+027.06E+002.30-2.04

    50123.7198.68-6.99586198.8031298977-2.016648458214.701.99E+027.00E+002.30-2.02

    39811.2198.979-7.19506199.1090438162-2.070907161814.601.99E+027.20E+002.30-2.07

    31634.1199.241-7.72258199.3906074088-2.219672959614.501.99E+027.72E+002.30-2.22

    25123.7199.352-8.58266199.5366682008-2.465220844314.401.99E+028.58E+002.30-2.47

    19954.4199.656-10.0152199.9070348113-2.871679872814.302.00E+021.00E+012.30-2.87

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    1995.37202.518-83.2837218.9742336662-22.354473016813.302.03E+028.33E+012.34-22.35

    1584.49203.751-104.814229.1297549359-27.222287193413.202.04E+021.05E+022.36-27.22

    1259.04206.128-131.571244.5397317922-32.54999088613.102.06E+021.32E+022.39-32.55

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    316.113289.367-508.128584.7455216357-60.339500034112.502.89E+025.08E+022.77-60.34

    251.185338.627-628.516713.9331953236-61.685443584812.403.39E+026.29E+022.85-61.69

    199.495413.459-770.704874.6044822072-61.787809851412.304.13E+027.71E+022.94-61.79

    158.533523.702-931.2891068.4395099045-60.649166723512.205.24E+029.31E+023.03-60.65

    125.911681.991-1103.441297.1860150653-58.28148950612.106.82E+021.10E+033.11-58.28

    100.009897.989-1270.191555.5599892711-54.74075182912.008.98E+021.27E+033.19-54.74

    79.40831173.9-1408.91833.8594329992-50.198829228711.901.17E+031.41E+033.26-50.20

    63.09711496.21-1492.722113.4941122464-44.933099075711.801.50E+031.49E+033.33-44.93

    50.00831844.9-1501.932378.9598010265-39.148977599111.701.84E+031.50E+033.38-39.15

    40.00592166.95-1437.762600.5434278435-33.56406598511.602.17E+031.44E+033.42-33.56

    31.57932463.89-1307.592789.3629273008-27.954957160211.502.46E+031.31E+033.45-27.95

    24.99112697.3-1141.972929.0822403784-22.946595616911.402.70E+031.14E+033.47-22.95

    20.00112863.18-974.263024.3978342804-18.792001903711.302.86E+039.74E+023.48-18.79

    15.79342986.47-805.6983093.2430114855-15.097971148411.202.99E+038.06E+023.49-15.10

    12.50023069.63-658.3263139.4301169442-12.104538829711.103.07E+036.58E+023.50-12.10

    9.99873123.19-536.2923168.8996332109-9.743403329211.003.12E+035.36E+023.50-9.74

    7.893893159.96-428.7353188.9121815793-7.726559017810.903.16E+034.29E+023.50-7.73

    6.316233182.48-344.913201.1157208854-6.185445132610.803.18E+033.45E+023.51-6.19

    4.997483198.12-274.5183209.8803196886-4.906089426110.703.20E+032.75E+023.51-4.91

    3.973023206.37-219.0753213.845427603-3.908655320910.603.21E+032.19E+023.51-3.91

    3.157183212.04-175.3313216.8217111244-3.124421475910.503.21E+031.75E+023.51-3.12

    2.510853215.93-138.9893218.9320755525-2.474721273810.403.22E+031.39E+023.51-2.47

    1.994893218.03-112.1073219.9821521786-1.995214887510.303.22E+031.12E+023.51-2.00

    1.583253220.5-88.66493221.7203035788-1.577035119610.203.22E+038.87E+013.51-1.58

    1.257293221.05-71.06623221.8338733216-1.263914820310.103.22E+037.11E+013.51-1.26

    1.000063217.46-56.09343217.9489307824-0.998797038210.003.22E+035.61E+013.51-1.00

    0.7945993217.62-45.2683217.9384170963-0.80602875581-0.103.22E+034.53E+013.51-0.81

    0.6310583218.34-35.34913218.5341251058-0.62929112491-0.203.22E+033.53E+013.51-0.63

    0.5011873219.14-28.91513219.2698586183-0.51463081491-0.303.22E+032.89E+013.51-0.51

    0.3982723218.14-22.85143218.2211306997-0.40683959411-0.403.22E+032.29E+013.51-0.41

    0.3162963219.25-18.29473219.3019831243-0.32560311051-0.503.22E+031.83E+013.51-0.33

    0.2512063218-13.47993218.0282328942-0.24000523961-0.603.22E+031.35E+013.51-0.24

    0.1994613219.26-11.63633219.2810301491-0.20709976071-0.703.22E+031.16E+013.51-0.21

    0.1585263217.97-9.395983217.9837173827-0.16729443361-0.803.22E+039.40E+003.51-0.17

    0.1258723218.18-6.954173218.1875136294-0.12381034341-0.903.22E+036.95E+003.51-0.12

    0.0999943217.38-6.493023217.3865518008-0.11562890841-1.003.22E+036.49E+003.51-0.12

    0.0794493217.88-3.846543217.8822990082-0.06848931681-1.103.22E+033.85E+003.51-0.07

    0.0631033217.83-3.955653217.8324313219-0.07043316031-1.203.22E+033.96E+003.51-0.07

    0.0501343216.94-2.613253216.941061424-0.04654366041-1.303.22E+032.61E+003.51-0.05

    0.0398093219.72-2.873763219.7212824865-0.05113931511-1.403.22E+032.87E+003.51-0.05

    0.0316263218.1-2.608923218.1010575281-0.0464497911-1.503.22E+032.61E+003.51-0.05

    0.0251163219.33-0.8284923219.3301066059-0.01474502271-1.603.22E+038.28E-013.51-0.01

    0.0199543219.96-1.529853219.960363427-0.02722205921-1.703.22E+031.53E+003.51-0.03

    0.0158533218.91-0.989633218.9101521272-0.01761516181-1.803.22E+039.90E-013.51-0.02

    0.0125933217.99-1.060473217.9901747359-0.01888149221-1.903.22E+031.06E+003.51-0.02

    0.0100013217.78-0.969163217.7801459502-0.01725685911-2.003.22E+039.69E-013.51-0.02

    0.0079433218.450.3937673218.45002408810.00700995421-2.103.22E+03-3.94E-013.510.01

    0.0063093216.06-1.625743216.0604109112-0.02896340011-2.203.22E+031.63E+003.51-0.03

    0.0050123217.62-2.130783217.620705525-0.03794254241-2.303.22E+032.13E+003.51-0.04

    0.0039823216.57-0.7117053216.5700787367-0.01267738371-2.403.22E+037.12E-013.51-0.01

    Pstat: pc4Start Frequency: 100000 HzEnd Frequency: 0.001 HzEOC: 0.000048359 VDC Pot.: 0 VAC Pot.: 10 mV(rms)Area: 1 cm2Electrode: 7.87 gm/cm3, 27.92 g/equivConditioning: OFFDelay: ON, s

    Notes:

    Results:

    EISCURVE

    &F

    Page &P

    Potentiostatic EIS 'Copeland_EIS_Dummy.DTA' 12/9/1999-10:41:55

    &F

    Page &P

    #REF!

    &F

    Page &P

    #REF!

    2.29667929095.0000303995-2.37040472215.0000303995

    2.296992984.9000042456-2.14235179974.9000042456

    2.29762593974.8000672121-2.03785220944.8000672121

    2.29842321754.700043122-2.01664845824.700043122

    2.29909098674.6000052684-2.07090716184.6000052684

    2.29970469644.5001554832-2.21967295964.5001554832

    2.30002271624.4000835989-2.46522084434.4000835989

    2.30082807744.3000386737-2.87167987284.3000386737

    2.30114580454.2001087201-3.4058420114.2001087201

    2.30168324794.0999291328-4.1097753494.0999291328

    2.30231951343.99992182-4.98162649773.99992182

    2.30336896663.8999626914-6.18921904173.8999626914

    2.3052277483.8001050617-7.69710648033.8001050617

    2.3077384883.6999400027-9.5711456883.6999400027

    2.3114364613.5999845411-11.86848085193.5999845411

    2.31698136073.500053879-14.77931873393.500053879

    2.32669931273.3998743849-18.23600787083.3998743849

    2.34039301513.3000234384-22.35447301683.3000234384

    2.36008149063.1998895026-27.22228719343.1998895026

    2.38834943153.100039528-32.5499908863.100039528

    2.4262002052.9999782847-38.28727123932.9999782847

    2.47535931722.8999096494-43.99260207422.8999096494

    2.53531569812.8000183469-49.35612930442.8000183469

    2.60525096122.7001003036-54.02195942312.7001003036

    2.68303787162.6001110714-57.75151350832.6001110714

    2.76696690442.4998423564-60.33950003412.4998423564

    2.85365757562.3999937011-61.68544358482.3999937011

    2.94181169872.2999320153-61.78780985142.2999320153

    3.02874993952.2001196781-60.64916672352.2001196781

    3.11300225792.1000636732-58.2814895062.1000636732

    3.19188676412.0000390847-54.7407518292.0000390847

    3.26336604351.8998658986-50.19882922871.8998658986

    3.32500104231.8000093991-44.93309907571.8000093991

    3.37638710351.6990420912-39.14897759911.6990420912

    3.41506411071.602124045-33.5640659851.602124045

    3.44550502451.499402499-27.95495716021.499402499

    3.46673156561.3977853723-22.94659561691.3977853723

    3.48063891841.3010538812-18.79200190371.3010538812

    3.49041404041.1984756349-15.09797114841.1984756349

    3.49685082021.0969169617-12.10453882971.0969169617

    3.50090848420.999943538-9.74340332920.999943538

    3.50364255950.8972910703-7.72655901780.8972910703

    3.50530137420.8004579361-6.18544513260.8004579361

    3.506488840.6987510647-4.90608942610.6987510647

    3.50702498520.5991207513-3.90865532090.5991207513

    3.50742699130.499299343-3.12442147590.499299343

    3.50771181250.3998207684-2.47472127380.3998207684

    3.50785346450.2999189533-1.99521488750.2999189533

    3.50808783410.1995494967-1.57703511960.1995494967

    3.50810314320.0994354614-1.26391482030.0994354614

    3.50757914750.0000260569-0.99879703820.0000260569

    3.5075777286-0.0998519858-0.8060287558-0.0998519858

    3.5076581182-0.1999307233-0.6292911249-0.1999307233

    3.5077573834-0.3000002024-0.5146308149-0.3000002024

    3.5076158821-0.399820225-0.4068395941-0.399820225

    3.5077617171-0.4999063003-0.3256031105-0.4999063003

    3.50758985-0.5999699918-0.2400052396-0.5999699918

    3.5077588905-0.7001420079-0.2070997607-0.7001420079

    3.5075838423-0.7998994986-0.1672944336-0.7998994986

    3.5076113455-0.9000708672-0.1238103434-0.9000708672

    3.5075032422-1.0000260585-0.1156289084-1.0000260585

    3.5075701548-1.0999115648-0.0684893168-1.0999115648

    3.5075634245-1.1999499933-0.0704331603-1.1999499933

    3.5074431041-1.2998676433-0.0465436604-1.2998676433

    3.5078182783-1.4000187317-0.0511393151-1.4000187317

    3.5075996781-1.4999557334-0.046449791-1.4999557334

    3.5077655111-1.6000495256-0.0147450227-1.6000495256

    3.5078505257-1.6999700321-0.0272220592-1.6999700321

    3.5077088546-1.7998885404-0.0176151618-1.7998885404

    3.5075847138-1.8998707966-0.0188814922-1.8998707966

    3.5075563677-1.9999565727-0.0172568591-1.9999565727

    3.5076467698-2.10001543750.0070099542-2.1000154375

    3.507324198-2.2000394726-0.0289634001-2.2000394726

    3.507534848-2.2999889377-0.0379425424-2.2999889377

    3.5073930177-2.3998987443-0.0126773837-2.3998987443

    Log Freq (Hz)

    Log Modulus (Ohm)

    Phase (Degree)

    Potentiostatic EIS 'Copeland_EIS_Dummy.DTA' 12/9/1999-10:41:55

    197.8375.00003039958.189465.0000303995

    198.0114.90000424567.40734.9000042456

    198.3134.80006721217.056424.8000672121

    198.684.7000431226.995864.700043122

    198.9794.60000526847.195064.6000052684

    199.2414.50015548327.722584.5001554832

    199.3524.40008359898.582664.4000835989

    199.6564.300038673710.01524.3000386737

    199.74.200108720111.88484.2001087201

    199.7864.099929132814.35514.0999291328

    199.8373.9999218217.41893.99992182

    199.9083.899962691421.67893.8999626914

    200.1233.800105061727.04743.8001050617

    200.2863.699940002733.77213.6999400027

    200.4713.599984541142.13073.5999845411

    200.6183.50005387952.92813.500053879

    201.5213.399874384966.39713.3998743849

    202.5183.300023438483.28373.3000234384

    203.7513.1998895026104.8143.1998895026

    206.1283.100039528131.5713.100039528

    209.4222.9999782847165.3162.9999782847

    214.9552.8999096494207.5262.8999096494

    223.4262.8000183469260.2722.8000183469

    236.7232.7001003036326.0842.7001003036

    257.1862.6001110714407.6392.6001110714

    289.3672.4998423564508.1282.4998423564

    338.6272.3999937011628.5162.3999937011

    413.4592.2999320153770.7042.2999320153

    523.7022.2001196781931.2892.2001196781

    681.9912.10006367321103.442.1000636732

    897.9892.00003908471270.192.0000390847

    1173.91.89986589861408.91.8998658986

    1496.211.80000939911492.721.8000093991

    1844.91.69904209121501.931.6990420912

    2166.951.6021240451437.761.602124045

    2463.891.4994024991307.591.499402499

    2697.31.39778537231141.971.3977853723

    2863.181.3010538812974.261.3010538812

    2986.471.1984756349805.6981.1984756349

    3069.631.0969169617658.3261.0969169617

    3123.190.999943538536.2920.999943538

    3159.960.8972910703428.7350.8972910703

    3182.480.8004579361344.910.8004579361

    3198.120.6987510647274.5180.6987510647

    3206.370.5991207513219.0750.5991207513

    3212.040.499299343175.3310.499299343

    3215.930.3998207684138.9890.3998207684

    3218.030.2999189533112.1070.2999189533

    3220.50.199549496788.66490.1995494967

    3221.050.099435461471.06620.0994354614

    3217.460.000026056956.09340.0000260569

    3217.62-0.099851985845.268-0.0998519858

    3218.34-0.199930723335.3491-0.1999307233

    3219.14-0.300000202428.9151-0.3000002024

    3218.14-0.39982022522.8514-0.399820225

    3219.25-0.499906300318.2947-0.4999063003

    3218-0.599969991813.4799-0.5999699918

    3219.26-0.700142007911.6363-0.7001420079

    3217.97-0.79989949869.39598-0.7998994986

    3218.18-0.90007086726.95417-0.9000708672

    3217.38-1.00002605856.49302-1.0000260585

    3217.88-1.09991156483.84654-1.0999115648

    3217.83-1.19994999333.95565-1.1999499933

    3216.94-1.29986764332.61325-1.2998676433

    3219.72-1.40001873172.87376-1.4000187317

    3218.1-1.49995573342.60892-1.4999557334

    3219.33-1.60004952560.828492-1.6000495256

    3219.96-1.69997003211.52985-1.6999700321

    3218.91-1.79988854040.98963-1.7998885404

    3217.99-1.89987079661.06047-1.8998707966

    3217.78-1.99995657270.96916-1.9999565727

    3218.45-2.1000154375-0.393767-2.1000154375

    3216.06-2.20003947261.62574-2.2000394726

    3217.62-2.29998893772.13078-2.2999889377

    3216.57-2.39989874430.711705-2.3998987443

    Log Freq (Hz)

    Real (Ohm)

    -Imag (Ohm)

    Potentiostatic EIS 'Copeland_EIS_Dummy.DTA' 12/9/1999-10:41:55

    Real (Ohm)

    -Imag (Ohm)

  • Bode PlotIndividual charge transfer processes are resolvable.Frequency is explicit.

    Small impedances in presence of large impedances can be identified easily.Nyquist PlotIndividual charge transfer processes are resolvable.Frequency is not obvious.Small impedances can be swamped by large impedances.

    Nyquist vs. Bode Plot

  • Analyzing EIS: ModelingElectrochemical cells can be modeled as a network of passive electrical circuit elements.A network is called an equivalent circuit.The EIS response of an equivalent circuit can be calculated and compared to the actual EIS response of the electrochemical cell.

  • Frequency Response of Electrical Circuit ElementsResistor Capacitor Inductor Z = R (Ohms) Z = -j/C (Farads) Z = jL (Henrys) 0 Phase Shift -90 Phase Shift 90 Phase Shift

    j = -1 = 2f radians/s, f = frequency (Hz or cycles/s)A real response is in-phase (0) with the excitation. An imaginary response is 90 out-of-phase.

  • EIS of a Resistor

    Graph

    00

    0.01745240640.0139619251

    0.03489949670.0279195974

    0.05233595620.041868765

    0.06975647370.055805179

    0.08715574270.0697245942

    0.10452846330.0836227706

    0.12186934340.0974954747

    0.1391731010.1113384808

    0.1564344650.125147572

    0.17364817770.1389185421

    0.19080899540.1526471963

    0.20791169080.1663293527

    0.22495105430.1799608435

    0.24192189560.1935375165

    0.25881904510.2070552361

    0.27563735580.2205098847

    0.29237170470.2338973638

    0.30901699440.2472135955

    0.32556815450.2604545236

    0.34202014330.2736161147

    0.35836794950.2866943596

    0.37460659340.2996852747

    0.39073112850.3125849028

    0.40673664310.3253893145

    0.42261826170.3380946094

    0.43837114680.3506969174

    0.45399049970.3631923998

    0.46947156280.3755772502

    0.48480962020.3878476962

    0.50.4

    0.51503807490.4120304599

    0.52991926420.4239354114

    0.5446390350.435711228

    0.55919290350.4473543228

    0.57357643640.4588611491

    0.58778525230.4702282018

    0.60181502320.4814520185

    0.61566147530.4925291803

    0.6293203910.5034563128

    0.64278760970.5142300877

    0.6560590290.5248472232

    0.66913060640.5353044851

    0.68199836010.545598688

    0.69465837050.5557266964

    0.70710678120.5656854249

    0.71933980030.5754718403

    0.73135370160.5850829613

    0.74314482550.5945158604

    0.75470958020.6037676642

    0.76604444310.6128355545

    0.77714596150.6217167692

    0.78801075360.6304086029

    0.798635510.638908408

    0.80901699440.6472135955

    0.81915204430.6553216354

    0.82903757260.663230058

    0.83867056790.6709364544

    0.84804809620.6784384769

    0.85716730070.6857338406

    0.86602540380.692820323

    0.87461970710.6996957657

    0.88294759290.7063580743

    0.89100652420.7128052194

    0.89879404630.719035237

    0.9063077870.7250462296

    0.91354545760.7308363661

    0.92050485350.7364038828

    0.92718385460.7417470837

    0.93358042650.7468643412

    0.93969262080.7517540966

    0.94551857560.7564148605

    0.95105651630.760845213

    0.9563047560.7650438048

    0.96126169590.7690093568

    0.96592582630.772740661

    0.97029572630.776236581

    0.97437006480.7794960518

    0.97814760070.7825180806

    0.98162718340.7853017468

    0.9848077530.7878462024

    0.98768834060.7901506725

    0.99026806870.792214455

    0.99254615160.7940369213

    0.99452189540.7956175163

    0.99619469810.7969557585

    0.99756405030.7980512402

    0.99862953480.7989036278

    0.9993908270.7995126616

    0.99984769520.7998781561

    10.8

    0.99984769520.7998781561

    0.9993908270.7995126616

    0.99862953480.7989036278

    0.99756405030.7980512402

    0.99619469810.7969557585

    0.99452189540.7956175163

    0.99254615160.7940369213

    0.99026806870.792214455

    0.98768834060.7901506725

    0.9848077530.7878462024

    0.98162718340.7853017468

    0.97814760070.7825180806

    0.97437006480.7794960518

    0.97029572630.776236581

    0.96592582630.772740661

    0.96126169590.7690093568

    0.9563047560.7650438048

    0.95105651630.760845213

    0.94551857560.7564148605

    0.93969262080.7517540966

    0.93358042650.7468643412

    0.92718385460.7417470837

    0.92050485350.7364038828

    0.91354545760.7308363661

    0.9063077870.7250462296

    0.89879404630.719035237

    0.89100652420.7128052194

    0.88294759290.7063580743

    0.87461970710.6996957657

    0.86602540380.692820323

    0.85716730070.6857338406

    0.84804809620.6784384769

    0.83867056790.6709364544

    0.82903757260.663230058

    0.81915204430.6553216354

    0.80901699440.6472135955

    0.798635510.638908408

    0.78801075360.6304086029

    0.77714596150.6217167692

    0.76604444310.6128355545

    0.75470958020.6037676642

    0.74314482550.5945158604

    0.73135370160.5850829613

    0.71933980030.5754718403

    0.70710678120.5656854249

    0.69465837050.5557266964

    0.68199836010.545598688

    0.66913060640.5353044851

    0.6560590290.5248472232

    0.64278760970.5142300877

    0.6293203910.5034563128

    0.61566147530.4925291803

    0.60181502320.4814520185

    0.58778525230.4702282018

    0.57357643640.4588611491

    0.55919290350.4473543228

    0.5446390350.435711228

    0.52991926420.4239354114

    0.51503807490.4120304599

    0.50.4

    0.48480962020.3878476962

    0.46947156280.3755772502

    0.45399049970.3631923998

    0.43837114680.3506969174

    0.42261826170.3380946094

    0.40673664310.3253893145

    0.39073112850.3125849028

    0.37460659340.2996852747

    0.35836794950.2866943596

    0.34202014330.2736161147

    0.32556815450.2604545236

    0.30901699440.2472135955

    0.29237170470.2338973638

    0.27563735580.2205098847

    0.25881904510.2070552361

    0.24192189560.1935375165

    0.22495105430.1799608435

    0.20791169080.1663293527

    0.19080899540.1526471963

    0.17364817770.1389185421

    0.1564344650.125147572

    0.1391731010.1113384808

    0.12186934340.0974954747

    0.10452846330.0836227706

    0.08715574270.0697245942

    0.06975647370.055805179

    0.05233595620.041868765

    0.03489949670.0279195974

    0.01745240640.0139619251

    00

    -0.0174524064-0.0139619251

    -0.0348994967-0.0279195974

    -0.0523359562-0.041868765

    -0.0697564737-0.055805179

    -0.0871557427-0.0697245942

    -0.1045284633-0.0836227706

    -0.1218693434-0.0974954747

    -0.139173101-0.1113384808

    -0.156434465-0.125147572

    -0.1736481777-0.1389185421

    -0.1908089954-0.1526471963

    -0.2079116908-0.1663293527

    -0.2249510543-0.1799608435

    -0.2419218956-0.1935375165

    -0.2588190451-0.2070552361

    -0.2756373558-0.2205098847

    -0.2923717047-0.2338973638

    -0.3090169944-0.2472135955

    -0.3255681545-0.2604545236

    -0.3420201433-0.2736161147

    -0.3583679495-0.2866943596

    -0.3746065934-0.2996852747

    -0.3907311285-0.3125849028

    -0.4067366431-0.3253893145

    -0.4226182617-0.3380946094

    -0.4383711468-0.3506969174

    -0.4539904997-0.3631923998

    -0.4694715628-0.3755772502

    -0.4848096202-0.3878476962

    -0.5-0.4

    -0.5150380749-0.4120304599

    -0.5299192642-0.4239354114

    -0.544639035-0.435711228

    -0.5591929035-0.4473543228

    -0.5735764364-0.4588611491

    -0.5877852523-0.4702282018

    -0.6018150232-0.4814520185

    -0.6156614753-0.4925291803

    -0.629320391-0.5034563128

    -0.6427876097-0.5142300877

    -0.656059029-0.5248472232

    -0.6691306064-0.5353044851

    -0.6819983601-0.545598688

    -0.6946583705-0.5557266964

    -0.7071067812-0.5656854249

    -0.7193398003-0.5754718403

    -0.7313537016-0.5850829613

    -0.7431448255-0.5945158604

    -0.7547095802-0.6037676642

    -0.7660444431-0.6128355545

    -0.7771459615-0.6217167692

    -0.7880107536-0.6304086029

    -0.79863551-0.638908408

    -0.8090169944-0.6472135955

    -0.8191520443-0.6553216354

    -0.8290375726-0.663230058

    -0.8386705679-0.6709364544

    -0.8480480962-0.6784384769

    -0.8571673007-0.6857338406

    -0.8660254038-0.692820323

    -0.8746197071-0.6996957657

    -0.8829475929-0.7063580743

    -0.8910065242-0.7128052194

    -0.8987940463-0.719035237

    -0.906307787-0.7250462296

    -0.9135454576-0.7308363661

    -0.9205048535-0.7364038828

    -0.9271838546-0.7417470837

    -0.9335804265-0.7468643412

    -0.9396926208-0.7517540966

    -0.9455185756-0.7564148605

    -0.9510565163-0.760845213

    -0.956304756-0.7650438048

    -0.9612616959-0.7690093568

    -0.9659258263-0.772740661

    -0.9702957263-0.776236581

    -0.9743700648-0.7794960518

    -0.9781476007-0.7825180806

    -0.9816271834-0.7853017468

    -0.984807753-0.7878462024

    -0.9876883406-0.7901506725

    -0.9902680687-0.792214455

    -0.9925461516-0.7940369213

    -0.99


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