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(c) Bob Cottis 1995 Corrosion Measurement Corrosion Measurement Techniques Techniques A copy of this presentation is A copy of this presentation is available in the CAL group in available in the CAL group in the computers in the Teaching the computers in the Teaching Lab, or via the WWW at Lab, or via the WWW at http://www.cp.umist.ac.uk/CPC/L_ http://www.cp.umist.ac.uk/CPC/L_ Notes Notes

TECNICAS PARA EVALUAR CORROSION

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(c) Bob Cottis 1995

Corrosion Measurement Corrosion Measurement TechniquesTechniques

A copy of this presentation is available A copy of this presentation is available in the CAL group in the computers in in the CAL group in the computers in the Teaching Lab, or via the WWW at the Teaching Lab, or via the WWW at http://www.cp.umist.ac.uk/CPC/L_Notehttp://www.cp.umist.ac.uk/CPC/L_Notess

(c) Bob Cottis 1995

Corrosion Measurement Corrosion Measurement TechniquesTechniques

Polarization curvesPolarization curvesLinear Polarization ResistanceLinear Polarization ResistanceOpen Circuit Potential DecayOpen Circuit Potential DecayAC Impedance MeasurementAC Impedance MeasurementElectrochemical Noise MeasurementElectrochemical Noise MeasurementWeight Loss MeasurementWeight Loss Measurement

(c) Bob Cottis 1995

Polarization CurvesPolarization Curves

Measurement methodsCell designPlotting dataInterpretation

(c) Bob Cottis 1995

Measurement MethodsMeasurement Methods

ObjectiveObjective– determine current density under determine current density under steady-steady-

statestate conditions as a function of potential conditions as a function of potential– not really practical, as this would strictly not really practical, as this would strictly

require one sample for each potentialrequire one sample for each potential– therefore compromise on ‘closeness’ to true therefore compromise on ‘closeness’ to true

steady-statesteady-state

(c) Bob Cottis 1995

Measurement MethodsMeasurement Methods

Potential controlPotential control

AE

RE

WE

Potentiostat

Working Electrode - metal being studied

Counter Electrode (or Auxilliary Electrode or Secondary Electrode) - provides current path

into solution

Reference Electrode - reference connection for potential measurement

Luggin Probe - allows potential to be detected close to metal surface

Potentiostat controls potential

Connect electrodes to corresponding terminals

on potentiostat

(c) Bob Cottis 1995

Measurement MethodsMeasurement Methods

Current controlCurrent control

AE

RE

WE

Potentiostat

Working Electrode

Counter ElectrodeR

Luggin Probe still needed to limit IR error

V

Current controlled by control of voltage across

resistor (I=V/R)

Current path

Reference Electrode -only used to monitor potential, not connected to potentiostat

(c) Bob Cottis 1995

Measurement MethodsMeasurement Methods

Swept potential or currentSwept potential or current– Use Use sweep generatorsweep generator to produce slowly to produce slowly

changing potentialchanging potential– Sweep generator output controls Sweep generator output controls

potentiostatpotentiostat– Record response on chart recorder (or use Record response on chart recorder (or use

computer monitoring)computer monitoring)– Swept current not often used, as it moves Swept current not often used, as it moves

through corrosion potential very quicklythrough corrosion potential very quickly

(c) Bob Cottis 1995

Measurement MethodsMeasurement Methods

Potential or current stepPotential or current step– Step potential or current from one value to Step potential or current from one value to

the next, allowing time to stabilise at each the next, allowing time to stabilise at each new valuenew value

– Record current or potentialRecord current or potential– May be manually controlled, or use May be manually controlled, or use

computer to step potential/current and take computer to step potential/current and take readingsreadings

(c) Bob Cottis 1995

Measurement MethodsMeasurement Methods

Sweep directionSweep direction– Aim to perform experiment in such an order Aim to perform experiment in such an order

that the initial polarization affects that the initial polarization affects subsequent results as little as possiblesubsequent results as little as possible

– OptionsOptions new specimen for each potentialnew specimen for each potential one specimen for cathodic polarization, and one one specimen for cathodic polarization, and one

for anodic, both start at corrosion potentialfor anodic, both start at corrosion potential one specimen, sweep from cathodic to anodicone specimen, sweep from cathodic to anodic

(c) Bob Cottis 1995

Measurement MethodsMeasurement Methods

Sweep rate (or step rate)Sweep rate (or step rate)– Ideal, all measurements made at steady-Ideal, all measurements made at steady-

statestate– Time-dependent effects include:Time-dependent effects include:

Charging of double layer capacitance (I = C dV/dt)Charging of double layer capacitance (I = C dV/dt) Mass transport effects (t Mass transport effects (t L L22/D)/D) Adsorbed species and surface films (Faraday’s Adsorbed species and surface films (Faraday’s

Law)Law)

– Typical sweep rates are of the order of 1 Typical sweep rates are of the order of 1 mV/s or lessmV/s or less

(c) Bob Cottis 1995

QuestionsQuestions Consider the corrosion of iron in aerated Consider the corrosion of iron in aerated

neutral solution, with the following neutral solution, with the following parameters:parameters:– CCdldl = 35 = 35 F / cmF / cm22 DDO2O2 = 1.2 x 10 = 1.2 x 10-5-5 cm cm22 /s /s

– Boundary layer thickness, Boundary layer thickness, = 100 = 100 mm– Number of iron atoms on surface Number of iron atoms on surface 2 210101919/cm/cm22

– Charge on the electron = 1.6 x 10Charge on the electron = 1.6 x 10-19-19CC CalculateCalculate

– Capacitive current at 1 mV/sCapacitive current at 1 mV/s– Characteristic diffusion timeCharacteristic diffusion time

– Limiting current density for OLimiting current density for O22 reduction (8 ppm O reduction (8 ppm O22))

– Time to oxidise Fe surface to FeOH (FeTime to oxidise Fe surface to FeOH (Fe++) at ) at iilimlim

(c) Bob Cottis 1995

Cell DesignCell Design

Working electrodeWorking electrode Reference electrodeReference electrode Counter electrodeCounter electrode SolutionSolution Mass transportMass transport

(c) Bob Cottis 1995

Working ElectrodeWorking Electrode

RequirementsRequirements– reproduciblereproducible– representativerepresentative– free of crevicesfree of crevices– free of edge effectsfree of edge effects– free of galvanic effectsfree of galvanic effects– free of water-line effectsfree of water-line effects

(c) Bob Cottis 1995

Working ElectrodeWorking Electrode

Epoxy embedded electrode:Epoxy embedded electrode:

Pretreat specimen for good adhesion

Apply thin layer of epoxy to minimise stress and risk

of crevice formation

Weld or solder connecting wire to specimenApply thick layer of epoxy

to seal connecting tube and for strength Carefully grind surface

to expose metal

Clean surface - don’t use acetone

(c) Bob Cottis 1995

Working ElectrodeWorking Electrode

Stern-Makrides Stern-Makrides electrodes:electrodes:

Metal rodRetaining nutWashers

Heavy-walledglass tube

PTFE Washer

Electrode

Lip sealbetween

PTFE case and electrode

(c) Bob Cottis 1995

Working ElectrodeWorking Electrode

Avesta cell:Avesta cell:

Specimen

NaClSolution

PureH2O feed

Filter paper

(c) Bob Cottis 1995

Reference ElectrodeReference Electrode

Commonly use Saturated Calomel Commonly use Saturated Calomel Electrode (SCE)Electrode (SCE)

Properties may degrade with time (and Properties may degrade with time (and misuse)misuse)– check one against another (should not be check one against another (should not be

more than 1 to 2 mV difference)more than 1 to 2 mV difference)– do do notnot pass current through the reference pass current through the reference

electrode (e.g. do not connect to working or electrode (e.g. do not connect to working or counter electrode)counter electrode)

– do not allow to dry outdo not allow to dry out

(c) Bob Cottis 1995

Reference ElectrodeReference Electrode

Solution in SCE (or Ag/AgCl electrode) is Solution in SCE (or Ag/AgCl electrode) is saturated KClsaturated KCl– beware of chloride contamination of test beware of chloride contamination of test

solution by Clsolution by Cl-- leaking from reference leaking from reference electrodeelectrode

– make sure solution remains saturatedmake sure solution remains saturated

(c) Bob Cottis 1995

Luggin ProbeLuggin Probe

A Luggin probe should be used A Luggin probe should be used whenever there is a significant current whenever there is a significant current applied to the electrodeapplied to the electrode

Luggin probe allows point at which potential is measured to be close to electrode surface

(around 3 times tip diameter is best)

Ele

ctro

de

(c) Bob Cottis 1995

Counter electrodeCounter electrode

Counter electrode should allow current to Counter electrode should allow current to pass with tolerable polarizationpass with tolerable polarization

Often claimed that counter electrode Often claimed that counter electrode should have much larger area than working should have much larger area than working electrode, but this is not often necessary electrode, but this is not often necessary for corrosion studiesfor corrosion studies

Usually use platinum or graphite, although Usually use platinum or graphite, although stainless steel can be used in some stainless steel can be used in some situations (e.g. where only anodic situations (e.g. where only anodic polarization of specimen is used)polarization of specimen is used)

(c) Bob Cottis 1995

SolutionSolution

Requirements:Requirements:– as high a conductivity as possible (add as high a conductivity as possible (add

supporting electrolyte, such as sodium supporting electrolyte, such as sodium perchlorate?)perchlorate?)

– remain the same (pH, composition) throughout remain the same (pH, composition) throughout the experiment - ensure that volume is the experiment - ensure that volume is adequateadequate

– oxygen concentration often critical - aerate by oxygen concentration often critical - aerate by bubbling air or Obubbling air or O22 or deaerate with N or deaerate with N22 or Ar or Ar

– most reactions temperature sensitive, so most reactions temperature sensitive, so control, or at least record, temperaturecontrol, or at least record, temperature

(c) Bob Cottis 1995

Mass transportMass transport

Methods of controlling mass transportMethods of controlling mass transport– rotating disk or cylinderrotating disk or cylinder– flow channelflow channel– jet impingementjet impingement– gas bubblinggas bubbling

(c) Bob Cottis 1995

Plotting of Polarization Plotting of Polarization CurvesCurves Comparison of log-Comparison of log-ii and linear- and linear-ii plots plots Identification of anodic and cathodic Identification of anodic and cathodic

regions on log-regions on log-ii plots plots Orientation of plotsOrientation of plots

(c) Bob Cottis 1995

E-i Plot

-1.000E-01

-8.000E-02

-6.000E-02

-4.000E-02

-2.000E-02

0.000E+00

2.000E-02

4.000E-02

6.000E-02

8.000E-02

1.000E-01

-1.500 -1.300 -1.100 -0.900 -0.700 -0.500 -0.300 -0.100 0.100 0.300 0.500

Potential

Cu

rre

nt

De

nsi

ty

Fe anodic H cathodic

O2 cathodic Net

(c) Bob Cottis 1995

E log |i| Plot

1.000E-06

1.000E-05

1.000E-04

1.000E-03

1.000E-02

1.000E-01

-1.500 -1.300 -1.100 -0.900 -0.700 -0.500 -0.300 -0.100 0.100 0.300 0.500

Potential

Cu

rre

nt

De

ns

ity

Fe anodic H cathodic O2 cathodic

Net anodic Net cathodic

(c) Bob Cottis 1995

log |log |i|i|

E

E log |E log |II| - old plotting method| - old plotting method

(c) Bob Cottis 1995

Interpretation of Interpretation of Polarization CurvesPolarization Curves Addition of reactions on log-I graphsAddition of reactions on log-I graphs Tafel regionsTafel regions Mass transport controlMass transport control Active-passive transitionActive-passive transition Transpassive corrosionTranspassive corrosion Pitting CorrosionPitting Corrosion

(c) Bob Cottis 1995

Tafel regionsTafel regions

A Tafel region is a straight line in the A Tafel region is a straight line in the E-log|E-log|ii| plot| plot

For a reliable Tafel slope:For a reliable Tafel slope:– the line should be straight for at least one the line should be straight for at least one

decade (in this context a decade (in this context a decadedecade implies a implies a change of current density by a factor of ten, change of current density by a factor of ten, i.e a difference of 1 in log i.e a difference of 1 in log i i ))

– the region should be next to Ethe region should be next to Ecorrcorr

(c) Bob Cottis 1995

E log |i| Plot

1.000E-06

1.000E-05

1.000E-04

1.000E-03

1.000E-02

1.000E-01

-1.500 -1.300 -1.100 -0.900 -0.700 -0.500 -0.300 -0.100 0.100 0.300 0.500

Potential

Cu

rre

nt

De

ns

ity

Fe anodic H cathodic O2 cathodic

Net anodic Net cathodic

(c) Bob Cottis 1995

Tafel ExtrapolationTafel Extrapolation

Extrapolate anodic or cathodic Tafel Extrapolate anodic or cathodic Tafel region, or both, back to Eregion, or both, back to Ecorrcorr, when the , when the current density is icurrent density is icorrcorr

In aerated neutral solutions, where In aerated neutral solutions, where mass transport limited oxygen mass transport limited oxygen reduction is the main cathodic reaction, reduction is the main cathodic reaction, the cathodic reaction does not have a the cathodic reaction does not have a valid Tafel slope, but the anodic slope valid Tafel slope, but the anodic slope can sometimes be usedcan sometimes be used

(c) Bob Cottis 1995

QuestionQuestion

How can we estimate the rate of How can we estimate the rate of hydrogen evolution during free hydrogen evolution during free corrosion?corrosion?

Estimate the value for the graph shown.Estimate the value for the graph shown.

(c) Bob Cottis 1995

E log |i| Plot

1.000E-06

1.000E-05

1.000E-04

1.000E-03

1.000E-02

1.000E-01

-1.500 -1.300 -1.100 -0.900 -0.700 -0.500 -0.300 -0.100 0.100 0.300 0.500

Potential

Cu

rre

nt

De

ns

ity

Fe anodic H cathodic O2 cathodic

Net anodic Net cathodic

(c) Bob Cottis 1995

Mass transport controlMass transport control

When the supply of a reactant becomes When the supply of a reactant becomes mass transport controlled, we observe a mass transport controlled, we observe a limiting current densitylimiting current density

The most common case occurs for oxygen The most common case occurs for oxygen as a cathodic reactant in neutral solutionsas a cathodic reactant in neutral solutions

NOTE - the diffusion of a reaction product NOTE - the diffusion of a reaction product away from the electrode will away from the electrode will notnot affect affect the rate of the the rate of the forwardforward reaction reaction

(c) Bob Cottis 1995

Solution Resistance Solution Resistance EffectsEffects At high currents the potential drop At high currents the potential drop

associated with the solution resistance associated with the solution resistance can be significantcan be significant

It is generally referred to as an It is generally referred to as an IRIR error error Gives a straight line on Gives a straight line on EE--ii plots plots

(c) Bob Cottis 1995

E log |i| Plot

1.000E-06

1.000E-05

1.000E-04

1.000E-03

1.000E-02

1.000E-01

-1.500 -1.300 -1.100 -0.900 -0.700 -0.500 -0.300 -0.100 0.100 0.300 0.500

Potential

Cu

rre

nt

De

ns

ity

Fe anodic H cathodic O2 cathodic

Net anodic Net cathodic

(c) Bob Cottis 1995

E-i Plot

-2.000E-02

-1.500E-02

-1.000E-02

-5.000E-03

0.000E+00

5.000E-03

1.000E-02

1.500E-02

2.000E-02

-1.500 -1.300 -1.100 -0.900 -0.700 -0.500 -0.300 -0.100 0.100 0.300 0.500

Potential

Cu

rre

nt

De

nsi

ty

Fe anodic H cathodic

O2 cathodic Net

(c) Bob Cottis 1995

Active-passive transitionActive-passive transition

As a passive film develops, it covers the As a passive film develops, it covers the surface and shuts off the dissolution surface and shuts off the dissolution reaction, leading to an active-passive reaction, leading to an active-passive transitiontransition

log |i|

E

(c) Bob Cottis 1995

Active-passive transitionActive-passive transition

For stainless steel we sometimes see For stainless steel we sometimes see two active-passive transitions, on for two active-passive transitions, on for Chromium, and one for IronChromium, and one for Iron

E

log |i|

(c) Bob Cottis 1995

Transpassive corrosionTranspassive corrosion

A passive metal (notably Cr and Fe) may A passive metal (notably Cr and Fe) may start to dissolve at a very positive potential start to dissolve at a very positive potential when a higher oxidation state (e.g. Crwhen a higher oxidation state (e.g. Cr6+6+ as as chromate) is formedchromate) is formed

This is known as transpassive corrosion, This is known as transpassive corrosion, and will give something like a second and will give something like a second activation-controlled reaction activation-controlled reaction

For alloys the behaviour will be For alloys the behaviour will be complicated by the differing behaviours of complicated by the differing behaviours of the alloy componentsthe alloy components

(c) Bob Cottis 1995

Anodic Polarization Curve Anodic Polarization Curve for Stainless Steelfor Stainless Steel

E

log |i|

Activation-controlled dissolution

Active-passive transition

Active peak for ironTranspassive

corrosion of CrOxygen reduction

Overall anodic curve

(c) Bob Cottis 1995

Pitting CorrosionPitting Corrosion

Pitting shows up as an increasing anodic Pitting shows up as an increasing anodic current before (at a less positive current before (at a less positive potential than) transpassive corrosion or potential than) transpassive corrosion or oxygen evolution, usually preceded by oxygen evolution, usually preceded by noisenoise

EE-log|-log|ii| plot does not follow same path if | plot does not follow same path if scan direction is reversed, but current is scan direction is reversed, but current is greater (since pit continues to grow)greater (since pit continues to grow)

(c) Bob Cottis 1995

Pitting CorrosionPitting Corrosion

E

log |i|

Noise spikes due to meta-stable pittingCurrent continues to increase after reversal of scan

Pit eventually re-passivates

(c) Bob Cottis 1995

What is going on?What is going on?

E

log |i|

Cathodic

Anodic

Cathodic

Anodic

Stainless Steel in Aerated Sulphuric Acid

(c) Bob Cottis 1995

Linear Polarization Linear Polarization Resistance MeasurementResistance Measurement

Theoretical basisTheoretical basis

Measurement methods

Interpretation

(c) Bob Cottis 1995

LPRM TheoryLPRM Theory

For an activation controlled reactionFor an activation controlled reaction

i

EEi

dE

di

EEii

oo

oo

exp

exp Exchange current density

Equilibrium potentialTafel slope based on

exponential (i.e. mV for a change of 1 in

ln(i))

(c) Bob Cottis 1995

LPRM TheoryLPRM Theory

Summing for two reactionsSumming for two reactions

Rearrange and convert to Rearrange and convert to bb rather than rather than

pca

cacorr

c

c

a

a

Ri

ii

dE

di

1

ca

ca

corrp bb

bbB

i

BR

3.2,

Anodic partial current density (=icorr)

Anodic Tafel slope (positive)

Cathodic partial current density (= -icorr)

Cathodic Tafel slope (negative)

Because c is taken as negative

Tafel slope based on a decade change in

current (i.e. a change of 1 in log i )

Stern-Geary coefficient

(c) Bob Cottis 1995

LPRM Measurement LPRM Measurement MethodsMethods Control variableControl variable WaveformWaveform Cell configurationCell configuration Sweep rateSweep rate

(c) Bob Cottis 1995

LPRM Control VariableLPRM Control Variable

Potential controlPotential control– potential range can be optimisedpotential range can be optimised

– problems with drift of Eproblems with drift of Ecorrcorr

Current controlCurrent control– potential range depends on potential range depends on RRpp

– measurement inherently centred about measurement inherently centred about ii = = 00

(c) Bob Cottis 1995

LPRM Measurement LPRM Measurement WaveformWaveform Triangle waveTriangle wave

– can measure can measure didi//dtdt at at ii = 0 = 0– requires relatively complex instrumentsrequires relatively complex instruments

Square wave (switch between +Square wave (switch between +ii and - and -i)i)– simple instrumentssimple instruments– simple to automatesimple to automate

Sine waveSine wave– simplest theory for frequency effectssimplest theory for frequency effects– complex to perform measurementcomplex to perform measurement

(c) Bob Cottis 1995

LPRM Cell ConfigurationLPRM Cell Configuration

Two electrodeTwo electrode– assume assume RRpp is the same for two similar electrodes is the same for two similar electrodes

and measure cell resistance (= 2and measure cell resistance (= 2RRpp + + RRsolsol))

– easy, no reference electrode requiredeasy, no reference electrode required Three electrodeThree electrode

– use conventional counter, reference and use conventional counter, reference and working electrodesworking electrodes

– provides lower solution resistance, therefore provides lower solution resistance, therefore better for low conductivity solutionsbetter for low conductivity solutions

– more complex instrumentationmore complex instrumentation

(c) Bob Cottis 1995

LPRM RecommendationsLPRM Recommendations

Use three electrode measurement with triangle Use three electrode measurement with triangle waveform for laboratory studieswaveform for laboratory studies

Use two electrode measurement with square Use two electrode measurement with square waveform for simple corrosion monitoring (use waveform for simple corrosion monitoring (use three electrodes for high resistance solutions)three electrodes for high resistance solutions)

Use potential control when Use potential control when iicorrcorr variation is large variation is large

Use current control when Use current control when EEcorrcorr varies a lot varies a lot

When both When both iicorrcorr and and EEcorrcorr vary use current control, vary use current control, but adapt current to keep potential range but adapt current to keep potential range reasonablereasonable

(c) Bob Cottis 1995

LPRM InterpretationLPRM Interpretation

Determination of B valueDetermination of B value– calculate from Tafel slopescalculate from Tafel slopes

– correlation with weight losscorrelation with weight loss– arbitrary valuearbitrary value

26 mV for activation control26 mV for activation control 52 mV for one reaction at limiting current52 mV for one reaction at limiting current

ca

ca

bb

bbB

3.2

(c) Bob Cottis 1995

LPRM Sweep RateLPRM Sweep Rate

Must be sufficiently slow for current Must be sufficiently slow for current charging double layer capacitance to be charging double layer capacitance to be much less than total currentmuch less than total current

Characteristic time given by Characteristic time given by RRctctCCdldl - cycle - cycle time should be at least 3 times thistime should be at least 3 times this

Need Need notnot be slow enough to allow be slow enough to allow diffusion processes to respond (as the diffusion processes to respond (as the basic theory is not valid for diffusion basic theory is not valid for diffusion processes)processes)

(c) Bob Cottis 1995

LPRM ProblemsLPRM Problems

Theoretically, either Theoretically, either – both reactions must be activation controlled, or both reactions must be activation controlled, or – one reaction must be activation controlled and one reaction must be activation controlled and

the other mass-transport limitedthe other mass-transport limited In practice it is rare for real systems to In practice it is rare for real systems to

meet these constraints, and application of meet these constraints, and application of LPRM is not theoretically justified LPRM is not theoretically justified

Solution resistance adds to measured Solution resistance adds to measured RRpp, , and produces lower apparent corrosion rateand produces lower apparent corrosion rate

(c) Bob Cottis 1995

Equivalent CircuitsEquivalent Circuits

An electrical circuit with the same An electrical circuit with the same properties as a metal-solution interfaceproperties as a metal-solution interface

The simplest circuit is a resistor, The simplest circuit is a resistor, RRctct, , corresponding to the polarization corresponding to the polarization resistance, in parallel with a capacitor, resistance, in parallel with a capacitor, CCdldl,, corresponding to the double layer corresponding to the double layer capacitancecapacitance

SolutionSolution

(c) Bob Cottis 1995

Equivalent CircuitsEquivalent Circuits

An electrical circuit with the same An electrical circuit with the same properties as a metal-solution interfaceproperties as a metal-solution interface

The Randles equivalent circuit adds a The Randles equivalent circuit adds a series resistor, corresponding to the series resistor, corresponding to the solution resistancesolution resistance

RctRct

RsolRsol

(c) Bob Cottis 1995

Time

i

E

Analysis of Solution Analysis of Solution ResistanceResistance If we analyse the full response to the LPRM If we analyse the full response to the LPRM

measurement, we can estimate measurement, we can estimate RRsolsol, , CCdldl and and RRctct

Vo=iRsol

V=iRct

The voltage across Rsol is given by Voexp(-t/RsolCdl)

When t= RsolCdl, V=Voexp(-1)

Estimate Cdl from the exponential decay. The time for V to fall to e-1 (37%) of the initial value is RsolCdl

(c) Bob Cottis 1995

Open Circuit Potential Open Circuit Potential DecayDecay Similar to analysis of LPRM Similar to analysis of LPRM

measurementmeasurement– charge double layer capacitance by charge double layer capacitance by

applying a current or potentialapplying a current or potential– disconnect charging currentdisconnect charging current– monitor decay of potentialmonitor decay of potential

(c) Bob Cottis 1995

Open Circuit Potential Open Circuit Potential DecayDecay

Time

E

Initial voltage drop = iRsol

Delayed voltage drop

= iRct

Charging at current i Disconnected

0.37iRct

Time = RctCdl

(c) Bob Cottis 1995