How batteries work Chapter 5 Polarization

Chapter 5

Galvanostatic cycling and polarization: overpotentials, hysteresis and heat generation

What happens when we charge a cell and use it, and why does it get warm?

Where we are. In chapter 4 we counted the charge your cell can hold and turned it into energy, with the cell mostly at rest. Now we let the current flow and ask what that costs. We start at the test bench, with how a charge and a discharge are actually done and recorded. Then we follow what current costs (voltage, capacity and heat), why some curves are flat and others slope, what happens inside real electrodes made of many particles, how a curve can be read for signs of ageing, and how many cycles a cell survives. From here on we follow one lithium ion, our ion, through the cell.

Figure 5.1 A cell wired to a cycler, with a meter showing the current, voltage, charge passed and time. Below, the voltage and the current against time through one full test: a discharge at constant current down to the lower cut-off voltage, a rest, a charge at constant current up to the upper cut-off, a constant-voltage step in which the current falls away to one twentieth of the 1C current, and a final rest. Electrons are drawn moving round the wire and lithium ions through the electrolyte, from the negative to the positive electrode on discharge and back on charge.
lithium ionelectroncut-off voltagesconstant-voltage step
One full test of your cell, as a cycler runs it. Try this: slide the current from C/10 to 5C and watch the time axis shrink, the discharge stop earlier and the constant-voltage step grow longer.
SourceGalvanostatic cycling, the 1C convention (the current that passes the full nominal charge in one hour) and a constant-current, constant-voltage protocol with a hold at 4.2 V until the current falls to C/20, after Olson, López and Dickinson 2023, section 4.1; Safari and Delacourt 2011 used a similar protocol (a 4.2 V vs Li hold, and a hold on discharge too) ending at C/25. Cut-offs after Winter and Brodd 2004. The cell is the illustrative cell of figure 5.3 with a steeper end of charge added so that the constant-voltage step can finish, on a nominal 2.0 Ah. The rests are drawn short, and the slow return of the voltage after the current stops (figure 5.4) is left out. Schematic, not experimental data.

Let us connect your cell to something that uses it. A discharge is the cell doing work: current flows out through a lamp, a phone or a motor. A charge is the reverse: a charger pushes current back in, in the opposite direction, and restores the cell to its charged condition.1 On discharge the negative electrode is oxidized and the positive electrode is reduced; charging runs both reactions backwards.1 In a lithium-ion cell that means lithium ions leave one electrode, cross the electrolyte and enter the other, while their electrons go round through the wire (chapter 2).

The test bench

To watch this happen we put the cell on a test bench. Cells are studied, and their capacities measured, on a cycler: an instrument that sets the current and records the voltage at regular intervals (once a second in one set of tests). Most tests hold the current constant. This is galvanostatic cycling, and its record, the voltage against the charge passed, is the raw material of everything else in this module.2

At constant current the charge passed is simply current times time, Q=It. Currents are quoted as C-rates: 1C is the current that passes the cell’s full nominal charge in one hour, so C/20 takes twenty hours. In one study, for example, a 210 mAh cell was recorded at C/20, that is at 10.5 mA.2

A step cannot run for ever. Each step stops at a cut-off voltage. Overcharging or overdischarging drives unwanted reactions that cost capacity.1 Charging often ends with a constant-voltage step: the cycler holds the upper cut-off, the current falls away by itself, and the step ends when it reaches a small limit. In two published protocols the cycler held 4.2 V until the current had fallen to C/20 in one and to C/25 in the other.2,3

Worked example. A 2.0 Ah cell discharged at 1C for 30 minutes. I=1C×2.0 Ah=2.0 A Q=It=2.0 A×1800 s=3600 C = 1.0 Ah, half the cell At C/20 the same half takes ten hours, at 0.10 A.
building a coin cell and running a cycling test
  1. Punch discs of both electrodes (1.27 cm in one published guide, the same size for both), dry the coated discs under vacuum overnight at 110 °C in the antechamber of an argon glove box, and assemble the cell inside it.4
  2. Stack from the bottom: the can, the positive electrode, the separator, the negative electrode (lithium foil in a half cell, graphite in a full cell), a spacer, a spring, the gasket and the cap.4
  3. Add the electrolyte drop by drop between the layers, a fixed amount for every cell: in that guide 19 drops, about 38 mg, with two thin separators, or 21 drops, about 42 mg, with one thick one.4
  4. Keep the discs exactly on top of each other. Place the spacer and spring from directly above, with a vacuum pen rather than tweezers: a slightly misaligned disc let lithium plate where the positive electrode was not covered by the negative, and three “identical” cells kept about 95, 90 and 80 % of their capacity after 100 cycles. One thick, compliant separator gave more even pressure than two thin ones.4
  5. Crimp the cell, then cycle it in a temperature-controlled box (30.0 ± 0.1 °C there): a slow formation cycle and then repeated cycles, C/10 followed by C/5 or C/5 followed by C/3, between fixed cut-offs, 3.0 to 4.3 V for NMC622 against graphite.4
  6. Charge at constant current, hold the upper cut-off until the current falls to a small limit such as C/20, and discharge at constant current; record the capacity and Coulombic efficiency of every cycle.2,5
  7. Read half-cell results with care: their excess of lithium masks the side reactions that consume lithium in a full cell. A lithium foil over 50 µm thick holds more than 10 mAh/cm2: against a positive electrode of about 3 mAh/cm2, a negative-to-positive capacity ratio above 3.3, against about 1.1 in a full cell.4,5
Setting the current: a positive electrode disc of 1.27 cm with a loading of 21.3 mg/cm2, taken here as the whole coating, of which 96 % is active material, holds 21.3×π(0.635)2×0.96=25.9 mg of active material.4 At an illustrative 170 mAh/g that is 4.40 mAh, so C/10 is 0.44 mA and C/3 is 1.47 mA.

A test is a recipe of currents, limits and rests; the cell answers with its voltage.

Figure 5.2 A cell with a negative electrode, an electrolyte with a separator and a positive electrode, connected through a lamp or a charger. A strip below shows the potential along the cell: at open circuit two jumps at the electrode boundaries and a flat electrolyte; on discharge both jumps shrink and the electrolyte slopes, so the terminal voltage falls below the open-circuit value; on charge the jumps grow and the slope reverses, so the charger must apply more.
our ion (Li+)electronpotential
Where the voltage goes when current flows. The open-circuit voltage is your cell’s, from the ladder of chapter 3. Try this: switch between Discharge and Charge and watch the two jumps on the strip shrink, then grow; then slide the current up and watch the slope steepen.
SourcePotential profile with current after Bard, Faulkner and White 2022, section 1.5.1 and Figure 1.5.2; V=VOC−η on discharge and VOC+η on charge after Goodenough and Park 2013, eqs. 1.1 and 1.2; the three parts of η after Winter and Brodd 2004, section 1.4. The sizes of the parts come from the illustrative cell model of figure 5.3 at half charge and are drawn four times larger than the open-circuit jumps so that they can be seen. How VOC splits between the two jumps cannot be measured and is drawn schematically.

The cycler shows us only the voltage at the terminals. Let us look along the cell to see where that voltage goes. At rest, a cell’s voltage is the sum of the potential jumps at its two electrode-electrolyte boundaries (chapter 3). Draw a current and two things change. The ions must be driven through the electrolyte, which has a resistance, so the potential now slopes across it, cations running down the slope and anions up it: the ohmic drop, IR.6 And each boundary has to be pushed off its equilibrium before its reaction keeps pace with the current; that extra push is the electrode’s overpotential.6

For the whole cell this comes down to two lines. On discharge the terminals give the open-circuit voltage minus a polarization η; on charge the charger must supply the open-circuit voltage plus it:7

Vdis=VOC−η,Vch=VOC+η

The polarization itself has three parts: activation, from the charge-transfer reactions at the interfaces; ohmic, from the resistance of the cell’s components and contacts; and concentration, from the limits of mass transport.1

Polarization or overpotential? You will meet both. Here polarization belongs to the whole cell, η=EOCV−ET, the open-circuit voltage minus the voltage under load; overpotential, η=E−Eeq, belongs to one electrode and has its own sign. We keep them apart.1,6

Why do practical values fall short of the theoretical ones a material promises? One textbook groups the causes into three: charge transfer and mass transport, in the bulk and at the interfaces; the structural and phase changes of the electrode as lithium leaves and enters it; and the way electrical and chemical effects are coupled in time.8 This chapter deals with the first; chapter 7 and chapter 16 with the other two.

Current costs voltage: a working cell gives less than its open-circuit voltage, and charging it takes more.

Figure 5.3 Voltage against state of charge for a discharge from full and a charge from empty at the chosen current, with the open-circuit curve dashed between them, the upper and lower cut-off voltages, red areas between each branch and the open-circuit curve for the energy turned into heat, and a panel that enlarges the polarization at a chosen state of charge, split into its ohmic, activation and concentration parts.
dischargechargeheat, ∫η dqohmicactivationconcentration
The voltage curve, computed, with the polarization drawn on the plot. The shape follows your cell: flat for LiFePO4, sloping for the oxides. Try this: put the probe at 90 % and slide the current from C/10 to 5C; watch which part in the right-hand panel grows fastest and where the discharge now stops.
SourceV=VOC∓η after Goodenough and Park 2013, eqs. 1.1 and 1.2. The three parts are computed from the cited forms: ohmic IRb (Winter and Brodd 2004, eq. 15); activation from the Butler-Volmer equation with α=0.5 (Bard, Faulkner and White 2022, eq. 3.4.11); concentration (RT/F)ln(C0/C) (Winter and Brodd eq. 16, with the Faraday constant restored). The open-circuit shapes, Rb giving a 50 mV ohmic drop at 1C (50 mΩ for a 1 Ah cell, 25 mΩ for 2 Ah), an exchange current equivalent to C/3.3 and a limiting current of 8C that falls to about 1.3C as the electrode being emptied or filled runs out are illustrative, not measured. The discharge starts from a full cell and the charge from an empty one. The cut-offs are illustrative; 4.2 V is the hold voltage of the protocols in figure 5.1. Heat is the area between each branch and the open-circuit curve. Schematic, not experimental data.

Now let us record the voltage while the current runs, all the way from full to empty: that record is the discharge curve. Its typical shape carries all three polarizations, and it tells us the capacity, the effect of rate and temperature, and the health of the cell.1 The upper end of the curve has its own story. A LiCoO2 electrode, for instance, gives up only about half its lithium.9,10 In practice that is about 135 to 140 mAh/g,11,12 because its structure degrades at high states of charge.10 Consumer cells have been pushing the upper cut-off voltage higher to win more capacity.11

Push harder and the curve pays twice. The polarization grows with the current, so the two branches move apart; and the discharge reaches its cut-off sooner, before all the charge is out. That missing capacity is not lost for good: at high current the transfer of ions becomes diffusion-limited, and what the cell could not deliver comes back at a gentler rate.7

Worked example. Let us price the ohmic part on its own. A 2 Ah cell with VOC = 3.8 V and an internal resistance of 50 mΩ, run at 1C, that is 2 A, counting only the ohmic part. η=IRb=2 A×0.050 Ω=0.10 V discharge: 3.8−0.1=3.7 V; charge: 3.8+0.1=3.9 V EoutEin=3.7×2 Ah3.9×2 Ah=0.95; the other 5 % leaves as heat Double the current and the ohmic part alone doubles: 0.2 V each way, and the share returned falls to 90 %.

The gap between the charge and discharge branches is energy paid as heat on every cycle, and it widens with current.

Figure 5.4 Voltage of a discharging cell before and after the current is switched off, on a logarithmic time axis from ten to the minus seven to one hundred seconds: an instant ohmic step, an activation part that relaxes between ten to the minus four and ten to the minus two seconds, and a concentration part that relaxes over seconds.
ohmicactivationconcentration
Switch the current off and the three polarizations leave on three different clocks. Try this: press Play and watch the time cursor; the curve stays put for a long while after the first jump and then moves twice more.
SourceTime ranges from Winter and Brodd 2004, section 1.4: ohmic polarization appears and vanishes within about 10−6 s, activation in 10−4 to 10−2 s, concentration in 10−2 s and longer. The sizes of the three parts come from the cell model of figure 5.3 at half charge; the ohmic part is drawn as an instant step, and the other time constants (10−3 and 1 s) are illustrative values inside those ranges, and the recovery is drawn as simple exponentials. Schematic, not experimental data.

If the three parts all add to one gap, how can we tell them apart? By their clocks. Ohmic polarization appears and disappears almost instantly, within about a microsecond. Activation polarization builds up and decays in about 10−4 to 10−2 s. Concentration polarization is slow: 10−2 s or longer.1 So let us switch the current off and time the recovery: the step is ohmic, the fast relaxation activation, the slow drift concentration. Interrupting the current in this way is how the polarizations are separated in practice.1

The middle clock is set by the charge-transfer reaction itself. A reaction that runs fast in both directions at equilibrium, one with a large exchange current, needs only a small overpotential to carry a given current, and battery reactions with exchange current densities around 10−2 A/cm2 are favoured.1,6 We meet the same clocks again in Lab 4, read with an alternating current.

Ohmic, activation, concentration: instant, milliseconds, tens of milliseconds and longer.

Figure 5.5 Heat released per ampere-hour of capacity against time through one discharge and one charge: an irreversible part that is always released and grows faster than the current, a reversible entropic part whose sign flips between discharge and charge, and their total.
irreversible, I·ηreversible, −I·T·dE/dTtotal
Where the heat comes from, through one cycle of your cell. Try this: flip the sign of dE/dT and watch only the green line change; then compare C/10 with 5C.
SourceHeat generation after the Bernardi form used by Jindal, Katiyar and Bhattacharya 2022, eq. 1: Q=I(U−V)−ITdU/dT, with I positive on discharge; the first term is irreversible, the second reversible (entropic). Signs and the Ni-Cd and lead-acid examples after Winter and Brodd 2004, section 1.3. The polarization comes from the cell model of figure 5.3; the size of dE/dT, ±0.2 mV/K, is illustrative. Heat per ampere-hour of low-rate capacity, at 25 °C. Schematic, not experimental data.

One cost of current is still to be followed: the energy lost on every cycle. The lost energy becomes heat, of two kinds. An irreversible part, often loosely called Joule heat, comes from the whole polarization: it is released whichever way the current flows. A reversible part comes from the entropy change of the reaction, and its sign follows the temperature coefficient of the voltage, dE/dT. If it is positive the cell heats on charge and cools on discharge, as nickel-cadmium does; lead-acid has a negative coefficient and does the opposite.1 Bernardi’s energy balance writes the two side by side:13

Q˙=I(U−V)−ITdUdT

The heat is released inside the cell, at the electrode surfaces, so a cell meant for high currents has to be built to shed it.1 Because the entropic term changes sign with the current, the heat it takes in on one half of a cycle it gives back on the other; the irreversible term only ever adds up.13

You can now explain how a cycler charges and discharges a cell and what a C-rate is, why a working cell gives less than its open-circuit voltage and why charging takes more, what the three polarizations are and how fast each relaxes, how a battery’s curve differs from a capacitor’s, why some curves are flat and others slope, why a flat curve tilts at high rate and can remember its past, why fast discharges leave material unused, how slopes of the curve reveal ageing, why 99.9 % per cycle is not enough, and where the lost energy goes.

So far we have watched the cell from outside, through its voltage. To see why the curve has the shape it has, let us follow our ion inside the cell.

Go deeper: the exchange current, the Butler-Volmer equation and the Tafel line

For one electrode, the current and the overpotential η=E−Eeq are linked by the Butler-Volmer equation. Written with reduction current positive, it reads6

i=i0[exp(−αfη)−exp((1−α)fη)],f=FRT

with α the transfer coefficient (between 0.3 and 0.7 in most cases) and i0 the exchange current. It holds while the current stays below about 10 % of the limiting current, so that mass transport does not interfere.6 Near equilibrium the exponentials can be expanded and the current is simply proportional to η: the electrode behaves like a resistor, the charge-transfer resistance Rct=RT/(Fi0) for a one-electron reaction. A large exchange current means a small Rct.6

Far from equilibrium one exponential wins and η grows with the logarithm of the current, the Tafel relation η=a+blogi, which holds once |η| exceeds about 118 mV at 25 °C; the same empirical form is quoted for batteries.1,6 In the battery literature it is written with oxidation current positive (the printed equation lost a bracket and a sign); the physics is the same with the roles of α and 1−α swapped.1,6

One caution about the heat equation. It takes the open-circuit voltage U from a slow measurement and dU/dT from small temperature steps, and in one set of measurements it estimated heat reasonably during continuous discharge but overestimated it during pulsed discharge, by up to 26 % for a LiFePO4 cell and 49 % for an NMC cell.13

The mathematics of this chapter

Q=∫Idt=It
I1C=Qnom1 hcharge passed at constant current, and the 1C current (figure 5.1)

Vdis=VOC−η(q,Idis)
Vch=VOC+η(q,Ich)the terminal voltage on discharge and on charge; η depends on the state of charge q and on the current (figures 5.2, 5.3)

η=EOCV−ET=ηohm+ηact+ηconcthe cell polarization, the open-circuit voltage minus the voltage under load, in three parts

ηohm=IRb
V=VOC∓IRbohmic polarization, with Rb the internal resistance

i=i0[e−αfη−e(1−α)fη],f=FRT
α=12:  η=−2RTFarsinhi2i0Butler-Volmer for one electrode, reduction current positive, and its inverse for α=0.5; its magnitude is the activation part used in figure 5.3

Rct=RTFi0
|η|=RTαFlnii0the charge-transfer resistance near equilibrium, and the Tafel limit (reduction branch; 1 − α for oxidation) for |η| above about 118 mV at 25 °C

ηconc=RTnFlnC0Cconcentration polarization when the concentration at the electrode falls from C0 to C (Winter and Brodd print it without F; the units need it)

U=Ueq
U=Uref+qCspthe ideal battery material and the ideal capacitor material (Olson and co-workers; figure 6.1)

v(t)=RHI0+RiI0(1−e−t/τi)+I0tC
vD(t)=v(t)−2u(t−t0)v(t−t0)Moya’s double-layer capacitor charged at I0 (his eq. 6), then discharged at −I0 from t0 (eq. 12, u the unit step); τi=RiCi, RESR=RH+Ri, Ci≈2C/7 (eq. 33)

F=C−P+2the Gibbs phase rule: two phases of two components at fixed temperature and pressure leave no freedom, so a plateau (figure 6.2)

Φ1,k=Φ1−Rc,kik
I=L∑kakik
Dk=1.184×10−18(1+y¯k)1.6 m2/sSafari and Delacourt’s resistive reactant (their eqs. 7, 8 and 12): each particle group k sees the matrix potential less its contact drop, the groups share the current, and diffusion slows as the particle fills (figure 6.3)

Sc=Rs2IDsF(1−ε)cTδc
Se=(δs+δc)2IDF(1−ε)cTδcDoyle, Fuller and Newman’s ratios of the times for diffusion in the solid and transport in the electrolyte to the time of discharge (eqs. 26 and 27; figure 11.6)

V=Upos−Uneg
dVdQ=dUposdQ−dUnegdQ
dQdV=(dVdQ)−1a cell’s voltage and differential voltage from its two electrodes, and the differential capacity (figure L1.8)

1−xpos=axneg+b
aa0=1−LAMNE1−LAMPE
1−b=1−LLI1−LAMPEthe charging line on the square plot and how ageing moves it, for a fresh cell with b0=0 (Olson and co-workers, eqs. 23, 48 and 49)

CE=100×QdisQch
retention after N cycles=CEN (idealized)Coulombic efficiency per cycle, and the idealized retention of figure 16.2

εE=100×∫Vdisdq∫Vchdqthe storage efficiency: the energy back over the energy in

Q˙=I(U−V)−ITdUdT
ΔS=nFdEdTheat generation, I positive on discharge: the irreversible term and the reversible, entropic term (figure 5.5); the entropy of reaction from the temperature coefficient of the voltage

References for this chapter

Numbers in the text refer to this list. All references of the course, with the chapters that use each one.

  1. 1 M. Winter, R. J. Brodd, "What Are Batteries, Fuel Cells, and Supercapacitors?", Chemical Reviews 104, 2004, 4245-4269. doi:10.1021/cr020730k
  2. 2 J. Z. Olson, C. M. López, E. J. F. Dickinson, "Differential Analysis of Galvanostatic Cycle Data from Li-Ion Batteries: Interpretative Insights and Graphical Heuristics", Chemistry of Materials 35, 2023, 1487–1513. doi:10.1021/acs.chemmater.2c01976
  3. 3 M. Safari, C. Delacourt, "Mathematical Modeling of Lithium Iron Phosphate Electrode: Galvanostatic Charge/Discharge and Path Dependence", Journal of The Electrochemical Society 158, 2011, A63–A73. doi:10.1149/1.3515902
  4. 4 V. Murray, D. S. Hall, J. R. Dahn, "A Guide to Full Coin Cell Making for Academic Researchers", Journal of The Electrochemical Society 166, 2019, A329-A333. doi:10.1149/2.1171902jes
  5. 5 G. G. Eshetu, H. Zhang, X. Judez, H. Adenusi, M. Armand, S. Passerini, E. Figgemeier, "Production of high-energy Li-ion batteries comprising silicon-containing anodes and insertion-type cathodes", Nature Communications 12, 2021, 5459. doi:10.1038/s41467-021-25334-8
  6. 6 A. J. Bard, L. R. Faulkner, H. S. White, "Electrochemical Methods: Fundamentals and Applications", Wiley, 3rd edition, 2022.
  7. 7 J. B. Goodenough, K.-S. Park, "The Li-Ion Rechargeable Battery: A Perspective", Journal of the American Chemical Society 135, 2013, 1167-1176. doi:10.1021/ja3091438
  8. 8 S. A. Arote, "Fundamentals and perspectives of lithium-ion batteries", chapter 1 of Lithium-ion and Lithium–Sulfur Batteries: Fundamentals to Performance, IOP Publishing, 2022, pp. 1-1 to 1-26. doi:10.1088/978-0-7503-4881-2ch1
  9. 9 J.-M. Tarascon, M. Armand, "Issues and challenges facing rechargeable lithium batteries", Nature 414, 2001, 359-367. doi:10.1038/35104644
  10. 10 J.-J. Marie, S. Gifford, "Developments in lithium-ion battery cathodes", Faraday Insights, Issue 18 (The Faraday Institution), 2023, September 2023. link
  11. 11 J. T. Frith, M. J. Lacey, U. Ulissi, "A non-academic perspective on the future of lithium-based batteries", Nature Communications 14, 2023, 420. doi:10.1038/s41467-023-35933-2
  12. 12 A. Manthiram, "A reflection on lithium-ion battery cathode chemistry", Nature Communications 11, 2020, 1550. doi:10.1038/s41467-020-15355-0
  13. 13 P. Jindal, R. Katiyar, J. Bhattacharya, "Evaluation of accuracy for Bernardi equation in estimating heat generation rate for continuous and pulse-discharge protocols in LFP and NMC based Li-ion batteries", Applied Thermal Engineering 201, 2022, 117794. doi:10.1016/j.applthermaleng.2021.117794
  14. 14 A. A. Moya, "A physically consistent electric circuit to interpret the galvanostatic charge-discharge curves of electric double layer capacitors", Journal of Power Sources 656, 2025, 238050. doi:10.1016/j.jpowsour.2025.238050
  15. 15 J. Landesfeind, J. Hattendorff, A. Ehrl, W. A. Wall, H. A. Gasteiger, "Tortuosity Determination of Battery Electrodes and Separators by Impedance Spectroscopy", Journal of The Electrochemical Society 163, 2016, A1373. doi:10.1149/2.1141607jes
  16. 16 M. Doyle, T. F. Fuller, J. Newman, "Modeling of Galvanostatic Charge and Discharge of the Lithium/Polymer/Insertion Cell", Journal of The Electrochemical Society 140, 1993, 1526–1533. doi:10.1149/1.2221597
  17. 17 M. M. Rahman, U. Nisar, A. Abouimrane, I. Belharouak, R. Amin, "Valuation of Anode Materials for High-Performance Lithium Batteries: From Graphite to Lithium Metal and Beyond", Electrochemical Energy Reviews 8, 2025, 14. doi:10.1007/s41918-025-00249-w
  18. 18 C. Villevieille, P. Novák, H. H. Girault, "Your battery may not be a battery", Nature Nanotechnology, 2026, Comment, published online 15 October 2026. doi:10.1038/s41565-026-02296-w

Cite this page

Using this page in teaching, a thesis or an article? Cite it as follows; the date is the last update of the course.

Shyma Sajeevan, A. (2026). Galvanostatic cycling and polarization: overpotentials, hysteresis and heat generation. How batteries work. https://asajeevan.github.io/batteries/polarization/

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