How batteries work Chapter 6 Voltage profiles

Chapter 6

The voltage profile: faradaic and capacitive storage, phase equilibria and many-particle electrodes

Why is the curve of LiFePO4 flat while that of LiCoO2 slopes?

Figure 6.1 Two views. First, two ideal limits side by side: an ideal battery material whose potential stays at one value while charge is stored, giving a single spike of charge per volt, and an ideal capacitor material whose potential rises in a straight line, giving a flat level of charge per volt. Second, a real supercapacitor charged and then discharged at constant current: an instant jump, a straight ramp, a drop of twice the jump when the current reverses, and a straight ramp down, with two zoomed panels showing the curve bending onto a straight asymptote just after each switch.
battery limitcapacitorstraight-line asymptote
Two ways to store charge, and the curves they draw. Try this: in the real supercapacitor, halve the current and check that the whole picture shrinks by half, the ramp included.
SourceIdeal battery and ideal capacitor materials (after Conway) and their potential-charge curves after Olson, López and Dickinson 2023, section 2.1 and Figures 2 and 3; schematic, no numbers. The supercapacitor is Moya’s capacitor C10 (nominal 1 F) with his fitted values in Table 2: RH = 0.31 Ω, Ri = 0.32 Ω, C = 0.94 F, τi = 0.095 s; the curve is his eq. 12 built from eq. 6, for a 10 s charge. Computed from the published circuit, not his measured points.

We have seen how current pulls the curve away from the open-circuit voltage. But why does a battery’s curve have the shape it has in the first place? Let us start from two extremes. An ideal battery material is one that supplies all its capacity through one uniform chemical process at one fixed potential. Its charge per volt, the differential capacity dq/dU, is a single spike, and the energy it stores is simply proportional to the charge.1

The opposite extreme is an ideal capacitor material: its differential capacity is constant, its specific capacitance Csp, so the potential rises in a straight line as charge goes in.1

battery:  U=Ueqcapacitor:  U=Uref+qCsp

“Ideal” here means a mathematical limit, not the best material for a job. And “capacitor material” covers more than capacitors: double layers, surface reactions, and insertion electrodes that take lithium as a single phase all behave this way to some degree.1 For any real material, a capacity only means something together with the voltage window it was measured in.1

A real supercapacitor

Real capacitors are not ideal either; let us see how one departs from the straight line. In one study, three commercial double-layer capacitors were charged and discharged at constant current and their curves read with a small circuit. It has a resistance RH in series with the capacitance C, plus an inner branch, a resistance Ri in parallel with a capacitance Ci, attributed to the diffusion of ions in the capacitor.2 For a current I0 switched on at t=0:2

v(t)=RHI0+RiI0(1−e−t/τi)+I0tC,τi=RiCi

So the voltage jumps by RHI0 the moment the current starts, then bends for a fraction of a second before settling to a straight line of slope I0/C. After about five time constants it runs along that straight asymptote, which starts from (RH+Ri)I0.2

Reverse the current and the voltage drops at once by 2RHI0, because the current has changed by twice I0. The bend that follows is twice as large as at the start, and the measurements showed that 2 : 1 ratio on all three capacitors (for the 1 F one at 10 mA, 0.34 mV on charge and 0.69 mV on discharge; the fitted circuit gives larger bends, so only the ratio is compared here). The inner capacitance came out close to two sevenths of the main one.2

Worked example. The 1 F capacitor of that study, with its fitted values, charged at 50 mA for 10 s. jump: RHI0=0.31 Ω×0.050 A=15.5 mV ramp: I0/C=0.050/0.94=53 mV/s after 10 s: (RH+Ri)I0+I0t0/C=0.032+0.532= 0.56 V on reversal: 2RHI0=31 mV down, then a ramp of −53 mV/s An ideal battery material at the same current would not move at all: it stores its charge in a reaction at one fixed potential.

A battery stores charge at a fixed potential, a capacitor in proportion to its voltage; real electrodes sit between the two.

Which behaviour a material shows can depend on its size. LiCoO2 behaves pseudocapacitively as a nanoscale material and as a conventional battery material in particles of micrometres and larger.3

Figure 6.2 Left: twelve particles of a LiFePO4 electrode taking up lithium, each drawn turning from lithium-poor to lithium-rich in turn, and below them the measured open-circuit voltage against lithium content, flat at 3.43 volts between about 0.15 and 0.85 and sloping at both ends. Right: twelve particles of a single-phase material filling evenly, with a schematic sloping voltage. Labels give the Gibbs phase rule count for each.
lithium in a particlewhere the electrode is on its curve
Plateau or slope: the Gibbs phase rule on a measured curve. Try this: press Step until the dot is on the plateau, then count the dark blue particles; keep stepping and watch the count rise while the voltage stays put.
SourcePhase-rule argument after Winter and Brodd 2004, section 2.4 and Figure 16. The LiFePO4 curve is the open-circuit fit of Safari and Delacourt 2011, eq. 10, made to the average of the charge and discharge curves at C/100; flat at 3.432 V for lithium contents of about 0.15 to 0.85 (our evaluation of their eq. 10). The 20 mV hysteresis at C/1000 is Dreyer and co-workers’ result as cited by Safari and Delacourt. The particles are schematic: each is drawn changing over as a whole, because how the two phases sit inside one particle is still studied, and Safari and Delacourt note that pictures with a lithium-rich shell round a shrinking core disagree with experiments that show the lithium moving along the phase boundary. The single-phase curve is schematic and carries no numbers.

Real battery electrodes sit between the two extremes, and their curves differ: some are flat, others slope. Why? The answer is the Gibbs phase rule, which counts the freedoms F left to a system of C components in P phases:4

F=C−P+2

When the reaction turns one solid phase into another, two phases coexist. With temperature and pressure fixed nothing is left free, so the voltage stays constant: a plateau. When lithium dissolves into a single phase, the composition stays free and the voltage slopes with it.4

A simple example makes the difference concrete. An electrode with the overall composition Li0.75M can be one uniform phase of that composition, or half Li0.5M and half LiM. In the second case, adding lithium only changes the proportions of two fixed phases; materials that split like this are often called staging materials.1

LiFePO4, measured

LiFePO4 is the classic two-phase case. Most of its range is a mixture of a lithium-poor and a lithium-rich phase.5 Its open-circuit curve, measured as the average of a charge and a discharge at C/100 and fitted, is flat at 3.43 V vs Li over the middle of the range and slopes at both ends, where the material takes lithium as a partial solid solution.5

Even this plateau is not perfectly single-valued. A gap of about 20 mV remains between the charge and discharge voltages even at C/1000. That is a true, thermodynamic hysteresis, traced to the many particles of the electrode.5

Other materials draw other shapes. The spinel LiMn2O4 shows steps near 4 V and a drop to 3 V.6 Which shape is better depends on the use: electronics prefer a flat, steady voltage, while a sloping curve makes it easy to read how much charge is left.4

Two phases pin the voltage to a plateau; one phase lets it slope.

Figure 6.3 Computed voltage of a lithium iron phosphate electrode against lithium, against the fraction of its theoretical capacity, for a charge or a discharge at the chosen rate with a slow C/25 run for comparison and the open-circuit curve dashed. Beside it four bars show the lithium content of four groups of particles that differ only in their contact resistance; the best-connected group fills first and the worst last. A third view compares two 1C charges from the same half-charged state reached by different routes.
Rate
dischargechargeC/25, and the open-circuit curve dashedlithium in each group
One electrode, many particles: why the plateau tilts with rate, and why the electrode remembers its past. Try this: run a 1C discharge and watch group 4; then press “Same 50 %, two histories” and compare where the two charges end.
SourceComputed on this page with the model and the case-1 parameters of Safari and Delacourt 2011: eqs. 2 to 12 and Table I (four particle groups of radius 36.5 nm, volume shares 25, 62, 8 and 5 %, contact resistances 0.08, 2.88, 7.81 and 36.99 Ω m2, electrode 70 µm thick, active volume fraction 0.43 from their 0.428 to 0.435, lithium counter electrode with 1.90 mA/cm2 exchange current). 1C here is the current that would pass the theoretical capacity in one hour, 18.4 A/m2. Rates stop at 1C because their rate-constant fit (eq. 11) covers C/25 to 1C. Our integration is simpler than theirs (explicit steps on the same equations), and the curves are not their published ones. At 50 % after C/25 it gives a spread of 0.09 in lithium content among the groups, against their 0.097 and 0.089.

That hysteresis is our first hint that the many particles matter, so let us now take them seriously. Real electrodes are not one particle but billions, and they do not all react together. LiFePO4 conducts electrons very poorly, about 10−9 S/cm on its own, and is made usable by coating its particles with carbon and making them small.5 Each particle then gets its electrons through its contact with the conductive carbon, and some contacts are better than others.5

A simple model builds that in, a resistive reactant: four groups of identical particles, differing only in their contact resistance. Its parameters were fitted to measured cells, and it does not try to describe the two phases inside a particle at all.5 With one set of parameters it reproduced those cells’ charge and discharge curves from C/25 to 1C, the first charge simulation of LiFePO4 its authors knew of.5

Why the plateau tilts

The measured curves tilt more steeply in the middle as the rate rises. Since the open-circuit curve there is flat, that means the cell’s resistance grows as the charge or discharge goes on.5 The model explains why: the particles with the lowest resistance react first, the worse-connected ones later, and the faster the current, the stronger the effect.5

The idea was tested directly. In LiFePO4 coin cells, adding one or two extra steel discs squeezed the electrode harder. The curves sloped less and the cells delivered more, and the fitted contact resistance of the worst group fell by up to 44 %.5

LiFePO4 also gives more on charge than on discharge at the same rate, by 8 % at 3C and 11.5 % at 5C in those cells. The model captures this with a solid diffusion coefficient that is lower when the particle is full of lithium.5

An electrode with a memory

The groups have one more surprise for us. Bring the electrode to half charge in two ways, by charging from empty or discharging from full, rest it for two hours, then charge it at 1C. The two charges differ: 3.96 % in delivered capacity in those cells. This is path dependence, and it fades at low rates.5

It has two ingredients. The particle groups end up holding different amounts of lithium, and nothing evens them out. Lithium moves between particles only where the open-circuit curve slopes, because only there does a fuller particle sit at a different voltage from an emptier one. On the plateau it barely moves, even after two hours of rest.5 Near 5 % charge, on the sloping end, the same experiment gave about a fifth of the effect.5

A flat curve hides a crowd of particles reacting in turn; how well each is wired decides the rate, and the past decides the present.

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 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
  2. 2 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
  3. 3 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
  4. 4 M. Winter, R. J. Brodd, "What Are Batteries, Fuel Cells, and Supercapacitors?", Chemical Reviews 104, 2004, 4245-4269. doi:10.1021/cr020730k
  5. 5 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
  6. 6 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

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). The voltage profile: faradaic and capacitive storage, phase equilibria and many-particle electrodes. How batteries work. https://asajeevan.github.io/batteries/voltage-profiles/

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