How batteries work Chapter 11 Composite electrodes

Chapter 11

Composite electrodes: percolation, porosity, tortuosity and transport limitations

Why is an electrode a porous coating of particles, carbon and binder?

Where we are. In chapter 9 we carried our ion across the electrolyte. On the far side it has to get into a particle of the positive electrode, one of the two hosts of chapter 7. In chapter 2 we saw that an electrode is not a solid block of that host but a coating of particles, carbon, binder and pores. In this module we ask what each of those ingredients is for, and what happens when one of them fails.

Figure 11.1 A cross-section of a composite electrode between an aluminium foil and the separator: twenty active particles linked to each other and to the foil by a carbon network, with the pores between them filled with electrolyte. Breaking carbon links or sealing pores leaves particles that electrons or ions can no longer reach; those turn pale and dashed, and a counter gives how many particles can still work.
active particle that worksdead particlecarbon networkelectrolyte in the poresaluminium foilseparator
Two roads to every particle: electrons through the carbon, ions through the pores. Try this: break carbon contacts several times and look for a particle that goes pale while its own links look intact: what counts is an unbroken path all the way to the foil.
SourceComposite of active particles, conductive diluent and binder on a current collector, about 30 % porous, after Winter and Brodd 2004, section 1.6. Which particles can work is computed from the network shown: a particle counts if the carbon links connect it to the foil and its pore is open. The geometry, the number of particles and the links are schematic, not to scale.

Let us look at the coating our ion is about to enter. Most battery electrodes are porous composites: particles of the active material, a conductive diluent such as carbon or a metal powder, and a polymer binder, coated on a current collector. About 30 % of the coating is pore space, so the surface where the reaction can happen is far larger than the electrode’s geometric area.1 A large surface lowers the current density at each point of it, and with it the polarization.1

Each ingredient, then, keeps one road open. The carbon carries electrons from the foil to every particle; the electrolyte in the pores carries ions from the separator to every particle; the binder holds the coating together and onto the foil. A particle that one of the two roads cannot reach stores nothing and only adds mass.

A picture to keep: a town where every house needs two roads, one for each of two deliveries that must arrive together, and is only lived in if both reach it. The picture breaks because here the roads also have tolls: the farther a delivery travels through the coating, the more voltage it costs.

A particle that the electrons or the ions cannot reach contributes nothing.

Figure 11.2 One electrode from current collector to separator, shaded by where the reaction happens, with a plot of the local reaction rate across its thickness. When the solid conducts much better than the electrolyte, the reaction crowds next to the separator; when the electrolyte conducts better, next to the collector; a thicker electrode makes the crowding stronger.
Where does the current go inside an electrode? Try this: set the two conductivities equal and the thickness to 0.2; then thicken the electrode and watch the middle go quiet.
SourceThe dependence on the two conductivities, exchange current, thickness and porosity after Winter and Brodd 2004, section 1.6 and Figure 9B. The profile is computed from Ohm’s law in the solid and in the electrolyte with the linearized kinetics of chapter 5 (Rct, Bard, Faulkner and White 2022, eq. 3.4.13): j(ξ) ∝ cosh(νξ)/κ + cosh(ν(1 − ξ))/σ, with ξ the distance from the collector as a fraction of the thickness and ν the thickness in units of the reaction depth; the cosh form and the depth scale are those analysed by Pathak and Bazant 2026. Schematic: no electrode is modelled.

The tolls decide where in the coating the work gets done. Inside a porous electrode the reaction is not spread evenly. Where it happens depends on the conductivities of the solid matrix and of the electrolyte, the exchange current, diffusion, the thickness of the electrode, its porosity, pore size and tortuosity; and a nonuniform current distribution wastes active material.1

The reaction concentrates where the electrons and the ions together have the shortest, cheapest way to meet. In the simplest theory the profile is a hyperbolic cosine with a characteristic depth.2 This reaction-limited picture is most realistic for thin electrodes, of order 10 μm, and needs corrections beyond about 100 μm.2

Part of the toll is paid in the pores themselves. The electrolyte inside the pores conducts worse than in bulk, because the ions have less room and a winding path. The usual correction is κeff = κεα, with ε the porosity and α = 1.5 (1.53 for LiFePO4 electrodes).3 Ionic transport is far slower than electronic transport, which is also why cells use thin electrodes of large area.4

Worked example. Let us put a number on it: how well does the electrolyte conduct inside a 30 % porous electrode? κeff/κ = 0.301.5 = 0.164 for a liquid of 10−2 S/cm: κeff = 1.6 × 10−3 S/cm, about a sixth Every micrometre of coating the ions must cross costs six times the voltage it would in the open electrolyte; another reason the reaction in figure 11.2 crowds toward the separator.1,3
Figure 11.3 Two particles of a poorly conducting active material. The left one touches carbon at one point and the reacted zone spreads from that point. The right one has either no carbon, and does not react, or a carbon coating all round, and reacts everywhere at once.
reacted materialactive material, not yet reactedcarbon
A poor conductor needs carbon. Try this: switch the right-hand particle to a carbon coating and compare how long each one takes to light up fully.
SourceUtilization beginning at the contact with the carbon additive and spreading from it, after Winter and Brodd 2004, section 2.4 and Figure 15E (MnO2); carbon coating of LiFePO4 after Goodenough and Park 2013. The speed of the spreading is schematic.

So far we have assumed that once electrons reach a particle, they reach all of it. For many active materials that is not so. An active material that conducts electrons poorly works only where it touches the conductive additive, and the reacted zone spreads outward from that contact, as in MnO2.1 LiFePO4 is the modern case: its poor electronic conductivity was overcome by coating the particles with carbon and making them small.4

The coating, introduced by Armand and co-workers, is made by decomposing a polymer precursor on the particles, which leaves a uniform layer of carbon.5 With it, a material that was once a dead end is now produced in thousands of tonnes a year.5,6

Figure 11.4 The same mass of active material as a few large particles and as many small coated particles. A bar shows that the time for an ion to cross a particle falls with the square of its size; a second, schematic bar shows that the small coated powder takes more room.
The nanoparticle trade-off: faster particles, emptier litres. Try this: shrink the particles to a tenth and read both bars.
SourceTime to cross a particle from the diffusion length of Bard, Faulkner and White 2022, eq. 4.4.3, t ≈ L2/2D, computed relative to the large particles. The loss of tap density and the difficulty of coating small particles uniformly after Goodenough and Park 2013; the size of the volume penalty is not given by the sources and the lower bar is schematic.

The other half of that remedy, making the particles small, has its own price. Small particles bring the ions closer to every atom: the time to cross a particle falls with the square of its size (chapter 7). But carbon-coated nanoparticles have a lower tap density, so the electrode holds less energy per litre, and coating them evenly gets harder as they shrink.4,7 Highly porous aerogel electrodes pay the same price, at a tap density of about 0.2 g/cm3.8

Figure 11.5 Left: a square slice through an electrode coating, particles in dark grey with the pores between them filled with blue electrolyte, drawn at the chosen porosity, with a blue line tracing the shortest route through the pores from the separator side to the current collector side beside a dashed straight line of the same thickness. Right: the share of the electrolyte's conductivity left in the pores against porosity, for the usual estimate, porosity to the power 1.5, dashed, and for fits to measurements on graphite and on two lithium iron phosphate electrodes, all of which lie below the estimate.
particlespores, filled with electrolyteshortest routeestimate ε1.5, dashedgraphiteLiFePO4, 5 % carbonLiFePO4, 15 % carbon
Porosity and tortuosity: how much room the ions have, and how far round they must go. Try this: at porosity 0.30, switch from round particles to flat plates and compare the blue route; then slide the porosity down and watch every curve on the right fall.
SourcePorosity, the MacMullin number NM = κ/κeff = τ/ε, the Bruggeman estimate τ = ε−0.5 (κeff = κε1.5) and the measured fits NM = 4.7ε−1.1 (graphite), 3.1ε−1.6 (LiFePO4 90/5/5 active/binder/carbon) and 3.0ε−1.2 (LiFePO4 70/15/15), drawn as 1/NM over the porosities measured, after Landesfeind et al. 2016, eqs. 1, 5, 6 and 8, Table I and Figures 19 and 20; the shaded band, porosity 0.30 to 0.35, is the range of commercial electrodes named there. The slice is a drawing: particles on a lightly jittered lattice, spaced so that their share of the slice’s area is 1 − ε. The blue route is the shortest path through its pores, found on a 1 px grid, and its length over the thickness is a geometric tortuosity of this 2-D sketch, not the effective tortuosity of a real electrode.

So far we have followed the electrons to each particle. The ions must reach the particles too, and for that we must look at the space between them. An electrode is not a solid block. It is a composite: particles of active material, often wired together by a conductive additive such as carbon,9 held by a binder, with about 30 % of its volume left as pores that the electrolyte fills.1 Every ion that reaches a particle has travelled through those pores, and two numbers describe them.

Porosity: how much room

The porosity ε is the fraction of the electrode’s volume that is pore. It is well defined and easy to find.10 We weigh a disc of the coating and measure its thickness. From the masses and densities of the solid ingredients we get the volume the solids fill, and whatever is left of the coating’s volume is pore.10

Freshly coated electrodes are then pressed, or calendered, to lower their porosity. That packs more energy into each litre and improves the electronic contact through the coating and to the current collector; typical lithium-ion electrodes end at roughly 30 %.10

Tortuosity: how far round

The pores do not run straight across. An ion has to wind round the particles, so its path is longer than the coating is thick, and the channels narrow and widen on the way. The tortuosity τ collects these effects. In its simplest, geometric sense it is the length of the path divided by the straight distance. The tortuosity that matters for transport, the effective tortuosity, also includes the changing cross-section and the rough surfaces; it is found by measurement, or by simulation on a three-dimensional image of the electrode.10

Together the two numbers set how well the electrolyte conducts inside the coating, κeff=κε/τ. The ratio NM=κ/κeff=τ/ε, the MacMullin number, says how many times worse the filled pores conduct than the free liquid.10 Battery models often take the tortuosity from the porosity alone, with the Bruggeman relation τ=ε−0.5, which gives κeff=κε1.5.10 The modelled cell in the next figure uses exactly this correction, for diffusion as well as for conduction.11

Measuring it, and why it matters

How is τ measured? One clean way builds a cell from two identical electrodes and fills it with a salt that cannot react with them or enter the particles, so that no charge crosses into the particles and the ions can only charge the pore walls. The impedance spectrum then follows a transmission line, a ladder of resistances along the pores and capacitances at their walls that we meet again in Lab 4. Fitting it gives the ionic resistance Rion of the pores, and for the pair of electrodes τ=RionAκε/(2d), with A the area and d the thickness of one electrode.10

Measured this way, real electrodes are worse than the estimate. In one systematic study, every electrode tested had a MacMullin number larger than the Bruggeman relation predicts, by about 1.5 to 3 times in the authors’ summary and by more in some of their fits (figure 11.5). The measured curves fall with porosity at roughly the same power as the estimate, exponents of 1.1 to 1.6 against 1.5, but start three to five times lower.10 A graphite electrode of porosity 0.43 gave τ=4.3±0.6, about three times the estimate of 1.5 (the fitted graphite curve in figure 11.5, drawn through electrodes of many porosities, runs a little lower than this single point). Graphite’s plate-like particles lie flat in the coating and hinder the way across: at similar porosities, electrodes of round NMC particles had MacMullin numbers of 10 to 11, against 18 to 19 for graphite. Particle size and the amount of conductive carbon matter too.10

Worked example. Let us find the porosity of a coating and what it leaves of the electrolyte’s conductivity. Take an illustrative NMC coating 50.0 µm thick with 15.6 mg of solids per cm2: 96 % NMC (density 4.7 g/cm3), 2 % carbon (2.2 g/cm3),10 and 2 % binder, for which we assume 1.8 g/cm3. solids per cm2: 15.6 mg×(0.964.7+0.022.2+0.021.8) cm3/g=3.50×10−3 cm3 coating per cm2: 50.0 μm×1 cm2=5.00×10−3 cm3 porosity: ε=1−3.505.00= 0.30 estimate: κeff/κ=ε1.5=0.301.5= 0.16 measured graphite: κeff/κ=ε/τ=0.43/4.3= 0.10, against 0.431.5=0.28 Even by the estimate the coating keeps only a sixth of the liquid’s conductivity; the measured graphite electrode keeps a tenth, not the 28 % the estimate gives it.
measuring the porosity and tortuosity of a coating

Porosity, from a balance and a thickness gauge.

  1. Punch discs of the electrode, 11 mm across in one study.10 Punch discs of the bare current-collector foil too: their mass and thickness are subtracted, because we want the coating alone.
  2. Weigh each disc on a microbalance (1 µg resolution in that study) and measure its thickness with a precision gauge. Thickness was the weaker measurement there: about ±2 µm, that is 4 to 10 % of a coating 20 to 50 µm thick, against ±0.01 mg/cm2 for the mass.10
  3. From the composition and the bulk density of each ingredient, compute the volume of the solids and the porosity, ε=1−mAd∑iwiρi, as in the worked example above.10

Tortuosity, from an impedance spectrum.

  1. Build a symmetric cell from two electrodes of closely matched loading and porosity, with a separator between them, in a T-cell or a pouch cell.10
  2. Fill it with a salt whose ions cannot react with the electrodes or enter them, such as tetrabutylammonium perchlorate in EC:DMC, so that no charge crosses into the particles. Measure the conductivity κ of this electrolyte separately at the same temperature.10
  3. Record the impedance around open circuit, for example from 200 kHz to 0.5 Hz with a 10 mV perturbation, at 25 °C.10
  4. Fit a transmission-line model with constant-phase elements, or extrapolate the low-frequency branch and the high-frequency intercept to the real axis: the gap between them is Rion/3.10
  5. Compute τ=RionAκε/(2d); the 2 is there because the spectrum holds both electrodes.10
Worked numbers: two graphite electrodes, each 63.2 µm thick with porosity 0.41 and an effective area of 2.37 cm2, in an electrolyte of 1.74 mS/cm, gave Rion=31.0 Ω for the pair.10 So τ=31.0×2.37×0.00174×0.41/(2×0.00632)≈4.1, with lengths in cm and κ in S/cm, close to the 4.3 ± 0.6 the study reports for graphite electrodes of porosity about 0.43.

This is why the two numbers matter. At high rates the performance of a cell is largely set by the ionic resistance of its electrodes and separator, filled with electrolyte, and porosity and tortuosity are the key to that resistance.10 Pressing a coating therefore cuts both ways: more material in each litre, but less room for the ions and a longer way round. The next figure shows what happens when the ions run short.

Porosity says how much room the ions have, tortuosity how far round they must go; together they divide the electrolyte’s conductivity inside the coating.

Figure 11.6 A cross-section of a lithium polymer cell: lithium foil, a polymer separator and a composite positive electrode of particles in electrolyte, drawn to scale in thickness (100 micrometres at porosity 0.30, 175 at 0.60, with the same amount of particles), then a current collector. During discharge the particles nearest the separator fill first and the reaction moves deeper. Below, the salt concentration in the electrolyte rises near the foil and falls towards the back of the electrode. A bar shows how much of the positive material has been used, with a mark where the cut-off voltage is reached for the chosen current and porosity.
lithium in a particlereacting fastest nowsalt concentrationlithium ionLi foilseparatoraluminium collector
Inside a porous electrode: why fast discharges stop early. Try this: switch from 10 to 20 A/m2 and watch the blue fill stay near the separator while the salt at the back runs out.
SourceAfter Doyle, Fuller and Newman 1993, a Li | PEO-LiCF3SO3 | TiS2 cell at 100 °C: the cut-off at about 30 % of the material at 20 A/m2 (Fig. 2), 84 % at 10 A/m2 with porosity 0.30 and 97 % with 0.60 (text), and the cut-off near 1.7 V are theirs. The particle filling and the concentration profile are drawn to follow the trends of their Figures 3, 5 and 6 (10 A/m2, porosity 0.30); they are schematic, not computed (the sketch does not conserve the total salt), and for the other two settings only the final percentages are from the paper. The electrode is drawn to scale in thickness, 1.9 px per µm: 100 µm at porosity 0.30 (their Table II) and 175 µm at 0.60, our arithmetic for the same theoretical capacity, 100 µm × 0.70/0.40, taking the solid as all active material, as their solid-diffusion number does. The particles are drawn far larger than the model’s 1 µm radius, but their share of the area is 1 − ε; separator and foil are not to scale.

Now let us run a discharge through such a coating and look across its whole thickness. One model treats a composite of TiS2 particles, themselves good electronic conductors, mixed with a polymer electrolyte, as two continua, the particles and the electrolyte, laid over each other. The modelled cell had a lithium foil, the same polymer electrolyte as separator and this TiS2 positive electrode.11

The computed discharges show the main lesson at once. The faster the current, the less of the material is used before the voltage collapses: at 20 A/m2 the voltage drops sharply when only about 30 % of the positive material has reacted.11

Where the reaction happens

Why does the material go unused? Let us follow where the reaction happens. At first it crowds near the separator. The particles carry electrons about 105 times better than the polymer carries ions, so the short ionic path wins. As the particles at the front fill up, the reaction moves towards the middle of the electrode.11

Meanwhile the salt piles up near the lithium foil and is drawn down at the back of the electrode, where it cannot be replaced fast enough. There the conductivity falls and the material is poorly used.11

Two dimensionless numbers decide which transport matters, each a ratio of a transport time to the discharge time. For diffusion inside the modelled particles, of 1 µm radius, it was only 0.0001, so solid diffusion hardly mattered: the particles would have had to be as large as the electrode is thick. For the electrolyte it was 0.15 at 10 A/m2, and at high currents it is the electrolyte that ends the discharge early.11

Worked example. Let us check the solid-diffusion number with the model’s parameters at 10 A/m2: particle radius 1.0 µm, solid diffusion coefficient 5.0 × 10−13 m2/s, porosity 0.3, maximum lithium concentration 29 000 mol/m3, electrode 100 µm thick. the number: Sc=Rs2IDsF(1−ε)cTδc top: Rs2I=(1.0×10−6)2×10=1.0×10−11 bottom: 5.0×10−13×96485×0.7×29000×1.0×10−4 bottom: =9.8×10−8 Sc=1.0×10−119.8×10−8≈ 1.0 × 10−4, far below 1 The units cancel, so the number is a pure ratio: diffusion inside a particle takes about a ten-thousandth of the time the discharge takes.

The design lesson follows. Keeping the theoretical capacity fixed, a more open electrode (porosity 0.60 instead of 0.30, and so thicker; in figure 11.6, 175 µm instead of 100 µm) raised the material used at 10 A/m2 from 84 % to 97 %; 0.60 was the optimum found. In general, thinner electrodes use their material better when the electrolyte is the bottleneck, at the cost of harder processing and the extra weight of more, thinner cells.11 A stronger starting salt solution was also suggested, because the 1 M presumably chosen for its conductivity is far weaker by the time it reaches the depths of the electrode.11

At high current the back of a thick electrode starves of ions, so the cell stops with material unused.

Figure 11.7 The eight steps of making a lithium-ion cell: mix a slurry, coat it on foil, dry it, calender it, cut and stack or wind, fill with electrolyte, form the SEI, and test, grouped into electrode making, cell assembly and cell finishing.
From powder to cell in eight steps. Try this: step through and find the two steps where the pores of figure 11.1 are made and then partly squeezed out.
SourceProcess chain, the share of active material, the solvents, drying, calendering, cell formats, formation and quality control after Fichtner et al. 2022, section 5.3. Schematic.

So far we have taken the coating as given. Let us see how it is made, because the making decides how well the two roads work. The composite of this module is made in a few industrial steps. The powders, about 90 % active material by mass with conductive carbon and a polymer binder, are mixed into a slurry, coated on the current-collector foil, dried so that the pores form, and pressed between rollers to an even, denser layer; the electrodes are then cut, stacked or wound with the separator, filled with electrolyte, and cycled for the first time to form the SEI.12

Every step leaves its mark on the roads of figure 11.1: where the carbon and binder end up, how open the pores stay, how well the electrolyte wets them.12

A textbook account adds detail to each step.13 The slurry is usually made with PVDF as the binder and N-methyl-2-pyrrolidone (NMP) as the solvent, and it must be uniform for the coating to be even. It is pumped to a slot die and coated on the foil on a roll-to-roll line, aluminium for the positive electrode and copper for the negative; drying in an oven evaporates the solvent, and it is during drying that the conductive additive and the binder form the continuous network that carries the electrons. Calendering then presses the coating to its final thickness, bonding it to the foil and setting its mass loading, porosity and conductivity. Many groups prefer two passes: a soft one that keeps the microstructure and the pores even, then a harder one that reaches the target density and thickness.13

Assembly follows. The electrodes and separator are cut to size by precisely calibrated slitting machines, with laser cutting the coming option; dried under vacuum to remove moisture, or under flowing argon; stacked or wound into a spiral; and fitted with tabs of the matching metal, aluminium on the positive and copper on the negative, by spot, ultrasonic, bolt or laser welding. The stack goes into its case, the electrolyte is filled in, and the case is sealed by laser welding or by heat. Formation, the first slow cycles of charge and discharge, activates the cell and stabilizes the SEI; following how the SEI affects ageing gives an early indication of capacity loss and of safety problems.13

You can now explain what each ingredient of an electrode coating is for, why a particle without both roads is dead weight, where the reaction happens inside a thick electrode and why, why poorly conducting materials need carbon, and what small particles gain and cost.

Our ion now has a round trip that works: across the electrolyte, into a particle, and back. Now let us push the electrodes further, starting with the negative one, and see how the round trip fails.

Go deeper: the porous electrode in one equation

Let us write the current in the solid i1 and in the electrolyte i2; their sum is the cell current, and the electrolyte current grows through the thickness by whatever the reaction converts, di2/dx = a j, with a the reacting area per volume. Ohm’s law in each phase, dφ1/dx = −i1/σ and dφ2/dx = −i2/κ, and the linearized kinetics of chapter 5, j = η/Rct, with η = φ1 − φ2 − U the local overpotential, give d2η/dx2 = (ν/L)2η, with ν2 = aL2(1/σ + 1/κ)/Rct, the same balance of current that underlies the porous-electrode equations.3,7 Its solution is the cosh profile of figure 11.2, and 1/ν is the reaction depth as a fraction of the thickness. This reaction-limited case and its depth scale have been analysed in full; when the depth is much larger than the electrode the overpotential is uniform.2

The mathematics of this chapter

ε=1−mAd∑iwiρi
κeff=κετ=κNM
τ=RionAκε2dporosity from the areal mass m/A of the coating, its thickness d, and the mass fraction wi and density ρi of each ingredient (our symbols for the procedure of Landesfeind and co-workers); the effective conductivity, tortuosity and MacMullin number; and the tortuosity from the ionic resistance of a symmetric pair of electrodes under blocking conditions (their eqs. 5 and 13); figure 11.5. The Bruggeman estimate is τ=ε−0.5, κeff=κε1.5 (their eq. 6), used in the model of figure 11.6.

κeff = κεα, Deff = Dε/τ, τ = ε1−α, α = 1.5the Bruggeman correction for transport in the pores of porosity ε and tortuosity τ (1.53 for LiFePO4)

j(ξ) ∝ cosh(νξ)/κ + cosh(ν(1 − ξ))/σthe reaction profile across an electrode with linear kinetics; ξ from the collector, ν the thickness over the reaction depth (figure 11.2)

ν2 = aL2(1/σ + 1/κ)/Rctwhat sets the reaction depth: reacting area per volume a, thickness L, the two conductivities and the charge-transfer resistance

t ≈ L2/2Dthe time to cross a particle of size L (figure 11.4)

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 S. Pathak, M. Z. Bazant, "Scaling and Analytical Approximation of Porous Electrode Theory for Reaction-limited Batteries", Journal of The Electrochemical Society 173, 2026, 160518. doi:10.1149/1945-7111/ae9229
  3. 3 B. Boz, T. Dev, A. Salvadori, J. L. Schaefer, "Review: Electrolyte and Electrode Designs for Enhanced Ion Transport Properties to Enable High Performance Lithium Batteries", Journal of The Electrochemical Society 168, 2021, 090501. doi:10.1149/1945-7111/ac1cc3
  4. 4 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
  5. 5 D. Shanmukaraj, P. Ranque, H. Ben Youcef, T. Rojo, P. Poizot, S. Grugeon, S. Laruelle, D. Guyomard, "Review—Towards Efficient Energy Storage Materials: Lithium Intercalation/Organic Electrodes to Polymer Electrolytes—A Road Map (Tribute to Michel Armand)", Journal of The Electrochemical Society 167, 2020, 070530. doi:10.1149/1945-7111/ab787a
  6. 6 H. Zhang, C. Li, G. G. Eshetu, S. Laruelle, S. Grugeon, K. Zaghib, C. Julien, A. Mauger, D. Guyomard, T. Rojo, N. Gisbert-Trejo, S. Passerini, X. Huang, Z. Zhou, P. Johansson, M. Forsyth, "From Solid-Solution Electrodes and the Rocking-Chair Concept to Today's Batteries", Angewandte Chemie International Edition 59, 2020, 534-538. doi:10.1002/anie.201913923
  7. 7 A. J. Bard, L. R. Faulkner, H. S. White, "Electrochemical Methods: Fundamentals and Applications", Wiley, 3rd edition, 2022.
  8. 8 J.-M. Tarascon, M. Armand, "Issues and challenges facing rechargeable lithium batteries", Nature 414, 2001, 359-367. doi:10.1038/35104644
  9. 9 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
  10. 10 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
  11. 11 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
  12. 12 M. Fichtner, K. Edström, E. Ayerbe, M. Berecibar, A. Bhowmik, I. E. Castelli, S. Clark, R. Dominko, M. Erakca, A. A. Franco, A. Grimaud, B. Horstmann, A. Latz, H. Lorrmann, M. Meeus, R. Narayan, F. Pammer, J. Ruhland, H. Stein, T. Vegge, M. Weil, "Rechargeable Batteries of the Future—The State of the Art from a BATTERY 2030+ Perspective", Advanced Energy Materials 12, 2022, 2102904. doi:10.1002/aenm.202102904
  13. 13 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

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). Composite electrodes: percolation, porosity, tortuosity and transport limitations. How batteries work. https://asajeevan.github.io/batteries/composite-electrodes/

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