How batteries work Chapter 9 Ion transport

Chapter 9

Ion transport and the electrode–electrolyte interface: diffusion, migration and transference

How does an ion cross the cell, and what does it meet at the electrode?

Figure 9.1 Left: a lithium ion in its shell taking a random walk through the solvent, its track drawn behind it; with the field switched on, the walk drifts slowly to the right. Right: two small panels showing the ion carried together with its shell, and the ion handed on from one shell of solvent molecules to another.
Li+ (our ion labelled)ECDMCfield
How an ion moves: a random walk, gently steered. Try this: switch the field on and off and compare the tracks.
SourceDiffusion as random motion and migration in a field after Bard, Faulkner and White 2022, chapter 4 (chapter 0); the two modes of lithium transport and their roughly equal share in EC : DMC with LiPF6 after Borodin 2014, p. 384. The walk is a random-number drawing device; the drift is drawn far stronger, relative to the wandering, than it is in a real cell.

Wandering

So far our ion has sat in its shell; now we set it moving. Even with no current, nothing in the liquid is still. Every ion is bumped continually by the molecules around it and wanders at random, in every direction alike. On average it goes nowhere; it only spreads out, a distance that grows as the square root of time, (2Dt)1/2, where D is its diffusion coefficient. This is diffusion, met inside the electrode particles in chapter 7.1

Drifting in a field

Draw current and a small electric field appears across the liquid. It does not replace the random wandering; it biases it: each step is pushed slightly one way, cations down the potential and anions up it. That steady bias is migration, and the Nernst-Planck equation adds the two together (chapter 0).1

What moves is the ion together with whatever stays bound to it, which is why a bigger moving object, or a thicker liquid, conducts less: u = |z|e/(6πηr).1 The lithium ion moves in two ways: it carries its first shell along with it, or it is handed on, leaving some solvent molecules behind and gaining others. In EC : DMC with LiPF6 the two contribute roughly equally; in short ethylene-oxide chains (oligoethers) the ion travels a distance of three to four solvent molecules before its shell is renewed, so there carrying dominates.2

A picture to keep: ionic conduction is a crowd crossing a corridor. More people, or people who move more freely, carry more across per second. The picture breaks in two places: the ions also push on each other through the field, and in a polymer the corridor itself has to keep moving for anyone to get through.

Worked example. Let us put numbers on it: how strong is the field, and how fast do the ions go? A current density of 1 mA/cm2 through a liquid of conductivity κ = 10−2 S/cm, with 1 mol/L of Li+. field in the liquid, from Ohm’s law: E = i/κ = 10−3 A/cm2 ÷ 10−2 S/cm = 0.1 V/cm in the steady state, when all the current is carried by lithium: mean velocity v = i/(Fc) = 10−3 ÷ (96 485 × 10−3) cm/s ≈ 0.1 µm per second to cross a 25 µm layer of liquid: about 4 minutes The field in the liquid is gentle and each ion drifts slowly; the current is large because there are so many ions, about 6 × 1020 in every cubic centimetre.1,3

An ion in a liquid wanders at random; a current only nudges the wandering, gently and all the same way.

Figure 9.2 An electrolyte between an electrode that releases lithium ions and one that takes them in. Under the field cations drift one way and anions the other; a bar shows the share of the current carried by lithium, the transference number; below, the steady-state salt concentration is high where lithium is released and low where it is taken in, and the difference shrinks as the transference number rises.
Li+ (our ion labelled)anionfield
The transference number, and the salt gradient it leaves behind. Try this: slide t+ from 0.3 to 0.6 and watch the gradient shrink by almost half; then all the way to 1.
SourceTransference number as the fraction of the ionic conductivity carried by Li+ (Boz et al. 2021, eqs. 6 and 7; Bard, Faulkner and White 2022, eq. 2.3.11). The steady-state difference Δc = (1 − t+) i L/(F D) follows from the salt balance of Boz et al., eq. 12, and is computed live with i L/(F D c0) = 0.8, an illustrative value; the profile is linear in this steady state. Values of t+ near 0.3 and 0.6 are those Tarascon and Armand 2001 give for polymer electrolytes without and with 10 % nanofillers. Schematic.

Conductivity is not the whole story, because both ions move but only one is wanted. The transference number t+ is the fraction of the ionic conductivity carried by Li+; in a dilute electrolyte it is the ratio of the lithium diffusion coefficient to the sum of both.1,4 (It is also called the transport number.5) In the organic liquid electrolytes of 1994 it was typically below 0.5.6

The rest of the current is carried by anions that no electrode can take in, so in a working cell salt builds up at one electrode and runs short at the other, and the steady-state difference grows with (1 − t+).4

That gradient is the concentration polarization of chapter 5, seen from the electrolyte. In polymer electrolytes t+ is about 0.3; about 10 % of ceramic nanofillers lifts it to about 0.6, raises the conductivity severalfold at 60 to 80 °C, and gives a stable interface with lithium.5

The logical end point is a single-ion conductor, in which the anions are tied to the polymer so that t+ approaches 1 and no salt gradient builds up; one block copolymer of this kind reached t+ above 0.85, though at only 1.3 × 10−5 S/cm at 60 °C, and conducting 10−4 S/cm at room temperature remains the challenge.7

Figure 9.3 Top: a to-scale close-up, 2.4 nanometres wide, of the electrolyte against a graphite electrode, with ball-and-stick EC, DMC, PF6 and lithium. Buttons set the electrode negative, uncharged or positive: at the negative electrode EC molecules touch the surface with their carbonyl oxygens turned away and a lithium ion in an EC shell sits against the surface; at the positive electrode PF6 anions sit at the surface and EC oxygens point towards it. Bottom: over the first 10 nanometres, the potential and the excess of lithium ions and anions, both fading with distance (stretched so they can be seen). A fourth view shows graphite end on with hydrogen-terminated edges and a lithium ion shedding first DMC and then EC as it enters the gap between two sheets, with the free-energy profile below.
lithiumcarbonoxygenfluorinephosphorus
Layer by layer, from the middle of the liquid to the electrode. Try this: switch from Negative to Positive and watch which way the red oxygens point in the first layer.
SourceClose-up drawn to scale with the computed molecular geometries of figure 8.1. Arrangements after the molecular-dynamics results in Borodin 2014, pp. 389 to 391: for the negative, uncharged and positive views, simulations at the basal face of graphite (EC replacing DMC on charged surfaces, carbonyls repelled at negative and gathered at positive, PF6− accumulating at positive, Li+ adsorbed and EC-solvated at negative); for Getting in, the desolvation model with graphite hydrogens at zero distance, the free-energy minimum of −0.5 kcal/mol about 5 Å out, the steep rise inside 4 Å to 12 to 19 kcal/mol and DMC shed before EC, a preliminary study without an SEI or applied voltage. Individual positions are illustrative, not a simulation snapshot. Lower plot: compact and diffuse layers, their extent and the screening of the electrode charge after Bard, Faulkner and White 2022, sections 1.6 and 2.2 to 2.3; curve shapes schematic; at 1 M the real diffuse layer is thinner than a nanometre (the simple Debye estimate gives about 0.3 nm), so the plot stretches it. Free-energy curve drawn through Borodin’s three numbers; its shape between them is schematic.

The bulk and the diffuse layer

We have followed the ion through the middle of the liquid. Now let us walk towards an electrode, because the electrolyte is not the same all the way across. Far from either electrode, in the bulk, every small volume holds as much positive as negative charge: a separation of charge there would set up a field large enough to erase it at once.1 The molecules point every way, and with no current the potential is flat. With current, the only field is the gentle ohmic one of the worked example above, about 0.1 V/cm.

A few nanometres from a charged electrode this changes. Ions of the opposite sign crowd towards the surface and solvent molecules turn: together they cancel the electrode’s charge, so its field does not reach the bulk.1 The excess of ions thins out over the diffuse layer, less than about 10 nm thick above 0.01 M; closest to the surface is the compact layer, one solvent shell thick.1

The field inside that thin step can reach 107 V/cm, a hundred million times the field in the bulk of the worked example.1 This is the electrical double layer of chapter 3, now seen molecule by molecule.

The first molecular layer

Closer still, we reach single molecules. Simulations show what the innermost layer looks like in EC : DMC with LiPF6. When the electrode is charged, either way, the more polar EC replaces DMC at the surface, an effect confirmed by experiment on LiCoO2. At a negative electrode the carbonyl groups are pushed away from the surface, and at the most negative potentials lithium ions sit against it, solvated mainly by EC, which fits the observation that EC reduction dominates the SEI formed in these mixtures. At a positive electrode the layer fills with carbonyl groups and PF6− anions accumulate, where they are expected to take part in oxidation reactions.2

Getting in

One step remains: our ion must get into the electrode. To enter graphite a lithium ion has to leave its shell. In preliminary simulations of bare graphite, with no SEI, a lithium ion rests about 0.5 nm from the surface, still solvated; closer than about 0.4 nm the free energy climbs steeply as it is forced to shed its solvent, consistent with DMC going first and EC last. The size of that climb agrees with measured activation energies of 16.4 to 17 kcal/mol, about 70 kJ/mol. The two contributions usually discussed for the resistance of this interface are the SEI and this desolvation, and experiments show that at low temperature the interface can resist more than the whole bulk electrolyte.2

Uniform in the middle, ordered in the last nanometre: the electrolyte’s structure, and almost all of its field, change within a few molecules of the electrode.

Go deeper: conductivity from the ions themselves

The conductivity of an electrolyte is the sum of what each ion contributes: κ = F Σ|zj|ujCj, with uj the mobility and Cj the concentration of ion j. The share carried by one ion is its transference number, tj = |zj|ujCj / Σ|zk|ukCk.1 For a salt of one cation and one anion this reduces to t+ = |z+|u+/(|z+|u+ + |z−|u−), and in the dilute limit to DLi/(DLi + D−).4 Chapter 0 gave the mobility of a sphere in a viscous liquid, u = |z|e/(6πηr): a bigger ion, or a more viscous liquid, conducts less.1

How independent are the ions? Measure each ion’s self-diffusion coefficient, D+ and D−, and compute the conductivity the ions would give if each moved on its own, λNE = e2(n+D+ + n−D−)/(VkBT), for n ions of each kind in a volume V. The measured conductivity divided by it is the ionicity, αd = λ/λNE: 0.62 for LiTFSI in PC and 0.11 in DMC, both at 20 solvent molecules per lithium and 298 K.2 Why about 1 mol per litre in the first place? Enough salt has to dissolve to provide plenty of charge carriers, and the conductivity is frequently found to peak near 1 M.8

Even water can be stretched. Water splits over a window of 1.23 V (chapter 3), but in “water-in-salt” electrolytes, with far more salt than water by weight, the salt crowds the electrode surfaces and its decomposition builds a LiF-rich interphase; with further measures to keep water off the negative electrode, aqueous electrolytes have reached stability windows above 4 V.9

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 A. J. Bard, L. R. Faulkner, H. S. White, "Electrochemical Methods: Fundamentals and Applications", Wiley, 3rd edition, 2022.
  2. 2 O. Borodin, "Molecular Modeling of Electrolytes", in T. R. Jow, K. Xu, O. Borodin, M. Ue (eds.), Electrolytes for Lithium and Lithium-Ion Batteries, Modern Aspects of Electrochemistry 58, Springer, New York, 2014, 371-401. doi:10.1007/978-1-4939-0302-3_8
  3. 3 M. Winter, R. J. Brodd, "What Are Batteries, Fuel Cells, and Supercapacitors?", Chemical Reviews 104, 2004, 4245-4269. doi:10.1021/cr020730k
  4. 4 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
  5. 5 J.-M. Tarascon, M. Armand, "Issues and challenges facing rechargeable lithium batteries", Nature 414, 2001, 359-367. doi:10.1038/35104644
  6. 6 K. Brandt, "Historical development of secondary lithium batteries", Solid State Ionics 69, 1994, 173-183. doi:10.1016/0167-2738(94)90408-1
  7. 7 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
  8. 8 W. A. Henderson, "Nonaqueous Electrolytes: Advances in Lithium Salts", in T. R. Jow, K. Xu, O. Borodin, M. Ue (eds.), Electrolytes for Lithium and Lithium-Ion Batteries, Modern Aspects of Electrochemistry 58, Springer, New York, 2014, 1-92. doi:10.1007/978-1-4939-0302-3_1
  9. 9 O. Borodin, X. Ren, J. Vatamanu, A. von Wald Cresce, J. Knap, K. Xu, "Modeling Insight into Battery Electrolyte Electrochemical Stability and Interfacial Structure", Accounts of Chemical Research 50, 2017, 2886-2894. doi:10.1021/acs.accounts.7b00486

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). Ion transport and the electrode–electrolyte interface: diffusion, migration and transference. How batteries work. https://asajeevan.github.io/batteries/ion-transport/

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