Deep divefrom Episode 4Batteries1 October 202621 min

Your Battery's Voltage Feels the Squeeze

A deep dive into one paper: what was known before, what the researchers did, what the numbers say and what they don't.

The paper: Battery open-circuit voltage is not purely chemical (arXiv)

Chapters

Transcript

The script the two AI voices read from. The times are worked out from the script, not heard, so a line can start a few seconds early or late.

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Cold open

Host 1You heard the trailer — this is the whole story. Welcome to the Hastrup Review of Science — where artificial intelligence reads real science, so you don't have to pretend you understood the paper. One paper today: the claim that a battery's resting voltage depends on how hard it is being squeezed, and on how its particles are packed.

Host 2The paper is Battery open-circuit voltage is not purely chemical, by Andrea Giudici, Christoph Pohl, Alberto Salvadori, Colin Please, Manuel Landstorfer and Jon Chapman. They are applied mathematicians and mechanicians from the University of Oxford, the Weierstrass Institute in Berlin and the University of Brescia. It is a short preprint on the arXiv with a long supplement, not yet peer reviewed, and it is entirely theory and simulation.

Host 1So no new battery was built.

Host 2No new cell was built or tested for this paper. There is one piece of real data, a re-analysis of an open dataset, and we will get to it. The rest is mechanics, asymptotic mathematics and computer models.

What the resting voltage is

Host 1Start with the basics. What is the open-circuit voltage?

Host 2As established background: a lithium-ion cell stores energy by moving lithium ions between two electrodes, usually graphite on one side and a metal oxide on the other. When no current flows and everything has settled, the voltage you read across the terminals is the open-circuit voltage. The paper defines it as the difference between the chemical potentials of lithium in the two electrodes, divided by Faraday's constant.

Host 1Chemical potential, in plain words?

Host 2Roughly, how much energy it costs to push one more ion into the material. If the anode makes lithium uncomfortable and the cathode makes it welcome, there is a large voltage between them. As the electrodes fill and empty, those costs change, and that traces out the open-circuit voltage curve.

Host 1And that curve is treated as a property of the material.

Host 2Yes. Each electrode gets its own half-cell curve, measured against lithium metal as a fixed reference, and it is written as a function of one number: the fraction of lithium sites that are filled. The paper points out that these curves are key inputs to battery models, and they are usually treated as purely chemical.

Host 1Why does anyone care about a few millivolts on that curve?

Host 2As background, battery management systems lean heavily on the open-circuit voltage curve to estimate how full a battery is, and models use it to predict performance and ageing. Where the curve is flat, a small voltage error turns into a large error in estimated charge. So the shape of the curve, especially near its ends, matters to people who build and run batteries.

Cracks in the chemical picture

Host 1What made the authors doubt that it was pure chemistry?

Host 2Two lines of earlier evidence, which they cite. Experiments have shown that cell voltage, impedance and ageing respond to the stack pressure applied to a cell. And direct measurements on insertion materials have shown that stress can shift the open-circuit voltage by several millivolts.

Host 1And the second reason?

Host 2Published curves for nominally similar electrodes disagree, sometimes substantially, and most of all near the ends of the usable range. The authors grant that material grade, ageing, hysteresis and measurement protocol can explain part of that spread. Their question is whether mechanics and microstructure explain another part.

Host 1Battery mechanics is not a new field, though.

Host 2No, there is a large literature on stress inside active particles. But as the paper describes it, most of those models treat each particle as mechanically isolated, so stress comes from uneven lithium concentration during charging, and it relaxes away once the particle settles. A model like that cannot produce a stress that survives at equilibrium, so it cannot shift the open-circuit voltage.

Host 1Some models do include outside stress.

Host 2Yes, continuum models have shown that imposed stress, or stress passed through the surrounding binder, can modify the equilibrium voltage. What this paper adds is direct particle-to-particle contact. In a densely packed electrode, particles push on their neighbours, and the authors argue that this contact network produces larger, nonlinear stresses whose strength depends on how the particles are arranged.

The physical idea

Host 1So what is the physical picture?

Host 2As background, many electrode materials swell when lithium goes in; graphite is the classic example, and its volume grows by roughly ten percent when fully loaded. Now pack those particles tightly between a separator and a metal foil, and squeeze the whole stack. When a particle swells, it presses against its neighbours, and those contacts carry compressive load.

Host 1And compression makes it harder to add lithium.

Host 2Exactly. Lithium takes up room, measured by what is called the partial molar volume, so inserting it into a compressed particle costs extra work against the pressure. That extra work shifts the chemical potential, and with it the voltage, even after everything has come to rest.

Host 1Which sounds obvious once you say it.

Host 2The direction is intuitive. The paper's contribution is working out how big the shift is, how it scales with filling, and why it depends on the geometry of the packing, and then carrying that up to a whole cell.

Method: idealised packings

Host 1How do they model a pile of particles?

Host 2They use idealised lattices of identical elastic spheres in three arrangements: simple cubic, body-centred cubic and face-centred cubic. These fill fifty-two, sixty-eight and seventy-four percent of space respectively when the spheres just touch. The point is not realism; the lattices let them isolate how the number and orientation of contacts change the answer.

Host 1How many contacts in each?

Host 2Simple cubic has six neighbours, body-centred cubic eight, and face-centred cubic twelve. Their orientation relative to the through-cell direction matters too, because that is the direction the stack gets squeezed. For simple cubic, two contacts point along that direction and four lie flat; in body-centred cubic all eight are diagonal; in face-centred cubic, eight are diagonal and four lie flat.

Host 1And inside each sphere?

Host 2Each particle is a small-strain, linear elastic solid that swells in proportion to its local lithium fraction. The chemical part of the chemical potential is the simplest textbook form, an ideal solution, and the mechanical part comes from the elastic energy. Lithium is allowed to redistribute inside the particle, so a stressed region can hold a little less of it.

Host 1What are the boundary conditions?

Host 2Contacts are frictionless, the rest of the surface is free, and in the reference state the particles are empty and just touching. Contact is enforced numerically with a penalty method. They solve the coupled mechanics and chemistry in three dimensions with finite elements, using symmetry to compute only one eighth of a particle.

First results: millivolts from packing

Host 1What did the simulations show first?

Host 2With particles clamped so their centres cannot move, contact produces residual stress and lithium redistribution even at equilibrium. The voltage correction from swelling alone comes out at several millivolts in their example. And it grows with state of charge to the power three halves.

Host 1Why three halves?

Host 2That is the fingerprint of Hertz contact, the nineteenth-century result for two elastic spheres pressed together. The force grows as the overlap to the power three halves, because the contact patch widens as you push. Since the overlap here grows in proportion to the lithium content, the voltage shift inherits the same exponent.

Host 1And the arrangement matters?

Host 2Yes: the three lattices give different curves for the same material at the same filling. The authors also bracket them with two extremes, a freely swelling particle with no correction and a fully constrained particle that cannot move at all, which gives the largest possible shift. Real packings sit in between.

Host 1What if the particles do not start out touching?

Host 2They test that by lowering the packed fraction by three hundredths with a stiffer material, so there is a gap at first. The curves then follow pure chemistry until swelling closes the gap, and only after that do they peel away. A gap shifts where the separation starts but not the three-halves exponent.

Host 1And squeezing the electrode from outside?

Host 2They let particle centres move in the through-cell direction only, because the current collector foils strongly restrict in-plane movement, and then impose a strain. At the same lithium content, changing that strain changes which contacts are closed, the stress, and the voltage. Their summary is that chemical composition alone is not sufficient to determine the open-circuit voltage.

The formula behind it

Host 1Simulations are one thing. Is there a formula?

Host 2That is the mathematical core. They assume the swelling is small and carry out what they call a weak-contact expansion, solving order by order. They also show they can average the three-dimensional problem over the sphere's surface, and the contacts then enter only through their total forces.

Host 1And what comes out?

Host 2First, free swelling on its own does not change the chemical potential at all; only contact does. Second, the leading correction equals the partial molar volume times the total compressive contact force per unit of particle surface. In averaged terms, the localised contacts act like a uniform hydrostatic squeeze on the particle.

Host 1So the details of where the lithium goes inside the particle drop out.

Host 2At leading order, yes. The redistribution of lithium changes the local fields but not the particle-averaged shift, which depends only on the closed contact loads. The formula splits into a material factor, a swelling-and-contact factor with the three-halves power, and a purely geometric factor that just counts contacts by orientation.

Host 1Does it match the finite-element results?

Host 2They report excellent agreement across all three lattices and a range of imposed strains, shown in one figure, which is a check of the asymptotics against their own numerics. It is not a check against an experiment. One more detail from the supplement: the derivation assumes the chemical part of the curve is not flat, so it applies away from a phase plateau or the very end of the range.

A word from our sponsor

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From particles to the stack

Host 1Back to the paper. How does this reach the scale of a whole cell?

Host 2First they average the contact forces crossing a plane to get each electrode's stress-strain law. Because of Hertz contact, that law is nonlinear: the stiffness grows as the particles press together. Contacts lying flat, perpendicular to the through-cell direction, still shift the voltage but carry no force through the stack.

Host 1And then the cell?

Host 2A cell is a cathode, a separator and an anode in series. Mechanical equilibrium means all three carry the same through-cell stress, and in a clamped cell their thickness changes have to add up to whatever the clamp allows. The separator is treated as a simple linear spring.

Host 1Which couples the two electrodes.

Host 2That is a key point of the paper. If one electrode swells, it raises the common stress, and that shifts the voltage contribution of both electrodes, not just its own. In their words, cell open-circuit voltage is a property of the whole battery architecture, not of chemistry alone.

Full-cell simulation

Host 1What numbers come out for a realistic cell?

Host 2They plug the model into PyBaMM, the open-source battery modelling package developed largely in Oxford, using its simplest model, the single-particle model, and a published parameter set called Mohtat twenty twenty. The stack is an anode of sixty-two micrometres, a separator of twelve and a cathode of sixty-seven, and the baseline open-circuit voltage runs from about two point eight one to four point two one volts. The cell is clamped at its thickness when fully discharged, and the mechanics is solved at a thousand points along the charge.

Host 1And the mechanical parameters?

Host 2Those are not in the parameter set, so the authors chose representative values. Partial molar volume of eight millionths of a cubic metre per mole in the graphite and one millionth in the cathode, stiffnesses of fifty and one hundred and fifty gigapascals, a separator modulus of one gigapascal, and the face-centred cubic packing as a highly coordinated network. They state plainly that these are modelling choices, not fitted to any cell.

Host 1Result?

Host 2The full-cell open-circuit voltage rises by tens of millivolts near full charge. The supplement gives the exact figures: the contact correction is minus zero point seven four four millivolts at empty, rising to a maximum of about forty-three point four millivolts. It is driven mainly by the graphite anode, because lithium takes up much more room in graphite than in the cathode.

Host 1Forty-three millivolts against four volts.

Host 2About one percent at the top of charge, under their chosen parameters, in a clamped cell. Small in absolute terms, but comparable to the differences people argue about when they compare open-circuit voltage curves.

Re-reading the experimental spread

Host 1You mentioned one piece of real data.

Host 2In the supplement, the authors re-analyse an open dataset from SINTEF, the Norwegian research institute: seven graphite, six NMC one-one-one and eight NMC five-three-two half-cells, with replicate cells from single electrode batches and four measurement protocols. To their knowledge, it is the only open record combining replicates, several protocols and an individually reported active mass. The question is how much the curves still differ once the obvious measurement problems are removed.

Host 1What are the obvious problems?

Host 2The biggest is side reactions. If you simply count charge, part of the current goes into parasitic processes such as growth of the protective film on graphite, and for one graphite cell the naive count says the electrode is one hundred and twenty-three percent full after five cycles, which is impossible. They correct this by assuming a constant parasitic current over those few cycles, a fraction of a microamp, about zero point three to zero point four percent of the pulse current for most cells.

Host 1And after the correction?

Host 2The cycles of each cell collapse neatly, with agreement to a median of zero point three millivolts, which is the measurement's own floor. Yet the corrected curves still differ from cell to cell for the same material. The authors take this as a suggestion that chemistry alone may not be enough, and that mechanical conditions or microstructure could play a significant role.

Host 1Could, not do.

Host 2Could. The dataset does not report the mechanical state or microstructure of those cells, so the analysis shows that a spread remains, not what causes it. The authors themselves write that the differences need not be mechanical, since ageing or composition changes can also reshape these curves.

Why it matters

Host 1Step back. Why does this matter?

Host 2The principle is that the resting voltage of an insertion electrode is a chemo-mechanical quantity. Even without any outside clamp, swelling particles that touch build up stress that persists at equilibrium, and that stress depends on how they are packed. So two electrodes made of identical material can, in principle, have different open-circuit voltage curves.

Host 1What changes, and for whom?

Host 2For people who measure these curves, the paper argues that stack pressure and electrode microstructure should be recorded and controlled, because they are hidden variables. For modellers, the voltage curve stops being a fixed lookup table and becomes something that depends on the mechanical state, and the authors give a closed-form estimate that can be added to standard models. And it gives electrode processing, which is known to change how particles touch, a route into equilibrium voltage.

Host 1And what does it not mean?

Host 2It does not mean chemistry is secondary; the correction is tens of millivolts on a cell curve that spans about one point four volts in their simulation. It does not mean the published spread between curves is explained, because nothing here measures how much of it is mechanical. And it does not tell anyone how to tune a real cell's pressure, since the numbers rest on idealised packings and chosen parameters.

Caveats

Host 1What are the honest caveats?

Host 2First, there is no experiment that tests the prediction. The key observable would be the same material measured under different stack pressures or packings, and the authors call for exactly that rather than providing it. The SINTEF re-analysis shows a residual spread, but it cannot attribute it.

Host 1Second?

Host 2The geometry is idealised: identical spheres on perfect lattices, frictionless contacts, small strains and linear elasticity, with particles assumed to just touch when empty. The authors themselves say more careful study is needed for realistic electrodes. They also assert that binder and conductive additives do not remove the dominant contact contribution, but they do not model them here.

Host 1Anything about the chemistry side?

Host 2The chemical part is an ideal solution, while real graphite charges through a series of flat plateaus. The asymptotic formula explicitly assumes the curve is not flat, away from a phase plateau or endpoint, so how this plays out in strongly phase-separating materials, and where curves are flattest, is left open. Hysteresis, which also separates charging and discharging curves, is not part of the model.

Host 1And the headline number?

Host 2The roughly forty-three millivolts comes from representative, unfitted mechanical parameters, a fully clamped cell and the most highly coordinated lattice. Change the partial molar volume, the stiffness or the packing and it moves; the half-cell example in the earlier figure was several millivolts. So treat tens of millivolts as an order of magnitude from one scenario, not a measured effect, and remember this is an unreviewed preprint.

What to watch

Host 1What should we watch for next?

Host 2Controlled experiments: the same electrode batch measured at several stack pressures, and electrodes of the same material processed to different densities. If the shift follows the three-halves power in state of charge and grows with compression, that would be a clean test. Measurements reported with the mechanical state recorded would help everyone comparing curves.

Host 1And on the modelling side?

Host 2Extending this to realistic, disordered electrodes, where the lattice counts become distributions of contact number, orientation and force, as the authors suggest. Watch also for whether pressure-dependent open-circuit voltage makes its way into tools like PyBaMM, and whether it improves state-of-charge estimates in cells that are clamped tightly in a pack.

Sign-off

Host 1That is the deep dive: Battery open-circuit voltage is not purely chemical, by Giudici, Pohl, Salvadori, Please, Landstorfer and Chapman, from Oxford, the Weierstrass Institute in Berlin and Brescia, a preprint on the arXiv. The link is in the show notes.

Host 2Every Friday this same feed carries the roundup episode: five new papers as trailers, and you pick which become deep dives. This has been the Hastrup Review of Science.