Intrinsic hysteresis behaviour of many-particleinsertion battery electrodes

Rok Kaufman

Mentor: prof. dr. Miran Gaberscek

(National Institute of Chemistry, Ljubljana)

June 6, 2018

Abstract

In this seminar, I will describe the phenomenon of intrinsic hysteresis behaviourwith separate equilibrium potentials on charge and discharge observed in LFP cath-ode system. The hysteresis can be described by coupling together single-particleequilibria, which have a non-monotonous dependence of chemical potential with re-gards to charge. A similar system of rubber balloons being filled with air, whichexhibits similar properties and serves as a good demonstration example, will also bedescribed.

Contents

1 Introduction 3

2 Electrochemistry: Theoretical background 42.1 Electrochemical work . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Electrochemical half-cell . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Nernst equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Lithium-ion batteries 73.1 Structure of LFP cathode . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Thermodynamics of LFP cathode 94.1 Single-particle system . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2 Many-particle system . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5 Analogous mechanical system – rubber balloons 13

6 Conclusion 16

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1 Introduction

During recent decades, lithium-ion (Li-ion) batteries have become one of the mostpopular types of rechargeable batteries owing to their high power and energy density.However, despite their widespread usage, the intricacies of their mechanism are stillnot fully understood. One of the observed phenomena that was a surprise to theresearchers is existence of a small potential gap between charging and dischargingplateau observed in Li-ion batteries which use LFP as the active cathode material,which turned out to be independent of battery construction and thus clearly a con-sequence of LFP material properties. While the phenomenon can be explained quiteeasily using many-particle equilibria, it is not a widely studied phenomenon and thusscientists may not recognise what’s going on at first glance.

Figure 1: Experimental evidence of a zero-current voltage gap in LFP cathodes. a;typical LFP cathode galvanostatic charge/discharge profile, at a low rate (C/20). b;galvanostatic partial charge/discharge profiles (within the blue region indicated ina) for a LFP cathode at rates ranging from C/10 to C/1000. c, a current–potentialplot generated from data read out from b. d, very slow charge/discharge experimenton LFP cathode where the cell was allowed to relax towards the equilibrium for along time, to show two distinct plateaus for charge and discharge. Taken from: [1]

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In this seminar, I will start by providing some theoretical and practical back-ground necessary to understand the phenomenon, followed by theoretical model de-scribing observed properties. Additionally, a different system of rubber balloonsexhibiting similar properties on air filling, one that is easier to grasp and observeas a demonstration experiment, will also be analysed as a way to demonstrate thisphenomenon in a way that is easier and perhaps more intuitive to grasp.

2 Electrochemistry: Theoretical background

Electrochemistry is the study of conversion between chemical and electric energy.This is achieved by a pair of chemical reactions, where one produces and the otherconsumes surplus charge. Separating the two reactions allows us to channel thecharge and use it to conduct electric work.

2.1 Electrochemical work

Consider a thermodynamic system which undergoes a reversible process at constanttemperature and pressure. From the definition of thermodynamic potentials it followsthat the change in Gibbs free energy of the system is equal to the non-mechanicalwork done by the system. In the case of an electrochemical system, this is electricwork:

∆G = Wel

Change in Gibbs free energy of the electrochemical system can therefore be di-rectly measured by measuring the electric work done.

Consider now an electrochemical system which operates at a potential differenceof E. The two terminals of the electrochemical system are connected through anexternal load of resistance R. The work done by the system is equal to the productof transferred charge q and the potential difference:

Wel = qE

The transferred charge is equal to the number of electrons transferred times ele-mentary charge, but in chemistry we usually operate using moles (dividing the countof particles by Avogadro number NA) to avoid unnecessarily large numbers. Chargeof one mole of electrons is defined as minus Faraday constant F whose value is about96 485 As. When n moles of electrons are transferred between the terminals of theelectrochemical system, the system has exerted work equal to:

Wel = −nFE

and in turn this must also be equal to the change in Gibbs free energy of thesystem:

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∆G = −nFE

Spontaneous process at constant temperature and pressure means decrease inGibbs free energy. Since there is ambiguity in sign when measuring the potential dif-ference between the terminals, we need a convention on polarity of this measurement.Consistently with this equation we decide that for spontaneous processes E > 0.

2.2 Electrochemical half-cell

An electrochemical system has two terminals, which correspond to the two reactionstaking place on two electrodes within the electrochemical cell. The positive electrodeis called the cathode and the reaction occurring is addition of electrons, or a chemicalreduction; the negative electrode is called the anode and the reaction occurring isremoval of electrons, or a chemical oxidation.

Since we technically have not one, but two complementary reactions taking placewithin the electrochemical cell, we can consider it as being composed of two half-cells. Due to the two reactions being separate, they do not affect each other and thefull cell can be considered as sum of both half-cells.

The potential difference E can be written as the difference between potentials ofthe two half-cells:

E = Ec − Ea

where Ec is the potential of the cathode half-cell and Ea is the potential of theanode half-cell. Note that what is the anode half-cell in one setting can also bethe cathode half-cell in another, in which case the same reaction occurs in oppositedirection.

Absolute potential, however, is not well-defined. But since only potential differ-ences will affect the behaviour of the cell, we are free to choose an arbitrary standardreference potential. By convention, the standard half-cell potential of 0 V is assignedto the standard hydrogen electrode (SHE)1. All other half-cells are measured againstthis reference half-cell, either directly by measuring the potential of an electrochem-ical cell containing it as one of the electrodes, or indirectly by measuring againstanother half-cell which has an established value against the hydrogen-platinum elec-trode.

1The standard hydrogen electrode is a electrode made of platinum, which reduces H+ ionspresent in the solution to molecular hydrogen H2: 2H+ + 2e− H2. The reaction ismade possible to occur in reverse direction by blowing hydrogen gas bubbles over the electrode.While potential of the SHE is taken to be 0 V at theoretical standard conditions (pH = 0), it is afunction of pH of the solution, decreasing with rising pH value. For that reason, the SHE is alsoused for measuring pH value of solutions.

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2.3 Nernst equation

The half-cell potentials are of course not constant, but depend on ambient conditionssuch as temperature and concentrations. To derive this, we need to consider the effectof entropy change during the electrochemical reaction.

Consider a generic chemical reaction:

aA + bB cC + dD

The entropy of each molecule is defined as:

S = kB ln Ω

where Ω is the number of available states. Since the number of available states islinearly correlated to the available volume of the system, we can write an expressionfor entropy of all molecules of a single species:

S ∝ NkB lnV = nR lnV

where we have written the expression both according to physical convention ofcounting single molecules and chemical convention of converting into moles and usingthe ideal gas constant R = kBNA. Additionally, concentration c of a species2 isinversely proportional to available volume, so for change of entropy as concentrationchanges form c1 to c2 we can write:

S2 = S1 − nR lnc2

c1

Summing together contributions from all four species our generic chemical reac-tion, taking n(X) = x for each of the species, and taking ∆SR as change of entropyunder standard conditions (by chemical convention, 1 atm for gases and 1 mol/L forsolutions), we get:

∆SR = ∆SR −R lncC

ccDd

cAacBb

The ratio under the logarithm is the reaction quotient, which can be furthergeneralised as:

Q =

∏j cj

νj∏i ci

νi

where ci and cj are concentrations of reactants and products respectively, and νiand νj corresponding factors in the reaction equation.

2We must note that due to interactions between molecules, activity of molecules is not directlyproportional to concentration, and we introduce a measure of effective concentration – thermody-namic activity a, for which derived thermodynamic equations are exact, but is a non-linear functionof concentration which needs to be determined experimentally.

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We have shown before that potentials are related to the Gibbs free energy. Thechange in Gibbs free energy can be expressed using the previous equations as:

∆G = ∆H − T∆S = ∆G +RT lnQ

where we have written change in Gibbs free energy under standard conditionsas ∆G. Inserting this into the equation for cell or half-cell potential, and writingpotential under standard conditions as E, we get:

E = E − RT

zFlnQ

the Nernst equation, which is perhaps the most important relation in electro-chemistry.

As a direct consequence, when batteries are cycled at currents high enough todeplete concentration of active species in vicinity of the electrode, the potentialsbecome shifted. This causes energy dissipation and batteries are charged at slightlyhigher voltage than they are discharged – batteries exhibit a voltage hysteresis.

3 Lithium-ion batteries

In a Li-ion battery, electrical energy is extracted during cycling by transferringlithium ions between the anode and the cathode.

Figure 2: Schematic representation of Li-ion battery mechanism. Takenfrom: https://www.androidcentral.com/sites/androidcentral.com/files/styles/

xlarge/public/article_images/2014/12/LIB.png

Liquid electrolytes used in Li-ion batteries generally consist of a lithium saltdissolved in organic solvents exhibiting high molecular dipole moment. Due to high

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operating voltage, organic solvents decompose in vicinity of the anode on initialcharging, forming a stable interphase layer which conducts lithium ions yet preventsfurther decomposition of the electrolyte. In addition, advances have also been madein using solid state electrolytes.

Anodes are generally constructed from carbon-based materials. The dominantcommercially-used material is graphite owing to its low cost and good conductivity,which can ideally bind one lithium atom per six carbon atoms. As battery discharges,lithium ions are released into the electrolyte, and one electron per lithium ion is re-leased into the circuit. Lithium is intercalated between graphene sheets constitutinggraphite, which means that all sites are not readily accessible, and that the materialexpands (by about 10%) on charging of the battery.

There is more variety in commercially used cathode materials. Cathode in Li-ionbatteries is composed of an ionic material, which can again intercalate lithium ions viareduction of transition metal ions into lower oxidation state, using one electron perlithium ion in the process. Common commercially used materials used are (in orderof application): lithium cobalt oxide (LCO, Li1−xCoO2), lithium manganese oxide(LMO, Li1−xMn2O4), lithium iron phosphate (LFP, Li1−xFePO4) and lithium nickelmanganese cobalt oxide (NMC, Li1−xNiyMnzCowO2, in various ratios y+z+w = 1),where x is the rate of delithiation of the material, which also corresponds to batterycharge level. Intercalation of lithium again causes changes in volume, as well ascrystal structure of cathode material.

3.1 Structure of LFP cathode

LFP was introduced as a cathode material in Li-ion batteries in 1996 by John Bannis-ter Goodenough of University of Texas. Its main advantages are natural abundanceof elements and consequent low cost, thermal and chemical stability, non-toxicity, lowexpansion on lithiation and good specific capacity (170 mAh/g). However, the maindisadvantage is its low electrical conductivity (10−9 S/cm), which was overcome bydecreasing size of LFP particles used, and coating them with a conductive material(usually carbon). Typically, dimensions of particles are on the order of 100 nm. ALFP cell operates at equilibrium voltage of about 3.42 V under usual conditions.

The reaction taking place on a LFP cathode can be written as:

Li+ + FePO4 + e− LiFePO4

A LFP cathode is a composite cathode, composed of active LFP particles, carboncoating which ensures electrical contact, and binder material. The cathode is manylayers of LFP materials thick, and needs to be described by a porous electrode model,where the electrochemical reaction isn’t confined to the flat surface but occurs withinthe electrode’s volume. For that reason, it is also clear that there exists a Li+

ion concentration gradient within the cathode while the battery is operating, andparticles don’t feel exactly the same conditions.

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Figure 3: Electron microscope picture of a typical LFP cathode, consisting of activenanoparticles in a conductive carbon matrix wetted by a liquid electrolyte (ionicconductor). Taken from: [1]

Pure LiFePO4 crystallises in olivine structure where Li+ ions are sitting in one-dimensional channels within a more tightly bound FePO4 structure. For that reason,mobility of Li+ ions is highly anisotropic and transport mainly occurs along thosechannels. In pure FePO4, the structure can be compared to α-quartz; the maindifference with LiFePO4 is that the channels are now vacant.

4 Thermodynamics of LFP cathode

Thermodynamics of LFP can be understood using a solid solution model, where sitesin channels within FePO4 can be either vacant or occupied by Li+ ions. In accordancewith x being the rate of delithiation let x be fraction of vacancies, so that 1 − x isfraction of Li+.

Specific free energy of Li1−xFePO4 as a function of x also contains the mixingentropy term and mixing enthalpy term:

g = xgFePO4+ (1− x)gLiFePO4

+RT (x lnx+ (1− x) ln (1− x)) + ∆h

Chemical potential is equal to the derivative of specific Gibbs free energy withrespect to fraction:

µ = gFePO4− gLiFePO4

+RT lnx

1− x+∂∆h

∂x

Using Flory–Huggins solution theory, we can write the mixing enthalpy term as:

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Figure 4: Crystal structure of LFP. Taken from: [3]

∂∆h

∂x= Ωx(1− x)

where Ω is a free parameter, positive for the case where A-B interactions areweaker than A-A and B-B and negative when they’re stronger3. In case of LFP,the Flory–Huggins parameter is large enough that specific Gibbs free energy has twominima, and for values of fraction x between those two values, spontaneous demixingoccurs into two separate lithium-rich and lithium-poor phases. For standard condi-tions, the lithium-rich phase has vacancy fraction of about 0.11, and the lithium-poorphase has vacancy fraction of about 0.95. At the same time, this means that chemicalpotential is a non-monotonous function of fraction x.

4.1 Single-particle system

Consider now a system, where a single LFP particle is being delithiated.As we slowly remove Li+ ions from the system (increasing x), we are randomly

voiding the sites in LFP crystal, and the Gibbs free energy is decreasing until wereach the first minimum. At this point, it is thermodynamically favourable for thesystem to start forming a lithium-poor phase with x equal to the second minimum,which then grows at expense of lithium-rich phase as Li+ ions are further removedfrom the system. The Gibbs free energy of the system stays on a common tangent.Once the entire particle is composed of lithium-poor phase, the delithiation onceagain starts to occur over the entire particle, theoretically until we remove the lastLi+ ion.

3Therefore, for negative Ω mixing is endothermic, and for positive Ω mixing is exothermic.

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Figure 5: Chemical potential of a single LFP nanoparticle as a function of fractionx. Horizontal axis is reversed so that higher state of charge is plotted to the right.Taken from: [5]

However, in order to form the lithium-poor phase, we need to locally overcomethe potential barrier. If we do not seed the lithium-poor phase, the particle willcontinue losing Li+ ions in a thermodynamically unfavourable way, particle-wide,as demixing by a small finite amount itself is still thermodynamically unfavourable,as chemical potential of the system is still increasing with x and spliting the systeminto two phases of infinitesimally different x increases the total Gibbs free energy dueto the function being convex. The system behaves similar to supercooled water inthat despite being in an unstable, thermodynamically unfavourable state, the phasetransition cannot occur. That is, until we reach the spinodal point – where chemicalpotential reaches a local maximum, and Gibbs free energy has an inflexion point fromconvex to concave. Once we reach that point, lithium-poor phase is spontaneouslyformed and filling continues along the common tangent. Random fluctuations andinhomogeneity of particles can cause the phase transition to trigger earlier, but itwill never trigger later than reaching the spinodal point.

4.2 Many-particle system

When measuring current-dependency of voltage gap between charge and dischargein a typical many-particle LFP battery system, it became apparent that even atextremely low currents (as low as C/10004), and when allowing considerable rest-time of several days for the system to equilibrate, the hysteresis doesn’t disappear,but instead it tends to voltage gap of about 8 mV between charge and discharge as

4The unit C is normalised to battery capacity. Current of 1 C means that a battery is chargedin 1 h.

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Figure 6: Chemical potential of a system consisting of multiple LFP nanoparticlesas a function of normalised charge q. For N = 2, 3, 5, 6 particles, all accessiblestates are plotted, and equilibrium branches are shaded. For N = 10, 30 particles,only equilibrium branches are plotted for clarity. Envelope of the plot defines thefull hysteresis. On partial cycling, we transfer between the upper and lower plateaualong one of the equilibrium branches. Taken from: [3]

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current tends to zero. Does irreversibility of the mechanism arise from the fact thatwe’re dealing with a many-particle system? How does being composed of a multitudeof nanoparticles even affect the charge/discharge mechanism?

Let us consider only the situations, where there is no phase border within singleparticles. The system can still break down into two separate phases, but in such away that we have two populations of particles at different x, close to lithium-richand lithium-poor equilibria. When one of the particles starts transitioning fromlithium-rich to lithium-poor phase, the released Li+ ions are absorbed by other par-ticles. Since we now have an additional free parameter – ratio between the sizesof both populations, we can have different states with different chemical equilibriacorresponding to a single x, corresponding to different ratios between sizes of popu-lations. Each of them is characterised by its own spinodal start and endpoints, wherethe equilibrium is broken and the system readjusts by changing some of the lithium-rich into lithium-poor particles, or vice-versa. If system doesn’t spontaneously jumpbetween different equilibria due to large enough fluctuations, delithiation will occurat the upper plateau and lithiation at the lower plateau voltage.

For now, we have only analysed the thermodynamic properties of the model. Inorder to understand a continuous process, we need to introduce kinetics. This isaccomplished by introducing two parameters determining the scale of the system.The first parameter, τ , determines the timescale. It is equal to the ratio betweenrelaxation time of the system and the filling time. For spatial scale of the system,we introduce ν2, which is defined as the ratio between thermal energy and changeof enthalpy between lithiated and delithiated LFP: ν2 = RT

n∆H. The parameter ν2 is

therefore related to how likely the system is tunnel through the potential barrier andinitiate phase transition before reaching the top of the branch.

For LFP cathode, typical values of those two parameters can be estimated asfollowing. Taking the value ∆H = 59 meV for a single Li+ ion from the literature,and assuming particles of size in the typical range of 10–100 nm, we obtain ν2 = 10−5–10−8. The timescale parameter can be estimated from the diffusion constant for Li+

ions in LFP, which is about 10−15–10−17 m2/s. If we are charging the battery withtypical current of C/3, and again for particles of size 10–100 nm, we obtain value ofτ = 10−3–10−7.

Graphs showing behaviour of the system with respect to different values of pa-rameters ν2 and τ are plotted in Figure 9.

5 Analogous mechanical system – rubber balloons

In general, the above theory can be applied to any system which exhibits a non-monotonous relation between filledness and potential, and is composed of manyparticles. For example, such hysteresis behaviour can be observed in a simple andeasily understandable system of interconnected rubber balloons. Here, fillednesscorresponds to the volume and potential to the pressure.

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Figure 7: Typical shape of radius-pressure curve, experimentally measured for asingle balloon. On first cycle, we observe a slightly different shape on filling due toinelastic conditioning of the balloon, similar to how the first charge of a battery givesa slightly different profile due to buildup of electrode interfaces. Taken from: [2]

The filledness of a balloon can be parametrised as radial strain λ = rR

, where rand R are radii of deformed and undeformed balloon respectively. We assume thatair is composed of two-atomic molecules of molecular mass mA

5, and the numberof molecules contained within a balloon is denoted by N , and we can write densityof air molecules nA = N

Vwhere V = 4π

3r3 is the volume. Values for undeformed

balloon can be written with an overline: N , V , and volume can be expressed also asV = λ3V . Let p be pressure within the balloon and p0 external pressure.

Free energy of a balloon as a function of radial strain is composed of two terms,one from air and another from the rubber membrane:

G(λ) = V

[λ3((ρg)A + p0) +

3

Rλ2(ρg)B

]where ρA and ρB are volume and surface mass densities of air and balloon rubber

membrane, and gA and gB are their specific free energies. Specific free energies canbe derived from:

dgA = −sAdT −p

mA

d1

nA

dgB = −sBdT −p− p0

ρBdr

where the entropy terms can be discarded since we take temperature T to beconstant. Free energy density of two-atomic ideal gas can also be given as:

(ρg)A = nAkBT lnnAnA

+ (ρg)A

5Atmospheric air is mostly a mixture of N2 and O2 with average molecular mass mA ≈ 28.97.

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Figure 8: Experimentally measured hysteresis for an ensemble of 8 balloons (sameas in Figure 7). On filling, each of the pressure maxima corresponds to the statejust before the next balloon starts inflating, as drawn in the panels below the plot.Taken from: [2]

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and free energy density of a rubber balloon can be described by a Mooney–Rivlinmodel up to 2nd order as

(ρg)B =1

2d[s+(2 + λ−6)− s−(λ2 + 2λ−4)

]where d denotes thickness of undeformed rubber membrane, and s+ and s− are

temperature-dependent parameters.Balancing out pressure, it follows that

p(λ) =2d

R

[s+(λ−1 − λ−7)− s−(λ− λ−5)

]+ p0

and number of contained air molecules as a function of radial strain is a monotonouslyincreasing function

N (λ) = Nλ3 p

p0

Again, as in the case of LFP cathode, we need to determine the parameters ν2

and τ . For balloons of initial diameter 0.1 m and filling time of 250 s, and takingthe characteristic relaxation speed to be equal to the speed of sound, 343 m/s, wearrive at value of τ = 10−6. The value of ν2 is much smaller than in the case of LFPnanoparticles, since the number of atoms in macroscopic balloons is much larger thannumber of atoms in LFP nanoparticles; it is on the order of ν2 = 10−22. However,the hysteresis will not be as smooth because the number of balloons in the system ismuch smaller, and we can actually observe how the system moves between differentequilibrium branches.

Graphs showing behaviour of the system with respect to different values of pa-rameters ν2 and τ are plotted in Figure 10.

6 Conclusion

We have observed that in cases, where potential of the system non-monotonouslychanges with a parameter of filling, we observe splitting of the equilibrium stateas a function of parameter of filling into N + 1 branches on coupling of N subsys-tems. This splitting leads to development of two distinct plateaus, one for fillingand one for emptying the system, as N →∞. This hysteretic behaviour is intrinsicto the system and is a consequence of thermodynamic, not kinetic properties. Thisphenomenon can be observed in many diverse systems, some of which are of techno-logical importance (LFP cathode) while others can serve as a great demonstrationtool (balloon cluster filling). Understanding this phenomenon can allow us to studysome properties of the system; for example, measuring the hysteretic behaviour ofLFP cathode as current tends to zero allows us to measure the chemical potentialof LFP as a function of charging parameter x, and consequently the Flory–Hugginsparameter of the system.

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References

[1] Wolfgang Dreyer, Janko Jamnik, Clemens Guhlke, Robert Huth, Joze Moskon,Miran Gaberscek: The thermodynamic origin of hysteresis in insertion batteries.Nature Materials 9(5), 448–453 (2010). doi:10.1038/nmat2730

[2] Joze Moskon, Janko Jamnik, Miran Gaberscek: In depth discussion ofselected phenomena associated with intrinsic battery hysteresis: Batteryelectrode versus rubber balloons. Solid State Ionics 238, 24–29 (2013).doi:10.1016/j.ssi.2013.02.018

[3] Wolfgang Dreyer, Clemens Guhlke, Robert Huth: The behavior of a many-particle electrode in a lithium-ion battery. Physica D 240(12), 1008–1019(2011). doi:10.1016/j.physd.2011.02.011

[4] Wolfgang Dreyer, Miran Gaberscek, Clemens Guhlke, Robert Huth, Janko Jam-nik: Phase transition in a rechargeable lithium battery. European Journal ofApplied Mathematics 22(3), 267–290 (2011). doi:10.1017/S0956792511000052

[5] Wolfgang Dreyer, Clemens Guhlke, Michael Herrmann: Hysteresis and phasetransition in many-particle storage systems. Continuum Mechanics and Ther-modynamics 23(3), 211–231 (2011). doi:10.1007/s00161-010-0178-1

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Figure 9: Kinetics of LFP battery filling, as a function of parameters ν2 and τ .There are three distinct regimes of behaviour. At high value of τ (marked A),the entire system fills simultaneously, and the profile appears similar to the single-particle system. At high values of ν2 (marked B), there is no observed barrier toinitiate filling of particles, leading to a simple profile without a hysteresis, similarto a homogeneous two-phase system. For low values of both τ and ν2 (marked C),we observe a hysteretic behaviour of the system, where particles fill and empty one-by-one on different plateaus maintained by energy barrier needed to initiate phasetransition in a single particle. Taken from: [5]

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Figure 10: Kinetics of rubber balloon filling, as a function of parameters ν2 and τ .Again we observe the same three regimes of behaviour as in Figure 9, but due tosmaller number of balloons in experimental setting the observed hysteresis is actuallysomewhere between regimes A and C, even though numerical parameters correspondto regime C. Taken from: [5]

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