The equations of the course

Everything below is the equation behind a figure above, in the form Bard, Faulkner and White give it. Quantities: q, q′ charges (C); r distance (m); ε₀ = 8.854 × 10⁻¹² C² N⁻¹ m⁻²; e = 1.602 × 10⁻¹⁹ C; F = 96 485.3 C/mol.

F = qq′ / (4πε₀r²)Coulomb’s law; positive means repulsion (figure 0.1)

E = F / q′ = q / (4πε₀r²)the field of a point charge: force per unit charge, pointing away from a positive q

φ(r) = −∫_∞^r E·dl = q / (4πε₀r)the potential: work per unit charge from infinity, path-independent (figure 0.2)

φ(B) − φ(A) = −∫_A^B E·dla potential difference is the field integrated along any path; in one dimension, E = −dφ/dx: the field is the slope of the potential

q_enclosed = ε₀ ∮ E·dSGauss’s law; with E = 0 inside a conductor at rest, the enclosed charge is zero and the excess charge sits on the surface

E = V / lthe uniform field between two parallel plates a distance l apart (figure 0.3)

ΔU = q Δφenergy change of a charge moved across Δφ; for q = e and Δφ = 1 V, ΔU = 1 eV = 1.602 × 10⁻¹⁹ J

F = N_A e = 96 485.3 C/molthe Faraday constant: one electron-volt per electron is F × 1 V = 96.5 kJ per mole

i = dQ/dtcurrent, in amperes = coulombs per second (figure 0.4); at 1 A, 1/e = 6.24 × 10¹⁸ electrons pass each second

V = IR, R = ρl/A, G = 1/R = κA/lOhm’s law; resistance grows with length and falls with cross-section; ρ resistivity, κ conductivity

P = IVpower delivered, in watts

|z|eE = 6πηrv ⇒ u = |z|e / (6πηr), v = uEan ion of charge number z in a liquid of viscosity η: the electric force balances the Stokes drag at the terminal velocity v; u is the mobility (figure 0.5)

κ = F Σ_j |z_j| u_j C_j, t_j = |z_j|u_jC_j / Σ_k |z_k|u_kC_kthe conductivity of an electrolyte adds up every ion’s charge, mobility and concentration; t_j, the transference number, is the share of the current that ion j carries

J_j = −D_j ∇C_j − (z_jF/RT) D_j C_j ∇φthe Nernst–Planck equation: an ion’s flux has a diffusion part (down the concentration gradient) and a migration part (along the field); with D_j = u_jRT/(|z_j|F) the two pictures agree. The modules on ion transport, still to come, are built on it.

Zn → Zn²⁺ + 2e⁻  Cu²⁺ + 2e⁻ → Cuthe two half-reactions; neither can run alone

Zn + Cu²⁺ → Zn²⁺ + Cuthe cell reaction, their sum; n = 2 electrons per zinc atom

E_cell = E°(Cu²⁺/Cu) − E°(Zn²⁺/Zn) = 0.340 − (−0.763) = 1.103 Vopen-circuit voltage at unit activities, from the standard potentials against the hydrogen electrode (figure 1.1)

E = φ^Cu′ − φ^Cu = Σ Δφ_interfacesthe voltage between two like leads is the sum of the potential steps met along the path through the cell: the two metal|liquid jumps plus the metal|metal contact potentials; only the sum is measurable (the strip under figure 1.1)

ΔG = −nFE = −2 × 96 485 × 1.103 = −213 kJ/molthe electrical work a mole of the reaction can do (chapter 3 derives it)

V_series = Σ V_i, Q_series = Qcells in a chain: potential differences along a path add; the same current threads every cell, so each coulomb passes all of them (figure 1.3)

V_parallel = V, I = Σ I_i, Q_parallel = Σ Q_icells side by side: one potential difference between the joined ends; the current, and therefore the charge, is shared

R_ion = l / (κA)the ionic resistance of the electrolyte layer between the electrodes: thin (small l) and wide (large A) is the whole design brief of figure 2.2. Winter and Brodd give κ of order 1 S/cm for aqueous and 10⁻² to 10⁻³ S/cm for the organic electrolytes of lithium cells.

κ_eff = κ ε^α, α ≈ 1.5inside a porous electrode only the pore fraction ε (about 0.3, figure 2.4) carries ions, along winding paths; the Bruggeman relation gives the effective conductivity of the electrolyte in the pores. Boz and co-workers quote α = 1.5, and 1.53 measured for LiFePO₄ electrodes.

D_eff = D ε/τ, τ = ε^1−αthe same correction for diffusion, with τ the tortuosity of the pore network

With ε = 0.3 and α = 1.5, κ_eff = 0.3^1.5 κ = 0.16 κ: the electrolyte inside a coating conducts about a sixth as well as the free liquid. The module on composite electrodes, still to come, builds on this.

μ̄_j = μ_j + z_jFφthe electrochemical potential of species j: its chemical potential plus its charge times the local potential (per mole)

μ_j = μ°_j + RT ln a_jthe chemical potential in terms of the activity a_j (roughly, concentration relative to a standard state)

μ̄_j^α = μ̄_j^βequilibrium of species j between phases α and β; the condition that stops the charge transfer described in figure 3.3

μ̄_e = μ°_e − Fφ, E_F = μ̄_e / N_Aelectrons in a metal (no activity term: their concentration never changes appreciably); the Fermi energy is the electrochemical potential of the electrons, per electron (figure 3.2). Making φ more negative by 1 V raises E_F by 1 eV.

φ^Zn − φ^Cu = (μ°_e^Zn − μ°_e^Cu) / Ftwo metals in contact exchange electrons until μ̄_e is equal: the contact potential

−nFE = n(μ̄_e^Cu′ − μ̄_e^Cu)write the equilibrium of the whole cell reaction in electrochemical potentials; the φ terms cancel everywhere except at the two copper leads, which is why only the whole-cell voltage is measurable

E = E° − (RT/nF) ln (a_products / a_reactants)expanding the chemical potentials gives the Nernst equation; at 25 °C, RT/F = 25.7 mV, so a tenfold change in an activity ratio shifts a one-electron E by 59 mV

V_OC = (μ_A − μ_C) / ethe same statement per electron, in Goodenough and Park’s notation: the open-circuit voltage is the difference in electron energy of the two electrodes (figures 3.4 and 3.5)

E = E° − 0.059 (q/n) pHa half-reaction with q protons per n electrons shifts 59 mV per pH unit; for water, E = 0 − 0.059 pH (hydrogen) and E = 1.229 − 0.059 pH (oxygen), so the window is 1.229 V wide at every pH and sits at 3.045 to 4.274 V versus lithium at pH 0 (figure 3.4)

ΔG = −nFE, ΔS = nF(∂E/∂T)_P, ΔH = ΔG + TΔSthe thermodynamics of the cell reaction (figure 3.7); the temperature coefficient of the voltage gives the entropy change, which a later module turns into reversible heat

q = C·E, C_H = εε₀/dthe double layer as a capacitor (figure 3.3); in the Helmholtz picture two sheets of charge a distance d apart in a medium of dielectric constant ε; measured values 10 to 40 µF/cm²

Q = nFN, m = ItM / (nF)Faraday’s law: the charge for N moles of reactant that give n electrons each; turned round, the mass transformed by a current I for a time t

Q_spec = nF / (3.6 M) mAh/gthe theoretical specific capacity of a material of molar mass M (g/mol); 3.6 converts coulombs per gram to milliampere-hours per gram (figure 4.1). Graphite as C₆: 372; LiFePO₄: 170; lithium metal: 3861.

energy = ∫₀^Δt IV dt = ∫₀^Q V(q) dq = 〈V〉 Qthe energy delivered is the area under the discharge curve, or the average voltage times the capacity (figure 4.2)

P = IV, specific energy = energy / mass, specific power = P / massthe two axes of the Ragone plot (figure 4.5), in Wh/kg and W/kg

η_Coulombic = 100 × Q_dis / Q_chthe share of the charge put in on charging that comes back on discharge; a later module follows it cycle by cycle

η_energy = 100 × ∫V_dis dq / ∫V_ch dqthe share of the energy that comes back; lower than the Coulombic efficiency because the charge branch sits above the discharge branch

$Q={\displaystyle \int I\text{d}t=It}$
$I_{1\text{C}}={\displaystyle \frac{Q_{\mathrm{n}\mathrm{o}\mathrm{m}}}{1\text{h}}}$charge passed at constant current, and the 1C current (figure 5.1)

$V_{\mathrm{d}\mathrm{i}\mathrm{s}}=V_{\mathrm{O}\mathrm{C}}-\eta (q,I_{\mathrm{d}\mathrm{i}\mathrm{s}})$
$V_{\mathrm{c}\mathrm{h}}=V_{\mathrm{O}\mathrm{C}}+\eta (q,I_{\mathrm{c}\mathrm{h}})$the terminal voltage on discharge and on charge; $\eta$ depends on the state of charge $q$ and on the current (figures 5.2, 5.3)

$\eta =E_{\mathrm{O}\mathrm{C}\mathrm{V}}-E_{\text{T}}={\eta }_{\mathrm{o}\mathrm{h}\mathrm{m}}+{\eta }_{\mathrm{a}\mathrm{c}\mathrm{t}}+{\eta }_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}}$the cell polarization, the open-circuit voltage minus the voltage under load, in three parts

${\eta }_{\mathrm{o}\mathrm{h}\mathrm{m}}=I{R}_{\text{b}}$
$V=V_{\mathrm{O}\mathrm{C}}\mp I{R}_{\text{b}}$ohmic polarization, with $R_{\text{b}}$ the internal resistance

$i=i_0[e^{-\alpha f\eta }-e^{(1-\alpha )f\eta }],f={\displaystyle \frac{F}{RT}}$
$\alpha =\frac{1}{2}:\eta =-{\displaystyle \frac{2RT}{F}}arsinh{\displaystyle \frac{i}{2i_0}}$Butler-Volmer for one electrode, reduction current positive, and its inverse for $\alpha =0.5$; its magnitude is the activation part used in figure 5.3

$R_{\mathrm{c}\mathrm{t}}={\displaystyle \frac{RT}{F{i}_0}}$
$|\eta |={\displaystyle \frac{RT}{\alpha F}}\ln {\displaystyle \frac{i}{i_0}}$the charge-transfer resistance near equilibrium, and the Tafel limit (reduction branch; 1 − α for oxidation) for $|\eta |$ above about 118 mV at 25 °C

${\eta }_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}}={\displaystyle \frac{RT}{nF}}\ln {\displaystyle \frac{C_0}{C}}$concentration polarization when the concentration at the electrode falls from $C_0$ to $C$ (Winter and Brodd print it without $F$; the units need it)

$U=U_{\mathrm{e}\mathrm{q}}$
$U=U_{\mathrm{r}\mathrm{e}\mathrm{f}}+{\displaystyle \frac{q}{C_{\mathrm{s}\mathrm{p}}}}$the ideal battery material and the ideal capacitor material (Olson and co-workers; figure 6.1)

$v(t)=R_{\text{H}}{I}_0+R_{\text{i}}{I}_0(1-e^{-t/{\tau }_{\text{i}}})+{\displaystyle \frac{I_{0}t}{C}}$
$v_{\text{D}}(t)=v(t)-2u(t-t_0)v(t-t_0)$Moya’s double-layer capacitor charged at $I_0$ (his eq. 6), then discharged at $-I_0$ from $t_0$ (eq. 12, $u$ the unit step); ${\tau }_{\text{i}}=R_{\text{i}}{C}_{\text{i}}$, $R_{\mathrm{E}\mathrm{S}\mathrm{R}}=R_{\text{H}}+R_{\text{i}}$, $C_{\text{i}}\approx 2C/7$ (eq. 33)

$F=C-P+2$the Gibbs phase rule: two phases of two components at fixed temperature and pressure leave no freedom, so a plateau (figure 6.2)

${\Phi }_{1,k}={\Phi }_1-R_{\text{c},k}{i}_k$
$I=L{\sum }_k{a}_k{i}_k$
$D_k={\displaystyle \frac{1.184\times {10}^{-18}}{(1+{\overline{y}}_k{)}^{1.6}}}{\mathrm{m}}^2/\mathrm{s}$Safari and Delacourt’s resistive reactant (their eqs. 7, 8 and 12): each particle group $k$ sees the matrix potential less its contact drop, the groups share the current, and diffusion slows as the particle fills (figure 6.3)

$S_{\text{c}}={\displaystyle \frac{R_{\text{s}}^{2}I}{D_{\text{s}}F(1-\epsilon )c_{\text{T}}{\delta }_{\text{c}}}}$
$S_{\text{e}}={\displaystyle \frac{({\delta }_{\text{s}}+{\delta }_{\text{c}}{)}^{2}I}{DF(1-\epsilon )c_{\text{T}}{\delta }_{\text{c}}}}$Doyle, Fuller and Newman’s ratios of the times for diffusion in the solid and transport in the electrolyte to the time of discharge (eqs. 26 and 27; figure 11.6)

$V=U_{\mathrm{p}\mathrm{o}\mathrm{s}}-U_{\mathrm{n}\mathrm{e}\mathrm{g}}$
${\displaystyle \frac{\text{d}V}{\text{d}Q}}={\displaystyle \frac{\text{d}{U}_{\mathrm{p}\mathrm{o}\mathrm{s}}}{\text{d}Q}}-{\displaystyle \frac{\text{d}{U}_{\mathrm{n}\mathrm{e}\mathrm{g}}}{\text{d}Q}}$
${\displaystyle \frac{\text{d}Q}{\text{d}V}}={\left({\displaystyle \frac{\text{d}V}{\text{d}Q}}\right)}^{-1}$a cell’s voltage and differential voltage from its two electrodes, and the differential capacity (figure L1.8)

$1-x_{\mathrm{p}\mathrm{o}\mathrm{s}}=a{x}_{\mathrm{n}\mathrm{e}\mathrm{g}}+b$
${\displaystyle \frac{a}{a_0}}={\displaystyle \frac{1-\mathrm{L}\mathrm{A}{\mathrm{M}}_{\mathrm{N}\mathrm{E}}}{1-\mathrm{L}\mathrm{A}{\mathrm{M}}_{\mathrm{P}\mathrm{E}}}}$
$1-b={\displaystyle \frac{1-\mathrm{L}\mathrm{L}\mathrm{I}}{1-\mathrm{L}\mathrm{A}{\mathrm{M}}_{\mathrm{P}\mathrm{E}}}}$the charging line on the square plot and how ageing moves it, for a fresh cell with $b_0=0$ (Olson and co-workers, eqs. 23, 48 and 49)

$\mathrm{C}\mathrm{E}=100\times {\displaystyle \frac{Q_{\mathrm{d}\mathrm{i}\mathrm{s}}}{Q_{\mathrm{c}\mathrm{h}}}}$
$\text{retention after }N\text{ cycles}={\mathrm{C}\mathrm{E}}^N\text{(idealized)}$Coulombic efficiency per cycle, and the idealized retention of figure 16.2

${\epsilon }_{\text{E}}=100\times {\displaystyle \frac{\int V_{\mathrm{d}\mathrm{i}\mathrm{s}}\text{d}q}{\int V_{\mathrm{c}\mathrm{h}}\text{d}q}}$the storage efficiency: the energy back over the energy in

$\dot{Q}=I(U-V)-IT{\displaystyle \frac{\text{d}U}{\text{d}T}}$
$\Delta S=nF{\displaystyle \frac{\text{d}E}{\text{d}T}}$heat generation, $I$ positive on discharge: the irreversible term and the reversible, entropic term (figure 5.5); the entropy of reaction from the temperature coefficient of the voltage

Li_xC₆, 0 ≤ x ≤ 1; Li_1−yCoO₂, y ≤ 0.5 in practicehow much lithium each host holds: up to one per six carbons, and about half the lithium of LiCoO₂ used in practice

2 EC + 2 Li⁺ + 2 e⁻ → Li₂EDC + C₂H₄↑the main SEI-forming reduction of ethylene carbonate, from about 0.8 V vs Li/Li⁺; the ethylene is among the gases let out during formation

LiCoO₂ + C₆ ⇌ Li_1−xCoO₂ + Li_xC₆the cell reaction, charge to the right: the carbon is made empty because Li_xC₆ is very reactive, and LiCoO₂ is the only source of lithium

N = Q/e = nN_A, n = Q/Fthe number of ions, and of moles, that carry a charge Q (worked example)

J = −D ∂C/∂x, Δ = (2Dt)^1/2, t ≈ L²/2DFick’s first law; the diffusion length in a time t; and, turned round, the order of magnitude of the time to cross a particle of size L

eV_OC = μ_A − μ_C, stable if μ_A < LUMO and μ_C > HOMOthe open-circuit voltage from the two electrodes’ electrochemical potentials, and the condition for the electrolyte to survive without a passivating layer

N_solvent/N_Li = (1/c) Σ φ_kρ_k/M_k, d = (1/(cN_A))^1/3solvent molecules per lithium ion, for volume fractions φ_k of solvents of density ρ_k (g/L) and molar mass M_k (g/mol), salt concentration c (mol/L) (salt volume neglected); the mean spacing of the ions (figure 8.2)

E = i/κ, v̄ = i/(Fc)the field in the bulk liquid from Ohm’s law; the mean velocity of the lithium ions once all the current is carried by lithium (worked example of figure 9.1)

κ = F Σ|z_j|u_jC_j, R = l/(κA)conductivity from the ions; the resistance of a layer of thickness l and area A (worked example)

t_j = |z_j|u_jC_j / Σ|z_k|u_kC_k, t₊ = D_Li/(D_Li + D₋) (dilute)the transference number, and its dilute-solution form

J_j = −D_j ∇C_j − (z_jF/RT) D_j C_j ∇φ, i = F Σ z_jJ_jNernst-Planck, as in chapter 0: each ion’s flux by diffusion and by migration in the field; the current density from the fluxes

Δc = (1 − t₊) i L / (F D)the steady-state salt difference across a layer of thickness L at current density i (figure 9.2)

$\epsilon =1-{\displaystyle \frac{m}{Ad}}{\sum }_i{\displaystyle \frac{w_i}{{\rho }_i}}$
${\kappa }_{\mathrm{e}\mathrm{f}\mathrm{f}}=\kappa {\displaystyle \frac{\epsilon }{\tau }}={\displaystyle \frac{\kappa }{N_{\text{M}}}}$
$\tau ={\displaystyle \frac{R_{\mathrm{i}\mathrm{o}\mathrm{n}}A\kappa \epsilon }{2d}}$porosity from the areal mass $m/A$ of the coating, its thickness $d$, and the mass fraction $w_i$ and density ${\rho }_i$ 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 $\tau ={\epsilon }^{-0.5}$, ${\kappa }_{\mathrm{e}\mathrm{f}\mathrm{f}}=\kappa {\epsilon }^{1.5}$ (their eq. 6), used in the model of figure 11.6.

κ_eff = κε^α, D_eff = Dε/τ, τ = ε^1−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)/R_ctwhat 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)

Q_pair = Q_posQ_neg/(Q_pos + Q_neg), E = Q_pair(V_pos − V_neg)specific energy of the two active materials (figure 12.1); as Q_neg → ∞, Q_pair → Q_pos

Li⁺ + e⁻ + xC ⇌ LiC_x; xLi⁺ + xe⁻ + M ⇌ Li_xM; M_xO_y + 2yLi⁺ + 2ye⁻ ⇌ yLi₂O + xMintercalation, alloying and conversion (figure 7.3)

Q_blend = (1 − w) × 372 + w × 3579 mAh/ga graphite/silicon blend with a mass fraction w of silicon (figure 12.3)

d_after/d_before = (1 + ΔV/V)^1/3how a volume change becomes a change of size: 3.8^1/3 = 1.56 for 280 %

η_E ≈ (V_pos − V_neg − h/2)/(V_pos − V_neg + h/2)round-trip energy efficiency with a hysteresis h on the negative electrode (figure 12.4)

C_n = CEⁿ (no excess); R = N(1 − E)capacity of an anode-free cell after n cycles, and Brandt’s excess needed for N cycles (figure 12.6)

ΔV = R_int ivoltage lost across an interface of area-specific resistance R_int at current density i (figure 14.4)

V_cell = V_pos − V_neg (both vs Li/Li⁺)reading a cell voltage off figure 12.1, as on the ladder of chapter 3

E_material = Q × V̄energy per gram of active material: 350 mAh/g × 3.0 V = 1050 mWh/g, about 1 kWh/kg (an oximate, figure 15.3); a cell holds much less once everything else is counted

retention = Q₃₀₀/Q₅capacity kept between cycle 5 and cycle 300 (figure 13.1)

L_SEI ∝ t^b, ΔQ ∝ t^b, b ∈ {0, 1/2, 1}SEI growth and the capacity it costs; b = 1/2 diffusion-limited, 1 reaction-limited or migration-limited on charge, 0 migration-limited on discharge

ΔQ(t₂) = ΔQ(t₁) (t₂/t₁)^bscaling a measured loss forward in time (worked example)

Q_{low rate} − Q_{high rate} = capacity hidden, not lostthe reversible part of figure 16.1, recovered by measuring slowly

E = Q ⟨V⟩the energy a short circuit can turn into heat: all the charge stored times the average voltage (chapter 4); a 3 Ah cell at 3.7 V holds 11.1 Wh, that is 40 kJ

Q̇ = I(U − V) − I T dU/dTthe heat a cell makes in normal use (chapter 5, figure 5.5); in runaway the side reactions add their own heat on top

L_SEI ∝ t^bthe ageing law of chapter 16 behind the advice to avoid long periods at full charge

ΔV/I = R₀ + Σ R_k(1 − e^−t/R_kC_k) + 2σ√(2t/π)the pulse resistance of figure L3.1: ohmic, R‖C pairs and a Warburg element of coefficient σ

D = (4/πτ)(m_BV_M/M_BS)²(ΔE_s/ΔE_t)²; for spheres the bracket is r/3Weppner–Huggins, valid for Dτ/L² below 0.25 (flat) or about 0.025 (spheres)

f(τ̂) = 3τ̂ + 1/5 − 2Σ e^−α_n2τ̂/α_n², with α_n = tan α_nsurface change of a sphere under constant flux, τ̂ = Dt/r²; 2√(τ̂/π) at short times (Nickol et al.)

V − V_eq ∝ √(t + τ) − √tthe rest after a pulse of length τ, t the time at rest; its slope gives D (Kang and Chueh)

Z_R = R, Z_C = 1/(jωC), Z_L = jωL, Z_CPE = 1/(Y₀(jω)ⁿ)impedances in series add; admittances in parallel add

R‖C: Z = R/(1 + jωRC), top at ω = 1/(RC)a semicircle of diameter R; two arcs resolve for τ₁ ≳ 100τ₂

Z_W = σω^−1/2(1 − j); tanh(B√(jω))/(Y₀√(jω)); coth(B√(jω))/(Y₀√(jω))semi-infinite, open (transmissive) and closed (reflective) diffusion

Z_d,3D = (R_d/3) tanh x/(x − tanh x), x = √(jωR_dC_d), D = r²/(R_dC_d)diffusion into a sphere (Abbas et al. eq. 6); DC limit R_d/15

Z = √(R_mz) coth √(R_m/z), z = R_ct‖C_dlthe porous electrode as a transmission line (Bisquert model)

i = i₀[e^{(1−α)Fη/RT} − e^−αFη/RT], R_ct = RT/(Fi₀)Butler–Volmer and its linear limit; harmonics measure the curvature

Z(ω) = R₀ + R_pol∫g(τ)/(1 + jωτ)dτthe distribution of relaxation times, found by regularized least squares