2.1.1 Double layers and charge neutrality

It has long been recognised that (see e.g. [Reference Newman and Thomas-Alyea69]), at the concentrations typically encountered in practical electrolytes, there is almost exact charge neutrality. This implies that there is a balance between the concentrations of positive and negative charges, throughout the vast majority of the electrolyte. The exception to this rule is in the so-called double layers which lie along the boundaries of the electrolyte region and are typically extremely thin, with widths less than a few nanometres. This observation can be justified mathematically by non-dimensionalising equations (2.2), (2.3) and (2.5) and conducting a boundary layer analysis in terms of the small dimensionless parameter which measures the ratio of the Debye length (i.e. the typical width of a double layer) to the typical dimension of the electrolyte.Footnote ¹ The result of such an analysis (see, e.g. [Reference Richardson78]) is that, with the exception of the double layers, $c_n$ must be almost exactly equal to $c_p$ . The physical meaning of this fact is that the attraction between charges is very strong compared to any space charge that the electric field may create. Therefore, a very good approximation to (2.2) is

(2.6) \begin{eqnarray}0= c_n - c_p,\end{eqnarray}

which is usually called the charge neutrality condition. As we shall discuss later, because (2.6) has neglected the derivatives that were in (2.2), the model needs fewer boundary conditions at the edges of the electrolyte (i.e. only two boundary conditions are required on the electrolyte ‘surface’ rather than the three needed for the full system).

2.1.2 The approximate equations

In line with the discussion above, we introduce a single concentration c by taking

(2.7) \begin{eqnarray}c_n=c_p=c, \end{eqnarray}

and substitute this into (2.3) and (2.5) to obtain the approximate charge-neutral equations:

(2.8a) \begin{eqnarray}\frac{\partial c}{\partial t} + \nabla \cdot \textbf{q}_n=0, \quad \mbox{and} \quad \textbf{q}_n= -D_n \left( \nabla c - \frac{F}{R T} c \nabla \phi \right), \end{eqnarray}

(2.8b) \begin{eqnarray}\frac{\partial c}{\partial t} + \nabla \cdot \textbf{q}_p=0, \quad \mbox{and} \quad \textbf{q}_p=-D_p\left( \nabla c + \frac{F}{R T}c \nabla \phi \right). \end{eqnarray}

A common approach to studying this problem is to assume that the ionic diffusivities $D_n$ and $D_p$ are constant and rewrite the system by adding the former equation in (2.8a) multiplied by $D_p$ to the former equation in (2.8b) multiplied by $D_n$ . On substituting for $\textbf{q}_n$ and $\textbf{q}_p$ , this yields a diffusion equation for c, of the form:

(2.9) \begin{eqnarray}\frac{\partial c}{\partial t} - {D_{\rm eff}}\nabla^2 c = 0\quad \mbox{with} \quad {D_{\rm eff}}=2\frac{D_n D_p}{D_n+D_p},\end{eqnarray}

where ${D_{\rm eff}}$ is termed the effective ionic diffusivity.

Figure 2. (a) Diffusion coefficient ${D_{\rm eff}}$ for LiPF $_{6}$ in 1:1 EC:DMC at $T=293$ K as a function of concentration and (b) electrolyte conductivity $\kappa$ for the same electrolyte. Lines represent the fit to to the experimental data (circles) taken from [Reference Valøen and Reimers98]. The fitted functions for the diffusivity and conductivity are given by ${D_{\rm eff}}(c)=5.3\times 10^{-10} \exp(-7.1\times 10^{-4}c)$ and $\kappa(c)=10^{-4} c (5.2-0.002c+2.3\times10^{-7}c^2)^2$ , respectively. We note that an exponential fitting function is ubiquitous throughout the literature and can be found in numerous other sources, for example, [Reference Stewart and Newman93].

Useful physical insight can be found using an alternative formulation of (2.8a) and (2.8b) by introducing the electric current density, $\textbf{j}$ , defined in terms of the ion fluxes by:

(2.10) \begin{eqnarray}\textbf{j}=F(\textbf{q}_p-\textbf{q}_n).\end{eqnarray}

Using this concept, a version of Ohm’s law may be obtained by subtracting (2.8a b) from (2.8a b), while a charge conservation equation may be found by subtracting (2.8aa) from (2.8b a). These may be written in the form:

(2.11a) \begin{eqnarray}{\textbf{j}=-\hat{\kappa}(c) \left[\nabla \phi - \frac{RT}{F} (1- 2t_+) \frac{\nabla c}{c} \right],}\end{eqnarray}

(2.11b) \begin{eqnarray}{\nabla \cdot \textbf{j}=0,}\end{eqnarray}

(2.11c) \begin{eqnarray} \mbox{where} \quad \hat{\kappa}(c)=\frac{F^2}{R T}(D_n+D_p) c \quad \mbox{and} \quad t_+ =\frac{D_p}{D_n+D_p}.\end{eqnarray}

Here $t_+$ is referred to as the transference number and $\hat{\kappa}(c)$ is referred to as the electrical conductivity of the electrolyte (c.f. the standard form of Ohm’s law is $\textbf{j}=-\kappa \nabla\phi$ ). We use the notation $\hat{\kappa}$ to distinguish the electrical conductivity in the dilute limit from the same quantity in moderately concentrated theory, see (2.43). The alternative formulation of (2.8a) and (2.8b) mentioned above is then (2.9), (2.11a) and (2.11b).

Assuming that the ionic diffusivities $D_n$ and $D_p$ are constant implies (i) that transference number $t_+$ is constant, (ii) the effective ionic diffusivity ${D_{\rm eff}}$ is constant and (iii) electrolyte conductivity $\kappa$ grows linearly with electrolyte concentration c. All three of these quantities are readily measured experimentally for real electrolytes. For most electrolytes, the transference number is usually found to remain close to a constant (with the exception of some polymer electrolytes e.g. [Reference Doeff, Edman, Sloop, Kerr and De Jonghe24, Reference Fauteux32]), electrolyte diffusivity usually decreases relatively weakly with concentration, except at very dilute concentrations, but the growth of electrical conductivity with concentration is far from linear. Examples of the experimentally measured concentration dependence of ${D_{\rm eff}}$ and $\kappa$ (from [Reference Valøen and Reimers98]) are plotted in Figure 2 for the battery electrolyte LiPF $_6$ . Notably at the typical concentrations used in batteries (roughly 1 molar for LiPF $_6$ ), electrical conductivity is nearly constant and often lies close to its maximum value and is thus not well approximated by the linear expression in (2.11c). The explanation given for this poor fit is usually that even at relatively dilute concentrations, there is a significant drag between ions of opposite charge. The reason for this is that two ions of opposite charge that lie close to each other experience a significant electrostatic attraction that negates, to a large extent, the effects of the global electric field which is trying to drive the ions in opposite directions. This observation has motivated Newman [Reference Newman and Thomas-Alyea69] to use Stefan–Maxwell theory for a multi-component solution to describe the charge transport behaviour of electrolytes. A summary of the modelling assumptions made in this theory is given in Section 2.2.

2.1.3 The dilute theory in terms of electrochemical potentials

The dilute model (2.8a)–(2.8b) can also be written in terms of the electrochemical potentials, $\bar{\mu}_n$ and $\bar{\mu}_p$ , of negative and positive ions, respectively. This is the preferred notation for the ion conservation equations in the electrochemical literature. For an electrolyte that is formed by ideal salt solution, the electrochemical potentials are given by:

(2.12) \begin{eqnarray}\bar{\mu}_n=\mu_n^0+ RT \log \left( \frac{c}{c_T} \right) - F \phi, \qquad \bar{\mu}_p=\mu_p^0+ RT \log \left( \frac{c}{c_T} \right) +F \phi,\end{eqnarray}

where $c_T$ is the total molar concentration of the electrolyte and, for a dilute solution, is approximately equal to the solvent concentration. The first term on the right-hand sides of both expressions in (2.12) is the standard state potential (per mole of the species), while the second is the entropy of mixing (per mole of the species) and the final term is the electrostatic potential (per mole of the species). Using this notation, the conservation equations (2.8a) and (2.8b) can be written in the form:

(2.13a) \begin{eqnarray}\frac{\partial c}{\partial t} + \nabla \cdot (c \textbf{v}_n)=0, \quad \mbox{and} \quad\textbf{v}_n= -\frac{D_n}{RT} \nabla \bar{\mu}_n,\end{eqnarray}

(2.13b) \begin{eqnarray}\frac{\partial c}{\partial t} + \nabla \cdot (c \textbf{v}_p)=0, \quad \mbox{and} \quad\textbf{v}_p=-\frac{D_p}{RT} \nabla \bar{\mu}_p.\end{eqnarray}

Here the average velocities of the two species, $\textbf{v}_n$ and $\textbf{v}_p$ , are obtained by multiplying the gradient of the electrochemical potentials by the species mobilities, ${D_n}/{RT}$ and ${D_p}/{RT}$ . This formalism extends to non-ideal salt solutions and to multicomponent systems. In Section 2.2, this approach of using electrochemical potentials is extended to moderately concentrated electrolytes.

2.1.4 The potential measured with respect to a lithium electrode

The dilute theory as formulated above is at odds with the electrolyte theory used by [Reference Newman and Thomas-Alyea69]; here, the factor in front of the concentration gradient in (2.11a), the constitutive law for the current, is $2({RT}/{F}) (1- t_+)$ rather than $({RT}/{F}) (1- 2 t_+)$ as above. As pointed out in [Reference Ramos76], this has generated some confusion in the literature. The explanation for this discrepancy (as initially demonstrated by [Reference Ranom77] and subsequently in [Reference Bizeray, Howey and Monroe8]) is that the theory in [Reference Newman and Thomas-Alyea69] is formulated in terms of $\varphi$ , the electric potential measured with respect to a reference lithium electrode, rather than $\phi$ , the true electric potential. In electrochemical applications, the potential in an electrolyte is typically measured by inserting a reference electrode of a pure compound. The potential measured depends on the composition of the reference electrode through its chemical potential. Since in lithium battery applications, the reference electrode used is nearly always made of lithium, and since much of the data used to calibrate battery models is collected using a lithium reference electrode, it makes sense to use the potential measured with respect to a lithium electrode. Note that $\phi$ , the true electric potential, is not a readily measured quantity. In order to switch between $\varphi$ and $\phi$ , we recall that there is a reversible reaction that occurs on the surface of the electrode between intercalated lithium in the electrode and lithium ions in the electrolyte, given by:

(2.14) \begin{eqnarray}\mbox{Li}^+_{(l)} + \mbox{e}^-_{(\rm ref)} \leftrightharpoons \mbox{Li}_{(\rm ref)}, \end{eqnarray}

where, since the lithium atom in the reference electrode is neutral, its electrochemical potential is equal to its chemical potential $\mbox{Li}_{(\rm ref)}$ . Here, the subscript (l) denotes a reactant within the electrolyte and $(\rm ref)$ one within the reference electrode. Typically, the current flow into a reference electrode can be assumed to be sufficiently small that this reaction is in quasi-equilibrium. It follows that the electrochemical potentials of compounds on both sides of this equation are equal, that is

(2.15) \begin{eqnarray}\bar{\mu}_{p}+ \bar{\mu}_{\mbox{e}^-_{\rm ref}}=\mu_{{Li}_{\rm ref}}. \end{eqnarray}

Since the potential within the lithium electrode is $\varphi$ the electrochemical potential of the electrons can be expressed as $\bar{\mu}_{\mbox{e}^-_{\rm ref}}=\mu^0_{\mbox{e}^-_{\rm ref}}-F \varphi$ , where $\mu^0_{\mbox{e}^-_{\rm ref}}$ is the work function of the lithium electrode. Furthermore, the lithium concentration within the electrode does not change, so that $\mu_{{Li}_{\rm ref}}$ is fixed, and equal to $\mu^0_{{Li}_{\rm ref}}$ . The electrochemical balance (2.15) thus implies that

\begin{eqnarray*}\left(\mu_p^0+ RT \log \left( \frac{c}{c_T} \right)+F \phi \right) +\left( \mu^0_{\mbox{e}^-_{\rm ref}} - F \varphi\right)=\mu^0_{{Li}_{\rm ref}}.\end{eqnarray*}

This leads to the following expression for the electric potential $\phi$ in terms of $\varphi$ :

(2.16) \begin{eqnarray}\begin{array}{l}\displaystyle \phi=\varphi +V^0_*- \frac{1}{F} \left(\mu_p^0+ RT\log \left( \frac{c}{c_T} \right)\right),\\\displaystyle \mbox{where} \quad V^0_*=\frac{1}{F} (\mu^0_{{Li}_{\rm ref}}-\mu^0_{\mbox{e}^-_{\rm ref}}).\end{array}\end{eqnarray}

Notably, with this new notation, the electrochemical potential of the two ion species now read

(2.17) \begin{eqnarray}\bar{\mu}_p=F (\varphi+V^0_*), \qquad \bar{\mu}_n=2 \mu_e(c) - F (\varphi+V^0_*)\end{eqnarray}

where $\mu_e(c)$ , the chemical potential of the electrolyte, is defined such that

(2.18) \begin{eqnarray}\mu_e(c) =\frac{\mu_n+\mu_p}{2}, \quad \mbox{or equivalently} \quad \mu_e(c) =\frac{\mu^0_n+\mu^0_p}{2}+RT \log \left( \frac{c}{c_T} \right).\end{eqnarray}

Using (2.16) to substitute for $\phi$ in (2.11a) yields the electrolytic version of Ohm’s law found in the Newman theory:

(2.19) \begin{eqnarray}\textbf{j}=-\hat{\kappa}(c) \left[\nabla \varphi -2 \frac{RT}{F} (1- {t_+}) \frac{\nabla c}{c} \right].\end{eqnarray}

The effect of the difference in the two different potentials is now readily seen by comparing (2.19) with (2.11a). Therefore (2.19), together with (2.9) and (2.11b) is a system of equations equivalent to the charge-neutral equations (2.8a) and (2.8b).

2.2.1 Stefan–Maxwell equations

We start by briefly considering the general case in which the electrolyte is comprised of N (ionic and solvent) species. Newman and Thomas-Alyea [Reference Newman and Thomas-Alyea69] uses the Stefan–Maxwell equations as the foundation of concentrated electrolyte theory. These relate the drag force acting on a component in a mixture to its relative velocity with the other components. It is by balancing these drag forces with gradients of electrochemical potential of a species and gradients in the fluid pressure that we obtain the average velocity of each species and in turn its flux. This combined with conservation laws for each species form the equations of moderately concentrated electrolyte theory. We note that other mechanisms in addition to the interspecies drag, electrochemical potential and pressure forces can also cause mass transfer and we briefly mention these without detailed dicussion.

In order to derive the force acting on each component of the mixture as a consequence of the pressure gradient, it is neccessary first to obtain an equation of state that relates the concentrations of the N species forming the mixture. According to [Reference Liu and Monroe61] for most electrolytes, it is usually a good approximation to assume that each species has constant molar volume. This is equivalent to the assumption that the volume occupied by one mole of a given species remains fixed whatever the composition of the mixture. On denoting the molar volume of the i’th species by $V_i$ , we obtain the following equation of state relating the molar concentrations:

(2.20) \begin{eqnarray}\sum_{i=1}^N V_i c_i=1. \end{eqnarray}

As we shall see, this relation allows us to write down the force on a species arising from the pressure gradient in the mixture. We note that electrolytes may have mechanical properties, such as acting as a viscous fluid or an elastic solid, which can create additional forces. We do not consider these but they can contribute significantly in certain situations.

The mutual friction force between species i and j is assumed to be proportional to the friction forces arising from velocity differences between the species. Furthermore, this force is proportional to the mole fraction, $\chi_k$ , of each species (see [Reference Bothe11]) as defined by:

(2.21) \begin{eqnarray}\chi_k=\frac{c_k}{c_T}, \quad\hbox{ for } k=1...N, \quad \mbox{where} \quad c_T=\sum_{k=1}^N c_k.\end{eqnarray}

Here, $c_k$ is the molar concentrations of species k and $c_T$ is the total molar concentration of all species in the solution. The Stefan–Maxwell equations give a relation between $\;\;{\hat{\textbf{d}}}_i$ , the drag force exerted on species i, per unit volume of mixture, and the velocities of the various species. In light of the above comments the drag force on the i’th species (per mole unit volume) is taken to depend linearly on the velocity differences between species, and modelled (see [Reference Bothe11]) by the expression:

(2.22) \begin{eqnarray}\;\;\hat{\textbf{d}}_i=RT c_i \sum_{\stackrel{j=1}{j \neq i}}^N k_{ij}\chi_j (\textbf{v}_j-\textbf{v}_i)=RT c_T \sum_{\stackrel{j=1}{j \neq i}}^N k_{ij} \chi_i\chi_j (\textbf{v}_j-\textbf{v}_i),\end{eqnarray}

where $\textbf{v}_k$ is the velocity of species k and $RT k_{ij}$ is the drag coefficient on one mole of species i moving through pure species j. Note that $k_{ij}$ is symmetric (i.e. $k_{ij}=k_{ji}$ ) because the drag exerted on species i by species j is equal and opposite to that exerted on species j by species i. Here the Maxwell–Stefan inter-species diffusivity is related to $k_{ij}$ by the Einstein relation so that $D_{ij}=1/k_{ij}$ .

The drag force, $\hat{\textbf{d}}_i$ , is balanced by motive forces (per unit volume) down gradients in the electrochemical potential $\bar{\mu}_i$ and down gradients in the pressure p:

(2.23) \begin{eqnarray}\hat{\textbf{d}}_i-c_i \nabla\bar{\mu}_i - V_i c_i \nabla p =\textbf{0}, \quad\hbox{ for } k=1...N.\end{eqnarray}

Here, the electrochemical potentials $\bar{\mu}_i$ may be rewritten in terms of the chemical potentials ${\mu}_i$ and the electric potential $\phi$ in the standard fashion:

(2.24) \begin{eqnarray}\bar{\mu}_i={\mu}_i + z_i F \phi, \quad\hbox{ for } k=1...N, \end{eqnarray}

where $z_i$ is the valence of species i. The force balance (2.23) equations are supplemented by the standard conservation equations, which can be written as:

(2.25) \begin{eqnarray}\frac{\partial c_i}{\partial t} + \nabla \cdot ( \textbf{v}_ic_i)=0 \qquad \mbox{for} \quad i=1,\cdots,N .\end{eqnarray}

A force balance on the entire mixture can be obtained by adding together the N relations (2.23) and noting that $\sum_{i=1}^N \hat{\textbf{d}}_i=\textbf{0}$ , to give

\begin{eqnarray*}\sum_{i=1}^N c_i \nabla\bar{\mu}_i + \left(\sum_{i=1}^N V_i c_i\right) \nabla p = \textbf{0}.\end{eqnarray*}

By substituting for $\sum_{i=1}^N V_i c_i$ from the equation of state (2.20) and for the electrochemical potential from (2.24), we obtain the following expression for the total force balance:

(2.26) \begin{eqnarray}\left(\sum_{i=1}^N c_i \nabla {\mu}_i \right) + \left(\sum_{i=1}^N F z_i c_i \right) \nabla \phi +\nabla p = \textbf{0}. \end{eqnarray}

It is straightforward to show from the Gibbs–Duhem relation between the chemical potentials, namely

\begin{eqnarray*}\sum_{i=1}^N c_i \frac{\partial {\mu}_i}{\partial c_p}=0,\end{eqnarray*}

as derived in Appendix A, that, since $\mu_i=\mu_i(c_1,c_2,\cdots c_N)$ ,

(2.27) \begin{eqnarray}\sum_{i=1}^N c_i \nabla {\mu}_i = \textbf{0}.\end{eqnarray}

This result implies that gradients in the chemical potential, in isolation, do not, as might be expected, lead to a net force on the mixture. In turn this means that the pressure equation (2.26) can be simplified to

(2.28) \begin{eqnarray}\nabla p = -\rho \nabla \phi \quad \mbox{where} \quad \rho=\sum_{i=1}^N F z_i c_i . \end{eqnarray}

in which $\rho$ represents the charge density of the mixture.

It is worth making some brief comments about (2.28) which gives an intuitively appealing balance between electrostatic forces and pressure forces acting on the mixture. By taking the curl of (2.28), it is clear that it can only be satisfied if $\nabla \rho \times \nabla \phi =\textbf{0}$ (i.e. the gradient of the charge density lies parallel to the electric field $\textbf{E}=-\nabla \phi$ ). There are two special cases where this relation is automatically satisfied which are particularly relevant here. The first of these is where $\rho =0$ (or is negligible), and in this instance no pressure gradient is required to balance the electric force on the mixture and so results in a spatially uniform pressure; this is the case that applies to charge-neutral elecrolytes and so is pertinent to battery modelling. The second special case is where the problem is strictly one-dimensional when the pressure gradient simply counteracts the electrical force on the mixture. This is the same situation as occurs in one-dimensional fluid flow where incompressibility dictates that the motion is determined by the concentrations without reference to any forces and is discussed in the context of other such multiphase systems in [Reference Drew27].

In more general cases, the force balances, described above in (2.23), are too naive and need to be supplemented by multiphase viscous dissipation terms as described in [Reference Drew27]. Such issues, however, are beyond the scope of this work, but they are addressed in this context in [Reference Liu and Monroe61].

A final comment about the general moderately concentrated problem is that the electrochemical potential of the i’th species $\bar{\mu}_i$ has essentially the same form as that written down for a dilute 1:1 solute in (2.12). The only modification is that the species mole fraction $c_i/c_T$ is replaced by its activity $a_i$ to reflect the fact that the solution is non-ideal. Hence, we find

(2.29) \begin{eqnarray}\bar{\mu}_i=\mu_i^0 + RT \log(a_i) + z_i F \phi ,\end{eqnarray}

where $z_i$ is again the charge state of the i’th species.

2.2.2 Stefan–Maxwell equations for a binary 1:1 electrolyte

We now take the general theory of Section 2.2.1 and restrict attention to the case of a 1:1 electrolyte comprised of a solution of Li $^+$ ions and a generic negative counter ion species dissolved in a single solvent species. Although battery electrolytes are often based on rather complex solvent mixtures, which are usually closely guarded industrial secrets, this approach provides a reasonable description of many battery electrolytes. In Figure 2, we parameterise the model against experimental data for the most common lithium ion electrolyte LiPF $_6$ in 1:1 EC:DMC from [Reference Valøen and Reimers98], treating the two component solvent (EC:DMC) as if it they were a single solvent.

In line with [Reference Newman and Thomas-Alyea69], we denote the three species making up the electrolyte, namely the solvent, the ${\rm Li}^+$ ions and the generic counterions, by the subscripts $i=0,p,n$ , respectively. We have therefore $z_0=0$ , $z_p=1$ and $z_n=-1$ . Combining (2.21)–(2.23) and expanding in component form yields

(2.30a) \begin{eqnarray}-c_p \nabla \bar{\mu}_p= \mathcal{K}_{pn}(\textbf{v}_p-\textbf{v}_n)+\mathcal{K}_{p0}(\textbf{v}_p-\textbf{v}_0)+V_p c_p \nabla p,\end{eqnarray}

(2.30b) \begin{eqnarray}-c_n \nabla \bar{\mu}_n=\mathcal{K}_{np}(\textbf{v}_n-\textbf{v}_p)+\mathcal{K}_{n0}(\textbf{v}_n-\textbf{v}_0)+V_n c_n \nabla p,\end{eqnarray}

(2.30c) \begin{eqnarray}-c_0 \nabla \bar{\mu}_0=\mathcal{K}_{0p}(\textbf{v}_0-\textbf{v}_p)+\mathcal{K}_{0n}(\textbf{v}_0-\textbf{v}_n)+V_0 c_0 \nabla p,\end{eqnarray}

where, by using (2.21) (i.e. $c_T=c_0+c_p+c_n$ ) and (2.22)–(2.23), the drag coefficients can be expressed in the form:

(2.31) \begin{eqnarray}\mathcal{K}_{ij}=RT\frac{c_i c_j}{c_T D_{ij}} = \mathcal{K}_{ji}, \qquad \mbox{with} \quad i=0,p,n,\quad j=0,p,n.\end{eqnarray}

We note that the three equations (2.30a)–(2.30c) are not independent, as can be seen by adding the three equations together. The result of this operation is an equation that can be shown to be always true by appealing to the Gibbs–Duhem equation, see for example [Reference Atkins, De Paula and Keeler2, Reference Newman and Thomas-Alyea69].

Henceforth, we can omit (2.30c), noting that the choice of physically realistic functions $\bar{\mu}_k$ (with $k=0,n,p$ ) ensures that this is satisfied. Using (2.29), the electrochemical potentials of the three species are

(2.32) \begin{eqnarray} \bar{\mu}_{n}&=&\mu_n^0 + RT \log(a_n) - F\phi, \qquad \bar{\mu}_p=\mu_p^0 + RT \log(a_p) + F\phi,\end{eqnarray}

(2.33) \begin{eqnarray} \bar{\mu}_0&=&\mu_0^0 +RT \log(a_0). \end{eqnarray}

Note that the pressure terms can be incorporated into the electrochemical potentials yielding modified chemical potentials (denoted by a prime) that read

(2.34) \begin{eqnarray}\bar{\mu}'_{\!\!n}&=&\mu_n^0 +V_n p+ RT \log(a_n) - F\phi, \qquad \bar{\mu}'_{\!\!p}=\mu_p^0 +V_p p+ RT \log(a_p) + F\phi, \end{eqnarray}

(2.35) \begin{eqnarray} \bar{\mu}'_{\!\!0}&=&\mu_0^0 + V_0 p + RT \log(a_0). \end{eqnarray}

These (modified) electrochemical potentials are actually those that should be used in the Stefan–Maxwell equations given in [Reference Newman and Thomas-Alyea69] (i.e. equations (2.30a)–(2.30c) from which the terms involving $\nabla p$ have been removed). Indeed the definition (2.34)–(2.35) are consistent with the true definition of the chemical potential, with a variable pressure (see for example p. 163 of [Reference Atkins, De Paula and Keeler2]), for which the thermodynamic relation $\partial \mu_i/\partial p = V_i$ is satisfied. However, since most people are more familiar with the definitions (2.32)–(2.33) for solutions, we stick with these.

Equations (2.30a)–(2.30c) and (2.32)–(2.33) give a consistent description of transport within the entire electrolyte, when coupled to Poisson’s equation which enables the electric potential to be determined from the charge density. However, in practice, throughout nearly all of the electrolyte, there is almost exact charge neutrality (a consequence of Poisson’s equation) which implies that to a very good approximation:

(2.36) \begin{eqnarray}c_n= c_p= c .\end{eqnarray}

The only exception to this occurs in the double layers, around the edge of the electrolyte. Since the width of these double layers is typically less than a couple of nanometers, they are usually neglected in the context of electrolyte modelling, their effects only making themselves felt through the phenomenological Butler–Volmer boundary condition. As discussed above, where the electrolyte is charge neutral (i.e. $c_n=c_p$ ), the solution to the pressure equation (2.28) is such that p is constant meaning that the pressure gradient terms in (2.30a)–(2.30c) vanish.

We now seek to find a constitutive equation for the current density $\textbf{j}$ . We will broadly follow the derivation given in [Reference Newman and Thomas-Alyea69] but attempt to clarify their argument. As in the dilute case, the current density is given by (2.10), which can be rewritten as:

(2.37) \begin{eqnarray}\textbf{j}=Fc(\textbf{v}_p-\textbf{v}_n) .\end{eqnarray}

Substitution of (2.37) into (2.30a) and (2.30b), taking account of (2.36) and the fact that the pressure gradient terms vanish, yields

(2.38a) \begin{eqnarray}-c\nabla \bar{\mu}_p=\mathcal{K}_{p0}(\textbf{v}_p-\textbf{v}_0)+\frac{\mathcal{K}_{pn}}{Fc} \textbf{j} ,\end{eqnarray}

(2.38b) \begin{eqnarray}-c\nabla \bar{\mu}_n=\mathcal{K}_{n0}(\textbf{v}_n-\textbf{v}_0)-\frac{\mathcal{K}_{np}}{Fc} \textbf{j} ,\end{eqnarray}

where

(2.39) \begin{eqnarray}\mathcal{K}_{pn}=\mathcal{K}_{np}=\frac{RT c^2}{c_T D_{pn}}, \quad \mathcal{K}_{p0}=\frac{RT c c_0}{c_T D_{p0}}, \quad \mathcal{K}_{n0}=\frac{RT c c_0}{c_T D_{n0}} \ \ \mbox{and} \ \ c_T=(c_0+2c).\end{eqnarray}

Equations (2.37), (2.38a) and (2.38b) can be re-arranged to give expressions for the ion velocities in terms of the solvent velocity:

(2.40a) \begin{eqnarray}\textbf{v}_p=\textbf{v}_0-\frac{c_T}{RT}\frac{D_{p0}}{c_0}\nabla \bar{\mu}_p-\frac{D_{p0}}{D_{pn}F c_0}\textbf{j} ,\end{eqnarray}

(2.40b) \begin{eqnarray}\textbf{v}_n=\textbf{v}_0-\frac{c_T}{RT}\frac{D_{n0}}{c_0}\nabla \bar{\mu}_n+\frac{D_{n0}}{D_{pn}F c_0}\textbf{j} .\end{eqnarray}

Subtracting (2.40b) from (2.40a) gives

(2.41) \begin{eqnarray}\textbf{v}_p - \textbf{v}_n=\frac{c_T}{RT c_0}(D_{n0}\nabla\bar{\mu}_n-D_{p0}\nabla\bar{\mu}_p)-\frac{D_{p0}+D_{n0}}{F c_0 D_{pn}}\textbf{j}. \end{eqnarray}

On substituting for $\textbf{v}_p-\textbf{v}_n$ (in terms of $\textbf{j}$ ) from (2.37), and for $\bar{\mu}_p$ and $\bar{\mu}_n$ from (2.32), and rearranging the resulting expression, we obtain the following expression for $\textbf{j}$ :

(2.42) \begin{eqnarray}\textbf{j} = -\kappa(c) \left( \nabla \phi + \frac{RT}{F} ( t_{+}^{0} \nabla \log(a_p) - (1-t_{+}^{0}) \nabla \log(a_n)) \right),\end{eqnarray}

where

(2.43) \begin{eqnarray}t_{+}^{0}=\frac{D_{p0}}{D_{p0}+D_{n0}}, \qquad \kappa(c)=\frac{F^2 c_T D_{pn} c (D_{p0}+D_{n0})}{RT(c(D_{p0}+D_{n0})+D_{pn} c_0 )},\end{eqnarray}

are the transference number of the (positive) lithium ions with respect to the solvent velocity, and the conductivity of the electrolyte as a function of the concentration. Note how this equation for $\textbf{j}$ , and the definition of transference number and conductivity compare to the dilute version given in (2.11a)–(2.11c).

We now rewrite equations (2.40a)–(2.40b) in a form that avoids the use of a chemical potential for each ion species and instead uses a chemical potential for the entire electrolyte $\mu_e$ . Without any loss of generality, we can express $\textbf{v}_p$ and $\textbf{v}_n$ in the form:

(2.44) \begin{eqnarray}\begin{array}{l}\displaystyle \textbf{v}_p=((1-\alpha) \textbf{v}_p+\alpha \textbf{v}_n) +\alpha(\textbf{v}_p-\textbf{v}_n), \\ \displaystyle \textbf{v}_n=((1-\alpha) \textbf{v}_p+\alpha \textbf{v}_n)-(1-\alpha)(\textbf{v}_p-\textbf{v}_n),\end{array}\end{eqnarray}

for any function $\alpha$ . Our procedure is to substitute $\textbf{v}_p-\textbf{v}_n=\textbf{j}/(Fc)$ in the final terms of these expressions and then to replace $\textbf{v}_n$ and $\textbf{v}_p$ , everywhere else on the right-hand side of these expressions, from (2.40a) and (2.40b). Choosing $\alpha=t_{+}^{0}$ , as defined in (2.43), allows the electric potential $\phi$ to be eliminated from the term $(1-\alpha) \textbf{v}_p+\alpha \textbf{v}_n$ . The expressions for $\textbf{v}_p$ and $\textbf{v}_n$ , in (2.44), can then be rewritten in the form:

(2.45a) \begin{eqnarray}\textbf{v}_p=\textbf{v}_0-\frac{c_T}{RT c_0}\frac{D_{n0}D_{p0}}{D_{p0}+D_{n0}}(\nabla \bar{\mu}_p+\nabla \bar{\mu}_n)+\frac{t_{+}^{0}}{Fc}\textbf{j} ,\end{eqnarray}

(2.45b) \begin{eqnarray}\textbf{v}_n=\textbf{v}_0-\frac{c_T}{RT c_0}\frac{D_{n0} D_{p0}}{D_{p0}+D_{n0}}(\nabla \bar{\mu}_p+\nabla \bar{\mu}_n)-\frac{(1-t_{+}^{0})}{Fc}\textbf{j} .\end{eqnarray}

These equations can be further simplified by introducing the electrolyte chemical potential $\mu_e(c)$ (a function of electrolyte concentration only) and the chemical diffusion coefficient $\mathcal{D}$ , as we did in (2.9), defined by:

(2.46) \begin{eqnarray}\mu_e=\frac{\mu^0_n+\mu^0_p}{2}+RT \log((a_n a_p)^{1/2}) \quad \mbox{and} \quad \mathcal{D}=\frac{2D_{n0}D_{p0}}{D_{n0}+D_{p0}}, \end{eqnarray}

and noting that, from (2.32), $\mu_e=\frac{1}{2}(\bar{\mu}_p+\bar{\mu}_n)$ . We then find that (2.45a)–(2.45b) can be rewritten in the form:

(2.47a) \begin{eqnarray}\textbf{v}_p &=&\textbf{v}_0-\frac{c_T}{c_0 RT} \mathcal{D}\nabla \mu_e+\frac{t_{+}^{0}}{Fc}\textbf{j}, \end{eqnarray}

(2.47b) \begin{eqnarray}\textbf{v}_n &=&\textbf{v}_0-\frac{c_T}{c_0 RT} \mathcal{D}\nabla \mu_e-\frac{(1-t_{+}^{0})}{Fc}\textbf{j}.\end{eqnarray}

Notably, $D_{n0}$ and $D_{p0}$ can vary with concentration independently, without affecting the preceding analysis. It follows that $\mathcal{D}$ and $t_{+}^{0}$ may also vary independently as functions of concentration.

The remaining equations governing the behaviour come from considering conservation of the ions and solvent as well as the volume they occupy. The mass conservation equations for the two ion species are the same as (2.25), namely

(2.48) \begin{eqnarray}\frac{\partial c}{\partial t}+\nabla\cdot(c\textbf{v}_p)=0, \quad\frac{\partial c}{\partial t}+\nabla\cdot(c\textbf{v}_n)=0.\end{eqnarray}

Taking the differences of these two equations, and substituting for $\textbf{v}_p-\textbf{v}_n$ from (2.37), yields an equation for current conservation:

(2.49) \begin{eqnarray}\nabla \cdot\textbf{j}=0.\end{eqnarray}

Substituting $\textbf{v}_p$ (from (2.47a)) in (2.48) yieldsFootnote ²

(2.50) \begin{eqnarray}\frac{\partial c}{\partial t} - \nabla\cdot\left( \frac{c_T}{c_0 RT}c\mathcal{D} \nabla \mu_e\right)+\nabla\cdot(c\textbf{v}_0)=-\frac{\nabla t_{+}^{0}\cdot \textbf{j}}{F},\end{eqnarray}

and this can be compared to its dilute theory counterpart given in (2.9). These equations couple to the mass conservation equation for the solvent which, from (2.25), is given by:

(2.51) \begin{eqnarray}\frac{\partial c_0}{\partial t}+\nabla\cdot(c_0 \textbf{v}_0)=0. \end{eqnarray}

Finally, we require an equation of state. On using the charge neutrality condition $c_n=c_p=c$ in (2.20), we find that this can be written in the form:

(2.52) \begin{eqnarray}V_c c+V_0 c_0 = 1,\end{eqnarray}

where $V_c=V_n+V_p$ .

In one dimension, we now have sufficient equations to specify the problem; these are composed of the five equations (2.42), (2.49), (2.50), (2.51) and (2.52) for the five variables c, $c_0$ , $\phi$ , $\textbf{j}$ and $\textbf{v}_0$ . Note that, in more than one dimension, these equations are not sufficient, at least not in general, as can be seen by multiplying (2.48) by $V_c$ and adding to (2.48) multiplied by $V_0$ . On using (2.52) to eliminate the time derivative from the resulting equation, this yields the scalar partial differential equation (PDE):

(2.53) \begin{eqnarray}\nabla \cdot ( c V_c \textbf{v}_p + c_0 V_0 \textbf{v}_0 )=0,\end{eqnarray}

which in multiple dimensions is insufficient to determine the vector $\textbf{v}_0$ . As has been described in [Reference Drew27], this conservation equation needs to be supplemented by a momentum equation, but this is beyond the scope of this work.

Having posed the governing equations, an additional simplifying assumption is commonly made, which appears to be adequate for most solvent-based battery electrolytes. This consists of assuming that the electrolyte is sufficiently dilute so that $c_0 \approx c_T$ in which case (2.52) can be approximated by $c_0 \approx 1/V_0$ and the solvent velocity is small (i.e. $|\textbf{v}_0| \ll |\textbf{v}_p|$ and $|\textbf{v}_0| \ll |\textbf{v}_n|$ ). This limit is equivalent to the approximation:

(2.54) \begin{eqnarray}\textbf{v}_0 \equiv \textbf{0} \end{eqnarray}

and only requires solution of the three equations (2.42), (2.49) and (2.50) for the three variables c, $\phi$ and $\textbf{j}$ (instead of five equations for the five variables in the full model). This assumption also avoids the complication, that arises in multiple dimensions, of requiring to be supplemented by a momentum equation. Note however that, even in this small concentration limit, we still include interphase drag between negative and positive ions. This is because interphase drag is very often significant even at quite at low concentrations (often as low as 0.1 molar which is usually a small mole fraction). It arises because positive and negative ions interact via a strong long-range force (the Coulomb force).

2.2.3 The potential measured with respect to a lithium electrode

As in the dilute case, it is useful to reformulate the model in terms of $\varphi$ , the potential measured with respect to a lithium electrode (i.e. $\varphi$ is the potential measured in the lithium reference electrode). Once again we assume that the reaction (2.14) occurring on the surface of the reference electrode is in quasi-equilibrium so that sum of the electrochemical potentials of compounds on each side of (2.14) are identical (i.e. $\bar{\mu}_{p}+ \bar{\mu}_{\mbox{e}^-_{\rm ref}}=\mu_{{Li}_{\rm ref}}$ ). However, owing to the slightly different definition of $\bar{\mu}_p$ in the moderately concentrated case (compare (2.32) with (2.12)), this leads to a modified version of the relation between $\phi$ and $\varphi$ , namely

(2.55) \begin{eqnarray}\phi=\varphi+V_*^0-\frac{1}{F} (\mu_p^0+RT \log(a_p)), \end{eqnarray}

where once again the potential shift is defined by $V_*^0=\frac{1}{F} (\mu^0_{{Li}_{\rm ref}}-\mu^0_{\mbox{e}^-_{\rm ref}})$ , as in (2.16). This may be compared to the equivalent formula (2.16) for the dilute electrolyte. On substitution of this expression for $\phi$ into (2.42), we can re-express the constitutive equation for the current density equation in the form:

\begin{eqnarray*}\textbf{j} =-\kappa(c)\left(\nabla \varphi - \frac{RT}{F}(1-t_{+}^{0})(\nabla \log(a_p)+\nabla\log(a_n))\right),\end{eqnarray*}

which, on referring to the definition of the electrolyte chemical potential in (2.46), can be re-expressed as:

(2.56) \begin{eqnarray}\textbf{j} =-\kappa(c) \left(\nabla \varphi-\frac{2}{F}(1-t_{+}^{0})\nabla \mu_e\right). \end{eqnarray}

This is a clearer way to write Ohm’s law than (2.42) because it allows $\textbf{j}$ to expressed solely in terms of the measurable quantities $\varphi$ and $\mu_e$ (notably the activities of the individual ion species $a_n$ and $a_p$ are not directly measurable). This expression differs slightly from that typically presented by Newman and co-authors because they tend to use the dilute limit of (2.56), namely (2.19).

Electrochemical potentials of the ion species.

By substituting (2.55) into (2.32), we obtain expressions for the electrochemical potentials of the two ion species in terms of $\varphi$ , the potential measured with respect to a lithium electrode, which are as before:

(2.57) \begin{eqnarray}\bar{\mu}_p=F (\varphi+V_*^0), \qquad \bar{\mu}_n=2 \mu_e(c) - F (\varphi+V_*^0). \end{eqnarray}

3.1.1 Microscopic lithium transport models in individual electrode particles

At its very simplest, transport of intercalated lithium within electrode particles is modelled by linear diffusion in an array of uniformly sized (radius a) spheres (see e.g. [Reference Doyle, Newman, Gozdz, Schmutz and Tarascon26]). Hence, the lithium concentration in an electrode particle at position x within the electrode, $c_s(r,x,t)$ , evolves according to

(3.2) \begin{eqnarray}\frac{\partial c_s}{\partial t}= \frac{1}{r^2} \frac{\partial}{\partial r} \left( D_s r^2 \frac{\partial c_s}{\partial r} \right), \qquad \end{eqnarray}

where r measures distance from the particle’s centre and $D_s$ is the solid phase Li diffusion coefficient. This model for lithium transport in the electrode particles is typically coupled to the processes taking place in the adjacent electrolyte through a Butler–Volmer relation, which gives the transfer current density $j_{\rm tr}$ flowing out through the surface of the particle, in terms of the lithium concentrations on the particle’s surface $c_s$ , the adjacent electrolyte concentration c and the potential difference between the electrolyte and the electrode particle $\varphi-\phi_s$ . Following Faraday’s laws of electrolysis, the flux of lithium on the particle surface is proportional to the current density, leading to the following boundary condition on the electrode particle surface:

\begin{eqnarray*} -D_s \frac{\partial c_s}{\partial r}= \frac{1}{F}\;j_{\rm tr} \qquad\hbox{ on } r=a.\end{eqnarray*}

The use of a linear diffusion model to describe lithium transport within electrode particles is extremely innacurate and it is much better to use a non-linear diffusion equation (with $D_s=D_s(c_s)$ ) instead, an approach that is adopted in [Reference Farkhondeh and Delacourt31, Reference Karthikeyan, Sikha and White50, Reference Krachkovskiy, Foster, Bazak, Balcom and Goward55], for example. Experimental work demonstrates the strong relationship between diffusivity and lithium ion concentration within certain materials, such as graphite [Reference Baker and Verbrugge4, Reference Levi, Wang, Markevich, Aurbach and Chvoj59, Reference Takami, Satoh, Hara and Ohsaki94, Reference Verbrugge and Koch99] and LiNi $_x$ Mn $_y$ Co $_{1-x-y} O_2$ , often referred to as NMC [Reference Ecker, Käbitz, Laresgoiti and Sauer29, Reference Ecker, Tran, Dechent, Käbitz, Warnecke and Sauer30, Reference Wu, Zhang, Song, Shukla, Liu, Battaglia and Srinivasan101], a positive electrode material discussed in detail in Section 3.3.2. We will also discuss in Sections 3.2 and 3.3 more complicated models of lithium transport applicable to cases where phase separation occurs or the behaviour is highly anisotropic.

3.1.2 The Butler–Volmer equation

The transfer current density $j_{\rm tr}$ is the normal component of the current density on the particle surface, directed from the electrode particle into the surrounding electrolyte and is usually expressed in terms of a Butler–Volmer equation (see, e.g [Reference Bockris and Reddy10, Reference Dickinson and Wain23]). This equation quantifies the transfer of lithium ions from the electrode particle material to the electrolyte via the reversible reaction:

(3.3) \begin{eqnarray}\mbox{Li}^+_{(s)} \begin{array}{c} {\scriptstyle v_f}\\ \rightleftharpoons \\{\scriptstyle v_r} \end{array} \mbox{Li}^+_{(l)}, \end{eqnarray}

where $\mbox{Li}^+_{(s)}$ denotes a lithium ion in the electrode material and $\mbox{Li}^+_{(l)}$ denotes one in the electrolyte. In the vicinity of the electrode–electrolyte interface, lithium ions move in an energy landscape with a form that is represented in Figure 3. In particular, both the forward (from electrode to electrolyte) and reverse (vice-versa) reactions have to overcome a potential barrier. As is typical in these scenarios, the reaction rates can be modelled by an Arrhenius equations (see, e.g. [Reference Bockris and Reddy10] for similar arguments applied to a generic charge transfer reaction). Referring to Figure 3(b), in which the (molar) energy barriers to the transitions from the electrode to the electrolyte ( $\Delta E_f$ ) and from the electrolyte to the electrode ( $\Delta E_r$ ) are both illustrated, we expect the forward and reverse lithium ion transfer velocities to have the forms:

(3.4) \begin{eqnarray}v_f=v_{f,0} \exp\left(-\frac{\Delta E_f}{RT} \right), \qquad v_r=v_{r,0} \exp\left(-\frac{\Delta E_r}{RT} \right),\end{eqnarray}

Figure 3. Schematic showing the energy landscape that a Li $^+$ ion transitions through as it moves from electrode to electrolyte (or vice-versa). (a) In the absence of an electrical potential difference between electrode and electrolyte. (b) With an electrical potential difference $\Delta\phi=\phi_s-\phi$ between electrode and electrolyte.

respectively. Since the lithium ion is positively charged, the size of these energy barriers is affected by the (Galvani) potential drop $\Delta \phi= \phi_s-\phi$ between the interior of the electrode and the interior of the electrolyte. Indeed it is clear that

(3.5) \begin{eqnarray}\Delta E_f-\Delta E_r=\Delta E_{f,0}-\Delta E_{r,0}-F \Delta \phi, \end{eqnarray}

where $\Delta E_{f,0}$ and $\Delta E_{r,0}$ are the energy barriers to the forward and reverse reactions when there is no potential difference between the electrode and the electrolyte (i.e. $\Delta \phi=0$ ). Notably, since $E_{s,0}$ and $E_{l,0}$ are the molar energies of lithium ions in electrode and electrolyte, respectively, we require that $E_{s,0}=E_{s,0}(c_s)$ and $E_{l,0}=E_{l,0}(c)$ (i.e. they depend only on the lithium concentrations in their respective materials). Furthermore, since $\Delta E_{f,0}=E_{\rm barr}-E_{s,0}(c_s)$ and $\Delta E_{r,0}=E_{\rm barr}-E_{l,0}(c)$ and since we expect $E_{\rm barr}$ (the molar energy of the lithium ions at the energy barrier) to be independent of the local lithium concentration (which is always very small there), it follows that $\Delta E_{f,0}=\Delta E_{f,0}(c_s)$ and $\Delta E_{r,0}= \Delta E_{r,0}(c)$ . Turning to the potential difference across the interface it is apparent, from (3.5), that the the potential difference across the interface $\Delta \phi$ is split between the two activation energies, this suggests partitioning it as follows:

\begin{eqnarray*}\Delta E_f=\Delta E_{f,0}(c_s)-(1-\beta) F \Delta \phi \quad \mbox{and} \quad \Delta E_r=\Delta E_{r,0}(c)+\beta F \Delta \phi,\end{eqnarray*}

where $\beta$ is some constant between 0 and 1 (usually $\beta=1/2$ ). We thus expect the transfer current $j_{\rm tr}$ to have the form:

(3.6) \begin{eqnarray}j_{\rm tr}=F \left(c_s v_f^*(c_s) \exp \left( \frac{F}{RT} (1-\beta) \Delta \phi \right) -c \left( 1- \frac{c_s}{c_{s,max}} \right) v_r^*(c) \exp \left(- \frac{F}{RT} \beta \Delta \phi \right) \right), \end{eqnarray}

(3.7) \begin{eqnarray}\mbox{where} \quad v_f^*(c_s)=v_{f,0} \exp\left(-\frac{\Delta E_{f,0}(c_s)}{RT} \right) \quad \mbox{and} \quad v_r^*(c)=v_{f,0} \exp\left(-\frac{\Delta E_{r,0}(c)}{RT} \right).\end{eqnarray}

Here the factor of $\left( 1- {c_s}/{c_{s,\max}} \right)$ in the final term in (3.6) is to account for the fact that the reverse transfer (from the electrolyte to the electrode) is dependent on the availability of sites in the electrode material and so cannot occur if $c_s=c_{s,\max}$ and all the sites in the electrode material are already occupied. We can rewrite (3.6) in the more usual form:

(3.8) \begin{eqnarray}j_{\rm tr}= i_0(c,c_s) \left( \exp \left( \frac{F}{RT} (1-\beta) \eta \right) - \exp \left(- \frac{F}{RT} \beta \eta \right) \right), \end{eqnarray}

(3.9) \begin{eqnarray}\mbox{where} \quad i_0(c,c_s) =F \left( c_s v_f^*(c_s) \right)^{\beta}\left( c v_r^*(c) \left( 1- \frac{c_s}{c_{s,\max}} \right) \right)^{1-\beta}, \end{eqnarray}

where the overpotential $\eta$ and the open-circuit potential are defined by:

(3.10) \begin{eqnarray}\eta=\phi_s-\phi -u_{eq}(c,c_s), \end{eqnarray}

(3.11) \begin{eqnarray}u_{eq}(c,c_s)=\frac{RT}{F} \log (c v_r^*(c))-\frac{RT}{F} \log\left(\frac{c_s v_f^*(c_s)}{\left( 1- \frac{c_s}{c_{s,\max}} \right)} \right), \end{eqnarray}

and we have used the fact that $\Delta \phi=\phi_s-\phi$ . We write it in this form to emphasise the fact that $u_{eq}(c,c_s)$ can be written in the form $u_{eq}(c,c_s)=\mathcal{G}(c)- \mathcal{H}(c_s)$ , where $\mathcal{G}$ is a function of c and $\mathcal{H}$ is a function of $c_s$ . In the special case where the Galvani potential difference between electrode and electrolyte $\Delta \phi=\phi_s-\phi$ is held at zero, an equilibrium between the electrode and the electrolyte (at which $j_{tr}=0$ ) is established when $u_{eq}(c,c_s)=0$ . At this equilibrium, the chemical potential of the lithium ions in the electrolyte $\mu_{p}(c)$ is equal to that in the electrode material $\mu_{(s)}(c_s)$ . Given that $u_{eq}(c,c_s)$ has the form $\mathcal{G}(c)- \mathcal{H}(c_s)$ , and accounting for the dimensions of chemical potentials and $u_{eq}$ , this strongly suggests that

(3.12) \begin{eqnarray}F u_{eq}(c,c_s)=\mu_{p}(c)-\mu_{(s)}(c_s).\end{eqnarray}

It then follows that the overpotential can be written in the form:

(3.13) \begin{eqnarray}\eta=\phi_s-\phi-\frac{1}{F} \left(\mu_{p}(c)-\mu_{(s)}(c_s) \right), \end{eqnarray}

or equivalently

(3.14) \begin{eqnarray}\eta=\frac{1}{F} \left(\bar{\mu}_{(s)}(c_s)-\bar{\mu}_{p}(c) \right),\end{eqnarray}

where $\bar{\mu}_{(s)}$ and $\bar{\mu}_{p}$ are the electrochemical potentials of lithium ions in the electrode material and the electrolyte, respectively. This is an intuitively appealing form for the overpotential because it shows that the Butler–Volmer equation (3.8) drives the system towards an equilibrium in which $\eta$ vanishes, and in which the electrochemical potentials of lithium ions on either side of the interface are equal and the transfer current $j_{\rm tr}$ switches off. This result is in agreement with the derivation presented in [Reference Latz and Zausch56] and also with the discussion in [Reference Schmickler and Santos83] who note that the “driving force” for the reaction is this difference in the electrochemical potentials. Furthermore, in this form we can substitute for $\bar{\mu}_{p}$ in terms of $\varphi$ , from (2.57), and obtain the expression for the overpotential that is typically used in the Butler–Volmer equation, namely

(3.15) \begin{eqnarray}\eta=\phi_{\rm s}-\varphi-U_{\rm eq}(c_s)\quad \mbox{where} \quad U_{\rm eq}(c_s)= V_*^0-\frac{1}{F} \mu_{(s)}(c_s). \end{eqnarray}

Notably, on using this definition of $U_{eq}(c_s)$ and the relation (3.12), we find the following relationship between $u_{eq}(c,c_s)$ and $U_{eq}(c_s)$ :

(3.16) \begin{eqnarray}U_{\rm eq}(c_{\rm s})=u_{\rm eq}(c,c_{\rm s})+V_*^0-\frac{\mu_p(c)}{F}, \end{eqnarray}

which shows that $U_{eq}(c_s)$ inherits the same singularities in $c_s$ that $u_{eq}(c,c_s)$ has. We point out that $u_{\rm eq}$ is the open-circuit potential and $U_{\rm eq}$ is the open circuit potential measured with respect to a lithium electrode.

The standard form of the Butler–Volmer equations.

The standard way of expressing the Butler–Volmer equation is in terms of the open-circuit potential $U_{eq}(c_s)$ , as defined in (3.15), and on referring back to (3.8)–(3.9) we see that it should be written in the form:

(3.17) \begin{eqnarray}j_{\rm tr}= i_0(c,c_s) \left( \exp \left( \frac{F}{RT} (1-\beta) \eta \right) - \exp \left(- \frac{F}{RT} \beta \eta \right) \right) \qquad \quad \mbox{where}, \end{eqnarray}

(3.18) \begin{eqnarray}\eta=\phi_s - \varphi -U_{eq}(c_s), \quad i_0(c,c_s) =F \left( c_s v_f^*(c_s) \right)^{\beta}\left( c v_r^*(c) \left( 1- \frac{c_s}{c_{s,\max}} \right) \right)^{1-\beta}. \end{eqnarray}

It is important to realise that, it is only by expressing the overpotential in terms of a difference between the true electric potential in the electrode $\phi_s$ and the potential measured with respect to a lithium electrode in the electrolyte $\varphi$ that we obtain an expression for the open-circuit potential $U_{\rm eq}$ that is a function solely of $c_s$ . Furthermore, it is relatively straightforward to obtain $U_{eq}(c_s)$ experimentally, simply by measuring the potential difference $\phi_s-\varphi$ , at steady state (with $j_{\rm tr}=0$ ) between the electrode material and a lithium reference electrode placed in the electrolyte. The standard expression for the exchange current density $i_0(c,c_s)$ is obtained by setting both $v_f^*$ and $v_r^*$ to be constant. Although this is not strictly correct, it does at least ensure that $i_0(c,c_s)$ has the correct singularities as $c \rightarrow 0$ , $c_s \rightarrow 0$ and $c_s \rightarrow c_{s,\max}$ . This is enough, as shown below to ensure reasonable agreement with the real behaviour because the exponential terms in (3.17) usually dominate the behaviour compared to the prefactor $i_0(c,c_s)$ , except where the latter tends to zero.

From a mathematical perspective, it is important to realise that the open-circuit potential measured with respect to a lithium electrode (OCP) should tend to minus infinity as $c_s \nearrow c_{\rm s,max}$ and tend to plus infinity as $c_s\searrow 0$ . This can be seen clearly from the relation between $U_{eq}(c_s)$ and $u_{eq}(c,c_s)$ contained in (3.16) and the definition of $u_{eq}(c,c_s)$ found in (3.11). It is this property of $U_{eq}(c_s)$ that stops $c_s$ reaching $c_{\rm s,max}$ or 0, rather than the factors of $(1-c_s/c_{\rm s,max})$ and $c_s$ that appears in the exchange current density $i_0$ (see (3.18b)). Thus, if the experimentally obtained plots of $U_{eq}(c_s)$ do not display these singularities (which is usually because the experiments do not explore the extreme ends of the range), they should be added in by hand. It is standard to plot the OCP as a function of ${y}=c_s/c_{\rm s,max}$ , where y is termed the lithium stoichiometry. The OCP varies widely depending on the electrode material and to illustrate this it is plotted in Figure 4(a) for the lithiated graphite Li $_x C_6$ while in Figure 4(b) and (c) it is plotted for lithiated (NLC) Li $_y$ (Ni $_{0.4}$ Co $_{0.6}$ )O $_2$ , and iron phosphate Li $_y$ FePO $_4$ , respectively. Note the OCP for Li $_y$ FePO $_4$ has a large almost flat plateau symptomatic of a two-phase state within the material. Note further that in all the experimental data, at least one of the singularities in $U_{eq}(c_s)$ is missing.

Figure 4. The OCP, $U_{\rm eq}$ , of (a) LiC $_6$ , from [Reference Ecker, Tran, Dechent, Käbitz, Warnecke and Sauer30], (b) LiFePO $_4$ , from [Reference Srinivasan and Newman92]) and (c) Li(Ni $_{0.4}$ Co $_{0.6}$ )O $_2$ , from [Reference Ecker, Tran, Dechent, Käbitz, Warnecke and Sauer30]. Each is shown as a function of Lithium stoichiometry.

A comment on the functional form for the transfer current.

For the sake of completeness, we write down the form of the transfer current $i_0(c,c_s)$ that is logically consistent with the derivation of the Butler–Volmer equation presented above. By comparison of (3.11) with (3.12), we see that $c v_r^*(c)={const.} \exp(\mu_p(c)/RT)$ and $c_s v_f^*(c_s)={const.} (1-c_s/c_{s,\max}) \exp(\mu_{(s)}(c_s)/RT)$ . In addition, on recalling that $\mu_p=\mu^0_p+RT \log (a_p)$ , where $a_p$ is the activity of the lithium ions, and that $\mu_{(s)}(c_s)$ is related to $U_{eq}(c_s)$ via (3.15), we see that these can be rewritten as:

\begin{eqnarray*}c v_r^*(c)=const.a_p, \quad \mbox{and} \quad c_s v_f^*(c_s)=const. \left(1-\frac{c_s}{c_{s,\max}} \right) \exp\left(-\frac{F}{RT} U_{eq}(c_s) \right).\end{eqnarray*}

It follows that we can rewrite the transfer current density in (3.18) as:

(3.19) \begin{eqnarray}i_0(c,c_s) =i^* \left( 1- \frac{c_s}{c_{s,\max}} \right) \exp\left(-\frac{F}{RT} \beta U_{eq}(c_s) \right)a_p^{1-\beta}.\end{eqnarray}

for a suitable constant $i^*$ with units of current density.

Behaviour of $\,\textbf{j}_{\textbf{tr}}$ close to the extreme cases.

Setting $v^*_{\rm r}$ to be constant (as mentioned above) but keeping $v^*_{\rm f}$ dependent on $c_{\rm s}$ (which seem to be necessary in order to be able to fit open-circuit potential data, measured with respect to a lithium reference electrode), from (3.10), (3.11), (3.16), (3.17) and (3.18) we have that

\begin{equation*}i_0(c,c_s) =K_0 \left( c_sv^*_{\rm f}(c_{\rm s})\right)^{\beta}\left( c \left( c_{s,\max}- c_s \right) \right)^{1-\beta},\end{equation*}

\begin{eqnarray*}j_{\rm tr} & = & i_0 \left( \exp \left( \frac{F(1-\beta)}{RT} \left( \phi_{\rm s}-\phi - u_{\rm eq}( c,c_{\rm s})\right) \right) - \exp \left(- \frac{F\beta}{RT} \left( \phi_{\rm s}-\phi - u_{\rm eq}( c,c_{\rm s})\right) \right) \right) \\[4pt] & = & i_0 \left( \exp \left( \frac{F(1-\beta)}{RT} \left( \phi_{\rm s}-\varphi - U_{\rm eq}( c_{\rm s})\right) \right) - \exp \left(- \frac{F\beta}{RT} \left( \phi_{\rm s}-\varphi - U_{\rm eq}( c_{\rm s})\right) \right) \right) ,\end{eqnarray*}

\begin{equation*}u_{eq}(c,c_s)=\frac{RT}{F} \log \left( \frac{K_1c (c_{\rm s,max} -c_{\rm s})}{c_{\rm s}}\right) + {\mathcal U}_{\rm eq}(c_{\rm s}),\end{equation*}

with ${\mathcal U}_{\rm eq} (c_{\rm s})= - \frac{RT}{F} \log \left( v^*_{\rm f}(c_{\rm s})\right)$ and, for ideal electrolytes (so that $a_p=\frac{c}{c_T}$ ),

(3.20) \begin{equation} U_{\rm eq} (c_{\rm s})= \frac{RT}{F} \log \left( \frac{K_2 (c_{\rm s,max} -c_{\rm s})}{c_{\rm s}}\right) + {\mathcal U}_{\rm eq}(c_{\rm s})\end{equation}

for suitable constants

\begin{equation*}K_0=\frac{F (v^*_{\rm r})^{1-\beta}}{(c_{\rm s,max})^{1-\beta}}, \ K_1=\frac{v^*_{\rm r}}{c_{\rm s,max}} \mbox{ and }K_2=\frac{v^*_{\rm r}c_{\rm T}\exp \left( \frac{F}{RT}V^0_*-\frac{\mu_p^0}{RT}\right)}{c_{\rm s,max}}.\end{equation*}

Therefore, when using the electric potential of the electrolyte ( $\phi$ ), we have

(3.21) \begin{equation} \begin{array}{l}{\displaystyle j_{\rm tr} = \frac{K_0}{K_1^{1-\beta}} c_sv^*_{\rm f}(c_{\rm s})\exp \left( \frac{F(1-\beta)}{RT} \left( \phi_{\rm s}-\phi \right) \right)} \\ {\displaystyle - K_0K_1^\beta c(c_{\rm s,max}-c_{\rm s})\exp \left( -\frac{F\beta}{RT} \left( \phi_{\rm s}-\phi \right) \right)}\end{array}\end{equation}

and, when using the electric potential of the electrolyte measured with respect to a lithium electrode ( $\varphi$ ), we have

(3.22) \begin{equation} \begin{array}{l}{\displaystyle j_{\rm tr} = \frac{K_0}{K_2^{1-\beta}} c_sv^*_{\rm f}(c_{\rm s})c^{1-\beta}\exp \left( \frac{F(1-\beta)}{RT} \left( \phi_{\rm s}-\varphi \right) \right)} \\ {\displaystyle - K_0K_2^\beta c^{1-\beta}(c_{\rm s,max}-c_{\rm s})\exp \left( -\frac{F\beta}{RT} \left( \phi_{\rm s}-\varphi \right) \right) .} \end{array}\end{equation}

We notice here four important features of this formulation:

1. (1). The open-circuit potential ( $u_{\rm eq}$ ) depends on c and $c_{\rm s}$ , while the open-circuit potential measured with respect to a lithium electrode ( $U_{\rm eq}$ ) only depends on $c_{\rm s}$ .

2. (2). If $c_{\rm s}\rightarrow 0$ , with $c\neq 0$ (constant) and $v^*_{\rm f}(c_{\rm s})$ is bounded and greater than a positive constant, then $i_0\rightarrow 0$ , $u_{\rm eq}\rightarrow \infty$ and $U_{\rm eq}\rightarrow \infty$ . Furthermore, as $c_{\rm s} \to 0$ , from (3.21) or (3.22), we can see that the forward (anodic) reaction is first order, while the reverse (cathodic) reaction is bounded and positive (like a zeroth-order reaction).

3. (3). If $c_{\rm s}\rightarrow c_{\rm s,max}$ , with $c\neq 0$ (constant) and $v^*_{\rm f}(c_{\rm s})$ is bounded and greater than a positive constant, then $i_0\rightarrow 0$ , $u_{\rm eq}\rightarrow -\infty$ and $U_{\rm eq}\rightarrow -\infty$ Furthermore, as $c_{\rm s} \to c_{\rm s,max}$ , from (3.21) or (3.22), we can see that the forward (anodic) reaction is bounded and positive (like a zeroth-order reaction), while the reverse (cathodic) reaction is first order.

4. (4). If $c\rightarrow 0$ , with $c_{\rm s}\neq 0$ , then $i_0\rightarrow 0$ and $u_{\rm eq}\rightarrow -\infty$ but $U_{\rm eq}$ remains constant. Furthermore, as $c \to 0$ , from (3.21) we can see that the forward (anodic) reaction is bounded and positive (like a zeroth-order reaction), while the reverse (cathodic) reaction is first order. Nevertheless, this property is not directly seen from (3.22) (i.e. when writing $j_{\rm tr}$ in terms of the electric potential of the electrolyte measured with respect to a lithium electrode), since that equation is based on the dependence of $\varphi$ on c, so that $\varphi \to -\infty$ as $c\to 0$ in a way consistent with the correct asymptotic behaviour.

These conditions ensure that the lithium flux is constrained to prevent concentrations in the solid being taken into non-physical regimes and also to ensure that, if the solid is nearly fully depleted (or filled) with lithium, that the transfer current is not artificially forced to zero. Thus, a Butler–Volmer relation of this form allows the flux to move the system away from these depleted (and filled) states. This gives a model which is capable of describing battery charge from a completely depleted state or discharge from a fully charged state.

When fitting the various functions characterising the Butler–Volmer equation (3.17) to the data, it is necessary to include the singular behaviours as $c\rightarrow 0$ , $c_{\rm s}\rightarrow 0$ and $c_{\rm s}\rightarrow c_{\rm s,max}$ as given previously. Following the approach explained above (see (3.20)), one can take

\begin{equation*} U_{\rm eq}(c_{\rm s}) = \frac{RT}{F}\log \left( \frac{c_{\rm s,max}-c_{\rm s}}{c_{\rm s}} \right) + f(c_{\rm s}; a_1, a_2,\cdots , a_N),\end{equation*}

where $f(\cdot ; a_1, a_2,\cdots , a_N)$ is some family of suitable functions, and $\{ a_1,\cdots a_N\}$ is a set of parameters to be chosen by fitting experimental data. Note this could also be expressed in terms of the non-dimensional lithium stoichiometry $y= {c_{\rm s}}/{c_{\rm s,max}}$ :

(3.23) \begin{equation} U_{\rm eq}(y) = \frac{RT}{F}\log \left( \frac{1-y}{y} \right) + f(y; a_1, a_2,\cdots , a_N).\end{equation}

3.2.1 Lithium transport in Li_yC₆ (negative electrode material)

Much of the early modelling work on LIBs, such as [Reference Arora, Doyle, Gozdz, White and Newman1, Reference Doyle, Newman, Gozdz, Schmutz and Tarascon26, Reference Fuller, Doyle and Newman39, Reference Fuller, Doyle and Newman40], modelled lithium transport within Li $_y$ C $_6$ electrode particles by a linear diffusion equation (3.2). However, both [Reference Krachkovskiy, Foster, Bazak, Balcom and Goward55, Reference Takami, Satoh, Hara and Ohsaki94, Reference Verbrugge and Koch99] suggest a very strong dependence of solid-state lithium diffusion coefficient $D_{\rm s}(c_{\rm s})$ with lithium concentration $c_{\rm s}$ in Li $_y$ C $_6$ particles, with a range of variation of up to about two orders of magnitude, depending on the exact form of carbon used. In both sets of experiments, the size of the carbon particles used was around 10 $\mu$ m and diffusion decreased markedly as lithium concentration $c_{\rm s}$ was increased. To complicate matters, further Li $_y$ C $_6$ is known to exhibit (at least) three phases as lithium stoichiometry y is increased. The presence of these phases can be seen inferred from colour changes to the electrode particles (dark blue–low lithium, red–intermediate lithium and gold–high lithium). In [Reference Harris, Timmons, Baker and Monroe45] optical microscopy, measurements are used to characterise the phase transitions occurring (with increasing lithiation) within a Li $_y$ C $_6$ half-cell anode subject to uniform charging. This shows different phases co-existing (in distinct graphite electrode particles) at different positions in the anode. However, the relevance of these results to commercial devices should perhaps not be overstated, because the width of the negative electrode used in [Reference Harris, Timmons, Baker and Monroe45] is particularly large (around 800 $$\mu $$ m) compared to the standard electrode size in commercial devices (around 100 $$\mu $$ m). For this reason, the charging process observed in [Reference Harris, Timmons, Baker and Monroe45] is likely to be limited by lithium diffusion within the electrolyte, as the electrode particles deplete the surrounding electrolyte of lithium ions. Thomas-Alyea et al. [Reference Thomas-Alyea, Jung, Smith and Bazant95] have also applied optical microscopy to graphite electrodes and observed considerable spatial non-uniformity even after the electrode was left quiescent for an extended period. They were able to predict such states by employing a Cahn–Hilliard phase field model which will be discussed further in Section 3.3.

Graphite reacts with the electrolyte, consuming lithium ions, to form a thin layer of solid material on the graphite which is referred to as the solid electrolyte interphase (SEI) layer and, as noted in [Reference Bruce, Scrosati and Tarascon12], is essential for maintaining the structural integrity of the electrode particles. However, the consumption of electrolyte by this reaction means that graphite electrode particle size cannot be reduced to the nanoscale (in an attempt to improve the charge–discharge rate of the battery) without severely compromising battery capacity [Reference Bruce, Scrosati and Tarascon12]; some lithium makes up the SEI layer and can no longer participate in the useful reactions that store charge. This SEI layer also forms a barrier to lithium ion (and current) transfer between the electrolyte and electrode which may have a significant bearing on electrode performance and has thus been incorporated into some models, for example [Reference Srinivasan and Newman91]. Diffusion of lithium along the graphene sheets of pure single-crystal graphite is extremely fast [Reference Persson, Sethuraman, Hardwick, Hinuma, Meng, van der Ven, Srinivasan, Kostecki and Ceder73] (diffusion coefficient of the order of $10^{-7}-10^{-6}$ cm $^2$ s⁻¹), so that, even in electrodes comprised of quite large electrode particles ( $\sim 100$ $$\mu $$ m) discharged (or charged) at very high rates, it should not significantly affect cell performance. However, [Reference Persson, Sethuraman, Hardwick, Hinuma, Meng, van der Ven, Srinivasan, Kostecki and Ceder73] demonstrates that diffusion perpendicular to the graphene sheets and along grain boundaries is many orders of magnitude slower (diffusion coefficient of the order of $10^{-11}$ cm $^2$ s⁻¹) and uses this to infer that this high degree of anisotropy can be used to explain the widely disparate measurements of diffusivity reported in polycrystalline graphite.

3.2.2 Lithium transport in NMC and LMO (positive electrode materials)

Lithium diffusion in NMC [Reference Wu, Zhang, Song, Shukla, Liu, Battaglia and Srinivasan101] is highly non-linear so that $D_{\rm s}(c_{\rm s})$ decreasing by about two orders of magnitude as lithium concentration within the material increases. Furthermore, NMC can be charged and discharged at high rates and the OCP is smooth without the stepped plateau features that usually characterise phase transitions. In contrast, [Reference Julien, Mauger, Zaghib and Groult48] notes that LMO undergoes a number of phase transitions as it charges and discharges, which are associated with plateaus in its OCP curve. Diffusivity of lithium in single crystals of LMO is about an order of magnitude lower than in multicrystalline particles [Reference Das, Majumder and Katiyar20], suggesting that grain boundaries form an easy pathway for lithium diffusion. Furthermore, LMO has the disadvantage of capacity loss and fade after repeated cell cycling. This capacity fade has been ascribed, by [Reference Das, Majumder and Katiyar20], to the formation of a SEI layer, and consequent loss of lithium mobility.

3.2.3 Lithium transport in LFP (positive electrode material)

Bruce et al. [Reference Bruce, Scrosati and Tarascon12] point out that intercalation in LFP involves a phase transition between FePO $_4$ and LiFePO $_4$ , which is reflected in its flat OCP curve (see Figure 4(b)). Kang and Ceder [Reference Kang and Ceder49] note that the transport of lithium is dominated by transport along channels in particular crystalline directions, the b-direction. Furthermore, transport in single-crystal nanoparticles is extremely rapid, so fast indeed that it is doubtful that lithium intercalation in LFP single-crystal nanoparticles will ever limit battery performance. This point is clearly made by [Reference Johns, Roberts, Wakizaka, Sanders and Owen46], who demonstrate that discharge in a half-cell nanoparticulate LFP cathode is limited by conduction and transport in the electrolyte. In larger LFP electrode particles, [Reference Jugović and Uskoković47] point out that performance is significantly impaired because lithium ion transport along the b-direction channels is easily obstructed by grain boundaries and crystal defects. This gives rise to an apparent lithium diffusivity in LFP that decreases sharply as the size of the particle increases, see [Reference Malik, Burch, Bazant and Ceder64]. Standard models of this material include the so-called shrinking core model, presented in [Reference Srinivasan and Newman92], which attempts to capture the phase transition by using a one-dimensional free boundary model of the phase transition (akin to a Stefan model, see e.g. [Reference Rubinštein81]) in a spherically symmetric electrode particle. This approach is used in a 3D-scale model, that captures agglomeration of LFP nanocrystals into agglomerate particles, by [Reference Dargaville and Farrell19] who show good agreement with experimental discharge curves for a wide range of discharge rates. However, the shrinking core model is known to predict distributions of the two phases that are not observed in practice and it is also not easy to implement in a form that allows numerous charge–discharge cycle because of the appearance of multiple free boundaries, as noted by [Reference Farkhondeh and Delacourt31]. A simpler alternative, suggested in [Reference Farkhondeh and Delacourt31], is to model lithium transport within LFP particles by a phenomenological non-linear diffusivity $D_{\rm s}(c_{\rm s})$ and this appear to fit discharge data well.

4.2.1 Remarks on the role of binder

In the discussion above, we have assumed that the electrode was formed solely from electrode particles bathed in electrolyte. While this is often a reasonable description of research cells, many commercial devices also incorporate a significant volume fraction of polymer binder material that acts both to enhance the structural integrity of the device and, in combination with a conductivity enhancer (such as carbon black), maintain good electrical contact between electrode particles (these are often poor conductors). Three-dimensional images of typical commercial electrodes using focused ion beam in combination with scanning electron microscopy (FIB-SEM) can be found in [Reference Foster, Gully, Liu, Krachkovskiy, Wu, Schougaard, Jiang, Goward, Botton and Protas35, Reference Gully, Liu, Srinivasan, Sethurajan, Schougaard and Protas41, Reference Liu, Foster, Gully, Krachkovskiy, Jiang, Wu, Yang, Protas, Goward and Botton60]. These show a porous binder material filling almost all the space between electrode particles with the exception of some linear features around the electrode particles where it appears that the binder has become delaminated from the electrode particles.Footnote ⁵ The porosity and pore size of the binder materials vary significantly between different electrode types. Typical pore sizes are usually in the range 10–500 nm, much smaller than typical electrode particle sizes which are usually at least micron sized. To account for these effects within an homogenisation approach, the analysis could be modified in two possible ways. Firstly, it could be performed directly on a microstructure in which all three constituents (electrode particle, electrolyte and binder) are resolved by the cell problem and would follow a very similar pattern to that described above in which the binder was treated as part of $\hat{\Omega}_{\mathrm{per}}$ with an interface with the electrolyte on which the transfer current density $j_{\rm tr} \equiv 0$ . Secondly, because obtaining a good representation of the pore geometry in the binder is challenging, even using modern high-performance microscopy, see [Reference Liu, Foster, Gully, Krachkovskiy, Jiang, Wu, Yang, Protas, Goward and Botton60], it is probably better to treat the electrolyte-permeated nanoporous binder as a single electrolyte material (albeit it one with reduced electrolyte volume fraction, electrolyte diffusivity and conductivity) and homogenise over the electrode particles and this composite material. Indeed this second approach is a standard way of treating such two phase (electrolyte/binder) materials in the literature which are often termed porous solid polymer electrolytes (see, e.g. [Reference Miao, Liu, Zhu, Liu, Li, Wang and Li66]).

4.2.2 Practical application of homogenisation

There are several important practical results that arise from this homogenisation procedure. Firstly, we learn that the same dimensionless permeability tensor appears in equation (4.7a) for the averaged ion flux $\langle \textbf{q}_p \rangle$ as appears in that for $\langle \textbf{j} \rangle$ , the averaged current (4.7b). Secondly, it is apparent that definition of the dimensionless permeability tensor $\underline{\underline{B}}$ , contained in (4.8) and (4.9), is equivalent to that which would be calculated in the homogenisation of Laplace’s equation:

(4.12) \begin{eqnarray}\nabla^2 c =0 \quad \mbox{in} \quad \hat{V}_{\mathrm{per}}, \quad \nabla c \cdot \textbf{n}|_{\partial \hat{V}_{\mathrm{per}}}=0, \end{eqnarray}

over the same domain $\hat{V}_{\mathrm{per}}$ . Indeed homogenising (4.12) results in the averaged equations:

(4.13) \begin{eqnarray}\nabla \cdot \langle \textbf{q} \rangle=0, \quad \langle \textbf{q} \rangle =- \underline{\underline{B}} \nabla c.\end{eqnarray}

Thus, from a practical perspective, the computation of the dimensionless permeability tensor $\underline{\underline{B}}$ can be accomplished by solving Laplace’s equation over a representative cell domain, $\hat{V}_{\mathrm{per}}$ , culled from real image data, and computing the averaged flux $\langle \textbf{q} \rangle$ that passes through this cell in response to a unit differences in concentration across the faces of the cell. This is the approach that has been adopted in a number of works, including [Reference Gully, Liu, Srinivasan, Sethurajan, Schougaard and Protas41, Reference Le Houx and Kramer57, Reference Le Houx, Osenberg, Neumann, Binder, Schmidt, Manke, Carraro and Kramer58, Reference Liu, Foster, Gully, Krachkovskiy, Jiang, Wu, Yang, Protas, Goward and Botton60, Reference Tjaden, Cooper, Brett, Kramer and Shearing96].

Bruggeman Relation.

For isotropic materials, for which $(\underline{\underline{B}})_{ij}={\mathcal B} \delta_{ij}$ where $ \delta_{ij}$ is the Kronecker delta tensor, a simple, but entirely ad hoc, way of computing the dimensionless permeability factor ${\mathcal B}$ is provided by Bruggeman relation, [Reference Bruggeman13], which states that

\begin{eqnarray*}{\mathcal B}(x)=(\epsilon_v(x))^{1.5}.\end{eqnarray*}

where $\epsilon_v$ is the electrolyte volume fraction.