Chemical capacitance matrices

We have been framing our electrochemical problems as 'voltage-controlled', and it's interesting to see how we naturally get a form of capacitance. We've already seen this in the Capacitance topic (and in device form in lithium-ion batteries), and now we're going to return to it in a more generic view:

• any number of ions
• any kinds of interactions, and
• explaining "chemical capacitance"

We are going to define two key kinds of chemical capacitance here:

• mutual chemical capacitance (describing bulk charge storage),
• internal chemical capacitance (for describing space charge regions),

and the relation between them.

Mutual chemical capacitance (bulk charge storage)

Thermodynamic setting

In our $V_i$ view, we work with a free energy function of the species voltages — a grand free energy we'll call $\Omega$:

$$\Omega(V_1, \ldots, V_N)$$

(I'm being a bit vague about the other control variables -- is it constant-volume with fixed solvent chemical potential or constant-pressure with fixed solvent mass? Is it isothermal or adiabatic? We'll leave that undecided for now but note that the choice does affect the capacitance values.)

When we change one of these $V_i$'s, the partial derivative gives us the charge of that species:

$$Q_i(V_1, \ldots, V_N) = -\frac{\partial \Omega}{\partial V_i}.$$

Each one of these $Q_i$ is a function of all the $V_i$ values. This has to be the case because the bulk is charge neutral: $0 = Q = Q_{\mathrm{fix}} + \sum Q_i$. So if $Q_1$ increases under some change of parameters then all the other $Q_i$'s have to decrease such that the net total change is 0. Note that there could be some dopants present which modify the free energy function, including by inducing a background charge $Q_{\mathrm{fix}} \neq 0$.

(And again, to remind, this is usually expressed in terms of electrochemical potential $\bar\mu_i = q_i V_i$ and particle number $N_i = Q_i / q_i$ but the voltage and charge units are more familiar in our context.)

Mutual chemical capacitance defined

Capacitance is a change in charge due to a change in some voltage difference $\mathrm{d}Q/\mathrm{d}V$. Beginners learn about linear plate capacitors with nice constant capacitance values ($Q/V$), but we are going to talk about capacitance in the generic differential sense: a change in some charge due to a change in some voltage.

In the case of charge storage in materials, we must be aware of these complications: we have 1) multiple control voltages, not just two, 2) multiple charge types, 3) nonlinearity, and 4) no parallel-plate geometry at all, it is simply a bulk (volumetric or gravimetric) capacitance.

We can define capacitance as a matrix, which I call the mutual chemical capacitance matrix:

$$\begin{aligned} \mathbf{C}^{\mathrm{mut}}_{ij} & = \left(\frac{\partial Q_i}{\partial V_j}\right)_{\{V_k\}_{k \neq j},\, Q=0} & (V_i~\text{definition}) \\ & = q_i q_j \left(\frac{\partial N_i}{\partial \bar\mu_j}\right)_{\{\bar\mu_k\}_{k \neq j},\, Q=0} & (\text{usual definition}) , \end{aligned}$$

where we added the subscript $Q=0$ to emphasize that this is for a charge-neutral bulk. As we'll see below, removing that $Q=0$ makes a huge difference.

This matrix $\mathbf{C}^{\mathrm{mut}}$ is truly a capacitance matrix in the mathematical sense: its units are farads, its rows and columns sum to zero, it is symmetric, and it is positive semi-definite. Normally we would use a capacitance matrix to describe the self- and mutual capacitance of a collection of conductors, but in this case the "conductors" are actually all overlapping in space. A regular geometrical capacitance matrix is highly sensitive to arrangement, shapes, and distances, while the mutual chemical capacitance is simply proportional to volume of solvent/medium (or mass, depending on what else we are holding fixed).

In general the mutual chemical capacitance is a dense matrix (all entries nonzero) and can be thought of in terms of a fully connected equivalent circuit:

A special case of mutual chemical capacitance is the two-carrier case, as found in battery electrodes. This results in $\mathbf{C}^{\mathrm{mut}}$ being a $2 \times 2$ matrix of the form $\big[\begin{smallmatrix} C & -C \\ -C & C \end{smallmatrix}\big]$ for some $C > 0$, which is just a single capacitor in the equivalent circuit:

(Often this two-species coupling $C$ value is called 'chemical capacitance' but that term is ambiguous. In other contexts it instead refers to the internal chemical capacitance defined below.)

If you only cared about two carriers, then the $\mathbf{C}^{\mathrm{mut}}$ matrix looks like overkill (why not just use $C$) but it is absolutely necessary in the case of more than two carriers. Note that some other multicarrier generalizations of chemical capacitance seen in the literature are not correct.^[1]

Mutual chemical capacitances can be very large, e.g. for a typical $1~\mathrm{mol/L}$ solution of $+e$ and $-e$ carriers, you'll have a $V_{\mathrm{cation}}$-$V_{\mathrm{anion}}$ mutual capacitance of order $2~\mathrm{kF}$ per $\mathrm{cm}^3$ of volume, far beyond supercapacitor levels of volumetric capacitance. This charge storage capacity is a key factor for both lithium ion batteries (in the electrodes) and lead-acid batteries (in solution). The ambipolar diffusion associated with mutual chemical capacitance is often too slow for capacitor use cases, though finely ground mixtures of charge storage materials can alleviate this and so some supercapacitors can be described in terms of mutual chemical capacitance (mutual chemical capacitance can be equally well used to describe mixtures of materials as long as they are reasonably homogeneous on the large scale).

We'll see below how mutual chemical capacitance naturally comes about even in ideal systems, and how the mutual chemical capacitance can be expressed in terms of ideal quantities.

Why mutual chemical capacitance matters

Charge storage

Integrated mutual chemical capacitance is the charge storage of a bulk material such as a battery electrode. Note that for $N$ independent charged species, the charge storage 'space' is $(N-1)$-dimensional. For all practical charge storage materials $N=2$ so the charging is one dimensional.

Rigour in nonideal solutions

Mutual chemical capacitance is a thermodynamic observable and directly relates to mean activities. E.g. Debye-Huckel or Pitzer models predict a specific mutual chemical capacitance matrix.

(In contrast, descriptions of electrolytes in terms of single-ion activities correspond to a description in terms of internal chemical capacitance $\mathcal{C}$ (see below). In nonideal solutions, $\mathcal{C}$ will be a dense matrix and also generally regarded as unmeasurable; different activity conventions will have different $\mathcal{C}$ matrices.)

Ambipolar diffusion

The volumetric mutual chemical capacitance, $\mathbf{c}^{\mathrm{mut}} = \mathbf{C}^{\mathrm{mut}}/V$ directly appears in the continuity equation in the quasi-charge-neutral regime, together with the regular conductivity equation:

$$\begin{aligned} \mathbf{c}^{\mathrm{mut}} \frac{\partial \mathbf{V}}{\partial t} & = -\nabla \cdot \mathbf{\vec{J}}, & \quad \text{(continuity)},\\ \mathbf{\vec{J}} & = -\boldsymbol{\sigma} \nabla \mathbf{V} & \quad \text{(conductivity)}, \end{aligned}$$

where conductivity $\boldsymbol{\sigma}$ is also possibly a dense matrix (for simplicity we'll assume scalar matrix entries, though it could be interesting to consider tensorial e.g. for magnetic field effects); cross-terms can represent e.g. 'ion drag'. You can't simply convert this into a diffusion equation because $\mathbf{c}^{\mathrm{mut}}$ can't be inverted (because of charge neutrality), so we have to proceed with a bit of care.

First, let's look at a homogeneous material, small-signal case. In this case both $\boldsymbol{\sigma}$ and $\mathbf{c}^{\mathrm{mut}}$ are constant in space, then this simplifies to $\mathbf{c}^{\mathrm{mut}} \partial_t \mathbf{V} = \boldsymbol{\sigma} \nabla^2 \mathbf{V}$, which reveals there are two distinct classes of modes:

• One charge mode of the form $\mathbf{V}(x,t) = v(x,t) \mathbf{1}_N$, which follows $0 = \nabla^2 v$. The charge mode is representative of the naive form of electricity: all $V_i$ values (and $\phi$ too) vary in unison, and so all ions conduct together as if they have a single common conductivity $\sigma_{\mathrm{tot}} = \sum_{ij} \sigma_{ij}$, i.e. a total current $\vec{J}_{\mathrm{tot}} = -\sigma_{\mathrm{tot}} \nabla v$ driven by the common gradient. And each ion has $\nabla \cdot \vec{J}_i = 0$ so there is no piling up (but see below about material boundaries).
• $N-1$ ambipolar modes of the form $\mathbf{V}(x,t) = v(x,t) \mathbf{a}$, where $\mathbf{a}$ is a generalized eigenvector satisfying $\mathbf{c}^{\mathrm{mut}} \mathbf{a} = D^{-1} \boldsymbol{\sigma} \mathbf{a}$ (often easy to solve^[2]) for eigenvalue $D^{-1}$. Each of these modes evolves with a diffusion equation: $\partial_t v = D \nabla^2 v$ with diffusion coefficient $D$, and there may be $N-1$ distinct ambipolar diffusion constants. These are neutral current modes: $\sum_i \vec{J}_i = 0$.

For example in the binary (two-ion) case of cation $\mathrm{M}$ and anion $\mathrm{X}$, assuming a diagonal conductivity then the diffusion mode has $D^{-1} = (\sigma_{\mathrm{M}}^{-1} + \sigma_{\mathrm{X}}^{-1})c^{\mathrm{mut}}$, where $c^{\mathrm{mut}}$ is the single degree of freedom in the $\mathbf{c}^{\mathrm{mut}}$ matrix. This is the most simple kind of ambipolar diffusion as usually considered in the literature, in fact one usually sees a further ideality assumption that $(c^{\mathrm{mut}})^{-1} = ((z_{\mathrm{M}} F)^2 c_{\mathrm{M}} / (RT))^{-1} + ((z_{\mathrm{X}} F)^2 c_{\mathrm{X}} / (RT))^{-1}$ for ion molarities $c_{\mathrm{M}}$ and $c_{\mathrm{X}}$.

It's worth noting however that when there are material boundaries (or any other reason for the matrices to vary, such as most nonlinearities), then there are no such simple modes overall: what is ambipolar in one region is not ambipolar in the next, and even the charge modes don't match up (the ions do pile up: $\nabla \cdot \vec{J}_i \neq 0$). Moreover, very close to the interfaces the charge neutrality assumption is often violated as well (within the screening length scale, see below), though often times this can be neglected. Besides that, even with a single material, the boundary conditions (e.g. one ion connected to electrodes but the other ion blocked) may mean that steady-state conduction need not follow the charge mode at all (however, when the bias is applied, the system will at first conduct according to the charge mode).

Internal chemical capacitance (space charge)

Near interfaces, impurities, and in/around depletion regions, there are variations in charge neutrality. We want to model these regions as having some continuum space charge $\rho$ and some continuum electrostatic potential $\phi$. To this end we make a "local density approximation", by declaring that each infinitesimal volume $\mathrm{d}\tau$ has a local free energy function where $\phi$ is now a control variable:

$$\Omega^\phi (V_1, \ldots, V_N, \phi) \propto \mathrm{d}\tau$$

It's worth reminding that we are abandoning thermodynamic rigour when asking for continuum thermodynamics to apply, especially in the case of space charge at microscopic scales. The local density approximation is justified in some idealized systems, but it is only approximately correct in reality.

We also now make the idealization that the medium is fixed, and does not expand at all due to the motion of the ionic/electronic solutes.

We're going to define internal chemical capacitance. It is exactly equivalent to quantum capacitance, though that is usually restricted to one carrier. Internal chemical capacitance can be defined in multiple equivalent ways:

$$\begin{aligned} \mathcal{C}_{ij} &= -\frac{\partial^2 \Omega^\phi}{\partial V_i \partial V_j} \\ &= \left( \frac{\partial Q_i}{\partial V_j} \right)_{\{V_k\}_{k \neq j},\, \phi} \\ &= q_i q_j \frac{\partial N_i}{\partial \mu^{\mathrm{int}}_j} \\ \end{aligned}$$

This is quite similar to mutual chemical capacitance above, but note we are fixing $\phi$ instead of fixing $Q=0$.

Note that "chemical capacitance" is ambiguous. Sometimes that refers to internal chemical capacitance as just defined,^[3] but often in ionics the term "chemical capacitance" refers to the 2-carrier mutual chemical capacitance described above.^[4]

Internal chemical capacitance is usually invoked in systems with a well defined mean field $\phi$, such as ideal-dilute solutes, ideal Fermi gases, and such. Consequently, it tends to be the case that $\mathcal{C}$ is diagonal:

$$\mathcal{C}^{\mathrm{ideal}} = \begin{bmatrix} \mathcal{C}_{11} & 0 & \cdots & 0 \\ 0 & \mathcal{C}_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \mathcal{C}_{NN} \end{bmatrix}$$

(And in fact in ideal cases we often see internal chemical capacitance defined as $\mathcal{C}_{ii} = q_i^2 (\partial \mu_i / \partial c_i)^{-1}$; this definition only makes sense however when the $\mathcal{C}$ matrix is diagonal.^[5])

For example:

• In a degenerate Fermi gas of electrons, $\mathcal{C}/V = e^2 g(E_F)$ for density of states $g(E_F)$ better known as quantum capacitance. For dilute (thermal) electrons or holes in semiconductors, $\mathcal{C}/V = e^2 n / (kT)$ for carrier density $n$.
• In ideal-dilute electrolytes, $\mathcal{C}_{ii}/V = (z_i F)^2 c_i / (RT)$ for ion molarity $c_i$, for each ion species.

Going beyond the ideal case, however, correlations will cause cross terms. And in fact charged solutes are quite eager to correlate with each other even when fairly dilute, because the microscopic electrostatic interactions between ions are so long ranged (this is the essence of Debye-Huckel effect).

Note the internal chemical capacitance matrix $\mathcal{C}$ is generally symmetric and positive definite, but it is not a proper capacitance matrix because it's not charge neutral: the rows do not add to zero.

Screening properties

We define a per-ion screening vector:

$$\begin{aligned} s_i &= -\left( \frac{\partial Q_i}{\partial \phi} \right)_{V_1, \cdots V_N} \\ &= \sum_j \left( \frac{\partial Q_i}{\partial V_j} \right)_{V_1, \cdots V_N,\phi} \\ &= \sum_j \mathcal{C}_{ij} \end{aligned}$$

where this step works because of gauge invariance: shifting $\phi$ up is equivalent to shifting all $V_i$ down. We can write this as $\mathbf{s} = \mathcal{C} \mathbf{1}_{N}$. In the ideal case, we have simply $s_i = \mathcal{C}_{ii}$.

We then have for all ion charges a full description of charge variations.

$$\delta\mathbf{Q} = \mathcal{C} \delta \mathbf{V} - \mathbf{s} \delta \phi$$

or in matrix form:

$$\begin{bmatrix}\delta Q_1 \\ \vdots \\ \delta Q_N \end{bmatrix} = \begin{bmatrix} \mathcal{C}_{11} & \cdots & \mathcal{C}_{1N} & -\mathbf{s}_1 \\ \vdots & \ddots & \vdots & \vdots \\ \mathcal{C}_{N1} & \cdots & \mathcal{C}_{NN} & -\mathbf{s}_N \\ \end{bmatrix} \begin{bmatrix}\delta V_1 \\ \vdots \\ \delta V_N \\ \delta \phi \end{bmatrix} .$$

Note also the total induced charge is:

$$\delta Q_{\mathrm{free}} = \sum_i(\delta Q_i) = \mathbf{s}^T \delta \mathbf{V} - \mathcal{C}_{\mathrm{tot}} \delta \phi$$

Where $\mathcal{C}_{\mathrm{tot}} = \sum_i s_i = \sum_{ij} \mathcal{C}_{ij}$ is the total screening power.

Screening is often derived laboriously in a particle-number basis. But in a $V_i$ basis we know simply that all $V_i$ are flat at equilibrium, and so the above equation means that at every point in space the induced free charge density from small variations $\delta \phi$ is $\delta \rho_{\mathrm{free}} = -\chi \delta \phi$ where $\chi = \mathcal{C}_{\mathrm{tot}} / \mathrm{volume}$ is the volumetric screening power — the same $\chi = -\mathrm{d}\rho_{\mathrm{free}}/\mathrm{d}\phi$ met in basic electrostatics — and again this is a sum of all the internal chemical capacitances, per unit volume.

The Poisson equation then leads exactly to the linear screening equation,

$$\varepsilon \nabla^2 \delta \phi = \chi \delta \phi - \rho_{\mathrm{imp}},$$

where $\varepsilon$ is the medium's absolute permittivity and $\rho_{\mathrm{imp}}$ is the impurity charge density. This means $\sqrt{\varepsilon/\chi}$ is precisely the screening length in the general case of mixed charge carriers with any statistics (the Debye length and Thomas–Fermi screening length are both special cases of this).

Mutual and internal chemical capacitance related

If we take our internal chemical capacitance and force the volume to be charge neutral ($\delta Q_{\mathrm{free}} = 0$), then $\phi$ must float to whatever value is necessary to get neutrality. We then get:

$$0 = - \mathbf{s}^T \delta \mathbf{V} + \mathcal{C}_{\mathrm{tot}} \delta\phi$$

$$\delta\mathbf{Q} = \mathcal{C} \delta \mathbf{V} - \mathbf{s} \delta \phi$$

and so we see that the mutual and internal chemical capacitance matrices are precisely related:

$$\begin{aligned} \mathbf{C}^{\mathrm{mut}} & = \mathcal{C} - \frac{\mathbf{s} \mathbf{s}^T}{\mathcal{C}_{\mathrm{tot}}} \\ \mathbf{C}^{\mathrm{mut}}_{ij} & = \mathcal{C}_{ij} - \frac{\sum_{kl}\mathcal{C}_{ik}\mathcal{C}_{jl}}{\sum_{kl}\mathcal{C}_{kl}} \\ \end{aligned}$$

So, $\mathbf{C}^{\mathrm{mut}}$ has strictly less information than $\mathcal{C}$. On the other hand, $\mathbf{C}^{\mathrm{mut}}$ is a general thermodynamic property that does not make any microscopic assumptions as are usually needed for $\mathcal{C}$, but only the latter can be used to model continuum space charges.

As for where each description applies: the mutual $\mathbf{C}^{\mathrm{mut}}$ alone works for microscopically messy materials, since it is $\phi$-agnostic; either description serves for long-range conduction, ambipolar diffusion, and ideal bulk charge storage; and only the internal $\mathcal{C}$ can describe interfaces, screening, and depletion/enhancement regions.

Extended chemical capacitance matrix (the Jamnik-Maier trick)

As mentioned above $\mathcal{C}$ is not a capacitance matrix, and that's because the capacitances therein are with respect to $\phi$, which is unusual for a capacitance since usually capacitance needs a balanced charge on both sides (and $\phi$ normally does not 'store charge'). At this point we can do a cool trick, following Jamnik and Maier:^[6] we define a 'displacement charge' to be the negative of the total charge. In effect this turns $\phi$ into a proper circuit node that contains a displacement charge.

$$\delta Q_{\mathrm d} = - \delta Q_{\mathrm{free}} = -\mathbf{s}^T \delta \mathbf{V} + \mathcal{C}_{\mathrm{tot}} \delta \phi$$

Together these give us a full system like so:

$$\begin{bmatrix}\delta Q_1 \\ \vdots \\ \delta Q_N \\ \delta Q_{\mathrm d} \end{bmatrix} = \begin{bmatrix} \mathcal{C}_{11} & \cdots & \mathcal{C}_{1N} & -\mathbf{s}_1 \\ \vdots & \ddots & \vdots & \vdots \\ \mathcal{C}_{N1} & \cdots & \mathcal{C}_{NN} & -\mathbf{s}_N \\ -\mathbf{s}_1 & \cdots & -\mathbf{s}_N & \mathcal{C}_{\mathrm{tot}} \\ \end{bmatrix} \begin{bmatrix}\delta V_1 \\ \vdots \\ \delta V_N \\ \delta \phi \end{bmatrix} .$$

And this large matrix I call the extended chemical capacitance matrix $\mathbf{C}^{\mathrm{ext}}$, in block matrix form:

$$\mathbf{C}^{\mathrm{ext}} = \begin{bmatrix} \mathcal{C} & -\mathbf{s} \\ -\mathbf{s}^T & \mathcal{C}_{\mathrm{tot}} \end{bmatrix}$$

The extended chemical capacitance matrix is a capacitance matrix!

In relation to this, the mutual chemical capacitance can be seen as elimination of the $\phi$ node from the $\mathbf{C}^{\mathrm{ext}}$ capacitance matrix (a Schur complement operation), which when viewing both $\mathbf{C}^{\mathrm{ext}}$ and $\mathbf{C}^{\mathrm{mut}}$ as capacitor networks is known as a Kron reduction or a star-mesh transform.

This makes $\mathbf{C}^{\mathrm{mut}}$ to be a dense matrix ('fully connected'), even when $\mathcal{C}$ is ideal and diagonal (so $\mathbf{C}^{\mathrm{ext}}$ is a 'star' topology).

We can use this to express the Jamnik-Maier equivalent circuit picture of charge conservation in transport and electrostatics. (Assuming that the movement of ions is not causing any expansion of the medium):

$$\frac{\mathbf{C}^{\mathrm{ext}}}{\mathrm{volume}} \frac{\partial}{\partial t} \begin{bmatrix} V_1 \\ \vdots \\ V_N \\ \phi \end{bmatrix} = - \begin{bmatrix} \nabla \cdot \mathbf{J}_1 \\ \vdots \\ \nabla \cdot \mathbf{J}_N \\ \nabla \cdot \mathbf{J}_{\mathrm{d}} \end{bmatrix}$$

where $\mathbf{J}_{\mathrm{d}} = \partial \mathbf{D} / \partial t$ is the displacement current. The last row is basically Gauss's law, $\rho_{\mathrm{free}} = \nabla \cdot D$ but in a time-derivative form!^[7] The other part of the Jamnik-Maier picture is the constitutive relations: $\mathbf J_i = -\sigma_i \nabla V_i$ (Ohm's law i.e. drift-diffusion; resistors in the equivalent circuit), and $\mathbf{D} = - \varepsilon \nabla \phi$ (capacitors between displacement nodes on the equivalent circuit). The Jamnik-Maier equivalent circuit is based on discretizing this into finite volumes,^[8] but in fact the above is an elegant restatement of the Poisson-Nernst-Planck equations (which are continuum).

Takeaways

This whole appendix is one object, the chemical capacitance, seen through two matrices. The mutual $\mathbf{C}^{\mathrm{mut}}$ is the thermodynamic, $\phi$-agnostic one: a true capacitance matrix (its rows sum to zero) that lives in the charge-neutral bulk, sets the volumetric charge storage of a battery electrode, and carries the ambipolar diffusion modes. Because it asks nothing of a mean field, it survives into microscopically messy, nonideal materials. The internal $\mathcal{C}$ is the extrathermodynamic partner: it needs a well-defined $\phi$, and its rows do not sum to zero, but it is the only one that can speak about interfaces, screening (the total $\chi = \mathcal{C}_{\mathrm{tot}}/\text{vol}$), and depletion.

The two are not rivals but the same information at different resolutions. Eliminate the $\phi$ node from $\mathcal{C}$ (a Schur complement, or a Kron reduction of the equivalent circuit) and $\mathbf{C}^{\mathrm{mut}}$ drops out: strictly less detailed, yet more robust. Push the construction one step further, with Jamnik and Maier's displacement node, and the matrix becomes a full equivalent circuit: a finite-volume restatement of the Poisson-Nernst-Planck equations, with one capacitor hung from every carrier's $V_i$ rail. The interfaces that only $\mathcal{C}$ can see are exactly where the inhomogeneities topic picks up, at the far end of a chapter that first puts $\phi$ itself under the microscope.

NEXT TOPIC: $\phi$ under the microscope