Mechanics of the Ideal Double-Layer Capacitor

The mechanical state within a parallel-plate electrolytic capacitor is examined by appending a local momentum balance to a quasielectrostatic theory that describes charge screening in both the electrolyte and the electrodes. A classical diffuse-double-layer model, which treats the capacitor ’ s separator as a dilute electrolytic solution, is augmented to include metal electrodes, modelled as electron gases. When accounted for in this way, the electrodes are found to impact the interfacial capacitance signi ﬁ cantly, as well as exerting compressive stress on the electrolyte. Nonlinear and quadratically perturbed theories are explored, the former around a single plate and the latter around the entire capacitor. Perturbation reveals several mechanical scaling laws generally applicable to capacitive metal/electrolyte interfaces. The two-plate model rationalizes the exponential decay of disjoining pressure between voltage-biased plates as their separation distance grows, as well as retrieving the well-known properties of a dielectric capacitor when the plate separation is small. This was Paper 1964 presented at the Dallas, Texas, Meeting of the Society, May 26-May 30, 2019. © 2020 The Author(s). Published on behalf of The Electrochemical Society by IOP Publishing Limited. This is an open access article under the terms of the Creative Commons Attribution 4.0 License (CC by/4.0/), which permits unrestricted reuse of the work in any medium, provided the original work is [DOI: 10.1149/ 1945-7111/ab6b04] This Paper is part of the JES Issue on Mathematical Modeling of Electrochemical Systems at Multiple in Honor of Richard Alkire.


Background
Among energy researchers, curiosity is growing about the mechanical responses that electricity can induce within electrolytic media. Our group has fleshed out a physico-chemical approach to such questions, based in classical equilibrium thermodynamics and the irreversible thermodynamics of Onsager 1 and de Groot. 2,3 The theory extends in principle to electrodynamic mass-transport phenomena that violate the condition of local electroneutrality. Other frameworks that describe equilibrium in electrically polarised, nonneutral solid ionic conductors have been put forward in the past by Kornyshev and Vorotyntsev,4 and more recently by Latz et al. 5 We have added to these earlier efforts by producing a general framework that also handles dissipative processes, 6 encompassing Newman's now-standard electrolytic transport equations, 7 but also accounting for mechanical dynamics via a coupled momentum balance. When applied to equilibrated systems, this dynamical model is consistent with Kornyshev-Vorotyntsev theory, as well as confirming results put forward by Latz.
Our model suggests that if a locally non-neutral electrolyte (i.e. a medium containing mobile charge carriers) is placed between blocking electrodes, and if a potential bias is applied between said electrodes, then double-layer formation is accompanied by a buildup of appreciable tensile stress in the bulk of the electrolyte relative to its edge. 8 In hopes of elucidating the practical problem of the "critical current", above which dendrites form in high-modulus ceramic lithium-ion conductors, I reported this observation in a recent seminar. 9 Dr. Biggins, now of Corpus Christi College, Cambridge, expressed surprise that tension developed in the ceramic, since it is obvious that the oppositely charged plates of a traditional dielectric capacitor, being electrostatically attracted, will exert a compressive stress on their separating medium. He also rightly pointed out that our model did not consider any action whatsoever by the capacitor plates; nor did it address how stresses from the plates would compare to any stresses that might develop in the separator due to space charges near its boundaries.
The question whether bulk tensile stress due to interfacial space charging is physical raised particular concerns with us. At low voltage bias, the electrochemical portion of our modelling framework produces a description of electrode/electrolyte interfaces more or less identical to Gouy-Chapman theory; 10 we merely used this electrochemical information to compute the local Lorentz body force in a relatively uncontroversial way. Gouy-Chapman theory is well established and experimentally validated, 11 and has been used for a century to explain capacitance due to diffuse double layers at the boundaries of liquid electrolytes.
Despite its age, the theory of diffuse double layers continues to advance. Great strides have been made recently by Bazant et al., 12 who created a variational analysis that addresses several pathological issues in the Gouy-Chapman model. Slightly later, Monroe et al. independently developed a similar variational method, using it to resolve specific problems that arise when applying Gouy-Chapman theory to parallel-plate electrolytic capacitors whose interelectrode spacing is small compared to the electrolyte's Debye length. 13 When electrodes are very close together, an analysis employing a traditionally defined Debye length, based on the electrolyte's average concentration, leads standard Gouy-Chapman models to produce concentration distributions that fail to conserve mass. While resolving this mass-conservation issue, Monroe and colleagues solved a toy model of a capacitive "nano-gap", b which could produce a classical dielectric response in the limit of infinite electrolyte dilution. Thus we have available to us an electrochemical model that exhibits double-layer capacitance or classical dielectric behavior in appropriate limits of the electrolyte concentration.
This essay sets out to model the voltage distribution and mechanical forces in a parallel-plate capacitor separated by an electrolyte whose Debye length can be either much smaller than or much greater than the plate separation, which we respectively call the electrolytic and dielectric limits of the capacitor's response. We shall determine the stress distribution in both the electrode and the separator regions, and examine this result in the electrolytic and dielectric extremes. Primarily, we aim to deliver suitable responses to Dr. Biggins's insightful comments, within the context of the general framework developed by Goyal and Monroe. electrolyte is added through a resealable access port to fill the interelectrode gap. The stiffness of the coil springs is such that L is negligibly affected by the hydrostatic force this added liquid exerts.
To run a thought experiment, we imagine connecting a potentiostat to the leads that protrude from the liquid-filled assembly and imposing a steady voltage bias between the electrodes. We expect this bias will cause charge accumulation in the metal at the electrode/electrolyte interfaces, which, being inert, block electrons from crossing. Since the ions in the liquid are mobile, and likewise cannot cross the interface to enter the metal, countercharges should simultaneously accumulate in the electrolyte near its two outer boundaries. In this way the whole device acts as a capacitor, whose differential capacitance can be measured by observing the change in surface charge on the metal with respect to the applied voltage bias, and should be determined by material properties peculiar to the electrodes and electrolyte. Because the charge distribution within the system exists in the presence of an electric field, applied voltage may also induce a mechanical force between the electrodes, which will be balanced by tension in the coil springs.
First we will observe the distributions of voltage, charge, and material that arise around one electrode of our hypothetical capacitor in response to the bias. A momentum balance will subsequently be exploited to elucidate the resultant mechanical forces. We shall solve this nonlinear problem exactly, but our solution requires a numerical computation of the potential at the metal/electrolyte interface. In the limit of small bias, a complete closed-form solution obtains. After these preliminaries we analyze the entire two-electrode system, also described by closed-form expressions when the voltage bias is small.
Governing equations.-Cauchy's equation expresses the local balance of momentum. No thermodynamic contradictions arise if the electric body force is included as a generation term within the momentum balance. 6 Everywhere within a quasielectrostatic stationary system like our capacitor, then,   s is the Cauchy stress tensor and  b quantifies local body forces exerted by external fields.
Note that the sign of our Cauchy stress opposes the one used by practitioners of pure mechanics. For consistency with typical practices in classical chemical thermodynamics and standard chemical-engineering transport theory, we use a Cauchy stress   s that follows the sign convention employed in the canonical textbook by Bird, Stewart, and Lightfoot. 14 In the BSL convention, the Cauchy stress is defined such that its average trace identifies with the external pressure p, ; 2 1 3 and the deformation-stress tensor   t is defined as the aspherical part of the total mechanical stress, in which   I is the identity tensor. Any components of a traction  t n that push on a surface with outward normal  n are defined to be positive, so that n and the surface integral produces the total external mechanical force  f that acts on a volume V bounded by surface ∂V.
Electromagnetic phenomena contribute to the body force  b . Analysis of the capacitor's mechanical state therefore requires an understanding of how space charge is distributed inside the metal electrodes, as well as within the electrolyte-we cannot use the standard postulates that the surface charges of metals lie in planes and that the potentials within bulk metals are uniform. Avoiding such assumptions is probably for the best anyway, because for them to hold, the Thomas-Fermi screening length of the metal must be negligible compared to the electrolyte's Debye length. These two characteristic lengths may be comparable, especially when electrolytes are moderately concentrated. 15 Everywhere within the experimental apparatus, the electric displacement field,  D, is governed by Poisson's equation, in which ρ e is the local excess charge density. When electrochemical equilibrium exists under a voltage bias-as in the electrolytic capacitor imagined here-charge density is usually taken to depend solely on the potential, but more complicated dependences can arise. For example, if materials in the capacitor were compressible, then the pressure distribution would impact both the charge-density distribution and the slab thickness, requiring Eqs. 1 and 6 to be solved simultaneously. It should be emphasized that Cauchy-Poisson analysis, based on Eqs. 1 and 6, is far from new. These simultaneous governing equations are commonly applied in electrohydrodynamics, particularly when describing colloids or liquid suspensions. 16 The novelty of Goyal and Monroe's work was its demonstration that Cauchy-Poisson analysis is consistent with Newman's electrochemical transport analysis, in the sense that no phenomenon is "double counted" if both theories are applied in tandem. Goyal and Monroe also dealt with the local constraints potentially imposed by thermodynamic state equations beyond the constitutive laws for electrochemical potential. For example, a local volumetric state equation derives from the Gibbs function and must be incorporated to ensure that the kinematic treatment of a material is consistent, 17 as well as to permit the study of systems in which pressure, temperature, etc. substantially affect local carrier concentrations or volume strain. Including the state equation for volume also helps when enforcing conservation of carrier mass, a constraint we will apply later when exploring the two-electrode system. Body-force constitutive laws.-The electrodes and electrolyte are assumed to be isotropic, so that the displacement field in medium k relates to its permittivity ϵ k through Introduction of the Maxwell stress defined in Eq. 10 allows the steady-state momentum balance to be written alternatively as This equation can, in principle, be integrated directly to set up local relationships between the Cauchy stress and the electric field.
Mechanical constitutive laws.-The region of primary interest within the device depicted in Fig. 1 comprises two solid electrodes separated by a liquid electrolyte. In a steady state under a voltage bias, both solids and the liquid will be static. Throughout the liquid phase, this implies that the deformation stress in Eq. 3 vanishes, i.e.     t = 0. In the solid phases,   t may vary; the electrodes can be idealized as purely elastic, so that there is a one-to-one relationship between   t and the instantaneous deformation state. As further idealizations, suppose that the electrolyte and electrodes both have very large bulk moduli, and also that the mass densities of both materials remain relatively constant with respect to carrier concentration. When considering Eq. 1, this allows one to suppose that any strain response due to stresses within the capacitor will be negligible. Thus the plate separation L (cf. Fig. 1) is independent of the capacitor's charge state, which decouples the momentum balance from the material and charge balances.
Electrochemical constitutive laws.-The decoupling of momentum and mass transport ensures that the electrical body forces depend solely on the distribution of voltage Φ, which will be determined below by solving isolated electrostatic problems. When phrasing these problems, we will think of the electrode interiors as ideal free-electron gases. Adopting a similar level of approximation, we take the electrolyte to be an ideal, nonreactive binary electrolytic solution, made up of a fully dissociated symmetric salt dissolved in a single dielectric-liquid solvent.
As discussed long ago by Rice, 19 the Sommerfeld-Fermi theory implies that the local molar concentration of electrons in a metal, c e , depends on the local potential as follows: 3 2 Here F is Faraday's constant, R the gas constant, and T the ambient temperature; c bulk represents the free-electron concentration in the electrically neutral bulk metal, which state resides at Fermi energy −F Φ bulk . Also, the dimensionless Sommerfeld-Fermi parameter 2 3 in which N A , h, and m e respectively represent Avogadro's number, Planck's constant, and the electron mass, expresses the thermal energy of the electron gas in units of its Fermi energy. Observe that γ is typically quite small: for gold, whose 98.0 M free-electron concentration at 25 • C is typical for metals under ambient conditions, 20 γ = 0.007. It follows from Eq. 12 and the definitions of c bulk and Φ bulk that the local excess charge density is Geometrical considerations.-The apparatus schematized in Fig. 1 is intended to behave as a one-dimensional system. By design, all material properties within the cylindrical shaft vary only with respect to the spatial variable x. Edge effects are neglected, and the housing is taken to be electrically insulating, so that the electric potential within and between the electrodes varies only with x, and is invariant with respect to translation or rotation in the plane normal to x, which we call the y-z plane for convenience.
Given a potential distribution that satisfies Eqs. 6, 7, and 9, the stress distribution can be found relatively easily after a few steps of c At any positive relative voltage c e > 0; if Φ − Φ bulk < −3RT/ 2γF, however, c e vanishes and ρ e becomes constant, equalling Fc bulk . We ignore this possible change in the metal's behavior at present because for a typical γ, the voltage threshhold is well outside the stability window of a typical liquid electrolyte at room temperature (−5.5 V for γ = 0.007). mechanical analysis that account for the one-dimensional nature of the apparatus. Since electric potential varies as ( ) F x , definition 9 shows that the only nontrivial component of Further, Eq. 10 shows that the only nontrivial component of the body-force vector Considerations of invariance in the y-z plane, along with an integration of the y and z components of Cauchy's Eq.
This scalar equation is readily integrable to yield the single stress component The normal stress σ xx is all that is needed to compute any net force between the capacitor's plates, which would be balanced by the coil springs shown in Fig. 1. When examining local property distributions, however, we find it more intuitive to cast results in terms of the thermodynamic pressure, p. The relationships between σ xx and p are straightforward, but differ between the liquid and solid phases in the experimental apparatus. In the elastic solid-electrode phases, the body force ( ) b x x induces a uniaxial stress in the x direction; perpendicular to the electric field, one therefore has that , where p θ stands for the ambient pressure. It follows from Eq. 3 that the deformation stresses in the y-z plane vary as . Consistency of the definition in Eq. 2 demands that the deformation-stress tensor be traceless, however, so these y-z deformations must bring about a normal deformation stress in the x direction: τ xx = −(τ yy + τ zz ). Thus the gauge normal stress is three times the gauge pressure, In the liquid phase,     t = 0 uniformly, so the stress tensor must be spherical. It follows that i.e. all three normal stress components match the total pressure.

The Semi-Infinite Capacitive Interface
First we analyze a plane interface at position x = 0, with metal at positions x < 0 (which we may call the left side of the interface) and electrolyte at positions x > 0 (the right). A bias ΔΦ relative to the bulk-solution potential Φ ref is applied, holding the Fermi level of the metal far to the left at Φ bulk = Φ ref + ΔΦ. In this situation Poisson's equation governs the field everywhere, and reduces to nonlinear ordinary differential equations in both media. It will be convenient below to use the transformation to cast all voltages dimensionlessly; for example, the dimensionless bias is denoted Δφ. We suppose the metal has constant permittivity ϵ M (at x < 0) and the liquid has permittivity ϵ (at x > 0). In the metal, whose potential is labeled with a subscript L, for "left", Eqs. 6, 7, 9, and 14 combine to give Here the Thomas-Fermi length λ M is defined in terms of the bulk free-electron concentration far to the left of the interface, can be reduced to a first-order ordinary differential equation by asserting that the local electric field in the metal, E L = −dΦ L /dx, depends exclusively on the local potential, so that ( ) f E L L . Direct integration, adding the requirement that the bulk Fermi level of the metal should correspond to a condition with no The sign function in this result selects the appropriate branch of the square root, assuming radicals are naturally positive. Branches were chosen based on the physical argument that there should be a positive surface charge on the metal when the interfacial potential is below Δφ. Through Eqs. 6 and 16, the electric potential in the liquid is governed by the classical nonlinear Poisson-Boltzmann equation sinh . 27 1 2 This was derived assuming that ( ) = E 0 0, because the liquid far to the right of the interface is electroneutral.
The potential is continuous across the interface between metal and liquid, requiring that ( ) Also stability requires the field to change monotonically as it crosses the interface, so the interfacial potential φ 0 must lie between 0 and Δφ. Introduce the bounded parameter 0 ⩽ α ⩽ 1 to represent the fraction of the total potential drop on the liquid side of the interface: φ 0 = α Δφ. Because conservation of charge requires the electric displacement to be continuous across the interface, then, or, inserting Eqs. 24 and 27, 5 2 0 0 2 in which we have squared both sides to eliminate the sign function and the square root, and then used a half-angle formula to remove the square of the hyperbolic sine. This provides an implicit equation whose solution establishes the function ( ) a f D . The partition coefficient α is bounded between 0 and 1, so Eq. 29 can be solved readily via a numerical binary search. Note that a regular perturbation analysis of Eq. 28 was used to identify the parameter which quantifies how material properties partition the bias in the limit as Δφ tends to 0. (Note that ϵ/λ has units of capacitance per area.) Figure 2 shows the fraction of the voltage drop across the liquid electrolyte as a function of voltage bias.
To provide a sense of how a typical electrode and electrolyte partition the bias voltage, consider a 0.05 M solution of sodium chloride in water (ϵ = 80ϵ 0 ), adjacent to gold. Kornyshev and Vorotyntsev suggest that for gold, ϵ M ≈ ϵ 0 . (Kornshev et al. state more precisely that 1 < ϵ M /ϵ 0 < 10 for most metals, and that experiments suggest the relative permittivity is usually closer to 1. 15 ) This makes the Thomas-Fermi length 0.59 Å-about a twentieth of the Debye length (13.7 Å). For this interface α 0 = 0.23, so most of the potential drop occurs on the metal side. Fits of differential capacitance data, such as the canonical results presented by Grahame, 21 might provide further insight into ϵ M and λ M . Absent precise values, we will present robust methods that allow computation of the partition coefficient ( ) a f D for any α 0 between 0 and 1. Note that for an ideal electrolyte, α 0 goes to unity at infinite dilution, and reaches a minimum at the salt's solubility limit.
One useful property of the ideal metal/solution interface is its differential areal capacitanceC, found by differentiating the surface charge density on the metal, Σ M , with respect to the total bias voltage: Here the last equality follows from displacement continuity, Eq. 28. Equation 27 can be inserted to get Albeit unwieldy, this is straightforward to compute once ( ) a f D is known from Eq. 29. Figure 3 presents the interfacial capacitance as a function of bias voltage for interfaces with a range of partition coefficients. The responses look generally like predictions of the Gouy-Chapman-Stern theory 22 -the capacitance saturates as the bias reaches large magnitudes, and dips near the potential of zero charge. Unlike the traditional Stern correction, however, the metal's screening behavior causes the capacitance signatures to be asymmetric between the high-bias extremes.
The potential distribution can be found through Eqs. 24 and 27, noting that the potential at x = 0 is αΔφ. Implement the method of Chapman 10 by separating and integrating Eq. 27 to find which describes the region to the right of the interface. Following a similar procedure, separate and integrate Eq. 24 to show that on the left of the interface (that is, when φ L /Δφ > α),  . Specific capacitance as a function of interfacial voltage drop for null-bias partition coefficients α 0 ranging from 0.1 to 0.9, in a system whose electrode has Sommerfeld-Fermi parameter γ = 0.007. d NB: A closed-form solution for ( ) f x L was found in terms of incomplete elliptic integrals, but the expression was cumbersome enough, and incomplete elliptic integrals sufficiently obscure, that we instead chose to compute the function numerically.
interface between gold and 0.05 M aqueous sodium chloride (γ = 0.007, α 0 = 0.23, λ M /λ = 0.042) are shown in Fig. 4. Now that the electrical problem is solved, we can proceed with the mechanical analysis. The Korteweg-Helmholtz contribution to normal stress can be considered in isolation, because the only spatial change in permittivity is a step from ϵ M to ϵ as x crosses zero. Since the dividing surface that separates the metal from the liquid is infinitesimally thin, and therefore contains no charge density, one has that represents the Heaviside step function. Korteweg-Helmholtz force accounts for a discontinuity in ( ) s x xx at x = 0, because Eqs. 28 and 36 combine to show that Insertion of the fields from Eqs. 24 and 27 into Eq. 38 yields explicit formulas for the pressure distribution, parameterized by the bias Δφ, the Sommerfeld-Fermi parameter γ, the null-bias partition coefficient α 0 , and the length ratio λ M /λ: Here, the scaling of p−p θ was selected to make the peak dimensionless pressure on the metal side exactly equal Δφ 2 in the limit of small bias.
We can now provide a clear response to Dr. Biggins' first comment. Figure 5 shows how the pressure distribution around the interface varies as the metal is raised to increasingly positive voltages. (The distributions remain qualitatively similar when the bias is negative.) Although the bulk of the electrolyte is under tensile stress relative to its edge, the bulk actually resides at ambient pressure. Due to attraction between the back-to-back space-charge layers in the metal and electrolyte, a compressive force concentrates in a very thin region near the interface, whose width is of the order of ( ) l l + and does not change appreciably with bias. The maximum gauge pressures on both sides of the interface are directly proportional to Δφ 2 when the bias is small, and remain of the order of Δφ 2 at higher voltages. On the liquid side, the scale of the peak pressure differs from the metal's by a factor of three times the dielectric-constant ratio, since α 0 λ M /(1 − α 0 ) λ = ϵ M /ϵ.

Semi-Infinite Analysis at Low Bias
Approximate solutions of the nonlinear model can provide a clearer quantitative understanding, delivering scaling laws that show  the orders of the forces concentrated around the double layer. We begin by analyzing the semi-infinite metal-electrolyte interface in the limit where the dimensionless bias Δφ is small-an effort that will help to illustrate methods for tackling the two-electrode problem.
A regular perturbation analysis can be performed with respect to the dimensionless bias by letting Functions ( ) f L 1 , φ (1) , ( ) f L 2 , and φ (2) are expected to be O(1). A key result of this section will be to uncover the natural scale of pressure at the interface by analyzing the perturbed momentum balance.
Perturbation of Eq. 38 requires that p (1) = 0, in line with the prior observation that interfacial pressure scales as Δφ 2 (cf. Fig. 5). The first significant electrostatic contribution to pressure is First-order electrical information suffices to establish the pressure distribution. The electrostatic analysis will still be carried out to second order, however, to explore how capacitance varies with bias. The left side of the interface is governed by Eq. 22 and the right satisfies Eq. 25. By ansatz, voltages φ and φ L are both of the order of Δφ. Thus, if the total bias is small, then the right side of Eq. 22 can be approximated by Maclaurin expansion in Δφ − φ L , as Here the zero-bias partition coefficient α 0 , introduced earlier to simplify Eq. 30, arises naturally from the matching boundary conditions. These potential distributions allow one to identify the interfacial capacitance through Eq. 31, Equation 50 indicates a possible route by which the Sommerfeld-Fermi parameter could be measured experimentally: γ determines the slope of the differential capacitance at the potential of zero charge. If the value of γ = 0.007 for gold is correct, one would expect this linear variation of C with bias to be very weak, however. Observe that the finite slope of˜( ) f D C at zero bias shows that the commonly assumed equivalence between the potential of zero charge and the global differential-capacitance minimum cannot be drawn. This raises a general philosophical question about how the potential of zero charge can be established experimentally, since most electrode/electrolyte interfaces are presumably both nonideal and inherently charged.
Insertion of the first-order terms from Eq. 49 into Eq. 41 shows that the second-order contribution to the pressure distribution is The analysis has thus led us to a new parameter p 0 , which expresses the natural scale of the gauge pressure: Further recalling that α 0 λ M /(1 − α 0 )λ = ϵ M /ϵ = 1/80, Eq. 52 shows the peak pressure on the gold side to rise at 0.27 kPa · mV −2 , whereas the aqueous peak pressure grows at · -10 Pa mV . 2 Equations 52 and 53 expose several mechanical rules of thumb applicable to any electrochemical interfaces that support space charging. First, as we set out to show, the gauge stress in the double layer is always compressive, regardless of the sign of the bias. Second, stress peaks at the interface, and falls off exponentially to the ambient pressure on either side, with a characteristic length half of that phase's screening length. Third, pressure at the interface scales with the square of the total interfacial capacitance, and inversely with the geometric mean of the bulk permittivities on either side-interfaces with higher total capacitance experience much higher peak pressures. Fourth, the peak stress is highest on the side with lower permittivity, and lower on the opposing side; the ratio of these maxima scales with the bulk permittivity ratio, and the difference between them owes to the Korteweg-Helmholtz force. Last, the mechanical impact of double-layer charging can be significant, despite the fact that it emerges only at second order in the governing equations-a bias of just 19 mV produces about 1 atmosphere of gauge pressure on the gold side of an aqueous NaCl/ gold interface.

The Parallel-Plate Double-Layer Capacitor
Although they led to several useful qualitative observations, the semi-infinite analyses-both nonlinear and regularly perturbed-left one of our key motivating questions unresolved. We originally sought to query the dielectric extreme of electrolyte behavior, to check consistency of the present model with the well-known expression for the force between plates in a planar dielectric capacitor. One should be able to explore this regime analytically by taking l  ¥, but the analysis up to now cannot produce sensible results in that limit because the plate separation L was not brought into account.
Setup.-We shall finally consider the entire system from Fig. 1, with two plane metal/electrolyte interfaces, one at position x = 0 and another at x = L. One metal electrode, which we shall still call the left electrode, denoting its potential φ L , resides at positions x < 0. The other metal electrode resides at positions x > L; we shall call this the right electrode, whose potential is φ R . We assume that the two electrodes are composed of the same metal, so that both have the same permittivity ϵ M and the bulk electron concentration on the left, -¥ c , equals the bulk concentration on the right, ¥ c ; since = ¥ -¥ c c , both metals have the Thomas-Fermi length λ M defined in Eq. 23.
Whereas the electron concentrations on the far left and the far right are similar, the bulk metal's electrostatic potentials (Φ bulk in Eqs. 12 and 14) may differ. The left electrode's dimensionless Fermi level relative to the electrolyte's potential of zero charge will be designated φ bL , and the right electrode's, φ bR : two limiting relationships, and lim , 54 serve as far-field boundary conditions. On either side of the electrolyte domain, the potential and displacement field satisfy the matching conditions Here we continue to use φ to represent the potential distribution in the separating electrolyte.
Auxiliary constraints.-Before stating the governing equations, it is worthwhile to take a step back and consider three auxiliary conditions that physically self-consistent solutions of the twoelectrode problem must meet. Most simply, the application of voltage bias by a potentiostat fixes the difference between the electrodes' Fermi levels, Also, whatever its electrical state, if an electrolyte domain of finite size has nonreactive boundaries, then the total numbers of anions and cations it contains must be conserved. The two additional auxiliary conditions associated with ion conservation under bias are more conveniently expressed as the conservation of net charge and the conservation of total salt mass.
In the liquid phase, global conservation of net charge requires that One can express this as a functional of φ by inserting Eq. 16 for ρ e . Alternatively, put Eq. 6 into condition 57, then apply the divergence theorem to produce a simpler relationship to apply in practice.
When describing the electrowetting of liquid/liquid interfaces on metal surfaces, Monroe and colleagues showed how charge conservation constrains the zero-charge potential Φ ref (cf. Eqs. 15-17) of a fixed-volume electrolytic droplet. 23 Within the variational approach of Bazant et al., Φ ref arises as a Lagrange multiplier associated with the application of constraint 57 while minimizing the free-energy functional that applies to the electrolyte phase. 12 These prior observations will manifest differently here because of the choice of reference state for the dimensionless potentials: since Eq. 21 defines the voltage relative to the liquid's potential of zero charge, Φ ref does not appear directly below. Instead, constraint 58 will force an additional relationship-beyond Eq. 56-between the metals' dimensionless Fermi levels.
Global conservation of salt mass in the liquid phase implies the condition This self-consistency constraint extracts the value of θ from the potential distribution, thereby determining the zero-charge To find the capacitor's equilibrium electrical state under a voltage bias Δφ, one should solve this system subject to far-field boundary conditions 54 and matching boundary conditions 55. After finding a general solution, self-consistent values of the bulk parameters φ bL , φ bR , and θ in terms of Δφ emerge from the application of auxiliary conditions 56, 58, and 62. Ultimately, the local mechanical state throughout the capacitor can be computed in terms of the potential distribution, through ; . As we found when analyzing the semi-infinite capacitive interface above, Korteweg-Helmholtz forces will account for pressure discontinuities at both liquid/metal interfaces.
Perturbation analysis.-This problem is amenable to analytical attack by regular perturbation in bias, following the method from the "Semi-infinite analysis at low bias" section. In the first instance take the total bias to be small, so that the right sides of Eq. 63 can be replaced with Maclaurin expansions like those laid out in Eqs. 42 and 43. This small-bias assumption also allows auxiliary condition 62 to be rewritten as auxiliary conditions 56 and 58 retain the same form whatever the size of the bias.
To the list of perturbation-expanded functions in Eq. 40, add to describe the potential distribution in the right electrode. Also, to incorporate the auxiliary conditions, include perturbed parameters Being linear, matching conditions 55 are identical at every order of perturbation, taking perturbed forms similar to Eq. 48. Upon perturbation, far-field conditions 54 become and lim , 70 with the constant bulk potentials determined at all orders by the perturbed auxiliary condition Global charge neutrality of the liquid phase implies a perturbed auxiliary condition that, like the matching conditions, holds at every order, A first-order perturbation of the conservation law for liquid-phase salt mass shows that θ (1) = 0. At second order, the remaining auxiliary condition becomes in which the final equality includes the first-order result. Solution of the perturbation problem produces results that involve material parameters γ and a á ñ 0 , with the latter put in triangular brackets to emphasize its basis on l á ñ (and hence á ñ c ), rather than λ (or c ref ). Also, a new dimensionless geometric parameter appears, which puts the electrode separation in units of two unbiased Debye lengths. We expect that the dielectric limit of behavior should attain when Λ = 1; the electrolytic limit holds when Λ ? 1.
Distributions.-We shall see that the perturbed potential distributions depend naturally on the total null-bias differential capacitance of the parallel-electrode cell, 2t a n h . Observe that in the limit L  ¥,˜  á ñ C C 1 2 0 -that is, when the electrode spacing is large, the capacitance approaches that of two semi-infinite double-layer capacitances in series. When Λ = 1, on the other hand, e Some additional analysis is needed to get θ (0) if the metal/electrolyte interfaces have inherent surface charges at unbiased equilibrium, a case where Δφ = 0 does not require that φ bL = φ bR = 0. Inherently charged surfaces also might require one to revisit matching conditions 55 and auxiliary condition 57. For simplicity we do not consider such situations now, but they may be more realistic.
If the electrode spacing is much smaller than the Debye length of the electrolyte, but much more than 2ϵ/ϵ M times the Thomas-Fermi length of the electrodes, the system will satisfy the classical expression for dielectric capacitors,˜ »  C L. Thus the electrostatic model accesses the dielectric limit defined in the preamble.
When electrolytes have large relative permittivity it may be hard to achieve the dielectric limit cleanly in practice, however. For an aqueous solution sandwiched between gold electrodes, the electrode spacing must be much larger than 9.4 nm for the cell capacitance to be unaffected by Thomas-Fermi screening; for the Debye length to greatly exceed 9.4 nm, the salt concentration in water at room temperature must be far less than 1 mM.
Interesting features specific to the parallel-plate configuration appear when Λ = O (1), a domain we shall explore after developing dimensionless forms of the potential distributions. To state these more simply it is helpful to introduce a parameter analogous to α, which quantifies the fraction of the potential drop that occurs across the capacitor's separator in the limit of zero bias: t a n h . As well as the interfacial property a á ñ 0 , which can be computed solely in terms of metal and liquid properties, the partitioning of bias in the two-electrode case also depends on the plate separation, embodied by the geometric quantity Λ. The first equality here emphasizes that the fraction of the potential drop partitioned to the liquid can be found by putting the total cell capacitance in units of two series Thomas-Fermi-layer capacitances.
Together, differential governing systems 68 and 69, accompanied by the perturbed forms of matching conditions 55, far-field conditions 70, and auxiliary conditions 71 to 73, determine the potential distribution throughout the parallel-plate capacitor up to order Δφ 2 . Incorporation of the relative-capacitance parameter α ∥ into the distribution shows that Here the first-order perturbations of the potentials are similar to those found in the "Semi-infinite analysis at low bias" section, with α 0 replaced by α ∥ and half the bias apportioned to each interface. f The second-order terms in the expansions differ substantially from expectations set by Eq. 49, however. Figure 6 shows potential distributions across the ideal parallelplate electrolytic capacitor for plate separations ranging from 2 Debye lengths (Λ = 1) up to 200 Debye lengths (Λ = 100). (A lower electrolyte concentration of 0.5 mM was used when making this figure, to partition more of the applied bias to the separator domain.) When the plate separation is much smaller than the Debye length, the number of charge carriers in the electrolyte is too low to screen the electrodes' surface charges. The field between the electrodes therefore remains essentially constant, so the potential varies linearly across the liquid phase, as one would expect in a dielectric capacitor.
With much larger plate separation, the number of carriers suffices to facilitate complete charge screening. The field decreases rapidly within thin layers near the electrode surfaces, leading to a corresponding steep change in potential, and vanishes in the bulk of the separator, where the potential plateaus to a constant value. The model interpolates naturally between dielectric and electrolytic extremes. When the plates are around six Debye lengths apart (L = » 10 3.2), there is a mixed response, where some screening effect is observable, but a field still penetrates through the point of zero excess charge density halfway across the cell.
Differences between second-order terms of the potential distributions in Eqs. 49 and 78 can be understood by examining the selfconsistent parameters obtained from the parallel-plate analysis: Constraints expressing the conservation of ions require that when applied voltage increases, the liquid's potential of zero charge must shift downward relative to the bulk Fermi levels of both metal electrodes, as shown by the second-order terms in φ bL and φ bR . When ions move away from the bulk to screen surface charges at the electrolyte's boundaries, the electrolyte's concentration of zero excess charge also falls, leading to the second-order term in θ. In contrast, the semi-infinite analysis provided no route by which the potential or concentration of zero charge could change, so higher-order screening effects in the metal had to be compensated by changing the shape of the potential distribution on the liquid side of the interface. The local pressure is found by inserting Eq. 78 into Eq. 64. Up to the second order of perturbation, ⎧ has been magnified to illustrate the behavior within the separator, since the scale of electrolyte pressure differs from the electrode pressure by a factor of 3ϵ M /ϵ. The maximum pressure, plotted in the inset panel, occurs on the gold sides of the gold/electrolyte interfaces. Although maximum pressure has a sigmoidal shape with respect to plate separation, it drops by less than a factor of 2 between the dielectric and electrolytic extremes.
Disjoining stress.-We are finally in a position to determine what force-if any-attracts the capacitor's plates together, i.e. the disjoining stress. Through Eq. 5, disjoining stress will manifest as a net force that pulls the electrodes together, which will be compensated by the coil springs in Fig. 1. To quantify this attraction, one must distinguish local forces, which squeeze space-charge layers together near each interface, from the externally observable force that draws one electrode toward the other. Charge screening generally reduces this external force; compressive stress concentrates near the interface, but does not penetrate across the electrolyte.
Only the unscreened portion of the field, that is, the portion of the field that exists along the surface of zero excess charge density within the separator, pulls the electrodes together. In the present situation, for reasons of symmetry, said surface resides at x = L/2, halfway across the cell. Bearing in mind Eq. 20, the gauge stress there is found through Eq. 80, after substantial rearrangement and simplification involving Eqs. 21, 23, 51, 53, 60, and 77, to be The first of these expressions is useful to assess asymptotic behavior when the Debye length l á ñ is very large compared to the plate spacing, i.e. in the limit Λ = 1; the second is useful when  l á ñ L, so that  L 1. Away from the regime where λ M = L and  l á ñ L , the dimensionless disjoining stress Π (2) can be written as This result confirms the observation by Israelachvili, and before him, Verwey and Overbeek, that the electrostatic interaction between two charged surfaces should decay exponentially with their separation distance; the rate of decay is determined by the Debye length. 24,25 Our model shows that at moderate separation distances (when Λ ∼ 1), the electrostatic disjoining stress can deviate from exponential decay because of Thomas-Fermi screening within the electrode itself. This effect is present even in the linear theory (first-order perturbation). The disjoining pressure generally depends on a á ñ 0 , and is illustrated as a function of plate separation in Fig. 8. Observe that a á ñ 0 tends toward unity when  l l á ñ M -in this instance, electrostatics suggests that higher-than-expected disjoining pressures should be observed. Upward deviation from exponential decay as plates grow nearer to each other is commonly seen in experiments with the surface force balance, 26  Outlook A few new phenomena and scaling laws have been elucidated by this extended electrostatic theory, which considers electronic  structure within the electrode surface as well as charge screening in the adjacent electrolyte. Several motivating questions about the interelectrode forces and stresses within electrolytic capacitors were resolved. The theory shows that charge accumulation in a double layer always produces compressive gauge stresses, which peak at the interface and decay with respect to distance on either side of it. Underscreening of the field in the separating electrolyte can result in an externally observable interelectrode force, which manifests as a disjoining pressure. In the limit that the Thomas-Fermi length is very small compared to the plate separation and the plate separation is very small compared to the Debye length, the disjoining pressure achieves the value expected for a traditional dielectric capacitor. Away from this regime, the disjoining pressure decays exponentially as the electrodes are drawn apart.
The idealization of the system presented here helped to illustrate basic scaling laws, and also allowed us to clearly outline a broadly applicable analytical approach. Applications to practical electrolytic capacitors probably require several more detailed considerations. Straightforward modification of the theory could be made to account for inherently charged interfaces, which would produce a repulsive contribution to the disjoining pressure in a symmetric cell. It should also be possible to include more realistic physics, such as detailed band structures of the electrodes and entropic effects arising from van der Waals forces, by accounting for nonideality in the electrochemical constitutive laws that relate carrier concentrations to potential (and possibly pressure). A coupled Cauchy-Poisson analysis, which considers the finite bulk moduli and compositiondependent density of the materials making up the capacitor, could be enlightening.