Chemical resistance and chemical capacitance

1 Introduction

Chemical reactions are probably the most important processes of our world. They range from life-enabling biochemical reactions to solid state reactions providing functional materials [1]. They represent examples in which generalized fluxes (reaction rate, mass flux, electric current, heat flux, strain rate) are driven by forces (affinity, concentration gradient, voltage, temperature gradient, stress) [2]. In all these situations dissipation occurs which is due to the corresponding resistances but also pure storage, which is due to the corresponding capacitances. Close to equilibrium, fluxes and forces are proportional to each other and coupled via friction parameters (or their inverse quantities such as exchange rate, diffusion coefficient, electric or thermal conductivities, elasticity modulus). These friction parameters are proportional to resistances, a special role in this context being played by a chemical resistance. Far away from equilibrium one can define analogous differential parameters, but then it is usually more straightforward to apply master equations, which in the case of chemical reactions means applying chemical reaction kinetics [3], [4].

In the case of pure transport phenomena, storage effects occur only if flux divergences are involved (i.e. influx≠outflux). In chemical reactions, storage (i.e. local concentration variations) naturally occurs unless the product is quickly transported away. This coupling is directly referred to by the continuity equation (total storage=storage via flux divergence+storage via reaction rate). In the case of chemical reactions, it is straightforward to use master equations that include such effects implicitly. Close to equilibrium, storage effects can be, however, easily accounted for by introducing generalized capacities such as a chemical capacitance.

Using both chemical resistors and chemical capacitors allows for a very simple description of reaction networks, and by combining them with electric circuit elements, a simple description of solid state chemical processes or electrochemical processes is enabled in which transport plays a significant role. Let us first discuss these elements and then address a few selected problems that reveal the relevance for chemistry. More detailed but less intuitive treatments have been given in some of our earlier work (see e.g. [5]).

2 Exchange reactivity

The term “reactivity” is typically used in a rather undefined sense usually meaning the forward rate (ℛ⇀) of the fastest reaction that a substance can undergo. If the term is used to describe a specific, concentration-independent property, it is rather to be identified with the corresponding rate constant (k⇀).

A very clear definition can only be given close to equilibrium where forward and backward reactions are not very different, and it then refers to the exchange rate [3].

The simplest example considers A⇌B with

The exchange rate, that can be identified with the exchange reactivity, is thus composed of the geometric mean of the involved equilibrium concentrations (equilibrium values are denoted by hat) and the geometric mean of the rate constants.

(For this first order reaction ℛ° can – owing to mass conservation – also be written as k¯〈c〉 where k¯ is the arithmetic mean of the two rate constants and 〈c〉 the harmonic mean of the two equilibrium concentrations [A]⌢ and [B]⌢ [3]). The similarity to a conductivity σ∝u⋅c is obvious, where the mobility u corresponds to the rate constant of the hopping process and σ to its exchange rate. If the hopping process is viewed as a reaction with zero affinity (hop from one side to the next equivalent one), σ can indeed be shown to be the exchange rate of this process [3].

It is straightforward to demonstrate (Appendix A) that close to equilibrium any chemical reaction (A) reduces to a linear flux–driving force law

where the affinity (negative reaction free enthalpy) plays the role of the driving force. Clearly the validity range of linearity is small in the case of chemical kinetics and very often exceeded.

3 Chemical resistance

A precise definition of the chemical resistance refers to eq. (2) and is possible via

(Do not confuse the different R’s: resistance, gas constant, reaction rate).

In a sequence of reactions corresponding to a sum of Rδ values, it is the slowest step, i.e. the step with the largest Rδ, that determines the overall rate. In a parallel network (inverse Rδ values are summed) it is the fastest step (smallest Rδ) that determines the overall rate. In the steady state of such a reaction sequence all rates are equal. (The validity of Kirchhoff’s laws follows directly from the isomorphicity of the flux-force relations). Then all the other steps with a higher exchange rate exhibit a rate that is much lower than their exchange rates. According to eq. (2), their affinities are virtually zero, meaning that all these steps are now in quasi-equilibrium (A≃0 i.e. ℛ⇀≃ℛ↼) (see Fig. 1). The involved concentrations obey the specific (but not the global) mass action law but are time-dependent and different from true equilibrium concentrations.

Fig. 1:

(a) The rate determining step (rds: ​B⇌C) is the bottleneck, leading in the steady state to pre- and post-equilibria. These are quasi-equilibria as concentrations fulfil the respective mass action laws, but with time-dependent non-equilibrium concentrations. The water analogue shows then equal but time-variant water levels (for A⇌B and C⇌D). Reprinted from ref. [3]; (b) The oxygen chemical potential of an oxygen incorporation reaction into a solid whereby the surface step is the rate determining step. Reprinted from ref. [6] with permission from Elsevier.

A special role is met in the case of a diffusion controlled reaction [7], there ℛ corresponds to a flux, ℛ° to an exchange flux and Rδ to an inverse conductance. In a typical chemical process, such as a diffusion controlled stoichiometry change, the overall rate is determined by electroneutrally coupled motion of two carriers. In the case of a diffusion-controlled simple redox reaction we refer to the coupled motion of ions and electrons (e.g. transport of Li⁺ and e⁻ to change the lithium content, or counter transport of O²⁻ and 2e⁻ to change the oxygen content), yielding

Far from equilibrium the chemical resistance has to be defined differentially

A highly interesting case is met if an autocatalytic reaction is involved. Such reactions lead to growth, instabilities, oscillations, and structure formation. An autocatalytic reaction involves a negative chemical resistance. This is completely analogous to negative differential electric resistances that play a paramount role in electric nonlinearities and electronic structure formation in semi-conductors [8].

Let us consider the growth reaction

where the X-population grows at the expense of the consumption of F (“food”). Here ℛ=k⇀[X][F]−k↼[X]2 and A=RT(lnK−ln[X][F]) (cf. Appendix B for more details).

In Fig. 2 the courses of the reaction rate and of the chemical resistance is shown for the autocatalytic reaction (6) and compared with the above considered conversion reaction (X+F⇌Z). Let us concentrate on the beginning of the reaction, here ℛ≃ℛ⇀=k⇀[F][X]=d[X]dt. δℛ/δ[X]>0, i.e. δ[X]•δ[X]>0. This is the criterion of positive feedback [9], [10]. The same can be concluded from δℛ⋅δA which is also positive. It can be shown that this is equivalent to a positive variation of the entropy production, signifying the tendency of driving away from equilibria ([2], see also [3]). If the sign is opposite, the system moves toward equilibrium. The same is concluded from δ[X]•δ[X]<0. Such an opposite sign develops when the back-reaction becomes significant.

Fig. 2:

Chemical resistances and reaction rates for two chemical reactions (left column: conversion reaction; right column: autocatalytic reaction), as discussed in the text. The shaded areas in the individual figures refer to the regimes of linearity (LHS) and positive feedback (RHS). (See Appendix B for technical details.)

Reference [5] gives a more detailed account and also shows why not only a chemical resistance but also a chemical capacitance is necessary for a network description of reactions. Table 1 refers to generalized currents driven by both electrical and chemical driving forces.

The electrical and chemical driving forces can be unified to gradients in the electrochemical potentials. Such a unification is generally not possible as far as dielectric and chemical capacitances are concerned, as here the mechanisms are different. As far as the dielectric effects are concerned (see Table 1), a displacement current (Idis) is defined which is driven by the time derivative of the electric field. The missing equation is the mass conservation (continuity) equation that has to be written in terms of the chemical potential rather than concentration. This conversion from ∇c to ∇μ∗ then necessarily introduces a chemical capacitance.

4 Chemical capacitance

As chemical capacitance we define the increase of matter on varying the chemical potential. Note that in purely electrical processes the capacity is the increase of charge with increasing electrical potential.

As shown in a previous paper, generalized treatment of generalized capacitances is possible in an μ1n2TP ensemble [11]. In other words: We consider at given T, P a binary AxB in which the A content is defined by the mole number of A (n₂), while the B content is determined via the chemical potential of B (i.e. μ1), e.g. by establishing a partial pressure of B in the environment.

In the literature, this ensemble is referred to as an irrelevant one, the reason being that the intensive parameters are interrelated by an equation state. Yet this is only so if non-stoichiometries are neglected. In a material like PbO, to take an example, it is well possible to fix the particle number of Pb and then to independently fix the exact oxygen content by PO2 corresponding to the homogeneity width (PbO_1+δ). The ensemble is particularly applicable to multinaries such as perovskites ABO₃, where the ratio of A to B is often frozen while the oxygen content is still reversible. This ensemble is not only practically relevant, it moreover delivers, according to Table 2, the wanted capacitances as second derivatives of its characteristic thermodynamic function which we term Γ. (Note that Γ is at a minimum in an equilibrium for constant temperature, hydrostatic pressure and partial pressure of species 1.)

Considering Table 1, we can explicitly define generalized capacities via ∂2Γ∂λi2 where λ is short for p, T or μ1, namely the thermal capacitance (specific heat CpT≡T∂S∂T=−T∂2Γ∂T2), the mechanical capacitance (compressibility χ≡−1V∂V∂P=−1V∂2Γ∂P2) and our chemical capacitance, here for the component 1 C1δ≡∂n1∂μ1=−∂2Γ∂μ12. Interestingly a thermodynamic stability analysis [13] which relies on the second order variation of the respective thermodynamic potential function demonstrates that the specific heats, the compressibility but also a quantity such as C1δ must be positive. One realizes that the definition is also analogous to the electrical capacitance Cq=∂q∂ϕ (q: charge, ϕ: electric potential) which can be introduced by generalizing the work term in the thermodynamic relations. As exemplified below, chemical capacitances play a prominent role in stoichiometric variations and more prominently in electrode storage. As used in Table 1, it connects concentration effects with the thermodynamic driving force

Since in a battery ∂μLi=∂V (V: electric voltage), there the chemical capacity translates into an electric effect as well. In usual battery language the storable mass ΔnLi is used as integral “capacity”. This is sloppy but justified as then ΔV corresponds to the covered voltage window which stays invariant.

The concept of chemical capacitance proved to be very helpful in the case of stoichiometric variation as here Cδ can be given explicitly. If the non-stoichiometry is established by neutral defects such as in alloys (CuZn1+δ) or in ionic crystals at low T (where ionic and electronic defects are strongly correlated, e.g. YBa₂Cu₃O_6+δ), it simply holds that

In ionic crystals where point defects are rather dissociated

The χ-factors are unity if trapping effects are not important, and cion,ceon are sums of point defect concentrations [14].

In dilute systems ionic and electronic defects (j) follow characteristic power laws (power N) [15]

In such cases, it follows

showing that Cjδ is only significant if both Nj and cj are large. If the storage is due to a redox process (e.g. oxygen incorporation in an oxide), this is realized in regimes where ionic defects are compensated by electronic defects.

5 The use of chemical resistance and chemical capacitance for solid state processes

In the solid state, diffusion processes are usually important and coupled to interfacial reactions. Some selected examples given now will shed light on the use of the above introduced parameters.

5.1 Protection layers

Here we consider layers of reaction products that slow down or inhibit further reactions [3], [7].

Let us consider Al as an example. In spite of the huge affinity to form Al₂O₃ under air, Al can be safely used as construction metal. This is due to the very low transport rates which limit the oxide layer thickness to very small values and prevents – even for such small thicknesses – oxygen from reaching Al and Al from reaching the oxygen side. The low conductivities correspond to a high chemical resistance [cf. eq. (4)]. In other cases the oxide layer grows either towards the gas phase or into the metal depending on whether metal ions or oxygen ions are more mobile in the layer (Fig. 3).

Fig. 3:

Transport controlled growth of a passivation layer on top of a metal M exposed to oxygen. Reprinted from [16] with permission from Wiley & Sons.

A similar situation is met in Li batteries [17] where the electrolyte is almost exclusively unstable with respect to the positive and/or the negative electrodes (e.g. Li). Here the formed protection layers named SEI (solid electrolyte interphase) are a prerequisite for battery stability. The difference to the Al₂O₃ example is that the ionic conductivity is (desirably) not small (and required), but the very low electronic conductivity suffices to depress Rδ [cf. eq. (4)].

An important prerequisite for all these cases is the formation of dense layers. Otherwise fast parallel transport takes over (see Fig. 2).

5.3 Transport kinetics if the surface reaction is sluggish

Figure 5 shows the equivalent circuit for an impedance measurement of a high-temperature fuel cell cathode whereby the rate of the oxygen reduction reaction is important, i.e. influences the current to be extracted. As a consequence the corresponding reaction resistance matters and can be detected by the electric measurement. Storage only occurs in the electrode, but not in the electrolyte whose chemical capacitance is hence neglected. For details see [19].

Fig. 4:

Equivalent circuit for a battery exhibiting electric and chemical circuit elements. The upper figure displays the electrode (M) with its electronic and ionic contacts. Reprinted from [18] with permission from The American Association for the Advancement of Science.

Fig. 5:

Surface reaction dominated impedance of a high-temperature fuel cell cathode. Adapted from ref. [19].

5.4 Kinetics of stoichiometric changes in a polycrystalline ceramic

The precise evaluation of such examples in composites can be very complex even in simple brick-layer morphologies (see Fig. 6).

Fig. 6:

Simple model of a polycrystalline oxide in which oxygen is dissolved: LHS: Grain boundaries quickly transport the oxygen to the grains. RHS: Grain-boundary transport is sluggish, but oxygen is finally dissolved in the grains [11].

Let us recall that in a single crystal the time constant for stoichiometric changes τδ is given by RδCδ∝L2Dδ.

Let us first consider the situation that the grain boundary transport is very fast (Fig. 5 LHS). Then the relaxation time is determined by the diffusion in a single grain (size d) as the transport to all individual grains is fast. If the effective diffusion coefficient Deffδ is defined by τδ∝L2Deff (as it were for a homogeneous situation) then obviously

(12)Deffδ=DδL2d2

where L is the total size. As the overall chemical capacitance is not varied by the grain boundary (as long as the storage in the boundaries is negligible) and the chemical resistance is very much reduced, the effective diffusion coefficient is strongly increased if d≪L.

If on the other hand the grain boundaries are hardly permeable, the ambipolar conductivity σδ≡(σeonσionσeon+σion) is determined by the grain boundaries, while Cδ is still determined by the bulk (i.e. by cδ). The simple result is

(13)Deffδ∝ddgbσgbδcδ∝Dδddgbσgbδσδ.

Unless the grain boundary thickness dgb is very small, Deffδ is largely diminished as σgbδ≪σbδ. Should the bulk be absolutely impermeable unlike the boundaries, the storage time is determined by the product of the Rgbδ values and the now very small values of Cgbδ. This little storage was neglected in eq. (11). (Ref. [20] gives technologically relevant examples of Li storage, and thus of a high chemical capacitance, at heterointerfaces).

The reader may imagine that the same formalism can be used to describe much more complicated reactions or reaction-diffusion networks without or with electric contributions [5].

6 Summary

Precise definitions of chemical resistance and chemical capacitance are given. The terms are shown to be useful in particular in complex systems. Differentially defined they do also great service far from equilibrium. Used as elements of electrochemical circuits, simplifications are easily possible and intuitive even for conditions where the analytical treatment is intricate or impossible.