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INDUCTIVE BEHAVIOR IN ELECTROCHEMICAL MECHANISMS. David A. Harrington Pauline van den Driessche. Chemistry and Mathematics Departments, University of Victoria, Victoria, B.C. Canada, V8W 3V6. email: funding: NSERC & Uvic.

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David A. Harrington

Pauline van den Driessche

Chemistry and Mathematics Departments,

University of Victoria,

Victoria, B.C.

Canada, V8W 3V6.


funding: NSERC & Uvic.

An annotated (but unanimated) version of a talk given at the 6th International Symposium

on Electrochemical Impedance Spectroscopy, Cocoa Beach, Florida, 20 May, 2004.


It has been known for a long time how to take kinetic equations and derive the

impedance or the resulting equivalent circuit. But the experimenter who finds an

impedance spectrum with certain features (e.g., number of time constants, presence

of an inductive loop) would like to know what classes of mechanisms might give

rise to this behavior, i.e., the inverse problem of impedance to mechanism is a

qualitative but important problem.


As an example, consider the H.E.R. In general we assume a series of elementary

reactions, and for simplicity we consider the case where surface reactions occur

without mass transport limitations. A complete list of assumptions are given later;

they are the same ones that most workers make. M here means the atoms in the

surface layer (say Pt for a Pt catalyst) and MH means an H atom adsorbed on an

M atom.


We classify the species in the mechanism. Of course in electrochemistry, electrons

have a special role. We have adsorbed species, and we will treat the reaction sites

like adsorbed species. We also have some species in solution whose mass transport

is assumed to be so fast that we consider their concentrations at the surface to be

constant. These are called “static” or “external” species.


The concentrations of the static species can be included in the rate constants, and so

they won’t play a role in determining the type of behavior (though of course the values

of the equivalent circuit elements will depend on them). We omit them. We can write the

kinetics in the usual way, assuming for simplicity Langmuir kinetics and Tafel rate

constants for electron transfer steps (only one shown for simplicity).


Note that there are the same number of M atoms on each side of the reaction,

i.e., M atoms are conserved. Writing 1-q as the coverage of sites is another

statement of the same thing. Most mechanisms we write down have at least one

conservation condition, and this fact turns out to be significant in constraining the

possible types of behavior.


The numbers in front of the species are the stoichiometric coefficients and they play

an important role in the theory.

By convention the stoichiometric coefficients of reactants are negative.


Now we construct a stoichiometric matrix. It has one row for each species,

electrons first, and one column for each reaction.

For example, the first column is -1, 1 and 1 because in the first step we lose one

electron, make one MH and lose one M. We notice that the columns of numbers

are related: column 3 = column 2 – column 1, so not all reactions are independent.


From two of the reactions we can create the third; here step 3 is the sum of step 2 and

step 1 written backwards, so only two of the three reactions are independent.

Mathematically, I is the rank of the stoichiometric matrix. I determines the complexity of

the circuit, so adding the third step doesn’t complicate matters. In general, adding more

steps to a mechanism need not increase the complexity of the equivalent circuit.


Now we show how to construct the impedance from a reaction mechanism. We first

construct a matrix for each elementary reaction step, an “elementary matrix”. We will

then add them all together to get an overall matrix Q. We illustrate this for the first step

of the hydrogen evolution reaction. We start by creating the column vector of

stoichiometric coefficients as we did before. We duplicate this as a row vector.


Now we multiply each entry of the row vector by a rate. For the electrons it is a special

rate v1e that is a combination of the forward and reverse reaction rates, weighted by

the symmetry factor for that step. For the other species, we multiply by the forward

rate v1f if the stoichiometric coefficient is negative and by the backward rate v1b if the

stoichiometric coefficient is positive.


Now we divide each entry of the row vector by the coverage of the corresponding

species. MH is divided by its coverage q, M by its coverage j = 1-q.

For the electrons, we divide by qe, which is a combination of constants including

the double-layer capacitance.


We multiply the two vectors together to give a matrix according to the rules of matrix

multiplication, e.g., the third entry in the second row is the product of the third entry of

the row vector and the second entry of the column vector.


We add the elementary matrices for each step to get an overall matrix Q for the

mechanism. For many of the results we obtain, we do not need to know the exact

values of the entries, only their signs. This means that the results derived here

assuming the Langmuir isotherm and Tafel potential dependence of rate constants

are true also under somewhat more relaxed conditions. Note that if we had only steps

1 and 3 we would know the signs for Q, but steps 1 and 2 have conflicting signs

– this leads to the potential inductive behavior.


The impedance Z is determined from the matrix Q as shown. This impedance includes

the double-layer capacitance parallel to the Faradaic impedance. The vertical bars

denote determinants, and the notation Q(1) means the matrix Q stripped of its first row

and column; note that Q(1) has no explicit information about where the electrons are

in the mechanism. I is an identity matrix of appropriate size and s is iw.





But this expectation is wrong. Let’s see why...

Now we give two simple examples that illustrate the point that stoichiometry matters.

The first is the adsorption of hydrogen from two possible proton sources in solution,

hydronium or bisulfate. We would expect that since these two reactions will have different

rates that we will see the equivalent circuit above, with separate time constants

from the two charge-transfer/pseudocapacitance combinations.



X = 0


Next we write out the electrons and the external species, and ask the question:

Can we make a balanced reaction from them that includes electrons?

Here the answer is no, because there are no changes of oxidation state in the different

static species. We denote this impossibility by saying that the parameter X=0. This means

that there will be no dc path through the circuit.

The circuit has only one resistor/pseudocapacitor: we cannot separate out the rates of

the two steps.

I = 1

First of all we remove the external species and see that the two reactions look just the

same: they are not independent and I = 1. This is the number of resistors in the circuit.



I = 2


This time when we take away the static species, we do not have the same reaction,

and I = 2. Therefore we will have two resistors in the circuit.

X = 2

Consider another example, in which we also have two ways of adsorbing a single

species. The presence of the 2 in front of the electrons will make a big difference.

And this time we can make a reaction out of electrons and static species, which

possibility we denote by X = 2. This means there will be a d.c. path through the circuit.

The circuit is quite different. A little thought shows that the overall reaction of an electro-

catalytic mechanism serves as the reaction to give X = 2. In this case reaction 1 going

backwards and reaction 2 going forwards effects the redox reaction in solution.



Some classical circuit theory tells us which type of circuit follows from which impedance

expression. Either kinetic impedances or circuit impedances may be simplified to ...

a ratio of two polynomials in s. The a values are the zeroes of the impedance and

the b values are the poles.

These poles and zeroes may be plotted in the complex s plane. Their locations

determine the type of circuit.


Resistors and

capacitors only


An RC circuit has alternating (or “interlacing”) poles and zeroes lying on the negative

real axis. A pole is nearest to the origin and may lie at the origin (this is when X = 0).

This is a rather demanding set of conditions. If any of the poles or zeros are not real

and negative, or if the interlacing fails, then we will have an inductor in the circuit (if a

circuit is possible at all). So as a general rule, we expect inductive behavior to arise in

more mechanisms than not.



(and RC)

Type I


If some zeroes are complex (but still in the left half plane), then we cannot have an RC

circuit and must have an inductor (and usually resistors and capacitors as well).

We will call this way in which an inductor arises, “Type I” inductive behavior.

The zeroes arise from the matrix Q(1) (they are the negatives of the eigenvalues of

Q(1)/Gm). Recall that this matrix doesn’t depend on where the electrons are in the

mechanism. So we can determine “Type I” inductive behavior from the mechanism

stripped of electrons.



(and RC)

Type II


“Type II” inductive behavior arises when the zeroes are real and negative, but the

interlacing property fails, either because a pole becomes complex or the poles stay

real but don’t alternate with the zeroes (as shown).

So in this case, knowledge of the zeroes alone is insufficient to determine whether or

not the circuit is inductive. The poles arise from the matrix Q (they are the negatives of

the eigenvalues of Q/Gm). This matrix has the electron row and column in and so

the location of the electrons is crucial in determining the inductive behavior.



circuit possible



For some pole-zero arrangements, it is impossible to find a circuit containing only

resistors, capacitors and inductors (with positive values) that has this pole-zero pattern.

This happens when a zero moves over into the right half plane, and is associated (at

least for potentiostatic control) with unstable behavior. Circuits also can’t be found if

poles move to the right half plane (unstable behavior under galvanostatic control),

or if the impedance spectrum moves over into its left half plane (has a negative real part).



Definition: Impedance that is realizable

as an equivalent circuit

containing an inductor.

It is evident that our definition of inductive behavior comes from circuit theory.

Usually, when the impedance goes below the axis in the Nyquist plot, then we have

inductive behavior in the sense of the definition above. But not always, since some

below-the-axis behavior comes from impedances that are not realizable as circuits at

all (e.g. if they move into the left half plane). Also, sometimes a circuit can have an

inductor in it which is swamped by the other circuit elements and the spectrum

doesn’t go below the axis. So we have a definition which is a bit more picky that just

“below the axis”.



SINGLE-STEP MECHANISM: cannot be inductive

MECHANISM AT EQUILIBRIUM: cannot be inductive

We can show that no single-step mechanism can be inductive (even several steps if

their stoichiometries without external species are the same or multiples of each other

like the X = 0 example considered earlier).

We can also show that mechanisms at equilibrium cannot be inductive (this conclusion

extends to the case where the solution species are diffusing and not external). In fact,

the impedance tends to have more structure at equilibrium, and so this the best place

to make measurements, if it is possible to do so.




need two electron transfer steps: one ODA + one RDA

For mechanisms with a single adsorbed species (that’s two if you count the sites as

adsorbed), to get inductive behavior at least two electron transfer steps are needed.

Steps 1 and 2 of the HER provide an example (above): step 1 is reducing in the

direction of adsorption (the forward direction) and step 2 is oxidizing in the direction of

adsorption (which is the backward direction).

So a metal on which the HER proceeds by steps 1 and 3 cannot give inductive

behavior. Observation of inductive behavior for the HER is simple, qualitative

evidence that step 2 must be occurring.



1-e TREE GRAPH MECHANISM: cannot be inductive.

There is another class of mechanisms that give RC circuits. These may have any

number of adsorbed species; they can have only one electron and the graph without

the electron is a tree graph. The graph is made by replacing each adsorbed species

by a vertex and the reaction arrows by an edge. The reactions must have only one

product species and one reactant species (2A->3B is allowed: A is the reactant and

B is the product). As shown over, a graph is a tree if there are no cycles. So the first

mechanism has an RC circuit no matter where the electron is. On the other hand, the

simple cycle mechanism can be inductive no matter what the electron arrangement is:

it is type I inductive.


e- +

e- +



(no cycles)




I didn’t have time to talk about this, but for these tree

graphs it is possible to write down a topological equivalent

circuit directly from the reaction graph. See JEC (2004) for details.



  • Series of elementary steps at the electrode/
  • solution interface.
  • Rapid diffusion of solution species
  • (static or external species).
  • Potential-dependent rate constants for
  • electron-transfer steps (Tafelian
  • or monotonic dependence).
  • Reactions occur in forward and reverse
  • directions.
  • No species on product and reactant side of
  • the same reaction.
  • [Langmuir (mass-action) kinetics
  • for adsorbed species.]
  • [A conservation condition exists
  • (of metal surface atoms)]

For the record, here are the assumptions we make. The last two are needed only for

some of the results.



  • Circuit complexity and structure are determined by stoichiometry.
  • Requirements for RC circuits are stringent.
  • There are two distinct types of inductive behavior.
  • Some rules about when inductive behavior occurs can be given.
  • The graph structure can be related to the impedance.
  • References:
  • D.A. Harrington and P. van den Driessche, Equivalent Circuits for Some Surface Electrochemical Mechanisms,
  • J. Electroanal. Chem., 567 (2004) 153-166. (Tree mechanisms, topological equivalent circuits).
  • J. D. Campbell, D. A. Harrington, P. van den Driessche and J. Watmough, Stability of Surface Mechanisms with Three Species
  • and Mass-Action Kinetics, J. Math. Chem., 32 (2002) 281-301. (Two adsorbed species, no electrons, i.e., zeroes only)
  • D.A. Harrington and P. van den Driessche, Stability and Electrochemical Impedance of Mechanisms with a Single Adsorbed Species,
  • J. Electroanal. Chem., 501 (2001) 222-234.
  • D.A. Harrington and P. van den Driessche, Impedance of Multistep Mechanisms: Equivalent Circuits at Equilibrium,
  • Electrochim. Acta, 44 (1999) 4321-4329.
  • D.A. Harrington, Electrochemical Impedance of Multistep Mechanisms:
  • Mechanisms with Static Species, J. Electroanal. Chem., 449 (1998) 29-37. A General Theory, J. Electroanal. Chem., 449 (1998) 9-28.
  • Mechanisms with Diffusing Species, J. Electroanal. Chem., 403 (1996) 11-24.

Postscript: if it’s bugging you that there isn’t X =1, that’s because I assumed fast

mass transport for the cases considered here. Here’s the full definition.