INDUCTIVE BEHAVIOR IN ELECTROCHEMICAL MECHANISMS

INDUCTIVE BEHAVIORIN

ELECTROCHEMICAL MECHANISMS

David A. HarringtonPauline van den Driessche

Chemistry and Mathematics Departments,University of Victoria,

Victoria, B.C.Canada, V8W 3V6.

email: [email protected]://surface.chem.uvic.cafunding: NSERC & Uvic.

An annotated (but unanimated) version of a talk given at the 6th International Symposiumon Electrochemical Impedance Spectroscopy, Cocoa Beach, Florida, 20 May, 2004.

It has been known for a long time how to take kinetic equations and derive theimpedance or the resulting equivalent circuit. But the experimenter who finds animpedance spectrum with certain features (e.g., number of time constants, presenceof an inductive loop) would like to know what classes of mechanisms might giverise to this behavior, i.e., the inverse problem of impedance to mechanism is aqualitative but important problem.

1. + +M(site) H+

2. + +MH(ads) H+

MH(ads)

3. 2MH(ads)

M(site) + H2

2 +M(site) H2

HYDROGEN EVOLUTION REACTION

As an example, consider the H.E.R. In general we assume a series of elementaryreactions, and for simplicity we consider the case where surface reactions occurwithout 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 thesurface layer (say Pt for a Pt catalyst) and MH means an H atom adsorbed on an M atom.

1. + +M(site) H+

2. + +MH(ads) H+

MH(ads)

3. 2MH(ads)

M(site) + H2

2 +M(site) H2


We classify the species in the mechanism. Of course in electrochemistry, electronshave a special role. We have adsorbed species, and we will treat the reaction siteslike adsorbed species. We also have some species in solution whose mass transportis assumed to be so fast that we consider their concentrations at the surface to beconstant. These are called “static” or “external” species.

SPECIES

Adsorbed species: MH, M

Static species (fast mass transport): , H 2 H+

The concentrations of the static species can be included in the rate constants, and sothey won’t play a role in determining the type of behavior (though of course the valuesof the equivalent circuit elements will depend on them). We omit them. We can write thekinetics in the usual way, assuming for simplicity Langmuir kinetics and Tafel rateconstants for electron transfer steps (only one shown for simplicity).

KINETICS

v k k1 1 -1 = - = -

v k k2 2 -2 = -

v k k3 3 -3 = - 2 2

k k F RT1 1= exp(- / )eq

1

1. +M(site)

2. +MH(ads)

MH(ads)

3. 2MH(ads)

M(site)

2M(site)


Note that there are the same number of M atoms on each side of the reaction,i.e., M atoms are conserved. Writing 1- as the coverage of sites is anotherstatement of the same thing. Most mechanisms we write down have at least oneconservation condition, and this fact turns out to be significant in constraining thepossible types of behavior.

1. +M(site)

2. +MH(ads)

MH(ads)

3. 2MH(ads)

M(site)

2M(site)


1. -1 +M(site)

2. -1 +MH(ads)

1MH(ads)

3. -2MH(ads)

1M(site)

2M(site)


The numbers in front of the species are the stoichiometric coefficients and they playan important role in the theory.By convention the stoichiometric coefficients of reactants are negative.

1. -1 +M(site)

2. -1 +MH(ads)

1MH(ads)

3. -2MH(ads)

1M(site)

2M(site)


1 2 3

-1 -1 0 1 -1 -2 -1 1 2

MHM

N =

STOICHIOMETRIC MATRIX

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 oneelectron, make one MH and lose one M. We notice that the columns of numbersare related: column 3 = column 2 – column 1, so not all reactions are independent.

I = maximum number of INDEPENDENT STEPS.

Steps are independent if none can be writtenin terms of the others, in the sense of Hess’slaw (with static species omitted).

= 2. I = rank( ) I N

3.

-1. 2.

M(site) +MH(ads) +

MH(ads)

2MH(ads)

M(site)

2M(site)

I

From two of the reactions we can create the third; here step 3 is the sum of step 2 andstep 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 ofthe circuit, so adding the third step doesn’t complicate matters. In general, adding moresteps 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 firstconstruct a matrix for each elementary reaction step, an “elementary matrix”. We willthen add them all together to get an overall matrix Q. We illustrate this for the first stepof the hydrogen evolution reaction. We start by creating the column vector ofstoichiometric coefficients as we did before. We duplicate this as a row vector.

1. M1 + 1MH(ads)

-1 1 -1

MHM

A =1

-1 1 -1

MH M

1. M1 + 1MH(ads)

-1 1 -1

MHM

A =1

-1v1e 1v1b -1v1f

MH M

Now we multiply each entry of the row vector by a rate. For the electrons it is a specialrate v1e that is a combination of the forward and reverse reaction rates, weighted bythe symmetry factor for that step. For the other species, we multiply by the forwardrate v1f if the stoichiometric coefficient is negative and by the backward rate v1b if thestoichiometric coefficient is positive.

v1e v v= - + (1 ) 1f 1b v1f k= 1 v1b k= -1

Now we divide each entry of the row vector by the coverage of the correspondingspecies. MH is divided by its coverage , M by its coverage = 1-.For the electrons, we divide by e, which is a combination of constants includingthe double-layer capacitance.

1. M1 + 1MH(ads)

-1 1 -1

MHM

A =1

-1 /v1e e 1v1b/ -1v1f/MH M

e dl m= /RTC F2

We multiply the two vectors together to give a matrix according to the rules of matrixmultiplication, e.g., the third entry in the second row is the product of the third entry ofthe row vector and the second entry of the column vector.

1. M1 + 1MH(ads)

-1 1 -1

MHM

A =1

-1 /v1e e 1v1b/ -1v1f/MH M

+ /v1e e

- /v1e e

+ /v1e e -v1b/

-v1b/+v1b/

+v1f/-v1f/+v1f/

Q = A + 1 A + A2 3

1. M + MH

++

2. MH + M 3. MH2 2M

=

Z =| |m

-1Q I(1) + s

C sdl m| | -1Q I+

We add the elementary matrices for each step to get an overall matrix Q for themechanism. For many of the results we obtain, we do not need to know the exactvalues of the entries, only their signs. This means that the results derived hereassuming the Langmuir isotherm and Tafel potential dependence of rate constantsare true also under somewhat more relaxed conditions. Note that if we had only steps1 and 3 we would know the signs for Q, but steps 1 and 2 have conflicting signs– this leads to the potential inductive behavior.

Q = A + 1 A + A2 3

1. M + MH

++

2. MH + M 3. MH2 2M

=

Z =| |m

-1Q I(1) + s

C sdl m| | -1Q I+

The impedance Z is determined from the matrix Q as shown. This impedance includesthe double-layer capacitance parallel to the Faradaic impedance. The vertical barsdenote determinants, and the notation Q(1) means the matrix Q stripped of its first rowand column; note that Q(1) has no explicit information about where the electrons arein the mechanism. I is an identity matrix of appropriate size and s is i.

STOICHIOMETRY MATTERS!

Cdl

M + +H O3+

MH + H O2

M + +HSO4-

MH + SO4

2-

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 differentrates that we will see the equivalent circuit above, with separate time constantsfrom the two charge-transfer/pseudocapacitance combinations.

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

X


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 = 1M + MH

M + MH

H O3+

H O HSO SO2 4 4- 2-

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 differentstatic species. We denote this impossibility by saying that the parameter X=0. This meansthat there will be no dc path through the circuit.

X = 0Cdl

The circuit has only one resistor/pseudocapacitor: we cannot separate out the rates ofthe two steps.


M + + + 2H ClO+

2

-MOCl + H O2

M + + + 22H ClO+

2 MOCl + H O2

Consider another example, in which we also have two ways of adsorbing a singlespecies. The presence of the 2 in front of the electrons will make a big difference.

ClO ClO2 2

-+

M + MOCl

M + 2 MOCl

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.

I = 2

X = 2Cdl

And this time we can make a reaction out of electrons and static species, whichpossibility 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 goingbackwards and reaction 2 going forwards effects the redox reaction in solution.

))...()((

))...()((

121dl

21

n

n

sssC

sssZ

i

11

11

21

1

dl

s

RsC

RsC

Z

POLES AND ZEROES

Z =| |m

-1Q I(1) + s

C sdl m| | -1Q I+

Some classical circuit theory tells us which type of circuit follows from which impedanceexpression. Either kinetic impedances or circuit impedances may be simplified to ...

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

These poles and zeroes may be plotted in the complex s plane. Their locationsdetermine the type of circuit.

POLES, ZEROES AND CIRCUITS

o x o xResistors and

capacitors only

An RC circuit has alternating (or “interlacing”) poles and zeroes lying on the negativereal 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 realand negative, or if the interlacing fails, then we will have an inductor in the circuit (if acircuit is possible at all). So as a general rule, we expect inductive behavior to arise inmore mechanisms than not.


If some zeroes are complex (but still in the left half plane), then we cannot have an RCcircuit 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 ofQ(1)/m). 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 mechanismstripped of electrons.

o x xo

o

Inductor(and RC)Type I


“Type II” inductive behavior arises when the zeroes are real and negative, but theinterlacing property fails, either because a pole becomes complex or the poles stayreal but don’t alternate with the zeroes (as shown).

So in this case, knowledge of the zeroes alone is insufficient to determine whether ornot the circuit is inductive. The poles arise from the matrix Q (they are the negatives ofthe eigenvalues of Q/m). This matrix has the electron row and column in and sothe location of the electrons is crucial in determining the inductive behavior.

o o x xInductor(and RC)Type II


For some pole-zero arrangements, it is impossible to find a circuit containing onlyresistors, 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 (atleast for potentiostatic control) with unstable behavior. Circuits also can’t be found ifpoles 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).

o x o x o oNo RLC

circuit possible(unstable)

X

WHEN DOES INDUCTIVE BEHAVIOR OCCUR?

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 haveinductive behavior in the sense of the definition above. But not always, since somebelow-the-axis behavior comes from impedances that are not realizable as circuits atall (e.g. if they move into the left half plane). Also, sometimes a circuit can have aninductor in it which is swamped by the other circuit elements and the spectrumdoesn’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 iftheir stoichiometries without external species are the same or multiples of each otherlike the X = 0 example considered earlier).

We can also show that mechanisms at equilibrium cannot be inductive (this conclusionextends 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 placeto make measurements, if it is possible to do so.


SINGLE ADSORBED SPECIES MECHANISM: need two electron transfer steps: one ODA + one RDA

For mechanisms with a single adsorbed species (that’s two if you count the sites asadsorbed), 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 thedirection 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 inductivebehavior. Observation of inductive behavior for the HER is simple, qualitativeevidence 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 withoutthe electron is a tree graph. The graph is made by replacing each adsorbed speciesby a vertex and the reaction arrows by an edge. The reactions must have only oneproduct species and one reactant species (2A->3B is allowed: A is the reactant andB is the product). As shown over, a graph is a tree if there are no cycles. So the firstmechanism has an RC circuit no matter where the electron is. On the other hand, thesimple cycle mechanism can be inductive no matter what the electron arrangement is:it is type I inductive.

TREE GRAPH MECHANISMS

TREE GRAPH(no cycles)

SIMPLE CYCLE

e- +

e- +

D

Cdl

1 2

3

A B C

CB2

D3

A 1

TOPOLOGICAL CIRCUITS

A B C

D

Cdl

CA RA CB RCCC

RD

CD

Re R2R1

R3

0

ReA ReB

RB

I didn’t have time to talk about this, but for these treegraphs it is possible to write down a topological equivalentcircuit directly from the reaction graph. See JEC (2004) for details.

ASSUMPTIONS• 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 forsome of the results.

CONCLUSIONS

• 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.

If a reaction can be made from and species then

= 2

otherwiseIf a reaction can be made from and

soluble species ( and ) then

= 1

otherwiseAny reaction with must have an

species and = 0.

X

X

X

static

static diffusing

adsorbed

What types of reactions can be made from and the other species?

X

Postscript: if it’s bugging you that there isn’t X =1, that’s because I assumed fastmass transport for the cases considered here. Here’s the full definition.

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INDUCTIVE BEHAVIOR IN ELECTROCHEMICAL MECHANISMS