Electrochimica Acta 49 (2004) 4005–4010

A stochastic model for electrode effects

Wüthrich∗, V. Fascio, H. Bleuler

Laboratoire de systèmes robotiques, EPFL, CH-1015 Lausanne, Switzerland

Received 16 October 2003; received in revised form 11 December 2003; accepted 11 December 2003

Available online 10 June 2004

Abstract

The mechanism leading to the onset of the electrode effects is still under discussion in the literature. In this contribution it is proposed thatthe main mechanism responsible of the electrode effects is the formation of a gas film isolating the electrode. This gas film is formed becauseof a high population density of bubbles on the electrode surface. A simple model considering the bubble evolution as a stochastic renewalprocess is presented. By introducing some phenomenological relations, the model allows to evaluate the critical voltage and current densityas well as the static current–voltage characteristics leading to the onset of the electrode effects.© 2004 Elsevier Ltd. All rights reserved.

Keywords:Electrode effects; Gas-evolving vertical electrodes; Stochastic processes; Telegraph noise equation

1. Introduction

Electrode effects (also called cathode effects or anodeeffects depending where the phenomenon takes place), areknown since 1844 when Fizeau and Foucault[1] mentionedthese effects for water electrolysis with very thin electrodes.They proposed the hypothesis that the electrodes are isolatedwith a gas film from the electrolyte.

As the applied voltage or the current in an electrolysiscell, containing either an aqueous solution or molten salt,is raised sufficiently, electrode effects occur. In current con-trol (charge transfer control), the electrolysis cell voltageincreases very rapidly to large values (from typically 5 to20 V in a few milliseconds) as soon as the current densityexceeds a so-called critical current density[2]. In voltagecontrolled cells, the electrolysis current vanishes if a criti-cal voltage is exceeded (typical values are between 20 and30 V depending on the electrolyte nature and concentration)[3–5]. In both cases the electrolysis collapses and dischargephenomena in a gas film formed around the electrode set in.

First considered as an academic phenomenon, elec-trode effects became a problem with technical, economical

∗ Corresponding author. Tel.:+41 21 693 3810;fax: +41 21 693 38 66.

E-mail address:rolf.wuethrich@a3.epfl.ch (Wüthrich).

and ecological consequences in the context of industrialaluminium production. It is known today, that in theHall-Héroult process greenhouse gases (perfluorocarbon)are produced when an electrode effect (called anode effectin this context) happens[6]. Therefore, intensive researchesare conducted in order to better understand the conditionsleading to electrode effects.

Several systematic studies have been carried out. The maindebated question is the mechanism of the transition fromelectrolysis to gas discharges. During the past 150 years,several authors tried to give an explanation of the physicaland chemical mechanisms initiating the electrode effects.The main admitted explanations are summarised as follows(not being exhaustive, see[7] for a detailed review):

• Electrically insulating layer of solid phase forms aroundthe working-electrode

This idea goes back to Bunsen[8] who supposed the ex-istence of an electrically insulating layer of silicon or limearound the working-electrode. More recent ideas supposethe existence of a AlF3 layer. However, the electrode ef-fects are observed at inert electrodes too and for this thehypothesis of a solid phase fails[7].

• Gaseous phase film forms around the working-electrodeThe existence of a gas film around the working-electrode

in the electrode effects is a generally accepted finding.The main question is how this film can be formed and

0013-4686/$ – see front matter © 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.electacta.2003.12.060

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what the conditions of its formation are. Various expla-nations are proposed:◦ Change in wettability of the electrode(Arndt and Probst

[9]): the electrode effects are interpreted as the con-sequence of insufficient wetting of the electrode. Theability of the bubbles to adhere to the electrode surfaceis increased and they grow larger. The bubbles can co-alesce and form a continuous gas film.

◦ Hydrodynamic instabilities(i.e. Helmholtz instability)are responsible for the onset of anode effect (Mazzaet al. [10]).

◦ Local Joule heating: the gas film is formed by localevaporation of the electrolyte by Joule heating due tothe increase of local current density because the bubbleselectrochemically formed are shadowing the electrode(Guilpin and Garbaz-Olivier[3]).

◦ Combination of wettablity and hydrodynamic effects:Vogt [7] proposes that “anode effect occurs wheneverthe distance between neighbouring bubbles contactingthe electrode has been diminished to such an extent thatthe bubbles are enable to coalesce”. He calculates anexpression for the critical current density in function ofthe electrolyte flow around the electrode and the contactangle of the adhering bubbles. His model shows that thecritical current density depends on several parameters(which are not all independent), such as:– wettability of the electrode;– electrode geometry (area and typical length);– thermodynamic state (temperature and pressure);– bubble geometry;– bubble removal rate.However, as pointed out by Vogt himself, this model is

not able to predict the critical voltage nor the voltage priorto the onset of the effect. A first attempt in this directionwas done by Vogt and Thonstad[2] at the end of 2002.

By studying Spark Assisted Chemical Engraving (SACE)process, a new way for analysing electrode effects wasopened[4,5,13,14]. The present authors propose that theonset of the electrode effects is linked to the formation ofa gas film around a electrode. The model proposed hereis as follows: bubble evolution is described as a stochas-tic “birth and death” process. The onset of the electrodeeffect is given by the formation of an isolating gas filmaround the electrode. This very simple model can not onlypredict the critical voltage and current density character-ising the electrode effect onset, but the complete staticcurrent–voltage characteristics leading to this effect. In afirst description the detail of bubble coalescence may evenbe omitted. Later on, it can be included in the frameworkof percolation theory. Additional effects such as Joule heat-ing or hydrodynamics need not to be considered at thisstage.

In this contribution the first level model (stochastic pro-cess only) will be presented in detail. The inclusion of per-colation theory is subject of other publications[5,11].

i=1,...,L

σ = 0,1i

Fig. 1. Stochastic model: the lateral working-electrode surfaceA is sub-divided in a lattice. In each lattice site a bubble is growing or not. Thecharacteristic functionσi of each site indicates if a bubble is growing ornot.

2. Theoretical model

2.1. Definition of the model

A geometry consisting of two concentric cylindrical elec-trodes is considered. The smaller cylinder, with radiusrw, iscalled the working electrode. The surrounding counter elec-trode with radiusrc is considered to be much larger. In thisconfiguration the electrode effects happen at the workingelectrode only. Furthermore, for simplification, we consideronly the bubble evolution on the vertical lateral surface ofthe working electrode, i.e. that the bottom surface of theelectrode, where bubbles could tend to stick, is much smallerthan the lateral one.

During electrolysis, the gas bubbles form a gas layeraround the electrodes, which can be subdivided in three re-gions according to Janssen[12]. In the adherence region thebubbles are growing and adhere to the surface. Once theyleave the electrode surface they move into the bubble diffu-sion region which is the region in which the bubble have avery high gas void fraction. The bulk regions, with very fewbubbles, surrounds the diffusion region.

The bubble coalescence takes place in the adherence re-gion. Therefore, it will be supposed that all relevant effectsleading to the formation of a gas film around the electrode,take place in the adherence region and the evolution of thebubbles in this region is studied only. The bubble evolutionis considered as a stochastic process in which bubbles arerandomly growing and detaching. The lateral working elec-trode surface is subdivided in a lattice withL lattice sitesi(seeFig. 1).

2.2. Bubble evolution in the adherence region

On each lattice sitei, the bubble evolution is consideredas a stochastic renewal process characterised by two transi-tion probabilitiesλ andµ. The probabilityλ indicates theprobability that a bubble is appearing per time at the siteiand the probabilityµ indicates the probability that a bubble

Wüthrich et al. / Electrochimica Acta 49 (2004) 4005–4010 4007

is leaving per time unit. The transition probabilitiesλ andµ are function of the voltageU between the electrodesand the mean nominal current densityj ≡ I/A on the lat-eral working electrode surfaceA. The expression of theseprobabilities will be discussed inSection 2.4below.

To each lattice sitei is associated its characteristic func-tion σi(t) indicating if a bubble is growing or not at thetime t (σi = 1 means that a bubble is growing andσi = 0that no bubble is growing). The conditional probabilityP(σi(t) = 1|σi(to)) gives the probability that a bubble isgrowing on the sitei at the timet knowing that at timetothe site was in the stateσi(to).

The following master equations can be written for theconditional probabilitiesP of the stochastic process:

∂tP(σi(t) = 1|σi(to)) = −µP(σi(t) = 1|σi(to))

+ λP(σi(t) = 0|σi(to)) (1)

∂tP(σi(t) = 0|σi(to)) = +µP(σi(t) = 1|σi(to))

− λP(σi(t) = 0|σi(to)) (2)

These master equations are known in literature as the tele-graph noise equations. The mean occupation probabilitypof the bubbles is given by:

p(t) = 1

L

L∑i=1

1∑j=0

P(σi(t) = 1|σi(to) = j)P(σi(to) = j) (3)

The masterEqs. (1) and (2)are solved using:

P(σi = 1, t|σ′i, to) + P(σi = 0, t|σ′

i, to) = 1 (4)

and the initial conditions:

P(σi(to) = 1) = 0 (5)

It follows:

P(σi(t) = 1|σi(to) = 0)

= λ

λ + µ{1 − exp[−(λ + µ)(t − to)]} (6)

Together with (3) the evolution of the mean occupation prob-ability p is given by:

p(t) = λ

λ + µ{1 − exp[−(λ + µ)(t − to)]} (7)

The stationary solution is:

p = λ

λ + µ(8)

which is rather an intuitive results considering the definitionof the transition probabilities.

2.3. The onset of the gas film formation

For the transition from the bubble layer to the gas filmaround the working electrode, not only the mean occupation

probabilityp of the bubbles is from interest, but its fluctua-tions�p too. These fluctuations are given by the fluctuationsof the characteristic functions�i of the sites:

(�p)2 = Var

[1

L

L∑i=1

σi

]= Var[σi] (9)

where the second equality could be written because the ran-dom variables�i are independent. Using the known varianceof the random telegraph noise process, it follows:

�p =√

µλ

λ + µ(10)

A gas film around the working electrode is formed if allsites of the lattice are occupied. This happens if the followingcondition is met:

p + �p = 1 (11)

i.e. even if p ≤ 1. This shows that for the onset of theelectrode effect, a completely built gas film is not necessary.The fluctuations in the system itself are enough to insulatethe electrode and start the electrode effect. Using the results(8) and (10) it follows, after simplifications:

λ = µ (12)

which means that at the onset of the gas film, the probabilityof bubble production is equal to the probability of bubbledeparture. The gas film can be built, because the fluctuationsof the system are high enough that at any time all the surfacecan be covered by bubbles. From (12) it follows using (8)and (10):

pcrit = 0.5 (13)

�pcrit = 0.5 (14)

This means that in average half of the electrode is coveredby a gas film at the onset of the effect. Supposing that mainlyonly the adherence region contribute to the inter-electroderesistanceR, it follows:

R = Ro

1 − p(15)

whereRo is the intrinsic electrolyte resistance. This rela-tion makes use of the assumptions that the adhering bubblesshadow electrically the fractionp of the electrode surface.From the result (13), the critical resistanceRcrit is estimatedto be:

Rcrit = 2Ro (16)

Taking into account the bubble coalescence improve thistheoretical result toRcrit = 2.5Ro [5,11]. This is confirmedexperimentally by Fascio[13].

The fact that such simple assumptions on bubble evolution(basically only supposing that bubbles are appearing anddetaching) leads to a prediction like (16) is seen by theauthors as a strong argument in favour of the hypothesis thatthe onset of the electrode effects is linked to the formation ofa gas film from the bubbles evolving around the electrode.

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2.4. Modelling the transition probabilities

In order to get more information about the critical voltageand current density, the explicit expression of the transitionprobabilities have to be known. The present authors proposetwo qualitative expressions for them. The justification of thetransition probabilities where already presented in[5,11]andare summarised in this section.

In the following it is supposed that the bubble growth isuniformly distributed over the whole vertical working elec-trode surface.

In the case hydrogen is produced at the working electrode,we propose to consider that the probability of bubble pro-duction is given by the probability that electrons combinewith H+ ions. Therefore the amount of produced bubbles isproportional to the product between the electron flow (givenby the mean current) and the ionic H+ flow (given by thenumber of H+ ions times the drift velocityvd proportionalto the local electrical fieldElocal as seen by the H+ ions). Itfollows:

λ ∝ jnH+vd ∝ jnH+Elocal (17)

The density of H+ ions is supposed to be proportional to theelectrolyte weight concentrationw (strong electrolyte). Thelocal electrical fieldElocal can be expressed for the cylindri-cal geometry considered here. It follows:

λ = C × wU − Ud

rwln (rc/rw) + Bwj = λo × j (U − Ud) (18)

This expression models the screening effect of the ionsfrom the electrolyte around the working-electrode. The con-stantB andC have to be determined experimentally but areindependent of the electrode geometry and the electrolytemass concentrationw.

The probability of bubble departureµ is evaluated by es-timating the mean bubble life time (given by he time neededto produce a large enough bubble to detach) and makingthe simplification that the mean bubble departure radius isindependent of the current density:

µ = µoj (19)

2.5. Current–voltage characteristics

The knowledge of the transition probabilities gives thepossibility to predict the current–voltage characteristics upto the onset of the electrode effects. For simplifications wesuppose that only the bubbles in the adherence region con-tribute to the inter-electrode resistance modification. Underthis assumption the resistance is written as (15) and the meancurrentI and the applied voltageU is linked by:

U − Ud = RI (20)

whereUd is the sum of the electrode potentials depending onthe electrolyte and electrodes. Using (15) and the cylindrical

Electrode radius r [µm]w

Fig. 2. Comparison between the predicted critical current densityjcrit

according to (22) in function of the working electrode radiusrw. Mea-surements done in a 30 wt.% NaOH.

geometry, (20) is rewritten as:

j = (1 − p)κ

rwln(rc/rw)(U − Ud) (21)

with κ the electrolyte conductivity. Using (13) it followsthat the critical current densityjcrit and critical voltageUcrit

characterising the onset of the electrode effects, are linkedby:

jcrit = κ

2rwln(rc/rw)(Ucrit − Ud) (22)

Fig. 2shows the comparison between measurements donein a 30 wt.% NaOH solution with a counter-electrode of40 mm radius. A good agreement is seen.

Combining (21) and (22) results in

j

jcrit= 2(1 − p)

U − Ud

Ucrit − Ud(23)

which suggests to introduce a normalised current densityJand a normalised voltageU as follows:

J = j

jcrit(24)

U = U − Ud

Ucrit − Ud(25)

The critical voltage can be determined straight forwardusing the criterion (12) together with the expression (18)and (19) for the transition probabilities:

Ucrit − Ud = µo

λo= µo

C

[ln

(rc

rw

)rw

w+ B

](26)

This relation shows the dependence of the critical voltagefrom the geometry of the electrode as well as from the elec-trolyte concentration and is shown onFig. 3. Replacing in(23) the explicit expression (18) and (19) for the transitionprobabilities and using (26) it follows:

J = 2U

U+ 1p (27)

Wüthrich et al. / Electrochimica Acta 49 (2004) 4005–4010 4009[V

]

Fig. 3. Comparison between the predicted critical voltageUcrit in func-tion of the electrolyte concentrationw according to (26) and the experi-mental values for a NaOH electrolyte with cylindrical working electrodes(rw = 270�m, µo/C = 0.8 V wt.%/mm andB = 21.1 mm/wt.%).

This relation shows that the normalised characteristicsis independent of the electrode geometry and the elec-trolyte properties. This is verified experimentally onFig. 4where the normalised characteristics is shown for vari-ous working-electrodes (radii from 50 to 300�m) andvarious NaOH solutions (mass concentration from 5 to40 wt.%). Relation (27) describes qualitatively in correctway the normalised characteristics, but not quantitatively.A better agreement with relation (27) can be found con-sidering the coalescence effect of bubbles and consideringthe contribution of the bubble diffusion region around theworking-electrode to the apparent inter-electrode resistance[5,11].

The normalised characteristics is not only interesting froma theoretical point of view, but also from a practical one.

Fig. 4. Comparison between the predictedJ–U characteristics of electrodeeffects in NaOH by (27) and the experiment. Measurements done withvarious working electrodes radii and surface roughness. The NaOH con-centration was varied between 5 and 40 wt.% and electrolyte temperaturewas 30◦C.

The measurement of the critical voltage together with thecritical current density is enough to know the complete char-acteristics of the system without accurate knowledge of theelectrolyte properties, which are in general difficult to deter-mine (especially the electrolyte conductivity which dependshighly of the electrolyte temperature and purity).

2.6. Discussion

The presented model make use of several simplificationsthat are summarised here:

• The bubble evolution (growth and departure) is uniformover the whole electrode surface

• Only the screening effect of adhering bubbles is consid-ered to express the change in the inter-electrode resistance(the effect of the bubbles in the bubble diffusion region isnot taken into account).

• The variation ofUd, the sum of the different overpoten-tials, is considered to negligible compared to the appliedvoltage between the two electrodes.

• The kinetic of the electrochemical reactions is modelledonly qualitatively.

• The coalescence effect of bubble is ignored.

If the first two assumptions seem acceptable, the threeother are more questionable. Vogt and Thonstad[2] pro-posed that the overpotential is growing significantly at theonset of the anode effect in aluminium electrolysis withthe Hall-Héroult process. That bubble are coalescing in therange of the current densities just lower than the critical onejcrit is known experimentally[4]. During the formation of acoalescence bubble, its shape will change very rapidly andcompletely irregularly (see[12], Fig. 4). In the gas film for-mation range, bubbles will coalesce frequently and it hastherefore to be asked if there is any sense to speak about thegrowth of single bubbles. This question is addressed usingpercolation theory[11].

It is clear that all these assumptions are simplifications.However, the interesting point is that even making all thesesimplifications and keeping only the main idea that bubblesare evolving stochastically on the electrode surface accord-ing to a renewal process, an interesting relation between theapparent interelectrode resistanceRcrit at the onset of theeffect and the intrinsic interelectrode resistanceRo can beexpressed. If moreover some simple assumptions about theprobability of bubble growth and departure are made, thecritical voltage and current density can be predicted. An im-proved model should try to give some more accurate expres-sions for these transition probabilities.

3. Conclusion

This paper proposes a new approach for describingthe electrode effects for hydrogen evolving electrodes.The key idea is to consider the bubble evolution on the

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electrode surface as a stochastic renewal process, it is possi-ble to explain the transition to a compact gas film formationwithout adding other effects like local Joule heating or hy-drodynamic instabilities. The condition of the onset of theelectrode effects is met once the statistical fluctuations inthe system become as large as the mean values itself. Thisapproach allows two central predictions. First it predictsthat the critical resistance (the inter-electrode resistance atthe transition to the gas film formation) is two times greaterthan the electrolyte resistance, independently of the usedNaOH electrolyte concentration or electrodes. Secondlyit predicts that the current–voltage characteristics of anelectrolysis cell can be described in one normalised char-acteristics (the normalisation being done according to thecritical voltage and current density). Both results are ver-ified experimentally. The model also allows the predictionof the critical voltage and current density.

Based on these results, the presents authors suggests toconsider that the main mechanism responsible for the onsetof the electrode effects is the formation of a gas film from thebubbles evolving around the electrode. A very similar con-clusion was made by Vogt[7] who proposed that the elec-trode effects starts once “the distance between neighbouringbubbles diminishes to such an extent that the bubbles con-tact each other tending to form a continuous gas film”.

Acknowledgements

The authors would like to thank the Swiss National Sci-ence Foundation (FNS grant 061533.00) for its financialsupport.

Appendix A. Nomenclature

A electrode surface (m2)B experimental constant (18) ( V m−1 wt.%)C experimental constant (18)

( V−1 m3 A−1 s−1 wt.%−1)E Electrical field ( V m−1)i index of the lattice siteI mean current (A)j mean nominal current density ( A m−2)J normalised current density (-)

L number of lattice sites (-)nH density of H+ ions (m−3)p lattice site mean occupation probability (-)P probability (-)r radius (m)R resistance (�)t time (s)U voltage between the working electrode

and the counter electrode (V)U normalised voltage (-)vd drift velocity ( m s−1)w electrolyte mass concentration ( wt.%)κ electrolyte conductivity (�−1 m−1)λ probability that a bubble is growing on the lateral

electrode surface per time and sitei (s−1)µ probability that a bubble is leaving the lateral

electrode surface per time and sitei (s−1)σi characteristic function of the lattice sitei (-)

Subscriptsc counter-electrodew working-electrode

Superscriptcrit critical

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