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A stochastic model for electrode effects
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 that the main mechanism responsible of the electrode effects is the formation of a gas film isolating the electrode. This gas film is formed because of a high population density of bubbles on the electrode surface. A simple model considering the bubble evolution as a stochastic renewal process is presented. By introducing some phenomenological relations, the model allows to evaluate the critical voltage and current density as 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 anode effects depending where the phenomenon takes place), are known since 1844 when Fizeau and Foucault [1] mentioned these effects for water electrolysis with very thin electrodes. They proposed the hypothesis that the electrodes are isolated with a gas film from the electrolyte. As the applied voltage or the current in an electrolysis cell, containing either an aqueous solution or molten salt, is raised sufficiently, electrode effects occur. In current control (charge transfer control), the electrolysis cell voltage increases very rapidly to large values (from typically 5 to 20 V in a few milliseconds) as soon as the current density exceeds a so-called critical current density [2]. In voltage controlled cells, the electrolysis current vanishes if a critical voltage is exceeded (typical values are between 20 and 30 V depending on the electrolyte nature and concentration) [3–5]. In both cases the electrolysis collapses and discharge phenomena in a gas film formed around the electrode set in. First considered as an academic phenomenon, electrode effects became a problem with technical, economical
∗ Corresponding author. Tel.: +41 21 693 3810; fax: +41 21 693 38 66. E-mail address: [email protected] (Wüthrich).
0013-4686/$ – see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.electacta.2003.12.060
and ecological consequences in the context of industrial aluminium production. It is known today, that in the Hall-Héroult process greenhouse gases (perfluorocarbon) are produced when an electrode effect (called anode effect in this context) happens [6]. Therefore, intensive researches are conducted in order to better understand the conditions leading to electrode effects. Several systematic studies have been carried out. The main debated question is the mechanism of the transition from electrolysis to gas discharges. During the past 150 years, several authors tried to give an explanation of the physical and 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 around the working-electrode This idea goes back to Bunsen [8] who supposed the existence of an electrically insulating layer of silicon or lime around the working-electrode. More recent ideas suppose the existence of a AlF3 layer. However, the electrode effects are observed at inert electrodes too and for this the hypothesis of a solid phase fails [7]. • Gaseous phase film forms around the working-electrode The 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
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what the conditions of its formation are. Various explanations are proposed: ◦ Change in wettability of the electrode (Arndt and Probst [9]): the electrode effects are interpreted as the consequence of insufficient wetting of the electrode. The ability of the bubbles to adhere to the electrode surface is increased and they grow larger. The bubbles can coalesce and form a continuous gas film. ◦ Hydrodynamic instabilities (i.e. Helmholtz instability) are responsible for the onset of anode effect (Mazza et al. [10]). ◦ Local Joule heating: the gas film is formed by local evaporation of the electrolyte by Joule heating due to the increase of local current density because the bubbles electrochemically formed are shadowing the electrode (Guilpin and Garbaz-Olivier [3]). ◦ Combination of wettablity and hydrodynamic effects: Vogt [7] proposes that “anode effect occurs whenever the distance between neighbouring bubbles contacting the electrode has been diminished to such an extent that the bubbles are enable to coalesce”. He calculates an expression for the critical current density in function of the electrolyte flow around the electrode and the contact angle of the adhering bubbles. His model shows that the critical 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 prior to the onset of the effect. A first attempt in this direction was 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 was opened [4,5,13,14]. The present authors propose that the onset of the electrode effects is linked to the formation of a gas film around a electrode. The model proposed here is as follows: bubble evolution is described as a stochastic “birth and death” process. The onset of the electrode effect is given by the formation of an isolating gas film around the electrode. This very simple model can not only predict the critical voltage and current density characterising the electrode effect onset, but the complete static current–voltage characteristics leading to this effect. In a first description the detail of bubble coalescence may even be omitted. Later on, it can be included in the framework of percolation theory. Additional effects such as Joule heating or hydrodynamics need not to be considered at this stage. In this contribution the first level model (stochastic process only) will be presented in detail. The inclusion of percolation theory is subject of other publications [5,11].
σ i = 0,1 i=1,...,L
Fig. 1. Stochastic model: the lateral working-electrode surface A is subdivided in a lattice. In each lattice site a bubble is growing or not. The characteristic function σ i of each site indicates if a bubble is growing or not.
2. Theoretical model 2.1. Definition of the model A geometry consisting of two concentric cylindrical electrodes is considered. The smaller cylinder, with radius rw , is called the working electrode. The surrounding counter electrode with radius rc is considered to be much larger. In this configuration the electrode effects happen at the working electrode only. Furthermore, for simplification, we consider only the bubble evolution on the vertical lateral surface of the working electrode, i.e. that the bottom surface of the electrode, where bubbles could tend to stick, is much smaller than the lateral one. During electrolysis, the gas bubbles form a gas layer around the electrodes, which can be subdivided in three regions according to Janssen [12]. In the adherence region the bubbles are growing and adhere to the surface. Once they leave the electrode surface they move into the bubble diffusion region which is the region in which the bubble have a very high gas void fraction. The bulk regions, with very few bubbles, surrounds the diffusion region. The bubble coalescence takes place in the adherence region. Therefore, it will be supposed that all relevant effects leading to the formation of a gas film around the electrode, take place in the adherence region and the evolution of the bubbles in this region is studied only. The bubble evolution is considered as a stochastic process in which bubbles are randomly growing and detaching. The lateral working electrode surface is subdivided in a lattice with L lattice sites i (see Fig. 1). 2.2. Bubble evolution in the adherence region On each lattice site i, the bubble evolution is considered as a stochastic renewal process characterised by two transition probabilities λ and µ. The probability λ indicates the probability that a bubble is appearing per time at the site i and the probability µ indicates the probability that a bubble
Wüthrich et al. / Electrochimica Acta 49 (2004) 4005–4010
is leaving per time unit. The transition probabilities λ and µ are function of the voltage U between the electrodes and the mean nominal current density j ≡ I/A on the lateral working electrode surface A. The expression of these probabilities will be discussed in Section 2.4 below. To each lattice site i is associated its characteristic function σi (t) indicating if a bubble is growing or not at the time t (σi = 1 means that a bubble is growing and σi = 0 that no bubble is growing). The conditional probability P(σi (t) = 1|σi (to )) gives the probability that a bubble is growing on the site i at the time t knowing that at time to the site was in the state σi (to ). The following master equations can be written for the conditional probabilities P of the stochastic process: ∂t P(σi (t) = 1|σi (to )) = −µP(σi (t) = 1|σi (to )) + λP(σi (t) = 0|σi (to ))
(1)
∂t P(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 telegraph noise equations. The mean occupation probability p of the bubbles is given by: p(t) =
L 1 1
L
P(σi (t) = 1|σi (to ) = j)P(σi (to ) = j)
(3)
i=1 j=0
The master Eqs. (1) and (2) are solved using: P(σi = 1, t|σi , to ) + P(σi = 0, t|σi , to ) = 1
(4)
and the initial conditions: P(σi (to ) = 1) = 0
(5)
It follows: P(σi (t) = 1|σi (to ) = 0) λ = {1 − exp[−(λ + µ)(t − to )]} λ+µ
(6)
Together with (3) the evolution of the mean occupation probability p is given by: p(t) =
λ {1 − exp[−(λ + µ)(t − to )]} λ+µ
(7)
The stationary solution is: λ p= λ+µ
probability p of the bubbles is from interest, but its fluctuations p too. These fluctuations are given by the fluctuations of the characteristic functions i of the sites: L 1 2 (p) = Var σi = Var[σi ] (9) L i=1
where the second equality could be written because the random variables i are independent. Using the known variance of the random telegraph noise process, it follows: √ µλ p = (10) λ+µ A gas film around the working electrode is formed if all sites of the lattice are occupied. This happens if the following condition is met: p + p = 1
which is rather an intuitive results considering the definition of the transition probabilities. 2.3. The onset of the gas film formation For the transition from the bubble layer to the gas film around the working electrode, not only the mean occupation
(11)
i.e. even if p ≤ 1. This shows that for the onset of the electrode effect, a completely built gas film is not necessary. The fluctuations in the system itself are enough to insulate the 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 probability of bubble production is equal to the probability of bubble departure. The gas film can be built, because the fluctuations of the system are high enough that at any time all the surface can 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 covered by a gas film at the onset of the effect. Supposing that mainly only the adherence region contribute to the inter-electrode resistance R, it follows: Ro R= (15) 1−p where Ro is the intrinsic electrolyte resistance. This relation makes use of the assumptions that the adhering bubbles shadow electrically the fraction p of the electrode surface. From the result (13), the critical resistance Rcrit is estimated to be: Rcrit = 2Ro
(8)
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(16)
Taking into account the bubble coalescence improve this theoretical result to Rcrit = 2.5Ro [5,11]. This is confirmed experimentally by Fascio [13]. The fact that such simple assumptions on bubble evolution (basically only supposing that bubbles are appearing and detaching) leads to a prediction like (16) is seen by the authors as a strong argument in favour of the hypothesis that the onset of the electrode effects is linked to the formation of a 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 voltage and current density, the explicit expression of the transition probabilities have to be known. The present authors propose two qualitative expressions for them. The justification of the transition probabilities where already presented in [5,11] and are summarised in this section. In the following it is supposed that the bubble growth is uniformly distributed over the whole vertical working electrode surface. In the case hydrogen is produced at the working electrode, we propose to consider that the probability of bubble production is given by the probability that electrons combine with H+ ions. Therefore the amount of produced bubbles is proportional to the product between the electron flow (given by the mean current) and the ionic H+ flow (given by the number of H+ ions times the drift velocity vd proportional to the local electrical field Elocal as seen by the H+ ions). It follows: λ ∝ jnH+ vd ∝ jnH+ Elocal
(17)
The density of H+ ions is supposed to be proportional to the electrolyte weight concentration w (strong electrolyte). The local electrical field Elocal can be expressed for the cylindrical geometry considered here. It follows: λ=C×w
U − Ud j = λo × j (U − Ud ) rw ln (rc /rw ) + Bw
(18)
This expression models the screening effect of the ions from the electrolyte around the working-electrode. The constant B and C have to be determined experimentally but are independent of the electrode geometry and the electrolyte mass concentration w. The probability of bubble departure µ is evaluated by estimating the mean bubble life time (given by he time needed to produce a large enough bubble to detach) and making the simplification that the mean bubble departure radius is independent of the current density: µ = µo j
(19)
2.5. Current–voltage characteristics The knowledge of the transition probabilities gives the possibility to predict the current–voltage characteristics up to the onset of the electrode effects. For simplifications we suppose that only the bubbles in the adherence region contribute to the inter-electrode resistance modification. Under this assumption the resistance is written as (15) and the mean current I and the applied voltage U is linked by: U − Ud = RI
(20)
where Ud is the sum of the electrode potentials depending on the electrolyte and electrodes. Using (15) and the cylindrical
Electrode radius rw [µm] Fig. 2. Comparison between the predicted critical current density jcrit according to (22) in function of the working electrode radius rw . Measurements done in a 30 wt.% NaOH.
geometry, (20) is rewritten as: κ j = (1 − p) (U − Ud ) rw ln(rc /rw )
(21)
with κ the electrolyte conductivity. Using (13) it follows that the critical current density jcrit and critical voltage Ucrit characterising the onset of the electrode effects, are linked by: κ j crit = (U crit − Ud ) (22) 2rw ln(rc /rw ) Fig. 2 shows the comparison between measurements done in a 30 wt.% NaOH solution with a counter-electrode of 40 mm radius. A good agreement is seen. Combining (21) and (22) results in j U − Ud = 2(1 − p) crit (23) j crit U − Ud which suggests to introduce a normalised current density J and a normalised voltage U as follows: j J = crit (24) j U=
U − Ud U crit − Ud
(25)
The critical voltage can be determined straight forward using the criterion (12) together with the expression (18) and (19) for the transition probabilities: rc rw µo µo crit U − Ud = ln +B (26) = λo C rw w This relation shows the dependence of the critical voltage from the geometry of the electrode as well as from the electrolyte concentration and is shown on Fig. 3. Replacing in (23) the explicit expression (18) and (19) for the transition probabilities and using (26) it follows: U J=2 p (27) U+1
[V]
Wüthrich et al. / Electrochimica Acta 49 (2004) 4005–4010
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The measurement of the critical voltage together with the critical current density is enough to know the complete characteristics of the system without accurate knowledge of the electrolyte properties, which are in general difficult to determine (especially the electrolyte conductivity which depends highly of the electrolyte temperature and purity). 2.6. Discussion The presented model make use of several simplifications that are summarised here:
Fig. 3. Comparison between the predicted critical voltage Ucrit in function of the electrolyte concentration w according to (26) and the experimental values for a NaOH electrolyte with cylindrical working electrodes (rw = 270 m, µo /C = 0.8 V wt.%/mm and B = 21.1 mm/wt.%).
This relation shows that the normalised characteristics is independent of the electrode geometry and the electrolyte properties. This is verified experimentally on Fig. 4 where the normalised characteristics is shown for various working-electrodes (radii from 50 to 300 m) and various NaOH solutions (mass concentration from 5 to 40 wt.%). Relation (27) describes qualitatively in correct way the normalised characteristics, but not quantitatively. A better agreement with relation (27) can be found considering the coalescence effect of bubbles and considering the contribution of the bubble diffusion region around the working-electrode to the apparent inter-electrode resistance [5,11]. The normalised characteristics is not only interesting from a theoretical point of view, but also from a practical one.
• The bubble evolution (growth and departure) is uniform over the whole electrode surface • Only the screening effect of adhering bubbles is considered to express the change in the inter-electrode resistance (the effect of the bubbles in the bubble diffusion region is not taken into account). • The variation of Ud , the sum of the different overpotentials, is considered to negligible compared to the applied voltage between the two electrodes. • The kinetic of the electrochemical reactions is modelled only qualitatively. • The coalescence effect of bubble is ignored. If the first two assumptions seem acceptable, the three other are more questionable. Vogt and Thonstad [2] proposed that the overpotential is growing significantly at the onset of the anode effect in aluminium electrolysis with the Hall-Héroult process. That bubble are coalescing in the range of the current densities just lower than the critical one jcrit is known experimentally [4]. During the formation of a coalescence bubble, its shape will change very rapidly and completely irregularly (see [12], Fig. 4). In the gas film formation range, bubbles will coalesce frequently and it has therefore to be asked if there is any sense to speak about the growth of single bubbles. This question is addressed using percolation theory [11]. It is clear that all these assumptions are simplifications. However, the interesting point is that even making all these simplifications and keeping only the main idea that bubbles are evolving stochastically on the electrode surface according to a renewal process, an interesting relation between the apparent interelectrode resistance Rcrit at the onset of the effect and the intrinsic interelectrode resistance Ro can be expressed. If moreover some simple assumptions about the probability of bubble growth and departure are made, the critical voltage and current density can be predicted. An improved model should try to give some more accurate expressions for these transition probabilities.
3. Conclusion Fig. 4. Comparison between the predicted J–U characteristics of electrode effects in NaOH by (27) and the experiment. Measurements done with various working electrodes radii and surface roughness. The NaOH concentration was varied between 5 and 40 wt.% and electrolyte temperature was 30 ◦ C.
This paper proposes a new approach for describing the 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 possible to explain the transition to a compact gas film formation without adding other effects like local Joule heating or hydrodynamic instabilities. The condition of the onset of the electrode effects is met once the statistical fluctuations in the system become as large as the mean values itself. This approach allows two central predictions. First it predicts that the critical resistance (the inter-electrode resistance at the transition to the gas film formation) is two times greater than the electrolyte resistance, independently of the used NaOH electrolyte concentration or electrodes. Secondly it predicts that the current–voltage characteristics of an electrolysis cell can be described in one normalised characteristics (the normalisation being done according to the critical voltage and current density). Both results are verified experimentally. The model also allows the prediction of the critical voltage and current density. Based on these results, the presents authors suggests to consider that the main mechanism responsible for the onset of the electrode effects is the formation of a gas film from the bubbles evolving around the electrode. A very similar conclusion was made by Vogt [7] who proposed that the electrode effects starts once “the distance between neighbouring bubbles diminishes to such an extent that the bubbles contact each other tending to form a continuous gas film”.
L nH p P r R t U
Acknowledgements
References
The authors would like to thank the Swiss National Science Foundation (FNS grant 061533.00) for its financial support.
[1] H. Fizeau, L. Foucault, Ann. Chim. Phys. XI (3) (1844) 370. [2] H. Vogt, J. Thonstad, J. Appl. Electrochem. 32 (2002) 241. [3] Ch. Guilpin, J. Garbaz-Olivier, J. Chim. Phys. Phys-Chim. Biol. 75 (1978) 723. [4] V. Fascio, H.H. Langen, H. Bleuler, Ch. Comninellis Electrochem. Commun. 5 (2003) 203. [5] R. Wüthrich, Dissertation Thesis (2776), Swiss Federal Institute of Technology, EPF Lausanne, 2003. [6] F. Nordmo, J. Thonstad, Electrochim. Acta 30 (1985) 741. [7] H. Vogt, J. Appl. Electrochem. 29 (1999) 137. [8] R. Bunsen, Ann. Phys. 92 (1851) 648. [9] K. Arndt, H. Probst, Z. Electrochem. 29 (1923) 323. [10] B. Mazza, P. Pedeferri, G. Re, Electrochim. Acta 23 (1978) 87. [11] R. Wüthrich, H. Bleuler, Electrochim. Acta 49 (2004) 1547. [12] L.J.J. Janssen, Electrochim. Acta 34 (1989) 161. [13] V. Fascio, Dissertation Thesis (2691), Swiss Federal Institute of Technology, EPF Lausanne, 2002. [14] V. Fascio, R. Wüthrich, H. Bleuler, Special Issue, Electrochim. Acta, submitted for publication.
Appendix A. Nomenclature A B C E i I j J
electrode surface (m2 ) experimental constant (18) ( V m−1 wt.%) experimental constant (18) ( V−1 m3 A−1 s−1 wt.%−1 ) Electrical field ( V m−1 ) index of the lattice site mean current (A) mean nominal current density ( A m−2 ) normalised current density (-)
U vd w κ λ µ σi
number of lattice sites (-) density of H+ ions (m−3 ) lattice site mean occupation probability (-) probability (-) radius (m) resistance ( ) time (s) voltage between the working electrode and the counter electrode (V) normalised voltage (-) drift velocity ( m s−1 ) electrolyte mass concentration ( wt.%) electrolyte conductivity ( −1 m−1 ) probability that a bubble is growing on the lateral electrode surface per time and site i (s−1 ) probability that a bubble is leaving the lateral electrode surface per time and site i (s−1 ) characteristic function of the lattice site i (-)
Subscripts c counter-electrode w working-electrode Superscript crit critical
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 that the main mechanism responsible of the electrode effects is the formation of a gas film isolating the electrode. This gas film is formed because of a high population density of bubbles on the electrode surface. A simple model considering the bubble evolution as a stochastic renewal process is presented. By introducing some phenomenological relations, the model allows to evaluate the critical voltage and current density as 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 anode effects depending where the phenomenon takes place), are known since 1844 when Fizeau and Foucault [1] mentioned these effects for water electrolysis with very thin electrodes. They proposed the hypothesis that the electrodes are isolated with a gas film from the electrolyte. As the applied voltage or the current in an electrolysis cell, containing either an aqueous solution or molten salt, is raised sufficiently, electrode effects occur. In current control (charge transfer control), the electrolysis cell voltage increases very rapidly to large values (from typically 5 to 20 V in a few milliseconds) as soon as the current density exceeds a so-called critical current density [2]. In voltage controlled cells, the electrolysis current vanishes if a critical voltage is exceeded (typical values are between 20 and 30 V depending on the electrolyte nature and concentration) [3–5]. In both cases the electrolysis collapses and discharge phenomena in a gas film formed around the electrode set in. First considered as an academic phenomenon, electrode effects became a problem with technical, economical
∗ Corresponding author. Tel.: +41 21 693 3810; fax: +41 21 693 38 66. E-mail address: [email protected] (Wüthrich).
0013-4686/$ – see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.electacta.2003.12.060
and ecological consequences in the context of industrial aluminium production. It is known today, that in the Hall-Héroult process greenhouse gases (perfluorocarbon) are produced when an electrode effect (called anode effect in this context) happens [6]. Therefore, intensive researches are conducted in order to better understand the conditions leading to electrode effects. Several systematic studies have been carried out. The main debated question is the mechanism of the transition from electrolysis to gas discharges. During the past 150 years, several authors tried to give an explanation of the physical and 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 around the working-electrode This idea goes back to Bunsen [8] who supposed the existence of an electrically insulating layer of silicon or lime around the working-electrode. More recent ideas suppose the existence of a AlF3 layer. However, the electrode effects are observed at inert electrodes too and for this the hypothesis of a solid phase fails [7]. • Gaseous phase film forms around the working-electrode The 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
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Wüthrich et al. / Electrochimica Acta 49 (2004) 4005–4010
what the conditions of its formation are. Various explanations are proposed: ◦ Change in wettability of the electrode (Arndt and Probst [9]): the electrode effects are interpreted as the consequence of insufficient wetting of the electrode. The ability of the bubbles to adhere to the electrode surface is increased and they grow larger. The bubbles can coalesce and form a continuous gas film. ◦ Hydrodynamic instabilities (i.e. Helmholtz instability) are responsible for the onset of anode effect (Mazza et al. [10]). ◦ Local Joule heating: the gas film is formed by local evaporation of the electrolyte by Joule heating due to the increase of local current density because the bubbles electrochemically formed are shadowing the electrode (Guilpin and Garbaz-Olivier [3]). ◦ Combination of wettablity and hydrodynamic effects: Vogt [7] proposes that “anode effect occurs whenever the distance between neighbouring bubbles contacting the electrode has been diminished to such an extent that the bubbles are enable to coalesce”. He calculates an expression for the critical current density in function of the electrolyte flow around the electrode and the contact angle of the adhering bubbles. His model shows that the critical 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 prior to the onset of the effect. A first attempt in this direction was 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 was opened [4,5,13,14]. The present authors propose that the onset of the electrode effects is linked to the formation of a gas film around a electrode. The model proposed here is as follows: bubble evolution is described as a stochastic “birth and death” process. The onset of the electrode effect is given by the formation of an isolating gas film around the electrode. This very simple model can not only predict the critical voltage and current density characterising the electrode effect onset, but the complete static current–voltage characteristics leading to this effect. In a first description the detail of bubble coalescence may even be omitted. Later on, it can be included in the framework of percolation theory. Additional effects such as Joule heating or hydrodynamics need not to be considered at this stage. In this contribution the first level model (stochastic process only) will be presented in detail. The inclusion of percolation theory is subject of other publications [5,11].
σ i = 0,1 i=1,...,L
Fig. 1. Stochastic model: the lateral working-electrode surface A is subdivided in a lattice. In each lattice site a bubble is growing or not. The characteristic function σ i of each site indicates if a bubble is growing or not.
2. Theoretical model 2.1. Definition of the model A geometry consisting of two concentric cylindrical electrodes is considered. The smaller cylinder, with radius rw , is called the working electrode. The surrounding counter electrode with radius rc is considered to be much larger. In this configuration the electrode effects happen at the working electrode only. Furthermore, for simplification, we consider only the bubble evolution on the vertical lateral surface of the working electrode, i.e. that the bottom surface of the electrode, where bubbles could tend to stick, is much smaller than the lateral one. During electrolysis, the gas bubbles form a gas layer around the electrodes, which can be subdivided in three regions according to Janssen [12]. In the adherence region the bubbles are growing and adhere to the surface. Once they leave the electrode surface they move into the bubble diffusion region which is the region in which the bubble have a very high gas void fraction. The bulk regions, with very few bubbles, surrounds the diffusion region. The bubble coalescence takes place in the adherence region. Therefore, it will be supposed that all relevant effects leading to the formation of a gas film around the electrode, take place in the adherence region and the evolution of the bubbles in this region is studied only. The bubble evolution is considered as a stochastic process in which bubbles are randomly growing and detaching. The lateral working electrode surface is subdivided in a lattice with L lattice sites i (see Fig. 1). 2.2. Bubble evolution in the adherence region On each lattice site i, the bubble evolution is considered as a stochastic renewal process characterised by two transition probabilities λ and µ. The probability λ indicates the probability that a bubble is appearing per time at the site i and the probability µ indicates the probability that a bubble
Wüthrich et al. / Electrochimica Acta 49 (2004) 4005–4010
is leaving per time unit. The transition probabilities λ and µ are function of the voltage U between the electrodes and the mean nominal current density j ≡ I/A on the lateral working electrode surface A. The expression of these probabilities will be discussed in Section 2.4 below. To each lattice site i is associated its characteristic function σi (t) indicating if a bubble is growing or not at the time t (σi = 1 means that a bubble is growing and σi = 0 that no bubble is growing). The conditional probability P(σi (t) = 1|σi (to )) gives the probability that a bubble is growing on the site i at the time t knowing that at time to the site was in the state σi (to ). The following master equations can be written for the conditional probabilities P of the stochastic process: ∂t P(σi (t) = 1|σi (to )) = −µP(σi (t) = 1|σi (to )) + λP(σi (t) = 0|σi (to ))
(1)
∂t P(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 telegraph noise equations. The mean occupation probability p of the bubbles is given by: p(t) =
L 1 1
L
P(σi (t) = 1|σi (to ) = j)P(σi (to ) = j)
(3)
i=1 j=0
The master Eqs. (1) and (2) are solved using: P(σi = 1, t|σi , to ) + P(σi = 0, t|σi , to ) = 1
(4)
and the initial conditions: P(σi (to ) = 1) = 0
(5)
It follows: P(σi (t) = 1|σi (to ) = 0) λ = {1 − exp[−(λ + µ)(t − to )]} λ+µ
(6)
Together with (3) the evolution of the mean occupation probability p is given by: p(t) =
λ {1 − exp[−(λ + µ)(t − to )]} λ+µ
(7)
The stationary solution is: λ p= λ+µ
probability p of the bubbles is from interest, but its fluctuations p too. These fluctuations are given by the fluctuations of the characteristic functions i of the sites: L 1 2 (p) = Var σi = Var[σi ] (9) L i=1
where the second equality could be written because the random variables i are independent. Using the known variance of the random telegraph noise process, it follows: √ µλ p = (10) λ+µ A gas film around the working electrode is formed if all sites of the lattice are occupied. This happens if the following condition is met: p + p = 1
which is rather an intuitive results considering the definition of the transition probabilities. 2.3. The onset of the gas film formation For the transition from the bubble layer to the gas film around the working electrode, not only the mean occupation
(11)
i.e. even if p ≤ 1. This shows that for the onset of the electrode effect, a completely built gas film is not necessary. The fluctuations in the system itself are enough to insulate the 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 probability of bubble production is equal to the probability of bubble departure. The gas film can be built, because the fluctuations of the system are high enough that at any time all the surface can 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 covered by a gas film at the onset of the effect. Supposing that mainly only the adherence region contribute to the inter-electrode resistance R, it follows: Ro R= (15) 1−p where Ro is the intrinsic electrolyte resistance. This relation makes use of the assumptions that the adhering bubbles shadow electrically the fraction p of the electrode surface. From the result (13), the critical resistance Rcrit is estimated to be: Rcrit = 2Ro
(8)
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(16)
Taking into account the bubble coalescence improve this theoretical result to Rcrit = 2.5Ro [5,11]. This is confirmed experimentally by Fascio [13]. The fact that such simple assumptions on bubble evolution (basically only supposing that bubbles are appearing and detaching) leads to a prediction like (16) is seen by the authors as a strong argument in favour of the hypothesis that the onset of the electrode effects is linked to the formation of a 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 voltage and current density, the explicit expression of the transition probabilities have to be known. The present authors propose two qualitative expressions for them. The justification of the transition probabilities where already presented in [5,11] and are summarised in this section. In the following it is supposed that the bubble growth is uniformly distributed over the whole vertical working electrode surface. In the case hydrogen is produced at the working electrode, we propose to consider that the probability of bubble production is given by the probability that electrons combine with H+ ions. Therefore the amount of produced bubbles is proportional to the product between the electron flow (given by the mean current) and the ionic H+ flow (given by the number of H+ ions times the drift velocity vd proportional to the local electrical field Elocal as seen by the H+ ions). It follows: λ ∝ jnH+ vd ∝ jnH+ Elocal
(17)
The density of H+ ions is supposed to be proportional to the electrolyte weight concentration w (strong electrolyte). The local electrical field Elocal can be expressed for the cylindrical geometry considered here. It follows: λ=C×w
U − Ud j = λo × j (U − Ud ) rw ln (rc /rw ) + Bw
(18)
This expression models the screening effect of the ions from the electrolyte around the working-electrode. The constant B and C have to be determined experimentally but are independent of the electrode geometry and the electrolyte mass concentration w. The probability of bubble departure µ is evaluated by estimating the mean bubble life time (given by he time needed to produce a large enough bubble to detach) and making the simplification that the mean bubble departure radius is independent of the current density: µ = µo j
(19)
2.5. Current–voltage characteristics The knowledge of the transition probabilities gives the possibility to predict the current–voltage characteristics up to the onset of the electrode effects. For simplifications we suppose that only the bubbles in the adherence region contribute to the inter-electrode resistance modification. Under this assumption the resistance is written as (15) and the mean current I and the applied voltage U is linked by: U − Ud = RI
(20)
where Ud is the sum of the electrode potentials depending on the electrolyte and electrodes. Using (15) and the cylindrical
Electrode radius rw [µm] Fig. 2. Comparison between the predicted critical current density jcrit according to (22) in function of the working electrode radius rw . Measurements done in a 30 wt.% NaOH.
geometry, (20) is rewritten as: κ j = (1 − p) (U − Ud ) rw ln(rc /rw )
(21)
with κ the electrolyte conductivity. Using (13) it follows that the critical current density jcrit and critical voltage Ucrit characterising the onset of the electrode effects, are linked by: κ j crit = (U crit − Ud ) (22) 2rw ln(rc /rw ) Fig. 2 shows the comparison between measurements done in a 30 wt.% NaOH solution with a counter-electrode of 40 mm radius. A good agreement is seen. Combining (21) and (22) results in j U − Ud = 2(1 − p) crit (23) j crit U − Ud which suggests to introduce a normalised current density J and a normalised voltage U as follows: j J = crit (24) j U=
U − Ud U crit − Ud
(25)
The critical voltage can be determined straight forward using the criterion (12) together with the expression (18) and (19) for the transition probabilities: rc rw µo µo crit U − Ud = ln +B (26) = λo C rw w This relation shows the dependence of the critical voltage from the geometry of the electrode as well as from the electrolyte concentration and is shown on Fig. 3. Replacing in (23) the explicit expression (18) and (19) for the transition probabilities and using (26) it follows: U J=2 p (27) U+1
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The measurement of the critical voltage together with the critical current density is enough to know the complete characteristics of the system without accurate knowledge of the electrolyte properties, which are in general difficult to determine (especially the electrolyte conductivity which depends highly of the electrolyte temperature and purity). 2.6. Discussion The presented model make use of several simplifications that are summarised here:
Fig. 3. Comparison between the predicted critical voltage Ucrit in function of the electrolyte concentration w according to (26) and the experimental values for a NaOH electrolyte with cylindrical working electrodes (rw = 270 m, µo /C = 0.8 V wt.%/mm and B = 21.1 mm/wt.%).
This relation shows that the normalised characteristics is independent of the electrode geometry and the electrolyte properties. This is verified experimentally on Fig. 4 where the normalised characteristics is shown for various working-electrodes (radii from 50 to 300 m) and various NaOH solutions (mass concentration from 5 to 40 wt.%). Relation (27) describes qualitatively in correct way the normalised characteristics, but not quantitatively. A better agreement with relation (27) can be found considering the coalescence effect of bubbles and considering the contribution of the bubble diffusion region around the working-electrode to the apparent inter-electrode resistance [5,11]. The normalised characteristics is not only interesting from a theoretical point of view, but also from a practical one.
• The bubble evolution (growth and departure) is uniform over the whole electrode surface • Only the screening effect of adhering bubbles is considered to express the change in the inter-electrode resistance (the effect of the bubbles in the bubble diffusion region is not taken into account). • The variation of Ud , the sum of the different overpotentials, is considered to negligible compared to the applied voltage between the two electrodes. • The kinetic of the electrochemical reactions is modelled only qualitatively. • The coalescence effect of bubble is ignored. If the first two assumptions seem acceptable, the three other are more questionable. Vogt and Thonstad [2] proposed that the overpotential is growing significantly at the onset of the anode effect in aluminium electrolysis with the Hall-Héroult process. That bubble are coalescing in the range of the current densities just lower than the critical one jcrit is known experimentally [4]. During the formation of a coalescence bubble, its shape will change very rapidly and completely irregularly (see [12], Fig. 4). In the gas film formation range, bubbles will coalesce frequently and it has therefore to be asked if there is any sense to speak about the growth of single bubbles. This question is addressed using percolation theory [11]. It is clear that all these assumptions are simplifications. However, the interesting point is that even making all these simplifications and keeping only the main idea that bubbles are evolving stochastically on the electrode surface according to a renewal process, an interesting relation between the apparent interelectrode resistance Rcrit at the onset of the effect and the intrinsic interelectrode resistance Ro can be expressed. If moreover some simple assumptions about the probability of bubble growth and departure are made, the critical voltage and current density can be predicted. An improved model should try to give some more accurate expressions for these transition probabilities.
3. Conclusion Fig. 4. Comparison between the predicted J–U characteristics of electrode effects in NaOH by (27) and the experiment. Measurements done with various working electrodes radii and surface roughness. The NaOH concentration was varied between 5 and 40 wt.% and electrolyte temperature was 30 ◦ C.
This paper proposes a new approach for describing the 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 possible to explain the transition to a compact gas film formation without adding other effects like local Joule heating or hydrodynamic instabilities. The condition of the onset of the electrode effects is met once the statistical fluctuations in the system become as large as the mean values itself. This approach allows two central predictions. First it predicts that the critical resistance (the inter-electrode resistance at the transition to the gas film formation) is two times greater than the electrolyte resistance, independently of the used NaOH electrolyte concentration or electrodes. Secondly it predicts that the current–voltage characteristics of an electrolysis cell can be described in one normalised characteristics (the normalisation being done according to the critical voltage and current density). Both results are verified experimentally. The model also allows the prediction of the critical voltage and current density. Based on these results, the presents authors suggests to consider that the main mechanism responsible for the onset of the electrode effects is the formation of a gas film from the bubbles evolving around the electrode. A very similar conclusion was made by Vogt [7] who proposed that the electrode effects starts once “the distance between neighbouring bubbles diminishes to such an extent that the bubbles contact each other tending to form a continuous gas film”.
L nH p P r R t U
Acknowledgements
References
The authors would like to thank the Swiss National Science Foundation (FNS grant 061533.00) for its financial support.
[1] H. Fizeau, L. Foucault, Ann. Chim. Phys. XI (3) (1844) 370. [2] H. Vogt, J. Thonstad, J. Appl. Electrochem. 32 (2002) 241. [3] Ch. Guilpin, J. Garbaz-Olivier, J. Chim. Phys. Phys-Chim. Biol. 75 (1978) 723. [4] V. Fascio, H.H. Langen, H. Bleuler, Ch. Comninellis Electrochem. Commun. 5 (2003) 203. [5] R. Wüthrich, Dissertation Thesis (2776), Swiss Federal Institute of Technology, EPF Lausanne, 2003. [6] F. Nordmo, J. Thonstad, Electrochim. Acta 30 (1985) 741. [7] H. Vogt, J. Appl. Electrochem. 29 (1999) 137. [8] R. Bunsen, Ann. Phys. 92 (1851) 648. [9] K. Arndt, H. Probst, Z. Electrochem. 29 (1923) 323. [10] B. Mazza, P. Pedeferri, G. Re, Electrochim. Acta 23 (1978) 87. [11] R. Wüthrich, H. Bleuler, Electrochim. Acta 49 (2004) 1547. [12] L.J.J. Janssen, Electrochim. Acta 34 (1989) 161. [13] V. Fascio, Dissertation Thesis (2691), Swiss Federal Institute of Technology, EPF Lausanne, 2002. [14] V. Fascio, R. Wüthrich, H. Bleuler, Special Issue, Electrochim. Acta, submitted for publication.
Appendix A. Nomenclature A B C E i I j J
electrode surface (m2 ) experimental constant (18) ( V m−1 wt.%) experimental constant (18) ( V−1 m3 A−1 s−1 wt.%−1 ) Electrical field ( V m−1 ) index of the lattice site mean current (A) mean nominal current density ( A m−2 ) normalised current density (-)
U vd w κ λ µ σi
number of lattice sites (-) density of H+ ions (m−3 ) lattice site mean occupation probability (-) probability (-) radius (m) resistance ( ) time (s) voltage between the working electrode and the counter electrode (V) normalised voltage (-) drift velocity ( m s−1 ) electrolyte mass concentration ( wt.%) electrolyte conductivity ( −1 m−1 ) probability that a bubble is growing on the lateral electrode surface per time and site i (s−1 ) probability that a bubble is leaving the lateral electrode surface per time and site i (s−1 ) characteristic function of the lattice site i (-)
Subscripts c counter-electrode w working-electrode Superscript crit critical