The motion of electrolytic gas bubbles near electrodes

Electrochimica Acta 48 (2002) 357 /375 www.elsevier.com/locate/electacta

The motion of electrolytic gas bubbles near electrodes Steven Lubetkin
Eli Lilly & Co, 2001 West Main Street, Greenfield, IN 46140, USA Received 10 January 2002

Abstract Several reports in the literature mention oscillatory bubble motions close to, or in contact with electrodes. Such behaviour is obviously related to a time-variable force, but it was not apparent what the origin of this variable force was. Recent advances in understanding of the surfactant behaviour of gases, and in particular the gases liberated during electrolysis, have cast a new light on the underlying mechanism of these oscillations. Other unexpected, but not necessarily oscillatory bubble behaviour is discussed, and the role of electrolytic gas acting as a surfactant illuminates these non-oscillatory motions as well. Finally, the paradox of bubbles remaining attached to electrodes with zero contact angle is resolved. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: Bubble; Attachment; Nucleation; Motion; Marangoni

1. Introduction 1.1. The motions analysed here Reports of unexpected and surprising motions of bubbles at or near electrodes have been appearing in the literature for about 80 years (the first reports appear to be from Coehn and Neumann [1] in 1923). These reports were prompted by early detailed observation of the bubbles formed during electrolysis, and the phenomena appeared more complex and less easily explained as more detail became accessible with better observational methods. The time resolution achievable with relatively modern high speed cameras allowed individual events on the sub-millisecond time scale to be observed, with size resolution limited by the magnifications available with the appropriate camera lenses, and the ability to get enough light into the field of view, without too much heating. The introduction of transparent electrodes [2,3] and lasers relaxed this latter requirement somewhat. There are four main motions, and two other phenomena, which we seek to explain in the present paper. The first was described by Westerheide and Westwater [4], and might be called ‘bubble jump-off and return’. A
Fax: /1-317-277-4167 E-mail address: [email protected] (S. Lubetkin).

coalescence event results in a bubble leaving the horizontal electrode, but instead of the expected gravitational (buoyant) rise, the bubble immediately returns to the electrode. This behaviour has also been reported by others including Janssen and Hoogland [5], and Sides and Tobias [3]. The second motion has been termed ‘specific radial coalescence [3]’, and consists of a central (relatively) large bubble acting as a collector for smaller bubbles in the neighbourhood. The smaller bubbles translate across the electrode and coalesce with the central bubble. The third is the oscillation of a bubble whilst still attached to the electrode, reported by the same authors. The fourth is the report by Janssen and Hoogland [5] that bubbles rising on (or adjacent to) a vertical electrode follow sinusoidally oscillating tracks. Oscillation implies a periodically varying force, and the origin of such variations is discussed for two cases considered here. The jump-off phenomenon is not in itself periodic, but is probably a limiting case of a more general periodic motion. Clearly, the specific radial coalescence is not oscillatory. In addition to the four main motions addressed here, a further two phenomena, the so-called ‘fountain effect’ or ‘rapid-fire’ emission of bubbles [1,6,7] is tentatively explained in similar terms. Finally, there is a paradox at the centre of a commonly reported bubble behaviour. If the electrode is clean and has a contact angle of zero (or close to zero), then

0013-4686/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 4 6 8 6 ( 0 2 ) 0 0 6 8 2 - 5

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bubbles should spontaneously leave the electrode shortly after their nucleation. That they remain attached until some usually substantial buoyancy force develops is unexpected, and is explained here for the first time. The interest in the oscillatory results is that none of the well-known forces involved in bubble attachment are periodically varying, and thus there are no obvious candidates to cause oscillatory motion. However, it is in those cases where no oscillation is involved, but where this new force increases the strength of bubble attachment, that this new force may be most significant from point-of-view of electrochemical process efficiency. The role of bubbles and their departure diameter in determining the overpotential of cells, has been reviewed by Sides [8]. It is clear that in general, an additional attachment force will have the effect of increasing bubble departure diameters. 1.2. The new force Among the more familiar forces which act on a bubble during detachment from a surface in quiescent conditions are the buoyancy (gravity), and the pressure inside the bubble acting on the electrode which tend to lift the bubble off the electrode, whilst the surface tension and the hydrodynamic drag tend to hold the bubble on the surface. Electrostatic interactions may be of either sign. Another force which may be important is that produced by a temperature gradient. Surface tension is a function of temperature, so a temperature gradient produces a surface tension gradient, and this exercises a force on the bubble. Such forces resulting from surface tension gradients are generally called Marangoni effects, and are well-known in the engineering literature. The thermal Marangoni effect so produced is also called the thermocapillary effect, and sometimes thermophoresis. In principle, the surface tension gradient could equally well be produced by an externally applied gradient of a surface active species (a surfactant), but as far as the author is aware, no such reports appear in the literature. Non-steady-state or temporal variability in the concentration of surface active species have been reported, however, both at liquid/liquid interfaces and at liquid/vapour interfaces [9 /11], and the phenomenon is called the solutal Marangoni effect. The situation is illustrated in Fig. 1. Here, a gradient of surface active material is perpendicular to the horizontal electrode surface, with the highest concentration at the electrode. The basis of the theory to be presented here is that a bubble placed in such a gradient experiences a Marangoni force. The origin of the force is that the concentration gradient (although distorted by diffusion effects near the interface) is still present at the liquid/gas interface, and these concentration differences give rise to surface tension differences. At equilibrium,

Fig. 1. Dynamic concentration gradient at the bubble interface. When the bubble is immersed in a liquid with a vertical concentration gradient, the concentration gradient is distorted by the superposition of the diffusion field at the interface, and by vapour phase transport (indicated by the arrow). The model is based on the assumption that the concentration gradient is not completely eliminated at the interface, and it is the effective concentration at the bottom of the bubble vs. that at the top, which produces the solutal Marangoni effect. Note that in typical cases, the concentration at the bottom of the bubble is of the order of 10 /100 / that at the top, and that the bubble is growing rapidly, thus driving the interface rapidly into the bulk solution.

of course, such concentration gradients could not exist. The author recognises that the presence of the concentration gradient at the interface is not a proven fact */it is an assumption of the model. Oscillations of the bubble can also give rise to Marangoni effects, since changes in the curvature must be accompanied by changes in pressure and so adsorption. Finally, thermal effects may interact with concentration effects near the electrode (and elsewhere), and may also give rise to Marangoni effects, and an example is discussed below. It has been known for some time (although it has not been generally recognised) that dissolved gases can have a significant surface activity, and that when the gas in question is being produced electrolytically, then a concentration gradient of the surfactant gas exists away from the electrode. We show here for the first time that a bubble in such a gradient experiences a force holding it to the electrode. We also show that these forces can be substantial, and thus may be expected to cause measurable effects.

2. Surface tension gradients 2.1. Comparison of thermal and concentration Marangoni effects Young, Goldstein and Block [12] examined the motion of a gas bubble at a distance z from a heated surface. By solving the linearised Navier /Stokes equation, with appropriate boundary conditions, the authors

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showed that the presence of a surface tension (g ) @g gradient, ½  caused by a temperature (T ) gradient @T @T ½ ; gave rise to a thermal Marangoni effect. The @z authors also demonstrated the effect experimentally for the case of an air bubble rising through polydimethylsiloxane (PDMS) */a result confirmed by Hardy [13] and Morick and Wo¨rmann [14], who calculated the force on the bubble of radius R :

 @g @T Fthermal  2pR2 (1) @T @z The temperature gradient merely acts as the causative agent producing the surface tension gradient. The same overall effect could in principle be produced by another agent acting in the same role. It is clear by analogy with expression {1} above that we can obtain:

 @g @c Fconcentration 2pR2 (2) @c @z where c is the concentration of surfactant material producing the surface tension gradient. From the point-of-view of experimental verification of this equation, the important physical precondition is to have a sufficiently long-lasting gradient of surface active species present to allow observation of the resulting motion. One situation where this precondition is satisfied occurs during electrolysis, where the release of the gas occurs at the electrode surface, so that the concentration of gas is highest and equal to ce, adjacent to the electrode. An unstirred layer of thickness d (calculable by use of the Nernst equation), gives way to the bulk solution, and the concentration of the gas falls approximately linearly to close to the bulk value, cb, the saturation concentration of the gas concerned in the appropriate aqueous electrolyte solution. Values of the supersaturation (ce/cb) are given by many authors. It is not a well-known fact, but many simple gases exhibit significant surfactant behaviour when dissolved in water, and obey the Gibbs adsorption equation. The common electrolytic gases H2 and O2 act in this way as surfactants. Taking these two observations together, we can see that the electrolytic gas released adjacent to the electrode, and diffusing away into the bulk solution provides a steady-state gradient of surfactant species, which by virtue of equation {2}, gives rise to a Marangoni force.

2.2. Other gradients Other gradients will also produce surface tension gradients. In fact, the equations {1 and 2} above are actually specific cases of a more general equation in terms of a function g/g (x) of the surface tension:

Fx 2pR

2







@g

@x

@x

359



(3)

@z

The most obvious examples for the function, g(x ), are a gradient of potential c, which alters the surface tension at the (usually polar, charged) interface, and where the force is predicted to be given by: 2

Fvoltaic  2pR




@g




@c

@c



@z

(4)

Another is where the surface tension is a function of the applied hydrostatic pressure, P , where the force is predicted to be: 2

Fpressure 2pR




@g




@P

@P @z



(5)

A third is where the surface tension is a function of the applied magnetic field, although in this case, the solution is not of the same simple form as {4 and 5}. The first of these is likely to be of significance in cases where a bubble approaches a charged interface (electrode or ionic solid, for example). The second may be important in volcanology, where bubbles in molten magma are responsible for eruptions. The behaviour of fumaroles and geysers may also be influenced by this effect. The third will be influential where large gradients of magnetic fields are present, in magnetically susceptible materials */for example magneto-rheological fluids. It is worth pointing out, too, that the effects are not restricted to bubbles. Any two fluid media fulfil all the requirements, so that liquid drops (emulsions) should also be subject to these forces. There is however, a caveat, to be discussed in more detail below, about the influence of conventional surfactants. For the purpose of this paper, it is convenient to separate two kinds of surfactant: (1) gaseous, being simple gases which when adsorbed at the gas/water interface significantly lower the surface tension, and (2) conventional surfactants, consisting of molecules with both hydrophobic and hydrophilic portions, whose oriented adsorption at the interface also results in substantial reductions in the air/aqueous solution surface tension. Generally speaking, conventional surfactant molecules are sufficiently strongly adsorbed that they rigidify the interface to a greater or lesser extent, and thus prevent the transmission of tangential stress across the interface. Since this is the mechanism which couples the surface tension gradient to the fluid flow, blocking it prevents the Marangoni effect. For reasons which are not theoretically fully apparent yet, but nonetheless are experimentally beyond doubt (Kelsall et al. [15]), gas surfactants do not do this. It is possible to speculate that the chief difference

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360

between conventional and gaseous surfactants is that conventional surfactants are oriented at the interface, and this gives the molecules the opportunity for enhanced lateral interactions (in particular, van der Waals attractions between the hydrophobic tails) which are much reduced or absent with simple gas molecules. Due to the sensitivity of the surface rheology to low concentrations of conventional surfactant species, the presence of very small amounts of surfactant in the system (including proteins and other natural products) can interfere with the Marangoni effect. This behaviour of conventional surfactants is not so important in the case of electrolytic bubbles. Bubble streams by their nature are an effective means of removing any adventitious surfactant species as shown by for example [15]. Electrolytic redox reactions can deactivate small amounts of impurities, and many electrochemical experiments start with deliberate efforts to remove impurities in this way. In systems where significant quantities of surfactant species are deliberately added (including emulsions, which cannot in practice be made without such conventional surfactants) it is likely that the effects described here will play a minor or insignificant role.

2.3. Magnitude of the new force The thermocapillary effect is reasonably well understood, and has been both qualitatively and quantitatively verified, although only for air bubbles in PDMS [12 /14]. Here we make a direct comparison of the expected magnitude of the new force with the thermocapillary effect. For two bubbles of the same size, R , the ratio of the magnitudes of the two forces is simply:



 @g @c @g @c 2 Fconcentration 2pR @c @z @c @z (6) 

 

 @g @T Fthermal 2pR2 @g @T @T

@z

@T

@z

This ratio is now evaluated for both the extreme (high and low) cases and for a typical case. Note that the comparison is based upon PDMS as the liquid for the thermally driven, but for water for the concentration driven Marangoni effect. Table 1 has the results. By inspection of the last row of Table 1 it can be seen that the concentration Marangoni effect can be as much as eight orders of magnitude bigger than the thermal

Table 1 A comparison of the thermal and concentration driven Marangoni effects, together with some key references Quantity @g @g or  /  / @c @p

@z
d

Unit

Reference

Low

Typical

High

N m or m

[43 /45]

0.025 (H2)

0.078 (O2)

0.779 (CO2)

m

[46]

1 10 5

5 10 5

5 10 4

2

20

200

4 103

4 105

2 107

102

3 104

1.6 107

2

2

@p

Nm

@c @p or  /  / @z @z

m 4 or N m 3

½

/

@g @p ½ / @p @z

See Table 2

N m 2

@g  / / @T

N m 1 K 1

[12 /14,44] p. 221

5.5 10 5

6.5 10 5

1.5 10 4

@T  / / @z

K m 1

[12 /14,41]

5 102

1 104

1.8 104

N m 2

0.027

0.67

2.7

None

3

5 104

6 108

½

/

@g @T ½ / @T @z

@g @p ½ ½  @p @z / / @g @T ½ ½  @T @z

Data referring to the concentration effect are for water, but for the thermal effect, for PDMS.

S. Lubetkin / Electrochimica Acta 48 (2002) 357 /375

Marangoni, and is typically four orders of magnitude greater. Even with the ‘low’ case choices (i.e. conditions chosen to maximise the thermal effect, and minimise the concentration effect), the concentration effect is about three times greater than the thermal Marangoni. These large differences are a direct consequence of the fact that the concentration gradient is developed over a short distance (the Nernst layer) and that the surface tension is more sensitive to the effects of surfactants than to the effects of temperature. Given the large size of the concentration Marangoni effect, significant and detectable responses to this force are to be expected. 2.4. The range of the new forces The Marangoni force is of restricted range, whether it is generated by temperature, concentration, potential difference, pressure or other agent. That range is defined by the distance scale of the agent producing the surface tension gradient. Consider first the upper limits to the length scales. It is relatively easy to impose a uniform temperature or potential gradient over significant macroscopic distances (of the order of centimetres) and for pressure gradients, simply using hydrostatics allows uniform pressure gradients of almost unlimited length scales, up to a few kilometres. For concentration gradients, however, the range is severely limited by what can be achieved using steady-state diffusion. For electrolysis this means the thickness of the unstirred Nernst layer. At the other extreme, the shortest length scales achievable are limited by the size of the bubble on which the gradient acts. If the bubble falls much below a micron in size, it may approach the critical bubble size [16] for the system. Below this size, the internal pressure in the bubble is incapable of successfully opposing the surface tension tending to collapse the bubble. This imposes a practical lower limit on the range of size for the bubble and hence the distance over which the concentration gradient is developed. There is an interesting corollary to this discussion. In the experiments to measure the thermal Marangoni effect [13,14] the authors noted that the solubility of atmospheric air in the PDMS was a function of temperature. These authors showed an oscillatory vertical bubble motion which they ascribed to volume changes due to the bubble growing or shrinking as a result of the changing solubility of the gas. It is likely that the changing solubility of the air in the PDMS will also cause changes of surface tension, so that there would be a second force in addition to buoyancy acting on the bubble. The magnitude of this additional force @g would depend on the value of ½  for air in PDMS, but @c this is unknown at present. The length scale of this

361

concentration gradient, is however, the same as the length scale for the thermal Marangoni, so the relative magnitude of the effects would be given simply by the @g ½  F @c : This ratio, being independent ratio: concentration  @g Fthermal ½  @T of the length scale, would provide a better basis for comparison of the relative size of the effects driven by thermal and concentration differences, but experimental data to evaluate this ratio do not exist at present. It should be remarked that in general, thermal Marangoni effects should be accompanied by concentration Marangoni effects since the solubility of gases depend on temperature, and the gases also produce changes in surface tension dependent upon their concentration. Such effects may have consequences for the interpretation of thermal diffusion cloud chamber experiments.

3. Bubble motion in a concentration gradient Given that the new force is potentially much larger than the thermocapillary effect, what are the expected consequences? And perhaps more interestingly, why has this substantial new force not been identified before? First, it is important to recognise that although the new force is indeed much greater than the thermocapillary effect, it can operate only where a sufficient gradient of gaseous surfactant species exist. Also, the presence of small amounts of conventional surfactants will strongly mask the effect. The range of this new force is restricted to the range of the surfactant gradient, which for electrolysis, is defined essentially by the Nernst thickness */or as we will see below, a few tens of microns. The thermocapillary effect, by contrast is weaker, but typically of much longer range. Usually, it operates on macroscopic length scales: in the cases of the experiments of Hardy, Morick and Young et al. the distances were of the order of millimetres. The thermocapillary effect will also suffer from the masking effect of surfactants, and the experimental confirmations of the effect have all been done with silicone oils, where the elimination of surfactants is a much easier task than it would be for a water-based experiment. 3.1. Bubble jump-off and return When two similar size bubbles are adjacent to each other on the horizontal electrode surface, and they meet and coalesce, the equilibrium centre of gravity of the new larger bubble will be some distance D further from the electrode surface than the position for either of the two original smaller bubbles. As the new larger bubble adjusts to its new equilibrium position, there will be an

362

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impulse normal to the surface of the electrode. Consider a single bubble formed by the coalescence of two bubbles, the second of which has a radius with a variable percentage of the first, which is taken to have a fixed arbitrary radius. The difference in height of the centre of gravity of the composite bubble, D% is compared with the height of the centre of the largest original bubble, and is plotted in Fig. 2. The distance D is maximised for bubbles of the same size. Notice also that bubbles freshly nucleated have a similar impetus to leave the surface. If the system is clean, and the contact angle, u , between the bubble and the electrode surface is zero (or even close to zero) it might be expected that the freshly nucleated bubble would rapidly leave the electrode surface. This is not normally observed: the departure diameter is generally much bigger than the critical nucleation diameter. This interesting fact is addressed in more detail below. It is important to recognise that a ‘clean’ electrode does not necessarily have a zero contact angle, but there are cases in aqueous systems where ‘clean’ and ‘zero contact angle’ do in fact coincide. The impulse accelerates the bubble away from the horizontal electrode. The resulting motion depends on the balance of all the forces then acting on the bubble. For the purposes of the present discussion, the interesting case arises when the impulse is sufficiently great to cause the bubble to leave the electrode. Note that this is not always the case. The forces acting to slow the bubble may include an electrostatic interaction, the hydrodynamic drag, and the Marangoni concentration force. In addition, if there is a vertical temperature gradient (which very often there will be, particularly at high current densities), then a thermal Marangoni effect may operate, too, as has been pointed out by Lubetkin [17]. At the time of their original report, in 1964, Westerheide and Westwater speculated that the return might be due to surface tension gradients or electrostatic interactions. Note that as the velocity falls to zero, the hydrodynamic drag also falls to zero: this force, therefore, cannot account for a bubble returning to the electrode. The electrostatic interaction depends upon the charge on the electrode and the bubble. Often these are of the same sign, (an example would be H2 evolution at pH/3 on Pt) and the interaction, is therefore, repulsive: clearly this cannot lead to bubble return. Another influence on the electrostatics is that the effect of the electrode falls off very rapidly with distance, since the electrode is effectively screened by the double layer. The range of the force, is therefore, of the order of (1/k ), where 1/k is the Debye length. In high ionic strength media, (including typical electrolyte solutions) the Debye length is very small (typically of the order of a nanometre or less). The Marangoni effects are the only remaining sources of force which might account for the return. As seen by inspection of Table 1 above, the magnitude of the

concentration Marangoni force is greater than for the comparable thermal Marangoni effect by a significant factor. This justifies taking only the former into account in the subsequent analysis, although it does not mean that the latter effect is absent. We have no means at present of evaluating a third possible Marangoni effect; that due to a voltage gradient near the electrode, although the electrocapillary effect in the case of mercury electrodes is well known. Notice that in reports of the electrocapillary effect for mercury electrodes, the discussion centres on the alteration of the interfacial tension of the mercury electrode in the presence of the electric field, but not on the effect on the gas/aqueous electrolyte (bubble) interface adjacent to it. We, therefore, focus on the concentration Marangoni effect as the most important contributor, and we attempt to define the parameters for the particular case described by Westerheide and Westwater [4]: a H2 bubble on a Pt microelectrode at intermediate current density, and a combined bubble about 50 mm in diameter. This is some way below the normal departure diameter under these conditions */this aspect is addressed in more detail in the discussion, below. 3.1.1. The factor

@g / @c

This quantity has been measured for a number of simple gases, usually by measuring the surface tension as a function of pressure, by a number of authors [18]. It is important to recognise that these pressures are sufficiently low that they have no direct influence on the surface tension, unlike the pressure P referred to in equation {5}. They act by increasing the solubility of the gases concerned. At low pressures, the concentration and pressure can be related through Henry’s law: the constant of proportionality is dependent on the system examined, but for any given gas/liquid combination: c pkH The values of the surface tension as a function of pressure for a variety of common gases in water are given in Table 3 [19]. The only gas not showing an effect is He, while both H2 and O2 should show the effect, (as judged by the first coefficient, b), O2 is predicted to show about three times as large an effect as H2. Unfortunately data is lacking for Cl2, but it is predicted that when the data is available, that Cl2 will indeed show a similar trend, with the expected coefficient being about three times greater for chlorine than for CO2. @g (which In summary, the data show that the factor @p @g is proportional to ); is smallest for H2 of the common @c electrolytic gases, whilst that for O2 is intermediate, and

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363

Fig. 2. The change in height (expressed as a percentage of the radius of the larger bubble) as two bubbles of various sizes coalesce. The function reaches a cusped maximum at R1 /R2, where the volumes of the two spheres are equal. Table 2 Experimental and theoretical values for the supersaturation, (a is the dimensionless saturation ratio, a is the dimensional form) with references to the literature Author(s) (a ) Sides Westerheide and Westwater Dapkus and Sides Shibata Glas and Westwater H2 (b ) 1.54 19.9

a (ce/cb) or a (cecb)

@g At low pressure, the linear approximation that b/ is good @p Gas

b

c

He H2 O2 N2 Ar CO CH4 C2H4 C2H6 C3H8 N2O CO2 nC4H10

0.0000 0.0250 0.0779 0.0835 0.0840 0.1041 0.1547 0.6353 0.4376 0.9681 0.6231 0.7789 2.335

/ / 0.000104 0.000194 0.000194 0.000239 0.000456 0.00316 0.00157 0.0589 0.00287 0.00543 0.591

d

Reference


1000, 100 8 /24

[8] [4]

9 /16 7 /70 (1, 5. . . 100 mA) See Table 2b below O2

[22] [24]

1.36 15.4

Table 3

[7] CO2

1.08 1.64

Cl2

1.018 1.324

/ / / / / / / / / 0.000040 0.000042 /

g g0bpcp2dp3 (p in bar) for aqueous solutions at 25 8C from [19]. Deviations from this linear relation are apparent at higher pressures.

that for CO2 is the largest. It is anticipated that the value for Cl2 when it is measured will be larger still than that for CO2. The appropriate values appear in Table 3. At relatively low pressures, the gradient can be taken to be a constant, with little loss of accuracy, however, in the specific case of CO2, more precise data at low pressures suggest that the linear approximation is not completely reliable: a better representation is given by: g5:93 ln(p)70:67 where p is the gauge pressure in bar [20].

(6?)

3.1.2. The factor

@c / @z

This factor is made up of two parts, the concentration term, @c, and the distance over which the gradient develops, @z. Dealing first with @c: for electrolytic generation of gases, the concentration adjacent to the electrode is the highest attainable, since this is where the gas is liberated. This concentration must be estimated, since there is no completely reliable method of actually measuring the concentration at the electrode surface.

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364

c The saturation ratio is conveniently defined as a / e ; c where ce is the concentration at the electrode, and cb bis the bulk value, although some authors prefer the dimensional version, (ce/cb). When microelectrodes are used in conjunction with rather soluble gases (CO2, Cl2) the bulk value of the concentration may fall some way short of the equilibrium saturation value, cb since the amount of gas generated at the microelectrode may be insufficient to compensate for the relatively rapid diffusion of the gas away from the electrode. In these circumstances, experimentalists generally saturate the aqueous solution with the gas before starting the experiment. In general, however, the bulk is quickly saturated with gas at approximately the cb value. Note that the quoted solubilities should be those in an aqueous solution of the same composition as the electrolyte, rather than for pure water.

3.1.3. Supersaturation near the electrode, (a): @c There are two main approaches to estimating the supersaturation near the electrode, one being theoretical, based on nucleation theory, and the other is experimental. Theoretical models either attempt to calculate critical supersaturations for appreciable bubble nucleation using so-called classical nucleation theory [21], or use finite element analysis to model the situation close to electrodes [22]. The experimentalists base their estimates either on measured rates of growth of bubbles still attached to electrodes [23], or transient kinetic measurement [24]. Using classical nucleation theory [21], an upper bound can be placed on the supersaturation, s. This limit can be estimated using equation {7}:   16pg3 J C exp (7) 3kT(sP)2 where g is 7.2 /102 N m 1. The values of the physical

parameters used in the calculations are listed in Table 4 below. For these values, a predicted supersaturation, s (s/ p c 1 /a /  e ) in the aqueous solution adjacent to the p0 c b electrode is of the order of 1400. There are some subtleties to do with the interfacial tension in the case of bubble nucleation which will be addressed in the discussion, but these effects all result in lower estimates for the supersaturation needed to cause massive bubble nucleation, and thus do not affect the estimates of the upper bound. There is a further question to be answered in any treatment of the bubble nucleation problem, and that is whether the nucleation is homogeneous (as assumed above) or heterogeneous, taking place with the involvement of the electrode surface. In cases of partial wetting, the contact angle, u , is a measure of the potency of the surface for catalysis of bubble formation, and this is introduced into the equation for the nucleation process as a factor of: F(u)

[2  3 cos2 (u)  cos3 (u)] 4

(8)

and the modified nucleation equation then becomes:   16pg3 F(u) J C exp (9) 3kT(sP)2 Perfect wetting (u /0) corresponds to a factor of F (u) /1, and this in turn leaves the numerator in the exponent in equation {9} identical with its value in the homogeneous case. In other words, when the electrode surface is perfectly wetted by the electrolyte, bubble nucleation should be as difficult as the homogeneous case. It is established for clean platinum that aqueous electrolytes are completely wetting (see for example, Brandon and Kelsall [25]), and other authors report

Table 4 Quantities used in the theoretical calculation of the homogeneous nucleation limit Parameter Value

Unit

Note This is the usual assumption in nucleation theory, equivalent to 1 cm 3 s 1 This value of C is typical for nucleation in aqueous systems */but its magnitude is unimportant for the calculation Assumed that of pure water at 293 K

J C

106 1035

m 3 s 1 m 3 s 1

g k

7.2 10 2 1.38 10 23

T s

293.15 To be determined 105

N m 1 J K 1 per molecule K Dimensionless

P

Pa

Close to normal atmospheric pressure

Using the values in the table, together with equation {7} gives a value of about 1400 for the supersaturation required to cause rapid bubble nucleation in water. This represents an upper bound on the achievable supersaturation. Note that the presence of a dissolved gas would generally lower the required supersaturation in two ways. The surface tension will generally be lower than that of the pure solvent, and the dissolved gas results in a smaller critical nucleus than for the pure solvent. There may also be effects from non-ideality in both the gas phase and in solution.

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clean or wetting conditions in their experiments. Thus the upper limit on the concentration quoted above is probably a good estimate, corresponding as it does to both homogeneous and probably essentially uncatalysed heterogeneous nucleation. Impurities may render the electrode surface more hydrophobic, as might certain surfactant species, and under those circumstances, lower energy barriers to nucleation might be expected, and thus smaller supersaturations might be necessary. It is also possible that nucleation sites may be composite in nature, with both hydrophilic and hydrophobic parts, as pointed out by Carr [26]. In the context of this discussion, it is not clear from the report of Westerheide and Westwater whether their electrode was completely clean. Close inspection of the photomicrographs in their paper suggest that the contact angles were very close to zero, so the conditions were probably clean. A minimum value of the supersaturation can be estimated from the experimentally determined rates of growth of bubbles whilst they are still adjacent to the electrode. These are expected to be minimum values particularly for microelectrodes because significant depletion of the supersaturation near the electrode is predicted as a result of the nucleation event itself. The experiments of Westerheide and Westwater [4] to measure the rate of growth of hydrogen bubbles at a platinum electrode gave a value of about 8 for s. Smaller values have been reported by the same authors, who found values as low as 2% supersaturation ( /(s/ 1)100) for Cl2 bubble growth, and 8% for CO2. If the area of discussion is widened somewhat to include measurements of bubble nucleation by other than electrochemical means, for example by supersaturations imposed by pressure release [21] or by reducing the pressure below atmospheric pressure for solutions of gases saturated at ambient pressure, or by generating supersaturation by chemical generation [27] of gases in situ, lower values seem to be the rule [28]. The CO2 bubbles nucleated in carbonated drinks typically appear at supersaturations below about 4. In summary, then, the upper bound on the supersaturation (above which massive, homogeneous bubble nucleation is predicted to occur, thus relieving the supersaturation) is about 1000, whilst the lower (experimental) bound is B/10, while for pressure release experiments, rapid nucleation of bubbles occurs at supersaturations B/5. The various values are summarised in Table 2. 3.1.4. The factor

@c ; (continued): (b) @z @z

The concentration decays (approximately linearly) over the thickness of the Nernst layer, d . We will identify d with @z. The boundary layer thickness is not a particularly well-defined quantity: first, the value is dependent upon the current density, becoming smaller

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as the current density is increased. Typical values quoted in the literature [29,5] vary from about 10 to 100 mm for current densities in the range 100 /10 mA cm 2. A second important source of variability in d at least as far as bubble evolving systems are concerned, is that the bubbles released can act as quite efficient stirrers, thus altering the ‘unstirred’ layer thickness quite considerably. Various semi-empirical relationships have been developed which attempt to allow for such effects [2]. It may be that the bubble itself distorts the concentration distribution near the electrode so that effectively, the whole concentration drop occurs over the length scale of the bubble diameter, regardless of the thickness of the unstirred layer. Such a suggestion has been made by Vogt [30] based on analogy with temperature distributions near bubbles on heated surfaces. This question is not settled, and for the present purpose, we adopt the Nernst model to evaluate the force on the bubble; the resulting prediction of the thicknesses are at most about 100 mm and at the least about 10 mm. Of course, Nernst layers can be thinner than 10 mm, but a bubble of comparable diameter to the layer thickness, d would greatly distort the unstirred layer. A minimum practical limit, is therefore, set by the size of bubble producing the effect we seek to explain, and about 5 mm has been chosen as a practical or working lower limit, rather than one with any strong theoretical underpinning.

Fig. 3. The isolation of the bubble at A implies a generally higher level of supersaturation (and therefore, a relatively lower surface tension) in the vicinity, than in the populated region near B, where the opposite applies. There is thus a surface tension gradient from high to low in the sense B 0/A, and a corresponding Marangoni force in this direction. Only one quadrant is shown of an essentially circularly symmetrical arrangement.

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3.2. Specific radial coalescence Referring to Fig. 3, a central relatively large bubble (A) acts as a focal point for smaller surrounding bubbles, which move towards A and coalesce with it, leaving a relatively clear area on the electrode. An explanation of this behaviour is that the aqueous solution in the region labelled B, because of its greater density of bubbles, each acting as a sink for the electrolytic gas, has a correspondingly lower average value of the gas concentration, and therefore, a higher ambient interfacial tension when compared with A. The horizontal (surface) concentration gradient will produce a Marangoni force causing the smaller bubbles around B to migrate in the direction of A. Other processes may well be involved here, too. Ostwald ripening is capable of causing transfer of gas from the smaller to larger bubbles, but cannot explain the movement of whole bubbles. 3.3. Sinusoidal bubble tracks on a vertical electrode H2 bubbles generated repetitively at specific sites at the base of a vertical Pt electrode were observed by Janssen and Hoogland [5] to follow oscillating paths as they rose. The amplitude of the oscillation was of the same order of size as the bubble diameter, and the frequency was about 16 Hz. The authors had no explanation for the phenomenon. The key to the understanding of this interesting oscillatory motion is to be found in the fact that as each bubble rises and grows, it sweeps out a vertical track in the supersaturated gas solution near the electrode, creating a depleted zone, roughly coincident with its oscillating path. In doing this, it leaves two adjacent zones, which are relatively rich in dissolved gas. The situation is illustrated schematically in Fig. 4. The next bubble to be released from the site will start its rise in the depleted zone, but will experience a lateral force due to the Marangoni attraction of either one of the richer adjacent zones. This lateral force will accelerate the bubble towards the centre of the rich zone, overshoot will occur, and the bubble will then experience a restoring force back towards the centre of the rich zone. Such a motion will potentially be simple harmonic (sinusoidally oscillating). All the while, the bubble will be rising, creating the oscillating track observed by Janssen and Hoogland, and at the same time depleting what was one of the two rich zones, and creating two new rich zones to either side. The process is thus selfsustaining. A Xerox copy of Fig. 5 from the published paper in Janssen and Hoogland [5] was enlarged, and was then digitised using a program [31] called ‘Clicker’. The tracks of 30 of the bubbles were analysed (out of a total of 73), the selection being made on the basis of which

Fig. 4. A nucleation site (A) at the bottom of the electrode has just released a bubble. The clear zone (C) is enriched in dissolved gas (previous adjacent bubbles having depleted the zones B on either side). At the end of its rise, this bubble will have depleted zone C, and zones B will now be relatively enriched. The next bubble to be released from A will probably rise in the right hand zone B. Note that the horizontal position of the nucleation site does not necessarily determine the centre line of the oscillating track.

tracks could clearly be distinguished from other nearby tracks. Representative digitised tracks are shown in Fig. 5. A few tracks were clearly not vertical; an example is shown as track 4. The size scale was established from the original image size [32]. The mathematical model is based on two assumptions. First, that the bubble experiences a force directed towards a region of high gas content (low surface tension). The regions are modelled as having a sharp boundary between them, as shown in Fig. 6. This does not correspond with the physical reality, but simplifies the mathematics considerably. The second is that as the bubble penetrates the concentration ‘wall’ as illustrated in Fig. 6, the force on the bubble is proportional to the area of that portion of the bubble yet to penetrate the ‘wall’, which is a spherical cap of height x and area A .

 @g @c FM(cap) A @c @z

(11)

where A is given by: A /2pRx , and thus the force is:

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Fig. 5. Twenty-three of the digitised tracks of bubbles taken from Janssen and Hoogland [5]. Roughly, the area shown is 1 mm2, with the bottom of the electrode coinciding with the X -axis. The original photograph had an exposure time of 1/4 s. The track labelled 4 is clearly not vertical.

x D exp

  h t sin(btb) 2

(12?)

and the period is given by: 4p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4K 2  h2

Fig. 6. A bubble of radius R is penetrating a region (to the right) of relatively high electrolytic gas content, from a region (to the left) of relatively low gas content. The step function for the concentration is not physically realistic */it would be smeared out, and would be expected to have an error function shape.



 @g @c FM(cap) 2pRx @c @z

(12)

where the force is directed towards the centre of the region, in the direction of negative x . Also, the frictional resistance or drag (proportional to the viscosity, h ) is such as to oppose the motion, with the drag being proportional to the velocity for low Reynolds numbers. The equation of motion is then: x2pRxh ¨ x˙ Letting K2 /2pR , we have: 2 xhx˙ xK ¨

When h2 B/4K2, the solution of this equation is:

(13)

For small values of h , the frequency (v /period 1), therefore, goes as K and so it goes as R1/2. An example of the fit obtained to this equation of the raw data accumulated from the Janssen and Hoogland Fig. 5 is shown in Fig. 7 below. The procedure was as follows. Each track was normalised to a standard initial amplitude of 9/1, then the phase was adjusted to bring the experimental and theoretical initial points close to registry. Two adjustable parameters (the period and the frictional coefficient) were then used in combination to minimise the sum of the squares of the differences at each point. About 30 oscillating tracks were analysed in this way, and some interesting conclusions may be drawn. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4K 2  h2 3.3.1. The frequency, v / 4p No account was taken of the growth of the bubble during the approximately 250 ms exposure time of the photo. Such a growth would be expected to cause an acceleration of the bubble corresponding to an increase in the terminal velocity. In turn, this would be expected to show as an increase in the period, or a decrease in the frequency of the oscillation of any individual track as the track extends vertically. This was not observed, and this means that to a first approximation, and for the purposes of this analysis, the bubble growth can be ignored.

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Fig. 7. The experimental data (squares) and the theoretical fit (circles) for the track numbered 21 in Fig. 5. The fit is obtained by setting the phase, and the friction (or the viscosity, h */see text) then adjusting the frequency to give the minimum total square deviation of the Y coordinates for each point. Fixing the friction at 0.1, and adjusting only the frequency gives very nearly the same result. The experimental data were obtained from digitised enlarged Xerox copies made from the original reference, and are thus subject to significant (but unquantified) errors. AU/ Arbitary Units.

3.3.2. The positions of the tracks Taking the X -axis in Fig. 5 as a measure of the horizontal position of the tracks, these X values have been analysed for the presence of a natural spacing, using a statistical technique due to Broadbent [33]. The technique reveals the presence of a natural spacing of about 9% of the full width of the electrode. This is in reasonable agreement with the visual impression that there are about ten vertical tracks spaced roughly equally horizontally along the electrode. It may well be that the mechanism for generating the oscillating tracks stably divides the available space into uniform regions, each dominated by one nucleation source, and that an even spread of tracks does not imply an even spread of nucleation sites.

3.3.3. The friction, h The friction, h is subject to only small variations in the data set examined here. The mean value9/standard deviation in the viscosity term used as a fitting parameter is 0.109/0.04. This level of spread suggests that the viscosity is probably constant. Physically, of course, this is as expected. Reanalysis of the data, using a constant value of h equal to the mean, shows that this is the case. The minor improvement in the fitting does not justify the inclusion of a second adjustable parameter.

3.3.4. The length of the tracks The tracks shown in Janssen and Hoogland Fig. 5, represent the distance travelled by each bubble during the period of the photographic exposure, which is about 250 ms. The length of the track, is therefore, a measure of the velocity (and so, the size) of the bubble, since generally speaking, in this size range (up to about 300 mm) the velocity is proportional to R2 (for Stokes 2R2 Drg ): Thus from the length of the spheres, n 9h tracks, it is possible to estimate the size of the bubble producing the track. In this way, the analysis shown in Fig. 8 was produced. The point of interest is that the oscillation frequency is predicted to be approximately proportional to R1/2, by equation {13} above. The correlation analysis shows that using the track length to estimate the value of R , accounts for approximately 73% of the variability, which considering the very approximate nature of the method used to assess the bubble size, is considered satisfactory. 3.4. Oscillation whilst attached to the electrode This is a special case of jump-off and return. If the coalescence event occurs at such a bubble diameter that the impulse and the buoyancy lift together are insufficient to overcome the downward forces, then oscillation damped by viscosity would be the natural result. In the special circumstances where the motion is overdamped, there would be no oscillation, just a slow approach to

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Fig. 8. The radius, R in arbitrary units, is taken to be proportional to the length of the track. The assumption is that the terminal velocity of the bubbles is proportional to R2, (by Stokes’ law) so that the length of the track allows an estimate of the bubble size. The theory implies that the motion will have a characteristic frequency proportional to R1/2, and the graph shows the correlation. Though the correlation is by no means good, this very crude model successfully accounts for 73% of the variability of the data, providing some support for the underlying theory.

two oscillation modes, as the bubble moves to one extreme of its motion, growing by diffusion all the while, it will be depleting that region of some of its dissolved gas. At the same time, the other extreme will be gaining gas concentration relative to the opposite node. As the bubble begins its motion back to the centre, it will be accelerated by the Marangoni force due to the gradient of dissolved gas. 3.5. Rapid-fire emission

Fig. 9. In the upper diagram, a bubble attached to the surface of the electrode (either pinned by a surface imperfection, or still attached to the nucleation site) coalesces with another. The centre of gravity is thus suddenly displaced, and an oscillation ensues. Since the bubble is growing (and thus depleting the adjacent solution of dissolved gas) at one extreme of its motion at A, it will have depleted the solution there. Both diffusion and convection as a result of the motion will have enhanced the concentration at B. The net effect is that a Marangoni force will increase the acceleration towards B, thus assisting the momentum in maintaining the motion. Another possible oscillation is shown in the lower diagram.

equilibrium. Such motion as this does not require the presence of the new force. The momentum generated by the coalescence event would be dissipated by the oscillation. However, the presence of the concentration Marangoni effect would assist in making the motion more long-lasting. As shown schematically in Fig. 9 for

In their original report, Westerheide and Westwater ascribe [4] the jump-off to the coalescence of two similar sized bubbles. The force causing jump-off in this model comes from the fact that the centre of gravity of the combined bubble is initially some distance below the ultimate height of the combined centre, and the ‘rebound’ provides the impetus for the bubble to leave, as discussed above. Some time dependent force of this sort is needed to explain jump-off. An interesting question arises, if the electrolyte is truly wetting on the electrode, why do bubbles stay in the immediate vicinity of the electrode for a prolonged period during their early evolution? Clearly, the reason they form there is to do with the supersaturation being greatest there. It seems that in the absence of a contact angle, and without the Marangoni force identified here, unless there was a strong electrostatic attraction, that the bubble should jump off the electrode as soon as it forms, since the basis for the argument about centres of mass applies a fortiori to the case of a bubble nucleating ‘from nothing’ at the

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electrode surface. Note, however, that the balance of inertial and viscous forces, versus surface tension forces, changes as the bubble size increases, and that departure at the smallest sizes might be slow for this reason. The rapid departure of a bubble might cause sufficient stirring to reduce the concentration gradient near the electrode, thus minimising the Marangoni force there. Subsequent bubbles if nucleated sufficiently quickly might then leave the electrode in rapid succession-giving rise to the ‘rapid-fire’ emission of bubbles also reported by Westerheide and Westwater. In short, the conditions required for this phenomenon to take place are rapid nucleation and zero contact angle. Composite nucleation sites would allow rapid nucleation to take place in a hydrophobic region (favourable to nucleation), followed by rapid growth out into a region where the contact angle is zero, or close to zero (favourable to detachment) and thus the bubble detaches. With a suitable geometry for the composite site such as the conical or slit-like structure suggested in Fig. 11, the bubble could be forcibly ejected as it grows more rapidly than the acute angled site can accommodate. The zero (or near zero) contact angle provides assurance that the bubble will suffer negligible friction from the walls of the nucleation site. The hydrophobic portion of the site (depending on the details of the contact angle, the critical bubble size, and the extent of the hydrophobic region of the nucleation site) might retain a portion of the gas phase, thus avoiding the need for a fresh nucleation event. This is not necessary for the rapid-fire emission of bubbles,

Fig. 11. A hypothetical composite conical or crack-like nucleation site. The region A is hydrophobic, possibly as a result of capillary condensation of a volatile organic material. This deposition mechanism would restrict its presence to the narrowest region of the pore. Nucleation is easy in this environment, with its high contact angle. As the nascent bubble grows past the hydrophobic region, and into the hydrophilic region B, a small remnant may be detached, remaining in the hydrophobic region. This residual gas would promote the instantaneous growth of the next bubble. As it grows, it will be shot out of the conical site, its zero contact angle ensuring the absence of friction. The rapidity of its departure might cause sufficient stirring in the vicinity C, to disrupt the Nernst layer, thus eliminating the concentration Marangoni effect, which would otherwise tend to hold the bubble near the electrode until it was larger. Alternatively, the rapid growth of the first bubble during its rise, and the immediate following of others might cause depletion of the Nernst layer in a narrow bubble chimney, again destroying the concentration gradient.

Fig. 10. The plot of log10[CO2] against the surface tension for CO2 dissolved in water. The concentration is measured by the applied gauge pressure, for pressures between 1 and 11 bar, showing the approximate linearity of the log plot.

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however. The nucleation event itself might be sufficiently rapid to account for the succession of bubbles, without the need to invoke a mechanism which avoids nucleation. Some of the issues have been discussed in Lubetkin [34]. 3.6. Detachment There is yet another possible ramification for the concentration Marangoni effect, and this may be the most significant result of such a force. The presence of a term due to the Marangoni force on the bubble may alter the bubble departure diameter significantly, and this will be particularly true for systems where the gas @g @c released has a large and a large gradient away @p @z from the electrode. Such alterations of the departure diameter will have significance for the efficiency of electrolytic cells. It has been pointed out that in the overall kinetics of bubble evolution, the kinetics of nucleation and the kinetics of detachment are intimately linked [35]. Contact angle conditions likely to cause rapid nucleation are exactly those required to cause slow detachment. Large values of u promote heterogeneous nucleation, which is energetically favoured over homogeneous nucleation, as seen by comparing equations {7 and 9}. Thus bubbles will tend under these conditions to remain occupying the nucleation site, thus masking any further nucleation. The bubble may even grow over adjacent sites, masking their activity too. When the contact angle is low (as is very commonly claimed in electrolytic experiments, where many authors mention the excellent wetting of the electrode surface as evidence of the cleanliness of the system) detachment would be easy, with the attachment force in the limit falling to zero as u 0/0; however, the nucleation rate is now essentially that due to homogeneous nucleation */ the surface ceases to be an effective catalyst for the nucleation process when u
/0. Thus it seems as if the evolution of gas bubbles is inhibited to some extent regardless of the exact value of the contact angle. The new force due to the concentration gradient near the electrode can only make detachment more difficult. The effect will be greatest for Cl2, and will be smallest for H2 and intermediate for O2 or CO2. In addition, the thermal Marangoni effect, which generally will be present (particularly at higher current densities) will increase this tendency. However, as the bubble grows larger than the thickness of the diffusion layer, the concentration Marangoni effect will become less important, so that there is a limit to the retardation of the detachment process. A circumstance where the potential gradient may be playing an important role is that reported by Dapkus

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and Sides [22] where the very carefully cleaned mercury electrode (contact angle
/08) still had bubbles attached to its surface. Note that when the mercury surface is truly clean, that there is at present no other known force to hold the bubble on the electrode surface. The high thermal conductivity and large thermal inertia of the mercury electrode make any thermal Marangoni effects unlikely, so that the bubble is probably held by a combination of the concentration and electrostatic Marangoni effects.

4. Gases as surfactants The surfactant property of simple gases in aqueous solutions is a well-established fact. A reasonably complete listing of the literature is given in references [18]. Reports of organic gas adsorption causing reduction of surface tension for non-aqueous solutions seem to be restricted to Lubetkin and Akhtar [20], although there are a number of measurements of the effect of N2 as the gas dissolved in various organic simple liquids. This is not an area which has been well-explored theoretically or experimentally. It can be deduced that when compared with the kinetics of adsorption of conventional surfactants, the kinetics of adsorption/ desorption for gases are very rapid both from the gas phase and from the liquid (diffusion being relatively fast for gases in liquids). The heat of adsorption is low compared with that of conventional surfactants, and unlike conventional surfactants, specific orientation at the interface is not required in general. It is expected dg that the Gibbs elasticity, o ð Þ of the interface d ln A will be small. There is a good correlation between the water solubility of the gas, and the degree to which it lowers the surface tension. Fig. 12 has the summary. In the figure, k is defined by: p kx where x is gas mole fraction in the aqueous phase, p is the partial pressure of the gas in atmospheres. Smaller values of k thus denote more soluble gases. The correlation between solubility and the b coefficient (a @g measure of ) is convincing, though by no means @p perfect. Thus, the largest lowering of the surface tension occurs for the most soluble gases: as an approximation, G (the surface excess of the gas surfactant) is roughly proportional to the number density of gas molecules in solution. There is a key question in connection with the behaviour of bubbles in the presence of surfactants, which must be addressed, and that is, why does the

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Fig. 12. The approximately linear relationship between log(k) and the jbj coefficient. Note that 1/k is proportional to the solubility, and the value of b @g :/ represents an approximation to the factor @p

adsorption of a surfactant (gas) not cause the interface to become rigid? Levich [36] addresses the question of the adsorption of an insoluble surfactant, showing that this produces a rigid interface. Such a rigid interface is incapable of transferring tangential stress, and of course this blocks the effects of the surface tension gradient at the interface. In other words, this is precisely the mechanism invoked in the standard explanation of the fact that the predicted enhanced rise rate of a bubble (the Hadamard /Rybczynski [37] description) is not generally observed experimentally. Notice that there is something of a dichotomy between the Levich model of a bubble rising through a liquid, accumulating surfactant at the rear as a cap, which implies a progressive shielding of the interface against tangential momentum transfer, and the Hadamard /Rybczynski description of an interface either with the possibility of interfacial momentum transport, or not */a sort of on/off mechanism. The creeping flow solution formulated by Boussinesq attributes the Marangoni flow to the gradient of surfactant concentration which develops as a result of the motion of the bubble (or drop)*/the surfactant concentration outside the interface (in the continuous phase) is uniform. The situation here is greatly different */a surfactant gradient exists independently of the motion of the bubble and would for example still produce a concentration Marangoni force in the hypothetical situation

where the buoyancy force and thermal Marangoni force were in balance, and the bubble would thus otherwise be motionless */the experimental situation described by Young et al. for the thermal Marangoni effect. This is clearly different from the Boussinesq/Levich descriptions. We are not in a position to fully explain the apparent dichotomy theoretically, but it is possible to give definitive experimental answers. Kelsall et al. [15] have reported observations of ultra-clean electrolytic O2 bubbles which were indeed observed to rise in aqueous solutions with the predicted rate 50% greater than the Stokes’ rigid sphere solution. This experiment is of prime importance to the interpretation of our data, since O2 occupies the middle ground in terms of the size of the concentration Marangoni effect. It shows conclusively that the presence of a surfactant gas (O2) does not (in this case at least) rigidify the interface, which would of course restore the Stokes law, rigid sphere predicted rise rate. Interestingly, the presence of high concentrations of negatively adsorbed ions does not appear to influence the motion of the bubbles either. The second experimental fact demonstrated by the data of Massoudi and King [38] (see Table 3) is that O2 is significantly surface active at the water/O2 interface. Taken together, these observations show that the presence of a significant concentration of surfactant gas does not rigidify the bubble interface, and thus does

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not prevent transmission of tangential stress across the interface. As a speculation, we believe that the key to this apparent conundrum is that gases are very fast to adsorb and to desorb. This is in part due to their small size, and relative ease of diffusion in the liquid state, in part to their lack of need of specific orientation at the interface to produce the surfactant effect (and thus to a rather small loss of entropy on adsorption) and in part to the small heat of adsorption of gases at liquid interfaces (see for example Hauxwell and Ottewill [39] for toluene vapour adsorbed onto water. Note that toluene is somewhat soluble in water to the extent of about 6000 ppm, and that it is not necessarily a good model for the simple gases because of its aromaticity, molecular shape, molecular weight, and lower symmetry when compared with simple gases). For most of the gases listed in Table 3, the variation of surface pressure P /g0/g , is approximately linear with the applied gas pressure, at least at low pressures. This is equivalent to the assertion that the quadratic (and higher terms) are negligible. Also at low pressures, G2,1, the amount adsorbed is usually approximately proportional to the applied pressure of gas: G2;1 

dg  bp d log10 p2

For CO2 at low pressure (B/11 bar) the constancy of dg is demonstrated by the good fit of G2,1 with d log10 p2 the data by a straight line, as shown in Fig. 10. Increasing the pressure, therefore, results in no significant additional adsorption in this case. The experimental verifications of the thermal Marangoni effect depend upon measurements conducted in PDMS. Such effects have not been observed in any aqueous system. Because of its high surface energy, water is very susceptible to surface contamination in this way, while PDMS with its rather low surface free energy is not. Thus it is relatively easy to obtain a clean (i.e. surfactant free) PDMS surface, but much more difficult for aqueous systems. Since C12E5 (C-12 alcohol 5 mol ethoxylate) is known to adsorb at the PDMS/air interface, this material should be tested experimentally for its ability to block the thermal Marangoni effect in PDMS. There is one more interesting aspect of the PDMS system, which is worthy of comment. In the experiments of Morick and Wo¨rmann, and in those of Hardy, an oscillation in the motion of the air bubble was reported. The authors ascribe the oscillation to the changing solubility of air in PDMS with temperature. Since there is a temperature gradient imposed on the sample, the amount of air dissolved in the liquid phase varies with height. The authors point out that this implies that the bubble will tend to grow in regions where the amount of

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air dissolved in the PDMS is reduced (i.e. hot regions) and will tend to shrink in regions where the air in the PDMS is undersaturated (i.e. cold regions). There is a corollary to this explanation. In the undersaturated regions, the surface tension will be somewhat lower than predicted by the thermal effect alone, and in the supersaturated (hot) regions, it will be somewhat greater than predicted. The presence of the thermal gradient introduces a concentration gradient, which opposes the thermal gradient, lessening its effect. The size of this effect is not known, but notice that the concentration gradient is now operating over similarly large distances as the thermal effect, and will therefore, be correspondingly much smaller than the estimates in Table 1. This is another way in which to produce a long-lasting concentration gradient, although only in the presence of a temperature gradient, which makes it impossible to study the thermal and concentration effects independently.

5. Conclusions Concentration gradients of simple electrolytic gases in an aqueous environment can produce large Marangoni forces. Because of the relatively short range of the gradient, such Marangoni forces also have a short range, restricting their influence to the Nernst layer adjacent to the electrode. The details of the mode of action of the electrolytic gases as surfactants have not been fully elucidated, and in particular, the interfacial rheology and adsorption dynamics at the interface have not yet been measured. In the close vicinity of the electrode this force may play an important role, and is implicated in several unexpected bubble motions. Oscillatory motions, which have no obvious explanation in the absence of such a force, are readily accounted for by the concentration Marangoni effect. Perhaps more importantly, the same force will increase the departure diameter of bubbles, and this may be relevant to the efficiency of electrolytic cells. A particularly important case is the common situation where bubbles remain attached to clean (u :/ 0) electrodes. In the absence of a Marangoni force, this behaviour is paradoxical. In those cases where no electrostatic attraction operates, bubble attachment is solely due to surface tension, which approaches zero as the contact angle approaches zero. Since in general the solubility of gases is a function of temperature, and gases act as surfactants, the present analysis predicts that thermal Marangoni effects will be accompanied by concentration Marangoni effects. In high surface free energy liquids, both mechanisms will be suppressed by conventional surfactant adsorption. It is possible, but unproven, that at sufficiently high gas pressures, gaseous surfactants could successfully com-

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pete with conventional surfactants for adsorption sites at the interface, and once again, the Marangoni effect would operate. Electrolysis produces large numbers of bubbles, and these act to reduce the concentration of conventional surfactant adjacent to the electrode, so that during electrolytic bubble production, the Marangoni effect is revealed. It is an open question whether a similar cleansing effect of large numbers of nucleate bubbles as a result of boiling operates near heat exchanger surfaces. The mode of action of gases as surfactants is not revealed by the present analysis. The fact that solubility and the lowering of surface tension appear to be strongly correlated is significant. The solubility itself correlates (although not perfectly) for simple gases with the degree of hydration of the gaseous species [40]. It is clear that underlying the role of gases as surfactants are van der Waals forces, probably implicating quadrupole and higher n -pole interactions. In order to elucidate the deeper nature of the forces involved, further experiments with liquids other than water are planned. Recently, the suggestion has been made that a thermal Marangoni flow may be responsible for the specific radial coalescence effect [41]. The postulated thermal gradients may or may not be present in the systems reported, so whether it is more likely that the effect is caused by thermal or concentration gradients must remain an open question pending definitive experiments. Thermal Marangoni effects cannot be invoked to explain the oscillating rise path of a bubble adjacent to a vertical electrode */the putative thermal gradient is in the wrong plane. By Occam’s razor, therefore, it seems probable that the radial coalescence phenomenon is not thermally driven either. Finally, it was mentioned in Sections 3.1.3 and 3.1.4 that the surfactant properties of electrolytic gases had some important implications for the nucleation of bubbles. The experimental results generally show that gas bubble nucleation occurs in practice at much lower supersaturations than any current theory of nucleation predicts. The dominant step in the kinetics of bubble nucleation is actually the formation of a critical nucleus [16,42], which is the smallest bubble which is in unstable equilibrium with the surrounding supersaturated solution. The pressure in this critical nucleus is governed by the curvature, increasing in inverse proportion as the bubble gets smaller. The curvature is determined by the balance of the internal pressure in the bubble and the interfacial tension. The adsorption of the electrolytic gas causes a lowering of the surface tension, and this of course results in a very significant lowering of the barrier to nucleation, since the interfacial tension appears cubed in the exponent, as shown in equation {7 or 9}. The important point to note is that the reduction in interfacial tension is not just that due to the applied pressure of dissolved gas, substantial though

that can be. It is actually the sum of the hydrostatic pressure and the excess internal pressure in the critical nucleus, which is a much greater quantity, dramatically increasing the nucleation rate. This effect on its own could account for the poor agreement between the theory and published data for bubble nucleation at low supersaturations.

6. Symbols A b c ce cb ce/cb ce/cb C D, b, b Fj g J K P, p pH R T x z D Dr F (u) G2,1 P a g x d h k u s v c

area of spherical cap of height x first coefficient in expansion of surface tension versus pressure concentration of dissolved species concentration adjacent to electrode concentration in bulk saturation ratio (dimensionless) supersaturation (dimensional) pre exponential ‘constant’ constants of the bubble motion force as a result of the Marangoni effect, due to gradient in j acceleration due to gravity nucleation rate (m 3 s 1) constant of the bubble motion, /(2pR )1/2 applied hydrostatic pressure, pressure Henry’s law constant bubble radius temperature distance of centre of bubble from the concentration boundary vertical spatial coordinate difference in height of centre of gravity of larger bubble from that of combined bubble density difference between gas and liquid function of contact angle adsorbed amount surface pressure/g0/g saturation ratio; ce/cb /s/a/1 surface tension generalised function of g thickness of unstirred Nernst layer Viscosity inverse of the Debye length contact angle Supersaturation, /a/1 Period Potential

S. Lubetkin / Electrochimica Acta 48 (2002) 357 /375

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