Surface Technology, 22 (1984) 165 - 174 165

DETERMINATION OF THE NERNST DIFFUSION LAYER THICKNESS IN THE HYDROSON AGITATION TANK

ROBERT WALKER and NICHOLAS S. HOLT

Department of Metallurgy and Materials Technology, University of Surrey, Guildford GU2 5XH (Gt. Britain)

(Received October 18, 1983)

Summary

The thickness of the Nernst diffusion layer in the Hydroson tank is cal- culated from the limiting current density for zinc electrodeposit ion from the sodium zincate bath. The value is determined using a modified form of the equation for a rotating cylindrical cathode and also from the Vielstich equa- tion for turbulent f low across a plane surface. Another estimation of the thickness is calculated from the change in the pH measured at a ca thode - electrolyte interface which involves the precipitation of magnesium hydrox- ide detected by electron spectroscopy for chemical analysis. From this work the thickness of the diffusion layer in the Hydroson tank at the focal point between two jets is considered to be 13.5 #m at the corresponding f low rate of 1.16 m s -1.

1. Introduct ion

The Hydroson system, which involves the use of a controlled multistage centrifugal pump to circulate solution together with generators which convert velocity to acoustic energy, has been described in earlier papers [ 1 - 5] and it is now widely used in commercial cleaning operations. It has been employed in the electrodeposit ion of zinc from sodium zincate solutions [ 1, 2]. The f low rate at the focal point between two jets in the tank has been measured as 1.16 m s -I and details o f this work have been given elsewhere [1].

The electrolytic diffusion layer is a thin layer of solution adjacent to an immersed surface through which transport of species occurs by diffusion to, or from, the surface. It consists o f electrons on the metal surface, a film of adsorbed ions and a diffuse double layer of an ionic atmosphere in which the ions of one sign are in excess o f their normal concentration. This outer layer may also contain solvated cations. The diffusion layer can also be divided into regions called the inner Helmholtz plane and the outer Helmholtz plane.

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The diffusion layer at a metal-electrolyte interface is of particular significance because it has considerable influence on the rates of the elec- trode reactions. The thickness of the layer is important because it has a major effect on the rate of a reaction and as it becomes thicker the rate of diffusion decreases so that the reaction becomes slower. Agitation of the solution is generally beneficial as it reduces the thickness. Because convec- tion generally gives laminar flow, the thickness of the layer is governed by the geometry of the surface, the coefficient of viscosity and density of the electrolyte, the diffusion coefficient and the velocity of the liquid.

The rate of an electrochemical reaction is given by the current density. The maximum or limiting current density of deposition of metal ions is very dependent on the thickness of the diffusion layer. This is illustrated by using a disc electrode which can be rotated during deposition. Bockris and Reddy [6] have shown that, for a process involving iodine, the limiting current density was 2.89 × 10 -s A dm -2 and the thickness of the diffusion layer was estimated as 500 #m for a stationary disc electrode but when the disc was rotated at 50 rev min -1 the values became 13.41 × 10 -s A dm -2 and 110 #m respectively and at the higher speed of 240 rev min -1 the values were 29.21 X 10 -s A dm -2 and 50 #m. Hence agitation has a marked influence on both the limiting current density and the diffusion layer thickness.

2. Calculation of the diffusion layer thickness from the limiting current d e n s i t y : t h e o r y

The thickness of the Nernst diffusion layer can be calculated from the limiting current density. This can be considered to be the value of the current density at which a change in polarization produces little or no change in the current density. Fick's first law of diffusion can be written [7] in the form

1 dC - J D = - - D - -

nF dx

where j is the current density, n is the number of electrons involved in the electrode process, F is Faraday's constant, JD is the diffusion flux, D is the diffusion coefficient and dC/dx is the concentration gradient at the inter- face.

The concentration profile can be considered to be linear within the small distance from the metal-electrolyte interface and then it asymptoti- cally approaches the bulk value. The linear section can be extrapolated until it intersects the bulk concentration at a distance ~ which is the thickness of the double layer (Fig. 1). Hence, across the double layer, the concentration gradient dC/dx is equal to the change Co --Cx in concentration from the bulk solution to the interface over the distance 8:

dC Co -- Cx

dx

T z 0 I - ~C ~c I - z Lu

z 0 t~

Cx,o

L I N E A R I Z E D C-X CURVE

i o ~--'~ ~ DISTANCE "~

Fig. 1. Concentration vs. distance from the electrode interface.

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Therefore

j Co -- Cx - J D = --D - -

nF

When the rate of a reaction as measured by the current density reaches the limiting value J l ~ , the concentrat ion Cx of ions at the interface is zero, C~ = O, and the relationship becomes

J~m DC0

nF 6

or

nFDCo 8 =

Jr, m

Therefore the thickness of the Nernst diffusion layer can be estimated if the value of the limiting current density is known.

The limiting current density for the electrodeposit ion of zinc from the sodium zincate bath has been determined in an earlier paper [1, 8]. The values are given in Table 1 together with the calculated thickness of the Nemst diffusion layer.

TABLE 1

Solution Limiting current density Diffusion layer thickness (A dm -2) (pm)

Still 2.5 97 Magnetically stirred 8.3 29 Hydroson agitation 18.2 13.5

The value of the thickness of the diffusion layer depends on many factors such as the composition of the electrolyte, the temperature, the current density, the viscosity, the diffusion coefficient and the electrode

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geometry. The value obtained in this work for the still solution is of the same magnitude as given in the literature, i.e. 50 #m [9, 10], 200 #m [11] and 250 gm [12].

3. Estimation of the diffusion layer thickness for the cathodic precipitation of magnesium hydroxide

The Eisenberg [13] equation for a rotating cylindrical electrode can be written as

jL.n = O.0791nFCoUO.' ( R )-°'3 ( V) -°'~4

where Jnm is the limiting current density, n is the number of electrons, F is Faraday's constant, Co is the bulk concentration, U (= WR) is the peripheral velocity, R is the cathode radius, W is the angular velocity, V is the kinematic viscosity and D is the diffusion coefficient.

Fick's first law of diffusion, as discussed above, can be written as

nFDCo ]rm~ = 8 Hence the Eisenberg equation can be rewritten as

nFDC o Jr~

_ nFDCo u_o.,(RI°'3(V) °'6'~ O.0791nfCo \V] \-D

= 12.64U-O.TRO.SDO.3S6vO.3'~ The hydrodynamic conditions in the Hydroson tank with the cathodes supported in the same position are the same for the electrodeposition of zinc from the zincate electrolyte and for the precipitation o f magnesium hydrox- ide from 0.01 M magnesium chloride solution. Therefore on the assumption tha t the Hydroson action approximates to the rotating cylindrical electrode the equation becomes

5 = constant × D°'3S6V °'3~

The value of the constant can be calculated as 2.432 X 106 using the values of 8 = 13.5 #m from the electrodeposition work, D = 6.8 X 10 - l° m 2 s -1 for the diffusion coefficient of zincate ions and V = 1.62 X 10 -6 m 2 s -1. This value of the constant can be used to calculate a value of 5 for the precipita- t ion of magnesium hydroxide for which the diffusion of hydroxyl ions is considered to be the rate~ontroll ing step. The diffusion coefficient for hydroxyl ions is 5.2 × 10 -9 m 2 s "1 and the kinematic viscosity is assumed to

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be 1.002 × 10 -6 m 2 s - l , the value for distilled water, because the concentra- t ion of magnesium chloride is very low (0.01 M). Using these values

5 = 2.432 × 106 × (5.20 X 10-9) °'as6 × 1.002 × 10-6) ° 'a~

= 23.5 #m

The Vielstich equation [14] for turbulent flow across a plane surface gives a different equation for the diffusion layer thickness:

~. LO.lv-O.9vO.S66DO.33a

where L is the distance from the edge along the direction of flow on the sur- face and V. is the flow velocity of the solution parallel to the surface in the direction of the coordinate L at infinite distance. No proportionality constant is given for turbulent flow so to calculate 8 the equation can be treated as the Eisenberg equation:

5 = constant × V°'S66D0"333

Using the values of V and D given above the constant is calculated as 2.888 × 106.

Hence for the magnesium hydroxide precipitation reaction, the Viel- stich equation becomes

= 2.888 X 10 6 X (1.002 X i0-6) 0"s66 X (5.20 X I0-9) 0"3a3

= 20 #m

Thus the value of the double layer at the cathode surface in a 0.01 M solu- t ion of magnesium chloride is estimated to be of the order of 20 - 23.5 #m.

4. Determination of thickness by pH change at a cathodic surface

4.1. Introduction The cathodic reduction of oxygen to produce hydroxyl ions at the

cathode surface gives an increase in the pH at the interface relative to that in the bulk solution according to the equation

02 + 2H20 + 4e > 4OH-

The extent of this change depends on the current density and the diffusion layer thickness. In a well-aerated solution, with an adequate supply o f dis- solved oxygen, the measurement of the pH shift at a known current density enables the thickness to be determined.

The relationship has been investigated by Castle and Tanner-Tremaine [15] who studied the precipitation of magnesium hydroxide from a dilute solution of magnesium chloride. This method was considered to be suitable for investigating the diffusion layer thickness in the Hydroson system. Harris [16] has listed the requirements for a mathematical interpretation of this type of experiment as follows: (a) a measured hydrogen evolution current; (b) detection of the onset of precipitation; (c) a hydrodynamical ly

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controlled exposure. Condition (a) was satisfied by using copper electrodes polarized to a potential below zero on the hydrogen scale which ensured that the only current carriers in the steady state were hydroxyl ions and hydrogen ions. Condition (b) allowed the assumption to be made that the net flux of magnesium ions to the surface and the loss of hydroxyl ions by precipitation of magnesium hydroxide are negligible. This was determined by electron spectroscopy for chemical analysis (ESCA) which detected the presence of magnesium ions on the surface after a current had been applied to the cathode. Condition (c) was satisfied when the cathodes were subjected to hydrosonic agitation and in the work of Castle and Tanner-Tremaine by magnetic stirring at a flow rate of 5 cm s -1 . In our earlier work the flow rate of liquid at the focal point between two jets in the Hydroson tank was measured as about 1.16 m s -1 .

4.2. Hydrodynamic model In view of conditions (a) and (b) the cathodic current density for the

oxygen reduction reaction can be considered to be the sum of the ionic current of the hydroxyl ions moving away from the electrode surface and the ionic current of the hydrogen ions moving towards the electrode surface, i.e. i

__ = F ( J o H - + J H + ) A

where i is the cathodic current, A is the area of the cathode, JOH- is the ionic flux of OH-, JH ÷ is the ionic flux of H + and F is Faraday's constant. Under steady state conditions, Fick's law can be applied for the diffusion of ions through the diffusion layer:

J dC nF - Jn = --D ~ix

where D is the diffusion coefficient and dC/dx is the concentration gradient. Combining these equations gives

i lDoH-(dC/dx)orI- A

Therefore

i = _ _ FA 8

{DoH- ( [ O H - ] ~ a --

+ Da+(dC/dX)H+8 1

[OH-l~ak) + Dri+([H+]bun~ - - [ H + ] s u ~ f ) }

At the critical current ip for precipitation, the concentration of hydroxyl ions at the surface is equal to the limiting value at which magnesium hydroxide is precipitated from the solution as given by the solubility product Ks, where

K s = [Mg 2+ ] [OH-] 2

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Substituting for [OH-] b~k using the ionic product Kw of water, where

Kw = [H +] [OH-]

the value of ip can be obtained from

Kw{[Mg2+ l ),,2

In this equation [H ÷] and [Mg 2÷] represent the bulk solution values and [Mg 2÷] is assumed to remain constant up to the metal surface. The value of ip was derived from this equation and plot ted as a function of pH using the following values: [Mg 2÷] = 0.01 tool l - l ; DH÷ = 0.93 X 10 -s m 2 s- l ; DOll- = 0.52 X 10 -s m a s- l ; KsMg(OH) 2 = 1.2 × 10-11; Kw = 1 0 -14 . These plots are shown in Fig. 2.

-2

T -3

? 6

( / ) z

i - z IJJ

ri- D 0 0-6 0 .-I

f

f

f

f

30/.,,.

Fig. 2. Current dens i ty at t he c a thode as a func t ion of pH.

The accuracy of the value of the thickness of the double layer was then tested experimentally by subjecting the copper cathodes in the hydrosoni- cally agitated magnesium chloride solution to conditions of current density and pH given by points A, B, C and D in Fig. 2. Points A and C lie on the theoretical curve given for a diffusion layer thickness of 10 #m and B and D of 30 #m. If precipitation were detected at points A and C but not at B and D, this would indicate a thickness value of between 10 and 30 #m.

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4. 3. Experimental details and results Flat elliptical copper specimens, which could be mounted directly into

the spectrometer, were supported on a brass rod and lacquered to give a cathode area of 1.4 cm 2. They were cleaned and held in the same position in the tank as used for the electrodeposition of zinc. The anode was platinum which was also thoroughly cleaned. The solution was made with distilled water and hydrochloric acid with 2.31 g of AnalaR grade magnesium chloride per litre (0.01 M) to give the desired pH. The tank contained 80 1 of solution so that any change in the bulk pH due to the formation of hydroxyl ions was negligible.

Electrolysis was carried out for 3 h at a bath temperature o f 20 + 1 °C so that equilibrium conditions were reached. Castle and Tanner-Tremaine calculated that an equilibrium at a pH of 6 and a current density of 10 -s A cm -2 would take 4.7 s to establish the level of pH 9.5 required for precipita- tion.

An ESCA 3 was used with A1 K~ radiation to detect the presence and thickness of magnesium hydroxide on the cathode surface by studying the ratios of the magnesium-to-copper signals. The intensity of the 2p3/2 line from the copper substrate is reduced by the presence of a layer of magne- sium hydroxide of thickness d by the relationship

Ic~ 0 X sin 0

where lcu0 is the signal from a clean copper surface,/Cud is the signal from a coated copper surface, k is the characteristic length related to the inelastic mean free path of the 2p3/2 electron in copper and 0 is the electron collec- t ion angle. A similar relationship exists for the increase in intensity of the KLL Auger line from the magnesium ion which increases with thickness:

IMg a _ 1 - - exp X 1 sin 0 /Mg ~o

where k ~ is the characteristic length related to the inelastic mean free path of the Mg KLL electron in magnesium hydroxide, IMg. is the signal from a layer of magnesium hydroxide which is thick compared with k I and IM~d is the signal from a coated surface. Combining these equations gives

IMgd _ I M ~ 1 - - e x p Xl exp I Cu---~ I ' -~ o s i n 0

A calibration curve for the right-hand side of this equation against the thick- ness d was plotted (Fig. 3) using the following values: IMglI~ 0 = 0.8 for copper meta l ; X 1 = 1.4 nm at a kinetic energy of 1180 eV; X = 0.8 nm at a kinetic energy of 550 eV; 0 = 45 °. By dividing the experimentally deter- mined ratio IMg d/Ic~ d by the ratio IMg ../Icu 0 a value for the right-hand side

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t 11

,, 8 X '" 7

~ 6

7 4 " 3 X w 2

0,2 0,6 1,0 1,4 THICKNESS d (nm)

Fig. 3. Cal ibra t ion curve for d e t e r m i n i n g t he t h i cknes s o f m a g n e s i u m h y d r o x i d e .

of the equation could be calculated. The thickness of the magnesium hydroxide was obtained from the calibration curve.

The thickness of the precipitate under the different conditions is given in Table 2.

T A B L E 2

Sample Current density pH Precipitate thickness ( A c m -2 ) ( n m )

A 1.86 x 10 -3 3 .75 1.36 B 1 . 7 4 × 10 -3 3 .25 0 C 7 . 1 0 × 10 - 4 4.25 0 .96 D 6 .30 × 10 -4 3 .75 0 .15

4.4. Discussion Because precipitation occurred at points A and C but not at B and D

the thickness of the diffusion layer must be in the range 10 - 30 ~m. A more accurate value of thickness could be obtained by the ESCA method by repeating the experiment at values nearer to the critical conditions, i.e. be- tween values A and B and C and D. The thickness is of the order of 10 - 30 pro, which is in good agreement with the estimated value of 20 - 23.5 ~m calculated from the zinc deposition data. These results give confidence in the value of 13.5 #m obtained f rom the limiting current density measurements for alkali zinc deposition under conditions of Hydroson agitation.

174

5. Conclusion

In the Hydroson tank the thickness of the Nernst diffusion layer was 13.5 ~um and the f low rate was 1.16 m s - l at an immersed surface posit ioned at the focal point between two jets. This value was confirmed by the ESCA method.

References

1 R. Walker and N. S. Holt, Surf. Technol., 17 (1982) 147. 2 R. Walker and N. S. Holt, Plat. Surf. Finish., 67 (5) (1980) 92. 3 R. Walker and N. S. Holt, Prod. Finish. (London), 33 (4) (1980) 14. 4 Nickerson Ultrason Ltd., Br. Patent 1,475,307, 1973. 5 I. A. Thomson, Prod. Finish. (London), 32 (2) (1979) 13. 6 J. O'M. Bockris and A. K. N. Reddy, Modern Electrochemistry, Vol. 2, MacDonald,

London, 1970, p. 1058. 7 J. O'M. Bockris and A. K. N. Reddy, Modern Electrochemistry, Vol. 2, MacDonald,

London, 1970, p. 1056. 8 N. S. Holt, Ph.D. Thesis, Surrey University, Guildford, 1981, p. 94. 9 R. E. Wilson and M. A. Youtz , Ind . Eng. Chem., 15 (1923) 603.

10 S. Glasstone, Trans. Electrochem. Soc., 59 (1931) 277. 11 E. H. Lyons, in E. A. Lowenheim (ed.), Modern Electroplating, Wiley, New York,

1974, p. 20. 12 H. A. Laitenen and J. M. Kolthoff, J. Phys. Chem., 45 (1941) 1061. 13 M. Eisenberg, C. W. Tobias and C. R. Wilke, J. Electrochem. Soc., 101 (1954) 306;

Inst. Chem. Eng. Syrup. Set. 16, 51 (1954) 1. 14 W. Vielstich, Z. Eiektrochem., 57 (1953) 646. 15 J. E. Castle and R. Tanner-Tremaine, Surf. Interface Anal., 1 (2) (1979) 49. 16 L. B. Harris, J. Electrochem. Soc., 120 (1973) 1034.