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The dendritic electrocrystallization of cadmium from the acid sulphate solution I: Granular cadmium substrate
Surface and Coatings Technology, 34 (1988) 219
-
229
219
THE DENDRITIC ELECTROCRYSTALLIZATION OF CADMIUM FROM THE ACID SULPHATE SOLUTION I: GRANULAR CADMIUM SUBSTRATE K. I. POPOV Faculty of Technology and Metallurgy, University of Belgrade (Yugoslavia)
M. I. CEKEREVAC Metalservis, Institute for Chemical Power Sources, Belgrade (Yugoslavia)
L. N. NIKOLH~ Kru~ik, Valjevo (Yugoslavia) (Received November 6, 1986)
Summary The effects of initial electrode surface coarseness and deposition overpotential on the initiation of dendritic growth are discussed. It is shown that, at the same deposition overpotential, initiation of dendritic growth is possible according to two mechanisms depending on the initial electrode surface roughness and on the deposition overpotential.
1. Introduction There are two kinds of cadmium dendrites: flat ones at lower and needles at higher overpotentials of electrocrystallization [1]. At the same time it was shown that dendritic growth can produce amplification of nondendritic surface coarseness controlled by spherical [2] or linear diffusion [3]. It seems that the shape of dendrites depends on the conditions under which dendritic growth starts. The purpose of this paper is to discuss this possibility.
2. Experimental details The electrolyte used throughout this work was CdSO4 (0.1 mol dm3) in H 3). The experimental conditions were the same as (0.5 mol dm[4]. Cadmium was deposited on to stationary copper those2S04 described earlier wire electrodes, modified by cadmium deposition from the same solution for 0257-8972/88/$3.50
© Elsevier
Sequoia/Printed in The Netherlands
220
40 mm at an overpotential of 30 mV, by linearly increasing the overpotential from 0 to 100 mV during 15 s and by deposition for 2 mm at 15 mV and linearly increasing the overpotential from 15 to 100 mV during 15 s. Further deposition was performed at different overpotentials corresponding to the limiting diffusion current density plateau.
3. Results and discussion A cadmium deposit obtained by deposition for 2 mm at an overpotential of 15 mV and by linearly increasing the overpotential from 15 to 100 mV during 15 s is presented in Fig. 1(a). Deposition at 100 mV for 1 and 2 mm on the substrate shown in Fig. 1(a) results in the deposit shown in Figs. 1(b) and 1(c) respectively. It is seen that the spherical diffusion layers are formed around the large grains and further deposition in their vicinity is not possible. The spherical diffusion layer is not formed around
(a)
(b)
~
(c)
Fig. 1. (a) Cadmium deposit obtained by deposition at an overpotential of 15 mV for 2 mm and by linearly increasing the overpotential from 15 to 100 mV during 15 s. (b) Cadmium deposit obtained by deposition at 100 mV for 1 mm on the substrate shown in Fig. 1(a). (c) The same as in Fig. 1(b) but for 2 mm of deposition.
221
smaller grains after deposition at 100 mV showing that they grow under linear diffusion control in the diffusion layer of the macroelectrode. It can be seen from Figs. 1(b) and 1(c) that dendrites of different shapes grow from different points on the electrode surface. It seems that needle dendrites grow from the points where spherical diffusion governs predendritic growth and flat dendrites grow from the points where linear diffusion controls that process. This can mean that both the above-mentioned theories of dendritic growth initiation are valid, and that different mechanisms produce differently shaped dendrites. This also explains the fact that different dendrites can be obtained at the same deposition overpotential on different substrates [1 5]. Cadmium deposits obtained by linearly increasing the overpotential from 0 to 100 mV during 15 s and by further deposition at 100 mV for 1 and 2 mm are presented in Fig. 2. Flat dendrites are also obtained by deposition on the substrate shown in Fig. 2(a) at 75 mV for 10 mm, as can be seen from Fig. 3. The roots of dendrites from Figs. 2 and 3 are shown in Figs. 4 -
(a)
(b)
10 ,.,,~
(c) Fig. 2. (a) Cadmium deposit obtained by linearly increasing the overpotential from 0 to 100 mV during 15 s. (b) Cadmium deposit obtained by deposition at 100 mV for 1 mm on the substrate shown in Fig. 2(a). (c) The same as in Fig. 2(b) but for a deposition time of 2 mm.
222
(a)
(h)
(c) Fig. 3. Cadmium deposit obtained by deposition at 75 mV for 10 shown in Fig. 2(a).
(a)
mm
on the substrate
(b)
Fig. 4. The roots of the dendrites shown in Fig. 2(c).
and 5. The absence of diffusion layers around the roots of dendrites indicates linear diffusion control. There is also an absence of any new nuclei on the “boulders” surrounding the roots of dendrites, but two kinds of flat dendrites are present. Deposition at 30 mV for 40 mm produces the deposit shown in Fig. 6(a), characterized by large protrusions around which spherical
223
(a)
(h)
Fig. 5. The roots of the dendrites shown in Fig. 3.
(a)
(b)
(c) Fig. 6. (a) Cadmium deposit obtained by deposition at 30 mV for 40 mm. (b) The needles obtained by deposition at 100 mV for 1 mm on the substrate shown in Fig. 6(a). (c) The same as in Fig. 6(b), but for a deposition time of 2 mm.
diffusion control can be established. The appearance of needles in further deposition at 100 mV supports the above conclusion. It may be noted that new nuclei are formed on this kind of protrusion as well as on large boulders in deposition at 100 mV. This can be seen from Figs. 1 and 6. Deposition at
224
(a)
(h)
(c) Fig. 7. (a) The dendrite obtained by deposition at 75 mV for 20 mm on the substrate shown in Fig. 6(a). (b) The root of the dendrite shown in Fig. 7(a). (c) The detail of a dendrite precursor shown in Fig. 7(b).
75 mV on substrate from Fig. 6(a) produces flat dendrites as shown in Fig. 7. In this case, dendritic growth starts from protrusions such as that in Fig. 7(c) where new nucleation is not observed. Electrodeposition on substrates shown in Figs. 2(a) and 6(a) at 130 mV produces needle dendrites as can be seen from Fig. 8. Their growth in both cases is related to new nucleation as can be seen from Fig. 9. Obviously, under suitable conditions, all kinds of dendrites can be obtained at the same time, as illustrated in Fig. 10. According to Diggle et al. [2] dendritic growth is possible if conditions of spherical diffusion are established around the tip of the protrusion growing inside the diffusion layer of the macroelectrode. A dendrite can be defined as an electrode surface protrusion which grows under activation or mixed control, while deposition onto the macroelectrode is under dominant diffusion control. The general polarization equation is given by
1=
1
(1) ~iofl/iL
225
(a)
(I))
Fig. 8. Cadmium deposits obtained by deposition at 130 mV for 2 strate shown in Fig. 2(a); (b) on the substrate shown in Fig. 6(a).
(a)
mm:
(a) on the sub-
(h)
Fig. 9. The details from (a) Fig. 8(a), and (b) Fig. 8(b).
Fig. 10. Cadmium deposit obtained by deposition at 110 mV for 3 mm shown in Fig. 2(a).
for 11 ~
JL,tip
on the substrate
f2 and for a flat macroelectrode.
The limiting diffusion current to the tip of a protrusion is given by nFDC0h+r / h\ JLIl~H r ‘, rj
(2)
226
This is because the distribution of concentration inside the diffusion layer is [2] C0h
(3)
where 0 ~ h ~ 6 and the concentration at the outer limit of the spherical diffusion layer is h +r (4) Equation (2) is valid if the protrusion height does not affect the local thickness of the macroelectrode diffusion layer, i.e. if 6 h. At the same time, the condition 6 > r must be satisfied because a spherical diffusion layer obviously cannot be established for r> 6. It follows from eqns. (1) and (2) that the tip current density is given by ~‘
Jt~P =
1
iof1 ~Jof1/JL(l
+
h/r)
(5)
It is obvious from eqns. (1) and (5) that at the same overpotential, the current density to the flat part of the electrode, given by eqn. (1), can be close to the limiting value, while deposition onto the tip of a protrusion, given by eqn. (5), can be under activation control if the ratio h/r is sufficiently high. Hence the protrusion, characterized by the tip radius r and the height h, grows under diffusion control up to some value of h/r at which growth becomes mixed or activation controlled. If protrusions are sufficiently far from each other and their diffusion zones do not overlap, then spherical control takes place and the conditions for dendritic growth initiation can be established. Around the independently growing grain a diffusion layer with a thickness equal to the radius of the grain is formed (Fig. 1). This means that under steady state conditions the maximum current to the growing grain is equal to the linear limiting diffusion current to the surface of the electrode formed by the covered surface of the grain and the diffusion zone around it, and is equal to 2lrjL (6) =
4r
for a hemispherical grain. Assuming that the maximum current density to each point of the growing grain is given by eqn. (5), the total possible current to the grain in mixed control deposition is then given by I
2
________________
=
1 +jofl/JL(1
+
h/r)
2r~
(7)
assuming a hemispheroidal shape for the growing grains. The growth of dendrites is characterized by a current density in mixed control deposition larger than the limiting diffusion current density to the flat surface [6]. Hence, dendritic growth starts if
227
(8) or at f
jL~lI(I) +1} 30{(h/r)
1S>
—
because of eqns. (6), (7) and (8). Equation (9) is obviously valid if h> r. The previous consideration is valid for those protrusions around whose tips a spherical diffusion layer can be established. If protrusions are close to each other the formation of the spherical diffusion layers around them is not possible because of the overlapping of diffusion zones [7]. In that case deposition to the tips of the protrusions is controlled by linear diffusion from the outer limit of the macroelectrode diffusion layer. Hence, if 6 ~ h the maximum possible current density is JL, but the limiting diffusion current density to the tip, JL tip, is given by eqn. (2). Dendritic growth in this case can be expected if JL< lou (10) 1 ~lof1IlL{(h/r)
+ 1}
and c L
/1
>
JL(l + nh)
(11)
.
Jo Equation (11) can be rewritten in the form IL>
JL
r
(12)
—
j~ h
for 6
r h. This equation is similar to that defined earlier (r instead of 6) for dendritic growth under similar conditions [3] and at present it seems that the equation ~‘
~‘
JL 6 f1L>_ Jo h
(13)
—
is valid if 6
—~
228
(a)
(b)
Fig. 11. Initiation of growth of flat dendrites (substrates from Fig. 2(a)): (a) deposition overpotential, 100 mV; (b) deposition overpotential, 80 mV.
It is obvious that the formation of nuclei around which spherical diffusion layers can be formed prevents linear diffusion control to the initial grains. It is also obvious that the formation of new nuclei on the tips of the grains decreases the effective tip radius relative to the radius of the new nucleus, while the height remains mainly that of the initial grain. In that way the ratio h/r becomes very large and needle dendritic growth can start. Finally, we can conclude that dendritic growth initiation according to two mechanisms is possible, depending on the kind of diffusion control of the electrodeposition during predendritic growth of surface protrusions. The shape of the needle dendrites but not of the flat dendrites is in accordance with the theory presented. For flat dendrites the explanation of the initiation of growths, their shape and growth mechanism ought to be found in a correlation of the specific characteristics of metal lattices and the preferred orientation of the crystal planes with diffusion conditions during electrodeposition (see Fig. 11). This aspect of dendritic growth of metal deposits certainly requires further investigation and explanation.
References 1 R. Barnard, G. S. Edwards, J. Holloway and F. L. Tye, J. Appi. Electrochem., 13 (1983) 765 - 773. 2 J. W. Diggle, A. R. Despié and J. O’M. Bockris, J. Electrochem. Soc., 116 (1969) 1503. 3 K. I. Popov, M. D. Maksimovié, J. D. Trnjav~evand M. G. Pavlovié, J. App!. Electrochem., 11 (1981) 239. 4 K. I. Popov and N. V. Krstaji~,J. Appi. Elect rochem., 13 (1983) 775. 5 M. Froment and G. Maurin, Electrodeposition Surf. Treat., 3 (1975) 245. 6 A. R. Despié and K. I. Popov, Mod. Aspects Electrochem., 7 (1972) 199. 7 B. Scharifker and G. Hills, Electrochim. Acta, 28 (1983) 879.
229
Appendix A: Nomenclature C0 11 [2 [is
1L
F h
j Jo JL JL, tip
r
bulk concentration exp(t~/r~00) exp(—fl/770,0) exp(ii/ri0,0) for spherical diffusion control exp(77~/?70,0)for linear diffusion control Faraday constant height of protrusion current density exchange current density limiting current density limiting current density to the tip of protrusion radius of protrusion tip
Greek symbols 6 diffusion layer thickness overpotential critical overpotential for dendritic growth initiation ~ 710,a slope of of anodic cathodicTafel Tafel line slope line
-
229
219
THE DENDRITIC ELECTROCRYSTALLIZATION OF CADMIUM FROM THE ACID SULPHATE SOLUTION I: GRANULAR CADMIUM SUBSTRATE K. I. POPOV Faculty of Technology and Metallurgy, University of Belgrade (Yugoslavia)
M. I. CEKEREVAC Metalservis, Institute for Chemical Power Sources, Belgrade (Yugoslavia)
L. N. NIKOLH~ Kru~ik, Valjevo (Yugoslavia) (Received November 6, 1986)
Summary The effects of initial electrode surface coarseness and deposition overpotential on the initiation of dendritic growth are discussed. It is shown that, at the same deposition overpotential, initiation of dendritic growth is possible according to two mechanisms depending on the initial electrode surface roughness and on the deposition overpotential.
1. Introduction There are two kinds of cadmium dendrites: flat ones at lower and needles at higher overpotentials of electrocrystallization [1]. At the same time it was shown that dendritic growth can produce amplification of nondendritic surface coarseness controlled by spherical [2] or linear diffusion [3]. It seems that the shape of dendrites depends on the conditions under which dendritic growth starts. The purpose of this paper is to discuss this possibility.
2. Experimental details The electrolyte used throughout this work was CdSO4 (0.1 mol dm3) in H 3). The experimental conditions were the same as (0.5 mol dm[4]. Cadmium was deposited on to stationary copper those2S04 described earlier wire electrodes, modified by cadmium deposition from the same solution for 0257-8972/88/$3.50
© Elsevier
Sequoia/Printed in The Netherlands
220
40 mm at an overpotential of 30 mV, by linearly increasing the overpotential from 0 to 100 mV during 15 s and by deposition for 2 mm at 15 mV and linearly increasing the overpotential from 15 to 100 mV during 15 s. Further deposition was performed at different overpotentials corresponding to the limiting diffusion current density plateau.
3. Results and discussion A cadmium deposit obtained by deposition for 2 mm at an overpotential of 15 mV and by linearly increasing the overpotential from 15 to 100 mV during 15 s is presented in Fig. 1(a). Deposition at 100 mV for 1 and 2 mm on the substrate shown in Fig. 1(a) results in the deposit shown in Figs. 1(b) and 1(c) respectively. It is seen that the spherical diffusion layers are formed around the large grains and further deposition in their vicinity is not possible. The spherical diffusion layer is not formed around
(a)
(b)
~
(c)
Fig. 1. (a) Cadmium deposit obtained by deposition at an overpotential of 15 mV for 2 mm and by linearly increasing the overpotential from 15 to 100 mV during 15 s. (b) Cadmium deposit obtained by deposition at 100 mV for 1 mm on the substrate shown in Fig. 1(a). (c) The same as in Fig. 1(b) but for 2 mm of deposition.
221
smaller grains after deposition at 100 mV showing that they grow under linear diffusion control in the diffusion layer of the macroelectrode. It can be seen from Figs. 1(b) and 1(c) that dendrites of different shapes grow from different points on the electrode surface. It seems that needle dendrites grow from the points where spherical diffusion governs predendritic growth and flat dendrites grow from the points where linear diffusion controls that process. This can mean that both the above-mentioned theories of dendritic growth initiation are valid, and that different mechanisms produce differently shaped dendrites. This also explains the fact that different dendrites can be obtained at the same deposition overpotential on different substrates [1 5]. Cadmium deposits obtained by linearly increasing the overpotential from 0 to 100 mV during 15 s and by further deposition at 100 mV for 1 and 2 mm are presented in Fig. 2. Flat dendrites are also obtained by deposition on the substrate shown in Fig. 2(a) at 75 mV for 10 mm, as can be seen from Fig. 3. The roots of dendrites from Figs. 2 and 3 are shown in Figs. 4 -
(a)
(b)
10 ,.,,~
(c) Fig. 2. (a) Cadmium deposit obtained by linearly increasing the overpotential from 0 to 100 mV during 15 s. (b) Cadmium deposit obtained by deposition at 100 mV for 1 mm on the substrate shown in Fig. 2(a). (c) The same as in Fig. 2(b) but for a deposition time of 2 mm.
222
(a)
(h)
(c) Fig. 3. Cadmium deposit obtained by deposition at 75 mV for 10 shown in Fig. 2(a).
(a)
mm
on the substrate
(b)
Fig. 4. The roots of the dendrites shown in Fig. 2(c).
and 5. The absence of diffusion layers around the roots of dendrites indicates linear diffusion control. There is also an absence of any new nuclei on the “boulders” surrounding the roots of dendrites, but two kinds of flat dendrites are present. Deposition at 30 mV for 40 mm produces the deposit shown in Fig. 6(a), characterized by large protrusions around which spherical
223
(a)
(h)
Fig. 5. The roots of the dendrites shown in Fig. 3.
(a)
(b)
(c) Fig. 6. (a) Cadmium deposit obtained by deposition at 30 mV for 40 mm. (b) The needles obtained by deposition at 100 mV for 1 mm on the substrate shown in Fig. 6(a). (c) The same as in Fig. 6(b), but for a deposition time of 2 mm.
diffusion control can be established. The appearance of needles in further deposition at 100 mV supports the above conclusion. It may be noted that new nuclei are formed on this kind of protrusion as well as on large boulders in deposition at 100 mV. This can be seen from Figs. 1 and 6. Deposition at
224
(a)
(h)
(c) Fig. 7. (a) The dendrite obtained by deposition at 75 mV for 20 mm on the substrate shown in Fig. 6(a). (b) The root of the dendrite shown in Fig. 7(a). (c) The detail of a dendrite precursor shown in Fig. 7(b).
75 mV on substrate from Fig. 6(a) produces flat dendrites as shown in Fig. 7. In this case, dendritic growth starts from protrusions such as that in Fig. 7(c) where new nucleation is not observed. Electrodeposition on substrates shown in Figs. 2(a) and 6(a) at 130 mV produces needle dendrites as can be seen from Fig. 8. Their growth in both cases is related to new nucleation as can be seen from Fig. 9. Obviously, under suitable conditions, all kinds of dendrites can be obtained at the same time, as illustrated in Fig. 10. According to Diggle et al. [2] dendritic growth is possible if conditions of spherical diffusion are established around the tip of the protrusion growing inside the diffusion layer of the macroelectrode. A dendrite can be defined as an electrode surface protrusion which grows under activation or mixed control, while deposition onto the macroelectrode is under dominant diffusion control. The general polarization equation is given by
1=
1
(1) ~iofl/iL
225
(a)
(I))
Fig. 8. Cadmium deposits obtained by deposition at 130 mV for 2 strate shown in Fig. 2(a); (b) on the substrate shown in Fig. 6(a).
(a)
mm:
(a) on the sub-
(h)
Fig. 9. The details from (a) Fig. 8(a), and (b) Fig. 8(b).
Fig. 10. Cadmium deposit obtained by deposition at 110 mV for 3 mm shown in Fig. 2(a).
for 11 ~
JL,tip
on the substrate
f2 and for a flat macroelectrode.
The limiting diffusion current to the tip of a protrusion is given by nFDC0h+r / h\ JLIl~H r ‘, rj
(2)
226
This is because the distribution of concentration inside the diffusion layer is [2] C0h
(3)
where 0 ~ h ~ 6 and the concentration at the outer limit of the spherical diffusion layer is h +r (4) Equation (2) is valid if the protrusion height does not affect the local thickness of the macroelectrode diffusion layer, i.e. if 6 h. At the same time, the condition 6 > r must be satisfied because a spherical diffusion layer obviously cannot be established for r> 6. It follows from eqns. (1) and (2) that the tip current density is given by ~‘
Jt~P =
1
iof1 ~Jof1/JL(l
+
h/r)
(5)
It is obvious from eqns. (1) and (5) that at the same overpotential, the current density to the flat part of the electrode, given by eqn. (1), can be close to the limiting value, while deposition onto the tip of a protrusion, given by eqn. (5), can be under activation control if the ratio h/r is sufficiently high. Hence the protrusion, characterized by the tip radius r and the height h, grows under diffusion control up to some value of h/r at which growth becomes mixed or activation controlled. If protrusions are sufficiently far from each other and their diffusion zones do not overlap, then spherical control takes place and the conditions for dendritic growth initiation can be established. Around the independently growing grain a diffusion layer with a thickness equal to the radius of the grain is formed (Fig. 1). This means that under steady state conditions the maximum current to the growing grain is equal to the linear limiting diffusion current to the surface of the electrode formed by the covered surface of the grain and the diffusion zone around it, and is equal to 2lrjL (6) =
4r
for a hemispherical grain. Assuming that the maximum current density to each point of the growing grain is given by eqn. (5), the total possible current to the grain in mixed control deposition is then given by I
2
________________
=
1 +jofl/JL(1
+
h/r)
2r~
(7)
assuming a hemispheroidal shape for the growing grains. The growth of dendrites is characterized by a current density in mixed control deposition larger than the limiting diffusion current density to the flat surface [6]. Hence, dendritic growth starts if
227
(8) or at f
jL~lI(I) +1} 30{(h/r)
1S>
—
because of eqns. (6), (7) and (8). Equation (9) is obviously valid if h> r. The previous consideration is valid for those protrusions around whose tips a spherical diffusion layer can be established. If protrusions are close to each other the formation of the spherical diffusion layers around them is not possible because of the overlapping of diffusion zones [7]. In that case deposition to the tips of the protrusions is controlled by linear diffusion from the outer limit of the macroelectrode diffusion layer. Hence, if 6 ~ h the maximum possible current density is JL, but the limiting diffusion current density to the tip, JL tip, is given by eqn. (2). Dendritic growth in this case can be expected if JL< lou (10) 1 ~lof1IlL{(h/r)
+ 1}
and c L
/1
>
JL(l + nh)
(11)
.
Jo Equation (11) can be rewritten in the form IL>
JL
r
(12)
—
j~ h
for 6
r h. This equation is similar to that defined earlier (r instead of 6) for dendritic growth under similar conditions [3] and at present it seems that the equation ~‘
~‘
JL 6 f1L>_ Jo h
(13)
—
is valid if 6
—~
228
(a)
(b)
Fig. 11. Initiation of growth of flat dendrites (substrates from Fig. 2(a)): (a) deposition overpotential, 100 mV; (b) deposition overpotential, 80 mV.
It is obvious that the formation of nuclei around which spherical diffusion layers can be formed prevents linear diffusion control to the initial grains. It is also obvious that the formation of new nuclei on the tips of the grains decreases the effective tip radius relative to the radius of the new nucleus, while the height remains mainly that of the initial grain. In that way the ratio h/r becomes very large and needle dendritic growth can start. Finally, we can conclude that dendritic growth initiation according to two mechanisms is possible, depending on the kind of diffusion control of the electrodeposition during predendritic growth of surface protrusions. The shape of the needle dendrites but not of the flat dendrites is in accordance with the theory presented. For flat dendrites the explanation of the initiation of growths, their shape and growth mechanism ought to be found in a correlation of the specific characteristics of metal lattices and the preferred orientation of the crystal planes with diffusion conditions during electrodeposition (see Fig. 11). This aspect of dendritic growth of metal deposits certainly requires further investigation and explanation.
References 1 R. Barnard, G. S. Edwards, J. Holloway and F. L. Tye, J. Appi. Electrochem., 13 (1983) 765 - 773. 2 J. W. Diggle, A. R. Despié and J. O’M. Bockris, J. Electrochem. Soc., 116 (1969) 1503. 3 K. I. Popov, M. D. Maksimovié, J. D. Trnjav~evand M. G. Pavlovié, J. App!. Electrochem., 11 (1981) 239. 4 K. I. Popov and N. V. Krstaji~,J. Appi. Elect rochem., 13 (1983) 775. 5 M. Froment and G. Maurin, Electrodeposition Surf. Treat., 3 (1975) 245. 6 A. R. Despié and K. I. Popov, Mod. Aspects Electrochem., 7 (1972) 199. 7 B. Scharifker and G. Hills, Electrochim. Acta, 28 (1983) 879.
229
Appendix A: Nomenclature C0 11 [2 [is
1L
F h
j Jo JL JL, tip
r
bulk concentration exp(t~/r~00) exp(—fl/770,0) exp(ii/ri0,0) for spherical diffusion control exp(77~/?70,0)for linear diffusion control Faraday constant height of protrusion current density exchange current density limiting current density limiting current density to the tip of protrusion radius of protrusion tip
Greek symbols 6 diffusion layer thickness overpotential critical overpotential for dendritic growth initiation ~ 710,a slope of of anodic cathodicTafel Tafel line slope line