Abstract
The application of a periodically changing current in metal electrodeposition practice leads to improvements in the quality of electrodeposits. Three types of current variation have been found useful: reversing current (RC), pulsating current (PC), and sinusoidal, alternating current superimposed on a direct current (AC) [1–12]. The schematic presentation of the different current regimes of electrolysis is shown in Fig. 4.1. Also, the beneficial effects of pulsating overpotential (PO) have also been discussed [3]. Even though this kind of electrodeposition at a periodically changing rate (EPCR) is important from a theoretical point of view and offers a variety of experimental possibilities, it is as yet not frequently used in metal electrodeposition practice.
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Keywords
- Pulsate currentPulsating Current
- Electron Transfer Process
- Reverse Current
- Current Density Distribution
- Cathodic Current Density
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
4.1 Introduction
The application of a periodically changing current in metal electrodeposition practice leads to improvements in the quality of electrodeposits. Three types of current variation have been found useful: reversing current (RC), pulsating current (PC), and sinusoidal, alternating current superimposed on a direct current (AC) [1–12]. The schematic presentation of the different current regimes of electrolysis is shown in Fig. 4.1. Also, the beneficial effects of pulsating overpotential (PO) have also been discussed [3]. Even though this kind of electrodeposition at a periodically changing rate (EPCR) is important from a theoretical point of view and offers a variety of experimental possibilities, it is as yet not frequently used in metal electrodeposition practice.
4.1.1 Reversing Current
Reversing current (RC) is represented schematically in Fig. 4.1a. It is characterized by the cathodic current density, i c, and the anodic current density, i a, as well as by the duration of flow of the current in the cathodic and the anodic direction, t c and t a, respectively. Naturally,
where T p is the full period of the RC wave.
The average current density is then given by:
and for \( {i}_{\mathrm{c}}={i}_{\mathrm{a}}={i}_{\mathrm{A}} \)
where
RC is used in the second and millisecond range [7, 12], and cathodic current density is taken as positive.
4.1.2 Pulsating Current
Pulsating current (PC) consists of a periodic repetition of square pulses. It is similar in shape to RC except for the absence of the anodic component, as is shown in Fig. 4.1b. The PC is characterized by the amplitude of the cathodic current density, i c, the cathodic deposition time, t c (on period), and the time interval t p, in which the system relaxes (off period).
The full period, T p, is given by:
and the average current density by:
or
where
It should be noted that rectified sinusoidal AC, especially half-rectified sinusoidal AC, often termed pulsating current in the literature, shows similar effects to those of the PC [7].
4.1.3 Alternating Current Superimposed on Direct Current
Sinusoidal AC superimposed on a direct cathodic current (DC) is represented in Fig. 4.1c. It is characterized by i dc, i p, and the frequency, which is usually 50 or 60 Hz. The resultant is termed an asymmetric sinusoidal current. The average current is equal to i dc.
At a given DC value, three different types of current can be obtained, which can be denoted as follows: i p < i dc “rippling current”; i p = i dc, “pulsating current ”; and i p > i dc, “current with an anodic component.” The last type is mainly used in plating practice.
4.1.4 Pulsating Overpotential
Pulsating overpotential consists of a periodic repetition of overpotential pulses of different shapes. Square-wave PO is defined in the same way as PC except that the overpotential pulsates between the amplitude value η A and zero instead of current density. Non-rectangular pulsating overpotential is defined by the amplitude of the overpotential, η A , frequency, and overpotential waveform [7].
4.1.5 Reversing Overpotential
Reversing overpotential is defined in the same way as RC except that overpotential pulsates between the cathodic amplitude value and anodic amplitude value.
Also, there are numerous other different current and overpotential waveforms used in EPCR [13, 14], but the most important have been mentioned above.
4.2 Surface Concentration of Depositing Ions in the Periodic Conditions
4.2.1 Electrodeposition with Periodically Changing Range in the Millisecond Range
Electrodeposition with a periodically changing rate can be described in terms of time-and distance-dependent concentrations:
where n, F, D, C 0, and δ have already mentioned meanings, while x is the coordinate in a horizontal direction and C is concentration in a time t.
Equations (4.9), (4.10), (4.11), and (4.12) are solved for different i(t) shapes and the solutions applied to different types of problems [7].
The current density i(t) is the periodic function of a time, which for periodic reverse currents is given by [15]:
for pulsating currents by [16]:
and for AC superimposed on DC by [17]:
In the case of pulsating overpotential , i(t) is given by [3]:
The surface concentration under periodic conditions can be evaluated as follows. For i(t) given by Eq. (4.13), the solution of Eqs. (4.9), (4.10), (4.11), and (4.12) for x = 0, t = [m + 1/(r RC + 1)] T p, and m → ∞, i.e., at the end of the cathodic pulses , under the periodic conditions is given by [15]:
where \( {\lambda}_{\mathrm{k}}=\frac{{\left(2k+1\right)}^2{\pi}^2D}{4{\delta}^2} \) and k = 0,1,2, .....
The surface concentration, C a, at the end of the anodic pulses under the same conditions, i.e., for x = 0, t = (m + 1) T p, and m → ∞ is given by:
For a sufficiently long period T p, (T p ≫ t 0 ), where t 0 = δ 2 /(π 2 · D), the system behaves as under DC conditions. For T p → ∞, and taking into account Eq. (1.14), Eqs. (4.17) and (4.18) become:
and
For a sufficiently small value of T p, (T p ≪ t 0):
For i(t) given by Eq. (4.14), solution of Eqs. (4.9), (4.10), (4.11), and (4.12) for x = 0, t = [m + 1/(p + 1)]T p, and m → ∞, i.e., at the end of the cathodic pulses under periodic conditions, is given by [16]:
The surface concentration at the end of pauses, C off, under the same conditions [x = 0, (m + 1)T p , m → ∞] is given by:
As in the previous case for T p ≫ t 0, the system behaves as under DC conditions where
and
For T p ≪ t 0, it follows from Eqs. (4.22) and (4.23) that
It is obvious from Eqs. (4.2), (4.4), and (4.7) as well as from Eqs. (4.21) and (4.26) that in both cases
taking into account also Eq. (1.14).
For i(t) given by Eq. (4.15), the surface concentration under periodic conditions is approximately given by [17]:
Hence, at sufficiently small value of T p and for not extremely high values of i p , C s, in AC will also be given by Eq. (4.27), implying that at sufficiently high frequencies, surface concentration is determined by the average current density regardless of the shape of the current wave.
In the case of a rectangular pulsating overpotential , η(t) as a function of a time is given by [18]:
where η A is the overpotential amplitude, and m = 0,1,2,.....
Assuming that the surface concentration is determined by the average current density, i av, Eq. (4.16) can be rewritten in the form
For a sufficiently high value of η A , Eq. (4.30) reduces during the on periods to:
and during the off periods to:
The average current density in the PO deposition can easily be determined by:
The overpotential amplitude is then given by:
and
where
and
Polarization curves for the average values for the copper deposition have been successfully calculated from the stationary polarization curve using Eq. (4.35) for i 0 ≪ i L [18]. This is good evidence that in PO deposition, the average current density also determines the surface concentration of the depositing ion.
The overpotential amplitude is larger than in the DC regime for one and the same average current density. Simultaneously, the diffusion overpotential remains constant, depending on the average current density only. Hence, the part of activation control in the overall amplitude overpotential increases with increasing pause to pulse ratio .
The situation is similar in pulsating or reversing current electrodeposition.
4.2.2 Capacitance Effects
From the above discussion, it can be concluded that the useful range of frequencies is limited by mass-transfer effects at low frequencies. At high frequencies, the useful range is limited by the effect of the capacitance of the electrical double layer [16]. This is shown here for the PC deposition. The time dependencies of the overpotential during the current pulses are shown in Fig. 4.2.
The time dependence of the overpotential during current pulses in PC copper deposition from 0.50 M CuSO4 in 1.0 M H2SO4: (a) t c = 10 s, p = 1, i c = 1.2 i L; x-axis, 2 s/div.; y-axis, 0.2 V/div., (b) t c = 10−4 s, p = 1, i c = 1.2 i L; x-axis, 5 × 10−5 s/div.; y-axis, 0.1 V/div., (c) t c = 10−2 s, p = 1, i c = 1.2 i L; x-axis, 5 × 10−3 s/div.; y-axis, 0.1 V/div., and (d) t c = 10−2 s, p = 9, i c = 6 i L; x-axis, 5 × 10−3 s/div.; y-axis, 0.2 V/div (Reprinted from Ref. [12] with kind permission from Springer and Ref. [16] with permission from Elsevier)
Mass-transfer limitations cause an increase in the overpotential at deposition times longer than the transition time as shown in Fig. 4.2a; the system enters full diffusion control at low frequencies if i c > i L . This is followed by an increase in the average overpotential [10, 16]. At high frequencies, the PC is used both for double-layer charging and discharging and for the deposition process, as illustrated by Fig. 4.3. The capacitance current during periodic charging and discharging of the double layer, at frequencies at which the effect of the double layer cannot be neglected, produces a smearing effect on the Faradic current wave, as illustrated by Fig. 4.2b.
Schematic representation of the effect of the double-layer capacitance on the faradic current during pulsating current electrodeposition (Reprinted from Ref. [12] with kind permission from Springer)
Hence, as the frequency increases, the faradic current wave flattens, approaching to a DC shape, and gives the same quality of deposit as DC even though the overall current appears to be pulsating one. This is also followed by an increase in the average overpotential. Hence, the minimum average overpotential is a good indicator of the optimum frequency range of pulsation in the PC deposition [3, 10, 16]. This range depends on the average current density and p, but, in general, the frequency lies in the range between 10 and 100 Hz, as illustrated in Fig. 4.2c, d.
In the PO deposition, the effect of the double -layer capacitance becomes less pronounced at higher frequencies compared to the other cases [10]. Also, at very high frequencies, the shape of the PO wave changes; for example, a square- wave PO becomes similar to a triangular one [10, 18].
4.2.3 Reversing Current in the Second Range
For T p close to t 0, the behavior of the system under RC conditions has to be analyzed using Eq. (4.18) [7]. In this case, the concentration distribution inside the diffusion layer at the end of the anodic pulse is close to that given by Eq. (4.10). It follows from Eq. (4.18) that this will occur at:
or
It is known [15] that for r rc T p/(r rc + 1) ≥ 1.5 t 0 , the series in Eq. (4.39) can be approximated using only the first term (k = 0). Hence, for i c = i a,
or
It is easy to show that for T p = 3 t 0, r RC = 0.7 and for T p = 16 t 0 , r RC = 0.2 by assuming that Eq. (4.40) is valid for T p > 3 t 0 and that for T p > 16 t 0 , the system behaves as under the DC conditions. The optimum ratio t c/t a is given by:
for periods T p such that
if t 0 = 1 s for δ = 10−2 cm and D = 10−5 cm2 s−1. A good agreement between the shape and the frequency of the RC calculated in this way and the literature data is obtained, because in practically all cases is according to Bakhvalov [2]:
On the other hand, the solution of Eqs. (4.9), (4.10), (4.11), and (4.12) for
is given by [19]:
It follows from Eq. (4.46) that the surface concentration of depositing ions for t ≥ t 0 can be given by:
The maximum amplitude of the current density variation, i A,max, corresponding to C s = 0 after a deposition time t c, is given by:
It is obvious that using Eqs. (4.1) and (4.4) and r RC = f (T p) given by Eq. (4.41), Eq. (4.48) can be rewritten in the form:
In this way, the complete RC wave can be estimated or precisely calculated without approximations using a computer.
4.3 Prevention of the Formation of Spongy Deposits and the Effect on Dendritic Particles
4.3.1 Basic Facts
EPCR is used in the charging of silver-zinc storage batteries, to prevent, or to delay, the formation of spongy and dendritic deposit s of zinc [20, 21]. It is impossible to obtain smooth deposits of zinc from alkaline zincate solutions during prolonged deposition at a constant rate due to formation of spongy deposits at lower and dendritic deposits at higher overpotentials [21, 22].
It is well known that the reversible potential of a surface with radius of curvature r cur would depart from that of a planar surface by the quantity [23]:
where γ is the interfacial energy between metal and solution. The filaments which form spongy deposits have extremely small tip radii. This makes the equilibrium potential of the spongy deposit to be 7–10 mV more cathodic than that of zinc foil [24, 25].
Spongy deposit formation can, however, be completely prevented by the PO deposition [21], as illustrated in Fig. 4.4.
Cross section photomicrographs of Zn deposits plated out from a 50 g dm−3 ZnO in 10 M KOH solution onto a copper wire: (a) constant overpotential, η = 22 mV, initial i av = 4.0 mA cm−2; deposition time: 40 s, and (b) PO, p = 1, η A = 40 mV, ν = 100 Hz, initial i av = 4.0 mA cm−2; deposition time: 4 h (Reprinted from Refs. [12, 21] with kind permission from Springer)
Obviously, more negative filaments dissolve faster during the off period than the flat surface, resulting in a smooth deposit. This is also valid for deposition using current or overpotential waveforms that are characterized by some anodic current flow [3, 7]. This means that the dissolution of a protrusion with tip radius r cur is faster relative to the flat surface or relative to a protrusion with a sufficiently large value of r cur. It is obvious that spongy filaments can be completely dissolved (Fig. 4.4), while dendrites with low tip radii can be either partially or completely dissolved during the pause (Fig. 4.5). This means that both branching of dendrites and the formation of agglomerates can be prevented in the square-wave pulsating overpotential deposition. In this way, even powder particle like that in Fig. 4.6 can be formed.
Pb deposits obtained from 0.10 M Pb(CH3COO)2 + 1.5 M NaCH3COO + 0.15 M CH3COOH on Cu substrate at a constant overpotential of: (a) 75 mV for 50 s, (b) the same as in (a) but for 350 s at 0 mV, (c) the same as in (a) but for 100 s at −10 mV (anodic), and (d) by PO with η A = 75 mV, pause-to-pulse ratio : 3; pulse duration, 0.5 s (Reprinted from Ref. [12] with kind permission from Springer and Ref. [26] with permission from Elsevier)
Powder particles obtained by a square-wave PO regime: (a) silver. Pulse-to-pause ratio of 1:5. Pulse duration: 50 ms. η A = 160 mV. Deposition was carried out from an electrolyte containing 10 g dm−3 AgNO3 in 100 g dm−3 NaNO3 onto a graphite electrode and (b) copper. Pulse-to-pause ratio of 1:5. Pulse duration: 5 ms. η A = 600 mV. Deposition was carried out from 0.10 M CuSO4 in 0.50 M H2SO4 onto a Pt electrode painted with shellac (Reprinted from Refs. [12, 27, 28] with kind permission from Springer)
The monocrystal surfaces of silver powder particles can be explained by the assumption that during the off period in the PO, the adatoms in nonstable positions will dissolve faster than atoms in a stable position in the lattice. The similar effect on the morphology of powder particles can be seen in the RC deposition [29, 30], which leads to the strong effect on the apparent density of copper powders [30].
4.3.2 Quantitative Treatment
4.3.2.1 One Electron Transfer Process
The one electron transfer processes were quantitatively discussed for the regime of the square-wave pulsating overpotential [31, 32]. The square-wave pulsating overpotential is described by Eqs. (4.9), (4.10), and (4.11), as well as by Eq. (4.51) [18, 31]:
In the case of a rectangular pulsating overpotential, η(t) as a function of a time is given by Eq. (4.29).
Assuming that at sufficiently high frequencies the surface concentration in the pulsating overpotential deposition does not vary with time, it is easy to show that response of the current density, i, to the input overpotential:
Equations (4.9), (4.10), (4.11), (4.12), (4.29), (4.51), and (4.52) are valid for the flat electrode surfaces or protrusions with sufficiently large tip radii where the surface energy term, γ, can be neglected [23]. If it cannot be neglected, then the surface energy term affects the reaction rate [3], and for one electron transfer process, it is valid Eq. (4.53):
where β, T, V, and R have already mentioned meanings.
The right-hand side of Eq. (4.51) should be transformed by taking Eq. (4.53) into account. The output current during pauses (η = 0) becomes:
if r cur → 0.
It is easy to show that the difference between the current density on the flat surface and at the tip of the dendrites during the “off” period is given by:
and, if i av ≈ i L, which leads to
because of
where h p is the height of protrusion, and h p,0 is the initial height of protrusion and taking Eq. (4.55) into account:
Equation (4.56) represents the change of the height of the protrusion with tip radius r cur relative to the flat surface or relative to the protrusion with a sufficiently large r cur. In square-wave PO electrodeposition [7, 21], the filaments on the growing grains formed in the spongy electrodeposition can be completely dissolved during the pause duration leading to the formation of the compact deposit . In powder electrodeposition by the same regime, the dissolution of branches on the dendrite stalk is also expected.
Hence, the larger the “off” period, the less dendritic particles are obtained. On the other hand, the current density during the “on” period on the tip of dendrites growing inside the diffusion layer is given by:
which is a somewhat modified Eq. (4.52) [33]. For the same “on” period, the particles will be more dendritic with increasing overpotential amplitude. In the millisecond range , the ratio between the overpotential corresponding to the bulk diffusion control and the activation overpotential can be reduced to the value corresponding to electrodeposition at the lower overpotentials in the constant overpotential regime. Hence, the deposits obtained in the PO regimes (at the same η A and with different p used) are more similar to those obtained in the constant overpotential regime (p = 0) at the lower overpotentials than those obtained in the constant regime at the overpotential corresponding to η A in the PO regimes. The degree of diffusion control decreases with increasing p, even at the limiting diffusion current density, and it can become sufficient to produce the quality of deposits corresponding to a mixed, activation , or a surface energy control. This will be discussed in more detail in the case of lead electrodeposition.
4.3.2.2 Two Electron Transfer Process
Equations (4.9), (4.10), (4.11), (4.12), (4.29), (4.51), and (4.52) are also valid for the two electron transfer processes. These equations are valid for flat electrode surfaces or protrusions with sufficiently large tip radii, where the surface energy term can be neglected [23]. If it cannot be neglected, the effect of the surface energy term on the reaction rate [3] for two electron reaction steps is described by Eq. (4.60) [32]:
The output current density, i, during pauses (η = 0) at the tip of the dendrite is presented by Eq. (4.61):
The corresponding output current density on the flat surface is given by Eq. ( 4.52b). The difference between the current density at the tip of the dendrite and on the flat surface during the “off” period is given by Eq. (4.62):
if i av ≈ i L, that is satisfied in most cases of dendrite growth.
Then, the change of height of surface protrusions with tip radius r cur relative to the flat surface is given by Eq. (4.63) [21]:
and finally, for the two electron reaction steps:
where
Equation (4.64) represents the height of the dendrite with tip radius r cur as a function of time, relative to the flat surface or relative to the protrusion with a sufficiently large r cur. It is obvious that dendrites with very low tip radii can be completely dissolved during the pause. This means that the branching of dendrites can be prevented in square-wave pulsating overpotential deposition. Obviously, the larger p, the greater the degree of dissolution, as followed from Eq. (4.65).
4.4 Compact Deposits
4.4.1 Surface Film
The first stage of metal film formation is nucleation on a foreign substrate. The nucleation rate in DC regime, J, is given by Eq. (2.86) and depends strongly on the deposition overpotential . The nucleation overpotential is larger than the stationary one in galvanostatic deposition, and the stationary values of the overpotential can be used in discussions of the effect of EPCR in galvanostatic as well as in potentiostatic deposition processes. It is obvious that in all types of EPCR, the overpotential amplitude, η A, is larger than in constant-current or constant overpotential deposition, η, for the same average current density [34].
Therefore,
where Δη > 0 and
Hence, for the same current density:
as illustrated in Fig. 4.7.
SEM photomicrographs of cadmium deposits obtained from 1.0 M CdSO4 in 0.50 M H2SO4 on a plane Cu cathode by the DC and the PC depositions at the different pause-to-pulse ratios . i av = 10 mA cm−2, t = 120 s: (a) DC, (b) PC; p = 4; t c = 10 ms, and (c) PC; p = 9; t c = 10 ms (Reprinted from Ref. [12] with kind permission from Springer and Ref. [34] with permission from Elsevier)
On the other hand, the increased amplitude on the current density leads to an increase in the ohmic potential drop during the pulses in EPCR relative to the constant regimes and Eq. (2.81) can be rewritten in the form:
It follows from Eqs. (2.81) and (4.69) that
because with increasing p
Hence, the increasing nucleation density is also due to the decreasing zero nucleation zone radii. This effect leads to an increased coverage of the foreign substrate by the same quantity of deposited metal and to a decreased porosity, a surface resistance and an increased density of a deposit. Also, it can be expected that increase in compactness of a deposit is associated with a decrease in internal stresses and increased ductility and hardness of metal deposits [7].
4.4.2 Electrode Surface Coarsening
The general equation of the polarization curve for the flat part of an electrode is given by Eq. (1.13) and by Eq. (2.23) for the tip of a protrusion (δ ≫ h p), around which the lateral diffusion flux can be neglected.
Using Eq. (4.27) in the form:
it is easy to show that
and
if f c ≫ f a, where C s and C s,t are the surface concentration of depositing ions on the flat electrode surface and on the tip of a protrusion , respectively.
The rate of growth of the tip of a protrusion relative to the flat surface is given by [35]:
or
after substitution of C 0 – C s,t and C 0 – C s from Eqs. (4.72) and (4.74) into Eq. (4.75) and further rearranging, assuming δ ≫ h p .
It was shown earlier that at sufficiently high frequencies, the average current density in electrodeposition at a periodically changing rate produces the same concentration distribution inside the diffusion layer as a constant current density of the same intensity. Hence, Eq. (4.76) is valid for all cases of electrodeposition at a constant and periodically changing rate at sufficiently high frequencies.
However, an increase in surface coarseness in a deposition using a rectangular pulsating overpotential or pulsating current is only possible during the pulses of current or overpotential [35] and the integral form of Eq. (4.76) can be written as:
and, if i = i av, and Q is given by Eq. (2.27).
Equation (4.77) is valid for pulsating current, pulsating overpotential, and reversing current in the millisecond range under the assumption that the entire surface dissolves uniformly during the pauses. The deposits obtained by constant and pulsating overpotential in the mixed control under the other conditions that are the same are shown in Fig. 4.8a, b. The deposit obtained by pulsating overpotential is considerably less rough.
Copper deposits obtained from 0.10 M CuSO4 in 0.50 M H2SO4 on a Cu wire electrode with a quantity of electricity of 20 mA h cm−2 in: (a) the constant potentiostatic regime; an overpotential: 210 mV; initial current density: 6.5 mA cm−2, as well as by the different pulsating overpotential (PO) regimes: (b) initial average current density: 6.5 mA cm−2; η A = 322 mV; pulse-to-pause ratio : 3, (c) initial average current density: 2.9 mA cm−2; η A = 210 mV; pause-to-pulse ratio: 3, and (d) initial average current density: 6.5 mA cm−2; η A = 322 mV; pause-to-pulse ratio: 3 (Reprinted from Ref. [12] with kind permission from Springer and Ref. [35] with permission from Elsevier)
The copper deposits obtained under activation and mixed controls as those shown in Fig. 2.4 are shown in Fig. 4.8c, d. A considerable decrease in the grain size of deposit obtained at the low current densities (in the activation controlled region ; Figs. 2.4ac and 4.8) due to the increase of the amplitude of the overpotential relative to the corresponding value in constant overpotential deposition can be seen. There is no qualitative change, however, in the structure of the deposit.
A qualitative change in the structure of the deposit appears in the mixed-controlled region (Figs. 2.4c and 4.8d). It can be seen that the protrusions caused by mass transport limitations are strongly reduced relative to the deposits shown in Fig. 2.4c, but the grain size is enlarged. It is obvious that the grains obtained by the PO regime with current densities belonging to the region of the mixed control (Fig. 4.8d) are almost regular in comparison with those deposited under the activation control (Fig. 4.8c). This is due to the increase of both the degree of activation control during the overpotential pulses and the grain size as a result of the selective dissolution during the “off” periods (in relation to those shown in Fig. 2.4b, c). The smaller nuclei formed during the overpotential pulse will be completely or partially dissolved during the overpotential pause, and the current density on the partially dissolved nuclei will be considerably lower during the next overpotential pulse than the one on larger ones because of their more negative reversible potentials. In this way, the growth of larger grains will be favored.
Then, the deposit shown in Fig. 2.4c changes and becomes like the one shown in Fig. 4.8d, which is formed by use of the PO deposition at the same average current density with the same quantity of deposited metal. It can be also seen from Fig. 4.8c, d that a good deposit can be obtained by the PO deposition over wide range of current densities. This means that in EPCR deposition, current density can be considerably increased relative to the DC regime. Nevertheless, it seems that the PO and PC regimes are not suitable for the prolonged depositions , due to the formation of large grains.
On the other hand, it is known that the orientation of nuclei strongly depends on the depositing overpotential and that the electrode reaction parameters can be different for different crystal planes. Therefore, it is not surprising that the effect of structure on EPCR has been reported for many cases. In some cases, deposits which behave as monocrystals [7] and deposits with improved crystal perfection can be obtained.
It is obvious that the same reasoning is valid for the RC in the millisecond range and the PC. Some different situations appear in the case of RC in the second range [36].
The surface concentration changes during the cathodic pulse in RC deposition in the second range according to Eq. (4.46).
where
In this case
is also valid and substitution of \( \frac{C_{\mathrm{s}}}{C_0} \) from Eq. (4.78) in Eq. (4.80) gives
for the tip of a protrusion and
for a position on a flat surface. If the whole surface is iso-potential, elimination of f c from Eqs. (4.81) and (4.82) after rearranging produces
The difference in the current densities between the tip of a protrusion and the flat portion of the electrode surface is then given by:
for δ ≫ h p.
Now, according to Eqs. (4.63) and (4.84) it follows:
or in the integral form:
where
for the first phase.
Assuming that the surface will dissolve uniformly during the anodic period (it is because f(0) = 0 and f(∞) = 1), the increase in the surface coarseness in the RC regime will be lower than in the DC regime until the condition
is satisfied. The above reasoning is valid if a polycrystalline deposit is obtained and the same derivation as for the case of electrodeposition at a constant overpotential can be used.
It can be seen from Fig. 4.9 that the structure of the deposit obtained by the RC in the second range is more similar to the one obtained in the DC than in the PC regime. However, the surface coarseness of this deposit is considerably lower than in the DC regime, and it is close to the surface coarseness obtained in the PC deposition. This is because in the RC deposition, there is a considerable concentration polarization, producing polycrystalline deposit.
4.5 Current Density and Morphology Distribution on a Macroprofile
The current density distribution on a macroprofile in EPCR has been treated in Ref. [7]. It seems that the current density distribution is improved by application of current waves with an anodic flow, but without this flow, it is worse than in the DC deposition. This behavior can be successfully explained [38].
Assuming that Eq. (3.49) is also valid for the flat electrode in a cell with a low anode polarization , the current densities in the middle, i, and at the edge, i e of a flat electrode in a cell with a low anode polarization can be related by:
Equation (4.89) is valid if the electrodeposition process is an under activation control in the Tafel region, the limiting diffusion current density is the same in both the middle and at the edge of the electrode, and I is the cell current corresponding to i. During the current pulses, the amplitude value s of current densities and current should be substituted in Eq. (4.89) producing
where i e, A and i c are the amplitude values of current densities at the edge and in the middle of electrode and I A is the amplitude of the current in the cell. On the other hand, the amplitude in pulsating current deposition and the average current density are related by Eq. (4.7), so Eq. (4.90) can be rewritten in the form:
assuming that i = i av, if
then i av,e > i av, meaning worse current density distribution in the PC than in the DC conditions.
The effect of reversing current on the current distribution at the macroprofile level can easily be discussed for the case of activation -controlled deposition if the Tafel slopes of the anodic and cathodic processes are different, as they are for copper deposition and dissolution from sulfate solutions. With the assumption that the current density in the RC deposition is sufficiently high so that the effect of the opposing processes can be neglected, the limiting diffusion current density is the same over all electrode surface. The difference between the current density at the edge and the one in the middle of the electrode in a cathodic deposition is:
and for an anodic case
since b a = b c/3 for copp er deposition, where I c and I a are the cell currents corresponding to i c and i a. It is obvious that for:
which occurs when
(if i = i a = i and I c = I a = I), deposits of equal thickness can be obtained at the edge and in the middle of the electrode. In this way, a completely uniform average current density distribution on the macroprofile level can be obtained in RC deposition. The diffusion limitations of the cathodic processes will improve the distribution in the RC, but this approach is sufficient to explain the essence of the effect, as illustrated in Fig. 4.10 [38].
Crosssection of copper deposits obtained from 1.0 M CuSO4 in 1.0 M H2SO4 at the edge of plane Cu electrodes, previously plated with bright nickel . i av = 10 mA cm−2, t = 4 h: (a) DC, (b) PC, p = 1, 50 Hz, and (c) RC, r RC = 1/7, T p = 8 s (Reprinted from Ref. [12] with kind permission from Springer and Ref. [38] with permission from Elsevier)
The best current distribution is expected in the case of PO if the whole electrode surface can be taken as an iso-potential. Under the assumption that the limiting diffusion current density does not vary over the electrode surface area, the same current density can be expected over all points of the electrode. A good approximation of the PO deposition can be the RC deposition by the current wave optimized relative to the current density distribution on both the microprofile and macroprofile [39]. Hence, it seems that the RC should be the optimum regime of EPCR. Besides, the crack-free chromium deposits with an improved current density distribution on both the microprofile and macroprofile and with practically no reduced hardness were obtained by the RC deposition [40, 41]. This means that the formation of unstable chromium hydride can also be prevented by the RC, but this phenomenon has been not tr eated semiquantitatively so far.
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Popov, K.I., Djokić, S.S., Nikolić, N.D., Jović, V.D. (2016). Electrodeposition at a Periodically Changing Rate. In: Morphology of Electrochemically and Chemically Deposited Metals. Springer, Cham. https://doi.org/10.1007/978-3-319-26073-0_4
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