Abstract
Full Text
Physical Chemistry
S. I. Krichmar
Approximate Calculation of the Leveling Effect in the Electrochemical Deposition of a Metal
(Presented by Academician A. N. Frumkin, May 21, 1963)
Recently, reports have appeared on the possibility of leveling the microprofile in the process of galvanic production of metallic coatings \((^1)\). This phenomenon is to a certain extent similar to smoothing during electrochemical polishing \((^2)\). It has been established that the effect is associated with the preferential adsorption of molecules of surface-active substances on the protruding regions of the microprofile \((^3)\).
If one takes into account that the rate of diffusion of molecules of surface-active substances through the diffusion layer to the protruding parts of the microprofile is greater than to the depressions, then the reason for their preferential accumulation on the protrusions is clear. On the other hand, it is known that an increase in the concentration of adsorbed molecules of surface-active substances produces a certain additional obstacle to the discharge of metallic ions. Thus, despite the fact that the transport of metal ions through the near-electrode diffusion layer to the protruding regions of the microprofile is greater than to the depressions, passivation of the former due to enhanced adsorption of foreign molecules can not only reduce the relative growth rate of a microprotrusion, but under certain conditions lead to inversion of the process.
For a quantitative estimate of this phenomenon, let us turn to the device used by us earlier to describe the mechanism of smoothing during electrochemical polishing \((^4)\). The rate of change of the microprofile \(dH/dt\) will, obviously, in this case also be proportional to the difference between the current densities of metal dissolution* at the highest \(i_\Lambda\) and lowest \(i_\nu\) points of the surface. And since at the upper and lower points of the microprofile the directions of the current vectors coincide with the normal to the plane of the electrode, the tangential components of these vectors are equal to zero, and therefore one may write
\[ \frac{dH}{dt}=\frac{M}{\rho nF}(i_\Lambda-i_\nu), \tag{1} \]
where \(M, \rho, n\) are respectively the molecular mass, density, and valence of the metal; \(i_\Lambda, i_\nu\) are the normal components of the current vector at the extreme points, equal to their vectors at these points.
Let us consider the case in which the height of the protrusion \(H<\delta \ll a\) (\(\delta\) is the effective mean thickness of the diffusion layer, \(a\) is the base of the protrusion). In the presence in the electrolyte of foreign molecules adsorbing on the electrode surface, the current density, other conditions being equal, will be a function of the distance from the given surface point to the outer boundary of the diffusion layer and of the surface concentration of foreign molecules \(s\). For a protrusion and for a depression we shall have, respectively:
\[ i_\Lambda=i(\delta-H,\,s+\Delta s);\qquad i_\nu=i(\delta+H,\,s-\Delta s). \tag{2} \]
Expanding in a series, we obtain:
\[ i_\Lambda=i(\delta_1s)-\frac{\partial i}{\partial \delta}H+\frac{\partial i}{\partial s}\Delta s-\ldots;\qquad i_\nu=i(\delta_1s)+\frac{\partial i}{\partial \delta}H-\frac{\partial i}{\partial s}\Delta s+\ldots \tag{3} \]
* In cases of gas evolution, the current efficiency should be taken into account.
Substituting (3) into (1), we shall have
\[ \frac{dH}{dt}=\frac{2M}{\rho nF}\left[\left(\frac{\partial i}{\partial s}\right)\Delta s-\left(\frac{\partial i}{\partial \delta}\right)H\right]. \tag{4} \]
According to what was said above, \((\partial i/\partial \delta)<0\) and \((\partial i/\partial s)<0\); therefore, in the case
\[
\left|(\partial i/\partial \delta)H\right|<\left|(\partial i/\partial s)\Delta s\right|
\]
the profile is leveled, and conversely.
For the case of concentration polarization, under the condition \(a \gg \delta\), one may put [^4]
\[ (\partial i/\partial \delta)\simeq -\,i/\delta, \tag{5} \]
Let us proceed to estimate the second term in equation (4). First we shall find \((\partial i/\partial s)\). We shall assume that the current density during crystallization is proportional to the probability of nucleus formation. For this case it has been shown [^5,^6] that it depends on the concentration of surface-active substances:
\[ i=i_0\exp\left(-4r_0\sqrt{sN}\right), \tag{6} \]
where \(r_0\) is the radius of the nucleus, \(i_0\) is the current density in the absence of surface-active substances, and \(N\) is Avogadro’s number. Differentiating (6), we obtain
\[ (\partial i/\partial s)=-2r_0N^{1/2}s^{-1/2}. \tag{7} \]
We shall find the quantity \(\Delta s\) in the following way. The flux of the surface-active substance \(f\) to the metal surface is equal to
\[ f=D_e(\partial c_e/\partial y)_{y=0}\simeq D_e(c_e-c_e^*)/\delta, \tag{8} \]
where \(c_e\) and \(c_e^*\) are the bulk concentrations of molecules, respectively, in the bulk electrolyte and at the metal surface, and \(D_e\) is the diffusion coefficient of the molecules of the surface-active substance. For not too large values of \(c_e\), the adsorption isotherm equation, giving the relation between the bulk and surface concentrations, may be written in the form
\[ c_e^*=Gs, \tag{9} \]
where \(G\) is the adsorption coefficient. Substituting (9) into (8), we obtain
\[ f=D_e(c_e-Gs)/\delta. \tag{10} \]
The rate of “burial” of molecules in the deposit, \(f_1\), under stationary conditions is
\[ f_1=isM'/\rho nFh \tag{11} \]
(where \(h\) is the linear size of foreign molecules) and is equal to their supply rate \(f\). Comparison of expressions (10) and (11) gives the relation between \(s\) and \(c_e\):
\[ s=c_e/(G+\mu i), \tag{12} \]
where \(\mu=M\delta/\rho nFD_eh\).
It is obvious that the sum of the molecular fluxes to the depression and to the protrusion is equal to twice the mean flux to the metal surface, i.e., \(f_v+f_\lambda=2f(\delta,s)\). Expanding the left-hand side of this equality in a Taylor series in powers of \(H\) and \(\Delta s\) and retaining terms up to the second degree, we obtain
\[ \frac{\partial^2 f}{\partial s^2}(\Delta s)^2 -2H\Delta s\frac{\partial f}{\partial s}\frac{\partial f}{\partial \delta} +\frac{\partial^2 f}{\partial \delta^2}H^2+\ldots=0, \tag{13} \]
whence, on the basis of (10), \((\partial^2 f/\partial s^2)=0\), \((\partial f/\partial s\,\partial \delta)=GD_e/\delta^2\), and \((\partial^2 f/\partial \delta^2)=2D_e(c_e-Gs)/\delta^3\), which after substitution into (13) gives
\[ \Delta s=H(c_e-Gs)/G\delta. \tag{14} \]
Comparing (14) and (12), we obtain
\[ \Delta s=c_eH\mu i/\delta G(G+\mu i). \tag{15} \]
Substituting relations (5), (8), (15) into (4), with account taken of (12), we obtain
\[ \frac{dH}{dt}=\frac{2MiH}{\rho nF\delta} \left(1-\frac{2r_0\mu i}{G}\sqrt{\frac{Nc_e}{G+\mu i}}\right) =\frac{2iM}{\rho nF\delta}\alpha H . \tag{16} \]
Equation (16) admits integration, as a result of which we obtain
\[ H=H_0\exp(-2Mi\alpha t/\rho nF\delta). \tag{17} \]
The latter, just as in the case of electropolishing \({}^{(2)}\), has an exponential form and differs only in the value of the quantity \(\alpha\).
Equation (17) satisfies the following conditions: 1) at \(i=0\), \(dH/dt=0\); 2) at \(c_e\to0\), \(dH/dt>0\), and in this case smoothing regions do not exist for any values of \(i\); 3) if \(c_e>0\), then at \(0<i<i_{\mathrm{n}}\) there lies the region of profile distortion \((dH/dt>0)\). Taking into account that, as the current density increases, the radius of the two-dimensional nucleus must decrease, at \(i_{\mathrm{n}}<i<\infty\) there lies the smoothing region \((dH/dt<0)\), where \(i_{\mathrm{n}}\) is the second root of equation (16), corresponding to the inversion point.
In order to carry out a more complete analysis of the relation obtained, let us estimate the order of magnitude of some of the quantities entering into it. Let us determine the order of \(\mu=M\delta/\rho_nFD_eh\). Since \(M/\rho\simeq10\ \text{cm}^3\), \(\delta\simeq10^{-2}\ \text{cm}\), \(D_e\simeq10^{-5}\ \text{cm}^2\cdot\text{cm}^{-1}\), \(h\simeq10^{-7}\ \text{cm}\), then \(\mu\simeq10^6\ \text{cm}^{-1}/\text{a}\). It may be assumed that the adsorption coefficient \(G\) has a value of \(10^6-10^4\ \text{cm}^{-1}\). Thus, the product \(\mu i\) reaches the value \(G\) at a current density of \(0.1—1.0\ \text{a}/\text{cm}^2\). Therefore, for values of \(G\) of the order of \(10^6—10^4\ \text{cm}^{-1}\) and \(i<1\ \text{a}/\text{cm}^2\), expression (16) may be simplified:
\[ \frac{dH}{dt}=\frac{2MiH}{\rho nF\delta} \left(1-2r_0\mu i\sqrt{\frac{Nc_e}{G^3}}\right). \tag{18} \]
Let us turn to the investigation of the dependence of the expressions obtained on the current density. The magnitude of the two-dimensional nucleus, as is known, is related to the overvoltage \(\eta\) by the relation
\[ r_0=\theta M/\rho nF\eta, \tag{19} \]
where \(\theta\) is the peripheral energy. Taking \(\theta\simeq10^3\ \text{erg}/\text{cm}\), \(M/\rho=10\ \text{cm}^{-3}\), we obtain \(r_0\simeq10^{-8}/\eta\) (where \(\eta\) is measured in volts). Since the overvoltage for the case of deposition of most metals at low densities is proportional to \(i\), then
\[ r_0=10^{-8}/k_1 . \tag{20} \]
Thus, for small current densities
\[ \frac{dH}{dt}=\frac{2MHi}{\rho nF\delta} \left[1-\frac{10^{-8}\mu}{k_1}\sqrt{\frac{Nc_e}{G^3}}\right]. \tag{18a} \]
Taking the overvoltage at current densities of the order of \(10^{-4}—10^{-5}\ \text{a}/\text{cm}^2\) to be equal to 10 mV, we obtain \(k_1=10^3—10^2\ \Omega\). Then the second term in brackets is equal to \(10^{-2}—10^{-3}\). Therefore, at low current values in the electrodeposition process only profile distortion is possible. On the other hand, at considerable current densities the dependence of \(\eta\) on \(i\) is
\[ \eta=k_2\lg bi \tag{21} \]
(where \(k_2\) and \(b\) are empirical constants). For the case of deposition of nickel from a sulfate electrolyte, one may take \(k_2=10^{-1}\ \text{V}\), \(\lg b=10\) \({}^{(7)}\). Substituting (20), we estimate the magnitude of the second term in (18) for this case. We obtain
\[ \left(\frac{\partial i}{\partial s}\right)\Delta s = \frac{10^{-8}\cdot10^6(6\cdot10^{23})^{1/2}}{10^{-1}(10+\lg i)} \frac{c_e^{1/2}}{G^{3/2}} . \]
Table 1 gives the values of \(i_{\mathrm{n}}\) at which \(dH/dt = 0\) (transition region) for different values of \((c_e G^3)^{1/2}\). The range of variation of the latter quantity was investigated for the range of changes in \(c_e\) from \(10^{-1}\) to \(10^{-4}\) mol/cm\(^3\) and \(G\) from \(10^3\) to \(10^6\) cm\(^{-1}\).
Table 1
| \((c_e G^3)^{1/2}\) | \(10^{11}\) | \(10^{10}\) | \(10^9\) | \(10^8\) | \(10^7\) |
|---|---|---|---|---|---|
| \(i,\ \mathrm{a/cm^2}\) | 10.3 | \(8.2\cdot10^{-1}\) | \(7.2\cdot10^{-2}\) | \(6.4\cdot10^{-3}\) | \(6.1\cdot10^{-4}\) |
As is seen from Table 1, real values for \(i\) are obtained at \((c_e/G^3)^{1/2}\) of the order of \(10^8\) and lower, which corresponds to high concentrations \(c_e\) (\(>10^{-4}\) mol/l) and high adsorbability (\(G \ll 10^3\) cm\(^{-1}\)).
In conclusion, we note that, as the calculations show, the exponential character of the dependence of the mean height of a microprofile irregularity on the electrolysis time is general in character for most cases of anodic and cathodic behavior of metals.
Dneprodzerzhinsk Branch
of the State Institute of the Nitrogen Industry
Received
13 V 1963
REFERENCES
- D. Whitwhan, P. Lelong, Metaux, 33, No. 391, 101 (1958); S. S. Kruglyshev, N. T. Kudryavtsev et al., ZhPKh, 35, No. 4 (1962).
- S. I. Krichmar, DAN, 100, No. 3, 48 (1955); ZhFKh, 31, No. 7, 1598 (1957).
- S. E. Beacom, B. J. Riley, Studies on the Influens of Organic Addition Agents on the Electrodeposition of Nickel, Michigan, 1961.
- S. I. Krichmar, DAN, 144, No. 4, 833 (1962).
- Erdey—Gruz, Volmer, Zs. Phys. Chem., 157, 165 (1931).
- S. I. Krichmar, ZhFKh, No. 1 (1962); DAN, 151, No. 3 (1963).
- V. I. Lainer, N. T. Kudryavtsev, Fundamentals of Electroplating, 1, Moscow, 1953, p. 419.