Growth of Nuclei

Early Growth of Nuclei
In this section, we examine initial growth of deposits from distinct nuclei. The distinguishing factor in early growth is that the nuclei are far enough apart to behave independently. Under such conditions, the total deposition current can be estimated by summing the contributions from the individual nuclei. Here we consider both kinetically limited growth at constant potential and, briefly, mass-transfer limited growth.

For kinetically limited growth, the key assumption is that the reaction rate is sufficiently slow that the concentration at the surface does not change with time. The potential of the surface and the temperature are also constant. Under these conditions, the reaction rate at the surface, characterized by the current density at the surface (isurf), is constant. The total current, however, changes with time as the nuclei grow. Considering a 2D single nucleus that is cylindrical in shape with a height h and a radius r, only the sides of the cylinder are active. We begin with a mass balance where we express the change in mass of the seed particle in terms of the current density at the surface. The 2D growth is characterized by an increase of the radius with time.

(13.21)
We can use this balance to solve for the rate of change of the radius:

(13.22)
Assuming that r ≈ 0 at t = 0, this expression can be integrated to yield

(13.23)
The current, from a single nucleus is simply the current density at the surface multiplied by the surface area of the growing nucleus (in this case a cylinder growing in the radial direction):

(13.24)
Following a similar procedure, the analogous expression for a 3D hemispherical nucleus growing under kinetic control can be derived:

(13.25)
Equations 13.24 and 13.25 show how the current from a single nucleus (2D or 3D) varies with time under kinetic control, assuming that the individual nuclei are sufficiently far apart to be considered independent.

In situations where the reaction rate at the surface is fast and the concentration of the depositing species in solution is low, it is possible for growth to be limited by mass transfer. The current associated with the growth of a single 3D hemispherical nucleus under mass-transfer control is given by

(13.26)
This equation assumes pseudo-steady state and a zero concentration of the depositing species at the surface. It has the square root of time dependence characteristic of diffusion-limited processes.

Now that we have expressions for the current as a function of time for a single nucleus, we are prepared to look at the combination of nucleation and growth. There are two important cases to consider.

For instantaneous nucleation, nucleation takes place much faster than growth. Under such conditions, the current from the entire surface is simply the product of the single-nucleus current (Equations 13.24–13.26) and the number of nuclei instantaneously formed, assuming, once again, that the nuclei are sufficiently far apart that they do not interact.

For progressive nucleation, nucleation and growth take place on a similar timescale. Therefore, the current due to independent nuclei increases as a function of time due to the increase in surface area of existing nuclei and the formation of new nuclei. For a known constant nucleation rate J, the relevant expressions follow. For 2D kinetically controlled growth and progressive nucleation,

(13.27)
Similarly, for a 3D hemispherical nucleus growing under kinetic control with progressive nucleation,

(13.28)
Finally, for mass-transfer limited growth with progressive nucleation,

(13.29)
Equations 13.27–13.29 are for the total current on the surface with a geometric area A and include the combined influence of nucleation and growth. The illustration below considers the difference between instantaneous nucleation and progressive nucleation for a situation where the total number of nuclei at the end of the time period of interest is the same. Although this is an artificial constraint, it illustrates the difference between the two situations.

Finally, we end this section by noting that the above expressions apply during the early stages of electrodeposition, where a combination of the observed time dependence of the growth and microscopic examination can be used in tandem to identify and quantify the processes that control deposit growth.

ILLUSTRATION 13.5
In this illustration, we compare the current that results from instantaneous and progressive nucleation for the growth of 3D hemispherical nuclei where the number of nuclei at the end of the growth period is the same. Assume that the nucleation rate, J, is 10,000 cm−2·s−1, and that growth takes place for 15 seconds. Assume that the geometric area (superficial surface area) is 1 cm2. The following additional parameters are known:

SOLUTION:
First, we need to determine the total number of nuclei at the end of the growth period so that we can use this same number of nuclei for instantaneous nucleation. For a nucleation rate of 10,000 cm−2·s−1, an area, A, of 1 cm2 and a growth time of 15 seconds, the number of nuclei at the end of the growth period assuming progressive nucleation is . We will use this same number of nuclei for the instantaneous calculation as follows:

where N0 is the number of nuclei formed instantaneously. For progressive nucleation, we use Equation 13.28, where nucleation takes place simultaneously during growth. The resulting currents are shown in the plot.

Even though the number of nuclei is the same after 15 seconds, the current is less for progressive nucleation because the nuclei are at all stages of growth, in contrast to the instantaneous situation where all of the nuclei have been growing for the entire 15-second period.

A quick check of the assumption of independent nuclei is appropriate. If we assume that the nuclei are evenly spaced over the 1 cm2 area, the spacing between nuclei is about . A nucleus that grew for the entire 15 seconds would have a radius of

which is much less than the average spacing between nuclei. Independent growth appears to be a reasonable assumption.

Interaction between Growing Nuclei
All of the above equations that describe current as a function of time for early deposit growth predict a continuing increase in the current due to an increase in the size of the nuclei and, for progressive nucleation, the addition of more nuclei. Such an increase, of course, is not sustainable since nuclei eventually overlap and coalesce. We illustrate the concept of overlap here by providing an expression for 2D deposit growth under conditions of both instantaneous and progressive nucleation. As you will remember, 2D growth occurs in a layer-by-layer fashion due to growth at the edges of the nuclei only (growth out, but not up); hence, growth stops once the layer is complete. Paunovic and Schlesinger (2006) provide expressions for 2D growth with overlap. For instantaneous nucleation,

(13.30)
Note that this is the total current since it has been multiplied by The analogous expression for progressive nucleation is

(13.31)
The current goes to zero for both of these expressions, consistent with the physical situation where a complete layer is formed and there are no edges left at which growth may occur. The current versus time plot for these two situations is shown in Figure 13.10. The total number of nuclei was artificially constrained to be equal at the end of 80 seconds in order to illustrate the difference between the two types of nucleation and growth with overlap. The nucleation rate was taken to be 100,000 cm−2·s−1. The other parameters are the same for both cases and correspond to Ag metal. The current initially varies linearly for instantaneous nucleation, as described by Equation 13.24. As before, progressive nucleation lags behind as expected. It peaks later and persists longer, but the magnitude of its peak current remains lower than that resulting from instantaneous nucleation.

Figure 13.10 The current as a function of time for 2D growth with overlap under the assumption of both instantaneous and progressive nucleation. J = 100,000 cm−2·s−1 for progressive nucleation. The total numbers of nuclei are equal and the area is1 cm2. Other parameters correspond to Ag.

Qualitatively similar behavior is expected for 3D nucleation, although prediction is much more difficult since the rates of nucleation and of growth in each of the coordinate directions are influenced by the different crystal planes and the relative growth rates of those planes, which make a variety of different morphologies or “textures” possible. Expressions that account for overlap under mass-transfer limited conditions are available in the literature, but are less important since electrodeposition is not typically performed under mass-transfer control.