The formation of new sites or nuclei from which deposition may occur is called nucleation, and it is a critical aspect of electrodeposition. The number and type of nuclei strongly influence the morphology of the deposit. Both the growth rate of the deposit and the rate of nucleation increase with increasing overpotential. If the nucleation rate is low relative to the growth rate, deposit growth will take place from relatively few nuclei and deposits with larger grains will be formed. In contrast, a high nucleation rate will lead to finer crystal grains and highly granular deposits. The manner in which the crystals grow also influences the appearance and structure of the deposit as discussed later in this chapter. The nuclei formed can be either 2D or 3D. The dimensionality depends to a large degree on the strength of the interaction between the substrate (the surface upon which the deposition is taking place) and the depositing metal, as well as the degree to which the crystal lattice of the substrate matches that of the depositing metal.
What are the factors that influence nucleation on a metal surface? The formation of a nucleus involves, of course, the transfer of metal ions from the solution to the surface. If just a single ion is transferred, it is in a higher energy state than the atoms incorporated into the metal lattice and does not remain on the surface long. In order for the metal to remain on the surface, it must move to a kink site, as discussed previously, or must combine with other adions to form a stable nucleus (the subject of this section). The new nucleus (crystal) formed will be stable and continue to grow only if it reaches a critical dimension. To determine that critical size and the factors that influence that size, we follow the work of Budevski et al. (1996) and begin with the associated Gibbs energy change, which consists of two principal parts:
(13.5)
where is the Gibbs energy of cluster formation for a cluster of N atoms. The first term on the right describes the change in the electron energy due to the difference in potential between the metal and the solution. It is always negative under conditions where deposition can occur. The second term, , is the excess energy associated with the formation of the cluster. It is always positive and represents the energy that is consumed as a new phase is formed with interface boundaries between (i) the cluster and the solution and (ii) the cluster and the substrate.
Three-Dimensional Nucleation
Let’s now consider the influence of the two terms in Equation 13.5 for 3D nucleation. In order to do this, we need an expression for . We begin with the following approximation:
(13.6)
As is the surface area of the 3D crystal and is the average specific surface energy [J·m−2]. The surface area depends on the size of the cluster and hence on N, the number of atoms in the 3D cluster; in contrast, is independent of N. Our next task is to express As as a function of N. The volume of a cluster is directly related to the number of atoms in that cluster , where Vm is the volume of one atom in the lattice. The surface area is also related to the volume. For a known 3D geometry, the relationship between As (proportional to L2) and (proportional to L3) can be written as . An example of B is shown in the following illustration for a spherically shaped cluster.
ILLUSTRATION 13.2
Please determine B for a spherically shaped cluster.
SOLUTION:
From above, . Applying this to a sphere,
Using the relationship between and , we can now write the desired relationship for :
(13.7)
With this, Equation 13.5 for 3D nucleation becomes
(13.8)
We can now plot the Gibbs energy of cluster formation as a function of the number of atoms in a cluster, Figure 13.6. This plot shows some interesting and important results. First of all, the surface energy term (Equation 13.7) is positive and dominates at low values of N, while the negative term associated with the difference in overpotential dominates at high values of N. This means that goes through a maximum at a particular value of N, which we will call Ncrit. For cluster sizes below Ncrit, an increase in the cluster size is accompanied by an increase in . The opposite is true for N values above Ncrit, where an increase in the cluster size reduces the free energy. Therefore, spontaneous growth is energetically favorable for cluster sizes greater than Ncrit, but not for cluster sizes smaller than the critical value. The two curves in Figure 13.6 correspond to two different values of the overpotential. As seen in the figure, at higher values of ηs the cluster energy curve shifts down and to the left, which corresponds to a lower value of Ncrit. In other words, the critical cluster size for stable growth decreases with increasing (negative) overpotential, and deposits can undergo stable growth from smaller seed clusters. Succinctly stated, nucleation is favored at higher overpotentials.
Figure 13.6 Gibbs energy of cluster formation as a function of the number of atoms in the cluster. Parameters: Ag atom (0.1444 nm radius), γ = 0.1 J·m−2, z = 1, atoms and clusters assumed spherical.
An expression for Ncrit can be derived by differentiating Equation 13.8, setting the derivative equal to zero, and solving for Ncrit:
(13.9)
For the parameters used in Figure 13.6, application of this equation yields Ncrit values of 83 and 10 atoms, respectively, consistent with the maximum value for each of the curves in the figure. The Gibbs energy of cluster formation at the critical cluster size is
(13.10)
This quantity plays an important role in determining the nucleation rate as we shall see below.
Two-Dimensional Nucleation
Let’s now consider the influence of the two terms in Equation 13.5 for 2D nucleation. In contrast to 3D nucleation where growth occurs in all directions, 2D growth takes place only on the edges of the nucleus so that the growth is outward and not upward. Therefore, the expression for involves the perimeter rather than the surface area as follows:
(13.11)
P is the perimeter of the 2D crystal and is the average specific edge energy. As is independent of N, our next task, analogous to the procedure followed above, is to express P as a function of N, the number of atoms in the 2D cluster. The area of the cluster is directly related to the number of atoms in the cluster , where Ω is the area of one atom on the surface. For a known 2D geometry, the relationship between P (proportional to L) and (proportional to L2) can be written as , where is the constant of proportionality. From this it follows that
(13.12)
With this, Equation 13.5 for 2D nucleation becomes
(13.13)
The corresponding 2D expressions for Ncrit and are as follows:
(13.14)
(13.15)
The behavior for two-dimensional nucleation is qualitatively similar to that shown above for 3D clusters. In two-dimensional growth, metal deposition takes place in the form of 2D monoatomic layers. This type of nucleation is important for deposition onto foreign metal substrates that have a strong metal–substrate interaction, and it can lead to underpotential deposition (UPD), which is 2D deposition at potentials below that needed for sustainable deposition onto a native metal substrate. UPD is also important in the absence of growth sites on single crystal faces of the native metal.
ILLUSTRATION 13.3
The calculations in Figure 13.6 assumed 3D spherical deposition of Ag. For the same conditions shown in Figure 13.6, r = 0.1444 nm, γ = 0.1 J·m−2, z = 1, what value of the overpotential would be required to reduce the critical cluster size to five silver atoms?
SOLUTION:
From Equation 13.9,
In this section, we explored the energy change associated with the formation of nuclei during electrodeposition. We have seen that a critical cluster or nucleus size is required for stability. That critical size is strongly dependent on the applied overpotential, where higher values of the overpotential reduce the cluster size required for stable growth. Having examined the conditions required for stability, we now explore to the rate at which nuclei are formed.
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