Above we treated the thermodynamics associated with stable cluster formation. We now turn our attention to the rate at which nuclei are formed. In doing this, we restrict ourselves to the classical expressions for the nucleation rate attributed to Volmer and Weber, noting that similar expressions can be derived from atomistic theory. Please refer to the references at the end of this chapter for additional information.

Nucleation is a probabilistic process that can be expressed as a function of the critical energy of formation for the cluster as follows:

(13.16)

where k is Boltzmann’s constant, and the pre-exponential factor, , is approximately constant. The nucleation rate, J, has units of cm⁻²·s⁻¹ (in practice, [cm⁻²·s⁻¹] are used rather than [m⁻²·s⁻¹]). It represents the rate per area of nuclei formed on the surface of interest. Substituting the expression for  from Equation 13.10, the nucleation rate becomes

(13.17)

The equivalent expression for 2D nucleation is

(13.18)

The dependence of the nucleation rate on the overpotential is evident from Equations 13.17 and 13.18, and it is different for 3D and 2D nucleation. In both cases, the nucleation rate is independent of time. This will be the case in practice as long as the nuclei are sufficiently separated that they do not influence the formation of additional nuclei. Once a sufficient number of nuclei are formed, growth at the existing nuclei will be favored over the formation of new nuclei from adions on the surface.

The strong impact of the overpotential is readily apparent from a plot of the nucleation rate as a function of . As shown in the Figure 13.7, the nucleation rate increases rapidly over a relatively narrow range of overpotentials (for this case, −0.080 to −0.105 V). In contrast, the nucleation rate at low overpotentials is essentially zero, and growth takes place only at existing sites as discussed previously. In practice, the value of the overpotential at which the nucleation rate rises sharply is sometimes referred to as . For our purposes, we define  to be the value of the overpotential at which the nucleation rate is equal to 1 (J = 1 cm⁻²·s⁻¹ or 10⁴ m⁻²·s⁻¹). It can be determined from Equation 13.17 or 13.18 if the parameters are known, or from experimental data.

Given that J, the rate of nucleation, does not change with time, the number of nuclei formed increases linearly with time. If Z_nucl is the number of nuclei per area, then

(13.19)

Therefore, J can be measured experimentally by counting the number of nuclei formed in a given period of time. One method of doing this is to apply an overpotential above  for a specific period of time and then drop the potential below  where the nucleation rate is essentially zero. The nuclei can then be grown to a size where they can be more easily seen and counted. This process can be repeated for various times at each of several values of the overpotential to yield curves such as those shown in Figure 13.8. Note that there is a short incubation period before the expected linear growth of the number of nuclei with time is observed.

With these data, assuming 3D nucleation, it is now possible to determine  as a function of the overpotential from a plot of ln J versus  as shown in Figure 13.9 and demonstrated in Illustration 13.4. A similar procedure can be done for systems that exhibit 2D nucleation. In addition, N_crit can be estimated from the following equation:

(13.20)

Physically, this represents the tangent to the curve at the point of interest since a plot of ln J versus is not linear. While this equation “stretches” the continuum approximation, since it does not account for the atomistic nature of the clusters that becomes increasingly important with decreasing size, it has been shown to give estimates of N_crit in good agreement with those from atomistic models.

In this section, we learned that the rate at which nuclei are formed is also strongly dependent on the overpotential. Above a certain critical overpotential, the nucleation rate increases very rapidly. This will be important as we examine deposition on a more macroscale. First, let’s investigate the early stage growth of nuclei.

ILLUSTRATION 13.4

The following data present the number of nuclei present as a function of time for mercury on a platinum substrate (Ber. Bunsenges. Phys. Chem. 73, 184 (1969).) Please use these data to determine the nucleation rate at each value of the overpotential. Then, fit the nucleation rate data to the expression for the 3D nucleation rate (Equation 13.17). Finally, estimate the critical value of the overpotential.

84 mV		86 mV		88 mV		90 mV		92 mV
t [ms]	Znuc [cm−2]	t [ms]	Znuc [cm−2]	t [ms]	Znuc [cm−2]	t [ms]	Znuc [cm−2]	t [ms]	Znuc [cm−2]
1.00	0.61	0.72	2.00	1.21	14.34	0.22	1.77	0.05	0.37
1.50	2.00	1.01	4.33	1.37	18.99	0.29	4.33	0.13	2.47
2.01	5.26	1.51	6.19	1.50	23.42	0.38	9.45	0.19	5.03
2.52	8.75	2.01	11.78	1.77	25.98	0.62	21.09	0.29	13.18
3.03	11.31	3.04	19.46	2.01	29.93	0.79	31.10	0.36	20.62
4.04	15.97	4.04	27.37	2.57	39.94	1.03	41.11	0.45	31.33
5.02	19.46	5.04	36.92	3.04	51.35	1.28	56.70	0.60	49.72
7.02	19.69	6.02	45.76	3.53	58.80	1.54	70.67	0.77	66.71

SOLUTION:

The rate of nucleation at each overpotential is obtained by fitting each dataset to a line. The nucleation rate, J, is the slope of the line (cm⁻²·s⁻¹). These are the data from Figure 13.8. The corresponding J values are shown in the table. Note that time was converted from “ms” to “s” to get the proper units on J. Also note the large magnitude of the nucleation rate.

ηs [V]	J [cm−2·s−1]
0.084	3,532
0.086	8,263
0.088	18,735
0.090	52,284
0.092	97,477

To fit the data to Equation 13.17, we plot ln J versus . The slope of this line is −0.1439, and the intercept is 28.512. This yields a value for A_3D of 2.41 × 10¹². Therefore, the expression for J is

Finally, η_crit can be found by taking the natural logarithm of both sides, setting J equal to 1, and solving for η_s.