Journal of Advanced Chemical Engineering

Electrochemical nucleation is usually described in the framework of the classical Gibbs theory of nucleation, which provides a simple expression for the critical radius of a growing nucleus (nanoparticles) [1-11]:

r_c=К(γ/μ) (1)

where γ is the specific surface energy, μ is the change in volume energy during a phase transition (e.g., nucleation), and K is the particle shape factor.

J ~ exp {-ΔE_с/ k_BT} (2)

where k_B is the Boltzmann constant, and T is the absolute temperature.

The dimensionless parameter G, which is also referred to as the Gibbs number, describes, equilibrium electrochemical nucleation and has the form [5]:

G=Δ E_с/kT (3)

Further, we briefly describe essential points of calculate ΔEc adopted in the classical Gibbs model first and then address modern theory of phase transitions.

Energy ΔE_с in the classical Gibbs nucleation theory (for a threedimensional case) was obtained by assuming that the nucleus is a sphere of radius r [1-9]:

(4)

where γ₃ is the specific surface energy (for the three-dimensional case), μ₃ is the change in volume energy during a phase transition (for the three-dimensional case).

The size of the critical radius of nucleation r_c is determined in Equation 1 (where for the three-dimensional case: K₃=2):

r_c=2γ/μ

Critical nucleus radius corresponding to maximum G₃ is as follows (Figure 1) [1-9]:

Figure 1: Qualitative kind diagram of the free energy barrier, ΔG₃. The crossover of the bulk -1 and surface-2 terms combined with their opposing signs lead to the formation of a free energy barrier.

(5)

For a two-dimensional case, we obtain [1-9]

where γ₂ is the specific surface energy (for the two-dimensional case), μ₃ is the change in volume energy during a phase transition (for the two-dimensional case).

The size of the critical radius of nucleation r_c is determined in Equation 1 (where for the two-dimensional case: K₂=1):

r₂=γ/μ

Critical nucleus radius corresponding to maximum G₂ is as follows [1-9]

The generalization can be obtained for a multidimensional system:

(7)

And

ΔG_n=G_np/n (8)

According to Equation 7, in one-dimensional case,

r_1c=0

because K₁=0.

This finding is consistent with the general theory. In terms of the droplet model, for convex surfaces, one can obtain the following expression:

(10)

To calculate energy barrier ΔG, it is necessary to know the surface energy and particle shape factor k (i):

(11)

Energy barrier ΔG can be assumed to be proportional to cathode overpotential η_k (see, e.g., [5,10,11]). This dependence can be presented as follows:

(12)

It was experimentally studied for the case of n=2 and 3, which follows from Equations 8-12 (see, e.g., [11]). Other general findings are given below. Here, we note that, if the short-range interaction covers the following focal areas, then radius r_nc increases to infinity, while the energy barrier vanishes. It was found that, for spinodal disintegration, the radius (which is similar radius r_nc) increases indefinitely, while the energy barrier decreases.

The Ginsburg–Landau–Abrikosov theory uses free energy (Hamiltonian) in form [12-15]

where order parameter Σ is a generalized spinor (linear term is omitted).

Function H(Σ) extreme condition allows deriving a differential equation. This equation was obtained in [14,15]:

where L determines the space dimensionality of spinor (nDdimensionality), and F[(a,ρ),cos{θ(ρ)}, sin{θ(ρ)}] is a polynomial (analytical function) that can be defined in each case.

Application of Equation 14 to a 2D problem was thoroughly considered in [15]. Below, we consider a 1D case (L=1), which corresponds to the Cahn-Hilliard-Hillert theory [16-22].

The resulting equation has the form

θ″(ρ)=(a/2)sin{2θ(ρ)} (15)

which is similar to previously reported equations [16-22] and according to which it is possible to obtain the solution [15]:

cos{θ(ρ)}=-th{aρ)}(16)

The size of the domain wall is an important parameter determining the nucleating seed size (Figure 2):

Figure 2: Schematic representation of the solution [16,17] in the form of a domain wall of energy vectors (generalized spinor).

(17)

Assuming that the domain wall size determines the critical radius of a nanoparticle, the difference between Equations 1 and 17 should be noted.

It can be shown that the energy of the domain wall is also different from the classical formulas given by Equations 5 and 8:

(18)

Agreement between the Ginsburg-Landau-Abrikosov (Cahn- Hilliard-Hillert) theories and the classical Gibbs nucleation theory is discussed below.

The simplest assumption regarding the relationship between the surface tension and the nanoparticle radius can be expressed as follows:

(19)

where Δ_in is the thickness of the surface layer of the nanoparticle and 1 is a parameter of length characterizing the nanoparticle size in terms of Δ_in .

If r_rel<1, then the Gibbs theory is not applicable. However, it can be applied in accordance with the Cahn-Hilliard-Hillert theory. In this case, a formula similar to Equation 1 (or Equation 7 without taking into account the k_n factor) can be written as follows:

(20)

It is not difficult to transit to the formulas in the form of Equation 17 from the Cahn–Hilliard–Hillert theory:

(21)

If

(22)

Thus, there is a correspondence between the Cahn-Hilliard-Hillert and Gibbs theories. It is of interest to show that there is no contradiction between the formulas for the activation energy in the Cahn-Hilliard- Hillert and Gibbs theories.

In fact, the order of accuracy for activation energy is determined by the energy (barrier) necessary for merging particles, which is a product of the particle chemical potential (-μ) and the defect size (linear case implied) and can be estimated via Equation 21.

Finally, it is not difficult to arrive at the following formula:

(23)

Therefore, there is no contradiction between the Cahn-Hilliard- Hillert and Gibbs theories.

The formation of electrochemical phase is a typical example of a first-order phase transition. Electrodeposition of nanostructures has been considered in terms of the classical Gibbs and the Cahn-Hilliard- Hillert theories. Agreement between the theories Kahn-Hilliard- Hillert and Gibbs was obtained. We found no contradiction between the theories Cahn-Hilliard-Hillert and Gibbs. We have reviewed the progress of the modern thermodynamic theory of the growth of nanoparticles. We note the existence and of other approaches of the theory, presented for example in [23-26].