 A method for interpretation of experimental data obtained from probabilistic experiments is proposed. This theory demonstrates the interrelation between the thermodynamic and the molecular kinetic approach to the nucleation phenomena and is extremely illustrative and useful from an educational point of view. The book is written as a textbook and is suitable not only for specialists in Electrocrystallization but also for graduate and PhD students, as well as for scientists from diverse but related fields like materials science, pure and applied electrochemistry, electrocatalysis, corrosion, electrochemical adsorption, crystal growth etc.

College and university lecturers might use the material involved in their own courses on Electrocrystallization.

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Kliment Ochridski" in In the same year he joined the Institute of Physical Chemistry of the Bulgarian Academy of Sciences where he is now a full professor and works in the field of Electrocrystallization of metals and alloys carrying out theoretical and experimental studies. He obtained his Ph. The same general formula for the nucleation work equation 1. Finally, equation 1. In what follows we shall show how this is done in the framework of the classical nucleation theory.

This 16 Chapter 1 outlines the region of validity of the classical theory of nucleation — it applies to sufficiently large clusters for which macroscopic concepts like surface and volume do have physical significance. That being the case, the size of the nucleus n can be considered as a continuous variable. For that purpose let us consider two finite solid bulk phases with equal size, and Figure 1.

The pressure P and the temperature T are kept constant and the two solid phases are polarised to the same potential E. Cleaving, reversibly and isothermally, the into two halves, A and B, results in the creation of two new dividing surfaces in the bulk of the electrolyte solution, each of them having a surface area S. If is the work done in this process, the work referred to unit surface area, is defined as the specific free surface energy, of the interface boundary.

Analogously, if is the work done to cleave into two halves A and B, the quantity is defined as the specific free surface energy, of the interface boundary. Note that the works and should be expressed, though formally, as and respectively, where and are the works done in cleaving and into two halves in vacuum and and are the works gained due to the solvation of the new interfaces in the electrolyte solution, the appearance of the electric double layer, etc.

It is the potential dependence of and that makes the specific free surface energies potential-dependent quantities. Since and are the specific free surface energies of and in vacuum, the conditions and are always fulfilled. Indeed, of silver measured in vacuum is [1. Suppose now that is a substrate and is a crystal that has to be deposited thereon. In doing this, two new identical interface boundaries each of them having a surface area S, are created Figure 1. The net work done in this process is: where the sum halves A and B and is the work done in cleaving and into is the work gained in creating the two contact 1.

Apparently, in the case of identical phases 1 and 2, and As expected, the last relation says that the work done in splitting two identical phases in parts equals exactly the work gained in rebuilding the phases to their initial state. In order to find the total surface free energy of the three phase electrochemical system droplet—solution—foreign substrate we shall proceed in the following way.

Thus for the difference giving the total surface free energy one obtains: 1. Introducing the adhesion energy equation 1. Equations 1.

For and which means that the droplet does not feel the presence of the foreign substrate. This is the case of complete non-wetting for which and the equilibrium form is a sphere Figure 1. For the droplet transforms into a monolayer disk Figure 1. For the equilibrium form is a hemisphere Figure 1. With decreasing the droplet size it is necessary to take into consideration the contribution of the specific free line 1.

## Electrocrystallization Fundamentals Of Nucleation And Growth

In this case the expression for the total surface free energy transforms into: where is the length of the three phase contact line Fig. For and 1. This should be a reasonable value for mercury droplets deposited on a platinum working electrode from a mercury nitrate solution Figure 1. The evaluation of is made bearing in mind that being the volume of a single mercury atom. For cm and the corresponding limiting wetting angle is This is, however, a physically unrealistic result since appears to be smaller than the diameter of the mercury atom Another peculiar effect caused by the line tension is that the droplets can have two different wetting angles for each curvature radius R.

For the two wetting angles are bigger than the macroscopic one since the line tension tends to reduce the length L of the three phase contact line. For the two wetting angles are smaller than the macroscopic one since for a negative line tension the length L increases thus diminishing the total surface free energy Concluding, we should emphasise that equations 1.

In order to give an idea of the order of magnitude, we shall use the Tolman formula [1. Assuming that the dependence could be neglected for radii for which for mercury one obtains The volume of such a spherical droplet is which corresponds to 1. There are 14 possible different arrangements of the crystal building particles in the three-dimensional space. They are known as spatial lattices of Bravais [1. Three co-ordinates, known as lattice constants fix the location of each lattice point: along the a-axis, along the b-axis and along the c-axis.

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Values of the lattice constants of some face-centred cubic fcc metal crystals are given in Table 1. The definition ratio is: 24 Chapter 1 1. Figure 1. If a crystal contacts a foreign substrate the total free surface energy of the system crystal—solution—foreign substrate is [1. Substituting the difference for according to equation 1. Making use of equations 1. Combining equations 1. The height of the crystal is For which is the case of complete non-wetting Figure 1. This is the limiting case of complete wetting when the height h equals the atomic diameter. In the last case the height of the crystal is 1. The derived theoretical formulae are used also for interpretation of experimental data obtained with crystalline phases and this brings up the question to what extent does the concept of wetting angle have any meaning with solid substances. The answer lies in the combination of equations 1. Note, however, that this is valid only for simple crystallographic forms.

The most inner from all obtained closed polyhedrons represents the equilibrium form Figure 1. In this case the condition for the equilibrium form reads: where is the total free edge energy of the two-dimensional crystal polygon. Note that the quantity is defined as the work done to create a unit edge length.

homogeneous nucleation example problem

It does not coincide with the line tension and can never be negative. In the case of two-dimensional crystals the condition for the equilibrium form leads to [1. However, what we can and do observe in growth experiments are forms of growth Figures 1. While liquid droplets grow, generally preserving the form of a spherical segment and it is the wetting angle that may change at the growth overpotential, the different crystallographic faces could have quite a different growth velocity and the growth forms differ significantly from the equilibrium ones. A most general rule says that faces having a higher growth rate disappear from the growing crystal and the growth forms contain simple, closely packed crystallographic faces that spread with a lower velocity.

Thus the equilibrium polyhedron from Figure 1. Due to their specific, potential dependent adsorption on the different crystallographic faces such substances may affect the growth rate in a different manner depending on the adsorption capacity of the particular crystallographic face. The situation is additionally complicated at the advanced stage of the growth process when the mass transport dominates the growth kinetics.

At that stage diffusion limitations may cause a non-uniform concentration distribution around the growing crystal thus compensating the anisotropy of the growth rate and leading to skeletal and dendritic modes of growth [1. In the case of electrocrystallization such a morphological instability was experimentally observed at relatively high densities of the growth current of silver and cadmium single crystals Figure 1. Without going into more details, we should stress that to construct a growth form is a difficult task which requires profound knowledge of the crystal structure, of the mechanism of growth, and last not least, of the values of the specific free surface and adhesion energies.

Unfortunately, the latter are not always known with a sufficient accuracy. In order to present the surface areas and as functions of n it is enough to recall the simple relation which combined with equation 1. A simple inspection of equation 1. Hence the first definition of the concept critical nucleus follows: 1 Cluster of the new phase formed in a supersaturated system with a maximal work 34 Chapter 1 We should remind, however, that the critical nucleus is defined as a cluster of the new phase having the equilibrium form. In the classical theory the number of atoms n is considered as a continuous variable, is a differentiable function and the condition for extremum applied to equation 1.

In order to demonstrate this we consider an electrolyte solution containing metal ions with electrochemical potential and an n-atomic cluster of the new metal phase formed on an working electrode with a Galvani potential The temperature T and the total number of particles in the whole system are kept constant.

Assuming that the cluster is a sufficiently large spherical segment its thermodynamic potential can be presented as a sum of a volume term and a surface term and the thermodynamic potential of the three phase electrochemical system electrode — electrolyte — n-atomic cluster is expressed as: Here is the Gibbs free energy of the foreign substrate equation 1. What is, then, the size of the cluster that remains in equilibrium with the ambient phase?

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We speak about unstable equilibrium because attaching more atoms from the parent phase the critical nucleus turns into a stable cluster and grows irreversibly. On the contrary, detachment of atoms from the critical nucleus leads to its irreversible decay. In the classical nucleation theory the definitions 1 and 2 are fully identical.

However, we shall show in Chapter 1. Correspondingly, the work for critical nucleus formation is [1. The above formulae allow us to calculate the work of formation of any n-atomic cluster of the new phase given the size of the critical nucleus and the work of critical nucleus formation.

We should emphasize that equations 1. Using the R n relationship equation 1. It gives the interrelation between the supersaturation and the radius of the homogeneously formed sphere that would be a critical nucleus in the case of a complete non-wetting when and the nucleation work is: Another useful expression of the Gibbs—Thomson equation can be obtained if the supersaturation is presented through the difference between the equilibrium potential of the bulk metal phase and the actual potential E of the working electrode where the critical nucleus is formed equation 1.

In that case equation 1.