Nucleation

=Proto-Crystal Nucleation: How it Works and Why it is Important=

Nucleation is the process by which a crystal of Barium Sulphate is initially formed out of solution. The nucleation mechanism itself involves some very complicated thermodynamic concepts. However, the underlying principles behind the nucleation process are not hard to understand. It is important to understand crystal nucleation, because many chemical inhibitors that are used industrially to manage Barium Sulphate scaling work through a nucleation inhibition mechanism.

=How Nucleation Works: The Thermodynamic Nuts & Bolts=

Introducing Mr. Gibbs:
The fundamental source of energy in the nucleation process is the **Gibbs free energy(G),** which involves the relationship between **Enthalpy (H)** and **Entropy(S).** The Gibbs free energy is a thermodynamic property, and is defined by the following equation:

** G = H - T*S ** The unit of the Gibbs free energy, like all energies, is the Joule. The important thing to remember here is that energy represents the capacity to perform **work** within a system.

A Simplified Model of Nucleation:
Consider a closed system consisting of two compounds: compound A and compound B.

I n our system, compound A is liquid water and compound B is a dissolved salt being held in a **supersaturated** aqueous solution. Together, these compounds make up phase 1.

Our initial system conditions are thermodynamically unstable. That is to say, there exists a **driving force** (the Gibbs free energy) that favors the formation of a second phase. However, in order for the second phase to form, an **activation energy barrier** must be overcome within the system.

Since phase 1 is initially supersaturated with compound B, the conditions within our system are such that if it were allowed to reach **thermodynamic equilibrium**at constant pressure and temperature, we would have a stable second phase present. This second phase would consist of solid crystalline compound B.

At these unstable initial conditions, if we were able to zoom in on a microscopic level we would observe small localized groups of molecules of compound B beginning to form microscopic aggregations of the crystal phase 2 within the existing bulk liquid phase 1. These molecules would have the same pressure and temperature properties that the stable crystal phase 2 would have at the equilibrium pressure and temperature. This results in a localized pressure and temperature fluctuation within the surrounding bulk phase 1. You will recall that the Gibbs free energy depends on **Enthalpy** (a function of pressure) and **Entropy** (a function of temperature). Therefore, at the **interface** between the bulk liquid phase 1 and the microscopic crystal phase 2 we will have a difference in Gibbs free energy (**ΔG = G2 – G1**).

As we just saw, when a molecule transfers from one phase to another it experiences a net change in Gibbs free energy (ΔG). Since a stable crystal phase 2 will exist at thermodynamic equilibrium, it will have a Gibbs free energy that is less than the Gibbs free energy of the initial supersaturated liquid phase 1. Therefore, the overall value of ΔG for a molecule transferring from phase 1 to phase 2 at initial conditions is negative.

Consider one small proto-crystal of compound B within the almost infinitely larger phase 1. The ratio of the surface area of this proto-crystal to its volume is very high when it initially forms. There will be a surface tension force (σ) that exists at the interface between phase 1 and the crystal phase 2. This **surface tension** force represents the main **energy barrier** that must be overcome before the crystal growth phase will occur **spontaneously**.

=**The Math Behind the Magic**=

The total work involved in creating a spherical proto-crystal of phase 2 within the bulk liquid phase 1 is expressed as follows:

(1)



It follows from (1) that there will exist some maximum value of the proto-crystal radius that will represent the maximum Gibbs free energy We find this value for the **critical radius** of the proto-crystal by setting the derivative of (1) equal to zero and solving for the critical radius, r*. We know that this will be a maxiumum, since the global minimum value of r is equal to zero.

(2)

=**LeChatelier's Roller-Coaster (Unstable Thermodynamic Equilibrium):**=

Since we have set (1) equal to zero, the value for the critical radius (**r***) occurs when the two phases are in thermodynamic equilibrium ( G1 = G2). However, since this point occurs at a maximum value in the total Gibbs free Energy, **the equilibrium will be unstable** as shown by the figure below:



Once the proto-crystal exceeds the critical radius, the change in chemical potential experienced by the total number (volume) of molecules in the proto-crystal leads to a decrease in the Gibbs free energy. This results in a **spontaneous growth-phase** for the crystal that will continue until a stable thermodynamic equilibrium is reached. As discussed earlier, this equilibrium condition will results in the stable co-existence of both a liquid phase (phase 1) and a crystal phase (phase 2). However, if the proto-crystal is not able to meet and exceed the critical radius, then it is doomed to be redissolved in the bulk mother phase.

**The animation below is a conceptual illustration of the nucleation process:**
media type="file" key="critical_radius.swf" width="390" height="390" align="center"

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