Nucleation Models

Classical Nucleation Theory

If the amount of a solute in solution increases, it is likely that solid will precip- itate once the conditions are such that solid phases are thermodynamically more favourable than solution. To link with the phase diagram discussion in the previous section, this is equivalent to passing a coexistence curve on a phase diagram de- fined by temperature and solution concentration/activity. From the earliest studies, classical nucleation theory (CNT) has been applied to understand the emergence of

solid phases from solution. Becker and D¨oring provided the first quantitative theory for classical nucleation as we know it today [Becker and D¨oring, 1935].

CNT stipulates that in a supersaturated solution, the growth of spherical particles is determined by,

∆G(r) = 4 3πr

3_{ρ∆µ+ 4πr}2_{γ, (1.1)}

where the change in free energy, ∆G, is a function of r, the particle radius. Figure 1.8 shows a general scheme for CNT. The first term in equation 1.1 is the volume bulk free energy, and is shown by the green curve in Figure 1.8, where ρ is the density of the emerging phase and ∆µ is the chemical potential difference between the solution and solid phases. The second term is the interfacial energy contribution, given by the red curve in Figure 1.8, and this is a function ofγ: the surface tension of the emerging interface. At small r, the interfacial term dominates, but as the particle size becomes large, bulk free energy governs particle growth, resulting in the typical blue curve (Figure 1.8). Density fluctuations in solution lead to the creation of particles, but these will probably re-dissolve unless r > rcrit, where rcrit is the critical size for spontaneous particle growth, and defines the radius where the bulk energy compensates the unfavourable interfacial energy.

Figure 1.8: Schematic diagram for classical nucleation theory; ∆Gex is the free energy barrier to nucleation and rcrit is the nuclei critical radius (if r > rcrit then growth is favourable). Taken and adapted from Gebauer et al. [Gebauer et al., 2014].

The bulk free energy can be determined by the affinity term,φ, provided in equation 1.2. Here, the Boltzmann constant, temperature, ionic activity product and bulk solubility are given by kB, T, IP and Ksp, respectively. Crucially, at IP/Ksp = 1 the solution is saturated. Further addition of solute will reduce the energy barrier to nucleation, ∆Gex, and the value of rcrit (rcrit ∼γ/φ).

φ=kBTln IP Ksp (1.2) At the heart of CNT is the capillary assumption which states that the emerg- ing phase from solution has the same properties and structure as the bulk phase. The values ofρ, ∆µ,γ andIP/Kspin the CNT equations are therefore often treated as parameters. The Gibbs–Thomson effect (which takes into account changes in in- terfacial energy with particle shape) can be incorporated into CNT to minimise error associated with spherical particles as opposed to bulk phases which often have well defined planar interfaces. However, at the smallest particle sizes, the shape and structure of particles is often unknown, which makes it difficult to determine these variables. For calcium carbonate, nanoparticles have been found to be amorphous in many experiments [Addadiet al., 2003; Gebauer and C¨olfen, 2011], for whichKsp is difficult to measure. This in turn can make it difficult to define calcium carbonate nucleation using CNT.

Phase Separation and Spinodal Decomposition

If theIP of solution is increased rapidly, then at a critical level, solid will crash out of solution. The limit of stability is known as the spinodal, and is preceded by anIP range where phase separation is likely to occur given enough time. To best explain this phenomenon Figure 1.9 shows the phase diagram for a generic two phase system. An arrow at constant temperature shows the evolution of the system in the direction of increasing solute concentration. Point A marks the region of composition where a stable homogeneous solution can be found. With increasingIP, the system reaches point B and the phase boundary (binodal), where the chemical potential of solution and solid phases are equal. On passing B, phase separation becomes likely. The metastable region labelled in Figure 1.9 represents a supersaturated solution, and is where CNT can take place. On reaching point C, the solution becomes unstable and no free energy barrier to phase separation exists (which is markedly different from CNT). The system instantly phase separates into solid and liquid. This process is known as spinodal decomposition and is marked by the red spinodal curve on the phase diagram (where ∂G/∂IP = 0).

Figure 1.9: A schematic for binodal demixing and spinodal decomposition for a generic system which separates into two phases. Taken and adapted from Gebauer

et al. [Gebaueret al., 2014].

In the case of calcium carbonate ions in solution, at the binodal curve an equilibrium exists between ions and solid mineral. However, the emerging solid may be of a metastable phase which goes through subsequent transformations (by Ost- wald ripening) to the most stable phase. Multiple local equilibria may therefore exist, in which case further curves representing the coexistence of different solid phases can be added to the region of metastability in Figure 1.9. Phase separation in the bionodal region is due to microscopic but large fluctuations in solution compo- sition (cf. CNT). However spinodal decomposition occurs because of system–wide, infinitesimal fluctuations in composition. With no free energy barrier, phase sepa- ration is diffusion dependent; Ostwald ripening can subsequently lead to reductions in interfacial energy.

As well as liquid–solid phase separation, it is also possible that the homo- geneous solution phase could separate into two liquids with high and low solute concentration. With reference to Figure 1.9, a binodal curve on a phase diagram for liquid–liquid separation refers to the coexistence of two liquids. Phase separation may occur by binodal demixing, as was the case in solid–liquid phase separation. The compositions of the two phases will be determined by the binodal curve for a given temperature (presuming temperature is constant). This is also true for sys- tems which reach the spinodal curve when the dense and lean solutions equilibrate.