nucleation
nucleation
nucleation
The driving force behind crystallization is supersaturation18. A solution in which the solute concentration exceeds the solution equilibrium concentration at a given temperature is termed a supersaturated solution. Supersaturated solutions are by their very nature metastable18.
Figure 1.4 Three states of an arbitrary crystallizing solution. (a) Refers to a solution which is termed stable (b) represent an unstable solution and (c) represents a solution which is metastable. Figure is adapted from
ref. 18.
By reference to Figure 1.4, one notes a cartoon representation of three arbitrary states of a crystallizing solution. A solution which is properly stable is depicted in Figure
1.4(a). The stable solution is at its lowest energy state and any perturbation will not shift the state of the solution. A solution which is in an unstable state is depicted in Figure
1.4(b). In this case, any small perturbation is able shift the state of the solution easily and hence it is unstable. A solution which is in a metastable state is depicted in Figure 1.4(c). The metastable solution is in a ‘quasi-minimum energy’ state and may be shifted into a lower, more stable energy state by a small but
finite
perturbation.As all supersaturated solutions are metastable, making a crystallizing solution more supersaturated may not immediately result in crystallization. To explain why a crystallizing solution does not immediately form a crystal one needs to first consider classical nucleation theory.
Classical nucleation theory (CNT) is a simple and widely used description of the nucleation process24. This thermodynamic description of nucleation was first developed by Gibbs at the end of the 19th century. Gibbs defined the Gibbs free energy change required for cluster formation, as the sum of the free energy change required for liquid- solid phase transformation (∆GV) and the Gibbs energy change for the formation of a
surface (∆GS)24.
Figure 1.5 Surface and volume contributions to the overall Gibbs energy of cluster formation. Figure adapted from ref. 25. If the overall Gibbs energy is negative, clusters will form and grow.
Since the solid state is more stable than the liquid state, ∆GV has a negative
contribution to the overall Gibbs energy change. The formation of a solid surface involves positive Gibbs energy changes which are proportional to the size of the cluster24.
“Competition” between the contributions of ∆GV and ∆GS will therefore dictate whether
a formed cluster will continue to grow or begin to ‘dissolve’ back into solution (Figure 1.5).
Chapter 1 Page 14 The situation is more easily described with the aid of a simple intuitive example25.
If one were to imagine a protein monomer which has been dissolved within an aqueous solvent and is free to move amongst the solvent molecules. The solution
composition is such that a crystallization event is thermodynamically favoured. Imagine also that protein monomers (represented for ease of visualization as balls,
vide infra
), have cubic symmetry (even though they are represented by balls in the diagram) and have 6 bonds which are normal to each face of the cubic growth unit.Within solution, a number of growth units will collide (inelastically) with one another and hence, on occasion, collisions will result in a ‘bonding’ event which will form clusters of an arbitrary size. The location of cluster formation is purely random and thus an exact location for the collision event is not given.
One finds that cluster lifetimes are governed by a balance between the cohesive forces between growth units (given by shared bonds between growth units) and the number of bonds shared between the growth units and the solvent itself. The number of shared bounds within the cluster is proportional to the volume (∆GV) whilst the number
of unshared bonds is proportional to the cluster surface area (∆GS).
Hence an energy balance for the simple cubic system, as just described, may be given by
V s
G G G
∆ = −∆ + ∆ (1.4)
For cubic systems of edge size 2 (Figure 1.6) which contain 8 growth units and 12 saturated bonds , the surface forces (24 unsaturated bonds) are in excess of the forces holding the cluster together and hence the cluster will fall apart.
At edge sizes of 3 or more, surface forces are less than volume forces and the cluster survives and grows spontaneously. At clusters of edge size of exactly three, both surface and volume forces are equal, and hence a critical nucleus is said to be formed (Figure 1.5).
Figure 1.6 Clusters of various sizes. The edge length of each cluster is depicted. Figure adapted from ref.25
The reason why supersaturated solutions are termed metastable is because the critical nucleus first needs to form before a macroscopic crystal can grow. The
metastability of the solution is inversely proportional to the level of supersaturation18. Every solution has a maximum amount that it can be saturated before it becomes unstable. The zone between the solubility curve and this unstable zone is termed the ‘metastable zone’.
Chapter 1 Page 16 This discussion refers to homogeneous nucleation, which is nucleation within a homogenous substance26. However, in practice, a much more frequently encountered form of nucleation (as explained via CNT) is that of heterogeneous nucleation.
Heterogeneous nucleation may be described as nucleation which has taken place on phase boundaries, surfaces (such as container walls) and impurities such as dust particles. The energy required for nucleation via a heterogeneous nucleation route is much lower than that required for homogenous nucleation26.
The existence of such heterogeneous surfaces promote nucleation at lower
energies due to an increased wetting effect. Particles can interact more favourably due to non-zero wetting angles.