1.3.5.1: Multi Step Nucleation Theories
The classical theory proposed by Gibbs describes nucleation on a phenomenological level. Its goal is to describe the phase transition or nuclei formation by means of macroscopic properties e.g. surface tension and density.
Local, short-lived temperature fluctuations induce the formation of nucleation clusters in a stochastic manner. At a given point, a cluster or fluctuation overcomes a certain size at which point energy is gained by forming a secondary phase. This outweighs the associated interfacial cost which limits the stability of smaller cluster.(33)
Over the years various shortcomings of this theory have come to light.(52) These include the overestimation of nucleation rates at low and high supersaturation and more troublesome inconsistencies between the formation of local periodic structure and density were detected. New microscopic nucleation theories were therefore proposed, including the density fluctuation theory (DFT). DFT basically describes a nucleus as a function of two structural parameters, namely a critical structured size and coinciding density. It does so by treating any nucleating system as having an inhomogeneous structure.(53)
Two Step nucleating theories work on those postulations. Here, density changes occur first, before a periodic structure develops in the crystallizing unit.(54) This can be imagined as a combination of two distinct steps, (1) the formation of short- lived, highly dense, disordered liquid droplets/ nucleation clusters and (2) internal rearrangement of those clusters to form the actual nuclei. The short-lived, liquid droplets formed in solution, with their high density of solute molecules and elevated local supersaturation levels therefore present a more favourable environment for nucleation to occur. The actual formation of a nucleus is concomitantly occurring through a reorganization of the cluster entities to give a structured object.(13, 55) Further, it is clear that we have here a process involving two energy barriers, corresponding to cluster formation and rearrangement. The critical free energy requirement for the phase transformation is divided into two quantities. The total energy requirement for a reaction can be assumed to be route-independent and is thus equal in both scenarios. The two-step nucleation theory should therefore predict higher nucleation rates than the CNT on a thermodynamic basis, and a cascading or catalytic effect can be imagined considering that a time
macroscopic scale was provided by Zang and Liu (56-58) who used an alternating electric field to induce colloidal particle precipitation via a two-step mechanism.
Going one step further, in recent experimental history two observations were made in mineralizing solutions of calcium carbonate - the detection of stable pre-nucleation clusters (PNC) and polymer induced liquid precursor phases (PILP) - which suggest that further alterations to, or a better description of the governing nucleation mechanism behind calcium carbonate nucleation is required.(59, 60) The reported existence of stable pre-nucleation clusters (PNC) compared to the short-lived clusters described above is currently suggested for a variety of organic and inorganic systems.(55, 59, 61-63) PNC refer to amorphous clusters of atoms ~ 1-2 nm in diameter, which are present in any solution prior to any nucleation event, even in undersaturated solutions. They are different to ion pairs in that they exhibit a particular meta-stability in solution, and sit in an ambiguous potential well. A refined view in the case of calcium carbonate mineralizing solutions refers to PNC as dynamically-ordered liquid-like oxyanion polymer (DOLLOP). These undergo constant change and thus remain in thermodynamic equilibrium with the surrounding solution, which suggests meta stability.(64) It has been suggested that PNC may provide a starting point in the formation chain of amorphous and crystalline calcium carbonate, Figure 1-9. Direct evidence of this or the involvement in any form of crystallisation has not as yet been obtained.
PNC and applicable experimental techniques for phase identification and compositional analysis are necessary limited in their ability to resolve short-lived species. Future studies will have to provide a conclusive answer as to the significance of the observations made.
Figure 1 - 9: An idealized sequential overview of calcium carbonate phase precipitation from supersaturated solution prior to crystalline calcium carbonate formation in the context of non-classical multi step nucleation theory. Shown is the position of PNC / DOLLOP and following liquid crystalline phases (LCP) e.g. PILP or liquid amorphous phases as a result of two phase segmentation of the crystalizing solution in the context of non-classical nucleation mechanism. The formation of secondary solid amorphous and crystalline phases is omitted. Further given are assumed relative activation barriers for phase transition. Reproduced from (64).
The detection of a polymer induced liquid precursor phases (PILP), and the subsequent identification of liquid–liquid phase separation in the absence of polymers brings a 2-step nucleation mechanism by means of spinodal decomposition into the picture.(25, 26, 33, 64, 65)
Spinodal decomposition. For reasons of entropy, a solution at a given composition is thermodynamically stable only at a particular composition. Away from this the solution splits into coexisting phases. Spinodal decomposition describes that process. Figure 1-10 provides a schematic representation of this, and shows the decomposition into a low and high density phase from an unstable solution past the bionodal or coexistence curve (- - - -) and the region of 2 phase liquid-liquid coexistence between spinodal (- - - -) and binodal. The area in which the nucleation of liquid CaCO3/ PILP is thought to occur is highlighted. Detailed formation of crystalline or solid amorphous phases following spinodal decomposition is omitted here for simplicity. See Figure 1-2 for the eventual phase/
polymorph transformations.
Figure 1 - 10: Schematic phase diagram of the CaCO3-H2O system assuming spinodal decomposition and liquid-liquid phase coexistence e.g. two or multi step nucleation. The olive line represents a constant temperature slice through the phase diagram as the saturation is increased. (SL) single solubility line for a given solid phase (calcite, aragonite, vaterite, and ACC). Blue undersaturated region.
“Indirect nucleation of the solid phases occurs to the high supersaturation of the dashed black liquid-liquid coexistence line (L-L). The bright yellow phase field bounded by the L-L line and the dashed red spinodal line (SP) indicates the conditions in which nucleation of the dense liquid phase is possible. In the region bounded by the spinodal line, the solution is unstable to fluctuations, and liquid-liquid separation proceeds.”(33) Image taken from (33).
1.3.5.2: Crystal Growth by Oriented Attachment and Mesocrystal Formation Over the last two decades numerous crystal growth studies have revealed that certain observations of the growth mechanism were not reconcilable with the classical ion by ion crystal growth mechanism, Figure 1-11 (a). This eventually lead to the proposition of two alternative, yet somewhat interchangeable, growth mechanisms in the formation of what commonly appear to be single crystals, (b) oriented attachment, or (c) polymer guided self-assembly of crystallites - Mesocrystals.(28, 66) The latter has recently been modified to refer to a structure classification rather than a formation mechanism per se.(67)
Figure 1 - 11: Classical and non-classical crystal growth mechanisms. Given are growth mechanism alternatives after nucleation of the primarily nanoparticles (~10-100 nm). (a) Represents the classical ion by ion growth pathway of nanoparticle amplification. (b) Oriented attachment of primary nanoparticles to form an oriented crystal, where the nanocrystalline building units can lock and fuse. (c) Mesocrystal formation primary nanoparticles covered by an additive (assembly enhancer/
enabler) undergo a mesoscale assembly (mesocrystal). The nucleation step may or may not involve the formation of an amorphous precursor liquid or solid and utilization of pre-existing nucleation cluster. Reproduced from (12).
The oriented attachment driven crystal growth mechanism is founded on observations made by Banfield of the self-assembly of iron oxide and titania nanoparticles in solution.(66, 68) Growth by oriented attachment describes the
nanoparticles) into superstructures with a common crystallographic orientation.(69) The result are “iso-oriented crystals” – which on the macroscopic level diffract ideally as single crystals, such as those found in biominerals such as nacre or sea urchins spines.(67, 70) Observations of such assembly at the nano-scale are currently limited to two dimensional observations synthetically.(71) A study of iron oxyhydroxide nanoparticle self-assembly revealed that particles undergo a continuous rotation and interaction until they find a corresponding lattice match. At this point direct interfacial contact is established and solidified via ion by ion addition around the contact point between the two nanoparticles.
Oriented attachment depends foremost on the existence of a metastable period of the nanoparticles in solution, which is sufficient to allow assembly to take place. The assembly process in itself can be a result of intra-molecular forces such as van-der-Waals forces, the isotropic structure of nanoparticles and statistical particle collision, and subsequent grain rotation.(72) The ultimate driving force underlying oriented attachment is assumed to be the minimization of surface free energy, as inferred from the phenomenon of Ostwald ripening. During oriented attachment, two particles/ nuclei fuse together causing two high energy crystal faces to disappear. In common with Ostwald ripening, oriented attachment must not be limited to supersaturated solutions, but can potentially also occur in saturated solutions.
Mesocrystals and the idea of mesocrystal formation is itself a sub classification of the oriented attachment growth mechanism.(37, 56) The initially, idealized mesocrystal formation concept introduced for calcium carbonate relied on the presence of nanoparticle stabilizing and self-assembly enabling factors such as dipole-dipole interactions, epitaxial growth (mineral bridge formation between two adjoining particles) or a constrained volume mediated by surface absorbed polymer species such as poly(styrenesulfonate) or poly(styrenesulfonate) maleic acid. The latter introduced the structural classification of a mesocrystal,(67) and states that mesocrystals are colloidal crystals made up of many particles which are ordered in a common crystallographic register. They therefore behave as highly ordered single crystals.(73-75)
So far, the only quantities used to define a mesocrystal are the total free surface area and scattering coherence length. The surface area should be significantly lager for a mesocrystal than for a single crystal of identical volume due to the internal surfaces between the singular building units. The coherence length
is expected to be much smaller than for a single crystal. Differentiation between iso–oriented crystals, mesocrystals and true single crystals is difficult as determination of crystalline coherence length is error prone and often not even considered, leaving judgement to be based on appearance and increased surface area.