1.3.2.1: Nucleation
Nucleation is the first irreversible step in the formation of a new phase. A nucleus is the smallest possible stable entity of the forming phase. The classical nucleation theory (CNT) was in part formulated by Volmer and is based on his observations of vapour condensation.(34)
CNT states that nucleation occurs as a result of stochastically addition/ aggregation of elemental building blocks (cluster) and is based on collision theory and local concentration fluctuations in an otherwise homogeneous supersaturated medium.
Concentration fluctuations are inherently unstable and “dissolve” rapidly beneath a critical cluster size in solution. Past this critical spatial dimension, the addition of elementary units to the formed “nuclei” is energetically more favourable than the subtraction of one.(34) Fluctuations can be imagined to be a result of Brownian motion or random walks of solutes in a solution. These occur independently of solution state. The probability of such fluctuations leading to a nucleus exceeding the critical size, naturally increases the more solutes are present and are out on a walk. Hence, nucleation is dependent on supersaturation.(35) Josiah Willard Gibbs laid the mathematical and physical foundations for such a description. He defined the free energy change associated with cluster formation as the sum of decreasing free energy as a new phase of volume x and decreased chemical potential (∆μ) is formed (𝐺𝑉(𝑟3)) and the increase in surface free energy owing to the creation of the solid-liquid interface surrounding the formed cluster (𝐺𝑆(𝑟2)), Equation 7-12. Here n is equal to the number of atoms associated with a cluster, Ω the atomic volume of solute inside the cluster, A the surface area and γ the interfacial energy of a cluster.(36)
∆𝑮 =𝟒𝝅𝒓³
𝟑𝜴 ∆𝝁 + 𝟒𝝅𝒓²𝜸 ( 11 )
Substituting (∆𝜇) with Equation 6.
∆𝑮 = −𝟒𝝅𝒓³
𝟑𝜴 𝒌𝑩𝑻 𝒍𝒏(𝑺) + 𝟒𝝅𝒓²𝜸 ( 12 )
From Equation 12, which is graphically represented in Figure 1-4 two observations spring to mind. The volumetric term drives the reaction towards its energy minimum and scales with the degree of supersaturation. The interfacial term destabilizes the forming cluster / nuclei increasing the total Gibbs free energy solely depended on cluster radius.
Based on geometrical constraints, as cluster volume is proportional to r3 and surface area to r2 there has to exist a supersaturation dependent critical cluster radius (rcrit) above which the volume associated gain in free energy outweighs surface - associated cost. Beyond this point, nuclei growth is self-perpetuating until a solution equilibrium state is re-established. The growth of nuclei can be controlled through either growth kinetics, or diffusion limitations rather than the thermodynamic driving force.(30)
Figure 1 - 4: Total free energy change as a function of nucleation cluster size.
Interfacial energy (GS) opposes nucleation and nuclei formation whereas the secondary phase formation (GV) promotes nucleation. As the cluster increases in size the interfacial term is outweighed by the volumetric term (r ≥ rcrit) leading to the formation of a stable nucleus.(36)
The above describes the general concept of homogeneous nucleation. This refers to nucleation in the absence of any secondary interfaces, such as substrate surfaces or suspended particles, that reduce the activation barrier for nucleation.
Of course this rarely occurs in practice. Nucleation in the presence of a catalysing interface is termed heterogeneous nucleation.(36) The reduction in activation energy or barrier for heterogeneously formed nuclei compared to homogenous formed nuclei is a result of the stronger interaction between the nucleating phase with the secondary interface than the bonds of solvation. As a result, a distinction has to be made between surfaces based on the degree of interaction with the nucleating phase, being either heterogeneously nucleating or crystallisation inert surfaces. On a macroscopic scale this is relatable to the wetting capability and contact angle of a solvent on specific substrates. Substrates that wet are likely to be good heterogeneous nucleators, whereas substrates with high contact angles are not expected to be.
1.3.2.2: Crystal Growth
Crystal growth is defined as the addition of elementary building blocks to an existing lattice structure after a crystalline nucleus has been formed. The conditions (most prominently the present degree of supersaturation, and agitation) under which crystal growth occurs determine not just the final particle size and/ or the number of crystals formed. Conditions may alter the limiting reaction steps and potentially change the underlying growth process altogether. This can have a direct influence over the final crystal morphology (Chapter 1.3.4).
The overall growth process can be broken down into the following successive steps.
I. Transport to and diffusion of building blocks through the diffusion boundary layer surrounding the growing crystal. The diffusion layer thickness is determined by the system’s Reynolds number.
II. Adsorption of building blocks onto the crystal surface (partial desolvation).
III. Building block surface diffusion to energetically favourable incorporation sites, from face to step to kink sites if kinetically feasible.
IV. Integration into the existing lattice structure, including complete desolvation of the building blocks.
V. Removal of the heat of crystallisation.
Each of these steps (I-V) has its own specific activation energy and kinetic factor, and is strongly dependent on the local surface and bulk supersaturation levels. As a general rule of thumb, the slowest process – which is generally associated with the highest activation energy - defines the growth rate, mechanism and type of crystal formed. In this manner, a distinction can be made between growth based on bulk kinetic limitations (i.e. a “shortage” of material supply to the crystal from the solution, where this takes place at low supersaturation levels and leads to steady-state crystal morphologies) and growth limited by building block integration and diffusion along the crystal surface (this occurs at high supersaturation levels and results in morphologies resulting from the imperfect incorporation of building blocks as material is supplied too fast from the solution). At even higher supersaturation values, crystal growth is offset by further nucleation events and polycrystalline materials can be obtained.(30)
On a crystal surface not all incorporation sites for new building blocks are equal. A differentiation between favourable and less favourable incorporation sites is based on the deduction that a 3D crystal can present up to 3 distinct environments to an incoming building block. Environments are based on the degree of interaction of a building bock with the crystal surface. These are in increasing order of accessible neighbouring lattice units - “unsaturated bonds” - face, edge and kink sites. See Figure 1-5 for a graphical illustration. As systems tend towards a state of lowest free energy, sites that maximize the interaction between elementary building blocks (offering the highest number of unsaturated bonds -kink sites) are energetically most favourable and will therefore be filled first, if reaction kinetics are sufficiently slow. In turn, those positions require the least amount of energy (supersaturation) to be filled and on this basis the differentiation between occurring growth mechanism is made.
In order of increasing supersaturation these crystal growth mechanism are:
I. Layer by layer or 1D growth. Growth units are added to a surface layer one by one. During the integration process blocks move from their original points of adsorption to energetically favourable position till a layer is completed and a new layer is nucleated by (III).
II. Screw dislocation driven growth. Dislocation provides self-perpetuating new kink sites for the integration of building blocks. Bypassing the need of nucleating a new layer by (III).
III. 2D island nucleation or birth and spread model. New layers are created on top of each other. Nucleation occurs on face sites, where this requires an elevated level of activation energy/ supersaturation. New layers are formed before the layer beneath is completed (insufficient surface diffusion kinetics).
The switch in the dominant crystal growth mechanism at a given supersaturation is similar to the formation of metastable polymorphs, and is a result of kinetic limitations in relaxing the supersaturation.
Figure 1 - 5: Possible building block adsorption sites on a growing crystal surface.
In order of decreasing free energy (i) face, (ii) step and (iii) kink site. Highlighted in grey are the connecting “surfaces” of elementary building blocks (pink) to a crystal surface (blue). Reproduced after (37).
Oswald ripening, the growth of a larger crystal at the expanse of smaller ones is another option of crystal growth. The presence of fewer larger crystal as compared to an equal volume of smaller crystals provides an energetically lower state of the system. This originates from the increasing surface to volume ratio as particles decrease in size, and the corresponding increasing number of unsaturated bonds across a given particle interface. If placed in equivalent solution the smaller particles dissolve while the already larger particles increase in volume by common crystal growth mechanisms.
1.3.2.3: Crystal Dissolution
Dissolution, the gradual release of elementary building blocks back into the solution in the form of solutes only occurs if a particular solution is unstable – undersaturated with respect to the dissolving phase. This originates from a negative state in chemical potential (Δμ). In analogy to the criterion for crystal growth, dissolution can be described as a direct reversal of crystal growth.(38) This also applies to the actual dissolution mechanism as seen in Figure 1-6. With increasing undersaturation the dissolution mechanism changes from 1D step (layer) retreat to etch pit formation above, or in the vicinity of, crystallographic defects/
dislocations. A further increase in undersaturation results in the 2D nucleation of vacancy islands. This is analogous to the growth mechanism – dissolution mechanisms requiring an enhanced degree of undersaturation are associated with faster reaction kinetics.
Figure 1 - 6: AFM images of SiO4 demonstrating the dominating growth and dissolution mechanism as function of super/ undersaturation. Scale bars 1μm.
Taken from (38).