(Even Beethoven is curious! Are you?)
(The maestro - Ludwig van Beethoven, clearly taking notes about nucleation! (Source))
According to the classical concept of nucleation, a cluster of solute molecules begins to form from the supersaturated solution, stimulating the formation of crystalline nuclei as liquid atoms are connected to the nuclei until they are depleted, resulting in a solid crystal.
To better comprehend the nucleation process, the particle's shape is usually assumed as a sphere similar to a ball. For spherical particles, the nucleation rate can be described by the total free energy necessary to stabilize nuclei and prevent them from dissolving in the solution. The sum of bulk-free energy or the volumetric free energy (∆Gv) (free energy between big particles and the solute in solution) and surface-free energy (∆Gs) (free energy between surface particles and the bulk particle) results in the development of a new phase containing the parent precursors through crystal growth. The supersaturation situation initiates nucleation, resulting in nucleus clusters that promote future crystal development. Notably, formation of the supersaturation in-turn depends on the critical nucleus (rc).
When nuclei are smaller than rc, they can only dissolve back into solution.
Impurities in precursor solutions, such as those from other phases, can lower the energy barrier and promote nucleation. Heterogeneous nucleation occurs more commonly and frequently compared to homogeneous nucleation. As ∆Gs is almost always positive and ∆Gv is negative, a stable nucleus requires maximum free energy to promote development. When the maximum value of free energy is reached, d∆G/dr should be zero, indicating the critical free energy (∆Gc*).
(Pictorial representation of the contact angle (θ) in Heterogeneous nucleation (Source))
Figure 3 depicts the interfacial energy diagram, which consists of three phases: liquid, solid, and contact. The terms γcl, γcs, and γsl reflect the interfacial energies between the liquid, crystalline, and solid surfaces, respectively. θ represents the contact angle on a solid surface. If θ < 180◦, nuclei and active centers have a strong affinity, lowering the energy barrier for nucleation. The process is ascribed to a significant decline in contact energy. The heterogenous nucleation barrier by virtue of the contact angle, θ and a new term, “shape factor (φ)” is altered. A parallel between the heterogeneous and homogeneous nucleation is given as follows:
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