10.2 Nucleation and crystal growth kinetics

Crystals form through nucleation, either primary (without existing crystals) or secondary (with existing crystals). Primary nucleation requires higher supersaturation, while secondary needs less. Understanding these mechanisms is crucial for controlling crystallization processes.

Crystal growth depends on factors like supersaturation, temperature, and impurities. These influence growth rate, crystal habit, and size distribution. Calculations for nucleation rates and growth kinetics help predict and optimize crystallization outcomes in industrial applications.

Nucleation Mechanisms

Primary vs secondary nucleation

• Primary nucleation occurs without existing crystals requiring higher supersaturation levels (supersaturation ratio > 1.5)
• Secondary nucleation happens with existing crystals needing lower supersaturation levels (supersaturation ratio 1.01-1.5)
• Primary types include homogeneous (pure solution) and heterogeneous (foreign surfaces)
• Secondary mechanisms involve contact nucleation (crystal-crystal collisions), fluid shear nucleation (fluid flow breaks crystal fragments), and attrition (mechanical breakage)

Homogeneous and heterogeneous nucleation

• Homogeneous nucleation forms spontaneously in pure solutions demanding high supersaturation levels (supersaturation ratio > 2)
• Gibbs free energy change for homogeneous nucleation: $\Delta G = -\frac{4}{3}\pi r^3 \Delta G_v + 4\pi r^2 \gamma$ balances volume and surface energies
• Heterogeneous nucleation occurs on foreign surfaces or impurities requiring lower supersaturation levels (supersaturation ratio 1.5-2)
• Contact angle factor for heterogeneous nucleation: $f(\theta) = \frac{(2+\cos\theta)(1-\cos\theta)^2}{4}$ determines nucleation barrier reduction

Crystal Growth and Kinetics

Factors in crystal growth

• Supersaturation drives crystal growth affecting rate and habit (needle-like, plate-like)
• Temperature influences solubility and diffusion rates impacting growth mechanisms (spiral growth, 2D nucleation)
• Impurities inhibit or promote growth on specific crystal faces altering morphology (cubic, octahedral)
• Fluid dynamics affects mass transfer and boundary layer thickness influencing growth rate and size distribution
• Crystal surface characteristics like roughness factor, kink and step densities determine growth sites and mechanisms

Calculations for nucleation rates

• Nucleation rate equation: $J = A \exp(-\frac{\Delta G^}{kT})$ where $\Delta G^$ is critical free energy for stable nucleus formation
• Crystal growth rate equations:
  1. Diffusion-controlled growth: $G = k_d (c - c^*)$
  2. Surface integration-controlled growth: $G = k_r (c - c^*)^n$
  3. Overall growth rate: $\frac{1}{K_G} = \frac{1}{k_d} + \frac{1}{k_r}$
• Supersaturation calculation:
  • Relative supersaturation: $\sigma = \frac{c - c^}{c^}$
  • Supersaturation ratio: $S = \frac{c}{c^*}$
• Induction time: $t_{ind} = \frac{1}{BJ}$ where B is shape factor and J is nucleation rate