Glossary

10.2 Nucleation and crystal growth kinetics

2 min read•july 24, 2024

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

Crystal growth depends on factors like supersaturation, , and impurities. These influence growth rate, crystal , 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

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  • 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

  • forms spontaneously in pure solutions demanding high supersaturation levels (supersaturation ratio > 2)
  • Gibbs free energy change for homogeneous nucleation: ΔG=−43πr3ΔGv+4πr2γ\Delta G = -\frac{4}{3}\pi r^3 \Delta G_v + 4\pi r^2 \gamma balances volume and surface energies
  • occurs on foreign surfaces or impurities requiring lower supersaturation levels (supersaturation ratio 1.5-2)
  • Contact angle factor for heterogeneous nucleation: f(θ)=(2+cos⁡θ)(1−cos⁡θ)24f(\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=Aexp⁡(−ΔG∗kT)J = A \exp(-\frac{\Delta G^*}{kT}) where ΔG∗\Delta G^* is critical free energy for stable nucleus formation
  • Crystal growth rate equations:
    1. Diffusion-controlled growth: G=kd(c−c∗)G = k_d (c - c^*)
    2. Surface integration-controlled growth: G=kr(c−c∗)nG = k_r (c - c^*)^n
    3. Overall growth rate: 1KG=1kd+1kr\frac{1}{K_G} = \frac{1}{k_d} + \frac{1}{k_r}
  • Supersaturation calculation:
    • Relative supersaturation: σ=c−c∗c∗\sigma = \frac{c - c^*}{c^*}
    • Supersaturation ratio: S=cc∗S = \frac{c}{c^*}
  • Induction time: tind=1BJt_{ind} = \frac{1}{BJ} where B is shape factor and J is nucleation rate

Key Terms to Review (19)

Aspect Ratio: Aspect ratio is the ratio of the dimensions of a crystal or particle, usually defined as the width to height. It is an important factor in nucleation and crystal growth kinetics because it influences the shape and growth rate of crystals, affecting their properties and behaviors during formation. Understanding aspect ratio helps in controlling processes like crystallization, where the morphology of crystals can affect their solubility and stability.
Avrami Model: The Avrami model is a mathematical framework used to describe the kinetics of phase transformations, particularly in crystallization processes. It provides insights into the mechanisms of nucleation and growth during the transformation and characterizes how the fraction of material transformed changes over time, ultimately helping to predict the microstructure of the resulting solid.
Classical theory of nucleation: The classical theory of nucleation describes the process by which a new phase or structure, such as a crystal, forms from a supersaturated solution or vapor. This theory emphasizes the importance of energy barriers and thermodynamic principles in determining whether nucleation occurs and how it influences crystal growth kinetics.
Concentration: Concentration refers to the amount of solute present in a given volume of solution, typically expressed in terms of molarity, mass percent, or other units. Understanding concentration is essential in various processes, as it directly affects the behavior of solute particles during crystallization and influences the efficiency and design of crystallizers. A precise control of concentration is crucial for optimizing crystal growth kinetics and achieving desired product qualities.
Free energy barrier: A free energy barrier is the energy threshold that must be overcome for a phase transition or reaction to occur, particularly in processes like nucleation and crystal growth. It represents the difference in free energy between the initial state and the transition state, indicating that a certain amount of energy is required to form a stable nucleus or crystal from a supersaturated solution. The concept is crucial for understanding how changes in temperature, concentration, and other conditions can influence the kinetics of phase transformations.
Gibbs-Thomson Effect: The Gibbs-Thomson effect describes the influence of curvature on the melting point of a solid phase, indicating that smaller particles or droplets will have lower melting points compared to larger ones. This phenomenon is crucial in understanding how nucleation and growth occur in materials, as it explains why smaller crystals can dissolve more easily and have different thermodynamic properties than larger crystals.
Growth stage: The growth stage refers to a phase in crystal formation where the size of existing crystals increases due to the addition of more molecules from the surrounding solution. This stage is crucial as it determines the final size, shape, and quality of the crystals, influencing their properties and applications in various fields.
Habit: In the context of nucleation and crystal growth kinetics, habit refers to the characteristic shape and morphology of a crystal as it grows. The habit of a crystal can influence its properties, such as solubility, stability, and packing efficiency. Different conditions during the crystallization process can lead to various habits, affecting the overall behavior and performance of materials in applications.
Heterogeneous nucleation: Heterogeneous nucleation is the process where the formation of a new phase, such as a crystal, occurs on the surface of an existing material or impurity rather than in a homogeneous environment. This process is critical in the context of crystal growth, as it significantly lowers the energy barrier for nucleation compared to homogeneous nucleation, allowing for more efficient and rapid formation of new phases in various materials.
Homogeneous nucleation: Homogeneous nucleation is the process where nucleation occurs uniformly throughout a supersaturated phase without any preferential sites or surfaces. This phenomenon is critical in understanding how crystals form from a solution or vapor, as it involves the spontaneous formation of stable nuclei in a homogeneous environment, allowing for crystal growth kinetics to be analyzed effectively.
Interstitials: Interstitials are foreign atoms or ions that occupy the spaces between the regular lattice sites in a crystal structure. These small atoms can distort the crystal lattice and significantly affect properties such as strength, ductility, and electrical conductivity, making them crucial in the processes of nucleation and crystal growth kinetics.
Layer-by-layer growth: Layer-by-layer growth is a process in crystal growth where new layers of atoms or molecules are deposited sequentially on a substrate, creating a highly ordered and uniform crystalline structure. This mechanism allows for the precise control of thickness and composition of the resulting films, making it crucial in applications like thin-film technology and semiconductor fabrication.
Non-classical nucleation theory: Non-classical nucleation theory is a framework that explains how nucleation processes can occur through mechanisms other than the traditional thermodynamic models, which typically focus on the formation of a critical nucleus. This theory incorporates the idea that intermediate structures, such as clusters or aggregates of molecules, can play a significant role in the early stages of nucleation, leading to diverse pathways for crystal growth. Understanding this theory is crucial for analyzing the kinetics of nucleation and crystal growth, as it shifts the focus from classical pathways to a broader range of possibilities that account for complex behaviors observed in various materials.
Nucleation Stage: The nucleation stage refers to the initial process of forming stable clusters, or nuclei, from a supersaturated solution or vapor, which leads to the growth of crystals. This stage is critical as it sets the foundation for subsequent crystal growth kinetics, determining how and when crystals will form in a material. During nucleation, small particles begin to aggregate, and the stability of these clusters determines whether they will grow or dissolve back into the surrounding medium.
Spiral Growth Mechanism: The spiral growth mechanism is a process of crystal growth where the addition of atoms or molecules occurs in a spiral pattern around a specific defect or step on the crystal surface. This mechanism is significant in explaining how crystals can grow at a rapid rate, particularly in systems where the availability of growth units is high, and it highlights the importance of kinetic factors in crystal formation.
Stirring speed: Stirring speed refers to the rate at which a mixing or stirring device operates, influencing the distribution and homogeneity of particles within a solution. This rate is crucial in processes like nucleation and crystal growth, as it can affect the size and shape of crystals formed during these stages, directly impacting yield and product quality.
Supersaturation: Supersaturation occurs when a solution contains more solute than it can normally dissolve at a given temperature and pressure, creating a state that can lead to crystallization or precipitation. This phenomenon is critical as it governs the formation of crystals and impacts the efficiency of various separation processes. Understanding supersaturation is essential for controlling crystallization kinetics, selecting the right crystallizer design, and ensuring optimal conditions for both nucleation and crystal growth.
Temperature: Temperature is a measure of the average kinetic energy of the particles in a substance, which influences various physical and chemical processes. It plays a critical role in determining the rates of diffusion, mass transfer, and reaction kinetics in separation processes, affecting how substances interact and separate under different conditions.
Vacancies: Vacancies refer to unoccupied lattice sites in a crystalline solid where an atom or ion is missing. This absence can significantly influence the physical properties of materials, including their mechanical strength, diffusion rates, and overall stability during processes like nucleation and crystal growth.
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