Study Notes on Diffusion and Irradiation Induced Defects in Nuclear Engineering

NC STATE UNIVERSITY - Department of Nuclear Engineering

Diffusion - Continued

Steady-State Diffusion

• Definition: ‘Steady-state’ refers to a situation in which the rate of diffusion is independent of time.

• Fick’s 1st Law: Used to describe steady-state diffusion, expressed as:

  $J = -D \frac{dC}{dx}$

• Simplification of Concentration Gradient: For a linear concentration gradient:
  $\frac{dC}{dx} \approx \frac{\Delta C}{\Delta x} = \frac{C<em>1 - C</em>2}{x<em>1 - x</em>2}$

• Units:

  • Flux, J: [mol/cm²s] or [kg/m²s]
  • Diffusion coefficient, D: [m²/s]
  • Concentration, C: [g/cm³]
  • Position, x: [cm]

Example Calculation (Steady-State Diffusion)

• Given:
  • Concentration gradient:
    $\frac{dC}{dx} = \frac{(0.005 - 0.02)}{(2 \times 10^{-3})} = -7.5 \text{ mol/m}^4$
  • Flux (J) calculation:
    $J = -(1 \times 10^{-10}) \cdot (-7.5) = 7.5 \times 10^{-10} \text{ mol/m}^2s$
• Physical Set-Up: Inner surface exposed to coolant (Material 1) and outer surface with fuel (Material 2).

Non-Steady State Diffusion

• Definition: In non-steady state diffusion, the concentration of diffusing species is a function of both time and position. Denoted as:
  $C = C(x,t)$

• Fick’s Second Law: Used for non-steady state, expressed as:

  $D \frac{\partial^2 C}{\partial x^2} = \frac{\partial C}{\partial t}$

Example of Copper Diffusing into Aluminum

• Conditions:
  • At time t = 0: Initial concentration, $C_0$ for $0 \leq x \leq \infty$.
  • At time t > 0: Surface concentration, $C_S$ at $x = 0$ (constant surface concentration).
  • At infinity ($x = \infty$): Concentration remains $C_0$.
• Situation: Copper atoms diffuse into the aluminum bar, starting from a pre-existing concentration.

Solution and Error Function for Non-Steady State Diffusion

• General solution:

  $C(x,t) = C<em>S - \left( C</em>S - C_0 \right) \text{erf}(z)$

• Where $z$ is defined based on diffusion parameters:
  $z = \sqrt{\frac{D t}{4}} \cdot \frac{x}{\sqrt{4 D t}}$

• Error Function (erf): Values are tabulated in Table 5.1 in the reference text.

Application Example: Lithium Diffusion into Cladding

• Context: In molten salt reactors (MSRs), lithium diffuses into structural alloys.
• Assumptions:
  • Treat solid as semi-infinite with constant surface concentration.
  • Initially, no lithium is present inside the material.
• Given Data:
  • Surface concentration: $C_S = 0.10 ext{ wt% Li}$.
  • Initial concentration: $C_0 = 0$.
  • Diffusion coefficient: $D = 5.0 \times 10^{-9} m²/s$.
  • Time of exposure: $t = 100$ hours = $3.6 \times 10^{5}s$.
  • Depth: $x = 0.2 mm = 2 \times 10^{-4} m$.
• Objective: Find concentration at depth $x$ using the diffusion equation and previously defined error function.

Irradiation Induced Defects

Definition of Irradiation Damage

• Irradiation Damage: Defined as structural and chemical changes due to high-energy radiation.
• Key Effects from Irradiation:
  • Atomic Displacements: Generate radiation-induced defects such as dislocations, dislocation loops, vacancies, and interstitials.
  • Phase Stability Changes: Alterations in the phase stability of materials.
  • Mechanical Property Degradation: Decrease in mechanical properties.

Mechanisms of Irradiation Damage

• Types of Radiation: Includes energetic neutrons, ions, and electrons that displace atoms.
• Primary Knock-on Atom (PKA): An energetic particle that displaces other atoms, leading to a phenomenon called secondary knock-on atoms (SKAs).
  • Displacement Cascade: Sequential atomic displacements caused by ballistic collisions.

Irradiation-Induced Microstructure Changes

• Point Defect Accumulation: Point defects such as vacancies and interstitials accumulate in the material.
• Dislocation Loop Formation: Formation of dislocation loops as a result of defects.
• Effects on Microstructure: Example observations include voids, precipitates, solute segregation, and helium bubbles at grain boundaries.
  • Typical scales of defects: 50 nm.

Effects of Irradiation on Material Properties

• Hardening & Embrittlement: Increased strength coupled with reduced ductility in materials.
• Swelling: Void formation leads to macroscopic expansion of the material.
• Phases Instability: Phase transformations can occur as a result of radiation-induced changes.
• Reduction in Thermal Conductivity: The presence of defects scatters phonons, leading to lower heat transport efficiency.

Mitigating Irradiation Damage

• Material Selection:
  • High-purity metals (e.g., tungsten, molybdenum) are preferred for their resistance to irradiation damage.
  • Radiation-resistant ceramics (e.g., SiC, ZrO₂) are advantageous in this context.
  • Self-healing materials: Alloys specifically designed to recombine defects post-damage.
• Temperature Control: Maintaining favorable conditions can support defect recombination.
• Ion Irradiation Testing: Conducting tests simulating radiation damage can aid in material screening for vulnerabilities.