JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II Year PHASE TRANSFORMATIONS Previous Question Paper (July/August - 2021 - R18)

Diffusion is the process by which matter is transported through solid materials by atomic motion. It is fundamentally driven by a concentration gradient and is described by two primary laws formulated by Adolf Fick.

Fick’s First Law: This law describes steady-state diffusion, where the concentration profile does not change with time. It states that the diffusion flux (J) is proportional to the concentration gradient (dC/dx). The mathematical expression is:

J = -D(dC/dx)

• J represents the diffusion flux, defined as the mass or number of atoms diffusing through a unit area per unit time (e.g., kg/m²⋅s).
• D is the diffusion coefficient or diffusivity, which is a material-specific constant dependent on temperature, the diffusing species, and the host lattice.
• dC/dx is the concentration gradient, representing the change in concentration per unit distance.
• The negative sign indicates that diffusion occurs from a region of high concentration to a region of low concentration (down the concentration gradient).

Fick’s Second Law: This law describes non-steady-state diffusion, where the concentration at any point within the solid changes with time. It is derived from the continuity equation and Fick's First Law, assuming the diffusion coefficient is independent of concentration:

dC/dt = D(d²C/dx²)

• dC/dt represents the rate of change of concentration with time at a specific position.
• d²C/dx² represents the curvature of the concentration profile.
• This partial differential equation allows for the prediction of how concentration profiles evolve over time in processes such as carburizing, decarburizing, or doping of semiconductors.

Applications and Limitations:

• Fick's laws assume an isotropic medium and are valid for dilute solutions where the diffusion coefficient is independent of concentration.
• In real metallic systems, D often follows an Arrhenius-type temperature dependence: D = D&sub0; exp(-Q&subd;/RT), where D&sub0; is a pre-exponential factor, Q&subd; is the activation energy for diffusion, R is the gas constant, and T is absolute temperature.
• These laws are essential for calculating case-hardening depths in steels and understanding homogenization treatments in alloys.

The diffusion coefficient (D) represents the rate at which atoms migrate through a crystal lattice. Its temperature dependence is described by the Arrhenius equation, which demonstrates that diffusion is a thermally activated process.

The mathematical relationship is given by:

D = D0 exp(-Qd / RT)

Where:

• D is the diffusion coefficient (m²/s).
• D₀ is the pre-exponential factor (m²/s), which is assumed to be independent of temperature.
• Q_d is the activation energy for diffusion (J/mol or eV/atom), representing the energy barrier atoms must overcome to jump from one lattice site to another.
• R is the gas constant (8.314 J/mol·K).
• T is the absolute temperature (K).

Graphical Representation:

To visualize this relationship, we take the natural logarithm of the Arrhenius equation:

ln D = ln D0 - (Qd / R)(1/T)

This equation follows the linear form y = mx + c, where:

• y = ln D
• x = 1/T
• m (slope) = -Q_d / R
• c (intercept) = ln D₀

By plotting ln D versus 1/T (an Arrhenius plot), the result is a straight line with a negative slope. The magnitude of the slope is directly proportional to the activation energy (Q_d). A steeper slope indicates a higher activation energy, meaning the diffusion rate is more sensitive to temperature changes.

Key takeaway: As temperature increases, the term (1/T) decreases, leading to a significant increase in the value of D. This occurs because higher thermal energy allows a greater fraction of atoms to possess sufficient energy to overcome the activation barrier (Q_d) for vacancy-mediated or interstitial diffusion.