JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II Year THERMODYNAMICS OF MATERIALS Previous Question Paper (July/August - 2021 - R16)

Fick's laws of diffusion describe the transport of matter due to concentration gradients. Fick's First Law states that the steady-state flux (J) of a diffusing species is proportional to the concentration gradient (dc/dx). Mathematically, it is expressed as:

In this equation, J represents the diffusion flux (amount of substance per unit area per unit time), D is the diffusion coefficient (diffusivity), and dc/dx is the concentration gradient. The negative sign indicates that diffusion occurs from regions of high concentration to low concentration.

Fick's Second Law describes non-steady-state diffusion, where the concentration changes with time. It is derived from the continuity equation and the first law, expressed as:

This second-order partial differential equation allows for the determination of concentration profiles over time in systems where the flux is not constant.

The applicability of these laws is defined by several key conditions:

• The laws are primarily applicable to interstitial and substitutional diffusion in solid solutions, liquids, and gases.
• They assume that the diffusion coefficient D is independent of concentration, which is a valid approximation for dilute solutions but requires modification for concentrated systems.
• The laws are strictly valid for isotropic media where the diffusion coefficient is a scalar constant rather than a tensor.
• They are widely used in materials science to model processes such as carburization, homogenization, and precipitation hardening, where the movement of atoms is driven by chemical potential gradients approximated by concentration gradients.
• In crystalline solids, the laws apply to vacancy-mediated diffusion, provided the concentration of vacancies remains in equilibrium.

To determine the temperature, we use Fick's second law for non-steady state diffusion in a semi-infinite solid, given by the equation:

Given values: C₀ = 0.1 wt%, C_s = 1.10 wt%, C_x = 0.30 wt%, x = 3.5 mm = 3.5 x 10^-3 m, and t = 48 h = 172,800 s.

Substituting the values:

Thus, erf(z) = 0.8. Using standard error function tables, the value of z corresponding to erf(z) = 0.8 is approximately 0.90. Since z = x / (2 * sqrt(D * t)), we solve for D:

The diffusion coefficient is related to temperature by the Arrhenius equation: D = D₀ * exp(-Q_d / (R * T)). For carbon diffusion in FCC iron (austenite), the pre-exponential factor D₀ is 2.3 x 10^-5 m²/s and the activation energy Q_d is 148,000 J/mol.