Jawaharlal Nehru Technological University Hyderabad B.Tech II THERMODYNAMICS OF MATERIALS Previous Question Paper (March 2022 - R16)

Darken's equation relates the intrinsic diffusion coefficients of individual species to the chemical diffusion coefficient in a binary alloy. For a binary system A-B, the flux of species A is given by J_A = -D_A(∂C_A/∂x) + C_Av, where v is the velocity of the lattice marker. Since the total flux J_A + J_B = C_totalv, and assuming a vacancy mechanism where J_A = -J_B, the velocity of the marker plane becomes v = (D_B - D_A)(∂C_A/∂x). Substituting this into the continuity equation, we derive the chemical diffusion coefficient D~:

where X_A and X_B are the mole fractions of components A and B. Incorporating the thermodynamic factor (1 + ∂lnγ_A/∂lnX_A), the final form is D~ = (X_BD_A + X_AD_B)(1 + ∂lnγ_A/∂lnX_A). The importance of this equation lies in its ability to describe how chemical concentration gradients drive diffusion while accounting for the differences in individual atomic mobilities and the thermodynamic non-ideality of the alloy.

Diffusion in metals occurs primarily through the movement of atoms between lattice sites, governed by the following mechanisms:

• Vacancy Mechanism: Atoms jump into adjacent vacant lattice sites. This is the dominant mechanism for self-diffusion and substitutional diffusion in metals, as the concentration of vacancies increases with temperature.
• Interstitial Mechanism: Small solute atoms (e.g., C, H, N) move between the interstitial voids of the host lattice. Because these atoms are small and do not require lattice vacancies, this process is significantly faster than substitutional diffusion.
• Interchange (Direct Exchange) Mechanism: Two adjacent atoms swap positions directly. This is rare in metals due to the extremely high activation energy required to distort the lattice to accommodate the exchange.
• Ring Mechanism: A group of three or more atoms rotate in a ring configuration. While theoretically possible, it is energetically unfavorable compared to the vacancy mechanism.
• Short-Circuit Diffusion: Atoms diffuse along high-diffusivity paths such as grain boundaries, dislocations, and free surfaces. These paths have lower activation energies than bulk lattice diffusion, becoming significant at lower temperatures.