Question

The Schiel equation models

• A. growth rate during dendritic solidification
• B. thermal diffusivity
• C. change in interface curvature
• D. solute distribution in solid along dendrite arm

Hint:

Contrast the two main solidification models:
- Equilibrium Solidification (Lever Rule) \(\rightarrow\) Assumes complete diffusion in both solid and liquid.
- Non-Equilibrium Solidification (Scheil Equation) \(\rightarrow\) Assumes zero diffusion in solid, complete diffusion in liquid.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
This question asks about the physical phenomenon modeled by the Scheil equation (often referred to as the Scheil-Gulliver equation) in solidification metallurgy.

Step 2: Key Formula or Approach:
The Scheil equation models non-equilibrium solute redistribution during solidification.
The key formula for solute concentration in the solid \( C_s \) as a function of fraction solidified \( f_s \) is:
\[ C_s = k C_0 (1 - f_s)^{k - 1} \]
where:
\( C_s \) is the solute concentration in the solid.
\( C_0 \) is the initial nominal alloy composition.
\( k \) is the equilibrium partition coefficient.
\( f_s \) is the fraction of solid formed.

Step 3: Detailed Explanation:

• Core Assumptions of Scheil Model:
1. Zero diffusion of solute in the solid phase (diffusion is extremely slow in solids).
2. Complete and instantaneous mixing (perfect diffusion) of solute in the liquid phase.
3. Local thermodynamic equilibrium is maintained at the solid-liquid interface.

• Solute Segregation Modeling:
- Because solute is rejected from the solid into the liquid (assuming \( k \lt 1 \)) and cannot diffuse back into the solid, the solid concentration increases continuously as solidification progresses.
- The Scheil equation accurately predicts this micro-segregation profile of solutes along a solidifying dendrite arm.
- It does not model thermal diffusivity, growth kinetics of dendrites directly, or interface curvature changes (which is modeled by the Gibbs-Thomson equation).


Step 4: Final Answer:
The Scheil equation is used to model solute redistribution and micro-segregation in the solid along a dendrite arm during solidification.
Therefore, the correct choice is option (D).