Question

Grain growth rate (G) is empirically related to time (t) by:

• A. G\(^n\) \(-\) G\(_0^n\) \(=\) kt
• B. G \(=\) k/t
• C. G \(=\) kT
• D. G \(=\) G\(_0\) \(+\) kt\(^2\)

Hint:

For highly pure metals:
- The grain growth exponent \( n \) approaches \( 2 \) (ideal parabolic growth).
- The rate constant \( k \) follows an Arrhenius relationship with temperature: \( k = k_0 \exp(-Q/RT) \).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This question asks for the empirical mathematical kinetic equation that describes grain growth in polycrystalline metals as a function of annealing time.

Step 2: Key Formula or Approach:
Grain growth is a thermally activated process driven by the reduction in total grain boundary surface area (and thus a reduction in grain boundary free energy).
The classic empirical relationship for grain size growth is:
\[ D^n - D_0^n = k t \]
where:
\( D \) is the average grain size at time \( t \) (represented as \( G \) in the options).
\( D_0 \) is the initial average grain size at \( t = 0 \) (represented as \( G_0 \)).
\( k \) is a temperature-dependent rate constant.
\( t \) is the annealing time.
\( n \) is the grain growth exponent.

Step 3: Detailed Explanation:

• Physical Interpretation:
- In an ideal, highly pure single-phase metal system, the grain growth exponent \( n \) is theoretically equal to \( 2 \) (parabolic growth, \( D^2 - D_0^2 = kt \)).
- This is derived assuming that the velocity of grain boundary movement is directly proportional to the driving force, which is inversely proportional to the grain boundary radius of curvature.
- In real commercial alloys, impurities, second-phase particles (which cause Zener pinning), and solute drag slow this process down, increasing the exponent \( n \) to values between \( 2 \) and \( 4 \).
- The formula \( G^n - G_0^n = kt \) is the standard generalized representation of this grain growth behavior.


Step 4: Final Answer:
The generalized empirical kinetic relationship for grain size growth is \( G^n - G_0^n = kt \).
Hence, option (A) is the correct choice.