Essentials Of Materials Science And Engineering
Essentials Of Materials Science And Engineering
4th Edition
ISBN: 9781337385497
Author: WRIGHT, Wendelin J.
Publisher: Cengage,
Question
Chapter 12, Problem 12.1P
Interpretation Introduction

Interpretation:

The critical nucleus size r needs to be designed.

Concept introduction:

Before the formation of the new phase, the clustering of atoms or molecules is done together which results into critical radius. This is of the minimum size. It is stable and growing in nature.

Nucleation is the process of stable nuclei formation.

By nucleation process, sometimes this precipitation rate is limited. Formation of dendrite occurs when the change of phase forms the crystallization structure of solid into a matrix of liquid.

The growth of crystal is in the form of three-dimensional form and the atoms are attached through the preferred directions. Usually, dendrite is formed along the crystal axis. Dendrites are tree-like structures.

The temperature of the new phase is less than the temperature of transformation. Due to this, the nucleus volume energy is lower than the initial phase volume.

Expert Solution & Answer
Check Mark

Answer to Problem 12.1P

The critical radius for homogeneous nucleation of β phase in α phase

  r*=2πσαβ( ΔG v( αβ )+ ε)

Explanation of Solution

Given Information:

Slope is zero when the total free energy change is the maximum

  (ΔG)r=0

Calculation:

The total energy for the nucleation process −

  ΔG=43πr3ΔGv(αβ)+4πr2σαβ+43πr3ε

Where,

Radius of spherical solid surface − r

Surface energy − σαβ

Interface − (α-β)

Strain energy − ε

Change in energy - ΔGv(αβ)

By rearranging the above equation,

  ΔG=43πr3(ΔGv( αβ)+ ε)+4πr2σαβ

Consider,

Critical radius for homogeneous nucleation − r

Slope is zero when the total free energy change is at the maximum.

So, the above equation gets partially differentiated with respect to radius r

  ( ΔG)r=r(43 πr3( ΔG v( αβ ) + ε) +4πr2σ αβ)( ΔG)r=43π( ΔG v( αβ )+ ε)( 3r2)+ 4πσαβ(2r)( ΔG)r= 4πr2( ΔG v( αβ )+ ε)+ 8πrσαβ

But,

  (ΔG)r=0

Put

  r = r*

Hence equation becomes -

  ( r *)2( ΔG v( αβ )+ ε)+ 8πr*σαβ=0( r *)2( ΔG v( αβ )+ ε)=8πr*σαβ( r *)2= 8πr*σ αβ( ΔG v( αβ ) + ε)r*= 2πσ αβ( ΔG v( αβ ) + ε)

Conclusion

Hence, by using free energy change expression, we get the critical radius for the nucleation of homogeneous mixture as −

  r*=2πσαβ( ΔG v( αβ )+ ε)

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Chapter 12 Solutions

Essentials Of Materials Science And Engineering

Ch. 12 - Prob. 12.11PCh. 12 - Prob. 12.12PCh. 12 - Prob. 12.13PCh. 12 - Prob. 12.14PCh. 12 - Prob. 12.15PCh. 12 - Prob. 12.16PCh. 12 - Prob. 12.17PCh. 12 - Prob. 12.18PCh. 12 - Prob. 12.19PCh. 12 - Prob. 12.20PCh. 12 - Prob. 12.21PCh. 12 - Prob. 12.22PCh. 12 - Prob. 12.23PCh. 12 - Prob. 12.24PCh. 12 - Prob. 12.25PCh. 12 - Prob. 12.26PCh. 12 - Prob. 12.27PCh. 12 - Prob. 12.28PCh. 12 - Prob. 12.29PCh. 12 - Prob. 12.30PCh. 12 - Prob. 12.31PCh. 12 - Prob. 12.32PCh. 12 - Prob. 12.33PCh. 12 - Prob. 12.34PCh. 12 - Prob. 12.35PCh. 12 - Prob. 12.36PCh. 12 - Prob. 12.37PCh. 12 - Prob. 12.38PCh. 12 - Prob. 12.39PCh. 12 - Prob. 12.40PCh. 12 - Prob. 12.41PCh. 12 - Prob. 12.42PCh. 12 - Prob. 12.43PCh. 12 - Prob. 12.44PCh. 12 - Prob. 12.45PCh. 12 - Prob. 12.46PCh. 12 - Prob. 12.47PCh. 12 - Prob. 12.48PCh. 12 - Prob. 12.49PCh. 12 - Prob. 12.50PCh. 12 - Prob. 12.51PCh. 12 - Prob. 12.52PCh. 12 - Prob. 12.53PCh. 12 - Prob. 12.54PCh. 12 - Prob. 12.55PCh. 12 - Prob. 12.56PCh. 12 - Prob. 12.57PCh. 12 - Prob. 12.58PCh. 12 - Prob. 12.59PCh. 12 - Prob. 12.60PCh. 12 - Prob. 12.61PCh. 12 - Prob. 12.62PCh. 12 - Prob. 12.63PCh. 12 - Prob. 12.64PCh. 12 - Prob. 12.65PCh. 12 - Prob. 12.66PCh. 12 - Prob. 12.67PCh. 12 - Prob. 12.68PCh. 12 - Prob. 12.69PCh. 12 - Prob. 12.70PCh. 12 - Prob. 12.71PCh. 12 - Prob. 12.72PCh. 12 - Prob. 12.73PCh. 12 - Prob. 12.74PCh. 12 - Prob. 12.75PCh. 12 - Prob. 12.76PCh. 12 - Prob. 12.77PCh. 12 - Prob. 12.78PCh. 12 - Prob. 12.79PCh. 12 - Prob. 12.80PCh. 12 - Prob. 12.81PCh. 12 - Prob. 12.82PCh. 12 - Prob. 12.83PCh. 12 - Prob. 12.84PCh. 12 - Prob. 12.85PCh. 12 - Prob. 12.86PCh. 12 - Prob. 12.87PCh. 12 - Prob. 12.88PCh. 12 - Prob. 12.89PCh. 12 - Prob. 12.90PCh. 12 - Prob. 12.91PCh. 12 - Prob. 12.92PCh. 12 - Prob. 12.93PCh. 12 - Prob. 12.94PCh. 12 - Prob. 12.95PCh. 12 - Prob. 12.96PCh. 12 - Prob. 12.97PCh. 12 - Prob. 12.98PCh. 12 - Prob. 12.99PCh. 12 - Prob. 12.100PCh. 12 - Prob. 12.101PCh. 12 - Prob. 12.102PCh. 12 - Prob. 12.103PCh. 12 - Prob. 12.104PCh. 12 - Prob. 12.105PCh. 12 - Prob. 12.106PCh. 12 - Prob. 12.107PCh. 12 - Prob. 12.108PCh. 12 - Prob. 12.109PCh. 12 - Prob. 12.110PCh. 12 - Prob. 12.111PCh. 12 - Prob. 12.112PCh. 12 - Prob. 12.113PCh. 12 - Prob. 12.114PCh. 12 - Prob. 12.115PCh. 12 - Prob. 12.116PCh. 12 - Prob. 12.117PCh. 12 - Prob. 12.118DPCh. 12 - Prob. 12.119DPCh. 12 - Prob. 12.120DPCh. 12 - Prob. 12.121DPCh. 12 - Prob. 12.122CPCh. 12 - Prob. K12.1KPCh. 12 - Prob. K12.2KP
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