Essentials Of Materials Science And Engineering
Essentials Of Materials Science And Engineering
4th Edition
ISBN: 9781337670845
Author: ASKELAND
Publisher: Cengage
Question
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Chapter 9, Problem 9.16P
Interpretation Introduction

Interpretation:

The need of undercooling for solidification process needs to be explained. An equation that shows the relation between the free energy change and undercooling when the nucleating solid have a critical nucleus radius r* needs to be derived.

Concept introduction:

Solidification is the process in which metal in the form of liquid starts to form crystal structure which is the solid phase.

Undercooling is necessary forsolidification.

Expert Solution & Answer
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Answer to Problem 9.16P

Undercooling for solidification is necessary because it is nothing but the difference in temperature which creates a force that helps in overcoming the obstacle in the formation of a solid. The expression for total free energy change is as follows:

  ΔG=[8πσslΔGvTm33HF(1ΔT)3]+8πσsl2Tm3ΔHF

Explanation of Solution

Given Information:

The equation relating the total energy and surface free energy is given as follows:

  ΔG=(43π3)ΔGv+4πr3σsl ------------------ (1)

Calculation:

The point at which the liquid phase starts to convert into a solid phase is called a freezing point. But the embryo must have a higherradius than a critical radius to start the stable formation of the nucleus. At freezing point embryos start to decrease the total available energy so the temperature should be decreased for the growth and stability of embryo.

Conversion of liquid to solid should be below freezing temperature. The temperature difference is known as undercooling.

For solidification, undercooling is required because undercooling is the difference in temperature that creates a driving force which helps in overcoming the resistance in solid formation. Two phenomena also hold true in liquid phase conversion to the gaseous phase.

From the given equation (1),

  ΔG=(43π3)ΔGv+4πr3σsl

Here,

   4 3 π 3 =volume of spherical solid. r=radius of the sphere Δ G v =free energy available per unit volume 4πr=surface area of spereσsl=free surface energy

Substitute the value of r in equation (1) as follows:

  r=2σslTmΔHFΔT

(Since r = critical energy)

In equation (1),

   ΔG=[ 4 3 π ( 2 σ sl T m Δ H F ΔT ) 3 ]Δ G v +4π ( 2 σ sl T m Δ H F ΔT ) 2 σ sl

   =[ 8π σ sl Δ G v 3Δ H F ( T m ΔT ) 3 ]+ 8π σ sl 2 Δ H F ( T m ΔT ) 2

   ΔG=[ 8π σ sl Δ G v T m 3 3Δ H F ( T m ΔT ) 1 ]+ 8π σ sl 2 T m 2 Δ H F ( 1 ΔT ) 2

Here,

   T m =Equilibrium solidification temperature Δ H f =Latent heat of fusionΔT=Undercooling temperature difference

Here,

Terms like ΔGv,ΔHf,σSlandΔTm are constant.

Then the equation becomes.

   ΔG= C 1 ( 1 ΔT ) 3 + C 2 ( 1 ΔT ) 2

   ΔG=α+(ΔT)

Where, C1andC2 are constants for a specific material.

Conclusion

Undercooling is required for solidification also with the help of free surface energy, free energy available per unit volume of the sphere can be written as the function of undercooling temperature.

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Chapter 9 Solutions

Essentials Of Materials Science And Engineering

Ch. 9 - Prob. 9.11PCh. 9 - Prob. 9.12PCh. 9 - Prob. 9.13PCh. 9 - Prob. 9.14PCh. 9 - Prob. 9.15PCh. 9 - Prob. 9.16PCh. 9 - Prob. 9.17PCh. 9 - Prob. 9.18PCh. 9 - Prob. 9.19PCh. 9 - Prob. 9.20PCh. 9 - Prob. 9.21PCh. 9 - Prob. 9.22PCh. 9 - Prob. 9.23PCh. 9 - Prob. 9.24PCh. 9 - Prob. 9.25PCh. 9 - Prob. 9.26PCh. 9 - Prob. 9.27PCh. 9 - Prob. 9.28PCh. 9 - Prob. 9.29PCh. 9 - Prob. 9.30PCh. 9 - Prob. 9.31PCh. 9 - Prob. 9.32PCh. 9 - Prob. 9.33PCh. 9 - Prob. 9.34PCh. 9 - Prob. 9.35PCh. 9 - Prob. 9.36PCh. 9 - Prob. 9.37PCh. 9 - Prob. 9.38PCh. 9 - Prob. 9.39PCh. 9 - Prob. 9.40PCh. 9 - Prob. 9.41PCh. 9 - Prob. 9.42PCh. 9 - Prob. 9.43PCh. 9 - Prob. 9.44PCh. 9 - Prob. 9.45PCh. 9 - Prob. 9.46PCh. 9 - Prob. 9.47PCh. 9 - Prob. 9.48PCh. 9 - Prob. 9.49PCh. 9 - Prob. 9.50PCh. 9 - Prob. 9.51PCh. 9 - Prob. 9.52PCh. 9 - Prob. 9.53PCh. 9 - Prob. 9.54PCh. 9 - Prob. 9.55PCh. 9 - Prob. 9.56PCh. 9 - Prob. 9.57PCh. 9 - Prob. 9.58PCh. 9 - Prob. 9.59PCh. 9 - Prob. 9.60PCh. 9 - Prob. 9.61PCh. 9 - Prob. 9.62PCh. 9 - Prob. 9.63PCh. 9 - Prob. 9.64PCh. 9 - Prob. 9.65PCh. 9 - Prob. 9.66PCh. 9 - Prob. 9.67PCh. 9 - Prob. 9.68PCh. 9 - Prob. 9.69PCh. 9 - Prob. 9.70PCh. 9 - Prob. 9.71PCh. 9 - Prob. 9.72PCh. 9 - Prob. 9.73PCh. 9 - Prob. 9.74PCh. 9 - Prob. 9.75PCh. 9 - Prob. 9.76PCh. 9 - Prob. 9.77PCh. 9 - Prob. 9.78PCh. 9 - Prob. 9.79PCh. 9 - Prob. 9.80PCh. 9 - Prob. 9.81PCh. 9 - Prob. 9.82PCh. 9 - Prob. 9.83PCh. 9 - Prob. 9.84PCh. 9 - Prob. 9.85PCh. 9 - Prob. 9.86PCh. 9 - Prob. 9.87PCh. 9 - Prob. 9.88PCh. 9 - Prob. 9.89PCh. 9 - Prob. 9.90PCh. 9 - Prob. 9.91PCh. 9 - Prob. 9.92PCh. 9 - Prob. 9.93PCh. 9 - Prob. 9.94PCh. 9 - Prob. 9.95PCh. 9 - Prob. 9.96PCh. 9 - Prob. 9.97PCh. 9 - Prob. 9.98PCh. 9 - Prob. 9.99PCh. 9 - Prob. 9.100PCh. 9 - Prob. 9.101PCh. 9 - Prob. 9.102PCh. 9 - Prob. 9.103PCh. 9 - Prob. 9.104PCh. 9 - Prob. 9.105PCh. 9 - Prob. 9.106PCh. 9 - Prob. 9.107PCh. 9 - Prob. 9.108PCh. 9 - Prob. 9.109PCh. 9 - Prob. 9.110PCh. 9 - Prob. 9.111DPCh. 9 - Prob. 9.112DPCh. 9 - Prob. 9.113DPCh. 9 - Prob. 9.114DPCh. 9 - Prob. 9.115DPCh. 9 - Prob. 9.116CPCh. 9 - Prob. 9.117CPCh. 9 - Prob. 9.118CPCh. 9 - Prob. K9.1KPCh. 9 - Prob. K9.2KP
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