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.

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:

  $\Delta G=[\frac{8\pi {\sigma }_{sl}\Delta G_v{T}_m{}^3}{3H_F}{(\frac{1}{\Delta T})}^3]+\frac{8\pi {\sigma }_{sl}{}^2{T}_m{}^3}{\Delta H_F}$

Explanation of Solution

Given Information:

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

  $\Delta G=(\frac{4}{3}{\pi }^3)\Delta G_v+4\pi r^3{\sigma }_{sl}$ ------------------ (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),

  $\Delta G=(\frac{4}{3}{\pi }^3)\Delta G_v+4\pi r^3{\sigma }_{sl}$

Here,

  $\begin{array}{l}\begin{array}{l}\frac{4}{3}{\pi }^3=\text{volume of spherical solid}.\hfill \\ r=\text{radius of the sphere}\hfill \\ \Delta G_v=\text{free energy available per unit volume}\hfill \\ 4\pi r=\text{surface area of spere}\hfill \end{array}\\ {\sigma }_{sl}=\text{free surface energy}\end{array}$

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

  $r=\frac{2{\sigma }_{sl}{T}_m}{\Delta H_F\Delta T}$

(Since r = critical energy)

In equation (1),

  $\Delta G=[\frac{4}{3}\pi {(\frac{2{\sigma }_{sl}{T}_m}{\Delta H_F\Delta T})}^3]\Delta G_v+4\pi {(\frac{2{\sigma }_{sl}{T}_m}{\Delta H_F\Delta T})}^2{\sigma }_{sl}$

  $=[\frac{8\pi {\sigma }_{sl}\Delta G_v}{3\Delta H_F}{(\frac{T_m}{\Delta T})}^3]+\frac{8\pi {\sigma }_{sl}{}^2}{\Delta H_F}{(\frac{T_m}{\Delta T})}^2$

  $\Delta G=[\frac{8\pi {\sigma }_{sl}\Delta G_v{T}_m{}^3}{3\Delta H_F}{(\frac{T_m}{\Delta T})}^1]+\frac{8\pi {\sigma }_{sl}{}^2{T}_m{}^2}{\Delta H_F}{(\frac{1}{\Delta T})}^2$

Here,

  $\begin{array}{l}\begin{array}{l}{T}_m=\text{Equilibrium solidification temperature}\hfill \\ \Delta H_f=\text{Latent heat of fusion}\hfill \end{array}\\ \Delta T=\text{Undercooling temperature difference}\end{array}$

Here,

Terms like $\Delta G_v,\Delta H_f,{\sigma }_{Sl}and\Delta T_m$ are constant.

Then the equation becomes.

  $\Delta G=C_1{(\frac{1}{\Delta T})}^3+C_2{(\frac{1}{\Delta T})}^2$

  $\Delta G=\alpha +(\Delta T)$

Where, $C_1\text{and}{C}_2$ 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.