Fundamentals of Materials Science and Engineering: An Integrated Approach, 3e

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2nd Revised Pages

11.3 The Kinetics of Phase Transformations • 405

0

+

1

1

radius, r

T 2 < T 1

at T 1

at T 2

r*

*

 G^ G*

r 2 *

2

G

Figure 11.3 Schematic free-

energy-versus-embryo/

nucleus-radius curves for two

different temperatures. The

critical free energy change (G∗)

and critical nucleus radius (r∗) are

indicated for each temperature.

whereHfis the latent heat of fusion (i.e., the heat given up during solidification),

andTmand the temperatureTare in Kelvin. Substitution of this expression forGv

into Equations 11.3 and 11.4 yields

r∗=

(

−

2 γTm

Hf

)(

1

Tm−T

)

(11.6)

Dependence of

critical radius on

surface free energy,

latent heat of fusion,

melting temperature,

and transformation

temperature and

G∗=

(

16 πγ^3 T^2 m

3 Hf^2

)

1

(Tm−T)^2

(11.7)

Activation free

energy expression

Thus, from these two equations, both the critical radiusr∗and the activation free

energyG∗decrease as temperatureTdecreases. (TheγandHfparameters in

these expressions are relatively insensitive to temperature changes.) Figure 11.3, a

schematicG-versus-rplot that shows curves for two different temperatures, illus-

trates these relationships. Physically, this means that with a lowering of temperature

at temperatures below the equilibrium solidification temperature (Tm), nucleation

occurs more readily. Furthermore, the number of stable nuclein∗(having radii greater

thanr∗) is a function of temperature as

n∗=K 1 exp

(

−

G∗

kT

)

(11.8)

where the constantK 1 is related to the total number of nuclei of the solid phase.

For the exponential term of this expression, changes in temperature have a greater

effect on the magnitude of theG∗term in the numerator than theTterm in the

denominator. Consequently, as the temperature is lowered belowTmthe exponen-

tial term in Equation 11.8 also decreases so that the magnitude ofn∗increases.

This temperature dependence (n∗versusT) is represented in the schematic plot of

Figure 11.4a.

There is another important temperature-dependent step that is involved in and

also influences nucleation: the clustering of atoms by short-range diffusion during

the formation of nuclei. The influence of temperature on the rate of diffusion (i.e.,

magnitude of the diffusion coefficient,D) is given in Equation 6.8. Furthermore, this

diffusion effect is related to the frequency at which atoms from the liquid attach

themselves to the solid nucleus,νd. Or, the dependence ofνdon temperature is the