Phase Transformation of Materials

Oct. 23, 2008

Phase Transformation of Materials

Nong-Moon Hwang

3.1 Interfacial Free Energy 3.2 Solid / Vapor Interfaces

3.3 Boundaries in Single-Phase Solids

Types of Interface

1. Free surface ( Solid / Vapor interface ) 2. Grain boundaries ( α / α interfaces ) 3. Interphase interfaces ( α / β interfaces )

3.1 Interfacial Free Energy

The Gibbs free energy of a system

containing an interface of area A ⁰ G=G + A

γ

F Ad dA

γ γ

= +

dG =

γ

dA+ Ad

γ

In case of a liquid film,

/ 0

dγ dA=

F γ

∴ =

F : a force per unit length: surface tension

γ γ dA Ad FdA = +

FdA = dG

In case of a solid, in general d / dAγ ≠0 At near melting temperature d / dAγ =0

→G_bulk+ G_interface

work done by the force →FdA

3.1 Interfacial Free Energy

3.2 Solid / Vapor Interfaces

Origin of the surface free energy?

Bond Strength : ε

Lowering of Internal Energy per Bond: ε/2 per atom → 3 ε/2

Energy per atom of a {111} Surface?

→Broken Bonds

# of Broken Bonds per atom? →3 per atom

Heat of Sublimation in terms of ^ε? →L_S = 12 N_aε/2 E_SV= 3 ε/2 = 0.25 L_S/N_a E_SVvsγ?

3.2 Solid / Vapor Interfaces

γ = E + PV - TS

γ_SV= 0.15 L_S/N_a J / surface atom

The measured values for pure metals near the melting temperature T ⎟_P =−S

⎠

⎜ ⎞

⎝

⎛

∂

∂γ

γ high L

high T

high

_m

→

_s

→

)

P

cf G S

T

⎛∂ ⎞ = −

⎜∂ ⎟

⎝ ⎠

SdT VdP

d γ

= −

1720

2

1084 γ

_Cu

= mJm at

^{− o}

C

L.E. Murr, Interfacial Phenomena in Metals and Alloys

2 1

/ 0.50

γ = − = −

^{− −}

d dT S mJm K

1

5 2

.

0 ^{− −}

= mJm K S

C?

84 at be

would

What γ

_Cu ^o

Which information do you need?

1720 1000 0.5 2220

2

84 γ

_Cu

= + × = mJm at

^{− o}

C

T⎟_P =−S

⎠

⎜ ⎞

⎝

⎛

∂

∂γ Temperature dependence of γ

Surface energy for high or irrational {hkl} index

A crystal plane at an angle θto the close-packed plane will contain broken bonds in excess of the close-packed plane due to the atoms at the steps.

(

^{cos a}

)( )

^1a

broken bonds out of the close -packed plane

θ

(

^{sin a}

) ( )

^1a broken bonds from the atoms on the step

θ

For unit length of interface in the plane of the diagram and unit length

out of the paper (parallel to the steps)

Attributing ε/2 energy to each broken bond,

The close-packed orientation (θ= 0) lies at a cusped minimum in the energy plot.

Similar arguments can be applied to any crystal structure

for rotations about any axis from any reasonably close-packed plane.

All low-index planes should therefore be located at low-energy cusps.

E-θplot cf) γ-θplot

Equilibrium shape of a crystal?

1 n

i j

i

Aγ Minimum

=

∑

=

Wulff Construction How is the

equilibrium shape determined?

γ-plot