Driving Force for SolidificationTopic 3

M.S DarwishMECH 636: Solidification Modelling

Definitions

• In the study of phase transformations we will be dealing with the changes that can occur within a given system e.g. an alloy that can exist as a mixture of one or more phases

• A phase can be defined as a portion of the system whose properties and composition are homogeneous and which is physically distinct from other parts of the system

• The components of a system at the different elements or chemical compound which make up the system

Gibbs Free Energy

G = H - T S

for condensed phases PV ≈0 and thus H ≈ E

Enthalpy TemperatureEntropy

H = E + PV

Internal Energy

Kinetic + Potential Energy

Pressure Volumemeasure of

heat content of the system

+=

A system is in equilibrium when it is at its most stable statewhen it has the lowest value of the Gibbs free energy

measure of randomness of

the system

dG = 0

Equilibirum

B A

Gibbs freeenergy, G

dG = 0

dG = 0

metastable equilibrium stable equilibrium

Any transformation that results in a decrease in Gibbs free energy is possible

∆G = G2 - G1 < 0

How fast does a phase transformation occur?

Free Energy Relations

Specific HeatCp

T(K)0

EnthalpyH

T(K)

298

0

slope = Cp

EntropyS

T(K)0

�

CP = ∂H∂T

⎛ ⎝ ⎜

⎞ ⎠ ⎟ P

�

H = CPdT298

T

∫

�

CP

T= ∂S

∂T⎛ ⎝ ⎜

⎞ ⎠ ⎟ P

�

S = CP

TdT

0

T

∫

EnthalpyH

T(K)

298

0

slope = Cp

G

TS

slope = -S

slope = -S

H

G

G = H - T S

Driving Force for Solidification

EnthalpyH

T(K)

298

0

slope = Cp

G

slope = -S

liquid stable

H (liquid)

G(liquid)

Tm

H (solid)L

G(solid)

solid stable

G

Temperature

Stable solid

Stable liquid

solid

Tm

Driving Force for Solidification

G

Temperature

Stable solid

Stable liquid

solid

liquid

Tm

GS

!G

GL

T

!T

GL = HL -TSL

GS = HS -TSS

∆G = ∆H -T∆S

∆H = HL - HS

∆S = SL - SSAt equilibrium ∆G = 0

∆G = ∆H -Tm∆S = 0

entropy of fusion (nearly a constant ≈ R)

for small ∆T we get

= LTm

∆Sf = ∆HfTm

∆G ≈ L -TLTm

∆G ≈ L ∆T

Tm

≈ ∆Sf ∆T

undercooling

Solidification

Growth: occurs as atoms from the liquid are attached to the tiny solid until no liquid remains

To get the solidification process started, the liquid phase must be undercooled, cooled to a temperature below the freezing point.

G

Temperature

Stable solid

Stable liquid

solid

liquid

Tm

GS

!G

GL

T

!T

Nucleation: occurs when a small piece of solid forms in the liquid and must attain a minimum critical size before it is stable

Controlled by:thermal conductivitiesrelative massesshapes of the melt, the solid, and mold

Once a nucleus forms, it can proceed to grow as fast as the latent heat of solidification and specific heat can be carried away.