PDF "Advanced Physical Metallurgy" - Seoul National University

1

Eun Soo Park

Office: 33-313

Telephone: 880-7221 Email: espark@snu.ac.kr

Office hours: by appointment

2019 Fall

10.10.2019

“Advanced Physical Metallurgy”

- Non-equilibrium Solidification -

2019 Fall

Q1: Theories for the glass transition

3

by thermodynamic origin, 2^ndorder transition

• Glass transition

H , V , S : continuous C_p α_T K_T : discontinuous

A. Thermodynamic phase transition

• Heat capacity dramatic change at T_g

B. Entropy

∫

= C d T S

_P

ln

• Description of glass transition by entropy (Kauzmann)

In fact, it appears on some evidences that the glass

transition is not a simple second-order phase transition.

The slow cooling rate, the lower T_g

Measurement of Kauzmann temp. is almost impossible.

( very slow cooling rate longer relaxation time crystallization )



T_K or T_g⁰

Theories for the glass transition

) 1

( ² ≠

∆

∆

= ∆

T P T

TV R C

α κ

: property of liquid-like structure suddenly changes to that of solid-like structure

C. Relaxation behavior

If (1) > (2) liquid // (1)~(2) glass transition// (1) < (2) glass (A concept of glass transition based on kinetic view point)

Below glass transition: frozen-in liquid

glass transition is observed when the experimental time scale

becomes comparable with the time scale for atom/molecule arrangement

(1) (2)

Theories for the glass transition

d. viscosity

• Viscosity (10¹⁵ centiPoise= 10^12-13 Pa s) at T_g

• most glass forming liquid exhibit high viscosity.

Fragility concept: Strong vs Fragile

• In glass transition region, viscosity suddenly changes. (fragile glass)

• Viscous flow Several atomistic model ^• absolute rate model

• free volume model

• excess entropy model

5 Fragile network glass : Vogel-Fulcher relation

Strong network glass : Arrhenius behavior

]

0 exp[

RT E_a η

η =

] exp[

0

0 T T

B

=η − η

< Quantification of Fragility >

Fragility

g n

g g T T

T n T

g d T T

T d

T T

d

T m d

=

=

=

= ( / )

) ( log )

/ (

) ( log

,

,

τ η

< Classification of glass >

Fragility ~ ability of the liquid to withstand changes in medium range order with temp.

~ extensively use to figure out liquid dynamics and glass properties corresponding to “frozen” liquid state

Slope of the logarithm of viscosity, η (or structural relaxation time, τ ) at T_g

6

T

_g

T increase

Supercooled liquid

10¹²~10¹⁴ poise

Q2: Glass formation

Glass Formation results when

Liquids are cooled to below T_m (T_L) sufficiently fast to avoid crystallization.

Nucleation of crystalline seeds are avoided

Growth of Nuclei into crystallites (crystals) is avoided

Liquid is “frustrated” by internal structure that hinders both events

“Glass Formation”

Critical cooling rate is inversely proportional to the diameter of ingot.

Critical Cooling Rates for Various Liquids

Nucleation and Growth Rates Control R

_c

 Nucleation, the first step…

 First process is for microscopic clusters (nuclei) of atoms or ions to form

 Nuclei possess the beginningsof the structure of the crystal

 Only limited diffusionis necessary

 Thermodynamic driving force for crystallizationmust be present

12

1.2.3 Driving force for solidification

*

2 r T

T

G L ^SL

m

= γ

= ∆

∆

Liquid Solid

solidification

SL V

r r G r

G π ³ 4π ² γ

3

4 ∆ +

−

=

∆

13 Fig. 4.5 The variation of r* and r_max with undercooling ∆T

∆T↑ → r* ↓

→ rmax↑

The creation of a critical nucleus ~ thermally activated process

of atomic cluster

* SL SL m

V V

2 2 T 1

r G L T

γ

^

γ

^

= ∆ =   ∆

Tm

T G = L∆

∆

∆TN is the critical undercooling for homogeneous nucleation.

The number of clusters with r^* at ∆T < ∆T_N is negligible.

③

14

θ

VA

VB

Barrier of Heterogeneous Nucleation

4

) cos

cos 3

2 ( 3

) 16 3 (

* 16

3 2

3 2

3

θ πγ θ θ

πγ − +

∆ ⋅

=

∆ ⋅

=

∆

V SL V

SL

S G G G

θ θ

 − + 

∆ = ∆  

 

3

* 2 3 cos cos

4

*

sub homo

G G

2 3 cos cos3

4 ( )

A

A B

V S

V V

θ θ θ

− +

= =

+

How about the nucleation at the crevice or at the edge?

* *

( )

hom

G

het

S θ G

∆ = ∆

Growth of crystals from nuclei

 Growth processes then enlarge existing nuclei

 Smallest nuclei often redissolve

 Larger nuclei can get larger

 Thermodynamics favors the formation of larger nuclei

16

Kinetic Roughening

Rough interface

-

Ideal Growth

Smooth interface - Growth by Screw Dislocation Growth by 2-D Nucleation

The growth rate of the singular interface cannot be higher than ideal growth rate.

When the growth rate of the singular Interface is high enough, it follows the ideal growth rate like a rough interface.

→ kinetic roughening

→ dendritic growth

→ diffusion-controlled

Large ΔT → cellular/dendritic growth Small ΔT → “feather” type of growth

Nucleation and Growth Control R

_c

 Poor glass formers:

 Liquids which quickly form large numbers of nuclei close to T_m

 That grow very quickly

 Good glass formers

 Liquids that are sluggish to form nuclei even far below T_m

 That grow very slowly

Nucleation and Growth Rates – Poor Glass Formers

 Strong overlap of growth and nucleation rates

 Nucleation rate is high

 Growth rate is high

 Both are high at the same temperature

T

_m

T

Rate Growth Rate (m/sec)

Nucleation Rate (#/cm

³

-sec)

Nucleation and Growth Rates – Good Glass Formers

 No overlap of growth and nucleation rates

 Nucleation rate is small

 Growth rate is small

 At any one temperature one of the two is zero

T

_m

T

Rate

Growth Rate (m/sec)

Nucleation Rate (#/cm

³

-sec)

Q3: Classical Nucleation Theory –TTT diagram

Nucleation Rate Theory

 Rate at which atoms or ions in the liquid organize into microscopic crystals, nuclei

 I = number of nuclei formed per unit time per unit volume of liquid

 Nucleation Rate (I) ∝ number density of atoms x fastest motion possible x

thermodynamic probability of formation x

diffusion probability

Nucleation Rate Theory

I = nνexp(-NW

^*

/RT)exp(-∆E

_D

/RT)

n = number density of atoms, molecules, or formula units per unit volume

= ρ N/Atomic, molecular, formula weight ν = vibration frequency ~ 10¹³ sec^-1

N = Avogadro’s number

= 6.023 x 10²³ atoms/mole

W^* = thermodynamic energy barrier to form nuclei

∆E_D = diffusion energy barrier to form nuclei

~ viscosity activation energy

Number density Fastest motion Thermodynamic probability Diffusion probability

Nucleation Rate – Thermodynamic barrier W*

 At r^*, (∂W(r)/ ∂r)_r=r* = 0

 r^* = - 2σ/ ∆G_cryst(T)

 W(r^*) ≡ W^* = 16π σ³/3(∆G_cryst(T))²

r

W

_S

= 4πr

²

σ, surface

σ is the surface energy

W

_B

= 4/3πr

³

∆G

_crsyt

(T), bulk

∆G_cryst(T), the Gibb’s Free-Energy of Cryst. per unit volume, V_m

W

_tot

= W

_S

+ W

_B

W

^*

r

^*

+

-

0

Nucleation Rate I(T)

 I = nνexp(-N 16π σ³/3(∆G_crsyt(T))² /RT)exp(-∆E_D/RT)

 ∆G_cryst(T) = ∆H_cryst(T_m )(1 – T/T_m)/V_m ≡ ∆H_cryst(T_m )(∆T_m/T_m)

∆G

_cryst

(T) 0

+

-

T

_m

Approx. for σ:

σ~ 1/3∆H

_melt

/N

^1/3

V

_m^2/3

note ∆H

_melt

= - ∆H

_cryst

 



 

 ∆ −



 





 

 



 





 ∆



 



 ∆

= RT

E T

T RT

n H

I

^{cryst m}

exp

^D

81 exp 16

π

2

ν

Liquid is Stable

Crystal is Stable

Liquid and Crystal are in equilibrium

Growth Rates - µ(T)

 Crystal growth requires

 Diffusion to the nuclei surface

 Crystallization onto the exposed crystal lattice

∆G

_cryst

∆E

_D

ν

_lc

= νexp(-∆E

_D

/RT)

ν

_cl

= νexp(-(∆E

_D

- ∆G

_cryst

) /RT) ν

_net

= ν

_lc

- ν

_cl

=

νexp(-∆E

_D

/RT) -

νexp(-(∆E

_D

- ∆G

_cryst

) /RT)

µ = a ν

_net

= a ν exp(-∆E

_D

/RT) x

(1 – exp(∆G

_cryst

) /RT)

Growth Rates - µ(T)

Diffusion coefficient, D

Stokes-Einstein relation between D and η

Hence:

 





 



 



 



 



 



  ∆



 



−  ∆

 

 



= 

m m

T T RT

H T

a N

T fRT 1 exp

) ( ) 3

(

₂

η µ π



 



= 



 ∆−

= exp 3 ( )

)

( ²

T a

N

fRT RT

a E T

D ^D

η

ν π

Nucleation and Growth Rates

Nucleation and Growth Rates for Water

0 5E+22 1E+23 1.5E+23 2E+23 2.5E+23 3E+23 3.5E+23 4E+23 4.5E+23

100 150 200 250 300

Temperature (K) Nucleation rate (sec-1 )

0.E+00 2.E-09 4.E-09 6.E-09 8.E-09 1.E-08 1.E-08 1.E-08

Growth rate (m-sec-1 )

Nucleation rate(sec-1) Grow th rate (m/sec)

Nucleation and Growth Rates

Nulceation and Growth for Silica

0.E+00 2.E+07 4.E+07 6.E+07 8.E+07 1.E+08 1.E+08 1.E+08

1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100

Temperature (K)

Nucleation Rate (sec-1 )

0 5E-26 1E-25 1.5E-25 2E-25 2.5E-25

Growth Rate (m-sec-1 )

Nucleation rate (sec-1) Growth rate (m/sec)

Time–Temperature–Transformation Curves (TTT)



How much time does it take at any one temperature for a given fraction of the liquid to transform

(nucleate and grow) into a crystal?



f (t,T) ~ π I(T) µ (T)

³

t

⁴

/3

where f is the fractional volume of crystals formed, typically taken to be 10

^-6

, a barely observable crystal volume.







 ∆−











 



 





 ∆



 



 ∆

= RT

E T

T RT

n H

I ^{cryst m} exp ^D

81 exp 16

π 2

ν 









 



 



 

 



 ∆



 



−  ∆



 



=

m m

T T RT

H T

a N

T fRT 1 exp

) ( ) 3

( ₂

η µ π

Nucleation rates Growth rates

Time Transformation Curves for Water T-T-T Curve for water

100 150 200 250

0 1 2 3 4 5 6 7 8 9 10

Time (sec)

Temperature (K)

Time Transformation Curves for Silica

T-T-T Curve for Silica

200 400 600 800 1000 1200 1400 1600 1800 2000

1 1E+13 1E+26 1E+39 1E+52 1E+65 1E+78

Time (sec)

Temperature (K)

TTT curves and the critical cooling rate, R

_c

T T

_m

time

R

_c

very fast R

_c

much slower

Poor glass former

Better glass former

T_rg 1/4 1/2 2/3 33

R_c = 10¹⁰ K/s

R_c = 10⁶ K/s

R_c = 3x10³ K/s R_c = 3.5x10¹ K/s