“ “ Amorphous Materials Amorphous Materials ” ”

1

Advanced Physical Metallurgy Advanced Physical Metallurgy

“ “ Amorphous Materials Amorphous Materials ” ”

Eun Eun Soo Soo Park Park

Office: 33-316

Telephone: 880-7221

Email: espark@snu.ac.kr

Office hours: by an appointment

2009 spring

03.11.2009

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What kind of randomness exist in amorphous materials?

Contents for previous class Contents for previous class

Crystal - disorder

Topological disorder Spin disorder

Substitutional disorder Vibrational disorder Random atomic structure

Crystal lattice

3

Quasicrystal: orientational order

long range order: quasiperiodic

-

-

two types of unit cell: acute & obtuse rhombi

Interpretation by penrose tiling

-

Contents for previous class

Contents for previous class

4

Perfect crystal disorder quasicrystal amorphous

crystalline lattice (b, c, d)

ex) icosahedral phase : no topological ordering (a)

What kind of randomness exist in amorphous materials?

Amorphous, non-crystalline, glass

- amorphous: do not possess long range order

= non-crystalline

- glass : amorphous materials which exhibits glass transition

Contents for previous class Contents for previous class

Formation of glass is not necessarily to be rapid quenching.

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Contents for today

Contents for today ’ ’ s class s class

• Glass transition

• Glass: Solid? or liquid?

• Free volume and the glass transition

- Classification of phase transition

1) Microstructural observation 2) Thermal analysis

- DSC (Differential Scanning Calorimetry)

• Amorphous vs Nanocrystalline

: Characterization of structure by pair distribution function

local clusters with atomic scale are difficult to identify by conventional observation tools of microstructure.

6

Glass

Glass : : subset of amorphous materials subset of amorphous materials

glass

glass Transition

reference?

7

Free volume and the glass transition Free volume and the glass transition

Free volume = specific volume (volume per unit mass) of glass - specific volume of the corresponding crystal

At the glass transition temperature, T

, the free volume increases

leading to atomic mobility and liquid-like behavior. Below the glass

transition temperature atoms (ions) are not mobile and the material

behaves like solid

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Silica - SiO

Amorphous silica Crystalline SiO

Si

O

9

Glass transition Glass transition

¾ Within the free volume theory, it is understood that with large enough free volume mobility is high, viscosity is low. When the temperature is decreased free volume becomes “critically” small and the system “jams up”.

¾ The glass transition is not first order transition (such as melting), meaning there is no discontinuity in the thermodynamic functions (energy, entropy, density).

¾ Typically T_g is ~ 50-60 % of the melting point

¾ The effective glass transition temperature is a function of cooling rate; higher rate → higher T_g. It is also called the fictive temperature.

¾ Sometimes the glass transition it is a first order transition, most prominently in Si where the structure changes from 4 coordinated amorphous solid to ~ six coordinated liquid. The same applies to water (amorphous ice).

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Classification of phase transition

Classification of phase transition

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The First-Order Transitions

G

T

solid

liquid

gas

P,N = const

(Pr. 5.9).

On the graph G(T) at P,N = const, the slope dG/dT is always negative:

N

T P

S G

,

⎟⎠

⎜ ⎞

⎝

⎛

∂

− ∂

=

G

T S

T

ΔS = L/T

T C_P

N P

P T

T S C

,

⎟⎠

⎜ ⎞

⎝

⎛

∂

= ∂ In the first-order transitions, the

G(T) curves have a real meaning even beyond the intersection point, this results in metastability and hysteresis.

An energy barrier that prevents a transition from the higher μ to the lower μ phase. (e.g., gas, being cooled below T_tr does not immediately condense, since surface energy makes the formation of very small droplets energetically unfavorable).

Water in organic cells can avoid freezing down to –20⁰C in insects and down to –47⁰C in plants.

Latent heat Energy barrier

Discontinuous entropy, heat capacity

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13

The Second Order Transition

As one moves along the coexistence curve toward the critical point, the distinction between the liquid phase on one side and the gas phase on the other gradually decreases and finally disappears at (T_C, P_C). The T-driven phase transition that occurs exactly at the critical point is called a second-order phase transition. Unlike the 1^st-order transitions, the 2^nd-order transition does not require any latent heat (L=0). In the second-order transitions (order-disorder transitions or critical phenomena) the entropy is continuous across the transition. The specific heat C_P =T(δS/δT)_P diverges at the transition.

Whereas in the 1^st-order transitions the G(T) curves have a real meaning even beyond the intersection point, nothing of the sort can occur for a 2^nd-order transition – the Gibbs free energy is a continuous function around the critical temperature.

G

T S

T

ΔS=0

Second-order transition

T C_P

No Latent heat

Continuous entropy

∞

⎟ →

⎠

⎜ ⎞

⎝

⎛

∂

= ∂

N P

P T

T S C

,

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Glass

Glass : undercooled ^: undercooled liquid with high viscosity liquid with high viscosity

A solid is a materials whose viscosity exceeds 10

A solid is a materials whose viscosity exceeds 10^14.614.6 poisepoise cf) liquid ~cf) liquid ~1010^--22 poisepoise

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Relaxation time of one day Relaxation time of one day

) /( dz G

dv

η =

Gx

dz dv

: Shear stress in x direction causing velocity gradient:

dz: thickness of element perpendicular to the applied stress

z

x

x

y

G

ex) 100 N applies for one day to 1 cm

yield deformation of 0.02 mm

small stress

just measurable

Solid

Solid

produces no permanent change.

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Amorphous vs Nanocrystalline

1) Microstructural observation XRD, (HR)TEM, EXAFS …

2) Thermal analysis

DSC (Differential Scanning Calorimetry)

cf) - glass nucleation & growth (perfect random)

- local clustering: quenched-in nuclei only growth - Nanocrystalline growth

local clusters with atomic scale are difficult to identify by conventional observation tools of microstructure.

: Measure heat absorbed or liberated during heating or cooling

: Characterization of structure by pair distribution function

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Glass

Glass : diffraction

¾ diffuse halo

¾ X-ray: electron diffraction

Crystalline

Crystalline :

sharp diffraction peaks

¾ grain size nanoscle

~ continuous ring pattern

~ sharp diffuse halo

5 nm

Formation of nano crystalline

Formation of nano crystalline (2~3 nm) : Laves phase

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[2211]_Laves (1120) (0132)

20

20 30 40 50 60 70 80

●

as-spun

★ ★

★

●

●

●

●

◆

◆

★

◆

●

CuTi

●

51Zr₁₄ Ti

◆ ₂Ni

510℃

560℃

640℃

680℃/

10 min annealed

as-spun Cu

Ti

Zr

Ni

Si

Relative Intensity (a.u.)

2

[deg.]

Crystallization after annealing

Crystallization after annealing

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Amorphous vs Nanocrystalline

1) Microstructural observation XRD, (HR)TEM, EXAFS …

2) Thermal analysis

DSC (Differential Scanning Calorimetry)

cf) - glass nucleation & growth (perfect random)

- local clustering: quenched-in nuclei only growth - Nanocrystalline growth

local clusters with atomic scale are difficult to identify by conventional observation tools of microstructure.

: Measure heat absorbed or liberated during heating or cooling

: Characterization of structure by pair distribution function

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Definitions

• A calorimeter measures the heat into or out of a sample.

• A differential calorimeter measures the heat of a sample relative to a reference.

• A differential scanning calorimeter does all of the above and heats the sample with a linear temperature ramp.

• Endothermic heat flows into the sample.

• Exothermic heat flows out of the sample.

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• Differential Scanning Calorimetry (DSC) measures the temperatures and heat flows associated with transitions in materials as a function of time and temperature in a

controlled atmosphere.

• These measurements provide quantitative and qualitative information about physical and chemical changes that

involve endothermic or exothermic processes, or changes in heat capacity.

DSC: The Technique

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Schematic of DSC Instrument Schematic of DSC Instrument

N₂ flow

Pt thermopile

Sample Reference

Pt thermopile

T₁ T₂

heater heater

Low mass 1 gram

ΔW

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Conventional DSC

Metal 1

Metal 2

Metal 1

Metal 2

Sample Empty

Sample

Temperature Reference Temperature

Temperature Difference =

Heat Flow

• A “linear” heating profile even for isothermal methods

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• Glass transitions

• Melting and boiling points

• Crystallization time and temperature

• Percent crystallinity

• Heats of fusion and reactions

• Specific heat capacity

• Oxidative/thermal stability

• Rate and degree of cure

• Reaction kinetics

• Purity

What Can You Measure with DSC?

What Can You Measure with DSC?

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79.70°C(I) 75.41°C

81.80°C

144.72°C 137.58°C 20.30J/g

245.24°C

228.80°C 22.48J/g

Cycle 1

-0.5 0.0 0.5 1.0 1.5

Heat Flow (W/g)

0 50 100 150 200 250 300

Temperature (°C)

Sample: PET80PC20_MM1 1min Size: 23.4300 mg

Method: standard dsc heat-cool-heat Comment: 5/4/06

DSC File: C:...\DSC\Melt Mixed 1\PET80PC20_MM1.001 Operator: SAC

Run Date: 05-Apr-2006 15:34

Instrument: DSC Q1000 V9.4 Build 287

4.2

Example DSC

Example DSC – – PET PET

T

T

T

28

6

DSC DSC Thermogram Thermogram

Temperature Heat Flowexothermic

Glass Transition

Crystallisation

Melting

Cross-Linking (Cure)

Oxidation

exo

endo

• dq_p/dt = heat flow

• dT/dt = heating rate

• (dq_p/dt) / (dT/dt) = dq_p/dT = c_p

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Glass Transition Glass Transition

• Step in thermogram

• Transition from disordered solid to liquid

• Observed in glassy solids, e.g., polymers, metallic glass

• T

Temperature, K Thermogram

dH/dt, mJ/s

Glass transition

T_g

30

6

DSC Thermogram

Temperature Heat Flowexothermic

Glass Transition

Crystallisation

Melting

Cross-Linking (Cure)

Oxidation

exo

endo

• dq_p/dt = heat flow

• dT/dt = heating rate

• (dq_p/dt) / (dT/dt) = dq_p/dT = c_p

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Crystallization Crystallization

• Sharp positive peak

• Disordered to ordered transition

• Material can crystallize!

• Observed in glassy solids, e.g., polymers, metallic glass

• T

• T

Temperature, K Thermogram

Crystallization

dH/dt, mJ/s

T

T

32

6

DSC Thermogram

Temperature Heat Flowexothermic

Glass Transition

Crystallisation

Melting

Cross-Linking (Cure)

Oxidation

exo

endo

• dq_p/dt = heat flow

• dT/dt = heating rate

• (dq_p/dt) / (dT/dt) = dq_p/dT = c_p

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Melting Melting

• Negative peak on thermogram

• Ordered to disordered transition

• T_s, soldus melting temperature

• T_l, liquidus melting temperature

Temperature, K Thermogram

Melting

dH/dt, mJ/s

T

T

6 34

Influence of Sample Mass Influence of Sample Mass

Temperature (°C) 150 152 154 156

0

-2

-4

-6

DSC Heat Flow (W/g)

10mg 4.0mg

15mg

1.7mg 1.0mg 0.6mg Indium at

10°C/minute Normalized Data

158 160 162 164 166

Onset not influenced by mass

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Effect of Heating Rate Effect of Heating Rate

on Indium Melting Temperature on Indium Melting Temperature

154 156 158 160 162 164 166 168 170 -5

-4 -3 -2 -1 0 1

Temperature (° C)

Heat Flow (W/g)

heating rates = 2, 5, 10, 20°C/min

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Amorphous vs Nanocrystalline

1) Microstructural observation XRD, (HR)TEM, EXAFS …

2) Thermal analysis

DSC (Differential Scanning Calorimetry)

cf) - glass nucleation & growth (perfect random)

- local clustering: quenched-in nuclei only growth - Nanocrystalline growth

local clusters with atomic scale are difficult to identify by conventional observation tools of microstructure.

: Measure heat absorbed or liberated during heating or cooling

: Characterization of structure by pair distribution function

37

Chen & Sapepen (Harvard,1988)

Glass :

glass nucleation & growth (perfect random)

Isothermal annealing

: rapid heating + maintain the temp.

) exp(

1 bt

x = − −

Corresponding heat release

)

1

( − ⋅

Δ

=

− H x n bt

dt dH

(ΔH: total transformation enthalpy)

crystallized volume fraction after time t

38

Fig. 1.4 Isothermal enthalpy release rates for crystallite nucleation and growth (solid line) and crystallite grain-coarsening mechanisms (dashed line)

Glass

: exothermic peak at non-zero time

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r

dt M

dr _{= ⋅} γ

(M: atomic mobility, γ : interficial surface tension) corresponding heat release

/

) 0 ( )

0

( ⋅ ⋅

=

− H r M r

dt

dH γ

(H(0): zerotime enthalpy of a grain size of r (0))

Nanocrystalline grain growth

Monotonically decreasing curve

40

Fig. 1.4 Isothermal enthalpy release rates for crystallite nucleation and growth (solid line) and crystallite grain-coarsening mechanisms (dashed line)

Glass

: exothermic peak at non-zero time

Nanocrystalline

(or quenched-in nuclei)

41

Amorphous vs Nanocrystalline

1) Microstructural observation XRD, (HR)TEM, EXAFS …

2) Thermal analysis

DSC (Differential Scanning Calorimetry)

cf) - glass nucleation & growth (perfect random)

- local clustering: quenched-in nuclei only growth - Nanocrystalline growth

local clusters with atomic scale are difficult to identify by conventional observation tools of microstructure.

: Measure heat absorbed or liberated during heating or cooling

: Characterization of structure by pair distribution function

42

Effect of quenched

Effect of quenched - - in in quasicrystal quasicrystal nuclei nuclei

Fully amorphous structure β-Zr particle(~70 nm) in amorphous matrix

50 nm 200 nm

(a)

β-Zr

(b)

(b) Zr₅₇Ti₈Nb_2.5Cu_13.9Ni_11.1Al_7.5 (a) Zr₆₃Ti₅Nb₂Cu_15.8Ni_6.3Al_7.9

2 mm rod

200 nm

I₅ I₃ I₂

I-phase

3 mm rod

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0 20 40 60

3.55 3.60 3.65 3.70

c

Isotherm in DSC

[b]

[b]

[a]

Effect of quenched

Effect of quenched - - in in quasicrystal quasicrystal nuclei nuclei

Isothermal annealing

5 nm

HRTEM image in [b] alloy

45

1 2 3 4 5 6

0.0 0.5 1.0 1.5 2.0

FT[k

3 ·χ(k)]

r( A )

alloy (2) rib.

alloy (2) 1 mm alloy (2) 2 mm alloy (2) 3 mm

(c)

1 2 3 4 5 6

0.00 0.05 0.10 0.15 0.20

r(A ) FT[k2 ·χ(k)]

alloy (2) rib.

alloy (2) 1 mm alloy (2) 2 mm alloy (2) 3 mm

(a)

Zr-K edge

Ni-K edge

1 2 3 4 5 6

0.0 0.5 1.0 1.5 2.0 2.5

r(A ) FT[k3 ·χ(k)]

alloy (2) rib.

alloy (2) 1 mm alloy (2) 2 mm alloy (2) 3 mm

(b)

Cu-K edge

EXAFS analysis

Distinctive structural change around Ni atom

Intensity change due to microstructural change

(b) Zr₅₇Ti₈Nb_2.5Cu_13.9Ni_11.1Al_7.5

Effect of quenched

Effect of quenched - - in in quasicrystal quasicrystal nuclei nuclei

46

Characterizing the structure

Characterizing the structure - - radial distribution function, radial distribution function

also called pair distribution function

Gas, amorphous/liquid and crystal structures have very different radial

distribution function

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Radial distribution function

Radial distribution function - - definition definition

1. Carve a shell of size r and r + dr around a center of an atom. The volume of the shell is dv=4πr

dr 2. Count number of atoms with

centers within the shell (dn) 3. Average over all atoms in the

system

4. Divide by the average atomic density <ρ>

dr

r

g(r) = 1

ρ

dn(r,r + dr)

dv(r,r + dr)

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Properties of the radial distribution function Properties of the radial distribution function

For gases, liquids and amorphous solids g(r) becomes unity for large enough r.

The distance over which g(r) becomes unity is called the correlation distance which is a measure of the extent of so- called short range order (SRO)

The first peak corresponds to an average nearest neighbor distance Features in g(r) for liquids and

amorphous solids are due to packing

(exclude volume) and possibly bonding

characteristics

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Radial Distribution Function

Radial Distribution Function - - Crystal and Liquid Crystal and Liquid

Q(r) = g(r) −1 ~ 1

r sin(r / d + ϕ )exp(r / λ )

-15 -10 -5 0 5 10 15

0 1 2 3 4 5 6 7

rQ(r)

r [σ]

crystal T=1000K fit numerical data

-6 -4 -2 0 2 4 6

0 1 2 3 4 5 6 7 8

rQ(r)

r [σ]

NaCl melt T=1000K ρ*=0.28 fit simulations

Liquid/amorphous g(r), for large r exhibit oscillatory exponential decay Crystal g(r) does not exhibit an exponential decay (λ → ∞)

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Radial distribution functions and the structure factor Radial distribution functions and the structure factor

• The structure factor, S(k), which can be measured experimentally (e.g. by X-rays such as EXAFS) is given by the Fourier transform of the radial distribution function and vice versa

Radial distribution functions can be obtained from experiment and compared with that from the structural model

S(k) =1+ 4 π ρ

k r[g(r) −1]

0

∞

∫ ^sin(kr)dr

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2 4 6 8 10 12

-4 -2 0 2 4

Cu₆₀Zr₄₀ Cu₅₀Zr₅₀

Cu_47.5Zr₄₀Be_12.5

k3 ·χ(k)

k(Å^-1)

Cu K-edge

1 2 3 4

0 1 2 3 4

Cu₆₀Zr₄₀ Cu₅₀Zr₅₀

Cu_47.5Zr₄₀Be_12.5 1/7 Cu foil

FT[k3 ⋅χ(k)]

r (Å)

Cu K-edge

Radial distribution functions

Radial distribution functions Structure factor Structure factor

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More detailed structural characterization

More detailed structural characterization -

-

Voronoi

Polyhedra

• Draw lines between a center of an atom and nearby atoms.

• Construct planes bisecting the lines perpendicularly

• The sets of planes the closest to the central atom forms a convex polyhedron

• Perform the statistical analysis of such constructed polyhedrons, most notably evaluate an average number of faces

54

More detailed structural characterization

More detailed structural characterization -

-

Voronoi

Polyhedra

• For no-directional bonding promoting packing number of faces is large ~ 13-14 (metallic glasses)

• For directional bonding (covalent glasses) number of faces is small

• Ionic glasses - intermediate

• In all cases the number of faces is closely related to the number of nearest neighbors (the coordination number)

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Medium range order and radial distribution function Medium range order and radial distribution function

Radial distribution functions (and also X-ray) of amorphous silicon and model Si with ~ 2 nm crystalline grains are essentially the same - medium range order difficult to see by standard characterization tools.

Such structure is called a paracrystal.

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Radial Distribution Function

Nanocrystalline material

Nanocrystalline materials shows clear crystalline peaks with some background coming from the grain boundary

0 2 4 6 8 10 12 14

0 0.3 0.6 0.9 1.2

G(r)

r [a0]

nanocrystalline 4nm grain size