Advanced Physical Metallurgy Advanced Physical Metallurgy

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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

04. 06. 2009

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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

Strong liquid vs. Fragile liquid

• Strong glass forming liquid covalent bond of SiO₂

small difference of C_p between SCL and glass at T_g

(small difference of structure)

SCL: relatively low entropy

• fragile glass forming liquid non-directional bonding

(Van der waals bonding)

large difference of C_p at T_g

SCL: relatively high entropy

(relatively large free volume)

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Use of Gibbs energy curves to construct a binary phase diagram of the eutectic type.

High temperature

low temperature

T₁ T₂ T₃

T₄ T₅

Liquid: metastable glass

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Endo.

Exo. heating

Glass SCL crystal Liquid

Glass Supercooled

liquid

liquid cooling

T_m T_g

T_x

T_p

T_s T_L

Glass transition

: Endo.

Crystallization

: Exo.

melting

: Endo.

C+L

< Thermal map >

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< Characteristics of metallic glass >

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Theories for the glass transition Theories for the glass transition

• • 4 viewpoints for Glass transition 4 viewpoints for Glass transition

Viscosity Viscosity Entropy Entropy

Free volume Free volume

Relaxation behavior Relaxation behavior

Kinetic variables depending on time

by thermodynamic origin, 2^nd order transition

H , V , S :

continuous

C

_p

α

_T

K

_T: discontinuous glass transition can be realized by evaluating one of factors.

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Theories for the glass transition Theories for the glass transition

by thermodynamic origin, 2^nd order transition But,

••

Glass transition Glass transition

H , V , S :

continuous

C

_p

α

_T

_K

_T_: discontinuous

1) Tg is dependent on thermal history of sample.

If 2^nd order transition,

Tg is not changed by kinetic factor.

A. Thermodynamic phase transition

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But,

2) Thermodynamic consideration

S , V : continuous

(a) continuity of S

S

₁

= S

₂ Entropies of the high- and low-temp. forms must be equal at the transition

dS₁

= dS

₂

with respect to temperature and pressure,

in terms of partial derivatives,

or

dP

P dT S

T dP S

P dT S

T S

T P

T

P

( ) ( ) ( )

)

(

¹ ¹ ² ²

∂ + ∂

∂

= ∂

∂ + ∂

∂

P T

P

T

V V T

V P

S T

T S

C ( ) , ( ) ( ) , 1 / ( )

∂

= ∂

∂

− ∂

∂ =

∂

= ∂ α

using one of Maxwell’s thermodynamic relations

Theories for the glass transition

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9 P

T P

P

T T

g

C TV C

C TV dP

dT

Δ

= Δ

−

= −

∴ α α α

) (

1 2

1

2 measureable

(1)

dP V

T dT dP C

V T dT

C

T p

2 2

1

+ α = − α

dP V

dT C

T 1 ( C

^P ^P

) (

^T ^T

)

2 1

2

1

− = α − α

Theories for the glass transition

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But,

2) Thermodynamic consideration (b) continuity of V

V

₁

= V

₂

& d V

₁

= d V

₂

T g T

dP dT

α κ Δ

= Δ

∴ ⁽²⁾

P dP dT V

T dP V

P dT V

T V

T P

T

P

( ) ( ) ( )

)

(

¹ ¹ ² ²

∂ + ∂

∂

= ∂

∂ + ∂

∂

T T

P

T

P

V V

T

V V 1 ( )

, ) (

/

1 ∂

− ∂

∂ =

= ∂ κ

α

dP V

dT V

dP V

dT

V α

_T₁

− κ

_T₁

= α

_T₂

− κ

_T₂

dP

dT

_T _T

T

) ( )

( α

₁

− α

₂

= κ

₁

− κ

₂

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Contents for previous class Contents for previous class

Eq. (1) & (2) should be proved experimentally.

T g T

dP dT

α κ Δ

= Δ

∴ ⁽²⁾

It is found by measuring the discontinuities Δα_T, ΔC_P, Δκ_T at the glass transition that Eq. (1) is almost always obeyed within experimental error, but that values for Δκ_T/Δα_T are generally appreciably higher than those of dT_g/dP (Eq. (2)).

Eq. (1) = satisfy Eq. (2) = dissatisfy :

T g T

dP dT

α κ Δ

< Δ

Therefore, it appears on this evidence that the glass transition is not a simple second-order phase transition.

P T P

P

T g T

C TV C

C TV dP

dT

Δ

= Δ

−

= −

∴ α α α

) (

1 2

measureable

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By combining both Ehrenfest equations without invoking a second-order thermodynamic transition,

Prigogine Defay Ratio:R

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) 1

(

²

=

Δ Δ

= Δ

T P T

TV R C

α κ

) 1 (

²

>

Δ Δ

= Δ

T P T

TV R C

α κ

T g T

dP dT

α κ Δ

= Δ

If a single ordering parameter determines

the position of equilibrium in a relaxing system, If more than one ordering parameter is responsible,

The latter case seems to describe most glasses.

Goldstein (1973) has suggested that

“ The specific volume V_g of the glass depends not only on the temperature,

being continuous through the transition, but also on the pressure of formation”

Jäckle (1989) has shown that

T

f g

T

g

V p

dP dT

α κ

Δ

∂

∂ +

= Δ (ln ) /

Additional consequence of the experimental verification,

“ Glasses prepared under high pressures to have higher than normal densities but normal entropies or enthalpies. ”

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Theories for the glass transition Theories for the glass transition

••

Description of glass transition by entropy Description of glass transition by entropy

((KauzmannKauzmann))

1) Heat capacity

dramatic change at Tg

B. Entropy B. Entropy

T/T

_g

1.0

Supercooled liquid

crystal

T < T

_g

crystal

glass P

P

C

C ≈

T > T

_g

crystal

SCL P

P

C

C >

( configurational degree of freedom in S.C.L. )

Q

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Theories for the glass transition Theories for the glass transition

2) The slow cooling rate, the lower Tg B. Entropy

B. Entropy

ideal glass transition temperature exist?

YES

∫

= C d T S

_P

ln

Entropy of fusion

The data are plotted against ln T so that integrated areas under the curves yield entropies directly, and the entropy of fusion is shown shaded in the upper part of the figure.

Heat capacities of glassy, liquid and

••

Description of glass transition by entropy ( Description of glass transition by entropy

(KauzmannKauzmann))

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Theories for the glass transition Theories for the glass transition

B. Entropy B. Entropy

••

Description of glass transition by entropy ( Description of glass transition by entropy

∫

= C d T

S

_P

ln

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Theories for the glass transition Theories for the glass transition

2) The slow cooling rate, the lower Tg B. Entropy

B. Entropy

The difference in entropy between liquid and crystalline phases as a function of temperature

The vanishing excess entropy temperature Is termed the “ideal’ glass transition temp.

T_oc(Wong and Angell 1976)

••

Description of glass transition by entropy ( Description of glass transition by entropy

Not satisfied with

third law of thermodynamics

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Variation of (a) C_p and (b) excess entropy, S depending on temp. for glass, crystal and liquid. Ideal glass transition temp, T_oc. is the temperature when excess entropy is disappeared.

• Ideal glass transition temperature (T

_oc

)

.lower temperature limit to occur glass transition thermodynamically

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T_K S_liq-S_cry

T_m S.C.L

••

Kauzmann’ Kauzmann ’s s paradox paradox

Measurement of Kauzmann temp. is almost impossible.

( very slow cooling rate longer relaxation time crystallization

Q

Thermodynamics: The configurational entropy

apparently extrapolates to zero at low temperatures.

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; the investigation of energy transformations

;

that accompany physical and chemical changes in matter

; use to evaluate the flow and interchanges of matter and energy

;

Thermodynamics

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How does thermodynamics different from kinetics?

Thermodynamics

says which process is possible or not and never says how long it will take.

The existence of a thermodynamic driving force does not mean that the reaction will necessarily occur!!!

There is a driving force for diamond to convert to graphite but there is (huge) nucleation barrier.

There is no time variable.

How long it will take is the problem of kinetics

^.

The time variable is a key parameter.

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The The second second law of thermodynamics law of thermodynamics

; The disorder of the universe always

; increase

; all chemical and physical processes occur spontaneously

;

only when disorder is increased.

S q

= T

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(1) All energy received as heat by a heat engine cycle cannot be converted into work (This means that no cycle can have a

thermal efficiency of 100%)

(2) The transformation of heat to work is dependent of a

temperature difference and on the flow of heat from a high temperature reservoir to a low temparature reservoir to a low temperature reservoir. (In other word heat must flow from hot to cold)

(3) It is impossible to construct an engine that, operating in a

cycle, will produce no effect other than the transfer of heat from

a low temperature reservoir to a hight temperature reservoir.

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; but it can be transformed from one into another. ;

; The total amount of energy in the universe is constant ;

for any physical and chemical change.

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(

_i _i

)

dE = δ Q − δ w d + ∑ μ N

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• In brief, this postulates that entropy is temperature

dependent and leads to the formulation of the idea

of absolute zero.

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