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Enthalpy Landscape Analysis for Calculating the Melting Temperature of a Material

Alex M. Nieves, Vaclav Vitek, and Talid Sinno

University of Pennsylvania

AIChE 2010 Annual Meeting

Salt Lake City, Utah

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Importance of Melting

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Process Conditions & Material Selection and Design

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simplystone.com.au

Theoretical Understanding

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HOW?

WHY?

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Thermodynamic definition of melting

Thermodynamics defines Melting as a first order phase transition:

Crystalline Phase long range

ordered state

Liquid Phaseshort range

disordered state

The Thermodynamic Melting Temperature is obtained by

comparing the Free Energy of the two phases:

SolidPhase

LiquidPhase

T

Fre

e E

nerg

y

TM

MH∆

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MSurfM TT <

Thermodynamic Melting “Heterogeneous Melting”

Heterogeneous melting mechanism at TM when a Liquid Phase already exits

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Homogeneous Melting

• Perfect bulk material:– No surfaces– No dislocations or grain boundaries– No missing or extra atoms

• No sources to nucleate liquid phase at TM. The material

stays in a metastable solid phase at T > TM – also called “superheating”

I-V PairIn perfect crystals, the “liquid phase” must be

generated from within the bulk lattice.

Frenkel Pairs

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• Lindemann Criterion:

• Born Criterion states that the shear moduli should vanish as TM is approached.

Empirical Observations of Material Behavior for Homogenous Melting

CNNamsd δ2≥

• Forsblom et al. [1] correlated melting to the formation of a critical sized aggregate of interstitials and vacancies.

Defect Cluster

[1] Forsblom et al., Nature Materials 4 (2005)

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Theories for the Limit of Superheating

Lu & Li, Phys Rev Lett 80 (1998)

En

tro

py

Cat

astr

op

he

(198

8)

Rig

idit

y C

atas

tro

ph

e (1

989

)

Vo

lum

e C

atas

tro

ph

e (1

989)

KTM 933=

Fetch and Johnson, Nature 334 (1988)

Molar Volume of the Crystalline Phase and

Liquid Phase are equal.

Rigidity of the Crystalline Phase at TS is equal to Rigidity of the Liquid

Phase at TM

Tallon, Nature 342 (1989)

Tallon, Nature 342 (1989)

Classical Nucleation Theory gives nucleation rate of a liquid sphere in a crystalline bulk material

Cla

ssic

al N

ucl

eati

on

Th

eory

(1

998)

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“Idealized” Energy Landscape

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The “Real” Energy Landscape

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Sampling the Energy Landscape

Discretization of continuous energy landscape into energy minima basins

known as Inherent Structures.

Sample energy landscape with MD. Histogram the visited basin minima

by quenching the system.

The energy of each basin is defined here by the formation energy of the inherent structure.

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Formation Energies

PD VVV −=∆Formation Volume of a Defect:

VPEHHH PD ∆+∆=−=∆Formation Enthalpy of a Defect:

=∆=∆

V

E

PD EEE −=∆Formation Energy of a Defect:

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∆E (eV)

Pro

b

0 10 20 3010-5

10-4

10-3

10-2

10-1

100

Calculation of the Density of States Using MD Simulations

( ) ( )

∆−∆≡∆kT

HHgHP exp

Density of States: Number of basins

with energy ΔH

Boltzmann Factor: Probability of being at

basin with energy ΔH at T

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Effect of density of states on probability distribution

0H

1H

2H

3H

( )Hg ∆

ΔH

If rate of increase of g(ΔH) is lower than rate of decrease of exp (-ΔH/kT)

ΔH

( )HP ∆

Negative Slope

If rate of increase of g(ΔH) is higher than rate of decrease of exp (-ΔH/kT)

( )EP ∆

Positive Slope

ΔH

If rate of increase of g(ΔH) is equal to rate of decrease of exp (-ΔH/kT)

( )HP ∆

Zero Slope

ΔH

Number of states increases

with enthalpy

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∆E (eV)

Ln

(g)

0 10 20 300

50

100

150

Extracting an Effective Temperature from the Density of States Function

Density of States Calculation:

∆

effkT

Hexp

( ) ( )

∆≈

∆∆=∆

effkT

H

kT

HHpHg expexp

Density of States Growth Rate can be approximated by an

exponential fit.

An effective temperature, Teff, can be extracted.

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( )

−∆

−≈

∆∂∆∂

effsim

effsim

effsim

effsim

TT

TT

k

H

TkT

TT

H

HPexp

Revisiting the Probability Distribution Curve

( )effsim TT

H

HP −∝∆∂∆∂

( ) ( )

−∆=

∆−

∆≈

∆−∆=∆effsim

effsim

simeffsim TT

TT

k

H

kT

H

kT

H

kT

HHgHP expexpexpexp

Giving:

Using this approximation for the density of states the probability becomes:

Slope of the Probability Distribution is:

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Determining Melting Temperature from Teff

( )effsim TT

H

HP −∝∆∂∆∂

∆E (eV)

Ln

(g)

0 10 20 300

50

100

150 effkT1

Assuming that the slope of g(ΔH) remains constant:

At Tsim > Teff, P(ΔH) becomes unbounded with increasing energy, giving access to liquid states.

Tsim > Teff

Tsim = Teff

Tsim < Teff

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Number of Atoms

T(K

)

10000 20000 300001500

1600

1700

1800

1900

2000

Melting Temperatures Calculated at Different System Sizes

MTMDTeffT

Number of Atoms

T(K

)

10000 20000 300001500

1600

1700

1800

1900

2000

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Effect of Pressure on Melting

Clausius-Clapeyron equation gives the slope of the melting curve:

M

M

VT

H

dT

dP

∆∆=

Young, Phase Diagrams of the Elements. (1991)

ΔVM > 0 for most materials ΔVM < 0 for the few materials

−∆

−∆

M

M

V

H Latent Heat of Melting

Change in Volume during Melting

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∆H (eV)

ln(g

)

5 10 15 200

50

100

150

Interpreting the Effect of Pressure on the Density of States

0>∆ MVAluminum:

P = 8GPa

P = 5GPa

P = 3GPa

P = 0GPa

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Interpreting the Effect of Pressure on the Density of States

0<∆ MVSilicon:

∆H (eV)

ln(g

)

12 14 16 18 20

80

100

120

P = 8GPa

P = 5GPa

P = 3GPa

P = 0GPa

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Effect of ΔVM on the Mechanism of Melting Initiated at a Void

Silicon (ΔVM < 0)

Aluminum (ΔVM > 0)

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Effect of Void Volume Fraction on the Superheating Melting Temperature

– Different behavior during melting depending on the sign of ΔVM.

– Similar effect of void volume fraction on the superheating melting temperature.

– Can density of states analysis provide us with insight on why the superheating melting mechanism does not appear to be affected by the sign of ΔVM?

Void Volume Fraction

TS*

0.00 0.01 0.02 0.03 0.041.00

1.05

1.10

1.15

1.20

1.25

1.30

Si - EDIP

Si - Tersoff

Al - EAM

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Summary

∆E (eV)

Ln

(g)

0 10 20 300

50

100

150 effkT1

Number of Atoms

T(K

)

10000 20000 300001500

1600

1700

1800

1900

2000

∆H (eV)

ln(g

)

5 10 15 200

50

100

150

∆H (eV)ln

(g)

12 14 16 18 20

80

100

120

Other ApplicationsPressure Effect on MeltingMelting Temperature from Density of States

Melting of:

• Alloyed Materials

• Interstitial Clusters

• Dislocation / Grain Boundaries

Tsim > Teff

Tsim = Teff

Tsim < Teff

MTMDTeffT

Number of Atoms

T(K

)

10000 20000 300001500

1600

1700

1800

1900

2000

Al

Si

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Acknowledgements

Funding from NSF-NIRT

Thanks to:

Advisors:

Prof. Talid Sinno

Prof. Vaclav Vitek

The Sinno group:

Sumeet Kapur PhD

Matthew Flamm

Xiao Liu

Yung-Chi Chuang

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System Size Effect

• Log decay on Teff due to system size effect.

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“Pretty” Picture of Melting

Crystal Phase

Liquid Phase

Real Picture of Melting

E∆

Crystal Phase

Liquid Phase

E∆

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Using the Enthalpy Landscape Analysis to Study Melting Stages of a Void

Teff

Teff

Teff homogeneous