A short introduction to the phase ﬁeld approach

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A short introduction to the phase field approach

Benoˆıt Appolaire

LEM – Onera/CNRS

2016, feb. 3rd

Benoˆıt Appolaire (LEM) Master DMS B3: Phase field 2016, feb. 3rd 1 / 54

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What is it for?

Karma et al. Khatchaturyan et al. Casademunt et al. Du et al.

For all free boundary problems (potentially)

Phase transformations (solidification, solid states, fluid . . . ) Fluid flow (free surfaces)

Membranes (biology) Cracks . . .

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

start play end

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

start play end

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

start play end

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What is it for?

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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The recipe

1

Identify the order parameter(s)

2

Build the most relevant thermodynamic functional

3

Derive the evolution equations

4

Relate the parameters to physical quantities

5

Solve (this afternoon)

6

Post-process (this afternoon)

(120)

The recipe

1

Identify the order parameter(s)

2

Build the most relevant thermodynamic functional

3

Derive the evolution equations

4

Relate the parameters to physical quantities

5

Solve (this afternoon)

6

Post-process (this afternoon)

(121)

The recipe

1

Identify the order parameter(s)

2

Build the most relevant thermodynamic functional

3

Derive the evolution equations

4

Relate the parameters to physical quantities

5

Solve (this afternoon)

6

Post-process (this afternoon)

(122)

The recipe

1

Identify the order parameter(s)

2

Build the most relevant thermodynamic functional

3

Derive the evolution equations

4

Relate the parameters to physical quantities

5

Solve (this afternoon)

6

Post-process (this afternoon)

(123)

The recipe

1

Identify the order parameter(s)

2

Build the most relevant thermodynamic functional

3

Derive the evolution equations

4

Relate the parameters to physical quantities

5

Solve (this afternoon)

6

Post-process (this afternoon)

(124)

The recipe

1

Identify the order parameter(s)

2

Build the most relevant thermodynamic functional

3

Derive the evolution equations

4

Relate the parameters to physical quantities

5

Solve (this afternoon)

6

Post-process (this afternoon)

(125)

1. What is an order parameter?

(126)

1. What is an order parameter?

1

Introduced in the context of phase transition (not only for phase field)

Physical quantity that discriminates different physical states (gas, liquid, solid, paramagnetic, ferromagnetic . . . )

2

Related to choice of phase diagram to display the different states Gas/liquid [Van der Waals, 1873!] : density

Ferromagnetic/paramagnetic [Weiss, 1907] : magnetization Order/disorder [Bragg-Williams, 1934] : order parameter Any phase transition! [Landau, 1937]

Liquid/solid [Kirkwood & Monroe, 1941] : density Ferroelectrics [Devonshire, 1949] : polarization

Superconductivity [Ginzbug & Landau, 1950] : density of Cooper pairs Spinodal decomposition [Cahn-Hilliard, 1958] : concentration

Ferroelastic materials (martensite) [F. Falk, 1980] : strains

(127)

1. What is an order parameter?

1

Introduced in the context of phase transition (not only for phase field)

Physical quantity that discriminates different physical states (gas, liquid, solid, paramagnetic, ferromagnetic . . . )

2

Related to choice of phase diagram to display the different states Gas/liquid [Van der Waals, 1873!] : density

Ferromagnetic/paramagnetic [Weiss, 1907] : magnetization Order/disorder [Bragg-Williams, 1934] : order parameter Any phase transition! [Landau, 1937]

Liquid/solid [Kirkwood & Monroe, 1941] : density Ferroelectrics [Devonshire, 1949] : polarization

Superconductivity [Ginzbug & Landau, 1950] : density of Cooper pairs Spinodal decomposition [Cahn-Hilliard, 1958] : concentration

Ferroelastic materials (martensite) [F. Falk, 1980] : strains

(128)

1. What is an order parameter?

1

Introduced in the context of phase transition (not only for phase field)

Physical quantity that discriminates different physical states (gas, liquid, solid, paramagnetic, ferromagnetic . . . )

2

Related to choice of phase diagram to display the different states Gas/liquid [Van der Waals, 1873!] : density

Ferromagnetic/paramagnetic [Weiss, 1907] : magnetization Order/disorder [Bragg-Williams, 1934] : order parameter Any phase transition! [Landau, 1937]

Liquid/solid [Kirkwood & Monroe, 1941] : density Ferroelectrics [Devonshire, 1949] : polarization

Superconductivity [Ginzbug & Landau, 1950] : density of Cooper pairs Spinodal decomposition [Cahn-Hilliard, 1958] : concentration

Ferroelastic materials (martensite) [F. Falk, 1980] : strains

(129)

1. What is an order parameter?

1

Introduced in the context of phase transition (not only for phase field)

Physical quantity that discriminates different physical states (gas, liquid, solid, paramagnetic, ferromagnetic . . . )

2

Related to choice of phase diagram to display the different states Gas/liquid [Van der Waals, 1873!] : density

Ferromagnetic/paramagnetic [Weiss, 1907] : magnetization Order/disorder [Bragg-Williams, 1934] : order parameter Any phase transition! [Landau, 1937]

Liquid/solid [Kirkwood & Monroe, 1941] : density Ferroelectrics [Devonshire, 1949] : polarization

Superconductivity [Ginzbug & Landau, 1950] : density of Cooper pairs Spinodal decomposition [Cahn-Hilliard, 1958] : concentration

Ferroelastic materials (martensite) [F. Falk, 1980] : strains

(130)

1. What is an order parameter?

1

Introduced in the context of phase transition (not only for phase field)

Physical quantity that discriminates different physical states (gas, liquid, solid, paramagnetic, ferromagnetic . . . )

2

Related to choice of phase diagram to display the different states Gas/liquid [Van der Waals, 1873!] : density

Ferromagnetic/paramagnetic [Weiss, 1907] : magnetization Order/disorder [Bragg-Williams, 1934] : order parameter Any phase transition! [Landau, 1937]

Liquid/solid [Kirkwood & Monroe, 1941] : density Ferroelectrics [Devonshire, 1949] : polarization

Superconductivity [Ginzbug & Landau, 1950] : density of Cooper pairs Spinodal decomposition [Cahn-Hilliard, 1958] : concentration

Ferroelastic materials (martensite) [F. Falk, 1980] : strains

(131)

1. What is an order parameter?

1

Introduced in the context of phase transition (not only for phase field)

Physical quantity that discriminates different physical states (gas, liquid, solid, paramagnetic, ferromagnetic . . . )

2

Related to choice of phase diagram to display the different states Gas/liquid [Van der Waals, 1873!] : density

Ferromagnetic/paramagnetic [Weiss, 1907] : magnetization Order/disorder [Bragg-Williams, 1934] : order parameter Any phase transition! [Landau, 1937]

Liquid/solid [Kirkwood & Monroe, 1941] : density Ferroelectrics [Devonshire, 1949] : polarization

Superconductivity [Ginzbug & Landau, 1950] : density of Cooper pairs Spinodal decomposition [Cahn-Hilliard, 1958] : concentration

Ferroelastic materials (martensite) [F. Falk, 1980] : strains

(132)

1. What is an order parameter?

1

Introduced in the context of phase transition (not only for phase field)

Physical quantity that discriminates different physical states (gas, liquid, solid, paramagnetic, ferromagnetic . . . )

2

Related to choice of phase diagram to display the different states Gas/liquid [Van der Waals, 1873!] : density

Ferromagnetic/paramagnetic [Weiss, 1907] : magnetization Order/disorder [Bragg-Williams, 1934] : order parameter Any phase transition! [Landau, 1937]

Liquid/solid [Kirkwood & Monroe, 1941] : density Ferroelectrics [Devonshire, 1949] : polarization

Superconductivity [Ginzbug & Landau, 1950] : density of Cooper pairs Spinodal decomposition [Cahn-Hilliard, 1958] : concentration

Ferroelastic materials (martensite) [F. Falk, 1980] : strains

(133)

1. What is an order parameter?

1

Introduced in the context of phase transition (not only for phase field)

Physical quantity that discriminates different physical states (gas, liquid, solid, paramagnetic, ferromagnetic . . . )

2

Related to choice of phase diagram to display the different states Gas/liquid [Van der Waals, 1873!] : density

Ferromagnetic/paramagnetic [Weiss, 1907] : magnetization Order/disorder [Bragg-Williams, 1934] : order parameter Any phase transition! [Landau, 1937]

Liquid/solid [Kirkwood & Monroe, 1941] : density Ferroelectrics [Devonshire, 1949] : polarization

Superconductivity [Ginzbug & Landau, 1950] : density of Cooper pairs Spinodal decomposition [Cahn-Hilliard, 1958] : concentration

Ferroelastic materials (martensite) [F. Falk, 1980] : strains

(134)

1. What is an order parameter?

1

Introduced in the context of phase transition (not only for phase field)

Physical quantity that discriminates different physical states (gas, liquid, solid, paramagnetic, ferromagnetic . . . )

2

Related to choice of phase diagram to display the different states Gas/liquid [Van der Waals, 1873!] : density

Ferromagnetic/paramagnetic [Weiss, 1907] : magnetization Order/disorder [Bragg-Williams, 1934] : order parameter Any phase transition! [Landau, 1937]

Liquid/solid [Kirkwood & Monroe, 1941] : density Ferroelectrics [Devonshire, 1949] : polarization

Superconductivity [Ginzbug & Landau, 1950] : density of Cooper pairs Spinodal decomposition [Cahn-Hilliard, 1958] : concentration

Ferroelastic materials (martensite) [F. Falk, 1980] : strains

(135)

1. What is an order parameter?

1

Introduced in the context of phase transition (not only for phase field)

Physical quantity that discriminates different physical states (gas, liquid, solid, paramagnetic, ferromagnetic . . . )

2

Related to choice of phase diagram to display the different states Gas/liquid [Van der Waals, 1873!] : density

Ferromagnetic/paramagnetic [Weiss, 1907] : magnetization Order/disorder [Bragg-Williams, 1934] : order parameter Any phase transition! [Landau, 1937]

Liquid/solid [Kirkwood & Monroe, 1941] : density Ferroelectrics [Devonshire, 1949] : polarization

Superconductivity [Ginzbug & Landau, 1950] : density of Cooper pairs Spinodal decomposition [Cahn-Hilliard, 1958] : concentration

Ferroelastic materials (martensite) [F. Falk, 1980] : strains

(136)

1. What is an order parameter?

1

But has been extended to some arbitrary indicator Solidification (A. Karma, J. Dantzig, I. Steinbach, M. Plapp . . . )

Crack (A. Karma, H. Levine, I. Aranson . . . )

Membranes (Q. Du, C. Misbah . . . )

Can be compared to level-set method: phase field can be considered as a physical regularization of the advective equation of the level-set In that case, must recover results from sharp interface descriptions

(137)

1. What is an order parameter?

1

But has been extended to some arbitrary indicator Solidification (A. Karma, J. Dantzig, I. Steinbach, M. Plapp . . . )

Crack (A. Karma, H. Levine, I. Aranson . . . )

Membranes (Q. Du, C. Misbah . . . )

Can be compared to level-set method: phase field can be considered as a physical regularization of the advective equation of the level-set In that case, must recover results from sharp interface descriptions

(138)

1. What is an order parameter?

1

But has been extended to some arbitrary indicator Solidification (A. Karma, J. Dantzig, I. Steinbach, M. Plapp . . . )

Crack (A. Karma, H. Levine, I. Aranson . . . )

Membranes (Q. Du, C. Misbah . . . )

Can be compared to level-set method: phase field can be considered as a physical regularization of the advective equation of the level-set In that case, must recover results from sharp interface descriptions

(139)

1. What is an order parameter?

Co-existence of two different states: the order parameter is a field (heterogeneous)

Two visions of the problem Either interface are sharp (macroscopic), so discontinuities of the field

Or diffuse, as proposed very early by van der Waals, 1893

What is important: sharp variation localized in the interface

(140)

1. What is an order parameter?

Co-existence of two different states: the order parameter is a field (heterogeneous)

Two visions of the problem Either interface are sharp (macroscopic), so discontinuities of the field

Or diffuse, as proposed very early by van der Waals, 1893

What is important: sharp variation localized in the interface

(141)

1. What is an order parameter?

Co-existence of two different states: the order parameter is a field (heterogeneous)

Two visions of the problem Either interface are sharp (macroscopic), so discontinuities of the field

Or diffuse, as proposed very early by van der Waals, 1893

What is important: sharp variation localized in the interface

(142)

1. What is an order parameter?

Co-existence of two different states: the order parameter is a field (heterogeneous)

Two visions of the problem Either interface are sharp (macroscopic), so discontinuities of the field

Or diffuse, as proposed very early by van der Waals, 1893

What is important: sharp variation localized in the interface

(143)

2. Build the relevant thermodynamic functional

(144)

2. Build the relevant thermodynamic functional

Same as usual macroscopic thermodynamics, with functionals rather than functions because variables are fields (depend on space . . . and time)

Isothermal condition and cst volume: Free energy F J.D. Van der Waals,

The thermodynamic theory of capillarity under the hypothesis of a continous variation of density

V.L. Ginzburg, L.D. Landau, On the theory of superconductivity

J.W. Cahn, J.F. Hilliard,

Free energy of a nonuniform system I. Interfacial free energy

Systems featuring heat conduction: entropy S

S.L. Wang, R.F. Sekerka, B. Wheeler, S. Murray, R. Corriel, G. McFadden

But other choices are in principle possible (equivalent but different advantages)

(145)

2. Build the relevant thermodynamic functional

Same as usual macroscopic thermodynamics, with functionals rather than functions because variables are fields (depend on space . . . and time)

Isothermal condition and cst volume: Free energy F J.D. Van der Waals,

The thermodynamic theory of capillarity under the hypothesis of a continous variation of density

V.L. Ginzburg, L.D. Landau, On the theory of superconductivity

J.W. Cahn, J.F. Hilliard,

Free energy of a nonuniform system I. Interfacial free energy

Systems featuring heat conduction: entropy S

S.L. Wang, R.F. Sekerka, B. Wheeler, S. Murray, R. Corriel, G. McFadden

But other choices are in principle possible (equivalent but different advantages)

(146)

2. Build the relevant thermodynamic functional

Same as usual macroscopic thermodynamics, with functionals rather than functions because variables are fields (depend on space . . . and time)

Isothermal condition and cst volume: Free energy F J.D. Van der Waals,

The thermodynamic theory of capillarity under the hypothesis of a continous variation of density

V.L. Ginzburg, L.D. Landau, On the theory of superconductivity

J.W. Cahn, J.F. Hilliard,

Free energy of a nonuniform system I. Interfacial free energy

Systems featuring heat conduction: entropy S

S.L. Wang, R.F. Sekerka, B. Wheeler, S. Murray, R. Corriel, G. McFadden

But other choices are in principle possible (equivalent but different advantages)

(147)

2. Build the relevant thermodynamic functional

Starting with the easiest model: Allen-Cahn

Oversimplified version (parameter free but unrealistic)

F (ϕ

(t,x)

) = Z

V

h

ϕ

⁴(t,x)

− ϕ

²(t,x)

+ |∇ ϕ

(t,x)

|

²

i dV

Where does all these terms come from?

1

Homogeneous system ∇ ϕ = 0: homogeneous free energy density I will consider two examples

1

Starting from macroscopic point of view: van der Waals fluid

2

Upscaling from a simplified atomic picture: Molecular field of P. Weiss

2

What is this gradient term?

(148)

2. Build the relevant thermodynamic functional

Starting with the easiest model: Allen-Cahn

Oversimplified version (parameter free but unrealistic)

F (ϕ

(t,x)

) = Z

V

h

ϕ

⁴(t,x)

− ϕ

²(t,x)

+ |∇ ϕ

(t,x)

|

²

i dV

Where does all these terms come from?

1

Homogeneous system ∇ ϕ = 0: homogeneous free energy density I will consider two examples

1

Starting from macroscopic point of view: van der Waals fluid

2

Upscaling from a simplified atomic picture: Molecular field of P. Weiss

2

What is this gradient term?

(149)

2. Build the relevant thermodynamic functional

Starting with the easiest model: Allen-Cahn

Oversimplified version (parameter free but unrealistic)

F (ϕ

(t,x)

) = Z

V

h

ϕ

⁴(t,x)

− ϕ

²(t,x)

+ |∇ ϕ

(t,x)

|

²

i dV

Where does all these terms come from?

1

Homogeneous system ∇ ϕ = 0: homogeneous free energy density I will consider two examples

1

Starting from macroscopic point of view: van der Waals fluid

2

Upscaling from a simplified atomic picture: Molecular field of P. Weiss

2

What is this gradient term?

(150)

2. Build the relevant thermodynamic functional

Starting with the easiest model: Allen-Cahn

Oversimplified version (parameter free but unrealistic)

F (ϕ

(t,x)

) = Z

V

h

ϕ

⁴(t,x)

− ϕ

²(t,x)

+ |∇ ϕ

(t,x)

|

²

i dV

Where does all these terms come from?

1

Homogeneous system ∇ ϕ = 0: homogeneous free energy density I will consider two examples

1

Starting from macroscopic point of view: van der Waals fluid

2

Upscaling from a simplified atomic picture: Molecular field of P. Weiss

2

What is this gradient term?

(151)

2. Build the relevant thermodynamic functional

Starting with the easiest model: Allen-Cahn

Oversimplified version (parameter free but unrealistic)

F (ϕ

(t,x)

) = Z

V

h

ϕ

⁴(t,x)

− ϕ

²(t,x)

+ |∇ ϕ

(t,x)

|

²

i dV

Where does all these terms come from?

1

Homogeneous system ∇ ϕ = 0: homogeneous free energy density I will consider two examples

1

Starting from macroscopic point of view: van der Waals fluid

2

Upscaling from a simplified atomic picture: Molecular field of P. Weiss

2

What is this gradient term?

(152)

2. Build the relevant thermodynamic functional

1.1. Homogeneous part: Van der Waals fluid

Equation of state for simple fluids: improvement wrt the ideal gas pv

m

= RT by accounting for finite volume of atoms/molecules and interactions (liquid can sustain negative pressures)

(p − a/v

²_m

)(v

m

− b) = RT

Features a transition between liquid and gas below a critical point given by ∂P/∂v

_m

= 0 and ∂

²

P/∂v

²_m

= 0

(153)

2. Build the relevant thermodynamic functional

1.1. Homogeneous part: Van der Waals fluid

Critical point p

^c

= a

27b

²

T

^c

= 8a 27bR v

_m^c

= 3b Reduced form

π = 8τ

3v − 1 − 3 v

²

with τ = T /T

^c

, π = p/p

^c

and v = v

m

/v

^c_m

(154)

2. Build the relevant thermodynamic functional

1.1. Homogeneous part: Van der Waals fluid

Critical point p

^c

= a

27b

²

T

^c

= 8a 27bR v

_m^c

= 3b Reduced form

π = 8τ

3v − 1 − 3 v

²

with τ = T /T

^c

, π = p/p

^c

and v = v

m

/v

^c_m ⁻ ⁰ ¹⁰

v

_m ²⁰ ³⁰

2 0 2 4 6

π

(155)

2. Build the relevant thermodynamic functional

1.1. Homogeneous part: Van der Waals fluid

Critical point p

^c

= a

27b

²

T

^c

= 8a 27bR v

_m^c

= 3b Reduced form

π = 8τ

3v − 1 − 3 v

²

with τ = T /T

^c

, π = p/p

^c

and v = v

m

/v

^c_m

[J. Ericksen, Equilibrium of bars, 1975]

(156)

2. Build the relevant thermodynamic functional

1.1. Homogeneous part: Van der Waals fluid

Free energy: integration of the law of state at constant temperature 1

RT

c

∂f

_m

∂v

τ

= − π Below the critical point, f

_m

features a non-convex part:

corresponds to states that cannot be observed Liquid ↔ Gas transitions

(157)

2. Build the relevant thermodynamic functional

1.1. Homogeneous part: Van der Waals fluid

Free energy: integration of the law of state at constant temperature 1

RT

c

∂f

_m

∂v

τ

= 8τ

3v − 1 − 3 v

²

Below the critical point, f

_m

features a non-convex part:

corresponds to states that cannot be observed Liquid ↔ Gas transitions

(158)

2. Build the relevant thermodynamic functional

1.1. Homogeneous part: Van der Waals fluid

Free energy: integration of the law of state at constant temperature f

m

RT

_c

= K(τ ) − 8

3 τ ln(3v − 1) − 3 v Below the critical point, f

m

features a non-convex part:

corresponds to states that cannot be observed Liquid ↔ Gas transitions

(159)

2. Build the relevant thermodynamic functional

1.1. Homogeneous part: Van der Waals fluid

Free energy: integration of the law of state at constant temperature f

m

RT

_c

= K(τ ) − 8

3 τ ln(3v − 1) − 3 v Below the critical point, f

m

features a non-convex part:

corresponds to states that cannot be observed Liquid ↔ Gas transitions

0 5 10 15 20 25 30

v

−9

−8

−7

−6

−5

−4

−3

−2

f /R T

c

− K ( τ )

(160)

2. Build the relevant thermodynamic functional

1.1. Homogeneous part: Van der Waals fluid

Equilibrium between liquid and gas: variational calculus

Minimum of F with constant total number of atoms (constraint) Miminize the Lagrangian L = F − λ R

V

ρdV − N

with ρ = 1/v

_m

δ L = Z

V

δ(ρf

_m

) − λδρ

dV = 0

δ L This is valid ∀ δρ

f

_m

+ ρ ∂f

_m

∂ρ = λ

(161)

2. Build the relevant thermodynamic functional

1.1. Homogeneous part: Van der Waals fluid

Equilibrium between liquid and gas: variational calculus

Minimum of F with constant total number of atoms (constraint) Miminize the Lagrangian L = F − λ R

V

ρdV − N

with ρ = 1/v

_m

δ L = Z

V

δ(ρf

_m

) − λδρ

dV = 0

δ L This is valid ∀ δρ

f

_m

+ ρ ∂f

_m

∂ρ = λ

(162)

2. Build the relevant thermodynamic functional

1.1. Homogeneous part: Van der Waals fluid

Equilibrium between liquid and gas: variational calculus

Minimum of F with constant total number of atoms (constraint) Miminize the Lagrangian L = F − λ R

V

ρdV − N

with ρ = 1/v

_m

δ L =

Z

V

δ(ρf

_m

) − λδρ

dV = 0

δ L This is valid ∀ δρ

f

_m

+ ρ ∂f

_m

∂ρ = λ

(163)

2. Build the relevant thermodynamic functional

1.1. Homogeneous part: Van der Waals fluid

Equilibrium between liquid and gas: variational calculus

Minimum of F with constant total number of atoms (constraint) Miminize the Lagrangian L = F − λ R

V

ρdV − N

with ρ = 1/v

_m

δ L =

Z

V

δ(ρf

_m

) − λδρ

dV = 0

δ L = Z

V

δρ f

_m

+ ρ δf

_m

− λδρ

dV = 0 This is valid ∀ δρ

f

_m

+ ρ ∂f

_m

∂ρ = λ

(164)

2. Build the relevant thermodynamic functional

1.1. Homogeneous part: Van der Waals fluid

Equilibrium between liquid and gas: variational calculus

Minimum of F with constant total number of atoms (constraint) Miminize the Lagrangian L = F − λ R

V

ρdV − N

with ρ = 1/v

_m

δ L =

Z

V

δ(ρf

_m

) − λδρ

dV = 0

δ L = Z

V

f

_m

+ ρ ∂f

_m

∂ρ − λ

δρ dV = 0 This is valid ∀ δρ

f

_m

+ ρ ∂f

_m

∂ρ = λ

(165)

A short introduction to the phase ﬁeld approach

A short introduction to the phase field approach

Benoˆıt Appolaire

2016, feb. 3rd

What is it for?

For all free boundary problems (potentially)

Phase transformations (solidification, solid states, fluid . . . ) Fluid flow (free surfaces)

Membranes (biology) Cracks . . .

What is it for?

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PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

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PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

What is it for?

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PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

What is it for?

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PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

What is it for?

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PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

What is it for?

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PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

What is it for?

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PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

What is it for?

Advertisement

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

What is it for?

Advertisement

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

What is it for?

Advertisement

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

What is it for?

Advertisement

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

What is it for?

Advertisement

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

What is it for?

Advertisement

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

What is it for?

Advertisement

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

What is it for?

Advertisement

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

What is it for?

Advertisement

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

What is it for?

Advertisement

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

What is it for?

Advertisement

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

What is it for?

Advertisement

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

What is it for?

Advertisement

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

What is it for?

Advertisement

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

What is it for?

Advertisement

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

What is it for?

Advertisement

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys

What is it for?

Advertisement

PhD A. Settefrati, 2012 on phase transformations in Ti-base alloys