Phase Field Simulations

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“ Fundamentals of Thermodynamic Modelling

of Materials ”

November 15-19, 2010 INSTN – CEA Saclay, France

Organized by

Bo SUNDMAN [email protected]

Constantin MEIS [email protected] PROFESSOR & TOPIC

Alphonse FINEL

ONERA, France

Phase Field

Simulations

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principles and applications

A. Finel

LEM ONERA-CNRS, France

Context Principles : Landau model Kim-Kim-Suzuki approach Superalloys Martensitic transition Conclusion

Plan

1 Context

2 Principles : Landau model

3 Kim-Kim-Suzuki approach

4 Superalloys

5 Martensitic transition

(3)

Plan

1 Context

4 Superalloys

6 Conclusion

Context

Origin of microstructures in alloys . . .

(4)

Context (cont)

Evolution of microstructures (thermomechanical treatments...) . . .

Ageing under constraints (creep, fatigue) . . .

Recrystallisation (static, dynamic) . . .

20 min. à 350°C

Croissance de grain Après déformation

3 min. à 350°C

Germination

Plan

1 Context

4 Superalloys

(5)

A simple phase separating system : segregation

Order parameters

x c(x)

c₁ c₂

→ Primary order parameter :

phase transition controlled by concentration

therefore, primary order parameter (OP) will be c(x)

→ Secondary order parameter :

if atomic sizes are slightly different :

precipitation _→ strain field _ij(x)

_ij(x) is a consequence of c(x) ; it does not drive the transition

hence, the notion of ”secondary” OP

A simple phase separating system : segregation

Order parameters

x c(x)

c₁ c₂

→ Primary order parameter :

phase transition controlled by concentration

therefore, primary order parameter (OP) will be c(x)

→ Secondary order parameter :

if atomic sizes are slightly different :

precipitation _→ strain field _ij(x)

_ij(x) is a consequence of c(x) ; it does not drive the

transition

(6)

A simple phase separating system : segregation (cont)

Thermodynamic model

Transition controlled by primary OP :

Ginzburg-Landau free energy on c(x)

F_GL(_{c(x)_}) =

d3x_{f_L(c(x)) +

1

2λ∇c(x)

2 }

f_L(c) = A(c −c₁)2(c −c₂)2

Contribution of secondary OP, i.e. strain

field _ij(x) :

atomic size difference is defined by an

eigenstrain 0

ij(x) = 0

ij [c(x)−c0]

if _ij(x) is the actual strain, elastic energy is given by (linear elasticity) :

E_strain(_{_ij(x)_}) =

d3x λ_ijkl(x) [_ij(x)−0

ij(x)] [kl(x)−0kl(x)]

A simple phase separating system : segregation (cont)

Thermodynamic model

Transition controlled by primary OP :

Ginzburg-Landau free energy on c(x)

F_GL(_{c(x)_}) =

d3x_{f_L(c(x)) +

1

2λ∇c(x)

2 }

f_L(c) = A(c −c₁)2(c −c₂)2

Contribution of secondary OP, i.e. strain

field _ij(x) :

atomic size difference is defined by an

eigenstrain 0

ij(x) =

0

ij [c(x)−c0]

if _ij(x) is the actual strain, elastic energy is

given by (linear elasticity) :

d3x λ_ijkl(x) [_ij(x)₋0

ij(x)] [kl(x)₋0

(7)

A simple phase separating system : segregation (cont)

Kinetic equations

Secondary order parameter kinetics :

elastic relaxation time characteristic time for diffusion

therefore, we suppose that strain field relaxes instantaneously

(quasi-static appr.)

Eeq

strain = min

{_ij(x)}

E_strain → Eeq

strain({c(x)}

with

E_strain =

d3x λ_ijkl(x) [_ij(x)−0_ij(c(x))] [_kl(x)−0_kl(c(x))]

Hence, thermodynamics of primary OP is given by :

F_tot(_{c(x)_}) = F_GL(_{c(x)_}) +Eeq

strain({c(x)}

A simple phase separating system : segregation (cont)

Kinetic equations

Secondary order parameter kinetics :

elastic relaxation time characteristic time for diffusion

therefore, we suppose that strain field relaxes instantaneously

(quasi-static appr.)

Eeq

strain = min

{_ij(x)}

E_strain → Eeq

strain({c(x)}

with

E_strain =

d3x λ_ijkl(x) [_ij(x)−0_ij(c(x))] [_kl(x)−0_kl(c(x))]

Hence, thermodynamics of primary OP is given by :

F_tot(_{c(x)_}) = F_GL(_{c(x)_}) +Eeq

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Coherent elastic energy

long-range elastic interactions

Strain energy :

d3x λ_ijkl(x) [_ij(x)₋0

ij(x)] [kl(x)−

0

kl(x)]

Stress : σ_ij(r) =

δ_E_strain(_{_ij(x)_})

δ_ij(r) = λijkl(x)

_kl(r)

− 0

kl(r)

Mechanical equilibrium : f_i(r) =

∂σ_ij(r)

∂_x_j = 0

If elastic constants do not vary spatially (homogeneous elasticity) : If elastic constants do not vary spatially (homogeneous elasticity) :

0_kl(r)

+

-“dipole-like”

r _δ

kl(r)_∼ 1 r3

interactions élastiques sont à longue portée

λij kl

∂ _kl₍r)

∂xj

=λij kl

∂ 0

kl(r)

∂xj

Residual strain energy :

E_strain = V

2

q=0 B(q) ||c(q)||

2

avec B(q) = λ_ijkl 0

ij

0

kl − qi σ0

ij Gjl(q) σ

0

lm qm

Microstructures and strain energy

no size effect

size effect inhomogeneous elasticity size effect

homogeneous elasticity

C₁₁ = 355 C₁₂ = 115 C₄₄ = 70 C₁₁ = 255

C₁₂ = 205 C₄₄ = 70 c₂ = 0.95

c₁ = 0.05

∆a

a = 0 .0028

d= 4.3 nm

σ= 10 mJm−2

D(T) =D₀ exp(₋∆ U kT ) D₀ = 1.45 1014 nm2s−1

∆U = 2.87 eV/at

precipitates

elasticity : cubic symmetry size effect : Vegard law

(9)

A simple phase separating system : segregation (cont)

Kinetic equations

c(x_,t) is a locally conserved field :

∂c

∂t

= ₋div j

Flux is non-zero only if the system is out of equilibrium :

_j ₌

−M(c)∇µ

where M(c) is a mobility and _µ the alloy chemical potential :

µ =

δF_tot(_{c_})

δc

If M independent of c, we get Cahn-Hilliard eq. (1958) :

∂c

∂t

= M_∇2

δF_tot(_{c_})

δc

••

A simple phase separating system : segregation (cont)

Kinetic equations

∂c

∂t =

−div j

_j ₌

−M(c)∇µ

µ =

δF_tot(_{c_})

δc

∂c

∂t

= M ∇

2δFtot({c_})

δc

(10)

A simple phase separating system : segregation (cont)

Kinetic equations

∂c

∂t

= ₋div j

_j ₌

−M(c)∇µ

µ =

δF_tot(_{c_})

δc

∂c

∂t

= M_∇2

δF_tot(_{c_})

δc

A simple phase separating system : segregation (cont)

Kinetic equations

∂c

∂t =

−div j

_j ₌

−M(c)∇µ

µ =

δF_tot(_{c_})

δc

∂c

∂t

= M ∇

2δFtot({c_})

(11)

A simple phase separating system : segregation (cont)

Kinetic equations

∂c(x,t)

∂t

= M∇2

δF_GL(_{c(x,t)_})

δc(x,t)

= M∇2( f(c(x,t))−λ∇2c(x,t) )

Easy to verify that CH eq. guaranties a decrease of total free

energy :

dF_GL(_{c(x,t)_}) dt

=

d3r

δF_GL(_{c(r,t)_})

δc(x,t)

∂c(x,t)

∂t

0

A simple phase separating system : segregation (cont)

Kinetic equations

If we need fluctuations, we add a Langevin noise _η(x,t) :

∂c(x,t)

∂t

= M∇2

δF_GL(_{c(x,t)_})

δc(x,t)

+ _η(x,t)

where the noise term must fulfilled the ”fluctuation-dissipation”

theorem :

< η(x,t) > = 0

< η(x,t) η(x,t) > = −2kTM∇2δ(x −x) δ(t −t)

in order to recover the correct configurational probabilities when

equilibrium is reached, i.e. :

P(_{c(x,t)_}) ∼ exp(−

F_GL(_{c(x,t)_}) kT

(12)

A simple phase separating system : segregation (cont)

→ But, this is not a well-controlled procedure ! ! !

This procedure is incoherent :

to add a noise to a pre-defined non-linear driving force is

inconsistent (fluctuations modify the equilibrium average

concentration)

fluctuations rˆole is to overcome energy barriers that result from the

metastable free energies...

However, these metastable free energies are not uniquely defined...

A simple phase separating system : segregation (cont)

→ What is a ”good” phase field theory with fluctuations ? :

Free energy density functional and fluctuations are inter-dependent

We cannot use a macroscopic free energy, i.e. a free energy that is

fitted to phase diagrams (such as Calphad free energies)

... nor a mean field free energy (which, by definition, neglect

fluctuations)

Fluctuations are dynamical quantities...

Kinetic equations and free energy functional must be derived

(13)

Alloys with ordering transitions

Order parameters

Ordering transition (1st or 2nd order)

A simple example : disorder _→ ”antiferromagnetic” order

variant 2 variant 1

We need to differentiate the different variants

Hence, the notion of ”long-range” order parameters (OP)

How to identify the long-range OP’s ?

Alloys with ordering transitions

Order Parameters (cont)

General method1 : concentration wave decomposition

c(x) = c +ηexpiq.x q = 2π

a

1 2

η = 0 : disorder

η > 0 : variant 1

η < 0 : variant 2

The symmetry change is controlled by _η : primary OP

Transition may be associated to a change in c : secondary OP.

(14)

Alloys with ordering transitions

Order Parameters (cont)

General situation : inhomogeneous distribution of the variants

η(x) la

variant 1 disorder variant 2

Order parameter _η(x) varies slowly with x, where x is the unit cell

coordinate

Concentration on site (x +u), where u is the coordinate inside the

unit cell at position x, is written as :

c(x +u) = c₀(x) +η(x) expiq.u

where c₀(x) and η(x) are the average concentration and

order-parameter in cell ”x”, respectively.

Alloys with ordering transitions

Thermodynamic model

Transition controlled by primary OP _η

Therefore, the non-linearity in Landau potential must be expressed

in terms of _η

Landau potential must be invariant if we exchange the 2 variants

Therefore :

if we have a 2nd order transition :

f_L(η) = −aη2+bη4 +λ∇η2

if we have a 1st order transition :

f_L(η) = −aη2 −bη4 +cη6 +λ∇η2

if we have a 1st order transition with a concentration change

(secondary OP) :

(15)

Alloys with ordering transitions

Thermodynamic model

in terms of _η

Therefore :

f_L(η) = −aη2+bη4 +λ∇η2

f_L(η) = −aη2 −bη4 +cη6 +λ∇η2

(secondary OP) :

f_L(η) = a(c −c0)2 +d(c₁ −c)η2 −bη4+c η6 +λ∇η2+β∇c2

Alloys with ordering transitions

Thermodynamic model

in terms of _η

Therefore :

f_L(η) = −aη2+bη4 +λ∇η2

f_L(η) = −aη2 −bη4 +cη6 +λ∇η2

(secondary OP) :

(16)

Alloys with ordering transitions

Thermodynamic model

in terms of _η

Therefore :

f_L(η) = −aη2+bη4 +λ∇η2

f_L(η) = −aη2 −bη4 +cη6 +λ∇η2

(secondary OP) :

f_L(η) = a(c −c0)2 +d(c₁ −c)η2 −bη4+c η6 +λ∇η2+β∇c2

Alloys with ordering transitions

Thermodynamic model

in terms of _η

Therefore :

f_L(η) = −aη2+bη4 +λ∇η2

f_L(η) = −aη2 −bη4 +cη6 +λ∇η2

(secondary OP) :

(17)

Alloys with ordering transitions

Kinetic equations

Underlying dynamics : thermally activated jumps (markovian

process)

Therefore, c and η follow Langevin equations :

concentration c(x,t) is conserved : Cahn-Hilliard eq. (with noise)

∂c(x,t)

∂t

= M∇2 δ

F_GL

δc(x,t)

+ _ξ(x,t)

order parameter _η(x,t) is not conserved : Allen-Cahn eq. (with noise)

∂ηi(x,t)

∂t

= −_Γ δ FGL

δηi(x,t)

+ _ζi(x,t)

Remark : secondary OP is not a ”slave”, i.e. is not quasi-static

w.r.t. the primary OP.

Alloys with ordering transitions

Kinetic equations

process)

∂c(x,t)

∂t

= M∇2 δ

F_GL

δc(x,t)

+ _ξ(x,t)

order parameter _η(x,t) is not conserved : Allen-Cahn eq. (with

noise)

∂ηi(x,t)

∂t

= −_Γ δ FGL

δηi(x,t)

+ _ζ_i(x,t)

(18)

Alloys with ordering transitions

Kinetic equations

process)

∂c(x,t)

∂t

= M∇2 δ

F_GL

δc(x,t)

+ _ξ(x,t)

noise)

∂ηi(x,t)

∂t

= −_Γ δ FGL

δηi(x,t)

+ _ζ_i(x,t)

Ordering alloys : the example of superalloys

γ

−

γ

FCC (_γ) _→ L1₂ (_γ)

L1₂ may display 4 different variants

we must be able to differentiate them

(19)

Ordering alloys : the example of superalloys

γ

−

γ

(cont)

→ General method 2 :

concentration wave decomposition : c(x) =

d η(q) expiq.x

Ex : 1 single homogeneous variant L1₂ :

c(x) = c₀ +η₁exp(₋iq₁.x) +η₂exp(₋iq₂.x) +η₃exp(−iq₃.x)

with :

q₁ = [100] q₂ = [010] q₃ = [001]

order parameters for the 4 different variants :

variant 1 : (_η₁,η₂,η₃) _∼ (111)

variant 2 : (_η₁,η₂,η₃) _∼ (1¯1¯1)

variant 3 : (_η₁,η₂,η₃) _∼ (¯1¯11)

variant 4 : (_η₁,η₂,η₃) _∼ (¯11¯1)

2

concentration wave decomposition, Krivoglaz (69), Khachaturyan(83)

Ordering alloys : the example of superalloys

γ

−

γ

(cont)

→ General method (cont) :

generic situation : inhomogeneous distribution of 4 variants

order parameters _η_i(R) varie slowly with R, where R is the unit

cell (fcc cube) coordinate :

η(x) la

variant 1 disorder variant 2

concentration on site of coordinate x = R +r, where r is the

coordinate inside the unit cell, may be written as :

c(R +r) = c₀(R) + η₁(R) exp(₋iq₁.r) +η₂(R) exp(₋iq₂.r) + _η₃(R exp(₋iq₃.r)

where c₀(R) and η_i(R) are the average concentration and order

(20)

Ordering alloys : the example of superalloys

γ

−

γ

(cont)

→ General method (cont) :

the ordering transition is controlled by the 3 order parameters

η1(R),_η₂(R),_η₃(R) : they play the rˆole of primary OP.

the concentration field c₀(R) does not control the transition : it is

a secondary OP.

If L1₂ and the disordered phase have different unit cell sizes _→ strain field _ij(x) as another secondary OP.

Ordering alloys : the example of superalloys

γ

−

γ

(cont)

We have seen that the transition FCC _→ L1₂ is characterized by 4 order parameters :

c₀(R) η₁(R) η₂(R) η₃(R)

which allow to describe locally any configuration :

c(R +r) = c₀(R) + η₁(R) exp(₋iq₁.r) +η₂(R) exp(₋iq₂.r) + _η₃(R exp(₋iq₃.r)

The primary OP’s _η_i(x) represent the amplitudes of the waves

with q-vectors (100),(010),(001).

We look now for a non-linear Landau functional in _η_i(x)

This potential must be invariant w.r.t. the symmetry operations

that leave the FCC lattice invariant, i.e. the space group

(21)

Ordering alloys : the example of superalloys

γ

−

γ

(cont)

Example :

FCC symmetry operations lead to sign changes and/or

permutation between order parameters

Look for polynomial components that are invariant w.r.t. these

operations

... and stop the expansion to the lowest order compatible with the

existence of a transition...

Ordering alloys : the example of superalloys

γ

−

γ

(cont)

→ Landau functional :

Here we get :

η1 +η2 +η3 : no

η₁2 +_η2₂ +_η₃3 : yes

η1η2 +η2η3 +η3η1 : no

η1η2η3 : yes

η₁2η2 +η1η₂2 +... : no

η₁4 +_η4₂ +_η₃4 : yes

(22)

Ordering alloys : the example of superalloys

γ

−

γ

(cont)

→ Landau functional (cont) :

In heterogeneous systems, order parameters and concentration

vary slowly with x.

Ginzburg-Landau functional has the following form :

F_GL =

V

d3x [ A

2(c −c0)

2 ₊ B 2(c

−c)(η 2

1 +η22 +η23

3 )

− Cη₁η₂η₃ + D

4(

η₁4 +_η4

2 +η34

3 ) ]

+ λ

2 ||∇c(x,t)||

2 ₊ βi

2 ||∇ηi(x,t)|| 2

Ordering alloys : the example of superalloys

γ

−

γ

(cont)

→ Kinetic equations :

process)

∂c(x,t)

∂t

= M_∇2 δ

F_GL

δc(x,t)

+ _ξ(x,t)

order parameter _η(x,t) is not conserved : Allen-Cahn eq. (with noise)

∂ηi(x,t)

∂t

= _−Γ δ FGL

δηi(x,t)

+ _ζi(x,t)

and the noise terms must fulfilled the fluctuation-dissipation

theorem :

< ξ(x,t) ξ(x••,t••) > = −2kTM_∇2δ(x ₋x••) δ(t ₋t••)

(23)

Ordering alloys : the example of superalloys

γ

−

γ

(cont)

→ Kinetic equations :

process)

∂c(x,t)

∂t

= M∇2 δ

F_GL

δc(x,t)

+ _ξ(x,t)

noise)

∂ηi(x,t)

∂t

= −_Γ δ FGL

δηi(x,t)

+ _ζ_i(x,t)

and the noise terms must fulfilled the fluctuation-dissipation

theorem :

< ξ(x,t) ξ(x,t) > = −2kTM∇2δ(x −x) δ(t −t)

< ζ_i(x,t) ζ_i(x,t) > = 2kTΓδ(x −x) δ(t −t)

Ordering alloys : the example of superalloys

γ

−

γ

(cont)

→ Application to Ni-Al :

Ginzburg-Landau potential fitted to the phase diagram (if no noise

terms...) :

and to the FCC/L1₂ interface free energy (taken from ab initio

(24)

Martensitic transition : square to rectangle

Order parameters

variant 1 :

¯ ¯ (1) 0 = λ 0 0 −λ

variant 2 :

¯ ¯

(2)

0 =

−λ 0

0 λ

Broken symmetry associated to ₁₁

− 22

Therefore, the primary OP is :

e₂(x) = ₁₁(x)₋₂₂(x)

However, due to the compatibility relations (lattice continuity), we

must take into account all the strain components.

Therefore, we have here two secondary OP’s :

e₁(x) = ( ₁₁(x) + ₂₂(x) )/2

e₃(x) = ₁₂(x)

Martensitic transition : square to rectangle

Order parameters

variant 1 :

¯ ¯ (1) 0 = λ 0 0 −λ

variant 2 :

¯ ¯

(2)

0 =

−λ 0

0 λ

− 22 Therefore, the primary OP is :

e₂(x) = ₁₁(x)

−22(x)

e₁(x) = ( ₁₁(x) + ₂₂(x) )/2

(25)

Martensitic transition : square to rectangle

Order parameters

variant 1 :

¯ ¯ (1) 0 = λ 0 0 −λ

variant 2 :

¯ ¯

(2)

0 =

−λ 0

0 λ

− 22 Therefore, the primary OP is :

e₂(x) = ₁₁(x)

−22(x)

e₁(x) = ( ₁₁(x) + ₂₂(x) )/2

e₃(x) = ₁₂(x)

Martensitic transition : square to rectangle

Thermodynamics

variant 1

λ 0

0 ₋λ

variant 2

−λ 0

0 λ

e₂ = ₁₁ ₋₂₂

e₁ = ( ₁₁ +₂₂ ) /2

e₃ = ( ₁₂ +₂₁ ) /2

The transition is controlled by primary OP e₂, therefore the

non-linearity in Landau potential must be written in terms of e₂

The Landau potential must be invariant by exchange of the 2

variants

The transition is 1st order

→ The non-linear terms have necessarily the following form :

a₂e2

2 −a4e 4

(26)

Martensitic transition : square to rectangle

Thermodynamics

variant 1

λ 0

0 ₋λ

variant 2

−λ 0

0 λ

e₂ = ₁₁ ₋₂₂

e₁ = ( ₁₁ +₂₂ ) /2

e₃ = ( ₁₂ +₂₁ ) /2

variants

a₂e2

2 −a4e24 +a6e26

Martensitic transition : square to rectangle

Thermodynamics

variant 1

λ 0

0 ₋λ

variant 2

−λ 0

0 λ

e₂ = ₁₁ ₋₂₂

e₁ = ( ₁₁ +₂₂ ) /2

e₃ = ( ₁₂ +₂₁ ) /2

variants

a₂e2

2 −a4e 4

(27)

Martensitic transition : square to rectangle

Thermodynamics

Secondary OP’s e₁ et e₃ must be taken into account :

they are linked to the primary OP e₂ by the compatibility relation

(lattice continiuity) :

∇2e₁−(∇2 x − ∇

2

y)e2 −2∇x∇ye3 = 0

they enter through the elastic properties of the system

To the lowest order, the Landau free energy density is therfore

given by :

F_GL = a₂e₂2 −a₄e₂4 +a₆e₂6 +b₂e₁2 +c₂e₃2 +_γ∇e₂2

Coe_fficients a₂,b₂ et c₂ are linked to the elastic constants

a₄,a₆ depend on the curvature of the phonon acoustic branche.

Martensitic transition : square to rectangle

Thermodynamics

∇2e₁−(∇2 x − ∇

2

y)e2 −2∇x∇ye3 = 0

given by :

(28)

Martensitic transition : square to rectangle

Thermodynamics

∇2e₁−(∇2 x − ∇

2

y)e2 −2∇x∇ye3 = 0

given by :

a₄,a₆ depend on the curvature of the phonon acoustic branche.

Dynamics of the (proper) martensitic transition

Dynamics is a priori inertial !

Kinetic energy density on the velocity field :

E_cin = 1 2ρ u˙i

2

where u_i is the displacement field.

Rayleigh dissipation on primary OP (because it drives the phase

transition) :

R = 1

2γ 2˙ 2

Ginzburg-Landau free energy :

F_GL = a₂e2

2 −a4e 4

2 +a6e 6

2 +b2e 2

1 +c2e 2

3 +γ∇e2 2

Lagrange equations, with L = T −F_GL :

d

dt

∂L

∂u˙_i

−

∂L

∂u_i

=

−

∂R

(29)

Dynamics of the (proper) martensitic transition

E_cin = 1 2ρ u˙i

2

transition) :

R = 1 2γ ˙2

2

F_GL = a₂e2

2 −a4e 4

2 +a6e 6

2 +b2e 2

1 +c2e 2

3 +γ∇e2 2

d

dt

∂L

∂u˙_i

−

∂L

∂u_i

=

−

∂R

∂u˙_i

Dynamics of the (proper) martensitic transition

E_cin = 1 2ρ u˙i

2

transition) :

R = 1 2γ ˙2

2

F_GL = a₂e2

2 −a4e 4

2 +a6e 6

2 +b2e 2

1 +c2e 2

3 +γ∇e2 2

d

dt

∂L

∂u˙_i

−

∂L

∂u_i

=

−

∂R

(30)

Dynamics of the (proper) martensitic transition

E_cin = 1 2ρ u˙i

2

transition) :

R = 1 2γ ˙2

2

F_GL = a₂e2

2 −a4e 4

2 +a6e 6

2 +b2e 2

1 +c2e 2

3 +γ∇e2 2

d

dt ∂L ∂u˙_i

−

∂L

∂u_i

=

−

∂R

∂u˙_i

Dynamics of the (proper) martensitic transition (cont)

Do we need an inertial dynamics ?

If 1D chain : _ρ ¨e = ∇2_{ dFL

de −λ ∇

2

e +γe˙_}

A linear analysis of the propagation of a perturbation within the

”austenite” leads to :

ω(q)

∼ vsq +iγ q2/2ρ

Therefore, in the limit q _→ 0, propagation time is relaxation

time :

1/v_sq ρ/γq2

In other words, long-wavelength waves propagate before they relax

Hence, the necessity to take into account inertia effects, i.e. the

fact that sound wave velocity v_s is finite

Therefore, we cannot use a purely dissipative dynamics, as the

’traditionnal” Phase field method (which corresponds to v_s

(31)

”Improper” martensitic transformation

cubic to tetragonal transition in _Zr02

”shu_ffle”_∆z = 0 → tetragonal symmetry

transition controlled by the shu_ffle, therefore, _∆z(x) is the primary

OP.

tetragonal symmetry _→ shape change associated to an

eigenstrain :

¯ ¯

₀ =

_λ ₀ ₀

0 λ 0 0 0 β

therefore, actual strain _ij(x) will play the rˆole of a secondary OP,

that enters into a strain energy in the framework of linear elasticity.

Plan

1 Context

4 Superalloys

(32)

The Kim-Kim-Suzuki phase field model

Interfaces as diffuse mixtures of phases at local equilibrium

based on ”artificial” fields

composition c(x)

if n phases : (n−1) parameter fields φi (2 in the following)

interface : each point within the interface is a mixture of both phases with = compositions c₁ and c₂

c = h(φ)c₁+ (1−h(φ))c₂

where h(φ) is a monotonous function from h(0) = 0 to h(1) = 1

ex : h(φ) = φ3(φ2 −15φ+ 10)

c₁(x) and c₂(x) spatially dependent within the interface

f₁(c₁) and f₂(c₂) free energy densities of phases 1 and 2

both phases at local equilibrium within the interface :

∂f1(c1) ∂c1 =

∂f2(c2)

∂c2 (therefore, c1 and c2 are functions of c)

The Kim-Kim-Suzuki phase field model (cont)

Total free energy

θ_{= 79}◦

0

0 1

C₁

C₂

φ

0

1 1

C₁

C₂

φ f(c,φ)

free energy density :

f(c,φ) = h(φ) f₁(c₁) + (1−h(φ)) f₂(c₂) +w g(φ)

where g(φ) is a double-well : g(φ) = φ2(1 −φ)2

total free energy :

F = d3x_{ f(c,φ) + 1

2λ∇φ 2

}

thermodynamic databases may provide a full description of the

(33)

The Kim-Kim-Suzuki phase field model (cont)

Kinetic equations

phase fields are not conserved _→ Allen-Cahn equations :

∂φ

∂t = −L

δF

δφ

concentration is conserved _→ Cahn-Hilliard equation :

∂c

∂t = ∇M(c)∇

∂f

∂c

mobility M(c) may be linked to interdiff. coef. ˜D(c) by :

˜

D(c) = M(c)∂

2_f

∂c2

Kinetic databases may provide interdiffusion coefficient ˜D(c) (or

mobilities)

Plan

1 Context

4 Superalloys

(34)

Superalloys : microstructures and creep

Motivation :

microstructure après service : mise en radeau

Fortes sollicitations

Mécaniques

Thermiques 650°C

1100°C

microstructure initiale : cuboïdes

Modeling :

Ginzburg-Landau model for _γ/_γ

strain energy due to elastic effects ( _γ and _γ _{unit cell sizes are}

different)

need to incorporate plastic relaxation :

Epishin et al. Acta mater. 49 (2001)

γ

_γ

Superalloys : thermodynamics

F_GL =

V

d3r A

2 ( c −c1 )

2 ₊ B

2( c2 −c )(

η₁2 + _η2

2 + η32

3 )

− Cη₁η₂η₃ + D

4(

η₁4 + _η4

2 + η34

3 )

+ λ

2 ||∇c(r,t)||

2 ₊ βi

2 ||∇ηi(r,t)|| 2

Parameters chosen to reproduce phase diagram and interface energy :

B = 2A(c

γ −cγ)

C = 6A(c

γ −cγ)(c0 −cγ)

D = 2A(c

γ −cγ)(cγ −cγ + 2c0)

(35)

Elastic effects

Coherent strain energy :

Elastic constants vary with local concentration (inhomogeneous elasticity) :

λ_ijkl(r) = λγ

ijkl(cγ −c(r)) +λ

γ

ijkl(c(r)−cγ)• •(cγ −cγ)

Eigenstrain :

0

ij(r) =

0

ij (c(r)−¯c)

Elastic strain :

el

ij (r) = ij(r)−

0

ij(r)−

p

ij(r)

with

_ij(r) : total strain

p

ij(r) : plastic strain

Elastic effects (cont)

Mechanical equilibrium :

Strain energy :

E_el = 1

2

V

λ_ijkl(r)_ij(r)₋0

ij(r)−

p

ij(r)

_kl(r)₋0

kl(r) −

p

kl(r)

dV

Local stress :

σ_ij(r) = δEel

δ_ij(r) = λijkl(r)

_kl(r)₋0

kl(r)−

p

kl(r)

Mechanical equilibrium :

∂σ_ij(r)

∂x_j

= 0

(36)

Kinetic path correctly reproduced by Phase field method

Phase field (with an atomic scale grid) G. (G. Boussinot, Y. Le Bouar, A. Finel, Acta Mater. (2010))

(isotropic elasticity)

Monte Carlo simulation : atomic scale (J.H. Choy, S.A. Hackney, J. K. Lee, MMMS (1996))

Elastic instability in an elastically inhomogeneous system

λM ij kl= 2λ

P ij kl

Plastic activity

Crystalline plasticity model (Cailletaud, 91) :

Resolved shear stress on glide system s :

τs = σ

∼

: _M ∼

s

M

∼

s ₌ 1

2 (m

s _⊗

ls +_ls _⊗_ms)

Schmid law (threshold) :

fs = _|_τs −Xs_|−Rs

where Xs : kinematic hardening et Rs : isotropic hardening

Norton type eq. for plastic flow :

˙ _p =

s

˙ γsM

∼

s

˙

γs = |τ

s ₋_Xs

| −Rs

k

n _sign(

τs −Xs)

(37)

Phase field method with elasticity and viscoplasticity

Coupling with microstructures by plastic relaxation of coherent strain

energy :

E_strain = 1

2

V

λ_ijkl(r)_ij(r)

−

0

ij(r)−

p ij(r)

_kl(r)

−

0

kl(r)−

p kl(r)

dV

Algorithm :

Equilibre mécanique

Equations viscoplastiques

Equations cinétiques

Algorithme de point fixe dans l'espace de Fourier

Schéma d'Euler explicite

Schéma semi-implicite dans l'espace de Fourier

Influence of coherency stresses on microstructures

Microstructures in Ni-Al alloys :

1 µ

m

(38)

Influence of coherency stress on microstructures

Microstructure 3D in Ni-Al under external stress :

δ = 2(aγ −aγ) a

γ +aγ

> 0

x y z

600 MPa compression

x

y z

600 MPa tension

Influence of thermal treatments on microstructures

Microstructures in Ni-Al alloys :

1 µm

(39)

Influence of plastic activity on microstructures

Creep : rafting in Ni-based superalloys

Phase field model with elasticity

Phase field model with elasticity 10µm

Phase field model with elasticity and viscoplasticity

t=0h t=5h t=10h t=20h t=40h

Influence of plastic activity on microstructures (cont)

(40)

Plan

1 Context

4 Superalloys

6 Conclusion

Martensitic transition and shape memory alloys

Martensitique

transition

austenite

martensite

Shape memory effect

cooling

loading heating

twinned martensite strained martensite austenite

Load

T

emperature

Elastic

accomodation

Ti50Ni25Pd25

R. Delville (2008)

1st order transition Athermale

(41)

Motivation 1 : origin of the stability of macrotwins in NiAl

collision between needle bending, tapering and ”splitting”

”laminates” in Ni₆₅Al₃₅

• • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • •• • • • ••••••••••••••••••••••••••••••••••••••••••••••••

Origin of macrotwin morphology ?

Simulations : nonlinear vs. linear geometry

Nonlinear

Initial conf.

Linear

t = 20 ×104 t = 50 ×104 t ∼ ∞

(42)

Simulation vs. experimental observation

Needles bending, tapering and ”splitting”

final state in nonlinear model macromacles in Ni-Al

θ _{= 78}◦

θ _{= 79}◦

θ

_{= 76}

◦

_±

₂

◦

Plan

1 Context

4 Superalloys

(43)

Conclusion

General :

mesoscopic scale 1_µm, late stage

very versatile

elasticity, plasticity, fracture..

Landau approach :

physically based

differences between various type of interfaces automatically

embedded...

link with atomistic model (coarse-graining approach)

less field variables than in KKS

Kim-Kim-Suzuki method :