Atelier Thermodynamique des Verres – 11 Octobre 2018

Campus de La Doua, Université Claude Bernard – Lyon 1

CINÉTIQUE DE CROISSANCE INSTATIONNAIRE

POUR SYSTÈME MULTICOMPOSÉ

AVEC DIFFUSION CROISÉE

–

SOLUTION ANALYTIQUE ET COUPLAGE CALPHAD

Charles-André GANDIN, Gildas GUILLEMOT

MINES ParisTech CEMEF, UMR CNRS 7635, Sophia Antipolis, France

2

X1

∞

Precipitate Matrix

r 0

X

V

X1

P/M

X2

P/M

X2

∞

X2

M/P

X1

M/P

R

Solute mass conservation in the matrix

Interfacial solute balance at r=R

Thermodynamic equilibrium at interface

Condition at infinity when r→∞

Initial conditions (t = 0 s)

GROWTH OF A SPHERICAL PARTICLE IN A MULTICOMPONENT ALLOY

V (Xi

M/P−Xi

P/M) = −∑

j=1

N

DM

ij ∂X

M

j

∂r|||P/M

+ R3

∂X

P/M

i

∂t

∂XM

i

∂t = ∑

j=1

N

D

M

ij

r2 ∂

∂r

r2 ∂X

M

j

∂r

limr→∞

XM

i (r,t) = X∞

i

MATHEMATICAL PROBLEM: ISOTHERMAL GROWTH IN A SEMI-INFINITE MATRIX PHASE

• Semi-infinite matrix phase M

• Spherical precipitate P, no curvature

• Isothermal system T

• N solute species

• Constant diffusion in M

• Homogeneous precipitate

• Constant and equal densities

XP

i P=X

P/M

i

T = FM/P

[ ] {Xi

M/P}1≤i≤N

Xi

P/M = i

P/M [ ]{Xi

M/P}1≤i≤N

r > R0, XM

i = Xi,0 , T = T0, R = R0

3

GROWTH OF A SPHERICAL PARTICLE IN A MULTICOMPONENT ALLOY

ANALYTICAL SOLUTION [GUILLEMOT GANDIN, Acta Mater. 97 (2015) 419]

The solute profile is a N-linear combination of the solution for the 1-component

expression [ZENER, J. of Appl. Phys. 20 (1949) 950] [AARON et al., J. of Appl. Phys. 41 (1970) 4404]

[HUNZIKER, Acta Mater. 49 (2001) 4191]

A unique solution for the interface position is given by .

l’ is determined by the N-interfacial solute balances:

The composition profile is computed when knowing the norm of the eigenvectors.

Kj are determined by solving the system:

XM

i (r,t) = X∞

i + ∑j=1

N

Kj Uij F

r

2

Bj t where F(u) = e

– u/4

/ u ‒ ( )/2 erfc ( )u/2

with Bj is the j-eigenvalue of matrix DM

Kj is the norm of the eigenvector Kj of matrix DM

, with i-component Kij = Kj Uij

Uij is the i-component of the j-unitary eigenvector Uj of matrix DM

R2 = l' t

( )XM/P

− XP/M

= (l') ( )XM/P

− X∞

where (l')ij = ∑k=1

N

Uik U-1

kj G

l'

Bk and G(u) = - 4

F'(u)F(u)

K = (l') ( )XM/P

− X∞

where (l')ij = U-1

ij / F( )l'/Bi

4

Physical parameters Thermodynamics database NI20 Thermo-Calc

Alloy composition XAl,0 7.56 at%

XCr,0 8.56 at%

Fixed temperature T0 600 ºC

Initial radius R0 0 / 0.8 nm “semi-infinite”/finite domain

Droplet radius Rf -- / 10.6 nm “semi-infinite”/finite domain

Interfacial energy sg/g’ 27∙10-3 J m-2 [BOOTH-MORRISON et al.,

Molar volume Vm 6.8∙10-6 m3 mol-1 Acta Mater. 56 (2008) 3422]

Numerical model Al Cr

FCC_L12#2 (g’) Dg’ Al 1 0 10-19 m2 s-1 taken artificially large

Cr 0 1

Initial radius R0 0 / 0.8 nm “semi-infinite”/finite domain

Droplet radius Rf 100 / 10.6 nm “semi-infinite”/finite domain

Mesh ng, ng’ 500 -

Time step Dt 5∙10-2 s

used as XAl∞

used as XCr∞

SINGLE g’ PRECIPITATE IN A g MATRIX - NI-AL-CR ALLOY

APPLICATION

[ ]

5

Al Cr

FCC_L12#1 (g) Dg

cro Al 20.8 8.59 10-21 m2 s-1 [LANGER et al.,

Cr 8.13 3.82 Acta Mater. 60 (2012) 1871]

Dg

inf Al 20.8 0 10-21 m2 s-1

Cr 8.13 3.82

Dg

sup Al 20.8 8.59 10-21 m2 s-1

Cr 0 3.82

Dg

dia Al 20.8 0 10-21 m2 s-1

Cr 0 3.82

SINGLE g’ PRECIPITATE IN A g MATRIX - NI-AL-CR ALLOY

[ ]

[ ]

[ ]

[ ]

APPLICATION

6

RESULTS

SINGLE g’ PRECIPITATE IN A g MATRIX - NI-AL-CR ALLOY

Analytical solutions ([A] diamonds

symbols) superimpose on the numerical

solution ([N] dotted curves) for all matrices

Strong effect of the g diffusion matrix on

the growth velocity of the g’ precipitate

The selected tie-line depends on Dg

Computation time: [N] 1 h vs. [A] few s

[10-21 m2 s-1]

l’cro 9.274

l’inf 30.59

l’sup 11.95

l’dia 15.10

7

SINGLE g’ PRECIPITATE IN A g MATRIX - NI-AL-CR ALLOY

RESULTS

The analytical solution ([A]

diamonds symbols) super-

imposes on the numerical

solution ([N] dotted curves)

Al depletes / Cr accumulates

ahead of the interface

A typical non monotonous

composition profile is found

for Cr (cross diffusion effect)

Validation of the N-component

analytical solution for the

growth of a single precipitate

with approximations:

- isothermal system

- semi-infinite matrix

- no curvature effect

Cr

Al

Dinf

8

SINGLE g’ PRECIPITATE IN A g MATRIX - NI-AL-CR ALLOY

COMPARISON WITH LITERATURE [CHEN et al. model, Acta Mater. 56 (2008) 1890]

[10-21 m2 s-1]

l’cro 1.452

l’inf 18.77

l’sup 10.42

l’dia 15.08

[10-21 m2 s-1]

l’cro 1.452

l’inf 18.77

l’sup 10.42

l’dia 15.08

The CHEN et al. solution ([CJA] open

diamonds symbols)

does not verify the exact solution,

is only an approximation that substantially

deviates from the exact solution (e.g., Dcro),

is only valid with no interaction between

species (i.e., Ddia),

provides different l’ growth parameters.

9

X1

∞

Precipitate Matrix

r 0

X

V

X1

P/M

X2

P/M

X2

∞

X2

M/P

X1

M/P

R

Global balances

total mass

solute mass

Providing the precipitate size R and its

average composition are known, the

average matrix composition can be

computed

Curvature

gg'

+ gg = 1 with g

g' =

R

Rf

3

EXTENSIONS

GROWTH IN A FINITE DOMAIN AND EFFECT OF CURVATURE

Rf

Rf

gg'

Xg'i

g' + g

g X

gi

g = Xi,0

• Semi-infinite matrix phase M

• Spherical precipitate P, no curvature

• Isothermal system T

• N solute species

• Constant diffusion in M

• Homogeneous precipitate

• Constant and equal densities

• Finite matrix phase M (limited by Rf)

• Spherical precipitate P with curvature

XP

i P=X

P/M

i

Xg2

g

Xg1

g

T = FM/P

[ ]{Xi

M/P}1≤i≤N ; DG

Xi

P/M = i

P/M [ ]{Xi

M/P}1≤i≤N ; DG

DG = 2 sP/M

Vm

P / R

10

Rf

EXTENSIONS

CHOICE FOR MATRIX COMPOSITION AT ∞ IN

Choice 1: use of the global mass balance

Choice 2: use of the analytical profile at Rf

Choice 3: integral method

Xi

g/g'

r 0 R

X

Xi

g'/g

Xg/g'

− Xg'/g

= (l')

Xg/g'

− X~∞

!?

Xgi (r,t) = X

~∞

i (t) + ∑j=1

N

Kj Uij F

r

2

Bj t

X~∞

i (t) = Xgi

g

Xgi

g

X~

i

∞

X~∞

i (t) = Xi

g(r=Rf ,t)

43

(Rf3-R3) X

gi

g =

R

Rf

Xgi (r,t) 4r2 dr

||

mathematical expression for X~∞

i (t) provided in

[GUILLEMOT GANDIN, Acta Mater. 97 (2015) 419]

Xi

g(Rf)

g’ g

11

Physical parameters Thermodynamics database NI20 Thermo-Calc

Alloy composition XAl,0 7.56 at%

XCr,0 8.56 at%

Fixed temperature T0 600 ºC

Initial radius R0 0 / 0.8 nm “semi-infinite”/finite domain

Droplet radius Rf -- / 10.6 nm “semi-infinite”/finite domain

Interfacial energy sg/g’ 27∙10-3 J m-2 [BOOTH-MORRISON et al.,

Molar volume 6.8∙10-6 m3 mol-1 Acta Mater. 56 (2008) 3422]

Numerical model Al Cr

FCC_L12#2 (g’) Dg’ Al 1 0 10-19 m2 s-1 taken artificially large

Cr 0 1

Initial radius R0 00 / 0.8 nm “infinite”/finite domain

Droplet radius Rf 100 / 10.6 nm “infinite”/finite domain

Mesh ng, ng’ 500 -

Time step Dt 5∙10-2 s

used as XAl∞

used as XCr∞

SINGLE g’ PRECIPITATE IN A g MATRIX - NI-AL-CR ALLOY

APPLICATION

[ ]

Vm

g'

12

GROWTH IN A FINITE DOMAIN

EXTENSIONS

The final fraction of the g’ phase tends toward

the equilibrium fraction (reached for 5.7 nm)

The analytical solution with choice 3 ([A]~

plain curves with + symbols) follows very well

the numerical solution ([N]Rf dashed curves)

during the 1st growth regime

Choice 3 is better than choice 1 ([A]¯ dashed

dot curves) for evaluating the far field

composition

An alternate growth/dissolution regime is

observed when approaching equilibrium (Dinf)

Extension of the Laplace solution for a full

diffusion matrix is used for dissolution as the

analytical solution is only valid for growth

The alternate growth/dissolution regime is

also computed with the analytical solution but

it differs from the numerical one

X~∞

i

13

EXTENSIONS

EFFECT OF CURVATURE

Curvature ([N]Rf plain curves) delays the

growth process for all diffusion matrices

The equilibrium volume fraction is smaller

The alternate growth/dissolution regime is

kept when approaching equilibrium

The same delay of the growth process is

observed with the analytical solution

including curvature ([A]~ plain curves with

× symbols)

The analytical solution follows very well the

numerical solution accounting for

curvature, except for the frequency of the

oscillation (due to the Laplace

approximation for the dissolution regime)

14

CONCLUSIONS

An analytical solution is developed to compute the interface velocity and the

composition profiles with cross diffusion between species in an infinite matrix

phase during growth of a single spherical precipitate [GUILLEMOT GANDIN, Acta

Mater. 97 (2015) 419].

Validations are achieved by comparison with numerical solutions.

The CHEN et al. [Acta Mater. 56 (2008) 1890] growth model, also used in ROUGIER

et al. [Acta Mater. 61 (2013) 6396] [Metall. Mater. Trans. 47A (2016) 5557], is not

recommended.

Extensions of the analytical model include

Method to approximate the far field composition when considering a finite size

spherical domain (i.e. interaction with other precipitates),

Laplace solution with cross diffusion used to approximate the dissolution regime (not

presented above),

Method to handle the curvature effect.

An oscillation regime is observed, with intensity depending on the diffusion

matrix.

15

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