Prediction of precipitation kinetics in Al alloys · New project ClaNG+ Collaboration ... • Alloy concentrations •Primary phase ... nearly always has the same shape and its inverse

Institute ofPhysical Metallurgy and Metal Physics

Prediction of precipitation kinetics

in Al alloys

GTT workshop / Herzogenrath (Germany) / June 20-22, 2007

E. Jannot, V. Mohles, G. Gottstein

2 GTT Workshop / June 20-22, 2007

New project ClaNG+

Collaboration between

• Hydro (Aluminum industry)

• GTT technologies

• IMM: Institut für Metallkunde und Metallphysik

Extension of the use of ClaNG (tool for the

prediction of precipitation kinetics)

• To other alloys extend the thermodynamic databank

(input of the model)

• To other process steps couple precipitation with

other phenomena (Deformation, RX)

Started 01/2007…


Aluminum sheet production

Evolution: start with a higher content of recycled products

Goal: control the end- properties of the material

Requirement: predict the precipitation state at each step

Casting - Homogenization - Hot Rolling - Cold Rolling - Annealing

break down tandem-mills


Contents

Description of ClaNG

Simulation results

Open questions


ClaNG model overview

INPUT

OUTPUTStarting state:

• Alloy concentrations

• Primary phase volume fraction (density & size)

• Dendrite arm spacing

Physical parameter:

• Phase diagram

• Diffusivity

• Interfacial energies

Matrix:

• Alloy concentrations

Primary & secondary phases:

• Volume fraction(density and size distribution)

• Composition

Material properties:

• Conductivity…

Industrial process:

• Time-Temperature curve

• (dislocation density)

Nucleation Growth &

Coarsening

Thermodynamic *Free Energy curves

Equilibrium calculation

Kinetic Diffusion

Phase

Composition

Update

State at time tTemperature, Alloy

conc…

State at time t + tNew conc., phase vol. frac…

* : performed using ChemApp from GTT technologies


Thermodynamic calculations (I)

At every interface, one assumes that a local equilibrium is achieved after a short time

The equilibrium conditions are calculated using the ChemApp application developed by GTT technologies

For these calculations, a thermodynamic databank is required

• The AlTT databank (Thermotech) is used for the moment

• Light version containing 8 elements (Al, Cr, Cu, Fe, Mg, Mn, Si, Ti)

For each possible phase (Mg2Si, Apha…),

perform the chemical reaction corresponding

the matrix decomposition occurring at the

interface between the matrix and the phase

focus

here


Thermodynamic calculations (II)

Use ChemApp:

• Set conditions (temperature, concentrations in the matrix)

• Enter 2 phases: matrix + another phase

• Perform equilibrium

• Extract information

Equilibrium concentrations ci in the Al matrix

Equilibrium concentrations ci in the phase

Chemical potentials of the elements

Derive the Chemical Driving Force gu

x Mole

Matrix 0

Gm0

Matrix 1

Gm1

1 Mole

Phase

Gm

(1-x) Mole

i

1

i

0

iiu )(cg


Nucleation

When gu is known, the critical radius rc can be derived

• User input = interfacial energy

The nucleation rate is then given by the classical theory of Becker & Döring

The nucleation rate is often underestimated

• Reason: the thermodynamic quantities (gu, ) are not precisely defined for nanoparticles

• corrected by a “tuning” factor

texp

Tk

rGfexpZNN

B

Ccorr0

u

mc

g

V2r

Usual value 1.0 0.1


Growth & Coarsening

Assumption: precipitate growth always diffusion controlled in Al alloys

ClaNG treats growth and coarsening in a single equation

• A particle above the critical radius grows

• A particle below the critical radius dissolves

The growth law used in ClaNG derives from Zener’s formulation

• All particles are considered spherical

r

D

)r(cc

)r(cc

dt

drv

/

/0

rTR

V2expc)r(c

g

m/

eq

/

c0

c

c/(r)

xr

Matrix

Gibbs-Thomson concentration

at the interface

/

.eqc

Phase

(sphere)


Evolution with time

particle distribution at t

radius of the particle

de

ns

ity

particle distribution at t+dt

radius of the particle

de

ns

ity+ t

g(ri)

g(ri+1)

ri

ri+1

For every radius ri at t+t, one determines its image g(ri) at t using a Runge-Kutta method

tNdr)t,r(fdr)t,r(f)t(N)tt(N nucl

i

r

)r(g

r

)r(g

ii

1i

1i

i

i

Ni(t+

t)

(Robson, Acta Mater. 51-2003)

Discretization in radius classes (1 nm width)


Phase composition update

During the heat treatment, the equilibrium phase composition changes

• The concentration in the precipitate should be consequently updated

Diffusion inside the precipitate does the work, but unfortunately the diffusion coefficient in the phase is generally unknown

• Hence this phenomenon can’t be accurately modeled

• Pragmatic modeling: use of a “tuning” factor to level diffusion strength


Conclusions

Each step is dependent on the results of an

thermodynamic equilibrium calculation

The classical nucleation and growth theory

provides the frame for the precipitation kinetics

calculation

However still hard to provide a 100% physical

based model

• “Tuning” parameters still required


Contents


Simulation results

Open questions


Microstructure after homogenization

AA 3104 investigated

Still inhomogeneous after homogenization

• Complex structure with 3 different regions

Primary phases or

constituents (3%)

Phase Beta = Al6(Fe,Mn)

Precipitation free zone

(30%)

Dendrites with secondary

phases or dispersoides

(67%)

Phase Alpha = Al12Mn3Si1.8


Formation of dispersoides

0

100

200

300

400

500

600

700

0 5 10 15 20 25 30

Zeit / h

Te

mp

era

tur

/ °C

2 m

2 m

2 m

2 m

2 m

2 m

2 m


Prediction quality

1.5

1.7

1.9

2.1

2.3

2.5

2.7

2.9

3.1

3.3

3.5

3.7

3.9

0 3 6 9 12 15 18 21 24

Homogenization time [h]

Re

sis

tiv

ity

[1

0^

-8 O

hm

.m]

-

100

200

300

400

500

600

Ho

mo

ge

niz

ati

on

te

mp

. [°

C]

Rsimu

Rexp_Center

Rexp_Edge

Temp.

-

20

40

60

80

100

120

0 3 6 9 12 15 18 21 24Homogenization time [h]

Av

era

ge

Ra

diu

s [

nm

]

-

100

200

300

400

500

600

Ho

mo

ge

niz

ati

on

te

mp

. [°

C]

Alpha

Mg2Si

Exp_Center

Exp_Edge

1.0E+18

1.0E+19

1.0E+20

1.0E+21

1.0E+22

0 3 6 9 12 15 18 21 24Homogenization time [h]

De

ns

ity

[m

^-3

]

-

100

200

300

400

500

600

Ho

mo

ge

niz

ati

on

te

mp

. [°

C]

Alpha

Mg2Si

Exp_Center

Exp_Edge

AA3104 Fe Mg Mn Si

Nom. [c] 0.45 1.02 0.85 0.18

Arm [c] - Start 0.02 1.03 0.75 0.20

Arm [c] - End 0.03 1.03 0.69 0.13

Alpha = 0.2 Jm-2 and fcorr(Alpha) = 0.12

Mg2Si = 0.2 Jm-2 and fcorr(Mg2Si) = 1.00

Simulation takes less than 30 min,


Conclusions

The model is able to deliver good predictions of the precipitation kinetics

• Used by Hydro since 2004

Acceptable sensitivity to empirical factors:

• Nucleation rates are very sensitive to their “tuning” parameters but you can’t do without

• Other factors allow simulations to 100% fit the experimental results

Results are however very sensitive to the starting conditions

• Model casting step…


Contents


Simulation results

Open questions


Problematic

The goal of the project is to extend the use of

ClaNG to different alloy systems

Hence, you need to have a robust growth law

which is correct for the whole range of alloys

ClaNG uses today a growth law based on the

one proposed by Zener

Question: Is this law the right one?

r

D

)r(cc

)r(cc

dt

drv

/

/0


Survey of possible growth laws

Many groups have proposed modified versions of this

growth law these last years

Robson et al. (good for Al3Sc vs. Al3Zr)

Kozeschnik et al. (derived from the extremum principle)

Hoyt, Morral, Purdy, Olson… (valid in case of coarsening)

r

D

)r(cc

)r(cc'r i

/

ii

/

i

0

i

'r)C[r

)C[]D[

ieq

i

/

ii

m

c

)r(cln.RTc

r

V2

+Differential equation

(r, r’) to solve

1

i i

0

i

20

ii

g

mu

Dc

cc

rTRr

V2g

'r

Originally developed for steels


GT effect in multicomponent alloy (I)

Problem: GT effect not correctly accounted when

multicomponent alloy or not pure solute phase

Solve it in the “static” case for a stoichiometric phase

Classical expression

i

/

i

0

i c)r()r(c)r(1c

Matrix0 Matrix1

1 Mole

Phase

1-ε(r)

ci0 ci

/(r) ciβ

GM0 GM1 G +

ε(r)

r

V2 m

ieq

i

/

ii

m

C

)r(cln.RTc

r

V2

“Master” equation for the GT effect

i0

i

iiu

m

c

c)r(1ln.RTcg

r

V2

Equation with only one unknown ε(r)

Equilibrium influenced by the radius r

rTR

V2expc)r(c

g

m/

eq

/


GT effect in multicomponent alloy (II)

Plotting as function of r is difficult but plotting r as

function of is straightforward

r() nearly always has the same shape and its

inverse (r) can then be extrapolated

a

ceq

r

r1)r(

The parameter a is easily

found by interpolation

When coarsening, a

reaches the limit 1.0

eq

ii

eq

i

0

ieq

cc

cc

Same value for all

components


Growth law N°1

Assumption: the slowest diffuser drives the dance

(among the elements contained in the phase)

a

ceq

SlowEltSlowElt1

r

r1

r

D)r(

r

Dvr

Problem: is it satisfactory?


Growth law N°2 (I)

Principle: the GT effect is a degree of freedom and the

system will use it to maximize the growth rate

oR

r

i

/

i

0

i dx)x()r(

)x(TR

)x(cD)x(j i

g

iii Generalized

1st Fick’s law

oo R

r i

2

i

ig

2R

r ii

ig

2

2/

i

0

i dx)x(cx

1

D

)r(jTRrdx

)x(cD

)x(jTR

x4

x4)r(

for steady-state diffusion, 4 x2 ji(x) = cst. for a spherical geom.

oR

r ii

ig/

i

0

i dx)x(cD

)x(jTR)r(

Do some

calculus

Assumption: coars. & GT effect small ci(x) cst. goes out of the integral

)r(cD

)r(jTRr

r

1

R

1

)r(cD

)r(jTRr)r(

/

ii

ig

o

/

ii

ig

2

/

i

0

i

If R0 >> r


Growth law N°2 (II)

)r(cD

)r(jTRr)r(

/

ii

ig/

i

0

i

)c)r(c(r)r(j i

/

ii

i

/

i

0

iim

u )r(cr

V2g

Stephan’s interface

mass balanceGT effect

i/

ii

/

iig

im

u)r(cD

))r(cc(rTRrc

r

V2g

1

0

i

i

i

i

g

mu

2 )1c

c(

D

c

rTRr

V2g

vr

The growth rate is only dependent on the intrinsic diffusion

coefficients (interdiffusion coefficients not necessary)

It is valid for coarsening & small GT effect ( r > 30 nm)

Very close to Kozeschnik

ci/(r) ci

0 near the

critical radius


Prediction quality

The 1st law provides the best results: kinetics given by the

diffusion of the slowest element (Mn)…

0.0E+00

5.0E-08

1.0E-07

1.5E-07

2.0E-07

2.5E-07

0.0 2.4 4.8 7.2 9.6 12.0 14.4 16.8 19.2 21.8

time [h]

pa

rtic

le r

ad

ius

[m

]

-

100

200

300

400

500

600

700

tem

pe

ratu

re [

°C]Law1

Law2

Exp

Temp


The end

Thank you for your attention,

A version of the model is available on SimWeb

• Contact: [email protected]

If you have questions…


Other slides


Calculation of the nucleation rate

Clear separation between the contributions

• Diffusivities are type dependant: hom. or het.

The effect on the energy of the critical radius is

accounted by a separate correction factor

texp

Tk

rGfexpZNN hom

B

C.corr

homhomhom

0

homhom

texp

Tk

rGffexpZNN het

B

Ccorr

hom

corr

hethethet

0

hethet

hethomtot NNN

HOMOGENEOUS NUCL. HETEROGENEOUS NUCL.

)m10(atomsofnumberN 3290

hom

)(.funct.geomN0

het

TR

QexpD)T(D

g

bulk0

bulkhom

TR

QfexpD)T(D

g

bulkpipe0

bulkhet

Usual value 0.6

Usual value ?


Dendrite flux

Dendrite arm focused modeling

• Start with the concentrations in the arm

The interactions between the center of the

dendrite (dispersoides) and the edge

(constituents) can be summarized by an

exchange of solutes ( ji )

Assumption: the concentrations near the

constituents are linked to the concentrations

in the dendrite arm by a simple proportional

law

Arm

iii

Arm

i

Edge

iiiii c

DccDcDj

Dispersoides Constituents

= dentride armspacing

ji

Arm

i

32

i c)1(3

4t)1(4j

Arm

ii

Edge

i c)1(c

Solute Flux:

Mass Conservation:tcDac Arm

iii

Arm

i

ai: empirical value, kept constant

during the simulation

Change of concentration in the

dendrite arm:

CiEdge

CiArm

.Nom

i

Arm

i cc


Efficiency (I)

temp [C]

0

100

200

300

400

500

600

700

- 3 6 9 12 15 18 21 24 27 30 33

time [h]

Rho [m^-2]

1.00E+09

1.00E+10

1.00E+11

1.00E+12

1.00E+13

1.00E+14

1.00E+15

1.00E+16

1.00E+17

- 3 6 9 12 15 18 21 24 27 30 33

time [h]

Elt. c [%w.]

Al 90.10

Cr 0.80

Cu 4.00

Fe 0.30

Mg 1.50

Mn 1.00

Si 0.80

Ti 1.50


Efficiency (II)

Number of precipitated phases: up to 15

Calculation takes less than a day

solute concentrations

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

0 3 6 9 12 15 18 21 24 27 30 33

Cr Cu Fe Mg Mn Si Ti

Documents

Prediction of precipitation kinetics in Al alloys · New project ClaNG+ Collaboration ... • Alloy concentrations •Primary phase ... nearly always has the same shape and its inverse