Multicomponent diffusion simulation

EPJ Web of Conferences 14, 04002 (2011) DOI: 10.1051/epjconf/20111404002

© Owned by the authors, published by EDP Sciences, 2011

“ Fundamentals of Thermodynamic Modelling

of Materials ”

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

Organized by

Bo SUNDMAN [email protected]

PROFESSOR & TOPIC

Lars HÖGLUND

MSE, KTH, Sweden

Multicomponent

diffusion simulation

Multicomponent diffusion

Basis and application

• Diffusion from latin ”spread out”

• Irreversible process no such thing as reversed diffusion, i.e. time has a certain direction!

dt

dm

_B

z

c

D

J

B

B B

:

law

(first)

Fick´s

A

dt

m

d

J

Flux

B B

1

:

volume

mass

mass

or

volume

mole

V

x

V

m

c

m B B

B

/

*

%

/

:

ion

Concentrat

A

Flux and concentration:

dt dn_B

is not necessarily constant

consider a planar case

in B

J J_Bout J_Bin

J_B

z

z

A

dt

dc

dt

dn

A

z

c

n

J

A

J

J

A

dt

dn

B B

B B

B out

B in

B B

(

)

Fick’s second law

Fourier law: J T z

Q

where J_Q is the heat flux, T the temperature and the heat conductivity.

Introduce the concentration of enthalpy as Hm /Vm, where Hm is the molar enthalpy:

J D H V

z

Q Q

m m

( / ₎

The heat diffusivity:

D_Q V_m/c_p

c_P ( H_m/T)_P is the heat capacity under constant pressure.

Ohm's law

I U /R

I is the electric current, U the voltage and Rthe resistance, may be rewritten in a similar form as Fourier's law

J z RA u z e ( / )

where Je I/ A is the electric charge flux, A the cross sectional area of the electric

conductor and z the distance over which the voltage is U. u is the electric potential. By introducing the electrical conductivity as

z/ (RA)

we have a form completely analogous with Fourier's law.

A

z

Introduce chemical potential:

z

c

c

d

d

L

z

L

z

c

D

J

c

f

B

B B B

B B

B B

B

B B

(

)

D

L

d

d c

B B

B

B

_B

_B0

RT

ln

a

_B

_B0

RT

ln

f

_B

RT

ln

c

_B

d

dc

RT

c

d

f

d

c

B B B B B

1

ln

ln

B B B B

c

d

f

d

M

RT

D

ln

ln

1

Ternary system A-B-C:

B

f

(

c

B

,

c

C

)

z

c

D

z

c

D

J

z

c

c

z

c

c

c

M

z

c

M

J

C BC B BB B C C B B B B B B B B B B

Example:

Frames of reference

•We have to measure flux relative something!

•Many choices possible!

•Example: Consider a composite of Cu and Cu-32% Zn:

56 days

0 -0. 1

0 8 16 24 3 2

D istance from interface [cm]

% Z n

-0. 18 w t% Z n C 0 6 days w t% Z n Cu Cu-32%Zn

If we are only interested in the mixing of Cu and Zn we may regard diffusion as if Cu and Zn exchange place: JCu=-JZn.

We then have:

~ : If and 0 : const 1 CuZn Zn Cu Zn Cu Cu Zn Zn Cu Cu Cu Zn Cu m m Zn m Cu Zn Cu Zn Zn Zn Cu Cu Cu D D D J J z c D J z c D J z c z c V V x V x c c z c D J z c D J

D_{is called, for example:}

chemical diffusion coefficient interdiffusion coefficient.

The corresponding frame of reference is usually called number fixed frame of reference because there is no net-flow of atoms:

Pfeil’s (1929) observations of the diffusion-controlled overgrowth of an oxide scale

Par t ic les

st e e l

Sca le

Fe

Scale

(a) (b) (c)

O₂

scal e

It seems as the particles have moved!

Kirkendall’s Diffusion Couple

70-30 Brass

2.5mm Mo w ires

0.13mm diameter

Cu h(t)

Schematic of the Smigelskas–Kirkendall diffusion couple

t K

h(0)

h(t)

z c₁

c₂

0

c*

Matano plane

2

1

0

c

c

zdc

Determining D with diffusion couple technique:

t c z

zdc c

D

c

c

2 /

*

1 *

~

Kirkendall and Smigelskas (1947):

Cu-Zn alloy Cu

Mo wires

d

Inert markers

Zn Cu

Matano plane

Zn Cu

Va

Volume-fixed frame

Lattice-fixed frame

Annihilation of lattice sites

56 days

0 -0. 1

0 8 16 24 32

Distance from interface [cm]

% Zn

-0. 18

w

t%

Z

n

C 0

6 days

wt% Zn

Zn concentration profile

Motion of Mo markers

If we express the fluxes relative the markers (fixed to

the lattice) we may evaluate one flux for Cu and one for Zn and in general J’_Cu + J’_{Zn 0. This is called the lattice-fixed}

frame of reference and we denote it with J’.

We may transform the number-fixed frame by means of:

) ' '

( '

) ' '

( '

Zn Cu Zn Zn Zn

Zn Cu

Cu Cu Cu

J J

x J

J

J J

x J

J

0 ) ' ' ( '

'

_{Zn Cu Zn Cu Zn}

Cu J J J J J

We thus find: CuZn Cu Zn Zn zn Zn Cu CuZn Cu Cu ZnZn Zn Cu CuZn Zn Cu ZnZn zn Cu ZnZn Cu CuZn Zn Cu ZnZn Cu ZnZn Cu ZnZn Cu Cu CuZn Cu Cu ZnZn Cu CuZn Cu Cu CuZn Cu CuZn

D

D

x

D

x

D

D

D

D

D

D

x

D

x

D

D

x

D

D

D

x

D

x

D

D

x

D

D

~

'

'

)

1

(

and

:

have

that we

Observe

'

'

)

1

(

)

'

'

(

'

'

'

)

1

(

)

'

'

(

'

The D´ coefficients are called individual diffusion coefficients.

For crystalline phases it’s usually believed that diffusion occurs through A vacancy exchange mechanism.

Assuming that there is a random distribution of vacant sites and that the number of vacncies is everywhere adjusted to equilibrium, it is possible to derive the following expression for the flux of k in a lattice-fixed frame of reference.

where M_kvais some kinetic factor which gives the rate of exchange if there is a vacancy adjacent to a k atom.

Atomistic treatment of diffusion

k kva

va k L

k

c

y

M

Phenomenological equations

z

L

z

P

L

z

T

L

z

L

J

i _{T P}

n i ki L k

1 1 1 1

They are called phenomenological since they stem from no model, but from the observed conditions of equilibrium.

If we choose to consider an isothermal, isobaric and isopotential system we have:

z

L

J

i n i ki L k

1

z

L

J

_kL _kk

k

Identification

Assuming that the vacancy exchange mechanism is predominant, and by comparing to the expression derived earlier under this assumption, we may identify:

kva va

k

kk

c

y

M

L

Transformation to a volume-fixed frame

z

L

V

c

z

L

J

V

c

J

c

J

J

i n i ki n k k k n i i ki n k L k k k L k k L k k

1 1 1 1

z

L

J

i n i ki k

1 ' or, where, ji n j j k jk

ki

c

V

L

L

1 '

)

(

Transformation to concentration gradients

Applying the chain-rule of derivation on the previous equation:

z

c

c

L

J

j j i n i ki k

1 '

Or equally if the unreduced diffusivities, D_kjare introduced:

z

c

D

J

j n j kj k

1 where, i n

L

Independent set of driving forces

There is a relation between the n concentration gradients, and it’s possible to eliminate one of them:

and thus, the flux is now expressed:

we may identify:

and finally obtain:

z

c

z

c

z

c

z

c

_{n n}

_{1 2 1}

.

...

z

c

D

D

J

j n j kn kj k

)

(

1 1 kn kj n

kj

D

D

D

z

c

D

J

j n j n kj k

1 1

Summary of steps taken when transforming from M’s to D’s

k

M

L

_ki

'

ki

L

D

_kjn

n kj i

D

kj i

D

kj

D

Identification Concentration gradients

Independent set of driving forces

Intrinsic diffusion coefficients Interdiffusion coefficients

Concentration gradients

Independent set of driving forces

From absolute reaction rate theory arguments the mobility coefficient for an element B, M_B, may be divided into a frequency factor, M_B0_and

an activation enthalpy Q_B, i.e.

B mg B

B

RT

RT

Q

M

M

0

exp(

)

1

Where mg_{is a factor taking into account the effect of the ferromagnetic}

transition. mg_{is a function of alloy composition and is represented by:}

)

exp(

)

6

exp(

RT

Q

_B

mg

Modeling of the atomic mobility

Frames of reference

Va n

K

k

J

J

1

0

1

k n

K

k

V

J

0

n

S K

k

J

Lattice-fixed frame of reference:

- Defined by the inert markers, or that There is no net flow of lattice sites.

Volume-fixed frame of reference:

- Defined in such a way that there is no net flow of volume.

DICTRA frame of reference:

When treating the composition dependency of the mobility it has been found superior to expand the logarithm of the mobility rather than the value itself, i.e.

mg

B B

B

RT

M

Q

RT

RTM

RT

ln

ln

0

ln

Modeling of the atomic mobility

A X_B

Because ln

RTMi

is often found to have a fairly linear composition dependency

RTM A

ln

Modeling of the atomic mobility

Both RTlnM_B0_{and -Q}

Bwill in general depend upon the composition,

the temperature and pressure.

In the spirit of the CALPHAD approach, the composition dependency of these two factors is represented with a linear combination of the values at each end-point of the composition space, and a Redlich-Kister expansion.

i j i

m

r

r j i j i B r j

i i

i B

i

x

x

x

x

x

0

,

B

(

)

RTM

_Al

RT

M

_Al

Q

_Al

RT

ln

ln

0

Example: Expression for the mobility of Al in a binary system Ni-Al.

Where,

Ni Al Al Ni

Al Ni

Al Ni

Al Al Al

Al

x

x

x

x

,

Al

Modeling of the atomic mobility

Ni X_Al Al

Al Al

Ni Al

Logarithm of mobility for Al in pure Ni

Logarithm of mobility for Al in pure Al

Modeling of the atomic mobility

Ni Al Al

,

Building a kinetic database

• Collect experimental information

Different levels of ambition!

In a database for A-base alloys:

1. Find Tracer or Dilute diffusivities of the elements (B,C,D…) in A. These can be entered directly in the database, assuming that the mobilities are concentration independent.

Example

PARAM MQ(FCC_A1&NI,NI:VA),, -287000+R*T*LN(2.26E-4)

Tracer diffusion coefficient of Al and Ni in pure Ni:

)

284000

exp(

10

5

.

7

4

*

RT

D

_Al

)

ln(

)

ln(

D

_i*

RT

RTM

_i

RT

The tracer diffusion coefficient is directly related to the mobility:

In the database:

)

287000

exp(

10

26

.

2

4

*

RT

D

_Ni

Building a kinetic database

Next level of ambition:

2. Find Tracer, Dilute and self diffusivities of the elements (A,B,C,D…) in the elements (A,B,C,D). These can be entered directly in the database, assuming that the mobilities are linear functions of composition.

Example

PARAM MQ(FCC_A1&AL,AL:VA),, -142000+R*T*LN(1.71E-4)

Tracer diffusion coefficient of Al and Ni in pure Al:

)

142000

exp(

10

71

.

1

4

*

RT

D

_Al

In the database:

)

145900

exp(

10

4

.

4

4

*

Building a kinetic database

High level of ambition:

3. Assess important systems A-B, A-C, B-C, A-B-C using ALL experimental information.

Possible experimental information

Intrinsic diffusion coefficient

DI(phase,k,j,n) LOGDI(phase,k,j,n)

Tracer diffusion coefficient

DT(phase,A) LOGDT(phase,A)

Interdiffusion coefficient

DC(phase,k,j,n) LOGDC(phase,k,j,n)

*

A

D

n kj i

D

n kj

Example of data

Let us start by considering the Ni-rich side of the system and

assume that the mobilities are independent of composition, i.e.

Ni Al Al

Al

Al

Then we need only two parameters for the database, i.e.

Ni Ni Ni

Al

and

4

10

5

.

7

ln

284000

ln

Ni_Al

RT

D

_Al

RT

These two parameters we can determine from the tracer

diffusion coefficients of Al and Ni in Ni, i.e.

4

10

26

.

2

ln

287000

ln

Ni_Ni

RT

D

_Ni

RT

Assessment of data: Binary Ni-Al system

Next step to improve the description would be to assume

a linear concentration dependence for the mobilities of Al

and Ni.

We then need the mobilities in pure Al as well, e.g.

4

10

71

.

1

ln

142000

ln

Al_Al

RT

D

_Al

RT

4

10

4

.

4

ln

145900

ln

Al_Ni

RT

D

_Ni

RT

Finally, in order to reach a good agreement with our experimental

data we would like to introduce two interaction parameters.

Ni Al Ni Ni

Al

Al

and

,

,

These will be fitted in an optimisation procedure in order to

give the best possible agreement with our experimental data

points.

Optimization software needed !!!

Assessment of data: Binary Ni-Al system

The Software

Software package for simulation of diffusion controlled reactions in multi-component alloys.

Simulation on geometries, which may be reduced to one dimension (e.g. planar, cylindrical, spherical)

Linked to Thermo-Calc, which provides all necessary thermodynamic properties.

The result of more than 20 years and 60 man-years R&D at:

Royal Institute of Technology in Stockholm, Sweden

Basic calculation procedure

J z t c

DATABASES

Kinetic Thermodynamics

Mobilities Gibbs Energy

Diffusivities

Solve Diffusion

where

2 2

c G

Boundary conditions

(External or Internal)

All simulations depend on assessed kinetic and thermodynamic data, which are stored in databases A numerical finite difference scheme is used for solving a system of coupled parabolic partial differential equations

z c D J

2 2

~ c

G M Dn

kj

Why a database with Mobilities?

Thermodynamic Databases

(

The CALPHAD approach)

Thermochemical measurements:

enthalpy, entropy, heat capacity, activity

Phase equilibria:

liquidus, solidus, phase boundary,

Gibbs Energy of Individual Phases

Applications

) , , (x T P f

Kinetic Databases

(

in a CALPHAD spirit

₎

Diffusion without a chemical gradient:

- Tracer diffusion coefficients

Diffusion under a chemical gradient:

- Chemical interdiffusion coefficients

- Intrinsic diffusion coefficients

Logarithm of the Atomic Mobility for Individual Elements

Applications

( , , ) ln RTM_B f x T P

Several application types

- Diffusion in single-phase systems

- Diffusion with moving interfaces (growth, dissolution of particles etc.)

- Cell calculations (particle distributions, immobile interfaces etc.)

Diffusion with a moving interface

C_k

z

k

J

k

J

k

c

k

c

k

k

k

k

c

)

J

J

(c

n-1 Flux Balance Equations: F-B Equations solved as:

2 1

n

k

k k

k

k c ) (J J )

(c

n-1 unknowns:

n-2 chemical potentials. Velocity of phase boundary,

Sharp interface with

assumption of local equilibrium

Diffusion with a moving interface

x

x

+

Binary example: Fe-C Ternary example: Fe-Cr-Ni

Some application examples

to transformations in steel

Growth or dissolution of carbides

Microsegregation during solidification

Nitriding of steels

Nitrocarburising of steels

-phase precipitation in stainless steels

Transient Liquid-Phase bonding of alloys

Sintering of cemented carbides

and much more …

Microsegregation during solidification Carbide dissolution

cementite

austemite

Purpose of DICTRA

Have a possibility to Teach, i.e. bring new insight into problems.

Purpose of DICTRA

Provide an engineering toolfor materials and process design.

…by allowing simulations to be performed with realistic conditions and data on alloys of practical importance.

Region

Grid

Distribution of node points for numerical calculations

Concentration Profile

Concentration, C_k of an element as a function of distance z

C_k

Global Conditions

T=1273K

P=1 atm

Conditions valid for entire system, T and P

Boundary Conditions

ac=1 (carburization)

Conditions that apply to region boundaries (could be functions of time and

Moving Phase Boundary Calculations

• Used for calculating growth or dissolution of a phase. • Assumptions:

• Local equilibrium holds at the phase boundary, i.e. concentrations at the boundary can be calculated from an equilibrium calculation in T-C.

• Diffusion controls the movement of the phase boundary •Application examples:

Carbide dissolution

Solidification

Growth of phase in a stainless steel

Moving phase boundary simulation

Solve diffusion equation in each phase

Calculate displacement of phase boundary

FCC BCC

C_k

z

v

k

Unknowns: Tie-line, specified by n-2 a_ior _i Velocity of phase boundary, v

Equations: n-1 flux-balance equations,

Solved as:

M-P-B theory

k

k k

k c ) J J

(c

0 ) J J ( ) c

(c_{k k k k}

C_k

z

k

J

k

J

k

c

k

c

v

Moving Phase Boundary

• Moving phase boundaries simulations may be setup in DICTRA in two different ways:

• Introducing two or more adjacent regions containing different phases

•Entering an inactive phase (formed when thermodynamically stable)

Phase 1 Phase 2

Phase 1

inactive phase

Assumptions – Sharp interface model

•local equilibrium

•profiles as piecewise linear functions

•volume is independent of composition

•volume fixed frame of reference

Calculation scheme - binary case

•determine tieline

•solve PDE

•solve flux balance equation

LE – Tieline in Binary system

+

c/ c/

c J

PDE

z

c

D

z

t

c

_k

k k

Flux balance equation

LE

from

c

,

c

_{k k}

)

,

(

p

k

k

k

k

J

J

c

c

k=1,2….n-1

)

LE

and

(

PDE

from

J

,

J

_{k k}

u-fraction

•mole fraction:

•mass fraction:

•u-fraction:

i

i i

k k k

m N

m N w

i i k k

N N x

s i

i k k

N N u

Calculation scheme

-multicomponent case

•fix activities/potentials

•determine tieline

•solve PDE

•solve flux balance equations

•guess new activities/potentials until

• flux balance equations are fulfilled

LE in ternary systems

A B

( c

_B

- c

_B

) = J

_B

- J

_B