1) Technische Universität Berlin Institut für Mechanik - LKM Einsteinufer 5

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Technische Universität BerlinFakultät für Verkehrs- und Maschinensysteme , Institut für Mechanik

Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller

Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: [email protected], 2008

A multi-component theory ofA multi-component theory of

solid mixtures with higher gradientssolid mixtures with higher gradients

and its application to binary alloysand its application to binary alloysby

A. Brandmair,1) T. Böhme,1),2) W. Dreyer,3) W.H. Müller1)

STAMM 2008 Symposium on Trends

in Applications of Mathematics to Mechanics

Levico Italy, September 24, 2008

1) 1) Technische Universität BerlinTechnische Universität Berlin Institut für Mechanik - LKM Institut für Mechanik - LKM Einsteinufer 5Einsteinufer 5 D-10587 BerlinD-10587 Berlin

German FederalEnvironmental Foundation

2) 2) ThyssenKrupp Steel AGThyssenKrupp Steel AG Werkstoffkompetenzzentrum Werkstoffkompetenzzentrum Physikalische Technik Physikalische Technik Kaiser-Wilhelm-Straße 100Kaiser-Wilhelm-Straße 100 D-47166 DuisburgD-47166 Duisburg

3) 3) Weierstraß-Institut fürWeierstraß-Institut für Angewandte Analysis Angewandte Analysis und Stochastik und Stochastik Mohrenstr. 39Mohrenstr. 39 D-10117 BerlinD-10117 Berlin

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Outline Introduction and motivation: Three types of microstructural change

An experimental investigation of spinodal decomposition and coarsening

Constitutive equations for diffusion flux and stress

Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients)

Reduction to the case of binary mixtures

Numerical simulation of spinodal decomposition and coarsening

Comparison with the experiment

Homogenization and effective properties

Conclusions and outlook

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4




SMT SnPb solder joints: formation of interface cracks spinodal decomposition

microstructural coarsening

Ball Grid Arrays and solder ball before and after 4000 temperature cycles aging at RT after (a) 2h, (b) 17d and (c) 63d

(a) after solidification, (b) 3h and (c) 300 h at 125°CMELF miniature resistor and solder joint before and after 3000 temperature cycles

Microstructural changes in solids I

Will and, if so, how will the microstructural changeWill and, if so, how will the microstructural change influence the material properties ? influence the material properties ?

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Introduction: Microstructural changes in solids II

after solidification after 2h after 20h after 40h

Ag-Cu: aging at 1000 Kelvin

Ag-rich

Cu-rich

cracks along

the phase

boundaries

Si-chipFlip-Chipunderfill

substrate

solder-ballsSnPb solder balls

with lead-free bumps

decomposition + coarsening in the bulk

Cu

Leadfree solder, e.g., AgCu28:

Formation of Inter-MetallicCompounds IMCs (Cu6Sn5, Ag3Sn)at the interface and in the bulk

Leadfree solder, e.g., SnAg3.8, SnAg3.8Cu0.7:

thor.inemi.org/webdownload/newsroom/Articles/Lead-Free_Watch_Series/Oct05.pdf


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• Triggered by different diffusion coefficients Cu atoms move to pile up IMCs.

• Kirkendall voids appear, e.g., between Cu substrate and thin Cu3Sn layer due to migration of Cu atoms from Cu3Sn to Cu6Sn5, which is much faster than Sn-diffusion from Cu6Sn5 towards Cu3Sn.

• Unbalanced Cu-Sn interdiffusion generates atomic vacancies at lattice sites which coalesce to voids.

• Model: vacancy diffusion

Introduction: Microstructural changes in solids III


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8




Experiments I: Setup & realization

material ► eutectic Ag-Cu temperature ► 970 K aging time ► 0 - 40 h etching (for Ag) ► solution of NH3-H2O2

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► instantaneous decomposition ► coarsening (Ostwald ripening)► light: α-phase (Ag-rich) , dark: β-phase (Cu-rich)

aftersolidification

after 2haging

after 20haging

after 40haging

Experiments II: Micrographs

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Experiments III: Image analysis determination of the β-areas and the total number of β-precipitates N by

means of Digital Image Analysis (DHSTM) Attention: 2D analysis of a 3D problem (observation of one cross-section) Solution: statistical averaging

• sufficient large areas of intersection • analysis of various micrographs at each coarsening stage

iA

N

iiA

NA

1

1

pictures

AA

6

1

π2

3 A

r

235.1

A

a

(individual photo)

(individual stage)

(spherical phases)

(oblate spheroids)

(cf., Underwood, 1970)

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faster coarsening rate for oblate spheroids with and

t 1/3- dependence well-known from LSW-theories

Experiments IV: Coarsening rates

baV 234 π

3/1ta 3/1tr

3/1024.0 tr 3/1032.0 ta

1/3

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Nomenclature I motion

displacements, velocity, deformation gradient

strains and stresses

),( tXx jii

,dd, txXxu iiiii j

iij

X

xF

0det ijFJ

gradientntdisplaceme:ijijj

iij F

X

uH

ijmjmiij cJFFC 3/2

strainslinearized:,,)( tensor)strain s(Green'21 jiijij CG

1det ijc

ij11 )()( jnmnimij FFJt

,: tensorstressCauchy

orGreen tens-Cauchyright:

shapeofn deformatiotensor,Green-Cauchyrightreduced:

tensorstressKirchhoffPiola2nd :

)(21 jiijij HH

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Nomenclature II

Primary variables:

Variables determined by partial mass balances (w/o chemical reactions), and balance of momentum and internal energy:

total mass balance

)(,)()( )def(

iiii

ii

i

i

Jx

J

xt

0,)( Def.

i

i

i

Jxt

j

iijii

i xqe

xt

e

,...,1with,)or(, euii

ijijj

i

xt

(no external forces)

constitutive equations for ijii qJ ,, sought

in a situation wheresolids separated byphase boundaries“move” toward equilibriumwhich, again, is characterized by a boundary→ higher gradienthigher gradienttheoriestheories required !

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Constitutive relations for the diffusion flux (w/o thermo-diffusion)

w/o gradients with density gradients(w/o strain gradients)

TxBJ

jiji

1

( : mobility)

F̂Def.

chemical potential

Helmholtzfree energy

density

δ

ˆδDef. TF

T

kl

klk

k

Def.

δ

δ

),,,,(ˆ ijiji cTF

ijB

functional derivative:

,,1 , ),,(ˆ ijcTF

16




Constitutive relations for the stress tensor

without gradients

Castigliano’s

2nd theorem

specific Helmholtz free energy

with density gradients(no strain gradients)

/2

31

Def. Fp kk pressure

),,,( ijccTF

),,,,,,( ijijiiji CcccTF

ij

ij

C

Ft

02

1,,1 , ),,(

ijCcTF

),,,,,,,( ijijiiji ccccTF

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Entropy principle I: Historic remarks

Entropy principles:

a) Clausius & Duhem (18th century)

b) Coleman-Noll (1963)

c) Green-Nagdhi (1967)

d) Müller (1968), Liu (1972)

Shortcomings of the above principles:

a) Entropy flux relation, application to mixtures of solids ?

b) Entropy balance global / local / principle for every constituent (too strong) ?

c) Lagrange multipliers in order to consider the

balances as constraints (some class of materials requires

“additional” constraints, e.g., materials with gradients as variables)

Therefore: Attempt to formulate a strategy based on commonly accepted points of the aforementioned principles

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Entropy principles:




d) Müller (1968), Liu (1972)








R. Clausius

P. M. M. Duhem

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Entropy principles:




d) Müller (1968), Liu (1972)









B.D. Coleman

W. Noll

A.E. Green

P.M. Nagdhi

I. Müller

I-S. Liu

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Entropy principles:




d) Müller (1968), Liu (1972)









Duhem: Tq ii /

radiation: Tq ii /3

4ideal gas:

)( 32 ijp

ijTqi j

mixtures:

iTT

qi Ji

1

22





Clausius: form?locald tS T

Q

Entropy principles:




d) Müller (1968), Liu (1972)








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Entropy principles:




d) Müller (1968), Liu (1972)








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Entropy principles:




d) Müller (1968), Liu (1972)








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0 z

zz DF

Entropy principle II: Formulation

Two constitutive quantities: entropy density and entropy flux .Constitutive relation of the entropy density:

Local balance for entropy density:

Entropy production positive definite & of the form fluxes x driving forces:

(Absolute) temperature defined as follows:

Stress tensor decomposed into an elastic and a dissipative part:

)( riablesprimary va theofsderivativeriables,primary vaS

i

ii

ixt

2nd law

ijijijdisselast

0eq

eT

Def.1

TTD ijii

z

1,,

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Entropy principle III: The evaluation procedure

H.W. Alt, I. Pawlow: On the entropy principle of phase transition models with a conserved order parameter. Advances in Mathematical Science and Applications, 6(1), pp. 291–376, 1996:

Balances interpreted as “evolution equations” for variables:

Balances viewed as a system of algebraic equations, i.e.,

choose right hand sides arbitrarily & calculate left hand sides.

Right hand sides of the balances + chain rule list of arbitrary terms:

Construct constitutive relations such that 2nd law is identically satisfied.

... t

ij

i

i

ij

ij

i

i

j

ii

i

i

i

i

ii

x

q

x

ee

xxxx

J

xx

,,,,,,,,,,,,

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28




Mixtures w/o higher gradients I: Choice of variables

Simple -component mixture without reactions / viscous effects:

Different representations of entropy density:

with a constant number of variables, e.g.:

Different representations useful under different circumstances, e.g.:

},{ ijij cC

ijijelast

ijtS

pS

S

S

stressesKirchhoffPiola2ndtheofDefinition:

pressuretheofDefinition:

,...,1,potentialchemicaltheofDefinition:ˆ

law2ndtheofonExploitati:~

1,,1,,,1

),,(),,,(),,(ˆ),,(~

ijijijij CcTSccTScTScS

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Mixtures w/o higher gradients II: Entropy production

Local entropy balance:

Chain rule for

Kinematic relation

i

i

i

i

ii

xxS

x

S

t

S ~~~

t

c

c

S

t

S

t

S

t

S ij

ij

~~~~

right hand side of balances

(note: balance of momentum

not required for exploitation,

since not in )

),,(~ ijcS

i

kl

kliii x

c

c

S

x

S

x

S

x

S

~~~~

,

)(3

2

d

d

d

d32

32

32 ikjliljk

j

i

i

iklkl

kl

FFFFx

Jx

CJCJtt

c

chain rule

kinematic relation

ii S

~

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Mixtures w/o higher gradients III: Entropy production

Substitution and rearrangement: a) separate terms into flux / force:

b) separate terms linear in (cf., arbitrary terms)

Definition entropy-flux (first parenthesis): Residual inequality:

klklij

klikjliljk

ij

j

i

ii

iii

ii

i

c

SCJ

S

TS

c

SFFFFJ

Tx

x

Tq

S

xJ

SJ

T

q

x~

3

2~

~~

/1~~

32

32

1

elast

11

z

zzDFji x /

1

Def.~S

JT

q ii

i

0.../1

~

1

j

i

ii

ii

xx

Tq

S

xJ

Q P

ij

ij ,0andinlinear

0Q0P

terms linear in drop out

Galileian invariance

i

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Mixtures w/o higher gradients IV: Heat & diffusion flux

0/1

~

1

i

ii

i

x

Tq

S

xJ

Helmholtz free energy density ),,(ˆ),,(~ ijij cTFcF

experimentallyinconvenient quantity

T

F

T

S

Def.ˆ1

~(chemical potential) Legendre Transformation

1

111

0

Tx

JTx

JJi

ii

ii Constraint

jij

jiji

x

TT

x

Tq

)(/1 (Fourier’s law of heat conduction)

1

1

1

1

,

iij

iji JJTx

BJ ( : mobility)

No coupling, quadratic form

ijB

Q-bit

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Mixtures w/o higher gradients V : Selected results

Pressure

0~

3

2~

~~

0... 32

32

1

elast

klklij

klikjliljk

ij

j

i

c

SCJ

S

TS

c

SFFFFJ

Tx

Legendre transform applied to ),,,( ijccTF

/2 F

p

Legendre transform applied to

1~

~)

~(

STST

F

kkp elast3

1

)pressureofdefinitionfromdirectlyfollows(

),,( ijCcTF

ijijij

C

F

C

FJt

/

2/

2 0

1

1p

P-bit

Gibbs-Duhem relation

2nd Piola-Kirchhoff stress tensor

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Mixtures with higher gradients I: Functional representations

functional representation of the entropy density

Note:- Choice of Higher Gradients depends on the problem

- Present choice: Convenient for diffusion problems & for definition of the chemical potential

111

,,,,,

,,,,,,,,,,,,,

,,,,ˆ,,,,~

ijijiijiijijiiji

ijijiijiji

CccceScccceS

cTSceS

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Mixtures with higher grad. II: Specialization to binary alloy

Now: Binary alloy A-B:

Relating difference of chem. pot. to derivative of Helmholtz free energy density:

with Helmholtz free energy density

diffusion flux for a binary alloy:

1

1

1

1

,)(

iij

iji JJ

xT

BJ

T

BB

xBJ

xBJ

ijAAij

jABiji

BjABiji

A

,)(

,)(

(isothermal case)

BAB ccc

F ,

δ

δ1

),,,,,,,( ijijiiji ccccTF

c

F

xTBJJ

jiji

Bi

δ

δ1)(

Def.

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lkij

kl

lkij

klklklijkl

ijij

x

c

x

cb

xx

caTcTC

F

2

local )Δ)(,(

Transformation to the reference configuration / re-definition of mobility

Problem with free energy density (more general case):

Decomposition & Taylor expansion

HGC can be identified & calculated by microscopic atomistic theories (e.g., EAM)

it follows (Böhme et al., 2007): (periodic arrangement of the lattice)

Approach for elastic contributions

ijijB 20

phase diagram ? ?

...))(()()(2

1),,(),(

),(),(

2

elast

cccc

Fc

c

Fc

c

FCcTFcTFF lk

kl

lkkl

kl

klk

k

k

ij

ijCcbijCcaL

0iL

,)Δ( Hooke

elast TF ijijij

leading term

2

1

Mixtures with higher grad. III: Specialization to binary alloy

c

F

XBJ

jiji

δ

δ0

),,,,,,,( ijijiiji ccccTFF

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Partial mass balance & mass concentration :

Bjij

i

i

ccc

F

XB

X

J

t

c

,

δ

δ

d

d00

/BBc

c

F

δ

δ

lk

mn

mn

kl

l

mn

k

op

mnop

kl

l

mn

kmn

kl

lk

kl

lkkl

XX

a

XX

a

XX

cA

X

c

X

c

c

A

XX

cA

c

FF

22

2elastdiagr.phase 22

)(

klkl

kl bc

aA

Material parameters:

c

baA

B

ij

ijijij

ij

, offunctionsas method

atom embeddedthebycalculatedbecanHGCs,:,,

t coefficiendiffusionthetolinkedbecanmobility,:

Mixtures with higher grad. IV: Specialization to binary alloy

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39




Application: Spinodal decomposition in AgCu I:

Simulations in 1D (no external stress)

Study concentration development along a line

40




Application: Spinodal decomposition in AgCu II:

Simulations in 1D (no external stress)

Fortran 95, FFTPack, explicit Euler scheme

s103Δ 7t

29.00 c

K1000T

256N

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Application: Spinodal decomposition in Ag-Cu II:

Simulations in 1D (tensile stress: 103 MPa)

Fortran 95, FFTPack, explicit Euler scheme

s103 7t

29.00 c

K1000T

256N

stresses

accelerate

coarsening

42




Application: Spinodal Decomposition in Ag-Cu III:

2D (no external stress)

initial 3000 time loops 6000 loops 60 000 loopssec10Δ 8t

29.00 cK1000T

Cu-rich phase

Ag-rich phase

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Experiment and simulation: Coarsening rates

By image analysis of experiments and computer generated microstructural evolution

3D simulations required ?

45













46




Average elastic properties by homogenization I

Homogenization performed by S.V. Sheshenin, M. Savenkova (FE-program “Elast”)

Boundary value problem (plane strain analysis) for a given micrograph (= RVE):

0ij ij

011 0

012 0

022 0

Loading sequences applied, e.g.:

etc.(0) 0i ij ju x

, ,

(0)

0, , , , , 1, 2

ijkl s k l j

i s i

C x ui j k l s

u x u

1( )dij ij s

V

x VV

1

( )dij ij s

V

x VV

effij ijkl klC eff 0

11 11/ , 1, 2ij ijC ij

eff 012 12/ , 1, 2ij ijC ij

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Average elastic properties by homogenization II

AgCu28 simulated temporal development of microstucture

1 2 3 4 5 6

Conclusion

material is cubic, just like its constituents Ag and Cu

changing microstructure leads to no change in elastic coefficients

48




Average elastic properties by homogenization III

SnPb37 effective elastic moduli of simulated microstucture

Conclusions

composite material is less “tetragonal” due to the slight difference between C1111 and C2222 for Sn and the presence of the cubic Pb

laminate theory gives similar results:

Tin (Sn) 29.751111 C 00.441122 C 52.952222 C 93.211212 C

Lead (Pb) 66.491111 C 31.421122 C 66.492222 C 98.141212 C

Attention

lead (Pb) is cubic = 3 elastic constants, tin (Sn) is tetragonal = 6 elastic constants

2D excerpt:

nkllmijmnkqqppllmijm

ijnkijnk

CCCCCCCC

CC

21

22221

22

1122

1222

)2(2

)1(1 ijklijklijkl CvCvC

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50




Conclusions and Outlook

Phase field / higher gradient models to be used for microstructural changes in multi- component alloys should not simply be postulated but rather based on balance laws for physical quantities.

Material theory should and can be used to derive the corresponding field equations.

Material parameters in these relations should not simply be “guessed,” rather they should be obtained from experiments that are independent of the to-be-described phenomenon and, eventually, also be obtained from atomic methods (e.g., embedded atom methchnique).

The spinodal decomposition observed in some solder/welding materials as well as the subsequent process of coarsening can be modeled quantitatively using such a strategy.

Homogenized elastic properties for experimentally observed as well as predicted micrographs showing microstructural change have been obtained.

Non-linear homogenized material properties were not obtained yet.

Documents

1) Technische Universität Berlin Institut für Mechanik - LKM Einsteinufer 5