16 - continuum mechanics of interfaces motivation phase...

me338 · continuum mechanics november 14, 2013

simulation of di↵usion processes

- continuum mechanics of the cahn hilliard equation –

some examples to adrian’s theory

• motivation

phase separation phenomena

• di↵usion equation

linear local · ficknonlinear local · flory huggins

nonlinear nonlocal · cahn hilliard• numerics

discontinuous galerkin method

mixed two-field formulation

• examples

ostwald ripening

mineral exsolution in perthite

• discussion

dani schmid · university of oslo · garth wells · tu delft · krishna garikipati · university of michigan

16 - continuum mechanics of interfaces 1


unmixing of salad dressing

spinodal decomposion of oil and vinegar

motivation · phase separation 2


This scanning-force-microscope image, 50-µm across, shows a unique structure

that can form when two organic polymers, blended from a common solution into

a thin film, are allowed to demix. Light colors represent higher profiles of

the film; dark colors, lower profiles. Cover · Physics Today



unmixing of two-phase colloid-polymer sample

nasa microgravity experiment on international space station

http://www.grc.nasa.gov



solid state di↵usion · mineral exsolution

perthite · alkali-feldspar unmixing - symplectite · finger print texture

thin sections by bjørn jamtveit · physics of geological processes · university of oslo

motivation · spinodal decomposition 5


further reading

• di↵usion equation

fick [1855] · van der waals [1893] · ostwald [1900] · flory [1942] · huggins [1942] · cahn &

hilliard [1958] [1959] · cahn [1959], [1961] · lifshitz & slyozov [1961] · wagner [1961] · langer[1971] · asaro & tiller [1972] · larche & cahn [1973] · srolovitz [1989] · govindjee & simo [1992]

· gurtin [1996] · naumann & balsara [1988] · phillips [2001] · naumann & he [2001]

• numerics

elliott & french [1989] · brezzi & fortin [1991] · barrett, blowey & garcke [1999] · ubachs,

schreurs & geers [2004], [2005] · bansch, morin & nochetto [2005] · kuhl & schmid [2005]

• discontinuous galerkin method

nitsche [1971] · douglas & dupont [1976] · arnold [1982] · engel, garikipati, hughes, larson,

mazzei & taylor [2002] · hansbo & hansbo [2002], [2004] · wells, garikipati & molari [2004]

· mergheim, kuhl & steinmann [2004], [2005] · wells, kuhl & garikipati [2005]

• mineral growth

waldbaum & thompson [1969] · puntis [1992] · aramovich, herd & papike [2002] · vernon [2004]

literature · spinodal decomposition 6


motivation

di↵usion equation

numerics

examples

discussion

simulation of di↵usion processes 7


fickian di↵usion

’... it was quite natural to suppose, that

this law for the diffusion of a salt in

its solvent must be identical with that,

according to which the diffusion of heat

in a conducting body takes place; upon

this law fourier founded his celebrated

theory of heat, and it is the same which

ohm applied with such extraordinary

success, to the diffusion of electricity

in a conductor...’ adolf eugen fick [1855]

�y

�t

= �k

�

2y

�x

2

di↵usion equation 8


fickian di↵usion · linear local

• evolution of concentration c

dtc = �r · j

• flux of concentrations j driven by gradients in the concentration rc

j = �M ·rc

• uniform concentration @equilibrium

dtc = r · (M ·rc)

for ideal mixtures · typical problem redistribution of concentrations of initially

perturbed system · parabolic equation · can be made nonlinear as M = M(c)· does not include nonlocal e↵ects



• fickian di↵usion · linear local

equilibration of concentrations in initially perturbed system

example · equilibration of concentrations 10


flory huggins di↵usion · nonlinear local

• evolution of concentration c

dtc = �r · j

• flux of concentrations j driven by gradients in the chemical potential rµ

j = �M ·rµ

• uniform chemical potential @equilibrium

dtc = r · (M ·rµ)

for non-ideal mixtures · redistribution of concentrations such that chemical

potential is uniformly distributed



flory-huggins free energy of mixing

’... it is customary to correlate the

thermodynamic properties of binary

liquid systems with the so-called ‘ideal’

solution laws resting fundamentally on

an entropy of mixing. when the disparity

between the sizes of the two components

is great, this expression gives entropies

differing widely from the classical

values, which accounts for the large

deviations of high polymer solutions from

‘ideal’ behavior...’ paul flory [1942]

con

=

Pi gi ci +

PiRT ci ln(ci) + exc(ci)



flory huggins di↵usion · nonlinear local

• chemical potential µ in terms of helmholtz free energy

µ = �c =

con

(c)

• configurational free energy

con

con

=

X

i

gi ci +X

i

RT ci ln(ci) + exc

(ci)

energy of components entropy of mixing nonideal mixture

two phase medium c1

= c and c2

= [ 1� c ] with 0 c 1

con

= g1

c+ g2

[ 1� c ] +RT c ln(c) +RT [ 1� c ] ln(1� c) + exc

(c)

• di↵usion equation rµ = @2

c conrc

dtc = r · (M · @2

c conrc)

for non-ideal mixtures · not able to capture phase separation · lack of a

surface free energy term · oscillating distributions



• alkali-feldspar unmixing · configurational energy

con

= g1

c+g2

[1�c]+RTc ln(c)+RT [1�c] ln(1�c)+�1c2

[1�c]+�2[1�c]2c

pure phase end members

potassium KAlSi3

O8

sodium NaAlSi3

O8

margules parameters

�1

= 32098� 16.1356T + 0.4690 p�2

= 26470� 19.3810T + 0.3870 p

uphill di↵usion against rc

phase separation in homogeneous mixture caused by thermal quench

example · mineral exsolution in perthite 14



con

= g1

c+g2

[1�c]+RTc ln(c)+RT [1�c] ln(1�c)+�1c2

[1�c]+�2[1�c]2c

700800

9001000

1100 00.2

0.40.6

0.81

500

1000

1500

2000

2500

3000

3500

4000

4500

concentration ctemperature

config

ura

tional e

nerg

y Ψ

con


potassium KAlSi3

O8

sodium NaAlSi3

O8

margules parameters

�1

= 32098� 16.1356T + 0.4690 p�2

= 26470� 19.3810T + 0.3870 p






con

= g1

c+g2

[1�c]+RTc ln(c)+RT [1�c] ln(1�c)+�1c2

[1�c]+�2[1�c]2c

700800

9001000

1100 00.2

0.40.6

0.81

500

1000

1500

2000

2500

3000

3500

4000

4500

concentration ctemperature

config

ura

tional e

nerg

y Ψ

con


potassium KAlSi3

O8

sodium NaAlSi3

O8

margules parameters

�1

= 32098� 16.1356T + 0.4690 p�2

= 26470� 19.3810T + 0.3870 p





alkali-feldspar unmixing · characteristic double-well potential

con

= g1

c+g2

[1�c]+RTc ln(c)+RT [1�c] ln(1�c)+�1c2

[1�c]+�2[1�c]2c

0 0.2 0.4 0.6 0.8 1

Binodal point

c

ci

cii

Free

Ener

gy

Spinodal pointFEM ModelFree energyCommon tangent

@equilibrium either of the two configurations defined by binodal points



• flory huggins di↵usion · nonlinear local

10x10elements

20x20elements

40x40elements

80x80elements

spurious alternations ·mesh dependency · internal length set by element size

example · phase separation 16


cahn–hilliard di↵usion

’... we would expect that the local free

energy in a region of nonuniform

composition will depend both on the

local composition and on the composition

of the immediate environment. we will

therefore attempt to express as the sum

of two contributions which are functions

of the local composition and the local

composition derivatives, respectively...’

john w. cahn & john e. hilliard [1958]

=

con

(c) + sur(rc)



cahn hilliard di↵usion · nonlinear nonlocal

• chemical potential µ in terms of helmholtz free energy

µ = �c =

con

(c) + sur

(rc)

• surface free energy

sur

gradient energy coe�cient

sur

=

1

2

rc ·rc

• variational derivative �c(•) = @c(•)�r · (@rc(•))µ = @c

con � r2c

rµ = @2

c con rc � r(r2c )

• equilibrium @complete phase separation

dtc = r · (M · [ @2

c conrc�r(r2c)) ] )

non-ideal mixtures · surface energy to capture phase separation



• cahn hilliard di↵usion · nonlinear nonlocal

10x10elements

20x20elements

40x40elements

80x80elements

no mesh dependency · initial bubble size

q�

@2

c con

determined by

example · phase separation 19


cahn hilliard di↵usion · nonlinear nonlocal

small · small interface tension · con

dominates

suf · many interfaces

example · influence of surface energy 20


choice of mobility tensor

constant isotropic mobility

M = M I

concentration dependent mobility

M = c [ 1� c ]M0

/RT

reduces to fickian di↵usion

in the case of ideal solution

direction dependent mobility

M = M iso

I +M ani

n⌦ n

with pronounced di↵usion direction n

mineral growth · direction dependent mobility induced by crystal lattice



cahn hillidard di↵usion · nonlinear nonlocal

anisotropy · direction dependent mobility · M = M iso

I +M ani

n⌦ n

example · anisotropic di↵usion 22


cahn hillidard di↵usion · nonlinear nonlocal

@@@I

@@@I

@@@I

@@@In

anisotropy · direction dependent mobility · M = M iso

I +M ani

n⌦ n

example · anisotropic di↵usion 22


motivation

di↵usion equation

numerics

examples

discussion



discontinuous galerkin method vs mixed two-field formulation

c=0.71

c=0.69

1.0

0.4

1.00.4

discontinuous galerkin

mixed formulation

11250 T6 elements

6400 Q1Q1 elements

elaboration of sensitivity to the nature of the discretization

example · model problem 24


discontinuous galerkin method vs mixed two-field formulation

dg11250T6

cg6400Q1Q1

insensitivity to the nature of the discretization

in collaboration with krishna garikipati & garth wells

example · model problem 25


• discontinuous galerkin method vs mixed two-field formulation

insensitivity to the nature of the discretization

example · numerical comparison 26


motivation

di↵usion equation

numerics

examples

discussion



mineral exsolution · perthite · alkali-feldspar unmixing

0 0.2 0.4 0.6 0.8 1

Binodal point

c

ci

cii

Free

Ener

gy

Spinodal pointFEM ModelFree energyCommon tangent

Spontaneous

demixing region

variation of initial concentration c · gradient energy · time t

example · parameter sensitivity studies 28



c=

0.40

c=

0.57

c=

0.68

c=

0.80

•

• • c0

= c +�c

particulate (bubble) and co-continuous (worm-like) morphologies

example · influence of initial concentration 29



=

0

=

fsp

=

4fsp

=

8fsp

particle/lamella size increases with increasing gradient energy parameter

example · influence of surface energy 30


ostwald ripening

’... diese aus der lehre von oberflachen-

energie zu ziehende schlussfolgerung

ist ja qualitativ seit langem durch

die bekannte kornvergrosserung belegt,

welche feinpulverige korper im laufe

der zeit unter ihrer gesattigten losung

erleiden... die alsdann erforderliche

ausgleichung der konzentration durch

diffusion nimmt indessen eine so lange

zeit in anspruch, dass ich bis jetzt

auch auf solche weise nicht viel weiter

gekommen bin...’ wilhelm ostwald [1900]

stage I min

con

spinodal decomposition · stage II min

sur

ostwald ripening




t⇤=

25

t⇤=

56

t⇤=

252

t⇤=

8050

two stages · minimization of con

and sur · clustering · grain coarsening

example · ostwald ripening 32


solid state di↵usion · mineral exsolution

pressure · temperature

6000 bar · 8000 kelvin

di↵usivity

10

�23

m

2

/s

gradient energy coe�cient

10

�17

m

2

normalized by RT

domain size

300 nm

simulation period

30 years

dimensional example · parameters from natural minerals

example · mineral growth induced by crystal lattice 33



afteroneyear

after30years

ostwald ripening · increase of lamella size



• solid state di↵usion · mineral exsolution

300nm

·after1year

1mm

·after?years

interpretation of geological history of specific regions

in collaboration with dani schmid



linear and nonlinear di↵usion

• linear local di↵usion fick

dtc = r · (M ·rc)

equilibration of concentrations in initially perturbed system

• nonlinear local di↵usion flory huggins

dtc = r · (M · @2c

conrc)

phase separation without surface tension · no internal length scale

• nonlinear nonlocal di↵usion cahn hilliard

dtc = r · (M · [ @2

c conrc� r(r2

c) ] )

phase separation with surface terms · gradients introduce internal length

discussion 36


• cahn hilliard di↵usion · nonlinear nonlocal •

3d di↵usion · couping to deformation · application to geophysics

in collaboration with dani schmid

outlook · future work 37

16 - continuum mechanics of interfaces motivation phase...

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