16 - continuum mechanics of interfaces motivation phase...
Transcript of 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
me338 · continuum mechanics november 14, 2013
unmixing of salad dressing
spinodal decomposion of oil and vinegar
motivation · phase separation 2
me338 · continuum mechanics november 14, 2013
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
motivation · phase separation 3
me338 · continuum mechanics november 14, 2013
unmixing of two-phase colloid-polymer sample
nasa microgravity experiment on international space station
http://www.grc.nasa.gov
motivation · phase separation 4
me338 · continuum mechanics november 14, 2013
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
me338 · continuum mechanics november 14, 2013
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
me338 · continuum mechanics november 14, 2013
motivation
di↵usion equation
numerics
examples
discussion
simulation of di↵usion processes 7
me338 · continuum mechanics november 14, 2013
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
me338 · continuum mechanics november 14, 2013
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
di↵usion equation 9
me338 · continuum mechanics november 14, 2013
• fickian di↵usion · linear local
equilibration of concentrations in initially perturbed system
example · equilibration of concentrations 10
me338 · continuum mechanics november 14, 2013
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
di↵usion equation 11
me338 · continuum mechanics november 14, 2013
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)
di↵usion equation 12
me338 · continuum mechanics november 14, 2013
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
di↵usion equation 13
me338 · continuum mechanics november 14, 2013
• 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
me338 · continuum mechanics november 14, 2013
• 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
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
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
me338 · continuum mechanics november 14, 2013
• 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
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
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
me338 · continuum mechanics november 14, 2013
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
example · mineral exsolution in perthite 15
me338 · continuum mechanics november 14, 2013
• flory huggins di↵usion · nonlinear local
10x10elements
20x20elements
40x40elements
80x80elements
spurious alternations ·mesh dependency · internal length set by element size
example · phase separation 16
me338 · continuum mechanics november 14, 2013
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)
di↵usion equation 17
me338 · continuum mechanics november 14, 2013
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
di↵usion equation 18
me338 · continuum mechanics november 14, 2013
• 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
me338 · continuum mechanics november 14, 2013
cahn hilliard di↵usion · nonlinear nonlocal
small · small interface tension · con
dominates
suf · many interfaces
example · influence of surface energy 20
me338 · continuum mechanics november 14, 2013
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
di↵usion equation 21
me338 · continuum mechanics november 14, 2013
cahn hillidard di↵usion · nonlinear nonlocal
anisotropy · direction dependent mobility · M = M iso
I +M ani
n⌦ n
example · anisotropic di↵usion 22
me338 · continuum mechanics november 14, 2013
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
me338 · continuum mechanics november 14, 2013
motivation
di↵usion equation
numerics
examples
discussion
simulation of di↵usion processes 23
me338 · continuum mechanics november 14, 2013
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
me338 · continuum mechanics november 14, 2013
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
me338 · continuum mechanics november 14, 2013
• discontinuous galerkin method vs mixed two-field formulation
insensitivity to the nature of the discretization
example · numerical comparison 26
me338 · continuum mechanics november 14, 2013
motivation
di↵usion equation
numerics
examples
discussion
simulation of di↵usion processes 27
me338 · continuum mechanics november 14, 2013
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
me338 · continuum mechanics november 14, 2013
mineral exsolution · perthite · alkali-feldspar unmixing
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
me338 · continuum mechanics november 14, 2013
mineral exsolution · perthite · alkali-feldspar unmixing
=
0
=
fsp
=
4fsp
=
8fsp
particle/lamella size increases with increasing gradient energy parameter
example · influence of surface energy 30
me338 · continuum mechanics november 14, 2013
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
di↵usion equation 31
me338 · continuum mechanics november 14, 2013
mineral exsolution · perthite · alkali-feldspar unmixing
t⇤=
25
t⇤=
56
t⇤=
252
t⇤=
8050
two stages · minimization of con
and sur · clustering · grain coarsening
example · ostwald ripening 32
me338 · continuum mechanics november 14, 2013
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
me338 · continuum mechanics november 14, 2013
mineral exsolution · perthite · alkali-feldspar unmixing
afteroneyear
after30years
ostwald ripening · increase of lamella size
example · mineral growth induced by crystal lattice 34
me338 · continuum mechanics november 14, 2013
• solid state di↵usion · mineral exsolution
300nm
·after1year
1mm
·after?years
interpretation of geological history of specific regions
in collaboration with dani schmid
example · mineral growth induced by crystal lattice 35
me338 · continuum mechanics november 14, 2013
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
me338 · continuum mechanics november 14, 2013
• cahn hilliard di↵usion · nonlinear nonlocal •
3d di↵usion · couping to deformation · application to geophysics
in collaboration with dani schmid
outlook · future work 37