Phase field modelling of phase separation using the Cahn ... · PDF filePhase eld modelling of phase separation using the Cahn-Hilliard equation Davide Fiocco January 25,
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Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Phase field modelling of phase separationusing the Cahn-Hilliard equation
Davide Fiocco
January 25, 2012
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Phase separation of partially miscible liquids
Typical phase diagram of an A+B fluid
Figure: A quench
I We take a configuration with aconcentration of A equal to φ0
I If we lower the temperature thesystem is unstable
I It prefers to separate in twophases and form interfaces
We want to study the dynamics of the process
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Phase separation of partially miscible liquids
Typical phase diagram of an A+B fluid
Figure: A quench
I We take a configuration with aconcentration of A equal to φ0
I If we lower the temperature thesystem is unstable
I It prefers to separate in twophases and form interfaces
We want to study the dynamics of the process
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Phase separation of partially miscible liquids
Typical phase diagram of an A+B fluid
Figure: A quench
I We take a configuration with aconcentration of A equal to φ0
I If we lower the temperature thesystem is unstable
I It prefers to separate in twophases and form interfaces
We want to study the dynamics of the process
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Phase separation of partially miscible liquids
Typical phase diagram of an A+B fluid
Figure: A quench
I We take a configuration with aconcentration of A equal to φ0
I If we lower the temperature thesystem is unstable
I It prefers to separate in twophases and form interfaces
We want to study the dynamics of the process
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Phase separation of partially miscible liquids
Typical phase diagram of an A+B fluid
Figure: A quench
I We take a configuration with aconcentration of A equal to φ0
I If we lower the temperature thesystem is unstable
I It prefers to separate in twophases and form interfaces
We want to study the dynamics of the process
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
How to do so?
The phase field method
I Coarse grained method
, (in this flavor) a coarsened version ofthe Ising model evolving with Kawasaki exchange kinetics
I Can be adapted to solve a much wider class of problems
I I’ll try to justify its use in this case
Maths needed. Hold on.There might be other movies
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
How to do so?
The phase field method
I Coarse grained method, (in this flavor) a coarsened version ofthe Ising model evolving with Kawasaki exchange kinetics
I Can be adapted to solve a much wider class of problems
I I’ll try to justify its use in this case
Maths needed. Hold on.There might be other movies
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
How to do so?
The phase field method
I Coarse grained method, (in this flavor) a coarsened version ofthe Ising model evolving with Kawasaki exchange kinetics
I Can be adapted to solve a much wider class of problems
I I’ll try to justify its use in this case
Maths needed. Hold on.There might be other movies
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
How to do so?
The phase field method
I Coarse grained method, (in this flavor) a coarsened version ofthe Ising model evolving with Kawasaki exchange kinetics
I Can be adapted to solve a much wider class of problems
I I’ll try to justify its use in this case
Maths needed. Hold on.
There might be other movies
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
How to do so?
The phase field method
I Coarse grained method, (in this flavor) a coarsened version ofthe Ising model evolving with Kawasaki exchange kinetics
I Can be adapted to solve a much wider class of problems
I I’ll try to justify its use in this case
Maths needed. Hold on.There might be other movies
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Dynamics
What makes the system evolve in time?
I For each volume element (restating 1st law)
ds =1
Tdu − µ
Tdφ
I If entropy has to change (2nd law), energy and matter mustflow
I Energy and matter flows are assumed to be linear in thegradients of the conjugate variables:
Ju = Luu∇(
1
T
)− Luφ∇
( µT
)Jφ = Lφu∇
(1
T
)− Lφφ∇
( µT
)(1)
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Dynamics
What makes the system evolve in time?
I For each volume element (restating 1st law)
ds =1
Tdu − µ
Tdφ
I If entropy has to change (2nd law), energy and matter mustflow
I Energy and matter flows are assumed to be linear in thegradients of the conjugate variables:
Ju = Luu∇(
1
T
)− Luφ∇
( µT
)Jφ = Lφu∇
(1
T
)− Lφφ∇
( µT
)(1)
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Dynamics
What makes the system evolve in time?
I For each volume element (restating 1st law)
ds =1
Tdu − µ
Tdφ
I If entropy has to change (2nd law), energy and matter mustflow
I Energy and matter flows are assumed to be linear in thegradients of the conjugate variables:
Ju = Luu∇(
1
T
)− Luφ∇
( µT
)Jφ = Lφu∇
(1
T
)− Lφφ∇
( µT
)(1)
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Dynamics
What makes the system evolve in time?
I For each volume element (restating 1st law)
ds =1
Tdu − µ
Tdφ
I If entropy has to change (2nd law), energy and matter mustflow
I Energy and matter flows are assumed to be linear in thegradients of the conjugate variables:
Ju = Luu∇(
1
T
)− Luφ∇
( µT
)Jφ = Lφu∇
(1
T
)− Lφφ∇
( µT
)(1)
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Dynamics: all the relevant equations
Uniform temperature case
If T is the same everywhere, the system evolves according tochemical potential gradients only (Fick’s law):
Jφ = −Lφφ∇( µ
T
)(1b)
Chemical potential as a functional derivative
µ(r) ≡ δF [φ]
δφ(r)(2)
Conservation of the order parameter requires
−∂φ∂t
= ∇ · Jφ (3)
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Dynamics: all the relevant equations
Uniform temperature case
If T is the same everywhere, the system evolves according tochemical potential gradients only (Fick’s law):
Jφ = −Lφφ∇( µ
T
)(1b)
Chemical potential as a functional derivative
µ(r) ≡ δF [φ]
δφ(r)(2)
Conservation of the order parameter requires
−∂φ∂t
= ∇ · Jφ (3)
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Dynamics: all the relevant equations
Uniform temperature case
If T is the same everywhere, the system evolves according tochemical potential gradients only (Fick’s law):
Jφ = −Lφφ∇( µ
T
)(1b)
Chemical potential as a functional derivative
µ(r) ≡ δF [φ]
δφ(r)(2)
Conservation of the order parameter requires
−∂φ∂t
= ∇ · Jφ (3)
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Dynamics: the Cahn-Hilliard equation
(1b), (2), (3) → Cahn-Hilliard equation
∂φ
∂t= ∇ · Lφφ∇
(1
T
δF [φ]
δφ(r, t)
)(C-H)
How about F [φ]?
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Dynamics: the Cahn-Hilliard equation
(1b), (2), (3) → Cahn-Hilliard equation
∂φ
∂t= ∇ · Lφφ∇
(1
T
δF [φ]
δφ(r, t)
)(C-H)
How about F [φ]?
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
The free energy functional
Landau-Ginzburg form of F [φ]
F [φ] =
∫dr f (φ(r)) +
ε
2(∇φ(r))2 (4)
I The free energy density f isthat of the uniform system with the same φ
I There’s a penalty for gradients in the order parameter(interfaces)
Landau-Ginzburg form of δF/δφ
(Definition (4) + differential operator algebra):
δF [φ]
δφ(r)=
df (φ(r))
dφ(r)− ε∇2φ(r) (5)
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
The free energy functional
Landau-Ginzburg form of F [φ]
F [φ] =
∫dr f (φ(r)) +
ε
2(∇φ(r))2 (4)
I The free energy density f isthat of the uniform system with the same φ
I There’s a penalty for gradients in the order parameter(interfaces)
Landau-Ginzburg form of δF/δφ
(Definition (4) + differential operator algebra):
δF [φ]
δφ(r)=
df (φ(r))
dφ(r)− ε∇2φ(r) (5)
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
The free energy functional
Landau-Ginzburg form of F [φ]
F [φ] =
∫dr f (φ(r)) +
ε
2(∇φ(r))2 (4)
I The free energy density f isthat of the uniform system with the same φ
I There’s a penalty for gradients in the order parameter(interfaces)
Landau-Ginzburg form of δF/δφ
(Definition (4) + differential operator algebra):
δF [φ]
δφ(r)=
df (φ(r))
dφ(r)− ε∇2φ(r) (5)
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
The free energy functional
Landau-Ginzburg form of F [φ]
F [φ] =
∫dr f (φ(r)) +
ε
2(∇φ(r))2 (4)
I The free energy density f isthat of the uniform system with the same φ
I There’s a penalty for gradients in the order parameter(interfaces)
Landau-Ginzburg form of δF/δφ
(Definition (4) + differential operator algebra):
δF [φ]
δφ(r)=
df (φ(r))
dφ(r)− ε∇2φ(r) (5)
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
The model (full glory)
And the mathematical problem is . . . (C-H) + (5)
∂φ(r, t)
∂t= Lφφ∇2
[1
T
(df (φ(r, t))
dφ(r, t)− ε∇2φ(r, t)
)](6)
Double well form of the free energy density
0.0 0.5 1.0φ
0.00
0.05
0.10
0.15
f/a
f is typically chosen to be of the form
f (φ) = aφ2(φ− 1)2 (7)
I so that’s a critical quench!
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
The model (full glory)
And the mathematical problem is . . . (C-H) + (5)
∂φ(r, t)
∂t= Lφφ∇2
[1
T
(df (φ(r, t))
dφ(r, t)− ε∇2φ(r, t)
)](6)
Double well form of the free energy density
0.0 0.5 1.0φ
0.00
0.05
0.10
0.15
f/a
f is typically chosen to be of the form
f (φ) = aφ2(φ− 1)2 (7)
I so that’s a critical quench!
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
How to treat the problem numerically?
A good recipe:
1. Recast (6) in Fourier space
∂φ̃(k,t)∂t = −Lφφk2
[1T
(d̃f (φ)dφ + εk2φ̃(k, t)
)]
2. Discretization in time, semi-implicit time-stepping
φ̃(k,t+dt)−φ(k,t)dt = −Lφφk2
[1T
(d̃f (φ)dφ + εk2φ̃(k, t + dt)
)]
Final formula
φ̃(k, t + dt) =φ(k, t)− Lφφk2
[1T
(d̃f (φ)dφ
)]dt
1 + εk4Lφφdt
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
How to treat the problem numerically?
A good recipe:
1. Recast (6) in Fourier space
∂φ̃(k,t)∂t = −Lφφk2
[1T
(d̃f (φ)dφ + εk2φ̃(k, t)
)]
2. Discretization in time, semi-implicit time-stepping
φ̃(k,t+dt)−φ(k,t)dt = −Lφφk2
[1T
(d̃f (φ)dφ + εk2φ̃(k, t + dt)
)]
Final formula
φ̃(k, t + dt) =φ(k, t)− Lφφk2
[1T
(d̃f (φ)dφ
)]dt
1 + εk4Lφφdt
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
How to treat the problem numerically?
A good recipe:
1. Recast (6) in Fourier space
∂φ̃(k,t)∂t = −Lφφk2
[1T
(d̃f (φ)dφ + εk2φ̃(k, t)
)]2. Discretization in time, semi-implicit time-stepping
φ̃(k,t+dt)−φ(k,t)dt = −Lφφk2
[1T
(d̃f (φ)dφ + εk2φ̃(k, t + dt)
)]
Final formula
φ̃(k, t + dt) =φ(k, t)− Lφφk2
[1T
(d̃f (φ)dφ
)]dt
1 + εk4Lφφdt
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
How to treat the problem numerically?
A good recipe:
1. Recast (6) in Fourier space
∂φ̃(k,t)∂t = −Lφφk2
[1T
(d̃f (φ)dφ + εk2φ̃(k, t)
)]2. Discretization in time, semi-implicit time-stepping
φ̃(k,t+dt)−φ(k,t)dt = −Lφφk2
[1T
(d̃f (φ)dφ + εk2φ̃(k, t + dt)
)]
Final formula
φ̃(k, t + dt) =φ(k, t)− Lφφk2
[1T
(d̃f (φ)dφ
)]dt
1 + εk4Lφφdt
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
How to treat the problem numerically?
A good recipe:
1. Recast (6) in Fourier space
∂φ̃(k,t)∂t = −Lφφk2
[1T
(d̃f (φ)dφ + εk2φ̃(k, t)
)]2. Discretization in time, semi-implicit time-stepping
φ̃(k,t+dt)−φ(k,t)dt = −Lφφk2
[1T
(d̃f (φ)dφ + εk2φ̃(k, t + dt)
)]
Final formula
φ̃(k, t + dt) =φ(k, t)− Lφφk2
[1T
(d̃f (φ)dφ
)]dt
1 + εk4Lφφdt
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Summary of the derivation
I Variation in time of the order parameter is due to currents
I Currents are generated by gradients of the chemical potential
I The chemical potential is a functional derivativeof the LG functional of the order parameter
I AfterI Calculation of the functional derivativeI Fourier transformI Clever discretization in time
we get an equation that can be iterated numerically
I Let’s solve it (in 2D)!This is done in detail in Phys. Rev. B 39, 11956-11964 (1989)
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Summary of the derivation
I Variation in time of the order parameter is due to currents
I Currents are generated by gradients of the chemical potential
I The chemical potential is a functional derivativeof the LG functional of the order parameter
I AfterI Calculation of the functional derivativeI Fourier transformI Clever discretization in time
we get an equation that can be iterated numerically
I Let’s solve it (in 2D)!This is done in detail in Phys. Rev. B 39, 11956-11964 (1989)
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Shouldn’t we talk about recent work?
I It’s fun to play with this advantage:
Figure: Computing power of the top 500 machines vs mine
I Dynamics of phase separation is still an active topic
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Shouldn’t we talk about recent work?
I It’s fun to play with this advantage:
Figure: Computing power of the top 500 machines vs mine
I Dynamics of phase separation is still an active topic
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Shouldn’t we talk about recent work?
I It’s fun to play with this advantage:
Figure: Computing power of the top 500 machines vs mine
I Dynamics of phase separation is still an active topic
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Python implementation
from p y l a b import ∗from numpy import ∗
N = 100 #l a t t i c e p o i n t s pe r a x i sdt = 1 #time s t epdx = 1 #l a t t i c e s pa c i n gt = a r a n g e ( 0 , 10000∗dt , dt ) #time s t e p sa = 1 . #c o e f f i c i e n t o f the Landau−Ginzburg f r e e ene rgy d e n s i t ye p s i l o n = 100 #i n t e r f a c e p e n a l t y
e v e r y = 100 #dump an image e v e r yp h i 0 = 0 . 5 #i n i t i a l mean va l u e o f the o r d e r paramete rn o i s e = 0 . 1 #i n i t i a l amp l i t ude o f the rma l f l u c t u a t i o n s i n the o r d e r paramete r
th = p h i 0∗ones ( (N, N) ) + n o i s e ∗( rand (N, N) − 0 . 5 ) #i n i t i a l c o n d i t i o n sx , y = meshgr id ( f f t f r e q ( th . shape [ 0 ] , dx ) , f f t f r e q ( th . shape [ 1 ] , dx ) )k2 = ( x∗x + y∗y )
d f = lambda th , a : 4∗a∗th ∗(1 − th )∗(1 − 2∗ th ) #d e r i v a t i v e o f fdef update ( th , dt , a , k2 ) : #per fo rm one s t ep o f semi−i m p l i c i t i t e r a t i o n
r e t u r n i f f t 2 ( ( f f t 2 ( th ) − dt∗k2∗ f f t 2 ( d f ( th , a ) ) ) / ( 1 + 2∗ e p s i l o n∗dt∗k2∗∗2))
f o r i i n r a n g e ( s i z e ( t ) ) :p r i n t t [ i ]i f mod( i , e v e r y ) == 0 :
imshow ( abs ( th ) , vmin =0.0 , vmax =1.0)c o l o r b a r ( )s a v e f i g ( ’ t ’+ s t r ( i / e v e r y ) . z f i l l ( 3 ) + ’ . png ’ , d p i =100)c l f ( )
th = update ( th , dt , a , k2 )
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Coarsening (φ0 = 0.5± 0.05)
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Getting (semi-) quantitative
Fourier transform in 2D
0 200 400 600 800
0
200
400
600
800
t = 10000
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
−0.4 −0.2 0.0 0.2 0.4
−0.4
−0.2
0.0
0.2
0.4
0
20
40
60
80
100
120
140
I From the radius of the ring one can extract a lengthscale (?)
I Any better ideas?
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Getting (semi-) quantitative
Length vs time
104 105
t
101
102
Typi
call
engt
hsca
leNumerical data
∝ t1/3
This is quite OK!Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Ostwald ripening (φ0 = 0.76± 0.05)
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Getting (semi-) quantitative
Analysis is easier
0 200 400 600 800
0
200
400
600
800
t = 345510
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 200 400 600 800
0
200
400
600
800
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
I Diameters can be easily measured in direct space
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Getting (semi-) quantitative
Size vs time
104 105 106
t
101
102A
vera
gepa
rtic
ledi
amet
er
Numerical data
∝ t1/4
This is wrong, but somewhat expected
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Details, and extensions of the model
Choice of the parameters
They can be chosen so to match the microscopic details of thesystem
Moving to more complicated systems
I More field variables can be introduced: F [φ, η, . . .] each fieldwith its one equation of motion (coupled to the other fields)
I Anisotropy in the interfacial energy can be introduced
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Conclusions
I We derived the Cahn-Hilliard equation to study the dynamicsof system with a conserved order parameter
I We implemented a not-so-naive solver in just a few lines ofcode
I We recovered power laws typical of coarsening/ripeningI What we saw is just an example of the phase field method:
I Describe the system with one parameterI Derive a PDE to make it evolve starting from a free energy
functional
I The method can be easily extended to other problems(see bibliography)
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Conclusions
I We derived the Cahn-Hilliard equation to study the dynamicsof system with a conserved order parameter
I We implemented a not-so-naive solver in just a few lines ofcode
I We recovered power laws typical of coarsening/ripeningI What we saw is just an example of the phase field method:
I Describe the system with one parameterI Derive a PDE to make it evolve starting from a free energy
functional
I The method can be easily extended to other problems(see bibliography)
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Conclusions
I We derived the Cahn-Hilliard equation to study the dynamicsof system with a conserved order parameter
I We implemented a not-so-naive solver in just a few lines ofcode
I We recovered power laws typical of coarsening/ripening
I What we saw is just an example of the phase field method:
I Describe the system with one parameterI Derive a PDE to make it evolve starting from a free energy
functional
I The method can be easily extended to other problems(see bibliography)
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Conclusions
I We derived the Cahn-Hilliard equation to study the dynamicsof system with a conserved order parameter
I We implemented a not-so-naive solver in just a few lines ofcode
I We recovered power laws typical of coarsening/ripeningI What we saw is just an example of the phase field method:
I Describe the system with one parameterI Derive a PDE to make it evolve starting from a free energy
functional
I The method can be easily extended to other problems(see bibliography)
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Conclusions
I We derived the Cahn-Hilliard equation to study the dynamicsof system with a conserved order parameter
I We implemented a not-so-naive solver in just a few lines ofcode
I We recovered power laws typical of coarsening/ripeningI What we saw is just an example of the phase field method:
I Describe the system with one parameter
I Derive a PDE to make it evolve starting from a free energyfunctional
I The method can be easily extended to other problems(see bibliography)
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Conclusions
I We derived the Cahn-Hilliard equation to study the dynamicsof system with a conserved order parameter
I We implemented a not-so-naive solver in just a few lines ofcode
I We recovered power laws typical of coarsening/ripeningI What we saw is just an example of the phase field method:
I Describe the system with one parameterI Derive a PDE to make it evolve starting from a free energy
functional
I The method can be easily extended to other problems(see bibliography)
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Conclusions
I We derived the Cahn-Hilliard equation to study the dynamicsof system with a conserved order parameter
I We implemented a not-so-naive solver in just a few lines ofcode
I We recovered power laws typical of coarsening/ripeningI What we saw is just an example of the phase field method:
I Describe the system with one parameterI Derive a PDE to make it evolve starting from a free energy
functional
I The method can be easily extended to other problems(see bibliography)
Davide Fiocco A simple example of application of the phase field method
Dynamics of phase separation The Cahn-Hilliard equation Numerical solution of the equation To the infinity and . . .
Bibliography
Numerical study of late-stage coarsening for off-criticalquenches in the Cahn-Hilliard equation of phase separation,Phys. Rev. B 39, 11956-11964 (1989)
http://flashinformatique.epfl.ch/spip.php?article2505
Phase-field models for microstructure evolution, AnnualReview of Materials Research, Vol. 32: 113-140 (2002)
Davide Fiocco A simple example of application of the phase field method
Bonus tracks
You might also like. . .
The Allen-Cahn equation
∂η(r, t)
∂t= − δF [η]
δη(r, t)
I Models system where the order parameter is not conserved
I The rate of change of the order parameter reflects how thefree energy functional is affected by a variation of it
Davide Fiocco A simple example of application of the phase field method
Bonus tracks
You might also like. . .
The Allen-Cahn equation
∂η(r, t)
∂t= − δF [η]
δη(r, t)
I Models system where the order parameter is not conserved
I The rate of change of the order parameter reflects how thefree energy functional is affected by a variation of it
Davide Fiocco A simple example of application of the phase field method
Bonus tracks
Spinodal decomposition
If we recast the Cahn-Hilliard equation in real space expanding atφ0 = 0.5 (φ = φ0 + ∆φ), using df (φ)
dφ from (7)
df (φ)dφ
∣∣∣φ0+∆φ
= −a∆φ0
Fourier transforming, with a bit of algebra, we get an evolutionequation something showing instability at low k :
∂∆̃φ∂τ = −k2(−a + ε′k2)∆̃φ
Davide Fiocco A simple example of application of the phase field method
Bonus tracks
Getting (semi-) quantitative
Highlight the interfaces!
0 200 400 600 800
0
200
400
600
800
t = 10000
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 200 400 600 800
0
200
400
600
800
t = 10000
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
I More clever idea to extract a lengthscale
Davide Fiocco A simple example of application of the phase field method
Bonus tracks
Getting (semi-) quantitative
Size vs time, highlighting the interfaces
103 104 105 106
t
10−6
10−5
Inve
rse
inte
rface
area
Numerical data
∝ t1/3
105 106
t
10−5
Inve
rse
inte
rface
area
Numerical data
∝ t1/3
Figure: Inverse interface area vs t in φ0 = 0.5; 0.76
Much better agreement w/ theory and literature on the same data
Davide Fiocco A simple example of application of the phase field method
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