Phase field modelling of phase separation using the Cahn ... · PDF filePhase eld modelling of...

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

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

Figure: A quench

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

How to do so?

I Coarse grained method, (in this flavor) a coarsened version ofthe Ising model evolving with Kawasaki exchange kinetics

How to do so?

Maths needed. Hold on.

There might be other movies

How to do so?

Dynamics

What makes the system evolve in time?

I For each volume element (restating 1st law)

Tdu − µ

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∇(

)− Luφ∇

)Jφ = Lφu∇

)− Lφφ∇

Dynamics

Tdu − µ

Ju = Luu∇(

)− Luφ∇

)Jφ = Lφu∇

)− Lφφ∇

Dynamics

Tdu − µ

Ju = Luu∇(

)− Luφ∇

)Jφ = Lφu∇

)− Lφφ∇

Dynamics

Tdu − µ

Ju = Luu∇(

)− Luφ∇

)Jφ = Lφu∇

)− Lφφ∇

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

Chemical potential as a functional derivative

µ(r) ≡ δF [φ]

δφ(r)(2)

Conservation of the order parameter requires

−∂φ∂t

= ∇ · Jφ (3)

µ(r) ≡ δF [φ]

δφ(r)(2)

−∂φ∂t

= ∇ · Jφ (3)

µ(r) ≡ δF [φ]

δφ(r)(2)

−∂φ∂t

= ∇ · Jφ (3)

Dynamics: the Cahn-Hilliard equation

(1b), (2), (3) → Cahn-Hilliard equation

∂t= ∇ · Lφφ∇

δF [φ]

δφ(r, t)

)(C-H)

How about F [φ]?

Dynamics: the Cahn-Hilliard equation

(1b), (2), (3) → Cahn-Hilliard equation

∂t= ∇ · Lφφ∇

δF [φ]

δφ(r, t)

)(C-H)

How about F [φ]?

The free energy functional

Landau-Ginzburg form of F [φ]

F [φ] =

∫dr f (φ(r)) +

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)

F [φ] =

∫dr f (φ(r)) +

2(∇φ(r))2 (4)

δF [φ]

δφ(r)=

df (φ(r))

dφ(r)− ε∇2φ(r) (5)

F [φ] =

∫dr f (φ(r)) +

2(∇φ(r))2 (4)

δF [φ]

δφ(r)=

df (φ(r))

dφ(r)− ε∇2φ(r) (5)

F [φ] =

∫dr f (φ(r)) +

2(∇φ(r))2 (4)

δF [φ]

δφ(r)=

df (φ(r))

dφ(r)− ε∇2φ(r) (5)

The model (full glory)

And the mathematical problem is . . . (C-H) + (5)

∂φ(r, t)

∂t= Lφφ∇2

(df (φ(r, t))

dφ(r, t)− ε∇2φ(r, t)

Double well form of the free energy density

0.0 0.5 1.0φ

f is typically chosen to be of the form

f (φ) = aφ2(φ− 1)2 (7)

I so that’s a critical quench!

The model (full glory)

And the mathematical problem is . . . (C-H) + (5)

∂φ(r, t)

∂t= Lφφ∇2

(df (φ(r, t))

dφ(r, t)− ε∇2φ(r, t)

Double well form of the free energy density

0.0 0.5 1.0φ

f is typically chosen to be of the form

f (φ) = aφ2(φ− 1)2 (7)

I so that’s a critical quench!

How to treat the problem numerically?

A good recipe:

1. Recast (6) in Fourier space

∂φ̃(k,t)∂t = −Lφφk2

(d̃f (φ)dφ + εk2φ̃(k, t)

2. Discretization in time, semi-implicit time-stepping

φ̃(k,t+dt)−φ(k,t)dt = −Lφφk2

(d̃f (φ)dφ + εk2φ̃(k, t + dt)

Final formula

φ̃(k, t + dt) =φ(k, t)− Lφφk2

(d̃f (φ)dφ

1 + εk4Lφφdt

A good recipe:

2. Discretization in time, semi-implicit time-stepping

Final formula

(d̃f (φ)dφ

1 + εk4Lφφdt

A good recipe:

)]2. Discretization in time, semi-implicit time-stepping

Final formula

(d̃f (φ)dφ

1 + εk4Lφφdt

A good recipe:

Final formula

(d̃f (φ)dφ

1 + εk4Lφφdt

A good recipe:

Final formula

(d̃f (φ)dφ

1 + εk4Lφφdt

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)

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)

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

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 )

Coarsening (φ0 = 0.5± 0.05)

Getting (semi-) quantitative

Fourier transform in 2D

0 200 400 600 800

t = 10000

−0.4 −0.2 0.0 0.2 0.4

−0.4

−0.2

I From the radius of the ring one can extract a lengthscale (?)

I Any better ideas?

Length vs time

104 105

leNumerical data

∝ t1/3

This is quite OK!Davide Fiocco A simple example of application of the phase field method

Ostwald ripening (φ0 = 0.76± 0.05)

Analysis is easier

0 200 400 600 800

t = 345510

0 200 400 600 800

I Diameters can be easily measured in direct space

Size vs time

104 105 106

Numerical data

∝ t1/4

This is wrong, but somewhat expected

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

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)

Conclusions

functional

Conclusions

I We recovered power laws typical of coarsening/ripening

I What we saw is just an example of the phase field method:

functional

Conclusions

functional

Conclusions

I Describe the system with one parameter

I Derive a PDE to make it evolve starting from a free energyfunctional

Conclusions

functional

Conclusions

functional

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)

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