Phase separation dynamics in a two-­‐dimensional magnetic mixture

Phase  separation  dynamics   in  a  two-­‐dimensional  

magnetic  mixture

DPG  Berlin,  Talk  CPP  10.8                                                                                 Fachverband  Chemische  Physik  und  Polymerphysik

March  27th,  2012        

K . Lichtner ¹ , A . J . Archer ² , S . H . L . Klapp ¹

1) 2)

Motivation

2

Soft Matter 2012

more examples:

V _core (r _i , r _j )+ V _mag (r _i , r _j , s _i , s _j )δ _α,B δ _{β ,B}

V _mag (r, r ^� , s ₁ , s ₂ ) = − J e ^{− ( | r − r}

^{| /σ − 1)}

| r − r ^� | /σ s ₁ · s ₂

A  simple  model:  Heisenberg  mixture

3

H ^int = 1 2

�

α,β

� N

i,j=1 i�=j

V ^αβ (r _i , r _j , s _i , s _j ) , where α, β = { A, B }

Interaction potential

εe ^{( −| r − r}

^{| )}

^/σ

Model

Part  I:  Equilibrium  Theory

µ _α = δ F [ { ρ _α } ] δρ _α

Central  quantity:

Density  functional  theory

5

Ω[ { ρ _α } ] = F [ { ρ _α } ] − �

α

�

dω

�

dr �

µ _α − V _ext (r, ω ) �

ρ _α (r, ω ) ρ _α (r, ω ) = ρ _α (r)h _α (r, ω )

F [ { ρ _α } ] = F ^id [ { ρ _α } ] + F ^ex [ { ρ _α } ]

Classical  grand   potential

Excess  free  energy:

F ^ex [ { ρ _i } ] = 1 2

�

α,β

� 1

0

dλ

�

dr

�

dr ^�

�

dω

�

dω ^� ρ ⁽²⁾ _αβ (r, r ^� , ω , ω ^� ; λ)V ^αβ ( | r − r ^� | , ω , ω ^� ).

:

Equilibrium  density  is  given  by:

Methods

δ Ω[ρ _α , h]

δρ _α (r)

� �

� �

ρ

(r)

= 0 , and δ Ω[ρ _α , h]

δ h _α (r, ω)

� �

� �

h

(r,ω)

= 0

0 0.2 0.4 0.6 0.8 1

x

0 1 2

3 4 5

!" 2

metastable unstable Curie line

(x ^II , ! ^II ) (x ^I , ! ^I )

O

ρ _c

Phase  diagram

Results  I:  Equilibrium  theory

µ ^I _α = µ ^II _α , P ^I = P ^II T ^I = T ^II

 Coexisting  states  with                                                                                  and

 Tricritical  point  with  stable  states  for  r  <  r_c

Phase  transitions   Isobars

First  order  demixing   transition  

    

Second  order  transition   for  the  magnetization

    

0 0.2 0.4 0.6 0.8 1

x

0 1 2 3 4 5

!"

metastable unstable Curie line

(x

, !

) (x

, !

)

O

ρ

10 20 30 40

0 1 2 3 4

!

"

P

= 96.85 P

= 71.47 P

= 62.39 P

= 53.96

10 20 30 40

z / "

1 2 3 4

!

"

P P

Inhomogeneous  densities

7

 Minimize:  

                     System  completely  equilibrated

 Full  information  of  system  after  phase  separation

                   Coexisting  bulk  densities  and  bulk  magnetizations                                

Results  I:  Equilibrium  theory

δΩ[ρ

, h]

δρ

(r)

� �

� �

ρ⁽⁰⁾_α (r)

= 0 , and δΩ[ρ

, h]

δh

(r, ω)

� �

� �

h⁽⁰⁾_α (r,ω)

= 0

10 20 30 40 z / !

0 0.2 0.4 0.6 0.8

m P

= 96.85

P

= 71.47 P

= 62.39 P

= 53.96

Magnetization

8

10 20 30 40

0 1 2 3 4

!

"

P

= 96.85 P

= 71.47 P

= 62.39 P

= 53.96

10 20 30 40

z / "

1 2 3 4

!

"

P P

 Inhomogeneous  magnetization  profile  for  species  B  

 No  magnetization  in  phase  I

 Softening  of  the  interface  can  also  be  seen  in  the   magnetization

Results  I:  Equilibrium  theory

Part  II:  Out  of  equilibrium

ρ _α (r)

Dynamical  density  functional  theory

10

One-body Smoluchowski equation for a binary mixture:

Γ ^{− 1} ∂ρ _α (r, t)

∂t = k _B T ∇ ² α ρ _α (r, t) + ∇ ^α [ρ _α (r, t) ∇ ^α V _ext (r, t)] + + 1

2 ∇ ^α �

αβ

�

dr ^� ρ ⁽²⁾ _αβ (r, r ^� , t) ∇ ^α v ₂ (r, r ^� )

Adiabatic approximation and sum rule:

Theory:  DDFT

DDFT key equation:

U. Marconi, P. Tarazona, JCP 110 8032,1999

Γ ⁻ _α ¹ ∂ρ _α (r, t)

∂t = ∇ ·

�

ρ _α (r, t) ∇ δ F [ρ _A (r, t), ρ _B (r, t)]

δρ _α (r, t)

�

Spinodal  decomposition

Results  II:  non-­‐equilibrium

 Study  states  where  dynamics  is  linearly  unstable          Density  variations  grow  slowly  over  time

0 0.2 0.4 0.6 0.8 1

x 0

1 2 3 4 5

!"2

metastable unstable Curie line

(x^II, !^II) (x^I, !^I) ^O

O

ρ_c

Nucleation  (2d)

12

 Exchange  of  energy,  volume  or  particles  such  that:

 Surface  tension  impedes  the  formation  of  droplets

 Supercritical  nuclei  for  R  >  Rc

Results  II:  non-­‐equilibrium

0 1 2 3

R / R

0

Ω

(R)

B

~R

~R

ρ _A

Nucleation  dynamics

Results  II:  non-­‐equilibrium

 Isotropic  particles  in  the  sea  of  magn.  particles

 Pathways  for  supercritical  and  subcritical  nuclei

0 0.2 0.4 0.6 0.8 1

x 0

1 2 3 4 5

!"2

metastable unstable Curie line

(x^II, !^II) (x^I, !^I) ^O

O

ρ_c

time

DDFT  free  energy  pathway

 EDFT  and  DDFT  consistent  in  predicting  nucleus  growth  or   shrinking,  but:  free  energy  pathways  are  different!

 Recall  that  EDFT  works  in  (T,V,      )  ensemble  and  DDFT  in   (T,V,N)  ensemble,  respectively

14

0 100 200

N

-30 -20 -10 0 10 20

ΔΩ / k

T

a

b c d

e

f

Results  II:  non-­‐equilibrium

µ

Summary

 Model  for  a  binary  colloidal  mixture

 Phase  diagram  and  equilibrium  (EDFT)                                 results  of  the  free  interface

 DDFT  study  with  consistent  results

 Free  energy  pathways  of  EDFT  and  DDFT

 Surface  tension

16

1 10 100

(P ^* -P _c ^* )

0 4 8 12

!" / (k B T)

Results  I:  Equilibrium  theory

P = P _T (z )ˆ e _x e ˆ _x + P _N (z )ˆ e _z e ˆ _z γ =

� _∞

−∞

dz (P − P _T (z ))

 Pressure

 Line  tension

0 0.2 0.4 0.6 0.8 1

x

0 1 2

3 4 5

!" 2

metastable unstable Curie line

(x ^II , ! ^II ) (x ^I , ! ^I )

O

ρ _c

Phase  separation  dynamics

17

Can  we  study  nucleation  with  DDFT?

Results  II:  non-­‐equilibrium

or  unstable  states  (spinodal  decomposition)?

 Full  dynamics  with  interactions                                                            

 Orientation,  hydrodynamics  etc.

 Difficult  to  solve

Methods

19

(I) Computer  simulations

noise forces

inertia friction

Methods

m(r, t)

(I) Computer  simulations

(II) Density  functional  theories

Static  DFT Dynamic  DFT

 Equilibrium  properties,   microscopic  description

 Phase  diagrams,  

interfaces,  nucleation   barriers

 States  out  of  equilibrium,   time-­‐dep.  order  parameter

 Relaxation  into  equilibrium   of                            ,  

noise forces

inertia friction

Methods

Methods

ρ(r, t)

Outline

21

(I)  Introduction

(II)  Equilibrium  theory  (EDFT)

• ^Model

• Phase  diagram

• Static  results

(III)  Out  of  equilibrium  (DDFT)

• Comparison  to  EDFT

• ^Nucleation

(IV)  Conclusion

DDFT key equation:

U. Marconi, P. Tarazona, JCP 110 8032,1999

DDFT  for  binary  mixtures

22

• we assume magnetic moments have very short relaxation time so that m can be minimized at each time instant: ^{∂ Ω[ρ α , h]}

∂h _α (r, ω)

� �

� �

h

(r,ω)

= 0

Γ ⁻ _α ¹ ∂ρ _α (r, t)

∂t = ∇ ·

�

ρ _α (r, t) ∇ δ F [ρ _A (r, t), ρ _B (r, t)]

δρ _α (r, t)

�

• we use the same Helmholtz free energy as above (EDFT):

F [ { ρ _α } ] = F ^id [ { ρ _α } ] + F ^ex [ { ρ _α } ] Theory:  DDFT

Note, that

Nucleation:  critical  droplet  theory  (2d)

23

 Exchange  of  energy,  volume  or  particles  such  that:

 surface  tension  impedes  the  formation  of  droplets:

Results  II:  non-­‐equilibrium

0 1 2 3

R / R

0

Ω

(R)

B

~R

~R

ρ _A γ

Nucleation  barrier

24

0 50 100 150 200

N

0

20 40

61 / k

T

x = 0.94 x = 0.96 x = 0.97

0.9 0.92 0.94 0.96 0.98

x

0 40 80 120

!"

/ k

T

DFT

 Results  consistent  for  concentrations  near  binodal

 CNT  is  weak  for  smaller  concentrations  (near  the  spinodal)            Reason:  CNT  is  clearly  a  macroscopic  theory

Results  II:  non-­‐equilibrium