lecture 12: domains, nucleation and coarsening outline: domain walls nucleation coarsening

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Lecture 12: Domains, nucleation and coarsening Outline: • domain walls • nucleation • coarsening

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Page 1: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

Lecture 12: Domains, nucleation and coarsening

Outline:• domain walls• nucleation• coarsening

Page 2: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

Domain walls

Below Tc: m0: uniform solution of

m = tanh βJm( )

Page 3: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

Domain walls

Below Tc: m0: uniform solution of

m = tanh βJm( )

or in Landau model: ϕ0 solves

dV

dφ= 0

Page 4: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

Domain walls

Below Tc: m0: uniform solution of

m = tanh βJm( )

or in Landau model: ϕ0 solves

dV

dφ= 0 ⇒ r0φ0 + u0φ0

3 = 0

Page 5: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

Domain walls

Below Tc: m0: uniform solution of

m = tanh βJm( )

or in Landau model: ϕ0 solves

dV

dφ= 0 ⇒ r0φ0 + u0φ0

3 = 0

⇒ φ0 = ±r0

u, r0 < 0

Page 6: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

Domain walls

Below Tc: m0: uniform solution of

m = tanh βJm( )

or in Landau model: ϕ0 solves

dV

dφ= 0 ⇒ r0φ0 + u0φ0

3 = 0

⇒ φ0 = ±r0

u, r0 < 0

Suppose we have boundary conditions

φ→r0

u, x → ∞

φ → −r0

u, x → −∞

Page 7: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

Non-uniform stationary solution

δE

δφ(x)=

δ

δφ(x)dd ′ x ∫ 1

2 r0φ2( ′ x ) + 1

4 u0φ4 ( ′ x ) + 1

2 (∇φ( ′ x ))2[ ]{ } = 0

Page 8: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

Non-uniform stationary solution

δE

δφ(x)=

δ

δφ(x)dd ′ x ∫ 1

2 r0φ2( ′ x ) + 1

4 u0φ4 ( ′ x ) + 1

2 (∇φ( ′ x ))2[ ]{ } = 0

rewrite this with

E[φ] = dd ′ x ∫ 14 u0 φ2( ′ x ) − φ0

2( )

2+ 1

2 (∇φ( ′ x ))2[ ]

Page 9: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

Non-uniform stationary solution

δE

δφ(x)=

δ

δφ(x)dd ′ x ∫ 1

2 r0φ2( ′ x ) + 1

4 u0φ4 ( ′ x ) + 1

2 (∇φ( ′ x ))2[ ]{ } = 0

rewrite this with

(differs only by an additive constant)

E[φ] = dd ′ x ∫ 14 u0 φ2( ′ x ) − φ0

2( )

2+ 1

2 (∇φ( ′ x ))2[ ]

Page 10: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

Non-uniform stationary solution

δE

δφ(x)=

δ

δφ(x)dd ′ x ∫ 1

2 r0φ2( ′ x ) + 1

4 u0φ4 ( ′ x ) + 1

2 (∇φ( ′ x ))2[ ]{ } = 0

rewrite this with

(differs only by an additive constant)

E[φ] = dd ′ x ∫ 14 u0 φ2( ′ x ) − φ0

2( )

2+ 1

2 (∇φ( ′ x ))2[ ]

⇒δE

δφ(x)= u0 φ2(x) − φ0

2( )φ(x) −

d2φ

dx 2= 0

Page 11: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

Non-uniform stationary solution

δE

δφ(x)=

δ

δφ(x)dd ′ x ∫ 1

2 r0φ2( ′ x ) + 1

4 u0φ4 ( ′ x ) + 1

2 (∇φ( ′ x ))2[ ]{ } = 0

rewrite this with

(differs only by an additive constant)

E[φ] = dd ′ x ∫ 14 u0 φ2( ′ x ) − φ0

2( )

2+ 1

2 (∇φ( ′ x ))2[ ]

⇒δE

δφ(x)= u0 φ2(x) − φ0

2( )φ(x) −

d2φ

dx 2= 0 (d = 1)

Page 12: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

Non-uniform stationary solution

δE

δφ(x)=

δ

δφ(x)dd ′ x ∫ 1

2 r0φ2( ′ x ) + 1

4 u0φ4 ( ′ x ) + 1

2 (∇φ( ′ x ))2[ ]{ } = 0

rewrite this with

(differs only by an additive constant)

E[φ] = dd ′ x ∫ 14 u0 φ2( ′ x ) − φ0

2( )

2+ 1

2 (∇φ( ′ x ))2[ ]

φ(x) = φ0 tanh xr0

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟solution:

⇒δE

δφ(x)= u0 φ2(x) − φ0

2( )φ(x) −

d2φ

dx 2= 0 (d = 1)

Page 13: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

Non-uniform stationary solution

δE

δφ(x)=

δ

δφ(x)dd ′ x ∫ 1

2 r0φ2( ′ x ) + 1

4 u0φ4 ( ′ x ) + 1

2 (∇φ( ′ x ))2[ ]{ } = 0

rewrite this with

(differs only by an additive constant)

E[φ] = dd ′ x ∫ 14 u0 φ2( ′ x ) − φ0

2( )

2+ 1

2 (∇φ( ′ x ))2[ ]

φ(x) = φ0 tanh xr0

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟solution:

⇒δE

δφ(x)= u0 φ2(x) − φ0

2( )φ(x) −

d2φ

dx 2= 0

“domain wall” (“kink”)solution

(d = 1)

Page 14: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

Non-uniform stationary solution

δE

δφ(x)=

δ

δφ(x)dd ′ x ∫ 1

2 r0φ2( ′ x ) + 1

4 u0φ4 ( ′ x ) + 1

2 (∇φ( ′ x ))2[ ]{ } = 0

rewrite this with

(differs only by an additive constant)

E[φ] = dd ′ x ∫ 14 u0 φ2( ′ x ) − φ0

2( )

2+ 1

2 (∇φ( ′ x ))2[ ]

φ(x) = φ0 tanh xr0

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟solution:

⇒δE

δφ(x)= u0 φ2(x) − φ0

2( )φ(x) −

d2φ

dx 2= 0

“domain wall” (“kink”)solutionlocalized: size r0

(d = 1)

Page 15: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

Non-uniform stationary solution

δE

δφ(x)=

δ

δφ(x)dd ′ x ∫ 1

2 r0φ2( ′ x ) + 1

4 u0φ4 ( ′ x ) + 1

2 (∇φ( ′ x ))2[ ]{ } = 0

rewrite this with

(differs only by an additive constant)

E[φ] = dd ′ x ∫ 14 u0 φ2( ′ x ) − φ0

2( )

2+ 1

2 (∇φ( ′ x ))2[ ]

φ(x) = φ0 tanh xr0

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟solution:

⇒δE

δφ(x)= u0 φ2(x) − φ0

2( )φ(x) −

d2φ

dx 2= 0

“domain wall” (“kink”)solutionlocalized: size r0

broad near Tc

(d = 1)

Page 16: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

kink energy calculation

E[φ] = dx 14 u0 φ2(x) − φ0

2( )

2+ 1

2 (∇φ(x))2[ ]−∞

Page 17: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

kink energy calculation

E[φ] = dx 14 u0 φ2(x) − φ0

2( )

2+ 1

2 (∇φ(x))2[ ]−∞

= dx 14 u0φ0

4 1− tanh2 xr0

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎝ ⎜ ⎜

⎠ ⎟ ⎟

2

+ 12 φ0

r0

21− tanh2 x

r0

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎢ ⎢

⎥ ⎥

2 ⎡

⎢ ⎢ ⎢

⎥ ⎥ ⎥

−∞

Page 18: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

kink energy calculation

E[φ] = dx 14 u0 φ2(x) − φ0

2( )

2+ 1

2 (∇φ(x))2[ ]−∞

= dx 14 u0φ0

4 1− tanh2 xr0

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎝ ⎜ ⎜

⎠ ⎟ ⎟

2

+ 12 φ0

r0

21− tanh2 x

r0

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎢ ⎢

⎥ ⎥

2 ⎡

⎢ ⎢ ⎢

⎥ ⎥ ⎥

−∞

= 12 r0 φ0

2 dx 1− tanh2 xr0

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎝ ⎜ ⎜

⎠ ⎟ ⎟

2

−∞

Page 19: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

kink energy calculation

E[φ] = dx 14 u0 φ2(x) − φ0

2( )

2+ 1

2 (∇φ(x))2[ ]−∞

= dx 14 u0φ0

4 1− tanh2 xr0

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎝ ⎜ ⎜

⎠ ⎟ ⎟

2

+ 12 φ0

r0

21− tanh2 x

r0

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎢ ⎢

⎥ ⎥

2 ⎡

⎢ ⎢ ⎢

⎥ ⎥ ⎥

−∞

= 12 r0 φ0

2 dx 1− tanh2 xr0

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎝ ⎜ ⎜

⎠ ⎟ ⎟

2

−∞

∫ = 12 r0 φ0

2 ⋅2

r0

du 1− tanh2 u( )2

−∞

Page 20: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

kink energy calculation

E[φ] = dx 14 u0 φ2(x) − φ0

2( )

2+ 1

2 (∇φ(x))2[ ]−∞

= dx 14 u0φ0

4 1− tanh2 xr0

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎝ ⎜ ⎜

⎠ ⎟ ⎟

2

+ 12 φ0

r0

21− tanh2 x

r0

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎢ ⎢

⎥ ⎥

2 ⎡

⎢ ⎢ ⎢

⎥ ⎥ ⎥

−∞

= 12 r0 φ0

2 dx 1− tanh2 xr0

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎝ ⎜ ⎜

⎠ ⎟ ⎟

2

−∞

∫ = 12 r0 φ0

2 ⋅2

r0

du 1− tanh2 u( )2

−∞

= 12 r0 φ0

2 ⋅2

r0

dy

1− y 2−1

1

∫ 1− y 2( )

2

Page 21: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

kink energy calculation

E[φ] = dx 14 u0 φ2(x) − φ0

2( )

2+ 1

2 (∇φ(x))2[ ]−∞

= dx 14 u0φ0

4 1− tanh2 xr0

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎝ ⎜ ⎜

⎠ ⎟ ⎟

2

+ 12 φ0

r0

21− tanh2 x

r0

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎢ ⎢

⎥ ⎥

2 ⎡

⎢ ⎢ ⎢

⎥ ⎥ ⎥

−∞

= 12 r0 φ0

2 dx 1− tanh2 xr0

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎝ ⎜ ⎜

⎠ ⎟ ⎟

2

−∞

∫ = 12 r0 φ0

2 ⋅2

r0

du 1− tanh2 u( )2

−∞

= 12 r0 φ0

2 ⋅2

r0

dy

1− y 2−1

1

∫ 1− y 2( )

2= 1

2 r0 φ02 ⋅

2

r0

⋅4

3

Page 22: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

kink energy calculation

E[φ] = dx 14 u0 φ2(x) − φ0

2( )

2+ 1

2 (∇φ(x))2[ ]−∞

= dx 14 u0φ0

4 1− tanh2 xr0

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎝ ⎜ ⎜

⎠ ⎟ ⎟

2

+ 12 φ0

r0

21− tanh2 x

r0

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎢ ⎢

⎥ ⎥

2 ⎡

⎢ ⎢ ⎢

⎥ ⎥ ⎥

−∞

= 12 r0 φ0

2 dx 1− tanh2 xr0

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎝ ⎜ ⎜

⎠ ⎟ ⎟

2

−∞

∫ = 12 r0 φ0

2 ⋅2

r0

du 1− tanh2 u( )2

−∞

= 12 r0 φ0

2 ⋅2

r0

dy

1− y 2−1

1

∫ 1− y 2( )

2= 1

2 r0 φ02 ⋅

2

r0

⋅4

3= 4

3 ε0ξ

Page 23: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

kink energy calculation

E[φ] = dx 14 u0 φ2(x) − φ0

2( )

2+ 1

2 (∇φ(x))2[ ]−∞

= dx 14 u0φ0

4 1− tanh2 xr0

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎝ ⎜ ⎜

⎠ ⎟ ⎟

2

+ 12 φ0

r0

21− tanh2 x

r0

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎢ ⎢

⎥ ⎥

2 ⎡

⎢ ⎢ ⎢

⎥ ⎥ ⎥

−∞

= 12 r0 φ0

2 dx 1− tanh2 xr0

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎝ ⎜ ⎜

⎠ ⎟ ⎟

2

−∞

∫ = 12 r0 φ0

2 ⋅2

r0

du 1− tanh2 u( )2

−∞

= 12 r0 φ0

2 ⋅2

r0

dy

1− y 2−1

1

∫ 1− y 2( )

2= 1

2 r0 φ02 ⋅

2

r0

⋅4

3= 4

3 ε0ξ________thickness

Page 24: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

kink energy calculation

E[φ] = dx 14 u0 φ2(x) − φ0

2( )

2+ 1

2 (∇φ(x))2[ ]−∞

= dx 14 u0φ0

4 1− tanh2 xr0

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎝ ⎜ ⎜

⎠ ⎟ ⎟

2

+ 12 φ0

r0

21− tanh2 x

r0

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎢ ⎢

⎥ ⎥

2 ⎡

⎢ ⎢ ⎢

⎥ ⎥ ⎥

−∞

= 12 r0 φ0

2 dx 1− tanh2 xr0

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎝ ⎜ ⎜

⎠ ⎟ ⎟

2

−∞

∫ = 12 r0 φ0

2 ⋅2

r0

du 1− tanh2 u( )2

−∞

= 12 r0 φ0

2 ⋅2

r0

dy

1− y 2−1

1

∫ 1− y 2( )

2= 1

2 r0 φ02 ⋅

2

r0

⋅4

3= 4

3 ε0ξ _____energydensity

________thickness

Page 25: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

kink energy calculation

E[φ] = dx 14 u0 φ2(x) − φ0

2( )

2+ 1

2 (∇φ(x))2[ ]−∞

= dx 14 u0φ0

4 1− tanh2 xr0

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎝ ⎜ ⎜

⎠ ⎟ ⎟

2

+ 12 φ0

r0

21− tanh2 x

r0

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎢ ⎢

⎥ ⎥

2 ⎡

⎢ ⎢ ⎢

⎥ ⎥ ⎥

−∞

= 12 r0 φ0

2 dx 1− tanh2 xr0

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎝ ⎜ ⎜

⎠ ⎟ ⎟

2

−∞

∫ = 12 r0 φ0

2 ⋅2

r0

du 1− tanh2 u( )2

−∞

= 12 r0 φ0

2 ⋅2

r0

dy

1− y 2−1

1

∫ 1− y 2( )

2= 1

2 r0 φ02 ⋅

2

r0

⋅4

3= 4

3 ε0ξ

=2 2

3

r03 / 2

u0

∝ Tc − T( )3 / 2

_____energydensity

________thickness

Page 26: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

Higher dimensions

energy of domain wall of size L is

2 2

3

r03 / 2

u0

Ld −1

Page 27: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

Higher dimensions

energy of domain wall of size L is

2 2

3

r03 / 2

u0

Ld −1

energy of a flipped domain is proportional to itssurface area

Page 28: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

Nucleation

E[φ] = 12 dd x∫ r0φ

2(x) + 12 u0φ

4 (x) + (∇φ(x))2 − 2hφ(x)[ ]With a field,

Page 29: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

Nucleation

Suppose we are below Tc (r0 < 0), ϕ = - ϕ0

E[φ] = 12 dd x∫ r0φ

2(x) + 12 u0φ

4 (x) + (∇φ(x))2 − 2hφ(x)[ ]With a field,

Page 30: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

Nucleation

Suppose we are below Tc (r0 < 0), ϕ = - ϕ0

Then make h > 0€

E[φ] = 12 dd x∫ r0φ

2(x) + 12 u0φ

4 (x) + (∇φ(x))2 − 2hφ(x)[ ]With a field,

Page 31: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

Nucleation

Suppose we are below Tc (r0 < 0), ϕ = - ϕ0

Then make h > 0€

E[φ] = 12 dd x∫ r0φ

2(x) + 12 u0φ

4 (x) + (∇φ(x))2 − 2hφ(x)[ ]With a field,

It is now favorable to change to ϕ = + ϕ0

Page 32: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

Nucleation

Suppose we are below Tc (r0 < 0), ϕ = - ϕ0

Then make h > 0€

E[φ] = 12 dd x∫ r0φ

2(x) + 12 u0φ

4 (x) + (∇φ(x))2 − 2hφ(x)[ ]With a field,

It is now favorable to change to ϕ = + ϕ0

but it costs energy to make a local region where ϕ(x) = - ϕ0

Page 33: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

Nucleation

Suppose we are below Tc (r0 < 0), ϕ = - ϕ0

Then make h > 0€

E[φ] = 12 dd x∫ r0φ

2(x) + 12 u0φ

4 (x) + (∇φ(x))2 − 2hφ(x)[ ]With a field,

It is now favorable to change to ϕ = + ϕ0

but it costs energy to make a local region where ϕ(x) = - ϕ0

energy of a spherical bubble of radius R of the + phase:

E[R] = −2hφ0 ⋅4π

3R3 + 4πR2 ⋅

2 2

3

r03 / 2

u0

Page 34: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

Nucleation

Suppose we are below Tc (r0 < 0), ϕ = - ϕ0

Then make h > 0€

E[φ] = 12 dd x∫ r0φ

2(x) + 12 u0φ

4 (x) + (∇φ(x))2 − 2hφ(x)[ ]With a field,

It is now favorable to change to ϕ = + ϕ0

but it costs energy to make a local region where ϕ(x) = - ϕ0

energy of a spherical bubble of radius R of the + phase:

E[R] = −2hφ0 ⋅4π

3R3 + 4πR2 ⋅

2 2

3

r03 / 2

u0

=8π

3

r0

u0

−hR3 + 2r0

u0

R2 ⎡

⎣ ⎢

⎦ ⎥

Page 35: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

a Kramers escape problem:

E(R) =8π

3

r0

u0

−hR3 + 2r0

u0

R2 ⎡

⎣ ⎢

⎦ ⎥

Page 36: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

a Kramers escape problem:

E(R) =8π

3

r0

u0

−hR3 + 2r0

u0

R2 ⎡

⎣ ⎢

⎦ ⎥

Page 37: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

a Kramers escape problem:

E(R) =8π

3

r0

u0

−hR3 + 2r0

u0

R2 ⎡

⎣ ⎢

⎦ ⎥

to find barrier height:maximize E(R)

Page 38: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

a Kramers escape problem:

E(R) =8π

3

r0

u0

−hR3 + 2r0

u0

R2 ⎡

⎣ ⎢

⎦ ⎥

to find barrier height:maximize E(R)

Eb =64π 2

81⋅

r0

7 / 2

u02h2

Page 39: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

a Kramers escape problem:

E(R) =8π

3

r0

u0

−hR3 + 2r0

u0

R2 ⎡

⎣ ⎢

⎦ ⎥

to find barrier height:maximize E(R)

Eb =64π 2

81⋅

r0

7 / 2

u02h2

τ nucl = exp(Eb ) = (prefactor) ⋅exp const ⋅r0

7 / 2

u02h2

⎝ ⎜ ⎜

⎠ ⎟ ⎟ nucleation time

Page 40: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

Spinodal decomposition

Ordering by nucleation is a transition from a metastable stateto a stable one.

In spinodal decomposition, one quenches to a temperature belowTc at zero magnetization, an unstable state. Local domains order and grow.

Page 41: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

Spinodal decomposition

Ordering by nucleation is a transition from a metastable stateto a stable one.

Page 42: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

Spinodal decomposition

Ordering by nucleation is a transition from a metastable stateto a stable one.

In spinodal decomposition, one quenches to a temperature belowTc at zero magnetization

Page 43: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

Spinodal decomposition

Ordering by nucleation is a transition from a metastable stateto a stable one.

In spinodal decomposition, one quenches to a temperature belowTc at zero magnetization, an unstable state.

Page 44: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

Spinodal decomposition

Ordering by nucleation is a transition from a metastable stateto a stable one.

In spinodal decomposition, one quenches to a temperature belowTc at zero magnetization, an unstable state. Local domains order and grow.

Ising model, T = 0

20 MC sweeps 200 MC sweeps

(from JSethna)

Page 45: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

coarsening by shrinkage of small domains

consider a droplet of size R

Page 46: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

coarsening by shrinkage of small domains

consider a droplet of size R, use

∂φ(x)

∂t= −

δE

δφ(x)

Page 47: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

coarsening by shrinkage of small domains

consider a droplet of size R, use

∂φ(x)

∂t= −

δE

δφ(x)

spherical symmetry:

∂φ∂t

=∂ 2φ

∂r2+

2

r

∂φ

∂r−

∂V (φ)

∂φ

Page 48: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

coarsening by shrinkage of small domains

consider a droplet of size R, use

∂φ(x)

∂t= −

δE

δφ(x)

spherical symmetry:

∂φ∂t

=∂ 2φ

∂r2+

2

r

∂φ

∂r−

∂V (φ)

∂φ

ansatz:

φ(r, t) = f r − R(t)( )

Page 49: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

coarsening by shrinkage of small domains

consider a droplet of size R, use

∂φ(x)

∂t= −

δE

δφ(x)

spherical symmetry:

∂φ∂t

=∂ 2φ

∂r2+

2

r

∂φ

∂r−

∂V (φ)

∂φ

ansatz:

φ(r, t) = f r − R(t)( )

⇒ − ′ f dR

dt= ′ ′ f +

2

r′ f − ′ V ( f )

Page 50: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

coarsening by shrinkage of small domains

consider a droplet of size R, use

∂φ(x)

∂t= −

δE

δφ(x)

spherical symmetry:

∂φ∂t

=∂ 2φ

∂r2+

2

r

∂φ

∂r−

∂V (φ)

∂φ

ansatz:

φ(r, t) = f r − R(t)( )

⇒ − ′ f dR

dt= ′ ′ f +

2

r′ f − ′ V ( f )

f’(x) is has a localized peak around x = 0 (the domain wall)

Page 51: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

coarsening by shrinkage of small domains

consider a droplet of size R, use

∂φ(x)

∂t= −

δE

δφ(x)

spherical symmetry:

∂φ∂t

=∂ 2φ

∂r2+

2

r

∂φ

∂r−

∂V (φ)

∂φ

ansatz:

φ(r, t) = f r − R(t)( )

⇒ − ′ f dR

dt= ′ ′ f +

2

r′ f − ′ V ( f )

f’(x) is has a localized peak around x = 0 (the domain wall)Multiply by f’ and integrate through the wall:

Page 52: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

coarsening by shrinkage of small domains

consider a droplet of size R, use

∂φ(x)

∂t= −

δE

δφ(x)

spherical symmetry:

∂φ∂t

=∂ 2φ

∂r2+

2

r

∂φ

∂r−

∂V (φ)

∂φ

ansatz:

φ(r, t) = f r − R(t)( )

⇒ − ′ f dR

dt= ′ ′ f +

2

r′ f − ′ V ( f )

f’(x) is has a localized peak around x = 0 (the domain wall)Multiply by f’ and integrate through the wall:

⇒dR

dt= −

2

R

Page 53: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

coarsening by shrinkage of small domains

consider a droplet of size R, use

∂φ(x)

∂t= −

δE

δφ(x)

spherical symmetry:

∂φ∂t

=∂ 2φ

∂r2+

2

r

∂φ

∂r−

∂V (φ)

∂φ

ansatz:

φ(r, t) = f r − R(t)( )

⇒ − ′ f dR

dt= ′ ′ f +

2

r′ f − ′ V ( f )

f’(x) is has a localized peak around x = 0 (the domain wall)Multiply by f’ and integrate through the wall:

⇒dR

dt= −

2

R

⇒ R2(t) = R2(0) − 4t

Page 54: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

coarsening by shrinkage of small domains

consider a droplet of size R, use

∂φ(x)

∂t= −

δE

δφ(x)

spherical symmetry:

∂φ∂t

=∂ 2φ

∂r2+

2

r

∂φ

∂r−

∂V (φ)

∂φ

ansatz:

φ(r, t) = f r − R(t)( )

⇒ − ′ f dR

dt= ′ ′ f +

2

r′ f − ′ V ( f )

f’(x) is has a localized peak around x = 0 (the domain wall)Multiply by f’ and integrate through the wall:

⇒dR

dt= −

2

R

⇒ R2(t) = R2(0) − 4t

=> disappearance and coalescence of domainsof size R(0) at time ~ ¼R(0)2

Page 55: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

scaling

remaining domains at time t have size ~ t½

Page 56: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

scaling

remaining domains at time t have size ~ t½

There is no other (long) length scale for correlations todepend on.

Page 57: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

scaling

remaining domains at time t have size ~ t½

suggests scaling of correlations

There is no other (long) length scale for correlations todepend on.

Page 58: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

scaling

remaining domains at time t have size ~ t½

suggests scaling of correlations

⇒ C(x − ′ x , t) ≡ φ(x, t)φ( ′ x , t) = φ02g

x − ′ x

t1/ 2

⎝ ⎜

⎠ ⎟

There is no other (long) length scale for correlations todepend on.

Page 59: Lecture 12: Domains, nucleation and coarsening Outline: domain walls nucleation coarsening

scaling

remaining domains at time t have size ~ t½

suggests scaling of correlations

⇒ C(x − ′ x , t) ≡ φ(x, t)φ( ′ x , t) = φ02g

x − ′ x

t1/ 2

⎝ ⎜

⎠ ⎟

There is no other (long) length scale for correlations todepend on.

g for Ising model: