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

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Presentation transcript:

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

Domain walls Below T c : m 0 : uniform solution of

Domain walls Below T c : m 0 : uniform solution of or in Landau model: ϕ 0 solves

Domain walls Below T c : m 0 : uniform solution of or in Landau model: ϕ 0 solves

Domain walls Below T c : m 0 : uniform solution of or in Landau model: ϕ 0 solves

Domain walls Below T c : m 0 : uniform solution of or in Landau model: ϕ 0 solves Suppose we have boundary conditions

Non-uniform stationary solution

rewrite this with

Non-uniform stationary solution rewrite this with (differs only by an additive constant)

Non-uniform stationary solution rewrite this with (differs only by an additive constant)

Non-uniform stationary solution rewrite this with (differs only by an additive constant) ( d = 1 )

Non-uniform stationary solution rewrite this with (differs only by an additive constant) solution: ( d = 1 )

Non-uniform stationary solution rewrite this with (differs only by an additive constant) solution: “domain wall” (“kink”) solution ( d = 1 )

Non-uniform stationary solution rewrite this with (differs only by an additive constant) solution: “domain wall” (“kink”) solution localized: size r 0 -½ ( d = 1 )

Non-uniform stationary solution rewrite this with (differs only by an additive constant) solution: “domain wall” (“kink”) solution localized: size r 0 -½ broad near T c ( d = 1 )

kink energy calculation

________ thickness

kink energy calculation _____ energy density ________ thickness

kink energy calculation _____ energy density ________ thickness

Higher dimensions energy of domain wall of size L is

Higher dimensions energy of domain wall of size L is energy of a flipped domain is proportional to its surface area

Nucleation With a field,

Nucleation Suppose we are below T c ( r 0 < 0 ), ϕ = - ϕ 0 With a field,

Nucleation Suppose we are below T c ( r 0 < 0 ), ϕ = - ϕ 0 Then make h > 0 With a field,

Nucleation Suppose we are below T c ( r 0 < 0 ), ϕ = - ϕ 0 Then make h > 0 With a field, It is now favorable to change to ϕ = + ϕ 0

Nucleation Suppose we are below T c ( r 0 < 0 ), ϕ = - ϕ 0 Then make h > 0 With a field, It is now favorable to change to ϕ = + ϕ 0 but it costs energy to make a local region where ϕ (x) = - ϕ 0

Nucleation Suppose we are below T c ( r 0 < 0 ), ϕ = - ϕ 0 Then make h > 0 With a field, It is now favorable to change to ϕ = + ϕ 0 but it costs energy to make a local region where ϕ (x) = - ϕ 0 energy of a spherical bubble of radius R of the + phase:

Nucleation Suppose we are below T c ( r 0 < 0 ), ϕ = - ϕ 0 Then make h > 0 With a field, It is now favorable to change to ϕ = + ϕ 0 but it costs energy to make a local region where ϕ (x) = - ϕ 0 energy of a spherical bubble of radius R of the + phase:

a Kramers escape problem:

to find barrier height: maximize E(R)

a Kramers escape problem: to find barrier height: maximize E(R)

a Kramers escape problem: to find barrier height: maximize E(R)  nucleation time

Spinodal decomposition Ordering by nucleation is a transition from a metastable state to a stable one. In spinodal decomposition, one quenches to a temperature below T c at zero magnetization, an unstable state. Local domains order and grow.

Spinodal decomposition Ordering by nucleation is a transition from a metastable state to a stable one.

Spinodal decomposition Ordering by nucleation is a transition from a metastable state to a stable one. In spinodal decomposition, one quenches to a temperature below T c at zero magnetization

Spinodal decomposition Ordering by nucleation is a transition from a metastable state to a stable one. In spinodal decomposition, one quenches to a temperature below T c at zero magnetization, an unstable state.

Spinodal decomposition Ordering by nucleation is a transition from a metastable state to a stable one. In spinodal decomposition, one quenches to a temperature below T c at zero magnetization, an unstable state. Local domains order and grow. Ising model, T = 0 20 MC sweeps 200 MC sweeps (from J Sethna)

coarsening by shrinkage of small domains consider a droplet of size R

coarsening by shrinkage of small domains consider a droplet of size R, use

coarsening by shrinkage of small domains consider a droplet of size R, use spherical symmetry:

coarsening by shrinkage of small domains consider a droplet of size R, use spherical symmetry: ansatz:

coarsening by shrinkage of small domains consider a droplet of size R, use spherical symmetry: ansatz:

coarsening by shrinkage of small domains consider a droplet of size R, use spherical symmetry: ansatz: f’(x) is has a localized peak around x = 0 (the domain wall)

coarsening by shrinkage of small domains consider a droplet of size R, use spherical symmetry: ansatz: f’(x) is has a localized peak around x = 0 (the domain wall) Multiply by f’ and integrate through the wall:

coarsening by shrinkage of small domains consider a droplet of size R, use spherical symmetry: ansatz: f’(x) is has a localized peak around x = 0 (the domain wall) Multiply by f’ and integrate through the wall:

coarsening by shrinkage of small domains consider a droplet of size R, use spherical symmetry: ansatz: f’(x) is has a localized peak around x = 0 (the domain wall) Multiply by f’ and integrate through the wall:

coarsening by shrinkage of small domains consider a droplet of size R, use spherical symmetry: ansatz: f’(x) is has a localized peak around x = 0 (the domain wall) Multiply by f’ and integrate through the wall: => disappearance and coalescence of domains of size R(0) at time ~ ¼R(0) 2

scaling  remaining domains at time t have size ~ t ½

scaling  remaining domains at time t have size ~ t ½ There is no other (long) length scale for correlations to depend on.

scaling  remaining domains at time t have size ~ t ½ suggests scaling of correlations There is no other (long) length scale for correlations to depend on.

scaling  remaining domains at time t have size ~ t ½ suggests scaling of correlations There is no other (long) length scale for correlations to depend on.

scaling  remaining domains at time t have size ~ t ½ suggests scaling of correlations There is no other (long) length scale for correlations to depend on. g for Ising model: