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: