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

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

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

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

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

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

7. Non-uniform stationary solution

8. rewrite this with

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

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

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

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

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

14. 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 )

15. 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 )

16. kink energy calculation

23. ________ thickness

24. kink energy calculation _____ energy density ________ thickness

25. kink energy calculation _____ energy density ________ thickness

26. Higher dimensions energy of domain wall of size L is

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

28. Nucleation With a field,

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

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

31. 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

32. 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

33. 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:

34. 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:

35. a Kramers escape problem:

37. to find barrier height: maximize E(R)

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

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

40. 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.

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

42. 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

43. 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.

44. 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)

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

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

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

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

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

50. 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)

51. 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:

52. 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:

53. 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:

54. 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

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

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

57. 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.

58. 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.

59. 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: