Collective diffusion of the interacting surface gas

1. Collective diffusion ofthe interactingsurface gas MagdalenaZałuska-Kotur Institute of Physics, Polish Academy of Sciences

2. Random walk Diffusion coefficientD

3. Collective diffusion + mass conservation local density

4. Equilibrium distribution c – microstate Local density The model –noninteractinglattice gas

5. Noninteracting system single particle result

6. Single particlediffusion – noninteracting gas. for small k Do=Wa2

7. Interacting particles

8. Interacting particles –2D system with repulsive interactions J’=3/4J Square lattice

9. Questions • How diffusion depends on interactions? • How minima of the density-diffusion plot are related to the phase diagram? • Where are phase transition points? • Are there some other characteristic points?

10. Example - hexagonal lattice - repulsion kT=0.25J kT=0.5J kT=J

11. J’=2J Attraction J<0 T=0.89Tc Tc=1.8|J|/k J’=J J’=0 J’=J J>0 J’=2J Repulsion J=0

13. Experimental results - Pb/Cu(100)

14. Simulation methods • Harmonic density perturbation • Step profile decay

16. kT=0.25J kT=0.5J kT=J

17. Profile evolution Boltzmann –Matano method

18. Definition of transitionrates

19. The model Equilibrium distribution c – microstate Detailed balance condition

20. Possible approaches - QCA Hierarchy of equations

21. X Analysis of microscopic equations. Local density X L - lattice sites + periodic boundary conditions

22. For N=2 when reference particle jumps =1 otherwise Fourier transformation of master equation.

23. Eigenvalue of matrix M Approximation: Eigenvalue Limit

24. one interaction constant J x - number of bonds Approximate eigenvector for interacting gas

25. ( ) Definition of transition rates in 1D system Possible transitions

26. Diffusion coefficient of 1D system Grand canonical regime Low temperature approximation

27. Diffusion coefficient - repulsive interactions p=2,10,100

28. Diffusion coefficient - repulsive - QCA p=2,10,100

29. Activation energy –repulsive interactions

30. Diffusion coefficient - attractive interactions p=0.5,0.3,0.1

31. Diffusion coefficient - attractive QCA p=0.5,0.3,0.1

32. Activation energy – attractive interactions

33. Eigenvector for random state Initial configuration

34. Repulsive far from equilibrium case θ ν p=100 θ

35. J’ J 2x2 ordering –definition of transitionrates M. A. Załuska-Kotur Z.W.Gortel – to be published

36. Equilibrium probability strong repulsion Diagonal matrix

37. * * Components of eigenvector Primary configurations: Secondary configurations (average of neighbouring primary ones):

38. Result Upper line: Lower line:

39. J’=3/4J Ordered phase

40. Other parameters – kT/J=0.3

41. Otherparameters – kT/J’=0.4

42. Other parameters – J’=0

43. Summary • New approach to the collective diffusion problem, based on many-body function description– analytic theory. • Exact solution for noninteracting system. • Collective diffusion in 1D system with nearest neighbor attractive and repulsiveinteractions. • Diffusion coefficient in 2D lattice gas of2X2 ordered phase with repulsive forces. • Agrement with numerical results • Numerical approaches: step density profile evolution and harmonic density perturbation decay methods

44. Possible applications Analysis of • Far from equlibrium systems. • More complex interactions – long range • Surfaces with steps • Phase transitions

45. ‘ J’=2J J=J’ J’=2J J=0

47. j x i Jak dyfuzja zależy od oddziaływań? Gaz cząstek na dwuwymiarowej sieci Szybkość przeskoków jednocząstkowych Einit,(i)- lokalna energia jednocząstkowa Ebar (ij) - energia cząstki w punkcie siodłowym

48. Analysis of microscopic equations. Local density

49. 1D -- z=2 for small k Do=Wa2

50. = n1 –n2 Calculation for s clusters Y: Łukasz Badowski, M. A. Załuska-Kotur – to be published