Collective diffusion of the interacting surface gas Magdalena Załuska-Kotur Institute of Physics, Polish Academy of Sciences

Random walk Diffusion coefficient D

+ mass conservation Collective diffusion local density

The model – noninteracting lattice gas Equilibrium distribution c – microstate Local density

single particle result Noninteracting system

D o =Wa 2 for small k Single particle diffusion – noninteracting gas.

Interacting particles

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

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?

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

Attraction J<0 T=0.89T c T c =1.8|J|/k J’=2J J’=J J’=0 J=0 J’=J J’=2J J>0 Repulsion

Experimental results - Pb/Cu(100 )

Simulation methods Harmonic density perturbation Step profile decay

kT=0.25J kT=0.5J kT=J

Profile evolution Boltzmann –Matano method

Definition of transition rates

The model Detailed balance condition Equilibrium distribution c – microstate

Possible approaches Hierarchy of equations - QCA

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

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

Eigenvalue of matrix M Approximation: EigenvalueLimit

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

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

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

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

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

Activation energy –repulsive interactions

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

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

Activation energy – attractive interactions

Eigenvector for random state Initial configuration

Repulsive far from equilibrium case θ θ ν p=100

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

Equilibrium probability strong repulsion Diagonal matrix

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

Result Upper line:Lower line:

J’=3/4J Ordered phase

Other parameters – kT/J=0.3

Other parameters – kT/J’=0.4

Other parameters – J’=0

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 repulsive interactions. Diffusion coefficient in 2D lattice gas of 2X2 ordered phase with repulsive forces. Agrement with numerical results Numerical approaches: step density profile evolution and harmonic density perturbation decay methods Summary

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

J=0 J’=2J J=J’ J’=2J ‘

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

Analysis of microscopic equations. Local density

1D -- z=2 D o =Wa 2 for small k

Calculation = n 1 –n 2 for s clusters Y: Łukasz Badowski, M. A. Załuska-Kotur – to be published

D o =Wa 2 Site blocking – noninteracting lattice gas Eigenvalue - For N=2