collective diffusion of the interacting surface gas magdalena załuska-kotur institute of physics,...
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Collective diffusion of the interacting surface gas
Magdalena Załuska-Kotur
Institute of Physics,
Polish Academy of Sciences
Random walk
DtR 42 Diffusion coefficient D
),(),( txPDtxPdt
d
rrDt
)(
DJ
+ mass conservation
Collective diffusion
local density
The model – noninteracting lattice gas
''' )],(),'([),(
ccccc tcPWtcPWtcP
t
Equilibrium distribution
c – microstate
Local density
k
knVcH )(
1)( cPeq
c
xi tcPnn ),(
1,0kn
ijij
jjii nnWnnWn
ti
)1()1(
ijjii nWnzWn
tsingle particle result
Noninteracting system
)()()(2)( keeWkWkt
ikaika
)()( xPekx
ixka
Do=Wa2for small k
20 Dk
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=0.89Tc
Tc=1.8|J|/k
J’=2J
J’=JJ’=0
J=0
J’=J
J’=2JJ>0
Repulsion
Experimental results - Pb/Cu(100)
Simulation methods
Harmonic density perturbation
Step profile decay
kT=0.25JkT=0.5J
kT=J
Profile evolution
Boltzmann –Matano method
Definition of transition rates
ks
jg
nJVV
nJVV
''
i
iijiij
iji nVnnJnH })({
The model
),(),'( '' tcPWtcPW eqcceqcc
k
kkl
lk nVnnJcH )(
''' )],(),'([),(
ccccc tcPWtcPWtcP
t
Detailed balance condition
)()( cHeq ecP Equilibrium
distribution
c – microstate
Possible approaches
0 11
2)()0(
)(2
1 N
ii
N
iieq tvvdt
ND
ijijij
jjijii nnWnnWn
ti
)1()1(
Hierarchy of equations
),(
kjijiji nnnnnfnnt
- QCA
X
Analysis of microscopic equations.
c
xx tcPn ),(Local density
],...,,[],[ 11 NmmXXc m
]5,3,1,[X
''' )],(),'([),(
ccccc tcPWtcPWtcP
t
L - lattice sites + periodic boundary conditions
X),( tXPm
Fourier transformation of master equation.
'
'' ),()()(m
mmmm tkPkMkP
),(),(1
tXPetkPL
x
ikxmm
ikacc ek )('F when reference particle jumps
=1 otherwise
)1()()( ''''
''' mmmmmm ccccc
cc WkWkM F
2)1(000
)1(400
0
004)1(
00)1(2
)(ˆ
ika
ika
ika
ika
eW
eWW
W
We
eW
kM
For N=2
Eigenvalue of matrix M
Approximation:
Eigenvalue
2)( Dkk dLN
L
/
0k
Limit
1)exp( ikaL
Approximate eigenvector for interacting gas
)(cHeq eP one interaction constant J
x - number of bonds xJxeq peP
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
AEeD )()(
VeWaD 20
||
)(||
2
)()0()( J
VE
Tk
J A
BeWaD
D
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
AEeD )()(
VeWaD 2)0(
||
)(||
2
)()0()( J
VE
Tk
J A
BeWaD
D
Eigenvector for random state
1)( iN Initial configuration
Repulsive far from equilibrium case
θ
θ
ν
VWW
JEA
)34)((
)34(
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
)]},()([exp{),( 0 jiEiEjiW barinit
i
iijiij
iji nVnnJnH })({
Gaz cząstek na dwuwymiarowej sieci
Einit,(i) - lokalna energia jednocząstkowa
Ebar (ij) - energia cząstki w punkcie siodłowym
Szybkość przeskoków jednocząstkowych
)/(1 TkB
Analysis of microscopic equations.
c
xx tcPn ),(Local density
''' )],(),'([),(
ccccc tcPWtcPWtcP
t
1D -- z=2
ii
ika nekn )(
Do=Wa2for small k
DtR 42 20 Dk
)()()(2)( kneeWkWnknt
ikaika
Calculation
ssNxeq CpppP
= n1 –n2
for s clusters
Y:
Łukasz Badowski, M. A. Załuska-Kotur – to be published
Do=Wa2
Site blocking – noninteracting lattice gas
Eigenvalue -
WeW
eWWW
W
WeW
eWW
kM
ika
ika
ika
ika
2)1(000
)1(400
0
004)1(
00)1(2
)(ˆ
For N=2