Fractal behavior of the BCC/FCC phase separation in iron-gold alloys Artur Błachowski 1, Krzysztof Ruebenbauer 1, Anna Rakowska 2,3 1 Mössbauer Spectroscopy.

Fractal behavior of the BCC/FCC phase separation in iron-gold alloys

Artur Błachowski 1, Krzysztof Ruebenbauer 1, Anna Rakowska 2,3

1 Mössbauer Spectroscopy Division,Institute of Physics, Pedagogical University, Kraków, Poland

2 Faculty of Materials Science and Ceramics, AGH University of Science and Technology, Kraków, Poland

3 Institute of Metallurgy and Materials Science, Polish Academy of Science, Kraków, Poland

Fe-Au Phase Diagram

BCC–Fe(Au)as-cast 3 at.% Auannealed ~0 at.% Au

FCC–Au(Fe)as-cast 33 at.% Feannealed 23 at.% Fe

Fe70Au30 and Fe50Au50

as-cast and annealed 250ºC/24h + 500ºC/48h

Nanoparticles of FCC-Au(Fe) and BCC-Fe(Au) phases

with diameter about 50 nm !!!Ultra high-resolution scanning electron microscope SEM-FEG + EDS (Nova 200 NanoSEM)

200x – 300000x

Fe70Au30 as-castEDS results (1 μm scale)

14 at.% Au in dark regions and 45 at.% Au in bright regions

Fe50Au50 as-castEDS results (1 μm scale)

49 - 56 at.% Au

Fe70Au30 annealedEDS results (1 μm scale)

14 at.% Au in dark regions and 45 at.% Au in bright regions

Fe50Au50 annealedEDS results (1 μm scale)

19 at.% Au in dark regions and 75 at.% Au in bright regions

What is fractal?

– self-similarity

– fractal dimension

1) mathematical fractalSierpiński carpet and triangle, Mandelbrot set, Julia set, Barnsley’s fern (atractor)

2) natural fractalmountain landscape, window frost, cauliflower surface, blood vessels,

exchange quotations

1

2

1

2

4

1

21

12 )2( )2( 4

N

n

M

m iL

inm xxMNR

Roughness method

Image were pixelized into square pixels with L = 8 bits per pixel representing linear gray scale with the highest value (255) representing “white”.

N = 2048 pixels vertical and M = 2375 pixels horizontal.

fractal dimension r - pixel edge size

rR loglog

lo

g R

log r

d = 1.903(4)

d = 1.8928

Test

diffusion limited aggregation (DLA) fractal d = 1.71

Results

500 μm 500 nm

Conclusions

– self-similarity

– fractal dimension similar to (hydrodynamic) DLA fractal