1. Introduction to fractal conceps
A very common deposition model: snow particles falling on a slanted glass window

2. Rough to the microscope”
“Smooth to the eye,
Rough to the microscope”
Different systems/objects look
very similar at different length
scales (surface roughness)
Roughness seems correlated
to the observation lengthscale
(zoom-in, zoom-out)

3. Scaling Concepts
Roughness increases with increasing zoom

4. Fractals
are mathematical
objects defined
by iteration
Rescaled replica
of a same element
Complexity
is generated in a
deterministic way

5. Ballistic deposition
Surface: set of particles at the top of each column at a given time t
Different grey intensity shows the arrival time of the particles ( different time intervals)
Roughness increases in time

6. Ballistic deposition: different plots
(logarithmic scale)
(logarithmic scale)

7. Ballistic depositions at different L
after rescaling

8. Self-affinity and surface roughness
Deterministic self-affine fractal Statistical self-affine fractal
DNA “walk” (i.e., sequence of “bases”):
Pyrimidines C,T : -1 (cytosine, thymine) Purines A,G : +1 (adenine, guanine)

9. Random walk and roughness
Random walk continuously zoomed in, non-uniformly:
zoom is quadratically faster in t (horizontally) than in x (vertically), because s(x)2t
Is there any parameter to “quantify the roughness” of random walk,
and the way it rescales under asymmetric transformations?

10. sx|t2-t1|a Random walk: roughness exponent =1/2 s(x)2t1-t2 i.e.
Brownian motion: roughness exponent as a variable
sx|t2-t1|a

11. Random deposition

12. Correlations: in Ballistic Deposition
the lateral growth is due to “correlations” among the surface sites, which is a consequence of surface potentials

13. Random Deposition with
Surface Relaxation

14. RD+SR: Surface tension
Smoothing

15. RD+SR: Surface tension
Wetting
Dewetting
 Adhesion Forces> Cohesive Forces
 Adhesion Forces< Cohesive Forces
 Spreading of the liquid on the surface of the solid.
 The liquid pulls itself together into the shape of a droplet.
 Contact Angle q : 0 <  q  <  p/2
 Contact Angle q :  p/2 <q < p

16. (Ballistic deposition)
KPZ: lateral growth
(Ballistic deposition)
Non-linear term
non-conservation of the number of particles

17. diffusion + desorption
Linear theory of MBE
Linear theory of MBE
diffusion + desorption
Chemical potential for diffusion
and local surface curvature:
Number of bonds (nearest neighbors)
Negative curvature

18. [EW] and [KPZ] equations relevant to desorption and diffusion
Linear theory of MBE
[EW] and [KPZ] equations relevant to desorption and diffusion