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

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)

Scaling Concepts Roughness increases with increasing zoom

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

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

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

Ballistic depositions at different L after rescaling

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)

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?

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

Random deposition

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

Random Deposition with Surface Relaxation

RD+SR: Surface tension Smoothing

RD+SR: Surface tension http://www.adhesives.org/adhesives-sealants/science-of-adhesion/wetting 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

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

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

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