Spring-block tessellations with n springs per block

Bruno Escribano Basque Center for Applied Mathematics Julyan Cartwright CSIC - University of Granada Outline

• 1. Motivation • 2. Introduction: – spring tessellations (tilings) – crystals, quasicrystals and amorphous solids • 3. Results • 4. Conclusions and outlook

Motivation: types of Ice

crystalline Ice on Earth hexagonal (snow, glaciers, icebergs…)

Amorphous ice Ice in the Universe and others… ???

Motivation: ice in the Universe

• Interstellar dust grains, asteroids, comets, surface of planets and moons… • Formed at very low temperatures • Unknown morphology • Unknown structure

Phase transition?

• Different amorphous ices depending on the deposition temperature.

130 K > Tm > 30 K → Low Density Amorphous (LDA)

30 K > Tm → High Density Amorphous (HDA)

Possible phase transition HDA → LDA ?

ESEM Setup

Environmental SEM Microscope + liquid He circuit. In low vacuum at temperatures 6-220 K

Grow ice in-situ injecting water vapor.

Mesoscopic morphologies

HDA T=6 K LDA T=77 K Hexagonal T=250 K

We still don't know about the molecular structure!

Characterizing amorphous solids

X-ray / neutron diffraction Radial distribution functions Jenniskens et al. APJ (1995) Finney et el. Phys. Rev. Lett. (2002).

What kind of structure would produce these results?

2D tessellations

3 regular tessellations

8 semiregular tessellations

Pentagonal tessellations

14 known pentagonal tessellations:

Beyond periodicity: quasicrystals

Penrose tilings (1976)

• Ordered but not periodic (no translational symmetry). • Defined by quasiperiodic functions. i.e. f(x)=cos(x)+cos(αx) where α is irrational

Can this happen naturally?

Beyond periodicity: quasicrystals

Shechtman et al. (1984)

AlCuFe icosahedral phase Beyond periodicity: quasicrystals

“Crystal is any solid which possesses an essentially discrete diffraction pattern.” International Crystallographic Union, 1991

Crystalline Amorphous

So we now define crystals (and quasicrystals) through an experimental technique, but...

Spring tessellations: the model

• Spring–block networks with n springs per block Parameters: – maximum coordination number (n) – total number of particles (N) equilibrium distance for springs (d ) – 0 – minimum and maximum radius for neighbor searching (r and r ) min max

Question: What kind lattice would produce the experimental values observed in amorphous ices?

Spring tessellations: the model

Damped harmonic oscillator

Critical damping

Example: n = 6

Results: Effect of n (number of springs per block)

n=2 n=3 n=4

n=5 n=6 Results: radial distribution functions

n=2 n=3 n=4

n=5 n=6 Results: XRD / FFT pattern (crystallization process)

Results: effects of rmax (generating deffects)

rmax =0.090 rmax =0.095 rmax =0.100 rmax =0.110

Conclusions • Experiments: - First electron microscopy images of amorphous ice thin films. - The proposed phase transition HDA-LDA shows no morphology change. • Model: - We can use Fourier transforms to measure crystallinity. - The triangular tiling is the the minimum energy configuration. Outlook

• 3D quasicrystals and amorphs. • Measure residual energies on each spring to find equilibrium configurations and classify according to crystallinity. • Introduce simulated temperatures...

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