Cellular Automata Models of Crystals and Hexlife

1. Cellular Automata Models of Crystals and Hexlife CS240 – Software Project Spring 2003 Gauri Nadkarni

2. Outline • Background • Description of crystals • Packard’s CA model • A 3D CA model • Hexlife • Summary

3. Background • What is a Cellular Automaton (CA)? • State • Neighborhood • Program • What are crystals? • Solidification of fluid, vapors, solutions • Relation of CA and crystals • Similar structure

4. History of Crystals • Crystals comes from the greek word meaning – clear ice • Came into existence in the late 1600’s • The first synthetic gemstones were made in the mid-1800’s • Crucial to semi-conductor industry since mid-1970’s

5. Categories of Crystals • Hopper crystals • Polycrystalline materials • Quasicrystals • Amorphous materials • Snow crystals and snowflakes

6. Hopper Crystals • These have more rapid growth at the edge of each face than at the center • Examples: rose quartz, gold, salt and ice

7. Polycrystalline materials • Composed of many crystalline grains not aligned with each other • Modeled by a CA which starts from several separated seeds • Crystals grow at random locations with random orientations • Results in interstitial region Growth process of polycrystalline materials

8. Quasicrystals • Crystals composed of periodic arrangement of identical unit cells • Only 2-,3-,4-, and 6-fold rotational symmetries are possible for periodic crystals • Shechtman observed new symmetry while performing an electron diffraction experiment on an alloy of aluminium and manganese • The alloy had a symmetry of icosahedron containing a 5-fold symmetry. Thus quasicrystals were born

9. Quasicrystals • They are different from periodic crystals • To this date, quasicrystals have symmetry of tetrahedron, a cube and an icosahedron Some forms of quasicrystals

10. Amorphous Materials • Do not have a well-ordered structure • Lack distinctive crystalline shape • Cooling process is very rapid • Ex: Amorphous silicon, glasses and plastics • Amorphous silicon used in solar cells and thin film transistors

11. Snow crystals • Individual , single ice crystals • Have six-fold symmetry • Grow directly from condensing water vapor in the air • Typical sizes range from microscopic to at most a few millimeters in diameter

12. Growth process of snow crystals • A dust particle absorbs water molecules that form a nucleus • The newborn crystal quickly grows into a tiny hexagonal prism • The corners sprout tiny arms that grow further • Crystal growth depends on surrounding temperature

13. Growth process of snow crystals • Variation in temperature creates different growth conditions • Two dominant mechanisms that govern the growth rate • Diffusion – the way water molecules diffuse to reach crystal surface • Surface physics of ice – efficiency with which water molecules attach to the lattice

14. Snowflakes • One of the well-known examples of crystal formation • Collections of snow crystals loosely bound together • Structure depends on the temperature and humidity of the environment and length of time it spends

15. Different Snowflake Forms Dendritic Sectored Plate Simple Sectored Plate Fern-like Stellar Dendrite

16. Packard’s CA Model • Computer simulations for idealized models for growth processes have become an important tool in studying solidification • Packard presents a new class of models representing solidification

17. Packard’s CA Model • Begin with simple models containing few elements.Then add physical elements gradually. • Goal is to find those aspects that are responsible for particular features of growth

18. Description of the model • A 2D CA with 2 states per cell and a transition rule • The states denote presence or absence of solid. • The rules depend on their neighbors only through their sum

19. Description of the model • Four Types of behavior • No growth • Plate structure reflecting the lattice structure • Dendritic structure with side branches growing along lattice directions • Growth of an amorphous, asymptotically circular form

20. Description of the model • Two important ingredients are: • Flow of heat – modeled by addition of a continuous variable at each lattice site to represent temperature • Effect of solidification on the temperature field – when solid is added to a growing seed, latent heat of solidification must be radiated away

21. Simulations • Temperature is set to a constant high value when new solid is added • Hybrid of discrete and continuum elements • Different parameters used • diffusion rate • latent heat added upon solidification • local temperature threshold

22. Different Macroscopic Forms Tendril growth dominated by tip splitting Strong anisotropy, stable parabolic tip with side branching Amorphous fractal growth

23. A 3D CA model of ‘free’ dendritic growth • Proposed by S. Brown and N. Bruce • A dendrite is a branching structure that freezes such that dendrite arms grow in particular crystallographic directions • ‘free’ dendrites form individually and grow in super-cooled liquid • Both pure materials and alloys can display free dendritic growth behavior

24. The CA Model • A 100x100x100 element grid is used with an initial nucleus of 3x3x3 elements placed at the center • Each element of the nucleus is set to value of 1 (solid) • All other elements are set to value of 0 (liquid) • Temperatures of all sites are set to an initial predetermined value representing supercooling.

25. Rules and Conditions • A liquid site may transform to a solid if cx >= 3 and/or cy >= 3 and/or cz >=3 • Growth occurs if the temperature of the liquid site < Tcrit • Tcrit = -γ ( f(cx) + f(cy) + f(cz) ) where f(ci) = 1/ ci ci >= 1 f(ci) = 0 ci < 1 (γ is a constant)

26. Rules and Conditions • If a liquid element transforms to a solid , then temperature of the element is raised to a fixed value to simulate the release of latent heat • At each time step, the temperature of each element is updated

27. Results and Observations • γ is set to value of 20 for all simulations • The initial liquid supercoolings are varied in the range –60 to –32 • Different dendritic shapes are produced • The growth is observed until number of solid sites grown from center towards the edge was 45 along any axes.

28. Results and Observations • With judicious choice of parameters , it is possible to simulate growth of highly complex 3-D dendritic morphology • For larger initial supercoolings, compact structures were produced • As the supercooling was reduced, a plate-like growth was observed • When decreased further, a more spherical growth pattern with tip-splitting was observed

29. Results and Observations • Results showed remarkable similarity to experimentally observed dendrites • Simulated dendrites produced, evolved from a single nucleus, but experimentally observed growth patterns comprised several interpenetrating dendrites

30. Hexlife • A model of Conway’s Game of Life on a hexagonal grid • Each cell has six neighbors. These are called the first tier neighbors. • The hexlife rule looks at twelve neighbors, six belonging to the first tier and remaining six belonging to the second tier

31. Hexlife V1 The first tier six neighbors are marked by ‘red’ color. The second tier six neighbors considered are marked by ‘blue’ color.

32. Hexlife - Rule • The live cells out of the twelve neighbors are added up each generation. • live 2nd tier neighbors are only weighted as 0.3 in this sum whereas live 1st tier neighbors are weighted as 1.0 • A cell becomes live if this sum falls within the range of 2.3 - 2.9, otherwise remains dead • A live cell survives to the next generation if this sum falls within the range of 2.0 - 3.3. Otherwise it dies (becomes an empty space)

33. Summary • Crystals have been known since the sixteenth century. • There are many different kinds of crystals seen in nature • It is very fascinating to see the different intricate and complex forms that one sees during crystal growth • CA models have been successfully used to simulate different growth behavior of crystals

34. Summary • Hexlife is modeled on Conway’s game of life on a hexagonal grid • Hexlife considers the sum of 12 neighbors as opposed to 8 neighbors considered on Conway’s game of life