cellular automata models of crystals and hexlife cs240 – software project spring 2003 gauri...
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Cellular Automata Models of Crystals and Hexlife
CS240 – Software Project
Spring 2003
Gauri Nadkarni
Outline
Background
Description of crystals
Packard’s CA model
A 3D CA model
Hexlife
Summary
BackgroundWhat is a Cellular Automaton (CA)?StateNeighborhoodProgram
What are crystals?Solidification of fluid, vapors, solutions
Relation of CA and crystalsSimilar structure
History of CrystalsCrystals 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
Categories of Crystals
Hopper crystals
Polycrystalline materials
Quasicrystals
Amorphous materials
Snow crystals and snowflakes
Hopper Crystals
These have more rapid growth at the edge of each face than at the center
Examples: rose quartz, gold, salt and ice
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
Quasicrystals
Crystals composed of periodic arrangement of identical unit cells Only 2-,3-,4-, and 6-fold rotational symmetries are possible for periodic crystalsShechtman observed new symmetry while performing an electron diffraction experiment on an alloy of aluminium and manganeseThe alloy had a symmetry of icosahedron containing a 5-fold symmetry. Thus quasicrystals were born
Quasicrystals
They are different from periodic crystals
To this date, quasicrystals have symmetry of tetrahedron, a cube and an icosahedron
Some forms of quasicrystals
Amorphous Materials
Do not have a well-ordered structureLack distinctive crystalline shapeCooling process is very rapidEx: Amorphous silicon, glasses and plasticsAmorphous silicon used in solar cells and thin film transistors
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
Growth process of snow crystals
A dust particle absorbs water molecules that form a nucleusThe newborn crystal quickly grows into a tiny hexagonal prismThe corners sprout tiny arms that grow furtherCrystal growth depends on surrounding temperature
Growth process of snow crystals
Variation in temperature creates different growth conditions
Two dominant mechanisms that govern the growth rateDiffusion – the way water molecules diffuse
to reach crystal surfaceSurface physics of ice – efficiency with
which water molecules attach to the lattice
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
Different Snowflake Forms
Simple Sectored Plate Dendritic Sectored Plate
Fern-like Stellar Dendrite
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
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
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
Description of the model
Four Types of behaviorNo growth Plate structure reflecting the lattice
structureDendritic structure with side branches
growing along lattice directionsGrowth of an amorphous, asymptotically
circular form
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
Simulations
Temperature is set to a constant high value when new solid is addedHybrid of discrete and continuum elementsDifferent parameters used diffusion rate latent heat added upon solidification local temperature threshold
Different Macroscopic Forms
Amorphous fractal growthTendril growth dominated by tip splitting
Strong anisotropy, stable parabolic tip with side branching
A 3D CA model of ‘free’ dendritic growth
Proposed by S. Brown and N. BruceA 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 liquidBoth pure materials and alloys can display free dendritic growth behavior
The CA Model
A 100x100x100 element grid is used with an initial nucleus of 3x3x3 elements placed at the centerEach 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.
Rules and Conditions
A liquid site may transform to a solid if cx >= 3 and/or cy >= 3 and/or cz >=3Growth 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)
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
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.
Results and Observations
With judicious choice of parameters , it is possible to simulate growth of highly complex 3-D dendritic morphologyFor larger initial supercoolings, compact structures were producedAs the supercooling was reduced, a plate-like growth was observedWhen decreased further, a more spherical growth pattern with tip-splitting was observed
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
Hexlife
A model of Conway’s Game of Life on a hexagonal gridEach 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
Hexlife
V1
The first tier six neighbors are marked by ‘red’ color. The second tier six neighbors considered are marked by ‘blue’ color.
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 1.0A 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)
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
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