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

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