Cellular Automata Models of Crystals and Hexlife

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Cellular Automata Models of Crystals and Hexlife

• CS240 Software Project
• Spring 2003
• Gauri Nadkarni

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Outline

• Background
• Description of crystals
• Packards CA model
• A 3D CA model
• Hexlife
• Summary

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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 1600s
• The first synthetic gemstones were made in the
  mid-1800s
• Crucial to semi-conductor industry since
  mid-1970s

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Categories of Crystals

• Hopper crystals
• Polycrystalline materials
• Quasicrystals
• Amorphous materials
• Snow crystals and snowflakes

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Hopper Crystals

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

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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
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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

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Quasicrystals

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

Some forms of quasicrystals
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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

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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

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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

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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

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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

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

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Packards 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

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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

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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

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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

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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

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

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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.

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Rules and Conditions

• A liquid site may transform to a solid if cx gt 3
  and/or cy gt 3 and/or cz gt3
• Growth occurs if the temperature of the liquid
  site lt Tcrit
• Tcrit -? ( f(cx) f(cy) f(cz) )
• where f(ci) 1/ ci ci gt 1
• f(ci) 0 ci lt 1
• (? is a constant)

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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

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Results and Observations

• ? 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.

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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

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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

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Hexlife

• A model of Conways 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

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

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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

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Summary

• Hexlife is modeled on Conways game of life on a
  hexagonal grid
• Hexlife considers the sum of 12 neighbors as
  opposed to 8 neighbors considered on Conways
  game of life