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
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
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
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 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
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
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
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 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
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
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 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
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 added
Hybrid of discrete and continuum
elements
Different parameters used
diffusion rate
latent heat added upon solidification
local temperature threshold
Different Macroscopic Forms
Amorphous fractal growth
Tendril 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. 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
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.
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(c
x
) + f(c
y
) + f(c
z
) )
where f(c
i
) = 1/ c
i
c
i
>= 1
f(c
i
)
=
0
c
i
<
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 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
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 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
Hexlife
V
1
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 1.0
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)
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
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