Cellular Automata Models of

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Dec 1, 2013 (3 years and 9 months ago)

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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



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




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