Pattern Formation in Physical Systems: From a Snowflake to an Air Bubble
Student: Vishesh Verma, Stevenson High School
Faculty Advisor: Professor Shuwang Li, Applied Math Department, IIT Chicago
Crystal Growth Problem
Crystal growth problem Is a classical example of a phase transformation
from the liquid phase to the solid phase via heat transfer, for example the
formation of snowflakes
Hypothesis
Conclusion
If we change the air pumping rate and the initial
shape, then we will get a variety of patterns.
The patterns found in physical structures depend
on
D
1. The initial shape
Variables
A.
Snowflakes are formed around a nucleus, such as a dust particle
B.
The snowflake forms when water VAPOR freezes around the nucleus, not from frozen
rain
C.
Snowflakes have six sides/branches because the water molecules form hexagonal
crystals when frozen
D.
Snowflakes have different parts (hexagonal pieces and branches) because of the
Mullins-Sekerta Instability
-The Mullins-Sekerta Instability explains how a not-so-smooth part of a crystal
grows into a noticeable bump/branch in the crystal
2. The air pumping rate or other similar driving
forces
Air flow rate, initial shape, number of computational
points, symmetry modes
3. The symmetry of the anisotropy mode
Materials
Implications
Computer cluster (for computation)
1.
Matlab (for post processing data)
Our research project has to understand the
formation of snowflakes and explains why
The shape of the snowflake is affected mainly by the air temperature and the amount of water in
the air (supersaturation)
-All snowflakes are different because no two snowflakes take the exact same path through
the atmosphere, leading them to experience slightly different conditions which make their
appearances different
snowflakes are all different.
Procedure
2.
Our research project also offers insights to
the oil-recovery application in petroleum
All branches grow the same way because “the local conditions are essentially the same for each
arm on a tiny snow crystal” – Kenneth G. Libbrecht
engineering.
INPUT
2 types of forces shape crystal growth
-Micro: surface tension (inversely related to curvature), kinetics
k = f’’(x) / [1+(f’(x))2]3/2
k=1/R
k means curvature, R = radius
-Macro: temperature, humidity
Driving forces for crystal growth
 Air flow rate
 Initial shape
 Number of
computational points
 Physical parameters
References
1. S. Li, J. Lowengrub, P. Leo, A rescaling scheme with
Purpose
application to the long time simulation of viscous
1. To understand the underlying physical mechanism governing the pattern formation
2. To design a method such that a crystal evolves to a desired or predetermined shape
3. To achieve this goal, physicists use a Hele-Shaw cell to study the pattern formation
fingering in a Hele-Shaw cell, Journal of Computational
TOOLBOX
Physics 225, p 554-567, 2007.
2. K. Libbrecht, Morphogenesis on Ice: The Physics of
Snow Crystals, Engineering and Science, p 10-19, 2001.
3. J. Adam, Flowers of Ice-Beauty, Symmetry, and
Complexity: A Review of The Snowflake: Winter’s
A Sister Problem: Hele Shaw bubbles
Secret Beauty, Notices of AMS 52, p 402-416, 2005.
4. E. Ben-Jacob, P. Garik, The formation of patterns in
non-equilibrium growth, Nature 343, p 523-530, 1990.
Results Explanation
Hele Shaw bubble
Set up: two plates with very little gap, viscous liquid (ex. oil) in
between
Drill hole in top plate, put tube in hole, and force air through
Forms a bubble in a shape representing a snow crystal (not
necessarily 6 sides though)
Viscous fingering pattern due to Saffman-Taylor instability
Snow crystals : Mullins-Sekerta Instability :: Hele Shaw
bubble : Saffman-Taylor instability
Side View
OUTPUT
 Different shapes (made
of a collection of points)
 Different fingering
patterns
In the first column, the initial
shape was a four fold star
shape.
We apply a time
decreasing pumping rate. We
observed a tip splitting. The
symmetry mode is four.
In the second column, we
started from a circle with a six
fold symmetry. The pumping
rate is constant
In the third column, we started
from a circle with four fold
symmetry. The pumping rate
was also constant.
Acknowledgements
Vishesh would like to thank the SPARK program
for this opportunity, the computer cluster resource
at IIT, and the Matlab tutorial at the Applied Math
Department of IIT.
S. Li would like to thank the National Science
Foundation for supporting the related research
projects.