Pattern Formation in Physical Systems: From a Snowflake to an Air BubbleStudent: Vishesh Verma, Stevenson High School

Faculty Advisor: Professor Shuwang Li, Applied Math Department, IIT Chicago

Pattern Formation in Physical Systems: From a Snowflake to an Air BubbleStudent: 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

D

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

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

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

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, humidityDriving forces for crystal growth

Purpose

1. To understand the underlying physical mechanism governing the pattern formation2. To design a method such that a crystal evolves to a desired or predetermined shape3. To achieve this goal, physicists use a Hele-Shaw cell to study the pattern formation

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

D

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

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

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

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, humidityDriving forces for crystal growth

Purpose

1. To understand the underlying physical mechanism governing the pattern formation2. To design a method such that a crystal evolves to a desired or predetermined shape3. To achieve this goal, physicists use a Hele-Shaw cell to study the pattern formation

A Sister Problem: Hele Shaw bubbles

Hele Shaw bubbleSet up: two plates with very little gap, viscous liquid (ex. oil) in betweenDrill hole in top plate, put tube in hole, and force air throughForms a bubble in a shape representing a snow crystal (not necessarily 6 sides though)

Viscous fingering pattern due to Saffman-Taylor instabilitySnow crystals : Mullins-Sekerta Instability :: Hele Shaw bubble : Saffman-Taylor instability

Side View

A Sister Problem: Hele Shaw bubbles

Hele Shaw bubbleSet up: two plates with very little gap, viscous liquid (ex. oil) in betweenDrill hole in top plate, put tube in hole, and force air throughForms a bubble in a shape representing a snow crystal (not necessarily 6 sides though)

Viscous fingering pattern due to Saffman-Taylor instabilitySnow crystals : Mullins-Sekerta Instability :: Hele Shaw bubble : Saffman-Taylor instability

Side View

References1. S. Li, J. Lowengrub, P. Leo, A rescaling scheme with

application to the long time simulation of viscous

fingering in a Hele-Shaw cell, Journal of Computational

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

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.

References1. S. Li, J. Lowengrub, P. Leo, A rescaling scheme with

application to the long time simulation of viscous

fingering in a Hele-Shaw cell, Journal of Computational

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

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.

Conclusion

The patterns found in physical structures depend

on

1. The initial shape

2. The air pumping rate or other similar driving

forces

3. The symmetry of the anisotropy mode

Implications

4. Our research project has to understand the

formation of snowflakes and explains why

snowflakes are all different.

5. Our research project also offers insights to

the oil-recovery application in petroleum

engineering.

Conclusion

The patterns found in physical structures depend

on

1. The initial shape

2. The air pumping rate or other similar driving

forces

3. The symmetry of the anisotropy mode

Implications

4. Our research project has to understand the

formation of snowflakes and explains why

snowflakes are all different.

5. Our research project also offers insights to

the oil-recovery application in petroleum

engineering.

ProcedureProcedure

HypothesisIf we change the air pumping rate and the initial

shape, then we will get a variety of patterns.

HypothesisIf we change the air pumping rate and the initial

shape, then we will get a variety of patterns.

MaterialsComputer cluster (for computation)

Matlab (for post processing data)

MaterialsComputer cluster (for computation)

Matlab (for post processing data)

VariablesAir flow rate, initial shape, number of computational

points, symmetry modes

VariablesAir flow rate, initial shape, number of computational

points, symmetry modes

INPUT

Air flow rate Initial shape Number of

computational points Physical parameters

 

 

TOOLBOX

OUTPUT

Different shapes (made of a collection of points)

Different fingering patterns

 

 

Results Explanation

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 constantIn 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.

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.