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 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 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, humidity Driving forces for crystal growth Purpose 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 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, humidity Driving forces for crystal growth Purpose 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 A Sister Problem: Hele Shaw bubbles 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 A Sister Problem: Hele Shaw bubbles 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 References 1. 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 , K. Libbrecht, Morphogenesis on Ice: The Physics of Snow Crystals, Engineering and Science, p 10-19, J. Adam, Flowers of Ice-Beauty, Symmetry, and Complexity: A Review of The Snowflake: Winter’s Secret Beauty, Notices of AMS 52, p , E. Ben-Jacob, P. Garik, The formation of patterns in non-equilibrium growth, Nature 343, p , References 1. 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 , K. Libbrecht, Morphogenesis on Ice: The Physics of Snow Crystals, Engineering and Science, p 10-19, J. Adam, Flowers of Ice-Beauty, Symmetry, and Complexity: A Review of The Snowflake: Winter’s Secret Beauty, Notices of AMS 52, p , E. Ben-Jacob, P. Garik, The formation of patterns in non-equilibrium growth, Nature 343, p , 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 1.Our research project has to understand the formation of snowflakes and explains why snowflakes are all different. 2.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 1.Our research project has to understand the formation of snowflakes and explains why snowflakes are all different. 2.Our research project also offers insights to the oil-recovery application in petroleum engineering. Procedure Hypothesis If we change the air pumping rate and the initial shape, then we will get a variety of patterns. Hypothesis If we change the air pumping rate and the initial shape, then we will get a variety of patterns. Materials Computer cluster (for computation) Matlab (for post processing data) Materials Computer cluster (for computation) Matlab (for post processing data) Variables Air flow rate, initial shape, number of computational points, symmetry modes Variables Air 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 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. 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.