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Modeling Spontaneous Three-Dimensional Polymerization

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Modeling Spontaneous Three-Dimensional Polymerization Michigan Technological University Digital Commons @ Michigan Tech Dissertations, Master's Theses and Master's Dissertations, Master's Theses and Master's Reports - Open Reports 2012 Modeling spontaneous three-dimensional polymerization Zhitong Zhao Michigan Technological University Follow this and additional works at: https://digitalcommons.mtu.edu/etds Part of the Mathematics Commons Copyright 2012 Zhitong Zhao Recommended Citation Zhao, Zhitong, "Modeling spontaneous three-dimensional polymerization", Master's Thesis, Michigan Technological University, 2012. https://doi.org/10.37099/mtu.dc.etds/208 Follow this and additional works at: https://digitalcommons.mtu.edu/etds Part of the Mathematics Commons MODELING SPONTANEOUS THREE-DIMENSIONAL POLYMERIZATION By Zhitong Zhao A THESIS Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE (Mathematical Sciences) MICHIGAN TECHNOLOGICAL UNIVERSITY 2012 c 2012 Zhitong Zhao This thesis, "Modeling Spontaneous Three-Dimensional Polymerization," is hereby approved in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE IN MATHEMATICAL SCIENCES. Department of Mathematical Sciences Signatures: Thesis Advisor Dr. Iosif Pinelis Committee Member Dr. Qiuying Sha Committee Member Dr. Yue Li Department Chair Dr. Mark S. Gockenbach Date To my adviser, teachers and friends who didn’t hesitate to criticize my work at every stage - without which I’d neither be who I am nor would this work be what it is today. Contents List of Figures ..................................... vi List of Tables ...................................... x Acknowledgments ................................... xi Abstract ......................................... xii 1 Background ..................................... 1 1.1 Introduction . ................................ 1 1.2Dipoles.................................... 4 1.2.1 Types Of Dipoles . ........................ 4 1.2.2 Dipole Moment ............................ 4 1.2.2.1 Electric Dipole Moment . ................ 4 1.2.2.2 Molecular Dipole Moment . ................ 5 1.2.2.3 Bond Dipole Moment . ................ 6 1.3 Spontaneous Polymerization . ........................ 7 1.4 Boltzmann Factor, Partition Function, And State Probabilities ........ 8 iv 2 Modeling ...................................... 11 2.1Ideal2DModel[1].............................. 13 2.2Ideal3DModel................................ 20 2.3Physical3DModel.............................. 26 2.4Range-adjustedPhysical3DModel...................... 32 3 Results & Conclusions ............................... 36 3.1Results..................................... 36 3.1.1 Physical 3D model with t=600 .................... 36 3.1.2 Physical 3D model with t=1500 . ................ 41 3.1.3 Range-adjusted physical 3D model with t=1500 . ........ 49 3.2Conclusions.................................. 57 3.2.1 Dominant Parameters . ........................ 57 3.2.2 The Role Of Time Limit . .................... 63 3.3FuturePlans..................................63 References ....................................... 65 A Matlab Code .................................... 67 B Generalized Dipole Moment And Potential Energy Of Dipole ......... 91 C Units Transformation ............................... 94 D Taylor Series Expansion Of The Configuration Factor cF ........... 98 v E Monotonic Kernel Fitting Function ........................100 vi List of Figures 1.1 Chemical bond examples. These figures are licensed under the Creative CommonsAttribution-ShareAlikelicense[2],[3]............... 7 (a) Covalent bond . ............................ 7 (b) Ionic bond ................................ 7 2.1 Four of adjacent 20 × 20 blocks, of which three are shown with oriented cells 14 2.2 Dipole distribution at the beginning of the polymerization process within a 20 × 20blockin2D............................... 19 2.3 Dipole distribution at the presumed end of the polymerization process within a 20 × 20 block in 2D with different parameter settings. 20 2.4 Dipole distribution at the beginning of the polymerization process within a 10 × 10 × 10 block in 3D. Note for colors of the small arrows: blue–right, cyan–left, white–up, black–down, red–front, green–back. ........ 24 2.5 Dipole distribution at the presumed end of the polymerization process within a 10 × 10 × 10 block in 3D when a = 0.5,β = 1, relatively small dipolesize,withhightemperature...................... 25 vii 2.6 Dipole distribution at the presumed end of the polymerization process within a 10 × 10 × 10 block in 3D when a = 0.5,β = 2, relatively small dipolesize,withlowtemperature....................... 26 2.7 Dipole distribution at the presumed end of the polymerization process within a 10 × 10 × 10 block in 3D when a = 0.9,β = 1, relatively large dipolesize,withhightemperature...................... 27 2.8 Dipole distribution at the presumed end of the polymerization process within a 10 × 10 × 10 block in 3D when a = 0.9,β = 2, relatively large dipolesize,withlowtemperature....................... 28 3.1 Average length of all chains in physical 3D model with t=600. Count both polymers and independent monomers as chains. ............ 37 3.2 Average length of nontrivial chains in physical 3D model with t=600. Count only polymers as chains. ........................ 38 3.3 Number of all chains in physical 3D model with t=600. Count both polymers and independent monomers as chains. ............ 39 3.4 Number of nontrivial chains in physical 3D model with t=600. Count only polymersaschains............................... 40 3.5 Average length of all chains in physical 3D model with t=1500. Count both polymers and independent monomers as chains. ............ 42 3.6 Number of all chains in physical 3D model with t=1500. Count both polymers and independent monomers as chains. ............ 43 viii 3.7 Average length of nontrivial chains in physical 3D model with t=1500. Count only polymers as chains. ........................ 44 3.8 Number of nontrivial chains in physical 3D model with t=1500. Count only polymersaschains............................... 45 3.9 Comparison between two measurements on average length of chains in physical 3D model with t=1500. .................... 46 3.10 Comparison between two measurements on number of chains in physical 3D model with t=1500 ............................ 47 3.11 Actual time of polymerization in physical 3D model with t=1500. 48 3.12 Average length of all chains in range-adjusted physical 3D model with t=1500. Count both polymers and independent monomers as chains. 50 3.13 Number of all chains in range-adjusted physical 3D model with t=1500. Count both polymers and independent monomers as chains. 51 3.14 Average length of nontrivial chains in range-adjusted physical 3D model with t=1500. Count only polymers as chains. ................ 52 3.15 Number of nontrivial chains in range-adjusted physical 3D model with t=1500. Count only polymers as chains. ................ 53 3.16 Comparison between two measurements on average length of chains in range-adjusted physical 3D model with t=1500. ............ 54 3.17 Comparison between two measurements on number of chains in range-adjusted physical 3D model with t=1500. ............ 55 ix 3.18 Actual time of polymerization in range-adjusted physical 3D model with t=1500. .................................... 56 3.19 Quadratic polynomial fitting using data in 5 ≤ γ < 20 in final 3D model. 59 3.20 Kernel function fitting using data in 5 ≤ γ < 20 in final 3D model. 60 3.21 Comparison between quadratic polynomial and kernel fitting functions when 5 ≤ γ < 20infinal3Dmodel...................... 62 x List of Tables 2.1 Full descriptive information of polymerization under different setting of parameters. NOD:total number of dipoles; NOA: number of all chains; ALOA: average length of all chains; NONT: number of nontrivial chains; ALONT: average length of nontrivial chains. ................ 33 2.2 γ interval and corresponding parameters based on physical criteria. 35 xi Acknowledgments I would like to thank all those who have helped me learn, understand and appreciate this subject. Without your help, this thesis can never be done. Particularly, I appreciate for all the guide and advice from my adviser during the last two years. I learned much more than I could have imagined. Not only in academic area, but also in the way of critical thinking. Besides, I would love to thank all my professors giving those useful statistics courses. Those courses provided me new perspectives in various disciplines, such as genetics and programming. Above all, I appreciate the opportunity I received at Michigan Technological University and really enjoy the life in the past two years. xii Abstract For human beings, the origin of life has always been an interesting and mysterious matter, particularly how life arose from inorganic matter through natural processes. Polymerization is always involved in such processes. In this paper we built what we refer to as ideal and physical models to simulate spontaneous polymerization based on certain physical principles. As the modeling confirms, without taking external energy, small and simple inorganic molecules formed bigger and more complicated molecules, which are necessary ingredients of all living organisms. In our simulations, we utilized actual ranges of parameters according to their experimentally observed values. The results from the simulations led to a good agreement with the nature of polymerization. After sorting out through all the models that
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