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Inverse Problems in Polymer Characterization Arsia Takeh Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2014 Inverse Problems in Polymer Characterization Arsia Takeh Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected] FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES INVERSE PROBLEMS IN POLYMER CHARACTERIZATION By ARSIA TAKEH A Dissertation submitted to the Department of Scientific Computing in partial fulfillment of the requirements for the degree of Doctor of Philosophy Degree Awarded: Summer Semester, 2014 Copyright c 2014 Arsia Takeh. All Rights Reserved. Arsia Takeh defended this dissertation on May 29, 2014. The members of the supervisory committee were: Sachin Shanbhag Professor Directing Dissertation William Oates University Representative Anke Meyer-Baese Committee Member Peter Beerli Committee Member Jim Wilgenbusch Committee Member The Graduate School has verified and approved the above-named committee members, and certifies that the dissertation has been approved in accordance with university requirements. ii Dedicated to: My mother, father and sister iii ACKNOWLEDGMENTS This work would not have been possible without the endless support, guidance and efforts of a lot of people. Foremost, I would like to express my profound gratitude to my advisor Prof. Sachin Shanbhag, who has been a constant source of guidance, patience and support over the last five years. I am deeply grateful for his patience during years of my research and for his trust that made me through PhD program in computational science, albeit my undergraduate degree in chemical engineering. For everything you’ve done for me, Prof. Shanbhag, I thank you and I am indebted. I am also extremely thankful to my committee members, Prof. Anke Meyer-Baese, Prof. William Oates, Prof. Peter Beerli and Dr. Jim Wilgenbusch for their continued faith and encouragement of my efforts. I would like to thank all the Scientific Computing staff at Florida State University and Cen- ter for Global Engagement staff for all their administration help through these years. I thank Research Computing Center staff at Florida State University for making high-performance and high-throughput computing resources available. I gratefully acknowledge the funding sources that made my Ph.D. work possible. Part of this work was funded by the National Science Foundation. I am indebted to many friends and family for their unconditional love and support. I am sin- cerely thankful to Dr. Shahriar Rouhani, Haleh Ashki, Yassi Ashki, Mehdi Shayan, Hoda Mirafzal, Asal Mohamadi who supported me during these years, and incented me to strive towards my goal. Thank you to my parents, lovely sister, Sepideh, for standing there for me all these years and supporting me through every single challenge of my life. Words alone cannot express the thanks I owe them. I would also thank my uncles, Dr. Alireza Sedghi, Dr. Ramin Berenji, and my aunt, Dr. Golnar Berenji for their constant support, love, encouragement and guidance. iv TABLE OF CONTENTS ListofTables.......................................... vii ListofFigures ......................................... viii ListofSymbols......................................... xii List of Abbreviations . xiv 1 Introduction 1 1.1 Polymers .......................................... 1 1.2 InverseProblems ................................... 3 1.2.1 Polyolefins...................................... 6 1.3 Rheology .......................................... 7 1.4 Branching......................................... 10 1.5 Polymer Characterization . 10 1.5.1 Chemical Composition Distribution of Random Copolymers . 10 1.5.2 Long-chain Branching Characterization . 17 1.5.3 Rheological Characterization . 20 1.6 Motivation and Scope . 22 2 Inferring Comonomer Content using Crystaf 24 2.1 CrystafModel...................................... 24 2.2 Dependence of Parameters on the Structure and Operating Conditions . 30 2.2.1 Classic and Expanded Parameters Sets . 30 2.2.2 Training and Testing Datasets . 31 2.3 ExperimentalData................................... 33 2.4 Inferring Average Comonomer Content . 34 2.5 Results.......................................... 34 2.5.1 Effect of Selection of Training Datasets . 38 3 LCB Detection and Measurement 42 3.1 Method ........................................... 45 3.1.1 ModelData..................................... 45 3.1.2 Rheological Probes . 45 3.2 Results.......................................... 49 3.2.1 LCB Detection . 49 3.2.2 LCB Measurement . 53 4 Determination of the Continuous and Discrete Relaxation Time Spectrum 57 4.1 RelaxationSpectra................................... 57 4.2 Method ........................................... 59 4.2.1 Continuous Relaxation Spectrum . 59 4.2.2 Discrete Relaxation Spectrum . 66 v 4.2.3 Error Estimation . 69 4.3 Program . 69 4.3.1 InputData ..................................... 72 4.3.2 Interfaces ...................................... 72 4.3.3 Functions ...................................... 73 5 Conclusion 80 5.1 Chemical Composition Distribution . 80 5.1.1 FutureWork .................................... 81 5.2 LCB Detection and Measurement . 81 5.2.1 FutureWork .................................... 82 5.3 Continuous and Discrete Relaxation Time Spectrum . 82 5.3.1 FutureWork .................................... 83 Bibliography .......................................... 84 BiographicalSketch ..................................... 94 vi LIST OF TABLES 2.1 Molecular characteristics of experimental data considered in this paper. 33 3.1 Stadler[101] rheological probes that are adopted in this work. 46 vii LIST OF FIGURES 1.1 a) A linear homopolymer, b) A diblock copolymer, and c) A random copolymer . 2 1.2 Architecture of a polymer chain: a) A linear chain, b) A branched chain, and c) A cross-linkedpolymer .................................. 3 [114] 1.3 The distribution of bm for HDB1 . Experimentally determined values are indicated by red lines, marginal distribution of bm from simulations in the absence of molecular weight information are indicated by the unfilled bars, and values from the simulations with prior molecular weight information are indicated by the blue filled bars. 6 1.4 Classification of polyethylenes according to density (and branching)[96] . 7 1.5 Time profile of (a) strain, (b) stress response in elastic solid, (c) stress response in viscous fluid, (d) stress response in viscoelastic material. ............. 8 1.6 CCD of a typical LLDPE[96] ................................ 12 1.7 Schematic diagram of a Crystaf vessel[6] .......................... 13 1.8 Effect of comonomer content on Crystaf profiles of ethylene/1-hexene copolymers.[6] . 15 1.9 Relationship between cooling rate and Crystaf peak temperature for ethylene/1-hexene copolymers[6] ......................................... 16 2.1 Integral Crystaf profile of a ethylene/1-hexene sample with φ = 1.50 mol % of 1- ◦ hexene, and Mn = 35, 000, at a cooling rate of 0.1 C/min. 25 2.2 Differential Crystaf profile of a ethylene/1-hexene sample with φ = 1.50 mol % of ◦ 1-hexene, and Mn = 35, 000, at a cooling rate of 0.1 C/min. 26 2.3 Illustration of ethylene sequences (ES), longest ethylene sequences (λ) in a copolymer sample. ............................................ 27 2.4 Weight distribution of longest ethylene sequence of a sample with φ = 1.50 mol % of 1-Hexene, and Mn = 35, 000. 28 2.5 X(λ, T ) for three fractions of λ = 100, 300, 1500 of a sample with φ = 1.50 mol % of −5 1-Hexene, and Mn = 35, 000, and model parameters n = 4.44, k = 1.0×10 , A = 85, and B = 630. 29 2.6 Estimated Gibbs-Thomson parameters (A and B) for the Crystaf model as a function of comonomer content.[3] .................................. 31 2.7 Estimated Avrami parameters for the Crystaf model. The parameter k is an average value. (The solid lines are only as an aid to eye. The dashed line is an average value viii for n). Comonomer content of the samples EH06, EH15, and EH31 is 0.68%, 1.51%, and 3.14% respectively.[3] .................................. 32 2.8 The function ǫ(φ) for dataset 17 from table 2.3 (φexp = 2.31) derived by using model parameters that described in the text. The φ which minimizes ǫ is denoted as φmodel. 35 2.9 Distribution of difference between comonomer content derived from model and exper- imental data for (a) 5 TDS with 9 parameters, (b) 10 TDS with 9 parameters, and (c) 5 TDS with 4 parameters. The main text describes how to interpret these box plots. 36 2.10 Fraction of samples with |∆φ| < 1.0............................. 38 2.11 Cost function Φ corresponding to the five different strategies for constructing training datasets. The blue symbols and lines denote the mean and standard deviation of the data, while the red lines extend from the minimum to the maximum values sampled. 40 2.12 |∆φ| corresponding to the five different strategies for constructing training datasets. The blue symbols and lines denote the mean and standard deviation of the data, while the red lines extend from the minimum to the maximum value sampled. 41 3.1 The storage and loss moduli of a linear sample with MW = 149.4kg/mol and a branched sample with the same molecular weight and average number of branches on a single molecule, bm = 0.74. 44 3.2 Complex viscosity of a linear sample with MW = 149.4 kg/mol and a branched sample with the same molecular weight and average number of branches on a single molecule, bm = 0.74. 47 ∗ 3.3 Loss tangent δ(|G |) of a linear sample with MW = 149.4 kg/mol and a branched sample with
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