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Molecular dynamics simulation of polymer crystallization process Triandafilidi, Vasilii 2015

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Molecular Dynamics simulation of polymercrystallization processbyVasilii TriandafilidiB. Applied Physics and Math, Moscow Inst of Phys and Tech, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMaster of ScienceinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Chemical and Biological Engineering)The University of British Columbia(Vancouver)September 2015c Vasilii Triandafilidi, 2015AbstractLarge scale molecular dynamics simulations were carried out to study the kinet-ics of polymer melt crystallization. A coarse-grained model CG-PVA developedby Meyer and Muller-Plathe [1] is applied. A new algorithm for analyzing crys-tallization is proposed. It is based on the alignment of individual chains whichspeeds up previous similar calculation by a factor of ten. Moreover, it is found tobe more suitable for investigating chain crystallinity in polydisperse systems.Different thermodynamic protocols of polymer crystallization were studied:deep quench, shallow quench and cooling with various rates, as well as polymerpre-stretching and consequent cooling and quenching. Cooling with the slow-est rate was shown to generate the highest terminal crystallinity values. Result-ing curves were fitted using the Avrami equation that showed good agreement atthe early stages of crystallization. As a result shorter chains were found to ex-hibit higher terminal crystallinity value than the longer ones. Pre-stretching andsubsequent quenching was found to have a minor effect on thefinal crystallinity,whereas pre-stretching followed by an intermediate rate cooling was found to in-crease the terminal crystallinity.The effect of polydispersity was modeled via two bidisperse melts compris-ing of different proportion of short and long chains. Due to the presence of tworelaxation times in the melt, initial stages of bidisperse polymers crystallizationwere found to be dominated by the short chains, whereas the final stages weredominated by the long ones. Further investigation concluded that the behaviorof bidisperse melts is governed by the proportion of short and long chains in theiimelt.When a critical fraction of the long chains was reached, they appeared to act asbaby nuclei for the short chains to attach themselves onto resulting in bundle-likefringed micelle structures. Otherwise, they acted as ”molecular traps” hinderingcrystallization of the short chains. When a critical fraction of the short chains wasreached, they were found to assist crystallization of the long chains at the initialstages of crystallization but impede crystallization dynamics at the final stages.iiiPrefaceThe thesis author was responsible for all of the numerical work as well as theanalysis of the results. Prof. Hatzikiriakos provided the initial idea for the projectand he gave suggestions for the analysis and interpretation of the results. Prof.Rottler assisted in the analysis of the numerical data and the results. The thesisauthor wrote the manuscript which was then edited by Prof. Hatzikiriakos andProf. Rottler.A manuscript for a peer refereed journal is being prepared from the results inchapter 3. The thesis author was responsible for all numerical work as well as theanalysis of the results.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Objectives and work outline . . . . . . . . . . . . . . . . . . . . . 82 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1 Polymer theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Polymer crystallization theories . . . . . . . . . . . . . . . . . . . 132.3 Molecular dynamics simulation . . . . . . . . . . . . . . . . . . . 172.4 Molecular dynamics of polymers . . . . . . . . . . . . . . . . . . 182.4.1 Creating polymer melts . . . . . . . . . . . . . . . . . . . 18v2.4.2 Equilibrating polymer melts . . . . . . . . . . . . . . . . 192.4.3 Running the simulation . . . . . . . . . . . . . . . . . . . 202.4.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1 Effect of molecular weight on polymer crystallization . . . . . . . 313.1.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3 Effect of molecular weight distribution on polymer crystallization 363.3.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.3.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 383.4 Effect of cooling protocol . . . . . . . . . . . . . . . . . . . . . . 413.4.1 Results and discussion . . . . . . . . . . . . . . . . . . . 413.5 Polymer crystallization of pre-stretched melt . . . . . . . . . . . . 433.5.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.5.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 474 Discussion and conclusion . . . . . . . . . . . . . . . . . . . . . . . . 494.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53A Supporting Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 57Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62viList of TablesTable 2.1 List of units for the quantities and their corresponding real units 20viiList of FiguresFigure 1.1 Polymer representation based on resemblance with cookedstrands of spaghetti . . . . . . . . . . . . . . . . . . . . . . . 2Figure 1.2 Characteristic length scales of polymer crystallization. Thesmallest length scales studied are 2nm crystalline structureson atomistic level. The second largest structures are foldedchains of 5nm size. Folded chains are part of crystalline lamel-lae 50nm which in turn form shperically symmetric spherulitestructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Figure 1.3 Main experimental result of Ergoz. Dependence of polymercrystallinity value for different molecular weights. Polymermelts relaxed at temperature higher than melting temperatureare suddenly quenched at lower temperature and isothermallyrelaxed. Shorter chains were found to correspond to higherdegree of crystallinity . . . . . . . . . . . . . . . . . . . . . . 5Figure 2.1 Schematic representation of a polymer chain. Two main char-acteristic values are shown: the end-to-end distance whichcorresponds to the vector connecting the first and the last beadsof the polymer, and the gyration radius which is a characteris-tic average size of the polymer chain. . . . . . . . . . . . . . 13viiiFigure 2.2 Schematic representation of evolution of Mean Square Inter-nal Distances (MSID) for equilibrated and non-equilibratedpolymer melts. Upon equilibration chain MSID saturates oncorrect End-to-End distances that can be directly obtained fromnumber of units in the chain, bond length and chain stiffness . 14Figure 2.3 Schematic representation of short molecules and polymer chainfree energy profile upon crystallizing. Increase in free energydue to the surface area and decrease due to the bulk free en-ergy. Free energy of formation of a primary nucleus built upthrough the addition of a large number of elements. DGa is theheight of the barrier for the addition of each element, i.e., theactivation energy. After enough elements have been added,free energy decreases with further growth. The free energy ofthe system, however, does not decrease until DG< 0. Here hcorrespond to the number of stems. . . . . . . . . . . . . . . . 16Figure 2.4 Schematic representation of the Molecular Dynamics algo-rithm. First initial positions are set up at time origin. Posi-tions on a new step are obtained after solving the Newton’sequations of motion. This cycle is repeated until the end ofsimulation time is reached. . . . . . . . . . . . . . . . . . . . 18Figure 2.5 Schematic explanation of CG-PVA forcefield. 7 particles + 23potentials are mapped onto 1 particle + 3 potentials. Excludedvolume interactions are modelled via repulsive only poten-tial. Bond interactions are represented as ideal springs. Angu-lar potential has three minima at gauche-gauche, gauche-transand trans-trans conformations favouring polymer crystallization. 21Figure 2.6 Schematic representation of different thermodynamic coolingprotocols used to study polymer crystallization. A ”hot” amor-phous polymer melt fully relaxed at temperature above melt-ing 0.82 undergoes several thermodynamic protocols. . . . . . 22ixFigure 2.7 Schematic representation of Individual Chain Crystallinity(ICC)algorithm used to calculate chain crystallinity. For every bondbi a director ni is calculated as an average of 10 neighbours.Bond is considered crystalline if cosa = (bi,ni)> 0.95; Crys-tallinity of melt is calculated as a fraction of aligned bonds. . . 25Figure 2.8 Schematic representation of Static Structure parameter (Sq).Upon crystallization a smoothly-oscillating black curve cor-responding to amorphous melt transforms into a spiky bluecurve of Bragg peaks corresponding to semi-crystalline struc-ture. Sharper peaks correspond to more crystallized struc-tures. Cylindrical symmetry of elongated chains raises hexag-onal symmetry in reciprocal space. . . . . . . . . . . . . . . 26Figure 2.9 Schematic representation of the angular heat maps. The x-axiscorresponds to the monomer index, y-axis - to time and colorrepresents local angle at a given monomer. Start of nucleationis indicated with a white line. . . . . . . . . . . . . . . . . . . 27Figure 2.10 Analogy of Avrami characterization of polymer crystalliza-tion with a wave ripple phenomenon. Expanding waves ona pond surface correspond to growing crystallites. The wavefronts are moving away from the centers with velocity v; theinteresting quantity is the probability that no wave has passeda particular point M at a time t . . . . . . . . . . . . . . . . . 29Figure 2.11 Schematic representation of the main parameters used for theAvrami fitting. Nucleation onset time t0 is the time momentwhen crystallization starts. The average crystallization time trepresents average time until crystal is crystallized by 1/e andcan be obtained via the Avrami fitting Equation 2.15. Satura-tion crystallinity value cs and terminal crystallinity values areobtained via linear fit and extrapolation. . . . . . . . . . . . . 30xFigure 3.1 Two snapshots of the long chain (monodisperse) polymer crys-tallization: amorphous melt(on the left) and semi-crystalline(on the right side). The amorphous melt was quenched and al-lowed to relax isothermally at lower temperature. Upon crys-tallization chains folded in random directions forming crys-talline parts separated by amorphous ”liquid-like” structure.Red color corresponds to the chains which are crystalline atthe final stages of polymer crystallization. Green color corre-sponds to the chains which are amorphous at the final stage. . 32Figure 3.2 Crystallinity evolution c(t) calculated using ICC algorithm isobserved for various chain lengths. Initial linear raise satu-rates and reaches plateau c(t)! cs . . . . . . . . . . . . . . 33Figure 3.3 The c0 and cs parameters representing terminal and satura-tion crystallinity for various chain lengths, see Figure 3.10.Two algorithms of calculating crystallinity are used: Individ-ual Chain Crystallinity(Individual Chain Crystallinity (ICC))and Yamamoto Crystallinity(Yamamoto Crystallinity (YC)).Quantative agreement between two algorithms is observed.Small level descripancy between c0 cs indicate that simula-tions were run long enough for crystallinity value to saturate. . 34Figure 3.4 Schematic representation of short and long polymer chainscrystallization.Bold lines representing straight, crystallized seg-ments.Short chains have chain stiffness high enough that al-lows them to crystallize and form ”whole chain crystals”, i.eno folding occurs. For the long chain crystallization, how-ever, stiffness of the chains is not high enough and thereforecrystallization is accompanied by chain folding. . . . . . . . . 35xiFigure 3.5 Crystallization kinetics of short (n= 40) and long chain (n=160) monodisperse melts and corresponding bidisperse blendscomprising of both long and short chains.The curve compris-ing of majority of short chains (bidisperse(1)) was ex-pectedly found to lay closer to the short-chain mono-dispersecurve. The curve comprising of equal proportion of each chainlength(i.e bidisperse(2))exhibits terminal crystallinity valueless than the arithmetic average of monodisperse cn=400 +cn=1600 /2. 37Figure 3.6 ICC algorithm applied for each individual chain length in thebi-disperse melt. Crystallinity of short chain component ofboth bi-disperse melts lays significantly lower than that in thepure mono-disperse melt. Terminal crystallinities of a longchain component of both bidisperse melts are found to beslightly lower than that of the corresponding mono-dispersecrystallization curve indicating small influence of short chainson the long time crystallization behavior of the bidisperse melts.The long chain crystallization curve of the bidisperse(1)is initially steeper than long chain mono-disperse curve indi-cating assistance of the short chains in the early stages of crys-tallization and impeding long chain crystallization in the finalstages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Figure 3.7 Crystallization snapshots of bidisperse(1) melt. Red chains -are the chains having n= 40 and green - chains with n= 160.Only chains that have at least one atom within certain distancefrom the origin are visualized. Since there are only few longchains, crystallization is governed by the short chains. Chainscrystallize in random directions anywhere within the domain.Some of the short polymers are being ”trapped” inside ”cage”formed by the long chains. . . . . . . . . . . . . . . . . . . . 40xiiFigure 3.8 Crystallization snapshots of bidisperse(2) melt.Red chains -are the chains with n= 40, green - chains with n= 160. Onlychains that have at least one atom within certain distance fromthe origin are visualized.Long chains in this case act as nu-cleating agents for the short chains and further growth ”short-chain crystals” occur only long-chain baby nuclei, formingbundle-like structures . . . . . . . . . . . . . . . . . . . . . . 41Figure 3.9 Angular heatmaps of polymer systems upon various thermo-dynamic protocols. Green paths indicate straightening of partsof a polymer chain. Nucleation in the case of deep quenchstarts instantly and along the whole length of the chain. Fi-nally for the case of gradual constant rate cooling an offset incrystallization is observed . . . . . . . . . . . . . . . . . . . 42Figure 3.10 Crystllinity curves obtained using the YC algorithm (symbols)and corresponding Avrami fitting curves c(t)/cs= 1exp(((tt0)/t)3) (continuous lines). Two thermodynamic protocolsplotted in Figure 2.6 are investigated: deep quench and slowcooling. Polymer melts that undergo deep instantaneous quenchexhibit instantaneous crystallization, whereas slow cooling ex-hibit a sigmoidal shape of the crystallization curve. . . . . . . 44Figure 3.11 Master crystallinity curves obtained using the YC algorithm(symbols)and corresponding scaled curves c(t)/cs((t  t0)/t)) (con-tinuous lines). Two thermodynamic protocols plotted in Fig-ure 2.6 are investigated: deep quench and slow cooling. Thepolymer melts that undergo deep instantaneous quench exhibitan instantaneous crystallization, whereas slow cooling exhibita sigmoidal-shaped behavior of the crystallization curve . . . . 45xiiiFigure 3.12 Saturation value crystallinity cs obtained from YC crystallini-ties curves. Errorbars are generated as a difference betweensauration value crystallinity and terminal crystallinity valueillustrated on Figure 2.11b. Shorter chains exhibit higher crys-tallinity value than long chains. For a given chain lengthslower cooling exhibits higher crystallinity value . . . . . . . 46Figure 3.13 Crystallinity evolution for presheared polymer melts . . . . . 47xivGlossaryICC Individual Chain Crystallinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9YC Yamamoto Crystallinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23MD Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17DPD Dissipative Particle Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57FIC Flow-Induced Crystallization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5SG Saddler-Gilmer model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15CG Coarse-Grained model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6CG-PVA Coarse-Grained Polyvinyl alcohol model . . . . . . . . . . . . . . . . . . . . . . . . . . 6LH Lauritzen-Hoffman model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15MSID Mean Square Internal Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19PE Poly-Ethylene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6SG Saddler-Gilmer model of polymer crystallization . . . . . . . . . . . . . . . . . . 15LH Lauritzen-Hoffman theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15LAMMPS LAMMPS Molecular Dynamics Simulator (seehttp://lammps.sandia.gov/) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19VMD Visual Molecular Dynamics (seehttp://www.ks.uiuc.edu/Research/vmd/) . . . . . . . . . . . . . . . . . . . . . . . 20MDANALYSIS Molecular Dynamics trajectory analysis (seehttp://www.mdanalysis.org/) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20xvAcknowledgmentsI would like to express my gratitude to my supervisor, Prof. Savvas G. Hatzikiri-akos, whose expertise, understanding, and patience, added considerably to mygraduate experience. I doubt that I will ever be able to convey my appreciationfully. The balanced freedom and guidance that he provided to me created theperfect zone for my development and growth as a researcher. His knowledge notonly supported me in my academic endeveours but also offered help in day to daydifficulties that an international graduate student may encounter.I would like to thank the other members of my committee: Prof. Joerg Rottlerand Dr. Antonios Doufas. Prof. Rottler combines vast knowledge in PolymerPhysics and Molecular Dynamics alongside great professionalism and dedicationto help. A great part of this thesis has been shaped by his insights proposed duringour discussions.I would also like to express my gratitude for my undergraduate supervisorsHenry Norman and Vladimir Stegailov without them I wouldn’t have chosen ca-reer in Molecular Dynamics. It was under their tutelage that I developed a focusand became interested in Molecular Dynamics (MD).Big thanks to my group mates and especially to Maziar Derakshandeh forsharing valuable comments and ideas throughout my research. To Amanda Parkerfor her specific help in Molecular Dynamics Simulation. I am also grateful toall of my dearest friends who were there for me making my last two years anunbelievable journey.Hat tips to Steve Plimpton and all the LAMMPS commonity, Oliver Beck-xvistein and MDAnalysis community helping me cope with many problems I haveencountered throughout my experience as a Molecular Dynamics researcher. I amvery grateful to WestGrid Canada network for provided computational facilitieshttps://www.westgrid.ca/.The most profound and loving thanks to my family, to my parents, to mybrother and grandmother who have given me the greatest love and support anyonecould ever expect. I am blessed to have them and I have missed them deeplyduring these years. They are my true inspiration.xviiChapter 1IntroductionPolymers chains represent strands of cooked spaghetti— Prof Savvas G. Hatzikiriakos,University of British Columbia, CanadaPolymers lack entropy— Prof Alexey Khohlov,Russian Academy of Science, RussiaPolymers play an essential and ubiquitous role in everyday life, see Piorkowskaet al. [2]. Polymers range from familiar synthetic plastics such as polyethylenes tonatural biopolymers such as DNA and proteins that are fundamental to biologicalstructure and function.The word polymer originates from Greek as something comprising of many”parts”; these elementary ”parts” are also called monomers. Monomers connect toeach other and form what is called a polymer chain. A single polymer chain mayconsist of tens of thousands of elementary units, see autoreffig:spagetti. How-ever, unlike simple liquids, elementary units within a single polymer chain areconstrained from moving freely in space, providing a basis for the popular say-ing ”polymers lack entropy”. The size of the chains and ”scarceness of entropy”produce unique physical properties such as viscoelasticity, phase separation, self-organization, tendency to form glasses and semi-crystalline structures.1Figure 1.1: Polymer representation based on resemblance with cookedstrands of spaghettiCrystallization of polymers is a process associated with partial alignment oftheir molecular chains. These chains fold together and form ordered regions calledlamellae, which compose larger spheroidal structures named spherulites, see Pi-orkowska et al. [2]. Polymers can crystallize upon cooling from the melt, mechan-ical stretching or solvent evaporation. Crystalline and rubbery amorphous phasesmay coexist within the same polymer resulting in remarkable ductility and tough-ness of these materials. Besides mechanical properties, crystallization affects op-tical, thermal and chemical properties of polymers resulting in high demand forthe investigation of polymer crystallization in industrial applications. Apart fromgreat industrial significance, polymer crystallization entails many peculiar prob-lems of academic interest. Self-organization of long soft molecules remains anunsolved puzzle for the scientific community.21.1 Related workIf I have seen farther it is by standing on the shoulders of Giants.— Sir Isaac Newton (1855)There are several length scales involved in the analysis of a polymer crystal-lization process, see Figure 1.2. To fully understand this phenomenon one needsto provide a model that can describe the behavior of polymer chains on all ofthese length scales. The most commonly experimentally observed superstructuresin melt-crystallized polymers are the spherulites. These experiments include dif-ferential calorimetry analysis, microscopy, x-ray scattering, rheological character-ization and others, see Piorkowska et al. [2]. The spherulites consist of radiatingarrays of periodic lamellar stacks, with size ranging from 1 to 100 µm.Frequently industrial applications are dealing with polymers of different molec-ular weight. Ergoz et al. [3] studied the effect of molecular weight (Mw) onquiescent polymer crystallization. They performed analysis of the evolution ofcrystallinity with time using Differential Scanning Calorimetry method (DSC).The system underwent rapid quench and was allowed to relax isothermally. Theyfound the degree of crystallinity to be strongly affected by the molecular weight ofthe polymer. Long polymer chains were found to exhibit a lower degree of crys-tallinity than the shorter ones, see Figure 1.3. Moreover they reported a plateau be-tween Mw = 101g/mole to 105 which corresponds to 10 entanglements. Then theplateau rapidly drops untilMw = 107 g/mole and saturates atMw = 107 g/mole.Unlike short molecules, long polymer chains require considerable undercool-ing below their melting temperature in order to crystallize. In 2000 Kraack et al.[4], using an advanced technique of droplet supercooling1, investigated crystal-lization of short alkane chains. The authors investigated two systems the most:short n-alkane waxes, comprising of tens of carbon atoms, and low molecularweight polymer chains, comprising of 600 units. Chains comprising of less than1A technique that allows to obtain highly pure low molecular weight polymer melts as well asvery short chain waxes and supercool them to temperatures significantly lower than their meltingtemperature3CCCCCCCCCCCCCCCCCrystalline order Folded chains Lamellae Spherulites< 2 nm < 5 nm < 50 nm > 500 nmCan be simulated by Molecular SimulationFigure 1.2: Characteristic length scales of polymer crystallization. Thesmallest length scales studied are 2nm crystalline structures on atom-istic level. The second largest structures are folded chains of 5nm size.Folded chains are part of crystalline lamellae 50nm which in turn formshperically symmetric spherulite structures15 units showed little undercooling; however, larger chains required significantundercooling up to 40 50 degrees. Systematic investigation of various chainlengths (n) revealed a transition in the degree of required undercooling at n= 25.Previously, the aforementioned transition was linked with chain folding. Thevalue n = 25, obtained by the authors, appears too short to facilitate chain fold-ing. Therefore, Kraack concluded some other mechanism to be responsible forthe transition. Based on these observations, as well as on calculations of surfacefree energies, the authors advocated for bundle-like ”fringed micelle” crystals tobe the dominant structures upon crystallization.4Degree of crystallinityMolecular Weight, g/mol105104 107Figure 1.3: Main experimental result of Ergoz. Dependence of polymercrystallinity value for different molecular weights. Polymer meltsrelaxed at temperature higher than melting temperature are suddenlyquenched at lower temperature and isothermally relaxed. Shorterchains were found to correspond to higher degree of crystallinitySeveral groups [2, 5] investigated the impact of polydispersity on polymercrystallization. Melts comprising blends of chains of different length were foundto behave in a rather different way than monodisperse ones. In general their rheo-logical behavior was found to depend strongly on the longest chain length presentin the melt, see [5].To mimic processing conditions a great amount of work has been done onflow-induced crystallization (Flow-Induced Crystallization (FIC)) which can befound in [2]. In [5, 6] authors analyze the effect of shear on the flowinducedcrystallization (FIC) of several polypropylenes using rheometry combined withpolarized microscopy. They found that generally, an increase in strain and strain5rate or decrease of temperature decreases the thermodynamic barrier for crystalformation and thus enhances crystallization kinetics at temperatures between themelting and crystallization points.As was mentioned above in the Figure 1.2, experimental knowledge availableis mostly macroscopic, i.e. it describes structures larger than 0.5 µm. Unfor-tunately, detailed molecular processes of polymer crystallization are not readilyaccessible by experiments. Computer simulations are used to fill that gap, seeFigure 1.2. Using the power of modern supercomputers, one could obtain infor-mation about polymer dynamics on the microscopic level. Molecular Simulationsof structural formation of polymer chain systems has recently become the focusof attention in physics, chemistry and material science, see [2, 7, 8]. In the presentwork polymer crystallization is studied using the method of Molecular Dynamics.To the best of our knowledge, the first paper addressing polymer crystallizationusing MD methods was published in 1988 by Rigby and Roe [9]. The authorssimulated crystallization of n-alkane polymer melt. Later on, several groups [10–12] modeled single polymer molecule crystallization from solvent. Kavassaliset al. [11] observed Poly-Ethylene (PE) chain crystallization, applying forcefieldparameters considerably stiffer than the realistic models of PE. They argued thestiffness of polymer chains to be essential for crystallization. This claim has laterbeen confirmed, see [8].Due to the computational limits all works mentioned above were forced tostudy systems comprising of only a few thousand particles (a single long chain ora polymer melt of short n-alkanes). Despite a great progress in computing power,physical systems remain unreachable for modern day computers. One possibilityto access realistic system sizes, is to use so-called coarse-grained (Coarse-Grainedmodel (CG)) models. One may attempt reaching larger length and time scales byusing so-called coarse-grained (CG) models. A Coarse-Grained Polyvinyl alcoholmodel (CG-PVA) model, developed by Muller-Plathe [13], may serve as a modelto simulate larger systems.In 2001, Meyer and co-workers used this forcefield to study systems of dif-6ferent chain lengths, see [14–16]. In this work, the authors simulated polymermelt crystallization of a system comprising of 200 k. They provided valuable in-sights into the dynamic behavior of various chain lengths under different coolingregimes. Results obtained by Meyer were found to be in a qualitative agreementwith experimental results obtained by Krack et al. [4].Later in 2006, Gee and colleagues [17] performed a first large scale 2 polymercrystallization simulation. The authors tried to find evidence of a spinodal phaseseparation. Although criticized, this paper holds a heroic simulation effort appliedfor investigating the polymer crystallization process. To our best knowledge it isthe largest system ever studied to address the problem of polymer crystallization.Polymer samples have a unique ability to keep a partial memory, i.e., a poly-mer sample may be completely molten according to the calorimetric or rheologicaldata and still keeps a partial memory, see [18].3 This feature raised a new waveof interest on a phenomenon of precursors states. There has been a large numberof investigations on the influence of shearing the melt for a defined time beforecooling. 4 Several groups [19–24] tried to investigate this phenomenon using themethods of Molecular Dynamics. In general shearing conditions were found toenhance the polymer crystallization.In [20], authors tried to investigate difference in crystallization kinetics fortwo different chain lengths. Polymer crystallization was characterized using twoparameters: dilatometric ad 5 and orientational parameter g2 based on mutualalignment of different chains 6. They found that in the quiescent regime, whenno shear applied, short chains exhibit increased crystallinity whereas long chains2A PE melt consisting of 4,000,000 atoms was investigated.3This can be seen by observation light microscope: the spherulites seen to dissapear afterheated, but re-emerge at the same places after being cooled again4This effect shows that direct correlations between polymers do not decay immediately butserve as precursors.5Dilatometric parameter ad represents as an estimate of the densification of the melt, i.e.,ad µ (r(t)r0)6Parameter g2 represents alignment value of a given chain bond with its neighbours averagedover all bonds in the melt, i.e., g2 µ cos(2fi j), where fi j = cos1(bi,bj), bi,bj are two given bondsin the melt.7almost none. The picture changed drastically when shear was applied. Uponshear, long chains showed a higher crystallinity value, while short chain crys-tallinity value remained the same as in the quiescent regime. The authors linkeddifference in short and long chains behavior with entanglements present in themelt. Obtained crystallinity values were fitted using the Avrami equation. Differ-ent crystallization parameters were found to exhibit different navrami 7 parameterranging from 1 to 3.1.2 Objectives and work outlineAlthough many studies [4, 15, 20] predict a higher crystallinity value for shorterchains, no systematic investigation has been done on the crystallinity as a functionof chain length n. In [20] authors show insignificant crystallinity value for the longchain polymer crystallization. One could question the system size that the authorsinvestigated, or forcefield parameters that were not stiff enough for a rigorousobservation of kinetics of polymer crystallization. To best of our knowledge, nowork has been done on systems comprising of various chain lengths8. In [15]authors attempted to compare crystallization kinetics for different thermodynamicprotocols, but due to the lack of computational resource this work has been doneonly for a single chain length. Present work attempts to systematically studycrystallization of various chain lengths across different thermodynamic protocols.It investigates the effect of polydispersity by focusing on the bidisperse melts.The effect of remaining correlations is studied upon pre-stretching of the polymermelt.The first chapter will cover the background information on polymer physicsand polymer crystallization, which are fundamental to this work. Then the method-ology is discussed which covers the simulation methods used for MD simulationsand the algorithms used for their analysis.Objective 1: How does molecular weight effect the kinetics of polymer7This parameter characterizes the dimensionality of crystal growth.8also known as polydisperse systems8crystallization?The first part of this thesis addresses the study of the effect of molecular weighton polymer crystallization. Different systems comprising of N chains of n unitswill be created and subsequently quenched under the melting temperature (Tm)and allowed to relax isothermally9. The resulting crystallinity curves are fittedby the Avrami equation to be easier compared with the experimental evidence.Individual chain behavior is compared using angular heat maps and trajectorysnapshots.Objective 2: How does molecular weight distribution affect the kineticsof polymer crystallization?Polymer melts in real life, are far from being monodisperse; therefore under-standing the behavior of polydisperse systems is of high importance.The secondpart of the thesis focuses on systems of various chain lengths n. In this section,two systems are studied: bidisperse (1)10 and bidisperse (2)11, with a polydis-persity index equal to 1.5. Far from describing polymer melts in real life, thischoice is the simplest model to describe the behavior of a polymer blend withvarious chain lengths. For polydisperse systems, we will take advantage of our”Individual Chain Crystallinity” (Individual Chain Crystallinity (ICC)) algorithmfor calculating crystallinity of the system. This approach allows to measure thecrystallinity of the system as a sum of crystallinities of individual chains. Ob-tained crystallinities will be compared with each other as well as with the bulkcrystallinities of corresponding monodisperse melts.Objective 3: How do different thermodynamic protocols affect polymercrystallization?A significant part of this thesis is devoted to analyzing the effect of differentthermodynamical protocols on the process of polymer crystallization. Polymer9By varying n, comprising the maximum of 200 atoms per chain, the molecular weight ofthe polymer melt will be varied. Henceforth designations n and Mw are considered to representnumber of monomer units per chain.10with a majority of the short chains11with an equal proportion of the short and the long chains9melts relaxed and equilibrated at T = 0.82 LJ , subsequently, undergo severalthermodynamic protocols. Three protocols are analyzed: deep quench, shallowquench and constant rate cooling with different rates.Objective 4: How does preliminary pre-shearing affect polymer crystal-lization kinetics?Crystallization and resulting morphology are strongly related to temperaturegradients, applied shear rate (strain rate) or a total strain. The last part of thisthesis examines the effect of the remaining correlations in polymer melts on thecrystallization process. These correlations are introduced via pre-stretching ofthe polymer melt along one direction. The computational experiment is set up inthe following way: a polymer melt comprising of chains of n = 160. 12 fullyrelaxed at temperature above crystallization point T = 0.82LJ > Tm undergoes auniaxial extension along a certain axis. Then the extended polymer chains un-dergo different thermodynamic protocols of cooling and quenching, to study theircrystallization behavior.12those are chains with lengths higher than the entanglement length10Chapter 2Background2.1 Polymer theoryIt doesn’t matter how beautiful your theory is, it doesn’t matter howsmart you are. If it doesn’t agree with experiment, it’s wrong—Richard FeynmanThe simplest model to describe polymer chains is the Freely Jointed Chain(FJC) model 1. In this simple model a polymer chain is represented as a succes-sion of monomer units that interact only by covalent binding forces, therefore thepotential energy of the polymer is taken to be independent of its shape. Thereforeat thermodynamic equilibrium, all of its shape configurations are equally likely tooccur as the polymer fluctuates in time, according to the Boltzmann distribution. As a result, the bond length is fixed, and the internal rotations are completelyfree2.Polymer configuration can be described by setting up the positions of eachmonomer: for example 3N Cartesian coordinates XYZ with respect to the labora-tory fixed frame. Since all of the configurations are random, the end-to-end vector,1Sometimes FJC model is also called an Ideal Chain Model2This makes possible to track the level of equilibration of polymer melts by tracking how idealthe chain configurations are11connecting the first and the last units of a chain, should have a zero average 3 anda certain variance.R(N) =i=NÂi=1ri (2.1)Evidently 4, a long freely jointed chain will obey Gaussian statistics: the probabil-ity to observe a certain end-to-end vector R is given by the Gaussian distributionfunction with zero average (centered at the origin) and mean squared value.DR(N)E= 0 (2.2)DR2(N)E⌘DR2(N)E= Nb2 (2.3)where b is the bond length and N is the number of bonds. Another value used inpolymer physics is the radius of gyration Rg which is defined by the Equation 2.4.R2g(N) =1NSi(ri rCM)2 (2.4)where ri is the position vector of bead number i in the chain and rCM is the positionvector of the center of mass of the polymer chain. Schematically it is representedin the Figure 2.1. For ideal chains, the radius of gyration (see Figure 2.1) can alsobe calculated by:< R2g(N)>=< R2(N)>6(2.5)Despite their simplicity, Equation 2.3 and Equation 2.5 are among the mostfundamental results of polymer science - they provide an estimate of the lengthscales in polymer melts, as well as serving as a bridge to connect experimentalresults to MD. In practice, chains are non-ideal, they interact and have internalstiffness. Moreover, since the number of beads N is usually a large number, no3After all different chains in the melt are averaged4According to Central Limit Theorem12RgReeR1R2R3Figure 2.1: Schematic representation of a polymer chain. Two main charac-teristic values are shown: the end-to-end distance which correspondsto the vector connecting the first and the last beads of the polymer,and the gyration radius which is a characteristic average size of thepolymer chain.distinction is made between the finite and the infinite number of bonds. In 1969Flory [25] used these ideas to define the characteristic ratioC• of a polymer as:C• = limN!•< R2(N)>Nb2By definition, CFJC• = 1. Values of C• larger than 1 occur when some of thedegrees of freedom are constrained. The C• can be used as a measure of stiffnessalong the polymer backbone. The valueC• is experimentally observable thereforeone can judge the level of equilibration of polymer melts by how well their MeanSquare Internal Distances(R2(n)/n) plot saturate on the value < R2(n! •) >!C•Nb2, explained in the Figure 2.2.2.2 Polymer crystallization theoriesQualitatively the polymer crystallization process can be view as follows: At tem-peratures lower than the melting temperature (Tm) trans-states become preferred13MSID <R2(i)/i>ii=1i=5not equilibrated meltwell equilibrated meltCorrect End-to-End distance according to FloryFigure 2.2: Schematic representation of evolution of Mean Square Inter-nal Distances (MSID) for equilibrated and non-equilibrated polymermelts. Upon equilibration chain MSID saturates on correct End-to-End distances that can be directly obtained from number of units inthe chain, bond length and chain stiffnessto the others. Stems are formed and repulsive interactions between polymers rise.Similarly to Onsager’s liquid crystal isotropic-nematic transition, polymer stemsorient along one direction due to the minimization of the excluded volume. Unlikethe isotropic-nematic transition: the stems in the case of polymer crystallizationdo not exist from the beginning, but emerge due to the occupation of the transstates, and orient and grow due to the repulsive interactions.The driving force of polymer crystallization is sacrificing entropy of a liquid-like disorder for enthalpy benefits that accompany the careful packing of moleculesinto a dense, regular crystalline form. This can be rewritten in terms of the Gibbsfree energy for a crystal which is usually divided into a ’bulk’ and a ’surface’contribution. For the case of an infinite molecular weight polymer melt, the end-14effects may be neglected, and the ’bulk’ free energy can be rewritten asDG(T ) = DH(T )TDS(T ) (2.6), where H(t) and S(T ) are the bulk enthalpy and entropy of melting per unit vol-ume. When T is decreasing, the entropy 5 decreases, therefore, G increases. Sim-ilarly, when T is decreasing polymers are packed into a dense structure, henceH and consequently G are decreasing. These two competing effects zero at thetemperatures corresponding to DG= 0. At the equilibrium melting point Tm(•) ofan infinitely large crystal, in which all surface effects are negligible, Equation 2.6can be rewritten asDH(Tm(•))/Tm(•) = DS(Tm(•)) (2.7)As a first approximation DH(T ) and DS(T ) do not vary at temperatures closeto Tm(•) therefore:DG(T ) = DH(Tm(•))(1T/Tm(•)) (2.8)Where DT = Tm(•)T is known as supercooling .From Equation 2.8 one can infer that polymer crystallization is frustrated bylarge free energy barriers which necessitaes the reorganization of polymer confir-mations. Unlike short molecules, depicted in the Figure 3.4, polymer crystalliza-tion requires a great degree of supercooling , therefore Tc < Tm by tens of degrees.There are several models addressing crystal growth of polymers of a finitemolecular weight, see [2]. One of them is the Lauritzen-Hoffman (Lauritzen-Hoffman theory (LH)) theory, see [8].Lauritzen and Hoffman limited factors that frustrate the polymer to the freeenergy barriers. The obtained model allowed the authors to effectively predictminimal thickness of stable lamellae and kinetics of polymer crystallization. An-other model that addresses polymer crystallization is the Saddler-Gilmer (Saddler-5which often could be connected to a conventional measure of disorder in the system15⌘G(a) Schematic representation ofsimple liquid free energy profile uponcrystallization.⌘G(b) Schematic representation ofpolymer chain free energy profileupon crystallizing.Figure 2.3: Schematic representation of short molecules and polymer chainfree energy profile upon crystallizing. Increase in free energy due tothe surface area and decrease due to the bulk free energy. Free energyof formation of a primary nucleus built up through the addition of alarge number of elements. DGa is the height of the barrier for the addi-tion of each element, i.e., the activation energy. After enough elementshave been added, free energy decreases with further growth. The freeenergy of the system, however, does not decrease until DG < 0. Hereh correspond to the number of stems.Gilmer model of polymer crystallization (SG)) model, see [8]. Saddler and Gilmerassumed that even the shorter(than stable) stems can attach themselves to growth,but pinning the short stems interrupts crystal growth. To resume the crystalliza-tion one needs to unpin the short(unstable) stem from the growth front, see [8].Therefore in the SG model the removal of the short stems controls polymer crys-tallization.Recently, based on various experimental and computational data Strobl pro-posed a new model where the mesomorphic precursor phase is first formed [2].Blocks of this mesomorphic state then attach to the growth front. This model,inspired by the observation of the hexatic phase in short n-alkanes is actively con-tested by the polymer community.162.3 Molecular dynamics simulationMolecular Dynamics - is not an experiment. It is modelling— Prof. Peter Englezos, University of British Columbia, CanadaMolecular Dynamics is a computational tool (see Figure 2.4) that treats matteras a system of interacting particles. In order to run a Molecular Dynamics Simu-lation one needs to provide the initial atomic positions r1,r2...rN, the inter-atomicpotential U(r) and a set of constraints also known as an ensemble for integratingthe equations of motion. Then using this information of the atomic positions, andinteracting potential; the Newtons equations of motion are used to calculate theatomic positions in the next time step, see Figure 2.4 6.After simulation is terminated one obtains the final positions r01,r02...r0N. Themacroscopic parameters can be calculated by integrating and averaging, e.g tem-perature can be calculated as a mean square kinetic energy, i.e velocity of thesystem.Molecular dynamics simulation provides realistic results as long as the inter-atomic potential of interaction of the system is valid and realistic ensembles areassumed. Therefore, the interatomic potential becomes the crucial part of simu-lation. Simple mono-atomic fluids can be adequately described by Lenard-Jonespotential Equation 2.9U(ri j) = 4e ✓sri j◆12✓sri j◆6!(2.9)It describes the attraction between particles when they are too far, and repulsionwhen they are too close. In real life, polymers exhibit complicated behavior and,therefore, need a complicated inter-atomic potential - also known as the forcefield. Generally, calculating inter-atomic potential could be very resource demanding6Molecular Dynamics (MD) simulations require to track and update information of every atomat every time step. This makes MD simulations extremely computationally challenging. Currently,the largest simulations that can be done are 1011 particles for tens of nanoseconds. These are heroicsimulations that utilize an immense amount of computational power17Set up initial positionsand topology Find interacting atomsProvide interatomic potential(Forcefield) Provide constraintsof integration(Ensemble)Calculate interaction forces:Fij = rUijSolve Newtons equations of motion,obtain new  positions:r¨i = ⌃nj=1Fij/mAtoms         51.6059   5.3279    5.6240    1         1         1         50.2541   7.4810    7.4995    1         1         2         51.2149   8.0236    8.2322    1         1         3End of time?No YesFjiFijCalculate macroscopic properties (P, T ...)Figure 2.4: Schematic representation of the Molecular Dynamics algorithm.First initial positions are set up at time origin. Positions on a newstep are obtained after solving the Newton’s equations of motion. Thiscycle is repeated until the end of simulation time is reached.and take up to 90 % of the computational time. Therefore in many cases coarse-grained potentials are used (see Figure 2.5).2.4 Molecular dynamics of polymersI have been impressed with the urgency of doing. Knowing is notenough; we must apply. Being willing is not enough; we must do.— Leonardo Da Vinci2.4.1 Creating polymer meltsThe MD scheme takes care of the time evolution of the model used to study aparticular system. Still, the first configuration (meaning positions and velocities18for all particles) has to be defined. For the case of short inorganic molecules,the initial positions can be set up ”by hand” on the vertices of a perfect crystal.This is not the case for the long chain high-temperature polymer melt. Therefore,one needs to start with a configuration that closely resembles an equilibrated, dis-ordered system. This can be achieved using a Monte-Carlo algorithm that willefficiently decorrelate an artificial configuration so as to let it acquire equilib-rium properties. For instance, in the present work melts were created using self-avoiding random walk via a chain.f tool provided by the LAMMPS MolecularDynamics Simulator (LAMMPS) package [26].2.4.2 Equilibrating polymer meltsAfter initial configurations are created, polymer chains need to be equilibrated.Unlike short molecules, long chain polymers require both thermodynamic andconfigurational equilibration. Configurational equilibration can be achieved whenthe Mean Square Internal Distance (MSID) parameter is equilibrated and corre-spond to pseudo-Gaussian chain, see Figure 2.2. This was done via Kremer-Grest equilibration process 7 using bead-springs polymer representation [27] Themethod used for configurational equilibration is a fast ’Dpd-push-off’ - it is a com-monly used way to prepare well-equilibrated melts. This method is an extensionof the slow push-off method developed by Auhl et al. [28]. The idea of appli-cation of soft repulsive potentials for equilibration of polymer melts is effectiveprovided the potential is applied to the initial configurations that closely matchequilibrium structures at large length scales. The detailes of the algorithm can befound in the appendix section, see Appendix A. The MSID plots, alongside plotsof the thermodynamic parameters evolution provide evidence on a fine quality ofequilibration achieved using the procedure explained above.7Equilibration process that utilizes bead-spring Kremer-Grest model for polymer chains19Quantity Units LJ Units reallength s 0.52 nmangle rad or degree rad or degreetime t 2 psenergy e 109 Kcal/moltemperature e 550 Kpressure e/s3 12662.5 PaTable 2.1: List of units for the quantities and their corresponding real units2.4.3 Running the simulationThe equilibrated polymer melt can be simulated over a longer time at constanttemperature and pressure to determine its properties with appropriate statistics. Inthe present work, the CG-PVA interatomic potential is used. The CG-PVA force-field (see Figure 2.5) was developed by Meyer and Mueller-Plathe using InverseBoltzmann method [1, 14, 15]. The model presented above has been found todescribe the properties of poly(vinyl alcohol) well, and in particular to allow oneto simulate its crystallization with high efficiency compared to atomistic or unitedatom models 8. Notably, the present forcefield assumes no attractive interatomicinteractions.All simulations were run using LAMMPS MD package [26], for details on sim-ulation please see Appendix A. Visualization and analysis have been done usingVisual Molecular Dynamics Visual Molecular Dynamics (VMD) and MolecularDynamics trajectory analysis (MDANALYSIS) packages, see [29–31]. Details onthe analysis and the vizualization can be found in my blog 9.Thermodynamic protocolVarious thermodynamic protocols are investigated:cooling with different rates, deep quenching and shallow quenching, as illustrated8Unlike the case of fine-grained simulations (run on the atomic level), it is not possible todefine all physical scales properly. For example there is a definite mapping for length scales, sincethe coarse-graining method explicitly specified the length scale that corresponds to 0.52 nm (theaverage distance between monomers). On the other hand, there exists no such mapping for timescales; therefore time values ( presented in Table 2.1) are only approximate9http://bazilevs31.github.io/tag-analysis.html20COHHH HCCOHHH HCCOHHH HCEnergy (kT)0.26 0.52420Distance (nm)1086420 80 100 120 140 160 180gggt ttBending energy (kT)Angle (degrees)Figure 2.5: Schematic explanation of CG-PVA forcefield. 7 particles + 23potentials are mapped onto 1 particle + 3 potentials. Excluded vol-ume interactions are modelled via repulsive only potential. Bond in-teractions are represented as ideal springs. Angular potential has threeminima at gauche-gauche, gauche-trans and trans-trans conformationsfavouring polymer crystallization.in Figure 2.6. Three types of cooling were used: slow cooling for 0.1 LJ unit in120 M steps (6105 LJ time steps), intermediate cooling with rate = 0.1 LJ/ 60Msteps and fast cooling 1.0 LJ/6M steps and consequent resting at T = 0.72 LJ10.Pre-stretch As was mentioned in Chapter 1, polymer chains in the melt ex-hibit a strong correlation behavior due to entanglements. These correlations play101 time step = 0.005 LJ time units which is approximately 1014s, therefore 60M timestepswould be 60⇤106 ⇤1014 = 0.5µs21Temperaturetime, usCooling x10 (fast)Cooling x1 (intermediate)Cooling x0.5 (slow)Instant deep quench0 0,3 0,6Instant shallow quench(underquench)Tm0.70.790.78amorphous melt(a) Instantaneous deep quenching from,shallow quenching(also called as underquench-ing) and cooling with various rates: slow withrate 55 K/µs, intermediate 110 K/µs and1100 K/µs fastTemperaturetime, usCooling x0.5 (slow)0 0,6Tm0.70.82 amorphous melt1,2heating x0.5 (slow)amorphous meltsemi-crystalline melt(b) Slow cooling with rate 55 K/µs and sub-sequent heating with 55 K/µsTemperaturetime, us0Tm0.70.820,3semi-crystalline meltCooling x1 (intermediate)uni-axial extensionL0L(t) = L0 et/t*(c) A short run of uni-axial extension of a”hot” amorphous polymer melt and subsequentquenching and cooling with an intermediate rate110 K/µsFigure 2.6: Schematic representation of different thermodynamic coolingprotocols used to study polymer crystallization. A ”hot” amorphouspolymer melt fully relaxed at temperature above melting 0.82 under-goes several thermodynamic protocols.22an important role in the crystallization of polymers. To study this effect a ”pre-stretch” protocol was used inspired by works of Jabbarzadeh and Yamamoto [20,24]. ”Pre-stretching” has been done by stretching an equilibrated melt along cer-tain axis before melt undergoes thermodynamic cooling (quenching), depicted inFigure 2.6c. True strain was used to stretch the melt. Our computational exper-iment consisted of a polymer melt fully relaxed at temperature well above thecrystallization point T = 0.82l j > Tm. Dimension of the box were altered at a”constant true strain rate” with e˙ = trate = 1/t⇤. 11 The box length L as a functionof time will change asL(t) = L0e(tt0)/t⇤2.4.4 AnalysisCalculating crystallinityIn the present work, crystallinity is defined as the overall number of straightsegments. This approach is inspired by [20, 24, 32]. Two algorithms are used todetermine the ”straightness” of the chains. First, the so called Yamamoto Crys-tallinity (YC) which is inspired by works of Yamamoto [8]. To calculate crys-tallinity using YC algorithm one needs to wrap a simulation box, using periodicboundary conditions. This is followed by dividing the simulation box into smallersub-domains. Subsequently, the alignment vector for each sub-domain is deter-mined as the eigen-vector ni corresponding to the largest eigen-value of the ori-entational tensor:Qi j =< bibjdij/3>where bi are the normalized bond vectors. Then, we check the alignment of eachbond of each sub-domain to its corresponding alignment. If a bond is within a11Note that this is not an ”engineering strain rate”, as the other styles are. Rather, for a ”true”rate, the rate of change is constant, which means the box dimension changes non-linearly withtime from its initial to final value. The units of the specified strain rate are 1/time. Tensile strainis unitless and is defined as d/L0, where L0 is the original box length and d is the change relativeto the original length.23certain threshold of alignment, i.e cos(bini) > 0.95 then the bond is consideredcrystalline, otherwise amorphous.Unlike YC method, which gets the alignment vector from the sub-domains,algorithm (ICC) attains alignment of a bond by its neighbors in the chain. For ev-ery monomer ten neighbor monomers are designated which define the local chordcolored in green inFigure 2.7. Then the alignment between the bond vector andthe green chord (corresponding to the local alignment vector) is calculated. Ifthe alignment is larger than 0.95, the bond is considered to be crystalline, other-wise it is counted as amorphous. By calculating total number of amorphous andcrystalline segments we define our crystallinity as:cicc =Number O f Crystalline BondsTotal Number o f Bonds(2.10)Comparison between the two discussed algorithms will be discussed in greaterdetail in Section 3.1. Overall identical, named ICC algorithm (see Figure 2.7)analyses every chain separately and is ten times faster and more memory efficient.Static structure parameterThe Static Structure Factor corresponds to a density fluctuation correlationfunction; thus we need to define the density fluctuations, which can be made de-pendent on the position of a particular monomer along the chain. With rai denotingthe position of monomer a of chain i, the density fluctuations of monomer a forthe wave vector q in reciprocal space can be expressed asrtot(q) =NtotÂj=1exp(iqrai ) (2.11)where Ntot total number of atoms, Therefore, the Static Structure Factor can berewritten asS(q) = 1nhrtot(q)⇤rtot(q)i (2.12)This parameter provides qualitative difference between the smoothly-oscillating24Kuhn length: n~10bibjbkninjnkbi niάicos άi =(bi,ni)bi nibi-5bi+5Figure 2.7: Schematic representation of Individual Chain Crystallinity(ICC)algorithm used to calculate chain crystallinity. For every bond bi adirector ni is calculated as an average of 10 neighbours. Bond is con-sidered crystalline if cosa = (bi,ni) > 0.95; Crystallinity of melt iscalculated as a fraction of aligned bonds.behavior of this quantity in the liquid and the Bragg peaks expected in the case ofa solid. However, as polymer melts are supposed to become semi-crystalline only,the structure factor measured for our systems at low temperature still exhibit char-acteristics of both liquid and crystalline states. For our coarse-grained models, weexpect to find crystals with a hexagonal symmetry since the (elongated) chainshave a cylindrical symmetry 12. The Bragg peaks should therefore correspond tothe possible values of the length of vectors of the reciprocal lattice, which is alsoa hexagonal lattice: behavior of this quantity in the liquid and the Bragg peaksexpected in the case of a solid, see Figure 2.8.12this is not obvious from the a priori knowledge, but lamellar and bundle-like structuresobserved in the melt upon crystallization allow this surmise251010.10 5 10 15 20S(q)q (2 π/σ)(1 0)(1 1) (2 0)(2 1) ( 3 0)ba(1 1)amorphous meltsemi-crystalline meltki kfq = ki - kfS(q)~I(q)Figure 2.8: Schematic representation of Static Structure parameter (Sq).Upon crystallization a smoothly-oscillating black curve correspondingto amorphous melt transforms into a spiky blue curve of Bragg peakscorresponding to semi-crystalline structure. Sharper peaks correspondto more crystallized structures. Cylindrical symmetry of elongatedchains raises hexagonal symmetry in reciprocal space.Angle heatmap of crystallizationAngle heatmaps provide information on the crystallization kinetics of a singlepolymer chain with time, see Figure 2.9. The x-axis of Figure 2.9 correspondsto the monomer index and the y-axis represents the time step; color depicts thelocal angle at a given monomer. The vertical line on this heatmap correspondsto an evolution of a single angle with time, horizontal line corresponds to a timesnapshot representing all angles at a given time. The white line indicates thestart of nucleation. Green ”paths” separated by the red zones indicate a stem of alamella.Avrami fittingAvrami analysis, used in the present work, can be introduced in analogy withthe ripple effect after a stone was dropped in the water. After N stones are dropped26Figure 2.9: Schematic representation of the angular heat maps. The x-axiscorresponds to the monomer index, y-axis - to time and color repre-sents local angle at a given monomer. Start of nucleation is indicatedwith a white line.in the water, N circular waves are generated with a radial symmetry. The probabil-ity that K waves have passed the point M during time t obeys Poisson distribution27and, therefore, can be described as:Pk(t) =Ekk!eE (2.13)where E is an expected number of waves at t. For the circular waves E can beconnected with I the rate of droplets per time per area, v is the velocity of eachwave:dE = I(t r/v)2prdrAfter integrating from 0< r < vt one obtains:E = pIv2t3/3This result could be generalized to 3D case E3D µ t4, and in general EnD µtn+1. Hence the parameter n correponds to the dimensionality of the problem.Waves expanding on a surface are mathematically identical to the crystallitesgrowing in a domain. The regions where waves have already come are equivalentto the crystallized areas, the regions where waves have not arrived yet are identicalto the amorphous regions. A probability that no crystallites have passed M at t isP0(t), i.e the probability that M is amorphous. Therefore the crystallinity can becalculated ascc = 1P0(t)Therefore crystallinity in 3D case can be calculated as:cc = 1 exp((t/t)n)where n= 3 corresponds to the dimensionality of the problem.Or equivalently, if we consider a non ideal case, where not all of the meltcrystallizes the general Avrami equation can be represented by:c(t) = c0(1 e(ktn)) (2.14)28MvFigure 2.10: Analogy of Avrami characterization of polymer crystallizationwith a wave ripple phenomenon. Expanding waves on a pond sur-face correspond to growing crystallites. The wave fronts are movingaway from the centers with velocity v; the interesting quantity is theprobability that no wave has passed a particular pointM at a time twhere k = tnA careful reader may have noticed that Avrami equation describes only growthof a formed nuclei, without saying anything about the formation of baby nucleiitself; therefore Equation 2.14 should be modified to consider time intervals afterbaby nuclei are formed. This is being done by shifting the time origin t⇤ = t t0 ,as shown in Figure 2.11b. Since the dimensionality of the simulations used in thepresent thesis is 3D, one would assume n= 3. The terminal crystallinity value inthe present work is defined as c0= c(t! t f inal)whereas the terminal crystallinityvalue in the Equation 2.15 implies cs = c(t ! •). To fill this gap a saturationcrystallinity value is introduced which is measured as a linear interpolation ofcs = c(1/t⇤) when 1/t⇤ ! 0. Therefore the final version of Avrami equationused in present work becomes:c(t) = cs(1 e(((tt0)/t)n)) (2.15)290 2 4 6 8 10 12 141/(t t0)0.00.10.20.30.40.5(1/(tt 0))(a) Linear extrapolation of c(1/t)! cswhen 1/t ! 0 to obtain csχtimet0 �χ0χs(b)Main parameters used for describingpolymer crystallizationFigure 2.11: Schematic representation of the main parameters used for theAvrami fitting. Nucleation onset time t0 is the time moment whencrystallization starts. The average crystallization time t representsaverage time until crystal is crystallized by 1/e and can be obtainedvia the Avrami fitting Equation 2.15. Saturation crystallinity valuecs and terminal crystallinity values are obtained via linear fit and ex-trapolation.30Chapter 3Results3.1 Effect of molecular weight on polymercrystallization3.1.1 ResultsSeveral different polymer melts were generated comprising of chains of variouslengths. All polymer melts were relaxed and equilibrated at T = 0.82 LJ,using theprotocol described in Section 2.4. They were quenched at T = 0.72 LJ and wereallowed to relax isothermally. Upon quenching several crystalline sub-domains(visualized as rectangular boxes in Figure 3.1) emerged in the initially amorphousmelt. Crystalline sub-domains were separated by amorphous ”liquid-like” struc-tures. Each sub-domain consisted of chains folded in a certain direction (alsocalled as ”an alignment director”) forming so-called lamellae structures. Chainfolding was guided by careful packing of molecules into a dense regular crys-talline form. Subsequently ICC and YC algorithms (described in Section 2.3)were used to investigate polymer crystallinity as a function of time c = c(t) (Fig-ure 3.2), only ICC curve is shown1).1henceforth Mw and n are equivalent terms used to describe number of units per chain31Figure 3.1: Two snapshots of the long chain (monodisperse) polymer crys-tallization: amorphous melt(on the left) and semi-crystalline (on theright side). The amorphous melt was quenched and allowed to relaxisothermally at lower temperature. Upon crystallization chains foldedin random directions forming crystalline parts separated by amorphous”liquid-like” structure. Red color corresponds to the chains which arecrystalline at the final stages of polymer crystallization. Green colorcorresponds to the chains which are amorphous at the final stage.Crystallinities obtained by both algorithms, grew linearly at the initial stagesand subsequently saturated at the value c02. To indicate the level of saturation, thesaturation crystallinity cs value (Figure 2.11a) described in Chapter 2 was used.To analyze the effect of the chain length on polymer crystallization values thec0,cs were plotted against the chain length (n). Both ICC and YC curves (Fig-ure 3.3) showed an identical decrease in the terminal crystallinity value implying2also called as the terminal crystallinity value320.00 0.05 0.10 0.15 0.20 0.25 0.30time, 106lj0.00.20.40.60.81.0Mw=20Mw=40Mw=80Mw=100 Mw=130 Mw=160 Mw=200Figure 3.2: Crystallinity evolution c(t) calculated using ICC algorithm isobserved for various chain lengths. Initial linear raise saturates andreaches plateau c(t)! csfast crystallization for short chains and slow for long ones. Discrepancy betweenvalues c0 and cs (Figure 3.3) indicate saturation of crystallinity value achievedduring the simulation time. Hereafter this discrepancy will be used as an errorestimate3. These results concur qualitatively with experimental works of [3], [4]and simulation results of [15]. Interestingly the ICC curve appears steeper than theYC curve. One could link it to the way alignment vector was calculated for eachof these algorithms4. The decrease saturated at around chain length of 130. Onecould surmise this value to be coupled with the start of entanglements in the melt.Later Avrami analysis was applied for obtained curves (see Figure 3.10a). Fit-ting revealed the start of nucleation time t0 to be close to 0 implying instantaneousstart of crystallization upon deep quench. Afterward an attempt was made to find a3i.e it can indicate when simulation was not long enough to observe the saturation of crys-tallinity value4ICC algorithm doesn’t consider ”bent” parts of lamellae-like chain stems to be crystalline andsince short chains have no ”bent” parts and they show higher crystallinity value. At the sametime YC struggles to provide accurate crystallinity value for the regions with mixed amorphous-crystalline polymer chains hence is less steep.330 50 100 150 200Mw0.00.10.20.30.40.50.60.70.80.90 YC xs YC x0ICC xsICC x0Figure 3.3: The c0 and cs parameters representing terminal and satura-tion crystallinity for various chain lengths, see Figure 3.10. Twoalgorithms of calculating crystallinity are used: Individual ChainCrystallinity(ICC) and Yamamoto Crystallinity(YC). Quantativeagreement between two algorithms is observed. Small level de-scripancy between c0 cs indicate that simulations were run longenough for crystallinity value to saturate.single master curve representing behavior of all chains at once (see Figure 3.11a).Using parameters obtained from the Avrami fitting (see Figure 3.10a), normal-ized crystallinity c(t)/cs was plotted as a function of ((t t0)/t). Longer chainsn> 80 were found to fold into a single master curve. Short chains were found tosomewhat deviate from this pattern.3.2 DiscussionThe behavior of polymers is usually governed by their relaxation times. Relax-ation times are connected to the chain mobility and are functions of the chainlength. Generally, the shorter the chains are - the smaller are the relaxation times.34(a) Long chain polymer melt. (b) Short chain polymer melt.Figure 3.4: Schematic representation of short and long polymer chainscrystallization.Bold lines representing straight, crystallized seg-ments.Short chains have chain stiffness high enough that allows themto crystallize and form ”whole chain crystals”, i.e no folding occurs.For the long chain crystallization, however, stiffness of the chains isnot high enough and therefore crystallization is accompanied by chainfolding.It is obvious that upon quenching system undergoes very high rate process; there-fore chains need to have very short relaxation times to be able to reorganize them-selves. Moreover, from the Figure 3.4b one could notice that chain stiffness ofthe short chains allows them to crystallize and form ”whole chain crystals”, i.e.,without any folding. For the long chain crystallization, however, the stiffness ofthe chains is not high enough and, therefore, crystallization is accompanied bychain folding. Chain folding and subsequent alignment may also be frustrated bythe entanglements present in the melt. All these factors combined result in fastercrystallization for short chains. For the polymers simulated in the present workthe typical chain length of the entanglements are around n = 130 and the plateauobserved in the Figure 3.3 for the chain lengths higher than 130, supposedly, isassociated with that. To verify this assumption, one needs to perform a carefulentanglement analysis which is beyond the scope of this work.In general deep quenching of polymers exhibited very fast straightening ofpolymer chains without any time for reorganization to happen. This could be ob-served from the angular heatmaps, plotted in Figure 3.9a. Angular heatmaps rep-resent a single chain evolution during the crystallization process. The chain that35underwent deep quenching shows instantaneous straightening of all of its parts.Therefore one can conclude, that short chains simply straighten and remain stilluntil the end of the simulation. Long chains, however, undergo straightening, thenfold to form a lamella and ”optimize chain ends”5 for the remaining simulationtime. This coincides with the discrepancy between behavior of short and longchains observed in Figure 3.11a. Short chain curves in Figure 3.11a lay separatefrom the long chain ones, indicating different mechanisms of crystal formation fordifferent chains lengths. The obtained results indicate that due to the rapid natureof quenching, any chain correlations present in the melt before quenching maynot have any effect on the crystallization. This phenomenon will be discussed ingreater detail in the Section 3.5.3.3 Effect of molecular weight distribution onpolymer crystallization3.3.1 ResultsTo investigate the effect of polydispersity on polymer crystallization two bidis-perse melts were generated. Bidisperse(1) - a melt comprising of 1300 shortchains (n = 40 units) and 200 long chains(n = 160 units)6 and bidisperse(2)melt comprising of an equal proportion of short and long chains (750 chains ofn=40 and 750 chains of n=160 each).All polymer melts were relaxed and equili-brated at T = 0.82 l j,using the protocol described in Section 2.4. Equilibrationwas followed by a deep quench to T = 0.72l j and isothermal relaxation. Thebulk crystallinity of polymer melts was analyzed by the ICC algorithm and wascompared with the corresponding monodisperse crystallinity curves in Figure 3.5.The resulting crystallinities were found to lay in between mono-disperse curves.This indicates that either short chains act as nucleating agents for long chains, or5to have the least surface energy6i.e with majority of short chains36long chains impede crystallization of the short chains. The curve comprising ofthe majority of short chains (bidisperse(1)) was expectedly found to laycloser to the short-chain mono-disperse curve. The curve consisting of equal pro-portion of each chain length(i.e bidisperse(2)) was found to exhibit terminalcrystallinity value less than arithmetic average of monodisperse cn=400 +cn=1600 /2which supports the idea of long chains restraining the crystallization of the shortones.0.00 0.05 0.10 0.15 0.20 0.25 0.30time, 106 lj units0.00.10.20.30.40.5(t)monodisperse, Mw=40bidisperse (1)bidisperse (2)monodisperse, Mw=1601500 of n=40750 of n=40 750 of n=1601300 of n=40 200 of n=1601500 of n=160Figure 3.5: Crystallization kinetics of short (n = 40) and long chain(n = 160) monodisperse melts and corresponding bidisperse blendscomprising of both long and short chains.The curve compris-ing of majority of short chains (bidisperse(1)) was expect-edly found to lay closer to the short-chain mono-disperse curve.The curve comprising of equal proportion of each chain length(i.ebidisperse(2))exhibits terminal crystallinity value less than thearithmetic average of monodisperse cn=400 +cn=1600 /2.To further confirm this claim individual chain crystallinity algorithm was ap-plied to each phase of certain chain length in the blend.These curves were com-37pared with corresponding bulk crystallinities of the monodisperse melts, see Fig-ure 3.6. The crystallinity of the short-chain component of both bidisperse meltswas found to lay significantly lower than the bulk crystallinity of a pure monodis-perse melt. Terminal crystallinities of a long chain component of the both bidis-perse melts appear to be slightly lower than the corresponding mono-dispersecurve suggesting a small influence of the short chains during the long time be-havior of bidisperse melts. Interestingly long chain curve of bidisperse(1)initially appears steeper than long chain mono-disperse curve, which indicates thaton the early stages nucleation was assisted by the short chains, whereas later longchains crystallization is hindered by the presence of the short chains.To illustrate the ideas above several snapshots were taken from the MolecularDynamics trajectories (see Figure 3.8,Figure 3.7). In Figure 3.7 one can observepolymer chains behavior in a bidisperse(1) melt, comprising of most shortchains. The rapid character of quenching, high chain stiffness and the lack ofentanglements of the short polymers resulted in a low chain mobility and crys-tallization in situ, anywhere in the melt in random directions. The long chainsacted as inhibitors, hindering the crystallization by trapping the short chains into”molecular cages”.When more long chains are added to the melt(bidisperse(2)) the crys-tallization behavior changes drastically. Some long chains still act as inhibitorstrapping short chains into the ”molecular cages”. However, some of the longchains act as ”baby nuclei” for the short chains to crystallize. Further growth inthese systems occurs by the chains attaching themselves to the nuclei formingstructures that remind fringed micelles (or bundle structures ) proposed by [4].3.3.2 DiscussionAs was mentioned in the Section 3.2, different chain lengths have different relax-ation times and, therefore, show different crystallization behavior upon quench-ing. The short chains align in random directions forming ”whole chain” crystalsand remain in place for the rest of the simulation time. The long chains, however,380.00 0.05 0.10 0.15 0.20 0.25 0.30time, lj0.00.10.20.30.40.5n = 40, bidisperse(1)n = 160, bidisperse(1)n = 160, bidisperse(2)n = 40, bidisperse(2)crossoverFigure 3.6: ICC algorithm applied for each individual chain length in thebi-disperse melt. Crystallinity of short chain component of both bi-disperse melts lays significantly lower than that in the pure mono-disperse melt. Terminal crystallinities of a long chain component ofboth bidisperse melts are found to be slightly lower than that of thecorresponding mono-disperse crystallization curve indicating smallinfluence of short chains on the long time crystallization behaviorof the bidisperse melts. The long chain crystallization curve of thebidisperse(1) is initially steeper than long chain mono-dispersecurve indicating assistance of the short chains in the early stages ofcrystallization and impeding long chain crystallization in the finalstagesundergo a multistep process of straightening, folding and, subsequently, form-ing lamella and optimizing the surface energy for the remaining simulation time.Bidisperse melts investigated in the present chapter comprise of both short andlong chains and exhibit somewhat a mixed and perplex behavior. Crystallizationat the first stages of a melt comprising of most short chains (bidisperse(1))is governed by the short chains. Some of the short chains crystallize in random39(a) Time t = t0 (b) Time t = t f inal/2 (c) Time t = t f inalFigure 3.7: Crystallization snapshots of bidisperse(1) melt. Red chains - arethe chains having n= 40 and green - chains with n= 160. Only chainsthat have at least one atom within certain distance from the origin arevisualized. Since there are only few long chains, crystallization is gov-erned by the short chains. Chains crystallize in random directionsanywhere within the domain. Some of the short polymers are being”trapped” inside ”cage” formed by the long chains.directions and act as nucleating agents for the long chains, as shown in Figure 3.7.Long chains act as constraints that hinder the movement of short chains by trap-ping them into ”molecular cages”. At the final stages of crystallization, shortchains are aligned and remain unmoved. Long chains, however, still evolve andmove, although their movement is inhibited by the crystallized short chains. Thisis demonstrated in the Figure 3.7c: the movement of the green long chains is ob-structed by the crystallized short chains (red). The difference in the behavior atthe onset and the terminating stages results in a crossover in the Figure 3.6.The crystallization mechanism is changed when the bidisperse melt consistsof a higher fraction of long chains (bidisperse(2)). At the early stages thecrystallization of short chains is restrained by the amorphous long chains andcrystalline structures form around a single fold of a long chain as see in Figure 3.8.The fold has a fluctuating origin and acts as a baby nuclei for the other molecules.40(a) Time t = t0 (b) Time t = t f inal/2 (c) Time t = t f inalFigure 3.8: Crystallization snapshots of bidisperse(2) melt.Red chains - arethe chains with n = 40, green - chains with n = 160. Only chainsthat have at least one atom within certain distance from the origin arevisualized.Long chains in this case act as nucleating agents for theshort chains and further growth ”short-chain crystals” occur only long-chain baby nuclei, forming bundle-like structuresAs a result, final structures appear to be bundle-like fringed micelles surroundedby amorphous ”liquid-like” long chains.3.4 Effect of cooling protocol3.4.1 Results and discussionPolymer crystallization is a non-equilibrium process and, therefore, the final crys-tallinity depends drastically on the thermal history. In the present work all poly-mer melts were first relaxed and equilibrated at T = 0.82 LJ. Subsequently, theeffect of several different thermodynamic protocols on the crystallization wereexamined: deep quench, shallow quench and gradual cooling with different rates,see Figure 2.6. As expected polymers behave differently upon different thermo-dynamic protocols. An approximate picture of the process can be obtained fromthe angular heatmaps Figure 2.9. The angular heatmaps allow to track the crys-tallization of each chain by tracking the behavior of each angle in the chain with41time and visualizing the evolution through a heatmap.For the case of the deep quench, the angular heatmaps exhibit instantaneousstraightening of the chain at various points of the chain as seen in Figure 3.9a.Constant rate cooling evinced certain onset shift in nucleation Figure 3.9c, whichis associated with the need of the certain degree of supercooling for the polymerto crystallize. The behavior of melts under shallow quench was found to lay inbetween Figure 3.9b.(a) Deep quench (b) Shallow quench (c) Constant rate coolingFigure 3.9: Angular heatmaps of polymer systems upon various thermody-namic protocols. Green paths indicate straightening of parts of a poly-mer chain. Nucleation in the case of deep quench starts instantly andalong the whole length of the chain. Finally for the case of gradualconstant rate cooling an offset in crystallization is observedThe YC crystallization analysis was applied to quantify the crystallinity inbulk. Obtained curves c(t), shown in Figure 3.10, were plotted and fitted usingthe Avrami fitting procedure described in Figure 2.11b. Quenching exhibited afast linear growth of crystallinity at the early stages and subsequent saturation.The slow cooling, depicted in Figure 3.10b, indicated a ”sigmoidal-like” behavior.Avrami fitting of the obtained curves showed a good agreement at the early stagesand subsequent deviation from the MD data. The deviation can be attributed to thelack of predictability of the Avrami equation at the final stages when crystallitesstart ”touching” each other to form larger crystals. The parameters obtained viaAvrami fitting were used for mapping the crystallization curves onto a mastercurve. Master curves are the plots of c(t)/cs((t t0)/t) and act as a litmus test42for the polymers to follow the same crystallization path.Short polymer chains upon quenching (see Figure 3.11a) lay outside of themain trend indicating a difference in their behavior upon deep quench. Due tothe rapid character of quenching, the short chains straighten and remain still uponcrystallization. The long chain, however, align, fold and minimize their surfaceenergy during the remaining time. The chains that undergo a slow cooling exhibita similar behavior and lay ”together” on their correspondent master curve (seeFigure 3.11b) indicating that upon slow cooling even short chains exhibit highmobility during crystallization. In Figure 3.12 one can observe a difference inthe final crystallinity values for various thermodynamic protocols. Constant ratecooling exhibits the highest final crystallinity possible whereas deep quench whichis the ultimate fastest cooling causes the lowest crystallinity.3.5 Polymer crystallization of pre-stretched melt3.5.1 ResultsThis chapter covers the efforts done on the investigation of the effect of the re-maining correlations in polymer melts on the crystallization process, or, in otherwords, the memory effect. These correlations were introduced via pre-stretchingof the polymer melt along one direction. A computational experiment is set up thefollowing way: a polymer melt comprising of chains of n = 160 7 fully relaxedat temperature above crystallization point (T = 0.82 > Tm) undergoes a uniaxialextension along certain axis. Upon stretching box, the box-size is governed byL(t) = L0ee˙(tt0), where e˙ is the true strain rate 8 and t is elapsed time.Subsequently, the polymer melts undergoes: quenching at T = 0.72 LJ fol-lowed by an isothermal relaxation and an intermediate rate cooling (see Figure 2.6for details). Crystallinity evolution was tracked via the ICC algorithm and thevalue c(t) was plotted in Figure 3.13. To understand the effect of pre-stretching,7which is higher than the entanglement length8also known as Hencky strain rate430.00 0.05 0.10 0.15 0.20 0.25 0.30t, 106 lj0.00.20.40.60.81.0(t)/sn=20n=40n=80n=100n=160n=200(a) Deep quench0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7t, 106 lj0.00.20.40.60.81.0(t)/sn=20n=40n=80n=100n=160n=200(b) Slow coolingFigure 3.10: Crystllinity curves obtained using the YC algorithm (symbols)and corresponding Avrami fitting curves c(t)/cs = 1 exp(((t t0)/t)3) (continuous lines). Two thermodynamic protocols plotted inFigure 2.6 are investigated: deep quench and slow cooling. Polymermelts that undergo deep instantaneous quench exhibit instantaneouscrystallization, whereas slow cooling exhibit a sigmoidal shape of thecrystallization curve.440.005 0.000 0.005 0.010 0.015 0.020 0.025tt00.00.20.40.60.81.0/s n=20n=40n=80n=100n=160n=200(a) Deep quench0.02 0.01 0.00 0.01 0.02 0.03 0.04 0.05tt00.00.20.40.60.81.0/sn=20n=40n=80n=100n=160n=200(b) Slow coolingFigure 3.11: Master crystallinity curves obtained using the YC algo-rithm(symbols) and corresponding scaled curves c(t)/cs((tt0)/t))(continuous lines). Two thermodynamic protocols plotted in Fig-ure 2.6 are investigated: deep quench and slow cooling. The poly-mer melts that undergo deep instantaneous quench exhibit an instan-taneous crystallization, whereas slow cooling exhibit a sigmoidal-shaped behavior of the crystallization curve450 50 100 150 200Mw0.30.40.50.60.70.8sdeep quenchshallow quenchcooling intcooling slowFigure 3.12: Saturation value crystallinity cs obtained from YC crystallini-ties curves. Errorbars are generated as a difference between saurationvalue crystallinity and terminal crystallinity value illustrated on Fig-ure 2.11b. Shorter chains exhibit higher crystallinity value than longchains. For a given chain length slower cooling exhibits higher crys-tallinity valuethe obtained curves are compared with the curves of polymers that undergo qui-escent crystallization9 . Pre-stretching and subsequent quenching did not alterthe degree of polymer crystallization significantly. Curves, corresponding to pre-stretched polymers, nearly match the quiescent crystallization curve. However,subsequent cooling exhibited 15% higher terminal crystallinity value when pre-strech is applied.9i.e without any mechanical impact460.00 0.05 0.10 0.15 0.20 0.25 0.30time, lj0.000.050.100.150.200.250.300.350.40pre-stretch + coolingcoolingpre-stretch + quenchquenchFigure 3.13: Crystallinity evolution for presheared polymer melts3.5.2 DiscussionIn the previous chapters, polymer behavior was described under certain thermo-dynamic treatment. Polymers of different chain length were subject to differentthermodynamic protocols of slow, intermediate, fast cooling and quenching. Itwas shown, that polymer crystallization due to its non-equilibrium nature de-pends strongly on the crystallization path. However, in Section 3.2 it was alsomentioned, that any chain correlations present in the melt before the quenchingoccurs, may not have any effect on crystallization due to the rapid character of thedeep quench crystallization.Pre-stretching of an entangled polymer melt orients polymer chains and, there-fore, introduces correlations between them. To relax and decorrelate themselvespolymers need time trelax (the longest relaxation time connected to the chain lengthand degree of stretching). Cooling with different rate introduces another timescalefor the process: tcooling connected to the cooling rate. When tcooling < trelax (poly-mers quenched rapidly) polymers don’t have time to rearrange and alignment47caused by pre-stretching, doesn’t have any effect. On the other hand, coolingafter pre-stretching gives the chains time to relax ( tcooling > trelax) and ”quasi-equilibrate” which results into a higher terminal crystallinity value. The inter-ested reader might refer to the works of Jabbarzadeh et al. [20], discussed in theliterature review section, for more details on the subject. They have reported thatpre-stretching causes higher crystallinity value only for long chains leaving shortchains intact.48Chapter 4Discussion and conclusion4.1 DiscussionUpon crystallization, polymer chains undergo an instantaneous straightening, form-ing blocks of a crystalline polymer. This can be explained via the free energyreasoning: when cooled stiffness of the chains increases, so polymers favor crys-tallizing (i.e., straightening) sacrificing raise of the entropy. This trade-off canalso be described as a competition between long chain nature of the crystallizingmolecule ( associated with enthalpy ) and its multiplicity of confirmations thatoccur in the disordered state ( associated with entropy ).The behavior of polymers is usually governed by their relaxation times. Re-laxation times are coupled with the chain mobility and are functions of the chainlength. The shorter the chains are - the smaller are the relaxation times tpolymer.At the same time, every thermodynamic protocol applied in the present work hasits own characteristic time tprocess as well. Predicting behavior of polymer chainsrequires careful capture of these two values.If tprocess < tpolymer (as in the case of quenching), chains don’t have enoughtime to reorganize themselves before they crystallize. This case is connectedwith the lower terminal crystallinity values. Short chains (having small tpolymerrelaxation time) form ”whole chain crystals” by aligning themselves in random49directions in a short period of time and remain still for the rest of the simulationtime. Long chains, however, undergo a multistep process of straightening, fold-ing, forming a lamella, terminate the process by optimizing the surface energyfor the remaining simulation time. This leaves short chains with higher termi-nal crystallinity values than the long chains. Long chain crystallization may alsobe frustrated by the presence of entanglements in the melt. For the polymerssimulated in the present work the typical length of the entanglements are aroundn= 130. The plateau observed on the Figure 3.3 for the chain lengths higher than130 , supposedly, connected to this effect. To verify this assumption, one needsto do a careful entanglement analysis which is beyond the scope of this work.When tprocess > tpolymer polymers have more time to reorganize themselves. Thisresults into a higher degree of crystallinity and the nucleation onset-shifts in thenucleation process depicted inFigure 3.10b.When bidisperse systems are investigated comprising of short and long molecules,the resulting systems have two timescales connected to the relaxation times ofeach chain length. The behavior of the each chain length scales differently in timeresulting in crossovers between the short-chain and the long chain dominatingparts. Further investigation concludes, that the behavior of the bidisperse meltsis governed by the proportion of the short and the long chains in the melt. Whenthe critical proportion of the long chains is reached, they start to act as nucleatingagents for the short chains. Upon crystallization, the short chains ”attach” them-selves to the folded long chain forming fringed micelles (bundle like structures).When the critical proportion is not reached the long chains act as ”moleculartraps” that restrain crystallization of the short chains. When the critical proportionof the short chains is reached, upon crystallization they act as nucleating agentsfor the long chains at the commencement of crystallization and start to inhibit thecrystallization of long chains at the terminating stages.Polymer samples have a unique ability to keep a partial memory, i.e., a poly-mer sample may be completely molten according to the calorimetric or rheologicaldata and still keeps the partial memory giving rise to a phenomenon of precursors50states. This memory memory effect can be described as a correlation betweenpolymers within the melt. These correlations could be obtained via pre-stretchingthe melt using a short time uniaxial extension. Upon crystallization, the memoryeffects were found to play a role, only if polymers undergo a slow process whichallows them to relax (like gradual cooling with slow rate). In the opposite case(like deep quench), correlations between polymers appear to have no significanteffect on their crystallization.4.2 ConclusionLarge-scale molecular dynamics simulations were carried out to study the kinet-ics of the polymer melt crystallization. Crystallization of polymers of differentmolecular weight was studied under a rapid quench. Upon the quench, shorterchains were found to align into straight segments and remain still for the rest ofthe crystallization process. Longer chains were found to exhibit a multistep pro-cess of aligning, folding and minimizing their surface energy for the remainingof the simulation time. As a result, shorter chains were found to exhibit higherterminal crystallinity valuse than the longer ones.The effect of polydispersity was modeled via two bidisperse melts compris-ing of a different proportion of the short and the long chains. The behavior ofpolymers of a different chain length scaled differently in time resulting in thecrossovers between the short-chain and the long chain dominant parts. Furtherinvestigation concluded that the crystallization behavior of the bidisperse melts isgoverned by the proportion of the short and the long chains in the melt.When a critical fraction of the long chains was reached they appeared to act asbaby nuclei for the short chains, resulting into bundle-like fringed micelle struc-tures; otherwise they acted as ”molecular traps” hindering crystallization of theshort chains. When the critical fraction of the short chains was reached, they werefound to assist the crystallization of long chains at the initial stages but impedecrystallization dynamics at the final stages.Different thermodynamic protocols of cooling were studied: deep quench,51shallow quench and cooling with various rates as well as polymer pre-stretchingand subsequent cooling and quenching. Cooling with the slowest rate revealedthe highest terminal crystallinity values. The resulting crystallization curves werefitted using the Avrami equation. They showed a good agreement at the earlystages and deviation at the later stages of crystallization, indicating a lack of pre-dicting power of Avrami analysis for the systems of ”touching” crystallites. Pre-stretching and subsequent quenching exhibited no significant impact on the finalcrystallinity whereas pre-stretching followed by the intermediate rate cooling wasfound to increase the terminal crystallinity.52Bibliography[1] Chuanfu Luo and Jens-Uwe Sommer. Coding coarse grained polymermodel for LAMMPS and its application to polymer crystallization.Computer Physics Communications, 180(8):1382–1391, August 2009. !pages ii, 20[2] Gregory C. Rutledge Ewa Piorkowska. Handbook of PolymerCrystallization. 2013. ! pages 1, 2, 3, 5, 6, 15, 16[3] E. Ergoz, J. G. Fatou, and L. Mandelkern. 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Journal of Molecular Graphics, 14:33–38, 1996. !pages 20[30] John Stone. An Efficient Library for Parallel Ray Tracing and Animation.Master’s thesis, Computer Science Department, University ofMissouri-Rolla, April 1998. ! pages[31] Naveen Michaud-Agrawal, Elizabeth J. Denning, Thomas B. Woolf, andOliver Beckstein. Mdanalysis: A toolkit for the analysis of moleculardynamics simulations. Journal of Computational Chemistry,32(10):2319–2327, 2011. ! pages 20[32] Chuanfu Luo and Jens-uwe Sommer. Molecular Dynamics ( MD )Simulations of Polymer Crystallization and Melting. (Md), 2010. ! pages23[33] Robert D. Groot and Patrick B. Warren. Dissipative particle dynamics:Bridging the gap between atomistic and mesoscopic simulation. TheJournal of Chemical Physics, 107(11):4423, 1997. ! pages 5756Appendix ASupporting MaterialsEquilibration Protocol for preparing well-equilibrated long chain polymer melts1. Generate the initial configurations that closely matchequilibrium structures at large length scales so theMD simulation is only needed to relax the short to intermediatelength scale configurations2. Perform short simulation runs with soft repulsion potentialused in Dissipative Particle Dynamics (Dissipative ParticleDynamics (DPD)), UDPD [33], on an ensemble of polymerchains with the correct end-to-end distance(generatedon the previous step). By this chains are allowed topass through each other to speed up the polymer dynamics.3. After a gradual increase of the strength of the DPDpotential, short simulation with target coarse-grainedpotential is performed4. Finish the simulation of the melts 1 with brute forceMD run, that will equilibrate the thermodynamic parameters.1The melts already have correct chain statistics57Lammps input file used for equilibrating polymer melts:# KremerGrest model .va r i ab l e Th equal 1.0 #> hotun i t s l ja tom sty le bondspec ia l bonds l j / cou l 0 1 1read data {0[ inda ta ] } . dataneighbor 0.4 b inneigh modi fy every 1 delay 1comm modify ve l yesbond s ty le fenebond coef f ⇤ 30.0 1.5 1.0 1.0dump mydump a l l dcd {0[ dumptraj ]} {0[ d c d f i l e ] } . dcdt imestep {0[ t imestep ]}thermo {0[dumpthermo ]}thermo modify norm nop a i r s t y l e dpd 1.0 1.0 122347pa i r c o e f f ⇤ ⇤ 25 4.5 1.0v e l o c i t y a l l c rea te ${{Th}} 17786140f i x 1 a l l nve / l i m i t 0.001run 500f i x 1 a l l nve / l i m i t 0.01run 500f i x 1 a l l nve / l i m i t 0.05run 500f i x 1 a l l nve / l i m i t 0.1run 500un f i x 1f i x 1 a l l nvev e l o c i t y a l l c rea te ${{Th}} 17786140run 50000wr i t e da t a {0[ tmpdata ] } . datap a i r c o e f f ⇤ ⇤ 50.0 4.5 1.0v e l o c i t y a l l c rea te ${{Th}} 15086120run 50pa i r c o e f f ⇤ ⇤ 100.0 4.5 1.0v e l o c i t y a l l c rea te ${{Th}} 15786120run 50pa i r c o e f f ⇤ ⇤ 150.0 4.5 1.058v e l o c i t y a l l c rea te ${{Th}} 15486120run 50pa i r c o e f f ⇤ ⇤ 200.0 4.5 1.0v e l o c i t y a l l c rea te ${{Th}} 17986120run 100pa i r c o e f f ⇤ ⇤ 250.0 4.5 1.0v e l o c i t y a l l c rea te ${{Th}} 15006120run 100pa i r c o e f f ⇤ ⇤ 500.0 4.5 1.0v e l o c i t y a l l c rea te ${{Th}} 15087720run 100pa i r c o e f f ⇤ ⇤ 1000.0 4.5 1.0v e l o c i t y a l l c rea te ${{Th}} 15086189run 100wr i t e da t a {0[ tmpdata ]}1 . datap a i r s t y l e hybr id / over lay l j / cu t 1.122462 dpd / t s t a t 1.0 1.0 1.122462 122347pa i r mod i f y sh i f t yesp a i r c o e f f ⇤ ⇤ l j / cu t 1.0 1.0 1.122462pa i r c o e f f ⇤ ⇤ dpd / t s t a t 4.5 1.122462v e l o c i t y a l l c rea te ${{Th}} 1508612013run 50v e l o c i t y a l l c rea te ${{Th}} 15021run 50v e l o c i t y a l l c rea te ${{Th}} 2086111run 50v e l o c i t y a l l c rea te ${{Th}} 126111run 50v e l o c i t y a l l c rea te ${{Th}} 1286111run 50v e l o c i t y a l l c rea te ${{Th}} 15021run 50v e l o c i t y a l l c rea te ${{Th}} 26111run 50v e l o c i t y a l l c rea te ${{Th}} 126111run 50v e l o c i t y a l l c rea te ${{Th}} 1286111run 50wr i t e da t a {0[ tmpdata ]} push . datav e l o c i t y a l l c rea te ${{Th}} 15086125rese t t imes tep 0run {0[ kremerequi l ]}wr i t e da t a {0[ outdata ] } . data59Interaction parameters in the simulations were set up using the following com-mands:pair_style lj96/cut 1.0188pair_modify shift yesbond_style harmonicangle_style table spline 181read_data polymer.datapair_coeff * * 0.37775 0.89bond_coeff * 1352.0 0.5angle_coeff * cgpva.table CG_PVAspecial_bonds lj 0.0 0.0 1.0Lammps program used for studying polymer crystallization:va r i ab l e Th equal 1.2va r i ab l e Tm equal 1.0va r i ab l e Tn equal 0.82va r i ab l e Tlunder equal 0.78va r i ab l e T l equal 0.72va r i ab l e Ps equal 8.0va r i ab l e dump tra j equal 15000un i t s l jboundary p p patom sty le anglep a i r s t y l e l j 9 6 / cu t 1.0188pa i r mod i f y sh i f t yesbond s ty le harmonicang l e s t y l e t ab l e sp l i ne 181read data . / f i g u r e s / polymer 0 . 8 . datap a i r c o e f f ⇤ ⇤ 0.37775 0.89bond coef f ⇤ 1352.0 0.5ang le coe f f ⇤ cgpva . t ab l e CG PVAspec ia l bonds l j 0.0 0.0 1.0neighbor 0.4 b inneigh modi fy every 1 once no c l u s t e r yes60t imestep 0.005thermo 300dump dump tra j a l l dcd 15000 quench . dcddump modify dump tra j sort i d unwrap yes# npt quenching a t T = 0.82 |||> 0.72 , P = 8.0f i x 1 a l l npt temp ${Tl} ${Tl} 10 iso ${Ps} ${Ps} 1000f i x 2 a l l momentum 10 l i n e a r 1 1 1 angularthe rmo s ty le custom step temp press dens i t y vo l pe ke epa i r ebond eangle e t o t a lr ese t t imes tep 0run 3000000 s t a r t 0 stop 60000000wr i t e da t a tmp . quench . ⇤ . dataBox pre-stretching was governed by this equation:L(t) = L0e(tt0)/t⇤and was implemented via the following Lammps command:fix 1 all npt temp ${Tn} ${Tn} 10 y 0 0 1000 z 0 0 1000 drag 2fix 2 all deform 1 x trate ${my_rate} units box remap xwhere (t t0) is the elapsed time (in time units). Thus if trate = R is specifiedas ln(1.1) and time units are picoseconds, this means the box length will increaseby 10% of its current (not original) length every picosecond. I.e. strain after 1picosecond = 0.1, strain after 2 picosecond = 0.21, etc. R= ln(2) or ln(3) meansthe box length will double or triple every picosecond. R= ln(0.99)means the boxlength will shrink by 1% of its current length every picosecond. Note that for a”true” rate the change is continuous and based on the current length, so runningwith R = ln(2) for 10 picoseconds does not expand the box length by a factor of11 as it would with erate, but by a factor of 1024 since the box length will doubleevery picosecond 2.2This is taken from LAMMPS manual61IndexAngle heatmaps, 26Avrami analysis, 26bidisperse, 36Bidisperse(1), 36bidisperse(2), 36Boltzmann distribution, 11bundle structures, 38CG-PVA potential, 20characteristic ratio, 13crystallinity definition, 23Dissipative Particle Dynamics, 57End-to-End distance , see end-to-end vec-torend-to-end vector, 11enthalpy, 49enthalpy of melting, 15entropy, 49entropy of melting, 15equilibration, 11fast cooling, 21forcefield, 17, 20Freely Jointed Chain, 11fringed micelles, 38Gyration radius, 13ideal chains, 12Individual Chain Crystallinity, 24individual chain crystallinity, 37intermediate cooling, 21isotropic-nematic transition, 14lamellae, 31LAMMPS, 20Lenard-Jones, 17master curve, 34models of polymer crystallization, 15Molecular Dynamics Simulation, 17Monte-Carlo, 19orientational tensor, 23polydispersity, 36polymer crystallization, 14polymer relaxation times , see also re-laxation times3462pre-stretch, 23quench, 20quiescent crystallization, 46radius of gyration, 12saturation crystallinity value, 29slow cooling, 21static structure factor, 24Strobl model, 16supercooling, 15Terminal crystallinity value, 29thermodynamic protocols, 20Visual Molecular Dynamics, 20Yamamoto Crystallinity, 2363

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