Die vorliegende Arbeit befasst sich mit dem Wachstum von semiflexiblen Polymeren in ungeordneten Medien. Um dieses Wachstum zu simulieren, kommen hoch entwickelte Monte Carlo Algorithmen zum Einsatz.
Eine Zahl von Polymeren wird mithilfe eines breadth-first (zuerst in die Breite gehenden) Algorithmus erzeugt. Dies geschieht im zwei- und dreidimensionalen Raum um die erhaltenen Ergebnisse zu validieren und zwischen beiden Dimensionen zu vergleichen.
Der verwendete Algorithmus wurde dahingehend modifiziert den Arbeitsaufwand, welcher durch das Umgehen der Hindernisse der Hintergrundunordnung entsteht, zu kompensieren, um die nötige Rechenzeit im Rahmen zu halten und somit dessen Effizienz zu erhöhen.
Es kommen verschiedene Typen von Unordnung zum Einsatz, nämlich die Gitterunordnung (einfache und korrelierte Unordnung) sowie kontinuierliche Unordnung. Die Erzeugung dieser ungeordneten Systeme hängt von folgenden Parametern ab, Dichte, Korrelationsstärke und Lennard-Jones-Wechselwirkungen.
Die wichtigste untersuchte Größe ist die Persistenzlänge der Polymere, welche aus der Tangenten-Tangentenkorrelation, gegeben durch die Konturvektoren der Polymere, berechnet wird.
Weitere Observablen sind der mittlere quadratische Ende-zu-Ende Abstand, die Ende-zu-Ende Abstandsverteilung und die Polymerform.
Ziel ist die Gegenüberstellung der Ergebnisse aus zwei und drei Dimensionen bezüglich des Einflusses der verschiedenen Hintergrundunordnungen auf die Renormalisierung der Persistenzlänge der Polymere.:1 Introduction and Motivation . . 1
1.1 Preamble . . . . . . 1
1.2 Objective . . . . . 2
2 Models . . . . . . 5
2.1 Polymers . . . . . . 5
2.1.1 Heisenberg chain model . . . . . . 5
2.1.2 From the Heisenberg chain to the worm like chain 6
2.1.3 Worm like chain model . . . . . . 7
2.1.4 Weakly bending rod 11
2.2 Disorder . . . . . . 12
2.2.1 Lattice disorder . . 13
2.2.2 Continuous disorder 14
2.3 Disorder creation . 14
2.3.1 Lattice disorder . . 14
2.3.2 Continuous disorder 16
3 Simulation methods . . . . . . . 19
3.1 Monte Carlo Method . . . . . . . 19
3.1.1 Monte Carlo simulation concept . 19
3.2 Growth algorithms . 22
3.2.1 Rosenbluth algorithm . . . . . . . 23
3.2.2 Off-lattice growth algorithm . . . 24
3.2.3 Perm algorithm . . 31
4 Data analysis . . . 33
4.1 Error estimation . . 33
4.2 Observables and simulation parameters . . . . . . 35
4.2.1 Mean square end-to-end distance and mean end-to-end distance . 35
4.2.2 End-to-end distance distribution function . . . . . 36
4.2.3 Tangent-tangent correlation function . . . . . . . 37
4.2.4 Persistence length renormalization 40
4.2.5 Polymer shape: prolateness and asphericity . . . 41
5 Numerical results 43
5.1 Simulation parameters . . . . . . . 43
vii
Contents
5.1.1 Length scales . . . 43
5.1.2 Disorder . . . . . . 44
5.1.3 Seeds . . . . . . . 44
5.1.4 Histograms . . . . 45
5.1.5 Polymer shape . . . 45
5.2 Free polymer . . . 46
5.2.1 Analysis . . . . . . 46
5.2.2 Comparison of 2D and 3D . . . . 46
5.2.3 Conclusion . . . . . 51
5.3 Lattice disorder . . 53
5.3.1 Analysis . . . . . . 53
5.3.2 Simple lattice . . . 53
5.3.3 Comparison of free polymer and simple lattice disorder . . . . . 63
5.4 Continuous disorder 65
5.4.1 Analysis . . . . . . 65
5.4.2 Comparison of polymers in continuous disorder to free polymers and lattice dis-
order 65
5.5 Correlated lattice disorder . . . . . 72
5.5.1 Conclusion . . . . . 78
6 Summary and Outlook . . . . . 79
Danksagung . . . . . . . 81
Bibliography . . . . . . . 83
A Appendix . . . . . A
A.1 Tangent-tangent correlation . . . . A
A.1.1 Simple lattice disorder . . . . . . . A
A.1.2 Correlated lattice disorder . . . . . C
A.2 Mean square end-to-end distance . D
A.2.1 Continuous disorder D
A.2.2 Correlated lattice disorder . . . . . F
A.3 End-to-end probability distribution L
A.3.1 Continuous disorder L
A.3.2 Correlated lattice disorder . . . . . N
A.4 Persistence length renormalization S
A.4.1 Correlated lattice disorder . . . . . S / The present thesis reports on the growth of semi-flexible polymers in disordered media. Highly advanced Monte Carlo algorithms are used to simulate the polymer growth.
A number of polymers is grown with the help of a breadth-first growing algorithm. This is done in two and three dimensions to validate and compare the results.
The algorithm was modified to be able to generate the desired number of polymers in decent computing time, by introducing a special guiding field to handle the additional workload introduced by the obstacles of the background disorder.
Different types of disorder are used, lattice disorder (simple lattice disorder and correlated disorder) and continuous disorder.
The creation of those disorder configurations is steered by certain parameters as density, correlation strength and interactions governed by the Lennard-Jones-potential.
The main value of interest is the persistence length, which is calculated from the tangent-tangent-correlation of the contour vectors of the polymers. Further observables which are investigated are the mean square end-to-end distance, the end-to-end distance distribution and the polymer shape.
The goal is to show the influence of the different disorder types on the persistence length renormalization will be shown, with special focus on differences and similarities between two and three dimensions.:1 Introduction and Motivation . . 1
1.1 Preamble . . . . . . 1
1.2 Objective . . . . . 2
2 Models . . . . . . 5
2.1 Polymers . . . . . . 5
2.1.1 Heisenberg chain model . . . . . . 5
2.1.2 From the Heisenberg chain to the worm like chain 6
2.1.3 Worm like chain model . . . . . . 7
2.1.4 Weakly bending rod 11
2.2 Disorder . . . . . . 12
2.2.1 Lattice disorder . . 13
2.2.2 Continuous disorder 14
2.3 Disorder creation . 14
2.3.1 Lattice disorder . . 14
2.3.2 Continuous disorder 16
3 Simulation methods . . . . . . . 19
3.1 Monte Carlo Method . . . . . . . 19
3.1.1 Monte Carlo simulation concept . 19
3.2 Growth algorithms . 22
3.2.1 Rosenbluth algorithm . . . . . . . 23
3.2.2 Off-lattice growth algorithm . . . 24
3.2.3 Perm algorithm . . 31
4 Data analysis . . . 33
4.1 Error estimation . . 33
4.2 Observables and simulation parameters . . . . . . 35
4.2.1 Mean square end-to-end distance and mean end-to-end distance . 35
4.2.2 End-to-end distance distribution function . . . . . 36
4.2.3 Tangent-tangent correlation function . . . . . . . 37
4.2.4 Persistence length renormalization 40
4.2.5 Polymer shape: prolateness and asphericity . . . 41
5 Numerical results 43
5.1 Simulation parameters . . . . . . . 43
vii
Contents
5.1.1 Length scales . . . 43
5.1.2 Disorder . . . . . . 44
5.1.3 Seeds . . . . . . . 44
5.1.4 Histograms . . . . 45
5.1.5 Polymer shape . . . 45
5.2 Free polymer . . . 46
5.2.1 Analysis . . . . . . 46
5.2.2 Comparison of 2D and 3D . . . . 46
5.2.3 Conclusion . . . . . 51
5.3 Lattice disorder . . 53
5.3.1 Analysis . . . . . . 53
5.3.2 Simple lattice . . . 53
5.3.3 Comparison of free polymer and simple lattice disorder . . . . . 63
5.4 Continuous disorder 65
5.4.1 Analysis . . . . . . 65
5.4.2 Comparison of polymers in continuous disorder to free polymers and lattice dis-
order 65
5.5 Correlated lattice disorder . . . . . 72
5.5.1 Conclusion . . . . . 78
6 Summary and Outlook . . . . . 79
Danksagung . . . . . . . 81
Bibliography . . . . . . . 83
A Appendix . . . . . A
A.1 Tangent-tangent correlation . . . . A
A.1.1 Simple lattice disorder . . . . . . . A
A.1.2 Correlated lattice disorder . . . . . C
A.2 Mean square end-to-end distance . D
A.2.1 Continuous disorder D
A.2.2 Correlated lattice disorder . . . . . F
A.3 End-to-end probability distribution L
A.3.1 Continuous disorder L
A.3.2 Correlated lattice disorder . . . . . N
A.4 Persistence length renormalization S
A.4.1 Correlated lattice disorder . . . . . S
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:81161 |
| Date | 10 October 2022 |
| Creators | Bock, Johannes |
| Contributors | Universität Leipzig |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/acceptedVersion, doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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