Polymers in Confined Geometry.pdf

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70 CHAPTER 5. SIMULATION RESULTS R R || Figure 5.10: Behavior of the end-to-end distances of flexible chains upon confinement. A representative example of these observables for a flexible chain ɛ = 10 is shown in figure 5.9, where the mean-square and mean-square parallel end-to-end distance, 〈R2 〉 and R2 || , are plotted as a function of the collision parameter c. The full end-to-end distance exhibits a ‘dip’ (local minimum) for c ≈ 1, when the chain begins to notice the tube through deflections. This dip is absent for R2 || . The two limits, c → 0 and c → ∞, are easily understood: In the limit of an unconfined chain (c < 1) the average of the z-component is exactly a third of the full average R2 2 || = 〈R 〉 /3. In the opposite limit, when the confinement becomes larger (c > 20), both observables, 〈R2 〉 and R2 || , behave identically since the polymer is oriented mainly along the tube. Therefore the end-to-end distance is nearly parallel to the tube axis. The dip—which has also been observed without explanation in [43]—can be explained as illustrated in figure 5.10. Upon confinement of the flexible chain, the initially randomly oriented end-to-end distance is hindered in the perpendicular directions. This implies that first the end-to-end distance is changed predominantly perpendicular to the tube axis, only decreasing the value of 〈R2 〉 but leaving R2 || unchanged. Formation of the dip should start, as seen in figure 5.9, at a value of c 1. Upon further confinement the end-to-end distance slowly starts giving way in the direction along the tube, finally also increasing the parallel component. The behavior of the end-to-end distances upon confinement can also be seen in the shape of the radial distribution functions (see next section 5.2.5). The dip is present in all end-to-end distances introduced in section 4.6, as it is absent for all parallel end-to-end distances. 5.2.5 Radial distribution function In this section we are going to investigate the probability distribution of the end-to-end distance or radial distribution function (RDF) (see section 3.3.3 for a discussion of the RDF in the unconfined case). Figures 5.11 show some representative numerical results for a chain with flexibility ɛ = 10 and for increasing collision parameter c. The RDF for the end-to-end distance R is shown in figure 5.11(a) and for δR || δR d
5.2. THE HARMONIC POTENTIAL 71 p(R) p(R ||) 4 2 0 0 4 2 0 0 c = 2 c = 3 c = 2 c = 3 0.2 0.2 0.4 c = 6 c = 10 R 0.6 (a) RDF for R = |R| 0.4 c = 6 c = 10 R || 0.6 (b) RDF for R || = |R ||| c = 15 c = 20 c = 15 c = 20 0.8 0.8 c = 60 c = 60 Figure 5.11: Comparison of the probability distributions of the end-to-end distances for a chain of flexibility ɛ = 10 and increasing collision parameter c. 1 1
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Diploma Thesis Polymers in Confined
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TMTOWTDI—There’s More Than One
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Contents 1 Introduction 1 2 Polymer
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Chapter 1 Introduction Polymers are
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(a) λ-DNA (b) T2-DNA Figure 1.2: A
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Chapter 2 Polymer models This chapt
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2.2. IDEAL PHANTOM CHAIN 7 configur
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2.3. SEMI-FLEXIBLE POLYMERS 9 In th
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2.3. SEMI-FLEXIBLE POLYMERS 11 to i
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2.3. SEMI-FLEXIBLE POLYMERS 13 for
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2.5. REAL CHAINS 15 no overhangs al
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2.5. REAL CHAINS 17 Figure 2.5: Sna
Page 27 and 28: 2.5. REAL CHAINS 19 ln R cases intr
Page 29 and 30: Chapter 3 Polymers in confined geom
Page 31 and 32: 3.2. SCALING RELATIONS 23 using eq.
Page 33 and 34: 3.2. SCALING RELATIONS 25 b a d R |
Page 35 and 36: 3.3. ANALYTICAL RESULTS 27 The mean
Page 37 and 38: 3.3. ANALYTICAL RESULTS 29 From thi
Page 39 and 40: 3.3. ANALYTICAL RESULTS 31 a consta
Page 41 and 42: 3.3. ANALYTICAL RESULTS 33 the proj
Page 43 and 44: 3.3. ANALYTICAL RESULTS 35 x 2 (s/
Page 45 and 46: Chapter 4 Simulation methods This c
Page 47 and 48: 4.3. SAMPLING THE CANONICAL ENSEMBL
Page 55 and 56: 4.4. SIMULATION OF UNCONFINED WORM-
Page 57 and 58: 4.4. SIMULATION OF UNCONFINED WORM-
Page 59 and 60: 4.5. SIMULATION OF CONFINED WORM-LI
Page 61 and 62: 4.6. SAMPLED QUANTITIES AND THEIR R
Page 63 and 64: 4.6. SAMPLED QUANTITIES AND THEIR R
Page 65 and 66: Chapter 5 Simulation results In thi
Page 67 and 68: 5.2. THE HARMONIC POTENTIAL 59 ˆt
Page 69 and 70: 5.2. THE HARMONIC POTENTIAL 61 5.2.
Page 71 and 72: 5.2. THE HARMONIC POTENTIAL 63 beha
Page 73 and 74: 5.2. THE HARMONIC POTENTIAL 65 R 2
Page 75 and 76: 5.2. THE HARMONIC POTENTIAL 67 R 2
Page 77: 5.2. THE HARMONIC POTENTIAL 69 R
Page 81 and 82: 5.3. THE HARD WALLS 73 〈Ra〉 1 0
Page 83 and 84: Chapter 6 Conclusion Emerging from
Page 85 and 86: Appendix A Discrete worm-like chain
Page 87 and 88: which, after taking all the derivat
Page 89 and 90: Appendix B Calculation for the conf
Page 91 and 92: B.2. END-TO-END DISTANCE CORRELATIO
Page 93 and 94: B.3. PARALLEL END-TO-END DISTANCE C
Page 95 and 96: Glossary L filament length lp persi
Page 97 and 98: Bibliography [1] M. Abramowitz and
Page 99 and 100: BIBLIOGRAPHY 91 [29] P. L’Ecuyer.
Page 101: :wq
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