Polymers in Confined Geometry.pdf

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72 CHAPTER 5. SIMULATION RESULTS the parallel end-to-end distance R|| in figure 5.11(b). For an unconfined chain (c 2), p(R) is peaked around R = 0.4 and shows Gaussian behavior, whereas p(R||) is peaked at R|| = 0. This is simply due to a phase space factor of 4πR 2 included in p(R) resulting from sampling all directions of R. Such a phase space factor is absent from p(R||) which samples the projection of R on the tube axis. For intermediate values of c (6 c 20) the RDF p(R) shows an interesting bimodal distribution. If the collision parameter c is increased from c ≈ 2 the peak is shifted towards smaller values of R such that the distribution is significantly skewed. Then, at about c ≈ 6 a second peak at large values of R develops, which then grows at the expense of the peak at low values of R. This feature is closely related to the ‘dip’ discussed in the previous section. The initial drift of the peak towards smaller values is due to the restriction of the polymers’ configuration perpendicular to the tube axis. Such a behavior is expected from a linear response argument. When the confinement becomes stronger, the end-to-end distance evades the transverse compression by also stretching along the tube axis. As a result the RDF p(R) exhibits two peaks, one at small and the other at large values of R. This quantifies the heuristic arguments for the dip of 〈R 2 〉 in the previous section. For comparison the parallel end-to-end distance distribution is shown in figure 5.11(b) for the same set of parameters. Obviously the average is growing monotonously to larger values, not showing any bimodal behavior. We have already eluded to the fact that p(R||) is peaked at R|| = 0 for weak confinement (due to phase space effects). Hence, increasing the confinement can only result in a shift of this peak to larger values. Finally for narrow ‘tubes’ (c 30) both RDFs show exactly the same shape since the end-to-end distance is now oriented parallel to the tube-axis. 5.3 The hard walls In the final chapter we turn to the investigation of preliminary simulation results concerning the conformations of polymers confined by hard walls. In particular, we are going to analyze the scaling of the end-to-end distance similar as in section 5.2.3. We study two different cross sections of the tube, a circular and square shaped channel, and leave other geometries for future investigations. Since we are mainly interested in comparing with the experimental data (cf. section 5.2.3) we concentrate on the mean apparent end-to-end distance 〈Ra〉. The simulation data for the two different channel cross sections are shown in figures 5.12 and 5.13. In both cases one finds scaling behavior with scaling variables (dɛ) −1 and (wɛ) −1 , respectively. Here d is the tube diameter for a circular cross section and w the channel width for a square cross section. Comparing both plots with each other and to figure 5.6 shows that the overall behavior is identical and that there appears to be no major influence of the chosen
5.3. THE HARD WALLS 73 〈Ra〉 1 0.8 0.6 0.4 0.2 0 0.01 0.1 1 (dɛ) −1 10 experimental data Odijk scaling (fit) 0.5 1 5 10 50 100 1000 10000 Figure 5.12: Mean apparent end-to-end 〈Ra〉 versus the scaling parameter (dɛ) −1 in the circular cross section tube with diameter d. Besides simulation data for various flexibilities ɛ, experimental data and the corresponding Odijk scaling fit are shown. 〈Ra〉 1 0.8 0.6 0.4 0.2 0 0.01 0.1 1 (wɛ) −1 10 experimental data Odijk scaling (fit) ɛ = 0.5 ɛ = 5 ɛ = 10 ɛ = 50 ɛ = 300 Figure 5.13: Mean apparent end-to-end 〈Ra〉 versus the scaling parameter (dɛ) −1 in the square cross section tube with width w. Besides simulation data for various flexibilities ɛ, experimental data and the corresponding Odijk scaling fit are shown. 100 1000
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Diploma Thesis Polymers in Confined
Page 3:
TMTOWTDI—There’s More Than One
Page 7 and 8:
Contents 1 Introduction 1 2 Polymer
Page 9 and 10:
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
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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 and 78: 5.2. THE HARMONIC POTENTIAL 69 R
Page 79: 5.2. THE HARMONIC POTENTIAL 71 p(R)
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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