Polymers in Confined Geometry.pdf

More documents

Recommendations

Info

62 CHAPTER 5. SIMULATION RESULTS 1 0.99 0.98 0.97 0.96 -0.5 φtt(0.5, s + 0.5)/ t 2 1 0.99 0.98 0.97 -0.5 -0.4 -0.3 -0.2 c = 0.1 c = 1 c = 3 c = 5 c = 10 c = 20 c = 100 -0.1 0 s − 0.5 0.1 (a) Normalized φtt for ɛ = 0.1 and a set of increasing collision parameters c indicated in the graph. -0.25 1 0.99 0.98 0.97 0.96 0.95 -0.5 0 -0.25 0 0.25 0.2 (b) same as in (a) for c = 5 · 10 −4 0.25 (c) same as in (a) for c = 3 0.5 1 0.99 -0.5 -0.25 0.3 0.5 0 0.4 0.5 0.25 (d) same as in (a) for c = 10 Figure 5.4: Normalized tangent-tangent correlation function along a chain with flexibility ɛ = 0.1. The reference point is fixed in the middle of the chain, therefore the graphs must be symmetric. N = 50 segments were used in the simulations. Additionally, in (b)–(d) the analytical result eq. (3.34) is shown. 0.5
5.2. THE HARMONIC POTENTIAL 63 behavior of the tangent-tangent correlation function. The actual polymer in the simulation has free ends and is therefore much less constrained in comparison to the analytical calculation where the fluctuation of the boundaries are suppressed. The influence of the free ends was already seen on the undulations in figure 5.3 as a large increase of the deviation from the analytical result. But also in figure 5.4(c) a reminiscence of anticorrelations of the segments can be seen as a small shoulder at |s − 0.5| ≈ 1/c = 1/3. Figure 5.4(b) shows, for a collision parameter c = 5 · 10 −4 , an effectively free polymer. The analytical result fits the data perfectly over the whole polymer length, except for a very small deviation at the ends. This behavior is expected since the analytical calculation is performed in the weakly-bending rod limit, which gives the correct result for unconfined stiff polymers up to first order in ɛ, as shown in the discussion leading to eq. (3.36). Therefore the small deviation are only due to the limits of the weakly-bending rod approximation (terms of order ɛ 2 ) and not due to boundary effects. Finite-size effects have only a very small influence on the results obtained for the tangent-tangent correlation. The overall shape is not affected by increasing the number of segments N. Only small global shifts of the whole correlation function can be observed. This has only an effect on integral quantities as the end-to-end distance, which will be discussed in the following section. After these preliminary investigations towards the validity of the dependence of the observables on the scaling parameter c/ɛ, we now turn to the systematic investigation using a scaling plot. 5.2.3 Scaling plot We have already discussed the influence of confinement on the undulations and the tangent-tangent correlation. Now we will turn to quantities, which are of particular interest in experiments, where it is easier to measure the end-to-end distance and related quantities than the very small undulation or a correlation function of the tangents. We want to present the data in a compact form, which works out the scaling properties expected by our investigation up to this point. In a suitable plot we expect a data collapse. Looking back at eqs. (3.12) and (3.38) we see that plotting the mean-square end-to-end distance 〈R 2 〉 versus c/ɛ is expected to show the data collapse. But before turning to the plot we first give an overview over the various parameter regimes and than try to find out in which range the scaling is expected to start, since it is not universal for all parameters as noted in the preceding section.
Page 1 and 2:
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
Page 11 and 12:
(a) λ-DNA (b) T2-DNA Figure 1.2: A
Page 13 and 14:
Chapter 2 Polymer models This chapt
Page 15 and 16:
2.2. IDEAL PHANTOM CHAIN 7 configur
Page 17 and 18:
2.3. SEMI-FLEXIBLE POLYMERS 9 In th
Page 19 and 20: 2.3. SEMI-FLEXIBLE POLYMERS 11 to i
Page 21 and 22: 2.3. SEMI-FLEXIBLE POLYMERS 13 for
Page 23 and 24: 2.5. REAL CHAINS 15 no overhangs al
Page 25 and 26: 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: 5.2. THE HARMONIC POTENTIAL 61 5.2.
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 and 80: 5.2. THE HARMONIC POTENTIAL 71 p(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
show all

Polymers in Confined Geometry.pdf

Create successful ePaper yourself

Delete template?

Save as template?