Polymers in Confined Geometry.pdf

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54 CHAPTER 4. SIMULATION METHODS For stiff and/or confined polymers this is also a quantity, which can be experimentally measured on fluorescently dyed polymers (e.q. [44, 38]). mean (square) parallel end-to-end distance 2 R|| , R|| : The parallel end-to-end distance is the component of the end-to-end distance along the confining ‘tube’ (here z-axis). So this only makes sense in connection with the simulation in confined geometries. In the unconfined case the simple relation R2 2 || = 〈R 〉 /3 must hold. The parallel end-to-end distance is given by R|| = N i=1 ˆti . (4.34) z This quantity should coincide with the end-to-end distance in the very strong confined limit, due to the orientation along the axis. It is more of theoretical interest, since it can be used to explain some interesting global features (cf. section 5.2.4). apparent end-to-end distances 〈Ra〉, 〈R2 a〉, 2 Ra|| , Ra|| : Another end-to-end distance is important, when experiments of flexible polymers in confined environments are investigated (e.g. [44, 38]). Since on fluorescently dyed polymers, the real ends are difficult to discern from the rest of the chain. Only a blurry picture is seen as depicted in figure 1.2. Therefore the apparent end-to-end distance is introduced, which is the maximal distance between two beads along the chain. The distance between two beads i, j at positions Ri, Rj is defined as dij = Rj − Ri (4.35) so that in the same spirit as before, apparent and the apparent parallel end-to-end distance are defined as mean-square (parallel) radius of gyration R 2 g Ra = max |dij| , (4.36a) i,j Ra|| = max i,j (dij)z. (4.36b) A scaling analysis and comparison to experimental data is provided in section 5.2.3. 2 , Rg|| : The radius of gyration can be calculated directly as given in eq. (2.7). Where the form with the center of mass Rcm is preferred since it only involves order 2N calculations not order N 2 .
4.6. SAMPLED QUANTITIES AND THEIR RELEVANCE 55 This value has importance for experiments on conformations of unconfined polymers. When e.g. a fluorescently dyed polymer shows spots of strong intensity (which are not resolved by the camera taking the picture), one can weighten the configuration with the intensity, leading to the radius of gyration obtained through the fluorescent “mass” (cf. [32]). mean-square undulations 〈x 2 (s)〉, 〈x 2 〉, 〈y 2 〉: This values for the undulation makes sense only in simulations of confined polymers, since it relies on having a finite system in the lateral direction to give finite averages. The sampling is straightforward by the perpendicular distance of the beads from the axis. The function 〈x 2 (s)〉 gives the mean-square undulation for each bead whereas the values 〈x 2 〉 and 〈y 2 〉 are additionally averaged over all beads. These results can also be compared with our results from chapter 3. This will be done in section 5.2.1. tangent-tangent correlation function ˆt(0.5) · ˆt(s + 0.5) : The tangent-tangent correlation function is sampled using a fixed reference point in the middle of the polymer, to reduce boundary effects as much as possible. We want to be able to compare the results with analytical results obtained for different boundary conditions (see section 5.2.2). The function is obtained directly by using the simulated tangents. radial distribution function p(R) = p(|R|): Instead of only sampling an average, sampling a probability distribution needs some additional steps. The range of possible values—here 0 ≤ R ≤ L = 1—has to be divided into bins. In each sampling step the value of R hits one of these bins, where the corresponding counter is increased. Since we do not obtain a whole distribution in one sampling step, we are forced to find a different way for the error calculation of the distribution points. Therefore, during each sampling step of τR MC-steps, 500 samples are taken for the averages to get a rough distribution. This is then used to obtain an averaged distribution with an error estimate. A value of 100 bins gives a smooth looking probability distribution. Only for strongly confined or very stiff polymers the number of bin should be increased around the fully stretched regime. The same sort of distribution function is obtained for R||. Both results are shown in section 5.2.5. In principle these distribution can be compared to experimental results. This work is currently in progress. All sampling is done using the recursive method introduced in section 4.3.4 in order to avoid large arrays of data or in-between storing and averaging to reduce
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: 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 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
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