Polymers in Confined Geometry.pdf

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84 APPENDIX B. CALCULATION FOR THE CONFINED POLYMER the first integral is calculated here, the others are evaluated in the same way, s s ′ s (τ−s ′ )/ld dτ dτ dx e −x sin x − π 4 s ′ 0 dτ ′ K(τ − τ ′ ) = −ld s ′ s = ld √2 = l2 d √2 = l2 d 2 dτ e τ/ld s ′ (s−s ′ )/ld τ−s′ − ld sin 0 1 √2 − e s−s′ ld sin s − l + e d sin s ld τ − s′ ld dy e −y sin y − τ − l − e d sin τ ld s ′ /ld s/ld ′ s − s + ld π 4 ′ s − π s′ − l − e d sin 4 ld dz e −z sin z − π . (B.9) 4 The trigonometric relations used can be found in [1]. Calculating the remaining integrals and combining the results gives 〈R(s) · R(s ′ )〉 = 1 − ld ss 2 lp ′ + l3 d 2 √ e 2 lp − |s−s′ | ′ |s − s | ld sin + ld π 4 s − s l − e d sin + ld π s′ ′ − s l − e d sin + 4 ld π + 4 1 (B.10) √ 2 valid for all s, s ′ . From this result the mean-square end-to-end distance is obtained in the lines following eq. (3.37). B.3 Parallel end-to-end distance correlation The mean-square end-to-end distance is calculated in detail, by inserting the derivatives of the average undulations eqs. (3.30) and (3.31) in eq. (3.40) ˙R||(s) ˙ R||(s ′ ) = d2 ds ds ′ R||(s)R||(s ′ ) = 1 − ld + 2 lp l2 d 16 l2 p + l2 d 8 l2 e p −2 |s−s′ | ld sin 2 ′ |s − s | − ld π 4 =〈 ˙r ||(s) ˙r ||(s ′ )〉 . (B.11) An integration for s ≥ s ′ yields the desired result for the projected length correlation R||(s)R||(s ′ ) s s ′ = dτ dτ 0 0 ′ R||(s) ˙ ˙ R||(s ′ ) = 1 − ld + 2 lp l2 d 16 l2 ss p ′ + l2 d 8 l2 s s ′ dt dt p 0 0 ′ K(|τ − τ ′ |), (B.12)
B.3. PARALLEL END-TO-END DISTANCE CORRELATION 85 with K(x) = e −2x/ld sin 2 (x/ld − π/4)). The integration over K(|τ − τ ′ |) has to be treated as in eq. (B.8). Evaluating only the first integral explicitly τ 0 dτ ′ K(τ − τ ′ ) = 1 2 τ dτ 0 ′ τ−τ′ −2 l e d 0 and performing the second integration gives s ′ 0 dτ τ 0 dτ ′ K(τ − τ ′ ) = ld 8 1 − cos ld ′ τ − τ 2 − ld π 4 = − ld dx e 4 2τ/ld −x (1 − sin x) = ld 2τ − l 1 + e d −2 + sin 8 2τ + cos 2τ , (B.13) s ′ 2τ − l dτ 1 + e d −2 + sin 0 2τ + cos ld 2τ ld = l2 2s ′ /ld d dτ 16 0 1 − 2e −x + e −x (sin x + cos x) = l2 d 16 2s ′ ld 2s′ − l + e d ld 2 − cos 2s′ ld ld 2s′ − l + e d 2 − cos 2s′ ld − 1 . (B.14) The remaining integrals are treated equivalently. Finally we obtain the following result valid for all s and s ′ R||(s)R||(s ′ ) = 1 − ld ss 2 lp ′ + r||(s)r||(s ′ ) = 1 − ld + 2 lp l2 d 16 l2 ss p ′ + l3 d 32 l2 min(s, s p ′ ) + l4 d 128 l2 − e p −2 |s−s′ | ld 2 − cos 2 |s − s′ | (B.15) ld 2s − l + e d 2 − cos 2s − 1 . This result is used in the lines following eq. (3.41).
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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
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2.5. REAL CHAINS 19 ln R cases intr
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Chapter 3 Polymers in confined geom
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3.2. SCALING RELATIONS 23 using eq.
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3.2. SCALING RELATIONS 25 b a d R |
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3.3. ANALYTICAL RESULTS 27 The mean
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3.3. ANALYTICAL RESULTS 29 From thi
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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 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: B.2. END-TO-END DISTANCE CORRELATIO
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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