Polymers in Confined Geometry.pdf

More documents

Recommendations

Info

82 APPENDIX B. CALCULATION FOR THE CONFINED POLYMER (−1 + i)l −1 d (−1 − i)l −1 d Im k (1 + i)l −1 d (1 − i)l −1 d path of integration for s > s ′ Re k path of integration for s < s ′ Figure B.1: Complex plane of the function ˜ f(k) with indicated poles. Inserting the residues, we obtain for s > s ′ 〈x(s)x(s ′ )〉 = kBT l3 ⎧ ⎫ s − s′ s − s′ ⎪⎨ exp i exp −i ⎪⎬ d ld ld s − s′ + exp − 8κ ⎪⎩ 1 + i 1 − i ⎪⎭ ld = kBT l3 d 4 √ 2κ exp ′ s − s′ s − s − cos − ld ld π (B.4) 4 which—by additionally closing the contour for s < s ′ in the lower half plane— results in 〈x(s)x(s ′ )〉 = kBT l3 d 4 √ 2κ exp − |s − s′ ′ | |s − s | cos − ld ld π . (B.5) 4 This is the result in eq. (3.26) used for further calculations. B.2 End-to-end distance correlation Here we present the details of the calculation for the mean-square end-to-end distance 〈R2 (L)〉. We start by calculating the distance between a point r(s) on the contour and the origin of the polymer r(0), given by the integral R(s) = r(s) − r(0) = s 0 dτ t(τ). (B.6)
B.2. END-TO-END DISTANCE CORRELATION 83 s ′ τ ′ £¤£¤£¤£¤£¤£ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ £¤£¤£¤£¤£¤£ £¤£¤£¤£¤£¤£ £¤£¤£¤£¤£¤£ ¥¤¥¤¥¤¥¤¥¤¥ ¥¤¥¤¥¤¥¤¥¤¥ ¥¤¥¤¥¤¥¤¥¤¥ ¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦ §¤§¤§¤§¤§¤§¤§¤§ ¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦ §¤§¤§¤§¤§¤§¤§¤§ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ 2 ¥¤¥¤¥¤¥¤¥¤¥ £¤£¤£¤£¤£¤£ 1 ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ £¤£¤£¤£¤£¤£ £¤£¤£¤£¤£¤£ £¤£¤£¤£¤£¤£ ¥¤¥¤¥¤¥¤¥¤¥ ¥¤¥¤¥¤¥¤¥¤¥ ¥¤¥¤¥¤¥¤¥¤¥ ¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦ §¤§¤§¤§¤§¤§¤§¤§ ¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦ §¤§¤§¤§¤§¤§¤§¤§ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¥¤¥¤¥¤¥¤¥¤¥ £¤£¤£¤£¤£¤£ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢ £¤£¤£¤£¤£¤£ £¤£¤£¤£¤£¤£ £¤£¤£¤£¤£¤£ ¥¤¥¤¥¤¥¤¥¤¥ ¥¤¥¤¥¤¥¤¥¤¥ ¥¤¥¤¥¤¥¤¥¤¥ ¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦ §¤§¤§¤§¤§¤§¤§¤§ ¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦ §¤§¤§¤§¤§¤§¤§¤§ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¥¤¥¤¥¤¥¤¥¤¥ £¤£¤£¤£¤£¤£ §¤§¤§¤§¤§¤§¤§¤§ ¦¤¦¤¦¤¦¤¦¤¦¤¦¤¦ Figure B.2: Integration ranges for the integral eq. (B.8). Using this, the end-to-end distance correlation function can be calculated by an integral over tangent-tangent correlation function, eq. (3.34), 〈R(s) · R(s ′ s )〉 = dτ = = = s 0 s 0 0 dτ dτ 1 − ld 2 lp 3 s ′ dτ 0 ′ t(τ) · t(τ ′ ) s ′ dτ 0 ′ 〈t(τ) · t(τ ′ )〉 s ′ dτ 0 ′ 1 − ld √2 − e 2 lp |τ−τ′ | ′ |τ − τ | ld sin − ld π + 1 4 ss ′ − ld s s ′ √2 dτ dτ lp 0 0 ′ K(|τ − τ ′ |). (B.7) The integral kernel has been abbreviated to K(x) = e −x/ld sin (x/ld − π/4). The integral over the absolute value in the last line has to be split. For s ≥ s ′ the integration range is shown in figure B.2 and is therefore calculated as s 0 dτ 1 s ′ dτ 0 ′ K(|τ − τ ′ s ′ τ |) = dτ dτ 0 0 ′ K(τ − τ ′ s ′ ) + s 0 + s ′ s ′ dτ dτ 0 ′ K(τ − τ ′ ) s τ 2 dτ s ′ dτ τ ′ K(τ ′ − τ) , (B.8) 3 where the numbers denote the corresponding areas in the figure. As an example
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 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: Appendix B Calculation for the conf
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?