Polymers in Confined Geometry.pdf

More documents

Recommendations

Info

78 APPENDIX A. DISCRETE WORM-LIKE CHAIN CORRELATIONS The persistence length lp was defined through the spatial tangent-tangent correlation function in eq. (2.12), which is for the discrete model ˆtm · ˆtn = exp −|m − n| l . (A.5) To derive the relation between k and lp the correlation function is calculated using eq. (A.4) 1 N ˆtm · ˆtn = dˆti ˆtm · ˆtn −βH({ki}) e Z({ki}) i=1 {ki=k} 1 N = dˆti ˆtm · ˆtm+1 ˆtm+1 · ˆtm+2 · · · Z({ki}) i=1 · · · ˆtn−1 · ˆtn −βH({ki}) e {ki=k} 1 ∂ = Z({ki}) |m−n| Z({ki}) ∂km · · · ∂kn−1 {ki=k} (A.3) (A.4) 1 = I0(k) N−1 ∂ |m−n| N−1 I0(ki) , (A.6) ∂km · · · ∂kn−1 i=1 {ki=k} where in the second line ˆti · ˆti+2 = ˆti · ˆti+1 ˆti+1 · ˆti+2 has been used, which can be seen by writing the product of tangents as the cosines, using trigonometric theorems and the properties of the averages. For the 3D case below it is shown in [24, p. 377]. Now using [1, (9.6.28)] in the form I ′ 0(k) = I1(k) gives |m−n| I1(k) ˆtm · ˆtn = I0(k) which results in the desired relation in 2D −1 I1(k) 2D: lp = −l ln I0(k) lp ! = (A.5), (A.7) . (A.8) For a given value of lp, k has to be determined numerically, since this formula is not invertible in a closed form. A similar calculation, following the above lines, can be done for three dimension. By substituting dˆti = dφi dcos θi, we obtain for the partition function Z3D({ki}) = (4π) N N−1 i=1 sinh ki , (A.9) ki
which, after taking all the derivatives of the partition sum and solving for the lp, results in 3D: lp = −l ln coth(k) − 1 −1 , k (A.10) for the desired relation. Some parts of the calculations are presented in [11] for the equivalent problem of the Heisenberg model for ferromagnets. This result must be inverted numerically as well. 79
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: Appendix A Discrete worm-like chain
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?