Polymers in Confined Geometry.pdf

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28 CHAPTER 3. POLYMERS IN CONFINED GEOMETRY Choosing the preferred direction of the parametrization along the symmetry axis of the tube, the harmonic potential is just a harmonic term on the undulations. With eq. (2.31) we arrive at the full Hamiltonian H[r⊥(s)] = κ 2 = κ 2 L 0 L 0 ds ¨r⊥(s) 2 + γ 2 L 0 ds r⊥(s) 2 ds ¨x(s) 2 + ¨y(s) 2 + γ 2 L 0 ds x(s) 2 + y(s) 2 , (3.20) where γ is the potential strength and characterizing the degree of confinement. 3.3.1 Averages by Fourier transformation We try to obtain analytical results by introducing the Fourier integrals in the Hamiltonian eq. (3.20) to get rid of the derivatives in the integral. This is an approximation, since the Fourier integral actually means dealing with an infinitely long polymer. It is ignored to introduce reasonable boundary conditions. We use the notation dk r⊥(s) = 2π ˜r⊥(k)e iks , (3.21a) ˜r⊥(k) = ds r⊥(s)e −iks , (3.21b) leading to the Hamiltonian dk 1 4 2 2 H = κk + γ |˜x(k)| + |˜y(k)| 2π 2 . (3.22) Using the equipartition theorem in Fourier space (cf. [26]) the result for the undulation correlation in Fourier space is obtained 〈˜x(k)˜x ∗ (k ′ )〉 = 〈˜y(k)˜y ∗ (k ′ )〉 = 2πδ(k − k ′ ) kBT κk4 . (3.23) + γ The x and y components are equivalent due to the cylindrical symmetry. A characteristic length scale can be identified as ld := 4 κ 1/4 . (3.24) γ Taking the factor 4 into the definition is an arbitrary choice since we are dealing with a scale. This scale is a measure of the confinement and can therefore be chosen as our Odijk deflection length ld. This will be more apparent when looking e.g. at the result in eq. (3.27). The correlation function is then rewritten to 〈˜x(k)˜x ∗ (k ′ )〉 = 2πδ(k − k ′ ) kBT κ(k4 + 4l −4 (3.25) d ).
3.3. ANALYTICAL RESULTS 29 From this equation the correlation function in real-space can be obtained by a Fourier back-transform. The details of the lengthy calculation can be found in appendix B.1. The result is 〈x(s)x(s ′ )〉 = kBT l 3 d 4 √ 2κ exp − |s − s′ | Reducing for s = s ′ to the mean-square undulation ld x 2 = kBT l3 d 8κ ′ |s − s | cos − ld π . (3.26) 4 1 = 8 l 3 d lp , (3.27) by using eq. (2.20). This is independent of the current position on the polymer, which is reassuring since we have assumed that the polymer is infinitely long and there is translational invariance along the symmetry axis of the tube (z-direction). Here the idea of the deflection length can be depicted a bit clearer. As already mentioned in section 3.2 on page 22, l3 d is approximately the volume occupied by segments between two deflections. The mean-square undulation scaling is then given by this volume divided by the length-scale on which the polymer is flexible lp. This gives the expected relation: 〈x2 〉 ∼ l3 d /lp. We want to use the harmonic potential to model—in first order—other types of confining potential, in particular a hard-wall tube model. The problem of comparing the two models is, that a harmonic potential only has an energy scale given by ld through the potential strength γ, but has no intrinsic length scale. In contrast, a hard-walled tube has a length scale given by the tube diameter d, but no energy scale, since the polymer is free, as long it does not hit the tube. Therefore only a heuristic length-scale comparison is possible, which is already indicated by eq. (3.1). From our current calculation it suggests itself to compare the mean-square undulations 〈r2 ⊥ 〉 with the length scale confining these undulations, the tube diameter d. Setting the radius of tube as the confining lateral distance 2 2 d r⊥ ≈ , (3.28) 2 we obtain the relation l 3 d ≈ d 2 lp, (3.29) to be compared to eq. (3.1) (cf. also [21, section 9.1]). In the following the derivatives of eqs. (3.26) and (3.27) along the contour will be needed. Therefore they are presented here: 〈 ˙x(s) ˙x(s ′ )〉 = − ld 2 √ e 2 lp − |s−s′ | ld sin |s − s ′ | ld − π 4 (3.30) ⇒ ˙x 2 = ld . (3.31) 4 lp
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: 3.3. ANALYTICAL RESULTS 27 The mean
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 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:
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