Polymers in Confined Geometry.pdf

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22 CHAPTER 3. POLYMERS IN CONFINED GEOMETRY d ld r Figure 3.1: The new scales for confined polymers. The Odijk deflection length ld measures the typical length between successive deflections along the contour. r denotes the distance between deflection along the tube. in confining fluorescently dyed λ-DNA strands into nano-sized channels seen in the figures 1.1. The channels have an aspect ratio between one and two and are of average size 440 nm down to 30 nm. These experiments provide a picture of increasingly stretched strands when the channel becomes smaller, which is seen for relatively long T2-DNA and the shorter λ-DNA in the figures 1.2. The value, which can be read off from the pictures by using appropriate fitting functions, is the apparent end-to-end distance 〈Ra〉, which is the maximal distance between two monomers along the chain. Here we will present first by scaling relations then by analytical approximations on the cylindrical symmetric harmonic potential 1 —modeling the tube— approaches to a solution of the problem of confinement and the identification of the relevant scaling parameters. This is investigated in detail—particularly for the relevant range of parameters for the experimentalists—by simulations in chapter 5. The resulting “scaling plot” in section 5.2.3 can serve as a gauge for converting measured apparent end-to-end distances into real contour lengths L. 3.2 Scaling relations Before giving a detailed analytical calculation, it is instructive to perform a scaling analysis starting from the relevant length and energy scales. Often it is relatively easy to find an intuitive scaling behavior. For an unconfined polymer the stiffness of the filament is characterized by the persistence length lp, and the thermal energies ∼ kBT determine the typical energy scale. Upon confining a polymer to a tube of lateral dimension d there is an additional scale, the Odijk deflection length ld (introduced in [37]). It measures the typical distance of successive collisions of the polymer with the walls of the confining tube. The new parameters d and ld are shown in figure 3.1. A relation between these values can be easily established. With r⊥ ∼ d and s ∼ ld the transverse and longitudinal length scales, respectively, we find kBT ∼ H ∼ (lpkBT ) d2 l 3 d ⇒ l 3 d ∼ d 2 lp , (3.1) 1 The harmonic potential being again and again the “laboratory animal” of the physicist. . .
3.2. SCALING RELATIONS 23 using eq. (2.20) and the Hamiltonian eq. (2.18). This relation may be explained heuristically: The length on which the polymer is stiff along the tube is the persistence length lp and the two perpendicular directions are confined by the diameter d. Hence a volume of roughly d2lp is occupied, which corresponds to the volume l 3 d occupied between two deflections. In the following and in particular for the simulation it is convenient to introduce dimensionless parameters ɛ = L which is a small parameter for stiff polymers, and lp c = L ld the flexibility, (3.2) the collision parameter, (3.3) being a measure for the number of collisions/deflections along the tube. 3.2.1 The flexible regime A flexible polymer is heavily coiled up in bulk solution. The effect of the confining tube on the end-to-end distance along the tube R||—which will be the main concern here—is therefore expected to be quite strong. The polymer conformation will be straightened and stretched due to excluded volume interactions between distant segments along the polymer backbone. Note that for an idealized phantom chain there would be no stretching at all. Ideal chains can be represented by a random walk on all scales (cf. section 2.2) where each of the possible spacial direction is completely independent of the others. Therefore the tube does not have an effect on the end-to-end distance along the tube. Merely the lateral directions are cut off, being independent of the the parallel. By introducing a little bit of stiffness—still in the range of high flexibility— the directions are not completely independent anymore. Then there is a visible influence of the tube which has to result from boundary layer effects. The polymer is stiff on the scale of some segment lengths. Close to the wall the orientation can happen to be towards the wall. Since this direction is forbidden the persistence forces the segment along the tube. This results in an increase of R||. For real self-avoiding chains we expect a strong influence of the confining tube. The polymer cannot be squeezed arbitrarily, due to steric constraints. When the polymer size is of the order of the confining tube Rg ≈ d, there are two simple ways to derive the scaling relation of the end-to-end distance R|| along the tube, as described in [8]. For tubes larger than the polymer size, it behaves as if it were in bulk solution. For smaller tubes the chain effectively stretches out and behaves as if it were stiff 2 . 2 There is still the possibility that loops and kinks occur. This is an interesting effect and is currently under investigation.
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: Chapter 3 Polymers in confined geom
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 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
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Glossary L filament length lp persi
Page 97 and 98:
Bibliography [1] M. Abramowitz and
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BIBLIOGRAPHY 91 [29] P. L’Ecuyer.
Page 101:
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