Polymers in Confined Geometry.pdf

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76 CHAPTER 6. CONCLUSION strong confining limit for semi-flexible to flexible polymers. In particular we could derive an explicit expression for the mean-square end-to-end distance 〈R 2 〉. It shows that instead of depending on both the persistence length and the Odijk length (or tube diameter), it depends on a single scaling parameter c/ɛ, the ratio of the confinement strength to the flexibility of the polymer, only. By off-lattice Monte-Carlo simulations we were able to verify this scaling behavior over a broad parameter range. This allowed us to verify the validity of the weakly-bending rod approximation in the strong confinement limit. It is found that polymers (for all flexibilities) stretch out to nearly their full contour length upon confinement to lateral dimensions smaller than their persistence length. These results obtained for the cylindrical harmonic potential were compared to experimental data obtained for λ-DNA confined in channels with a square cross section. The overall trend of the data for the apparent end-to-end distance is well described by the Monte-Carlo data. There are some systematic deviations hinting at additional effects not accounted for by a worm-like chain model. Further preliminary simulation for different geometries with hard-walls were performed as well. The results show a similar behavior as those for the harmonic potential. This implies that the universal behavior of the end-to-end distance is not sensitive to the specific details (e.g. channel shape) of the confining geometry used. As the key features of the confining potentials we have identified the collision parameter characterizing the lateral extension of the channel and, of course, the overall symmetry of the channel. There are also some limits of our investigations, clearly seen in the deviations of the simulation data from the experimental results in the weak to intermediate confinement regime for long semi-flexible polymers. This hints towards additional physical effects not contained in the worm-like chain model, e.g. the polyelectrolyte nature of DNA and resulting self-avoidance effects along the backbone. We were able to rule out that effects of simple excluded volume could explain the deviations. Modeling electrostatic effects leads to the theory of polyelectrolytes, which ask for special sophisticated simulation techniques not easily implemented. The development of an effective algorithm to simulate polyelectrolytes in nanochannels is an interesting task for future projects. In addition, there is room for improving the algorithm by introducing more complex Monte-Carlo moves. This would enable us to explore the behavior of chains with a larger number of segments or even different topologies of the confining environment.
Appendix A Discrete worm-like chain correlations For the continuum limit of the WLC we were able to relate the interaction energy κ and the persistence length lp in eq. (2.20). Now we will try to obtain similar relations for ε and lp for the discrete WLC in two- and three-dimensional embedding space. The Hamiltonian for the discrete WLC is is given by the sum in eq. (2.13). Setting k = βεl2 , the configuration part of the partition sum is calculated by N N−1 Z(k) = dˆti exp k ˆti · ˆti+1 . (A.1) i=1 To solve this in two dimensions we use the inextensibility constraint and introduce the angle between two successive segments as ˆti · ˆti+1 = cos θi. Then the integration variable changes dˆti → dθi Z2D(k) = +π = 2π dθN −π +π −π N−1 i=1 dθi exp k i=1 dθ exp (k cos θ) N−1 i N−1 cos θi . (A.2) Transforming this using [1, (9.6.19)] the integral is just a modified Bessel function of the first kind I0(x) Z2D(k) = (2π) N I0(k) N−1 . (A.3) Introducing local stiffnesses ki instead of a uniform stiffness k the calculation can be repeated to give N−1 βH({ki}, {ˆti}) = − ki ˆti · ˆti+1 ⇒ Z2D({ki}) = (2π) N i=1 77 N−1 i=1 I0(ki). (A.4)
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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.
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: Chapter 6 Conclusion Emerging from
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: :wq
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