Polymers in Confined Geometry.pdf

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74 CHAPTER 5. SIMULATION RESULTS geometry. The main features of the behavior of the polymer in the confining ‘tube’ are universal. The relevant ingredient is the rotational symmetry around the tube axis, but the details, continuous symmetry for the circular tube and harmonic potential or fourfold discrete symmetry for the square tube, are irrelevant. The only relevant scale is the deflection length ld! The difference in slope between the data in the harmonic potential scaling plot and the data presented here is only due to the relation converting the relevant “input parameters” d and ld by eq. (3.1). The details in the shape of the scaling function for hard and soft walls are, however, quite different. For soft walls discussed in section 5.2.3 we were able to fit the Odijk scaling result to both the experimental data and the Monte-Carlo results. Here this seems not to be possible. The Odijk curve fits the experimental data but misses the sharp increase of the Monte-Carlo data for hard walls in the limit of strong confinement. This raises the question on the actual form of the interaction potential between the polymer an the confining walls. Using a hard wall potential one assumes that there are steric interactions only. This is not necessarily true since we are dealing with a charged system. DNA is a polyelectrolyte and therefore we can only hope to describe this problem by an effective potential. If such a construction of an effective potential is not possible, an explicit simulation of the charged polymer and the charges in solution must be performed. This is an interesting future project but beyond the scope of this thesis. With the simulation program used to obtain all data here, many interesting properties can be discussed. Here our first task was to have a comparison to the analytical theory to validate the simulation and then to obtain experimentally relevant data presented here. It provides a solid basis for future work.
Chapter 6 Conclusion Emerging from bio- and nanotechnology, the interest in polymers in confining geometries is rapidly growing. The property of tube-like environments of lateral dimension on the nanometer scale forces polymers to extend to a substantial fraction of its contour length, which has direct applications in bioengineering, e.g. for the measurement of contour lengths. The goal in this thesis was to get a deeper understanding of the conformational behavior of polymers in confining environments, in particular tube like structures. Along with the exploration of analytically feasible limits of the polymer conformation, Monte-Carlo simulations were essential to investigate the experimental relevant parameter regimes, in particular for experiments on DNA. With a better understanding of the subtle interplay between chain flexibility and degree of confinement this should lead to the identification of universal scaling properties. Investigations of the scaling properties of confined polymers were pioneered by de Gennes. He considered the combined effect of confinement and self-avoidance on the conformation of flexible polymers. The limit of stiff chains was later explored by Odijk. Further theoretical progress in analytical and numerical studies was mainly on the confinement free energy, not directly relevant to current experimental investigations in nanofluidics. This lack of theoretical analysis as well as recent experimental results of the Austin group at Princeton motivated this thesis. We were able to get an advanced understanding of the behavior of polymers in confining geometries, extending the established scaling arguments to the limits of stiff chains and strong confinement relevant for recent experimental investigations. The main focus, however, was on finding a quantitative description of the conformational behavior of polymers using both analytical treatment if feasible and simulation techniques to fully explore parameter space. This lead to the following specific results. Analytical expressions were derived for the cylindrical symmetric harmonic potential in the weakly-bending rod approximation. By construction this approximation is valid for stiff polymers. But, it turns out to be valid also in the 75
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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
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: 5.3. THE HARD WALLS 73 〈Ra〉 1 0
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: :wq
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