Polymers in Confined Geometry.pdf

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50 CHAPTER 4. SIMULATION METHODS 4.4.1 Initial part of the simulation Before actually starting the main Monte-Carlo sampling loop first a set of things have to be performed in the initial phase of the program: • As mentioned before the acceptance rate has to be adjusted such that one is sampling the whole configuration space (making the steps sufficiently large) but still having the steps accepted frequently enough (making sufficiently small steps for ∆e not becoming too large). This has to be tuned for each simulation since the rate depends on the parameters of the system. The variable to be tuned here is the maximal rotation angle δα. In our simulations this is done in the first 1000N 2 steps, where this angle is de- or increased to have the acceptance rate as desired. A value of 25-50% seems to be an reasonable range. Fixing the acceptance rate is the first important step in a simulation. • Indicated in step 4 sampling should be done in effective steps—not taking too many samples which do not increase the accuracy of the final averages. We therefore have to know an estimate for the correlation time τR of the observables. Then samples can be taken in steps of this correlation time which assures, by eq. (4.19), to improve the averages. This value of τR is finally used to extract the error estimate by the same formula. Only one correlation time τR will be used as step size. We calculate it, as indicated, for the end-to-end distance since our main interest is in this value or other ‘types’ of end-to-end distances (see section 4.6), which are assumed to show correlations of the same order. As shown in section 4.3.3 (page 45) we need the values of 〈R〉, 〈R 2 〉 and 〈(δR) 2 〉 to obtain τR. This is the second task to be fulfilled in our simulation. Since we do not know the correlation time beforehand—which actually must be determined self-consistently—the mentioned averages are sampled in 1000 independent (uncorrelated) runs by restarting with different random initial configurations. In each run 10N 2 MC-steps are performed, where samples are taken in steps of N (since the correlation time somehow depends on N, the number of segments 7 ). We are interested in the static equilibrium properties of the system, so attention must be paid to the initial transient phase, where the polymer relaxes from the arbitrary initial non-equilibrium configuration to the equilibrium behavior—which is measured by a different (non-equilibrium) correlation time. We observed this relaxation to be very fast. To be on the sure side 10N 2 MC-steps are performed after the initial condition has been chosen, before the sampling is started. 7 Actually the number of possible configurations scales exponentially ∼ A N , assuming each segment to have A possible configurations.
4.5. SIMULATION OF CONFINED WORM-LIKE CHAINS 51 • Now all prerequisites are obtained to start the main simulation. For that purpose a new random initial configuration is chosen and 2000 τR MC-steps are waited for the chain to equilibrate, after that the main sampling loop as described above can be started. A final remark concerning the numerics: Due to a possible accumulation of numerical inaccuracies during the floating point operations on the tangents, it must be remembered to renormalize the tangents from time to time. 4.4.2 Self-avoiding chains A step to more realistic simulation has also been performed by introducing excluded volume effect. This has been done in two ways. The first straightforward way is to put a sphere, of diameter corresponding to the polymer diameter, around the bead positions, which are interacting by e.g. a hard-core potential. The problem here is, that by using very thin polymers as a model, the discretization N must be chosen very large for not having “empty spaces” in between the beads (the model without spacings between the beads is called shish-kebab model; cf. [10]), which would not take self-avoidance rigorously into account. Besides the normal increase of the number of configurations by increasing N the computational time increases additionally ∼ N 2 (for a naive algorithm. For smarter algorithms there is at least a N log N dependence.) due to the number of tests for overlaps. A way to keep the number of required segments smaller is to use tube-like segments of the the size of the polymer diameter as basic segments. Then selfavoidance is always rigorous. To check if two segments intersect in a MC-step, an algorithm is needed which calculates the minimal distance between to line segments. These kind of problems are common in 3D graphics programming (i.e. for computer games). A fast algorithm can be found in [42] which we implemented. In this model the polymer is a space curve with thickness, therefore we called it spaghetti model. It turned out that self-avoidance effects are not observable for the chains simulated. This is due to the relatively small number of segments N used. As described in section 2.5 on page 18, there is a critical number Nc below which, even for flexible polymers, self-avoidance is irrelevant. For stiff/semi-flexible polymers self-avoidance should not have an influence at all, since backbends are not likely. In the future, we are able to perform simulation of longer chains. With these consideration, we therefore have the basic algorithm already at hand. 4.5 Simulation of confined worm-like chains As it is the main subject of this thesis we now turn to the simulation of worm-like chains in confining tube-like environments. The ‘environment’ discussed in most
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: 4.4. SIMULATION OF UNCONFINED WORM-
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: :wq
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