Polymers in Confined Geometry.pdf

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8 CHAPTER 2. POLYMER MODELS chain, which is possible only when it is long enough. The end-to-end distance for this class of polymers scales with a power law in the number of segments N with an exponent ν = 1/2. This model can be used for analyzing scattering experiments. In particular in small angle scattering the mean-square radius of gyration R2 g can be directly measured. It is defined by R 2 g = 1 N N i=1 (Ri − Rcm) 2 ⇐⇒ R 2 g = 1 2N N i=1 N (Ri − Rj) 2 , (2.7) where Rj = j i=1 ti is the position vector of a joint/bead along the chain and Rcm = 1 N N j=1 Rj is the ‘center of mass’. The meaning of R2 g is that of the mean-square distance of the monomers to the center of mass or the mean-square distance between all the beads. It is a general measure for the size of a polymer because 〈R 2 〉 is not useful for e.g branched polymers. It scales in the same way with the number of segments as the end-to-end distance. Upon identifying ti with the spin vector si and an external force F with a magnetic field H, one can take over the results from the statistical mechanics of a paramagnet to calculate the force extension relation. An interesting result is, that a polymer behaves as an entropic spring with a spring constant proportional to kBT (see eq. (2.11)) which reflects the entropy dominated behavior of the system. In comparison to a solid which expands upon heating, the polymer chain stiffens. This allows us to write down the free energy F of the ideal chain j=1 F ∼ kBT R2 , (2.8) Nb2 which is derived from the end-to-end vector distribution, eq. (2.5), in [39]. The free energy increases when the end-to-end distance becomes lager, since the number of allowed configurations and with it the entropy are reduced. For mathematical convenience another, more abstract, model is introduced. The model generalizes the long-scale Gaussian behavior to all scales. For this Gaussian chain the distance between any two points along the chain is assumed to be Gaussian distributed. The probability distribution of the end-to-end vector becomes a path-integral in the limit of small bond length b → 0 but fixed total length L = Nb r(L)=R 3L p(R) = D[r(s)] exp 2 〈R2 〉 r(0)=0 L 0 2 ∂r(s) ds , (2.9) ∂s r(s) denoting the trajectory of the polymer curve. The solution is, of course, eq. (2.5) which is also the solution of a corresponding free particle diffusion equation.
2.3. SEMI-FLEXIBLE POLYMERS 9 In this path-integral representation the weight of a particular polymer configuration is described by an effective Hamiltonian Hgauss[r(s)] = 3kBT L 2 〈R 2 〉 L 0 ds ˙r(s) 2 . (2.10) Here a dot indicates the derivative along the contour. This effective Hamiltonian contains only entropic contributions, which govern the behavior of the FJC (cf. [9]). It can be interpreted as the continuum limit of the energy of a chain of entropic springs with spring constant (cf. [10]) k = 3kBT 〈R2 . (2.11) 〉 The central limit theorem does not provide any hint when the Gaussian distribution is applicable, i.e. it does not tell us how large N must be chosen. It certainly holds only if the total length L of the polymer is large compared to the correlation between the segments, which is often not the case for biopolymers. But still universal features can be extracted by using a different model. This is the topic of the following section. 2.3 Semi-flexible polymers 2.3.1 Kratky-Porod model The basic model for polymers showing stiffness along the backbone chain was introduced by Kratky and Porod in [20]. The stiffness can be included by a geometric hindrance on the orientation of segments which introduces a preferred direction. One of the simplest realizations of a constraint is the freely rotating chain, where the angle between successive segments is fixed, but the segments can still rotate freely around one axis. One of the central quantities of interest is the tangent-tangent correlation function. It is always of exponential form (cf. [39]) φtt(s, s ′ ) = ˆt(s) · ˆt(s ′ ) = exp − |s − s′ | (2.12) for a polymer in bulk solution. Through this formula the persistence length lp is introduced as the main length scale. It depends on the details of the underlying stiffness mechanism. For lp ≪ L the FJC is recovered—which shows that even semi-flexible polymers ‘behave’ Gaussian on large length scales, if L is large. The segments of the freely-rotating chain are essentially uncorrelated on length scales much larger than lp. On the other hand for lp ≫ L the polymer is essentially stiff over the whole length. It then resembles a stiff rod. In the next section 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: 2.2. IDEAL PHANTOM CHAIN 7 configur
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 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
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5.2. THE HARMONIC POTENTIAL 59 ˆt
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5.2. THE HARMONIC POTENTIAL 61 5.2.
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5.2. THE HARMONIC POTENTIAL 63 beha
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5.2. THE HARMONIC POTENTIAL 65 R 2
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5.2. THE HARMONIC POTENTIAL 67 R 2
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5.2. THE HARMONIC POTENTIAL 69 R
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5.2. THE HARMONIC POTENTIAL 71 p(R)
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5.3. THE HARD WALLS 73 〈Ra〉 1 0
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Chapter 6 Conclusion Emerging from
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Appendix A Discrete worm-like chain
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which, after taking all the derivat
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Appendix B Calculation for the conf
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B.2. END-TO-END DISTANCE CORRELATIO
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B.3. PARALLEL END-TO-END DISTANCE C
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Glossary L filament length lp persi
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Bibliography [1] M. Abramowitz and
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BIBLIOGRAPHY 91 [29] P. L’Ecuyer.
Page 101:
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