Polymers in Confined Geometry.pdf

Diploma Thesis
Polymers in Confined Geometry
Frederik Wagner
December 31, 2004
Supervisor: Prof. Dr. Erwin Frey
Freie Universität Berlin
Fachbereich Physik
Hahn-Meitner-Institut Berlin
Abteilung Theorie (SF5)

Contact: frederik.wagner@gmx.de

TMTOWTDI—There’s More Than One Way To Do It
—Larry Wall

Thanks to. . .
First of all I have to thank Prof. Dr. Erwin Frey arousing my interest in biological
physics and giving me the opportunity to write my diploma thesis in his groups
at the Hahn-Meitner-Institut Berlin. Also for very interesting discussions—and
some nice chats in between—and the huge amount of help by writing this thesis.
A big thanks to the small Italian. . . please, don’t kick me. . .
That is, a big thanks to Dottore Gianluca Lattanzi for: introducing and helping
me with my first steps in Monte-Carlo simulations, for being allowed to drop in
constantly and going onto his nerves, for encouraging me, for just having fun and
having some good talks. . . Hope to see you again soon 1 . . .
Thanks to Sebastian Fischer as a ‘Leidensgenosse’ and for some interesting
new ideas during our canteen-boycott.
And the rest of SF5 for all the nice discussions and coffee breaks. You know
who you are.
For the collaboration on the experimental side, I have to thank Walter Reisner
of Bob Austins groups at Princeton.
Thanks to Dr. Axel Pelster for many interesting lectures and the support
during my studies at the Freie Universität Berlin.
Finally, I have to thank my parents for so much. . . !
1 “God willing!”
i

Contents
1 Introduction 1
2 Polymer models 5
2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Ideal phantom chain . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Semi-flexible polymers . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.1 Kratky-Porod model . . . . . . . . . . . . . . . . . . . . . 9
2.3.2 Worm-like chain model . . . . . . . . . . . . . . . . . . . . 10
2.4 Stiff polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4.1 Weakly-bending rod approximation . . . . . . . . . . . . . 14
2.5 Real chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Polymers in confined geometry 21
3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Scaling relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.1 The flexible regime . . . . . . . . . . . . . . . . . . . . . . 23
3.2.2 The stiff regime . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Analytical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.1 Averages by Fourier transformation . . . . . . . . . . . . . 28
3.3.2 Averages by Fourier series . . . . . . . . . . . . . . . . . . 34
3.3.3 Radial distribution function . . . . . . . . . . . . . . . . . 36
4 Simulation methods 37
4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 The Monte-Carlo method . . . . . . . . . . . . . . . . . . . . . . 38
4.3 Sampling the canonical ensemble . . . . . . . . . . . . . . . . . . 39
4.3.1 Markov chains . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.3.2 Metropolis algorithm . . . . . . . . . . . . . . . . . . . . . 41
4.3.3 Correlations and error calculation . . . . . . . . . . . . . . 42
4.3.4 Recursive averaging and error calculation . . . . . . . . . . 46
4.3.5 Random number generators . . . . . . . . . . . . . . . . . 47
4.4 Simulation of unconfined worm-like chains . . . . . . . . . . . . . 47
4.4.1 Initial part of the simulation . . . . . . . . . . . . . . . . . 50
iii

iv CONTENTS
4.4.2 Self-avoiding chains . . . . . . . . . . . . . . . . . . . . . . 51
4.5 Simulation of confined worm-like chains . . . . . . . . . . . . . . . 51
4.5.1 Harmonic potential . . . . . . . . . . . . . . . . . . . . . . 52
4.5.2 Hard-wall tubes . . . . . . . . . . . . . . . . . . . . . . . . 53
4.6 Sampled quantities and their relevance . . . . . . . . . . . . . . . 53
5 Simulation results 57
5.1 The unconfined chain . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2 The harmonic potential . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2.1 Undulations . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.2.2 Tangent-tangent correlation . . . . . . . . . . . . . . . . . 61
5.2.3 Scaling plot . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2.4 End-to-end distances . . . . . . . . . . . . . . . . . . . . . 69
5.2.5 Radial distribution function . . . . . . . . . . . . . . . . . 70
5.3 The hard walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6 Conclusion 75
A Discrete worm-like chain correlations 77
B Calculation for the confined polymer 81
B.1 Undulation correlations . . . . . . . . . . . . . . . . . . . . . . . . 81
B.2 End-to-end distance correlation . . . . . . . . . . . . . . . . . . . 82
B.3 Parallel end-to-end distance correlation . . . . . . . . . . . . . . . 84
Glossary 87
Bibliography 89

Chapter 1
Introduction
Polymers are one of the main constituents of cells. The cytoskeleton is formed by
a set of protein filaments (microtubules, actin filaments, intermediate filaments).
It provides mechanic stability and a railway network for intracellular transport.
Genetic information is also stored in a macromolecule, deoxyribonucleic acid
(DNA). As the basic ‘cookbook’ of life it provides the recipe for sequencing all
the other biopolymers.
DNA is not only carrier of genetic information, but its mechanic properties
(such as bending and torsional stiffness) are also essential for how this information
is processed. For example some processes in gene expression are regulated by loop
formation and other conformational changes of DNA.
These few examples are supposed to give a flavor of the manifold role biopolymers
play in cellular processes (cf. [3, 5, 6, 14]).
Biopolymers differ from synthetic polymers in one important aspect. They are
stiff on length scales relevant for the biophysical processes they are involved in.
This stiffness is characterized in terms of the persistence length which measures
the length scale (along the backbone) over which correlations of the tangents
(describing the polymer configuration) decay. For DNA the persistence length is
approximately lp ≈ 50 nm. This is much larger than a typical microscopic length
scale of DNA, e.g. the diameter h ≈ 2 nm:
lp ≫ h semi-flexible polymer.
F-Actin is even stiffer, with lp ≈ 17 nm and h ≈ 7 nm. This unique feature
gives rise to a multitude of interesting phenomena genuinely different from those
found for their synthetic cousins (such as polyethylene), where lp is comparable
to the microscopic scale h (cf. [17, 31]).
Our focus in this thesis will be on biopolymers in confined geometry. This is
motivated by rapid growing interest in observing and manipulating singe polymer
chains in biotechnological applications using micro- and nanofluidic devices.
Using optical tweezers, it is possible to exert small forces on polymers. By
1

2 CHAPTER 1. INTRODUCTION
(a) 100 nm channel width (b) 200 nm channel width
Figure 1.1: Scanning electron micrograph (SEM) picture of nanochannels. These channels
are used in experiments to study the behavior of DNA under increasing confinement. Courtesy
of W. Reisner et al. [38].
real time optical microscopy on fluorescently dyed polymers a direct observation
method of polymer chains has become available.
One important task is the localization of binding sites for restriction and
other DNA binding enzymes. The restriction sites are traditionally determined
by measuring the length of restriction fragments by gel-electrophoresis. Alternatively,
they can be located by using optical mapping of stretched DNA molecules
trapped on a surface. To measure the contour-length of a single molecule using
optical techniques directly, it is necessary to extend the polymer such that
a one-to-one mapping can be established between the spatial position along the
polymer and the position within the genome.
Chip-based devices, that cannot only detect and separate single DNA molecules
by size but are able to sequence at the single molecule level, would have the
potential to revolutionize biology.
Confinement elongation of genomic-length DNA has several advantages over
alternative techniques for extending DNA, such as flow stretching and/or stretching
relying on a tethered molecule. Confinement elongation does not require the
presence of a known external force because a molecule in a nanochannel will remain
stretched in its equilibrium configuration. Second, it allows for a precise
measurement of length at the single molecule level (cf. [44]).
While a number of approaches have been proposed, all of them have in common
the confinement of DNA to nanometer scales, typically 5-200 nm. Confinement
alters the statistical mechanical properties of DNA. A DNA molecule in a
nanochannel (cf. figures 1.1) will extend along the channel axis to a substantial
fraction of its full contour length (cf. figures 1.2). While the study of confined
DNA is interesting from a physical perspective, it is also critical for device design,
potentially leading to new applications of nanoconfinement, for example, the use
of nanochannels to pre-stretch and stabilize DNA before threading through a

(a) λ-DNA (b) T2-DNA
Figure 1.2: Average intensity of fluorescently dyed DNA (persistence length of undyed DNA
lp ≈ 52 nm and contour-length of λ-DNA L ≈ 16.5 µm and T2-DNA L ≈ 55.8 µm) in nanochannels
of decreasing size (aspect ratio near one, width 440 nm down to 30 nm). The DNA stretches
to near full its length upon confinement to sizes around its persistence length. Courtesy of W.
Reisner et al. [38].
nanopore (cf. [38]).
To turn this idea into a useful device a quantitative theoretical understanding
of polymer conformations in confined geometry is needed. Previous models (discussed
in more detail in chapter 2 and section 3.2) were unable to account for the
effect of varying confinement over the entire range of scales used in nano-devices.
Only asymptotic results valid in special limits were available. In this thesis we
try to close this gap.
We will proceed as follows:
• A general introduction to specific models and universal behavior of unconfined
polymers is given in chapter 2. We present a derivation of exactly
known results as well as some scaling arguments. This chapter also serves
as a concise introduction into the field of study and introduces our basic
polymer model, the worm-like chain, used in the subsequent analysis.
• In chapter 3 we begin our discussion of polymer configurations confined to
tubes (represented in terms of a harmonic potential with cylindrical symmetry).
After discussing some scaling arguments we present an analytical
calculation in the limit of strong confinement. This allows us to give explicit
expressions for the correlation function and end-to-end distances.
• Since it is not possible to obtain analytical expressions for all relevant parameter
ranges it becomes necessary to perform Monte-Carlo simulations
in order to arrive at quantitative results. Chapter 4 will give a general introduction
to the basic principles needed for the simulations and a detailed
3

4 CHAPTER 1. INTRODUCTION
description of the methods used for simulating polymer chains.
• The discussion of the simulations performed in the context of this thesis are
finally discussed in chapter 5. The algorithm developed in the preceding
chapter is validated on known results for the unconfined polymer. A general
scaling behavior is worked out by the systematic investigation of the
simulation data and a comparison to the analytical results of chapter 3
is given. Different confining geometries are investigated in comparison to
experimental data.
• Finally a summary and a short outlook is given in our final chapter 6.

Chapter 2
Polymer models
This chapter introduces some of the basic models used in polymer physics. After
a short motivation of why using a minimal model—only keeping the most important
features—is possible, we turn to a detailed description of specific realizations
of these models. We present both heuristic scaling arguments and analytical calculations
in limiting cases. We conclude by a short discussion of when and how
to include more realistic effects.
2.1 Motivation
Condensed matter physics has a long standing tradition of reductionists approaches.
In order to describe the generic properties of a system one tries to
skip as much details as possible and keep only the essential features. How many
details have to be kept for a specific description depends on the length-scale on
which a problem is investigated and is not always obvious to decide.
Polymer physics is a paradigmatic example for such an approach. A biopolymer
consists of a large number of backbone monomers, i.e. molecules (e.g. DNA
base-pairs) normally consisting of several atoms (e.g. carbon, hydrogen). These
atoms again consist of electrons and nuclei build of quarks and so on. Naively
one might think that an exact model should actually start at the lowest possible
level. But this is neither feasible nor necessary since there is usually a separation
of time- and length scales between a microscopic description and the mesoscopic
level we are interested in.
For many applications it is even possible to disregard all atomistic details and
consider each monomer as a structureless segment of a given length l. This is
motivated by the fact that molecular processes within each monomer relax on
a much shorter time scale than the overall polymer conformations. Interactions
between successive segments are modeled such that its essential properties are
kept, e.g. bending and torsional stiffness of the thread. The basic models used in
polymer physics are distinct at this level, for a thorough discussion we refer to
5

6 CHAPTER 2. POLYMER MODELS
i = 0
R
(a) Chain of segments
i = N
s = 0
R
(b) Smooth space curve
Figure 2.1: The two basic polymer models for chains with stiffness.
s = L
one of the standard text books, e.g. [9].
How should one then picture a polymer if no unnecessary details are kept?
This depends on the mathematical description one would like to use for the
theoretical analysis. For computer simulations a chain of segments with some
prescribed interactions between the segments is most appropriate (see fig. 2.1(a)).
For analytical treatment one might want to use a continuous instead of a discrete
model. Here we have to be very careful how to perform such a limit.
Naively on might expect that upon taking the number of segments to infinity
(N → ∞) and the size of the segments to zero (l → 0) one arrives at a continuous
smooth space curve. But, this is not true in general! As we will discuss in more
detail in the following section, the proper continuum limit of a flexible chain
(when the segments are connected by “free hinges”) is not a smooth space curve
but a fractal which is continuous but non differentiable at any point along the
chain. The proper way to describe such a trajectory in space has been introduced
by Wiener in the aftermath of Einsteins discussion of random walks (see e.g. [16]).
It is only if one assigns some internal stiffness to a chain that a typical conformation
of the polymer will look like a smooth space curve as depicted in figure
2.1(b). This will lead us to the model of a worm-like chain in section 2.3.2.
This short introduction already shows that even on this abstract level the
problem is very complex. This thesis will be limited only to the static properties
of polymer systems. The area of dynamic properties reveals other interesting and
complex properties and will be the subject of future studies.
2.2 Ideal phantom chain
The most basic model describing the universal properties of a large class of polymers
is the freely jointed chain (FJC) model. It is a chain consisting of successive
segments ti of length |ti| = l connected by free hinges. As a first step all correlations
between the segments are ignored, which is reasonable only for very flexible
polymers.
One of the main quantities of interest is the end-to-end distance of a chain

2.2. IDEAL PHANTOM CHAIN 7
configuration
R =
N
ti, (2.1)
i=1
with N ≫ 1. We will be mainly interested in the limit of long chains.
The first two moments of the end-to-end vector distribution are obtained
quite readily. The mean of the distribution 〈R〉 must be zero because of spacial
isotropy. The mean square end-to-end distance 〈R 2 〉 is obtained in a straightforward
calculation due to the absence of correlations
R 2 =
N
i=1
N
〈ti · tj〉 =
j=1
N
i=1
N
j=1
δij
t 2 = N t 2 = Nl 2 , (2.2)
where we have used the fact that subsequent segments are uncorrelated. In
summary we have:
〈R〉 = 0, (2.3)
R 2 = Nl 2 . (2.4)
From eq. (2.4) we note that a proper continuum limit of the FJC needs to take
the limits N → ∞, l → 0 such that Nl 2 stays finite.
Once we know these moments we can already write down the full probability
distribution of the end-to-end distance by exploiting the central limit theorem.
It states that the limiting distribution of a random variable, which is the sum of
independent random variables, is a Gaussian 1 with mean 〈R〉 and width 〈R 2 〉
p(R) =
3
2π 〈R2 3/2
exp −
〉
3R2
2 〈R2
. (2.5)
〉
This result brings us back to the random walk since eq. (2.5) may be interpreted
as the Greens function of the diffusion equation with the initial condition p(R, t =
0) = δ(R).
Due to the central limit theorem, universal behavior for a larger class of
flexible polymers is expected. As long as correlation between the segments along
the chain decay on a scale small compared to the total length these correlation
can be subsumed in an effective segment length or Kuhn segment b, which is
defined by the universal relation to hold:
R 2 = b 2 N ⇒ R ∼ bN 1/2 . (2.6)
For the FJC, b = l the Kuhn length is identical to the segment length. For an
actual chain the correlations are renormalized by looking at a larger scale at the
1 Of course the probability distribution function can also be obtained by calculation from
statistical mechanics.

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

10 CHAPTER 2. POLYMER MODELS
we will give an explicit derivation of these results. Both regimes are convenient
for approximations as we already saw for the Gaussian chain and will see in the
section 2.4.1.
The correlation function, eq. (2.12), can be used to calculate the mean-square
end-to-end distance 〈R 2 〉. The details will not be presented here (cf. [15] and
[39]) since we will discuss the worm-like chain model in more detail, which will
be used in the rest of this thesis. Both models give equivalent results.
2.3.2 Worm-like chain model
Unlike the Kratky-Porod model the worm-like chain (WLC) introduces the stiffness
in a discrete model by an interaction energy ε between neighboring bonds
N−1
H ({ti}) = −ε ti · ti+1 with: |ti| = l|ˆti| = l. (2.13)
i=1
This model is identical to a one-dimensional Heisenberg model for ferromagnets,
where the spins ˆsi replace the tangents and the exchange interaction J the interaction
energy: HHeisenberg = −J N−1 i=1 ˆsi · ˆsi+1. In the WLC model non-parallel
orientation of successive segments are penalized by a bending energy.
The discrete model is ideally suited for Monte-Carlo simulations on polymer
systems; see later section 4. In order to be able to compare the discrete model
with the continuum limit, the tangent-tangent correlation function has to be
calculated to give a relation between the interaction energy ε an the persistence
length lp. This is shown in appendix A for two- and three-dimensional embedding
space.
Here we will first focus on an analytical treatment of the WLC model. For
that purpose it is advantageous to take the continuum limit to arrive at a real
‘worm-like’ continuous space curve as in figure 2.2. This continuum description
is a convenient mathematical description on a coarse-grained scale. For the continuum
limit the length of the chain segments must tend to zero, l → 0, as the
number of segments tends to infinity, N → ∞. The interaction energy must tend
R(s) = s
ds t(s)
0
r(s)
t(s)
R(L)
Figure 2.2: The WLC as continuous space curve r(s).

2.3. SEMI-FLEXIBLE POLYMERS 11
to infinity as well, ε → ∞, such that the the energy per segment stays fixed
˜ε = ε
N
The total length L of the polymer must be finite as well:
!
= fixed. (2.14)
L = Nl ! = fixed. (2.15)
Further, of course, derivatives are defined as the continuum limit of a finite difference
∂t(s) ti+1 − ti
= lim
.
∂s l→0 l
(2.16)
Here the arc length s, which replaces the index i, is labeling the position along
the contour of the chain. We use partial derivatives to indicate that, in general,
the tangent vector may also depend on time. But, since we are only concerned
with static properties, we will ignore this in the following.
Let us start off the Hamiltonian in eq. (2.13) which gives after a simple transformation,
where −ti+1 · ti = 1
2 [(ti+1 − ti) 2 − 2l 2 ],
N−1
εl
H[t(s)] = lim
l→0 2
ε,N→∞ i=1
2 ti+1 − ti
l
, (2.17)
l
by dropping irrelevant constants. These would only shift the energy zero in the
continuum limit. The sum finally transforms into an integral N−1 i=1 l → L
0 ds,
such that
H[t(s)] = κ
L 2 ∂t(s)
ds =
2 ∂s
κ
L 2 ∂ r(s)
ds
2 ∂s2 2 . (2.18)
0
The constraint |ti| = l transformed into |t(s)| = 1 2 , which means that the space
curve is locally inextensible 3 . The bending modulus is defined as κ = εl = ˜εNl =
˜εL.
2Arc length parametrization enforces the constraint |t(s)| = 1 since
ds 2 ∂x 2 2
2
∂y ∂z
= + + ds
∂s ∂s ∂s
2 ⇒ |t(s)| 2 = 1
=|t(s)| 2 ŕ ŕ
ŕ
= ŕ
∂r(s) ŕ2
ŕ
ŕ ∂s ŕ
3 When the inextensibility constraint is relaxed, then a potential energy for the stretching
of segments must be introduced, e.g. when the force-extension relation for very large forces
is investigated, the segments will start stretching [33]. Normally the backbone segments are
very rigid in length so that stretching requires considerably more energy than bending, therefore
there is a time-scale separation for the relaxations times, where the stretching modes are relaxing
much faster. That is why normally one can focus on an inextensible chain [2].
0

12 CHAPTER 2. POLYMER MODELS
We have now arrived at a description of a stiff polymer in terms of a continuous
(and inextensible) space curve r(s). For a given conformation the Hamiltonian
penalizes high absolute curvatures ¨r(s). One may also interpret the polymer
configuration as a trajectory of a particle which is ran through with constant
speed. The position on the curve corresponds to the time.
An alternative way to derive eq. (2.18) is by symmetry considerations. Assuming
the polymer can be represented by a space curve, the form of the Hamiltonian
has to reflect all symmetries of the system, i.e. spatial isotropy, homogeneity and
time-reversal (not in static problems). Therefore the second derivative of the
trajectory along the curve squared is the first allowed term. The disadvantage of
this derivation is that it does not explain the microscopic relevance of the bending
modulus κ.
The relation between the bending modulus κ and the persistence length lp is
obtained by calculating the tangent-tangent correlation function 〈t(s) · t(s ′ )〉. As
shown in [40] this is done by exploiting the similarity between the path-integral of
a free particle in quantum mechanics and the conformational part of the partition
function Z represented by the path-integral
Z = D[t(s)] δ(|t(s)| − 1) exp{−βH[t(s)]}, (2.19)
where the inextensibility constraint |t(s)| = 1 introduces an additional complication.
By analogy with quantum mechanics one arrives at a diffusion equation
for the partition sum. Due to the inextensibility constraint this corresponds to
diffusion on the unit sphere. This is similar to the path-integral representation
of the Gaussian chain in eq. (2.9), which was related to the diffusion of a free
particle. Now—in contrast to the purely entropic Hamiltonian eq. (2.10)—the
bending energy is the dominating contribution in the Hamiltonian eq. (2.18).
The relation between the bending modulus and persistence length is obtained
by an normal-mode analysis
lp = βκ or κ = lpkBT , (2.20)
which is valid in three dimensional embedding space4 .
With eq. (2.12) the second moment or mean-square end-to-end distance 〈R2 〉
and radius of gyrations R2
g can be calculated by straightforward integration (all
odd moments are zero because of the isotropy of the problem). The exact result
4 As shown e.g in [19], for arbitrary embedding space dimension d the relation is
lp =
2κ
kBT (d − 1) .
In the same reference the general relations for R 2 and R 4 can be found—calculated by
Green’s-functions—which have been used to test the simulations (cf. section 5.1).

2.3. SEMI-FLEXIBLE POLYMERS 13
for the mean-square end-to-end distance is (cf. [24])
L
2
2
R (L) = ds t(s)
L
0
L
ds
0
′ 〈t(s) · t(s ′ )〉
= ds
0
= 2 l 2
L
p − 1 + e
lp
−L/lp
= L 2
L
fD , (2.21)
where the Debye-function fD(x) = 2
x2 (x − 1 + e−x ) has been used. This function
is shown in figure 5.1 together with simulation results.
There are two interesting limits
2
R
2Llp for L ≫ lp
=
L2 . (2.22)
for L ≪ lp
The first line is the flexible limit in which the Gaussian model is valid. The
relation is identical to the result obtained in eqs. (2.4) and (2.6). For the WLC
the Kuhn segment length has to be set to
lp
b = 2lp. (2.23)
The other limit is that of a stiff rod, where the result is the fully stretched polymer
of length L.
The radius of gyration is calculated with r(s) − r(s ′ ) = s
s ′ dτ t(τ) using the
continuum version of eq. (2.7)
2 1
Rg =
2L2 = l 2
p
=
L
0
fD
2lpL
6
L2 12
L
ds ′
r(s)
′ 2
− r(s )
ds
0
L
− 1 + L
lp
= bL
6
3lp
(2.24)
for L ≫ lp
, (2.25)
for L ≪ lp
and again by the relation b = 2lp for the Kuhn segment the result for a flexible
limit is recovered as well as the result for a stiff rod.
Other interesting results can be calculated exactly: The forth moment 〈R4 〉
in [40], the pair correlation function 〈r(s) · r(s ′ )〉 in [2] and the projection of the
end-to-end vector on the direction of the first tangent
R · ˆt1 as well as the
details for mean-square radius of gyrations R2
g in [15].
Other observations concerning the radial distribution function p(R) are contained
in section 3.3.3.

14 CHAPTER 2. POLYMER MODELS
0
s − r ||(s)
r(s)
s
r⊥(s)
z-axis
L
Figure 2.3: Parametrization in the weakly-bending rod approximation. The symbols are
explained in the text.
2.4 Stiff polymers
Up to now we have dealt with the short list of exactly solvable quantities for the
minimal model, eq. (2.18). The model is called minimal since only one material
parameter, the bending stiffness, is used for the description for a complex molecule
like F-Actin. The existence of torsion along the chain is neglected (i.e. assumed
to be equilibrated) and possible interactions between torsion and bending are not
considered.
Having the minimal model described by the Hamiltonian eq. (2.18) still does
not allow us to solve problems analytically for all stiffnesses, therefore one has to
resort to approximations in the limit of very stiff polymers. In this case instead
of beginning with a flexible chain the stiff rod limit is used as a starting point.
2.4.1 Weakly-bending rod approximation
In a first approximation, we introduce a stiff rod with small fluctuations around
the fully stretched configuration, therefore the name weakly-bending rod approximation
is used. We choose a preferred direction ˆn and parameterize the curve
in the following way (see figure 2.3)
r(s) = r⊥(s), s − r||(s) . (2.26)
Fluctuations perpendicular to ˆn are termed undulations and are parametrized by
r⊥(s) = (x(s), y(s)). The parallel component r||(s) is the stored length describing
how much the length parallel to ˆn is reduced from the arc length s.
The inextensibility constraint |t(s)| = |˙r(s)| = 1 implies a relation
2 ˙r||(s) = ˙r⊥(s) 2 + ˙r||(s) 2 . (2.27)
Now the approximation of small changes in the undulations |˙r⊥(s)| ≪ 1 is introduced,
which allows us to neglect the term ˙r||(s) 2 in the above relation, since it
is already fourth order in the undulation change. The approximation states that
the polymer stays essentially stretched as depicted in figure 2.3.
This leads to the approximative relation
˙r||(s) ≈ 1
2 ˙r⊥(s) 2 . (2.28)

2.5. REAL CHAINS 15
no overhangs allowed
|r⊥(s)| too large
z-axis
Figure 2.4: Overhangs are and large changes in |r⊥(s)| are not allowed
Note that this implies that ˙r|| is non-negative or in other words that overhangs,
as shown in figure 2.4, are not allowed. The derivative
¨r||(s) 2
1 d˙r⊥(s)
≈
2
2 2 =
ds
˙r⊥(s) · ¨r⊥(s) 2 (2.29)
is already anharmonic and can therefore be neglected. Why this term does not
contribute is easily seen in Fourier space. Due to the derivatives this term depends
on the wave-vector as k 6 . Since in stiff polymers high frequency modes are
suppressed by the bending energy, this term can be neglected. The term ¨r||(s) in
the relation ¨r(s) 2 = ¨r⊥(s) 2 + ¨r||(s) 2 can be dropped
¨r(s) ≈ ¨r⊥(s). (2.30)
This leads to the bending energy Hamiltonian in the weakly-bending rod approximation5
Hwb[r⊥(s)] = κ
2
L
0
ds ¨r⊥(s) 2 = κ
2
L
0
ds ¨x(s) 2 + ¨y(s) 2 . (2.31)
Now the inextensibility constraint is implicitly, i.e. automatically, fulfilled. The
path-integral for the partition sum, eq. (2.19), has been simplified, since the
inextensibility constraint in weakly-bending rod approximation leaves only two
independent components.
This approximation is also called Monge-representation to indicate the chosen
parametrization. The single valued contour is parametrized in reference to a
reference line by the undulations 6 .
2.5 Real chains
We have dealt so far with idealized chains represented by segments or a continuous
space curve, allowed to interpenetrate each other. Therefore these models are
5In classical elasticity theory the energy of a weakly-bending rod has exactly the same form
as this Hamiltonian (cf. [23]).
6In differential geometry the Monge-representation is used to parametrize single-valued
planes by a ‘hight’ over a reference plane.

16 CHAPTER 2. POLYMER MODELS
frequently called phantom chains. But in reality polymers are charged objects in
a solution surrounded by charges. Therefore the interaction between the polymer
segments is very complex and depends on the solution. But often one is entitled
to replace these interaction by a simplified hard core interaction. The next step in
the direction of a more realistic model is then to incorporate self-avoidance effects
as a steep short-ranged repulsive potential between the segments (of the order of
the microscopic polymer thickness). The polymer is not allowed to intersect with
itself 7 . For detailed description of models with ‘realistic’ features confer e.g. [39].
The influence of the self-avoidance/excluded volume is visible in particular
for flexible/long (i.e. a large ratio of L/lp) chains since they fold back and forth
freely, therefore crossing themselves very often. As a result the polymer will tend
to be more extended than a phantom chain due to the reduced number of allowed
configurations (entropic penalty) in the strongly collapsed state. In contrast for
stiff polymers the effect is mostly negligible since it is unlikely for them to fold
back onto themselves.
At variance with the description of the ideal phantom chain as a random walk,
the self-avoiding random walk can be used to investigate the properties of these
chains. But we want to focus on the derivation of the basic scaling relation for
the end-to-end distance.
The considerations of the “entropic penalty” can be used to derive a scaling
relation similar to eq. (2.6) for the ideal chain. This argument dates back to P.
Flory and is described in detail in [8] and [39]. It considers a correction to the
free energy of the ideal chain, eq. (2.8), that is due to the excluded volume which,
in turn, depends on the number of monomer-monomer contacts. Let v be the
mean volume occupied by a monomer depending on the segment length b and
the thickness of the chain d. The monomer density φ of a chain of linear size R
which occupies a sphere of about the volume R 3 is
φ ∼ Nv
. (2.32)
R3 The entropy loss for an exclusion is of order kBT for each degree of freedom
(equipartition theorem). Since there are N monomers the free energy correction
is
F ∼ kBT v
N 2
. (2.33)
R3 Additionally using the free energy of the ideal chain, eq. (2.8), one can minimize
the sum of these contributions with respect to the end-to-end distance R. We
obtain the scaling result
R ∼ (b 2 v) 1/5 N 3/5 . (2.34)
7 There exists a so called θ-temperature (depending on the solution) where the segmentsegment
interaction is exactly balanced by effects from the surrounding solution. Then the
description of an ideal chain is valid.

2.5. REAL CHAINS 17
Figure 2.5: Snapshot of fluorescently labeled DNA molecules adhering to a two-dimensional
lipid membrane. Courtesy of J. Rädler and B. Maier [32].
The scaling exponent changes to ν = 3/5 for the number of segments. In comparison
to the ideal chain result of ν = 1/2 the polymers swells. The value for
the real chain is in good agreement with experiments and simulations which show
a scaling exponent ν ≈ 0.558.
For polymers restricted to an embedding space of two-dimension this scaling
exponent is found to be ν = 3/4 (e.g. by investigating a self-avoiding random
walk). In figure 2.5 snapshots of a fluorescently labeled DNA absorbed on a twodimensional
lipid membrane are shown (cf. [32]). With this experimental set up
measurements on the scaling exponent were performed, which showed a value of
ν = 0.79 ± 0.04 in good agreement with the exact result.
As explained in the references this good result is only due to the lucky cancellation
of errors made by naively summing up the scaling form of the free energy.
The number of monomer-monomer contacts as well as the entropic energy are
overestimated. The good result is destroyed if one tries to improve these energy
estimations. This argument should only be used “as it stands”.
Depending on the value of the monomer volume v inserted in eq. (2.34) three
different prefactors can be obtained. Using the the Kuhn segment length b and
a segment thickness/diameter h
v1 ∼ b 3
sphere, (2.35a)
v2 ∼ b 2 h disc, (2.35b)
v3 ∼ bh 2
cylinder. (2.35c)
The first relation assumes that the segments, on average, occupy a sphere of
diameter b, the second assumes a disc of diameter d and thickness h while the
third assumes a cylinder of length b and diameter d. As mentioned in [41] the
right average excluded volume to use for a segment is v2. Intuitively this can be
understood by ruling out the other relations: A cylinder would mean that the
segment is strongly fixed and in average barely moves around its position. In contrast
a sphere would mean that the segment, fixed in the chain, in average covers
all orientations. Therefore the usage of a disc is intuitively a good intermediate
approach.
Despite these facts, the sphere-picture is normally used in the literature as a
general assumption. Inserting in eq. (2.34) we obtain the standard result
R ∼ bN 3/5 . (2.36)

18 CHAPTER 2. POLYMER MODELS
We will see in the following that the different possible volumes vi can give significantly
different results for the relevance of self-avoidance as well as the scaling
in a confining tube. In this context the important parameter is the relation of
segment length to thickness b/h, called aspect-ratio.
Relevance of self-avoidance in static problems
In static problems the self-avoidance does not play a role for all flexible polymers 8 :
• For a system in θ-condition we can use the ideal chain model.
• Whether or not self-avoidance has to be taken into account depends on
the number of binary contacts Nφ between the segments. Only if this
number is large an effect is to be expected. Therefore there exists a critical
number of segments Nc where a cross-over between the description as ideal
phantom chain and as self-avoiding chain is expected. This cross-over will
be investigated in the following.
The scaling relation for the end-to-end distance of a free flexible polymer, eq.
(2.6), is valid for a specific range of values of N. But N must not be too small
since we still need L ≫ lp. Starting from a specific N ≈ Nc self-avoidance starts
playing a role. Using eq. (2.6) a scaling relation for Nc can be derived.
The number of binary contacts is given by (using eq. (2.32))
1/2 v
Nφ ∼ N
b3 !
= B, (2.37)
with an arbitrary constant B > 1. The number of binary contacts should be
large for a significant influence of self-avoidance on the conformation. Solving for
N yields a scaling relation for the critical number of segments:
Nc ∼ b6
. (2.38)
v2 For N > Nc self-avoidance is relevant.
The N-dependence of R can be split in two regimes with a cross-over point
at Nc which is sketched in figure 2.6.
As mentioned above the scaling of Nc depends crucially on the average volume
occupied by a monomer v. The following three results are obtained for the three
8 For the dynamic properties there is an additional effect of dynamic traps. By ergodicity all
configurations are allowed, but some can be very difficult to reach. E.g. when the chain would
have to cross through itself these conformations are close in configuration space, but since the
segments are impenetrable the polymer would have to uncoil and coil up again to make the
conformation change. This difficulty does not occur for static problems!

2.5. REAL CHAINS 19
ln R
cases introduced above
1
2
phantom chain
Nc
3
5
cross-over
self-avoiding chain
ln N
Figure 2.6: Cross-over from ideal-chain to real-chain behavior.
Nc1 ∼ 1, (2.39a)
Nc2 ∼
b 2
, (2.39b)
h
Nc3 ∼
b 4
, (2.39c)
h
except for the case of a sphere Nc1, Nc depends on the aspect-ratio b/h.
For a sphere self-avoidance is always important. But following [41] the dependence
of Nc2 on the aspect-ratio squared is the most reasonable. E.g. for DNA
the aspect-ratio is about 30 so that Nc2 ≈ 500 − 1000. This value is in the range
of the number of segments used in the simulations. We do not know the prefactor
in the scaling relations hence we cannot be sure whether self-avoidance must be
taken into account. We will see in chapter 5, that up to N = 500 segments we
could not detect any influence of self-avoidance on the static properties.

20 CHAPTER 2. POLYMER MODELS

Chapter 3
Polymers in confined geometry
In this chapter we are going to investigate the conformational behavior of polymer
chains in a confining geometry. First a short experimental motivation is given,
after which heuristic scaling relations are derived. This is followed by a more
detailed analytical calculation, starting from the limit of stiff chains in the weaklybending
rod approximation. In particular we investigate the cylindrical harmonic
potential as a generic model for confining environments with similar a symmetry.
3.1 Motivation
The interest in confinement of polymers into nano-sized structures emerges mainly
from biotechnology. As discussed in the introduction, there are various applications
encouraging our work on the problem of confinement.
One direct application is that of confinement elongation. The advantage is
that upon confining the chain stretches already in its equilibrium conformation.
There is no need of a stretching flow or elongating external force, which drives the
polymer out of its static conformation into a stretched non-equilibrium conformation.
Here we can try to understand the mechanisms leading from the unconfined
to the confined static equilibrium conformation.
An application with a large potential impact is the determination of the
contour-length L of a polymer by measuring its equilibrium apparent end-toend
distance in an confining tube of the order of the persistence length or less.
Up to now gel-electrophoresis is the tool for this sort of measurements, but it has
its problems. It is slow—a measurement takes in the order of a day—and it is
relatively imprecise. The advantage of the idea of the confinement measurement
seems therefore a faster way to obtain the data. It is a matter of minutes to
do the actual measurement and it is in principle possible to make the results
arbitrarily accurate, by just taking more pictures of equilibrium configurations.
The basic experiments are performed in nano-sized channels with a near
square geometry in the group of Bob Austin [38, 44]. They have succeeded
21

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.

24 CHAPTER 3. POLYMERS IN CONFINED GEOMETRY
d
blobs
Figure 3.2: ‘Blob’ picture scaling: Inside the blobs of size d the polymer behaves as if it were
free. The self-avoidance is taken into account by treating the blobs as hard-walled, stacking
them linearly.
The derivation of the intermediate scaling regime can be done most illustrative
by the “blob picture” introduced by de Gennes. For a tube of diameter d the
polymer can expand up to this diameter as if it were free, using g < N monomers.
Using the real-chain end-to-end scaling, eq. (2.34), g can be calculated (cf. figure
3.2)
d ∼ (b 2 v) 1/5 g 3/5 ⇒ g ∼ d 5/3 (b 2 v) −1/3 . (3.4)
The key assumption of the de Gennes scaling argument is, that, due to selfavoidance,
the ‘blobs’ behave as hard spheres. Hence N/g-‘blobs’ must be stacked
linearly along the tube:
R|| ∼ N
d ∼ Nv1/3
g
2/3 b
. (3.5)
d
Using the three different monomer volumes vi introduced in section 2.5 we obtain
R||1 ∼ L
b 2/3
, (3.6a)
d
R||2 ∼ L bh
d2 1/3 , (3.6b)
R||3 ∼ L
h 2/3
. (3.6c)
d
Again the relation R||1 is mostly found in the literature, but the disk-like distribution
of monomers R||2 is actually the right one to use for polymers with thickness
h = 0.
Actually R|| should better be called apparent parallel end-to-end distance Ra||,
as explained in section 4.6, because as the figure 3.2 suggests, the scaling is valid
for the maximal parallel extension 3 .
Approximation for strong confinement
When looking at very strong confinement, a point is reached where the tube
diameter d is smaller than the segment length b = L/N. In this case, when
3 Since we are not going to include self-avoidance effects, for the reasons discussed on page
18 and in section 4.4.2, we do not expect to see the scaling derived here. When stretching for
flexible chains is seen, then due to finite-size effects.

3.2. SCALING RELATIONS 25
b
a
d
R ||
Figure 3.3: Minimal extension R || of a chain with stiff segments of size b in a tube of diameter
d.
starting from a totally stretched conformation and not allowing ‘directional flips’
between fore and aft—which are very unlikely for chains with persistence—the
polymer can take a configuration with a minimal end-to-end distance by making
a zig-zag configuration along the tube axis. This is calculated by using the
Pythagoras’ theorem (cf. figure 3.3)
where
Hence:
a = √ b 2 − d 2 = b
min R|| ≈ Na, (3.7)
1 − 1
2
min R|| = √ b 2 − d 2 ≈ L
2 d
+ O
b
1 − 1
2
4
d
. (3.8)
b
2
d
, (3.9)
b
for d ≪ b. This result is true even, when dealing with an ideal phantom chain,
which should not show any influence of a confining environment. This argument
is only due to finite-size effects and particularly interesting in connection with
simulations (cf. section 4.4).
3.2.2 The stiff regime
Now turning to the limit of stiff polymers we can use again a ‘blob’-picture like
argument to arrive at a scaling relation. We know that the end-to-end distance
of a stiff polymer scales as
R
L ∼
⎧
⎨1
for strong confinement d → 0
⎩1
− L
for no confinement d → ∞ and
6lp
L , (3.10)
≪ 1
lp
where the second line is the square root of the series expansion of eq. (2.21).

26 CHAPTER 3. POLYMERS IN CONFINED GEOMETRY
In the spirit of de Gennes’ “blob picture” for flexible chains, we take the
segments between two deflections as a ‘blob’. We assume that the polymer has
a contour length of ld between two deflections separated a distance r along the
tube (cf. 3.1). Inside these ‘blobs’ the segments behave free as in eq. (3.10) with
the substitutions R → r and L → ld, leading to
r
ld
∼ 1 − α ld
. (3.11)
An arbitrary factor α has been introduced, which cannot be calculated in this
scaling argument, but can be used as a fit parameter. Again assuming the ‘blobs’
to be hard-walled due to self-avoidance, leads to a stacking of L/ld pieces
R ∼ L
ld
r ⇒ R
L
lp
∼ 1 − αld
lp
∼ 1 − α ′
d
lp
2/3
, (3.12)
using eq. (3.1), which is also the reason for the use of a second unknown parameter
α ′ . This is our scaling result, which suggests a data collapse for all polymers with
the same ratio of ld/lp or d/lp, respectively.
Staying in the current order of d/lp we can rewrite this result by adding the
higher order terms in a geometric series, obtaining a result which is positive over
the whole range of the parameters and giving a heuristic interpolation formula
R
L ∼
1 + α ′
d
lp
2/3 −1
. (3.13)
By chance investigating the wrong limit of a flexible chain lp → 0—the scaling
argument is actually only valid for the stiff confined chain — we get, coincidently,
the same scaling behavior as derived with the right assumption by de Gennes (cf.
eq. (3.6a))
R|| → L
2/3 lp
. (3.14)
d
This coincidence is artificial and has no real importance, since the resummation
is not well controlled.
Alternatively the scaling can be obtained in a bit sturdier way, using the
asymptotic expansions for a free stiff polymer derived by Yamakawa and Fujii in
[48]. This is valid as long as s ≪ lp (ˆt(0) is assumed to be fixed along the z-axis)
2 2
R (s) = s 1 − s
3lp
R(s) · ˆt(0) 2
= z 2 (s) = s 2
+ O s 2 l −2
p
1 − s
lp
, (3.15)
+ O s 2 l −2
p
. (3.16)

3.3. ANALYTICAL RESULTS 27
The mean-square average deviation from the z-axis (undulation) is thus given by
r⊥(s) 2 = x 2 (s) + y 2 (s) = 2 x 2 (s) = 2
3
s 3
lp
. (3.17)
We can again introduce the length scale ld, now following directly the line of
reasoning of Odijk in [37]. When confining the chain into a cylindrical tube of
diameter d ≪ lp, with its axis along the z-direction, the contour length ld can be
calculated by using eq. (3.17).
As a measure for the distance on the contour between two deflection points,
ld can be defined by assuming that the mean-square average lateral deviation
should be of the same order as the radius of the tube (d/2) 2
r⊥(ld) 2 ∼
2 d
2
⇒ l 3 d ∼ d 2 lp. (3.18)
In this way we reobtained the important relation eq. (3.1). By this, the scaling for
the end-to-end distance in the strongly confined regime can be obtained, stacking
linearly L/ld pieces of the polymer of length 〈z(ld) 2 〉 gives
R
L ∼ 〈z(ld) 2 〉/ld ∼ 1 − α ′
d
lp
2/3
, (3.19)
where α ′ is some unknown constant (cf. eq. (3.12)).
The results obtained here should be valid for all flexibilities as long as the
confinement is very strong. But for the intermediate regime where d ≈ R and
the polymer is semi-flexible lp ≈ L simulations become essential.
3.3 Analytical results
After this preparing scaling analysis, we address the task of analytical approximations.
In the context of analytical solutions one has to think about in what
complexity a problem is feasible to solve. Every reasonable behaving symmetric
potential is expandable in a power series, the first non-trivial term being harmonic.
These quadratic terms often result in Gaussian integrals, which can be
handled analytically. Already the introduction of slightly more complicated term
renders the integrals unsolvable and forces to use stronger approximations.
We are going to confine a polymer in a cylindrical symmetric environment,
hence we introduce a cylindrical symmetric harmonic potential. In the limit of
the weakly-bending rod, which is only justified when the tube is very narrow
(ld ≪ L and/or lp ≫ L, cf. 2.4.1), some calculation are rendered possible since
the inextensibility constraint has not to be taken into account explicitly.

28 CHAPTER 3. POLYMERS IN CONFINED GEOMETRY
Choosing the preferred direction of the parametrization along the symmetry
axis of the tube, the harmonic potential is just a harmonic term on the undulations.
With eq. (2.31) we arrive at the full Hamiltonian
H[r⊥(s)] = κ
2
= κ
2
L
0
L
0
ds ¨r⊥(s) 2 + γ
2
L
0
ds r⊥(s) 2
ds ¨x(s) 2 + ¨y(s) 2 + γ
2
L
0
ds x(s) 2 + y(s) 2 ,
(3.20)
where γ is the potential strength and characterizing the degree of confinement.
3.3.1 Averages by Fourier transformation
We try to obtain analytical results by introducing the Fourier integrals in the
Hamiltonian eq. (3.20) to get rid of the derivatives in the integral. This is an
approximation, since the Fourier integral actually means dealing with an infinitely
long polymer. It is ignored to introduce reasonable boundary conditions.
We use the notation
dk
r⊥(s) =
2π ˜r⊥(k)e iks , (3.21a)
˜r⊥(k) = ds r⊥(s)e −iks , (3.21b)
leading to the Hamiltonian
dk 1 4 2 2
H = κk + γ |˜x(k)| + |˜y(k)|
2π 2
. (3.22)
Using the equipartition theorem in Fourier space (cf. [26]) the result for the
undulation correlation in Fourier space is obtained
〈˜x(k)˜x ∗ (k ′ )〉 = 〈˜y(k)˜y ∗ (k ′ )〉 = 2πδ(k − k ′ ) kBT
κk4 . (3.23)
+ γ
The x and y components are equivalent due to the cylindrical symmetry.
A characteristic length scale can be identified as
ld := 4 κ
1/4 . (3.24)
γ
Taking the factor 4 into the definition is an arbitrary choice since we are dealing
with a scale. This scale is a measure of the confinement and can therefore be
chosen as our Odijk deflection length ld. This will be more apparent when looking
e.g. at the result in eq. (3.27). The correlation function is then rewritten to
〈˜x(k)˜x ∗ (k ′ )〉 = 2πδ(k − k ′ )
kBT
κ(k4 + 4l −4
(3.25)
d ).

3.3. ANALYTICAL RESULTS 29
From this equation the correlation function in real-space can be obtained by a
Fourier back-transform. The details of the lengthy calculation can be found in
appendix B.1. The result is
〈x(s)x(s ′ )〉 = kBT l 3 d
4 √ 2κ exp
− |s − s′ |
Reducing for s = s ′ to the mean-square undulation
ld
x 2 = kBT l3 d
8κ
′ |s − s |
cos −
ld
π
. (3.26)
4
1
=
8
l 3 d
lp
, (3.27)
by using eq. (2.20). This is independent of the current position on the polymer,
which is reassuring since we have assumed that the polymer is infinitely long and
there is translational invariance along the symmetry axis of the tube (z-direction).
Here the idea of the deflection length can be depicted a bit clearer. As already
mentioned in section 3.2 on page 22, l3 d is approximately the volume occupied by
segments between two deflections. The mean-square undulation scaling is then
given by this volume divided by the length-scale on which the polymer is flexible
lp. This gives the expected relation: 〈x2 〉 ∼ l3 d /lp.
We want to use the harmonic potential to model—in first order—other types
of confining potential, in particular a hard-wall tube model. The problem of
comparing the two models is, that a harmonic potential only has an energy scale
given by ld through the potential strength γ, but has no intrinsic length scale.
In contrast, a hard-walled tube has a length scale given by the tube diameter d,
but no energy scale, since the polymer is free, as long it does not hit the tube.
Therefore only a heuristic length-scale comparison is possible, which is already
indicated by eq. (3.1).
From our current calculation it suggests itself to compare the mean-square
undulations 〈r2 ⊥ 〉 with the length scale confining these undulations, the tube
diameter d. Setting the radius of tube as the confining lateral distance
2 2 d
r⊥ ≈ , (3.28)
2
we obtain the relation
l 3 d ≈ d 2 lp, (3.29)
to be compared to eq. (3.1) (cf. also [21, section 9.1]).
In the following the derivatives of eqs. (3.26) and (3.27) along the contour will
be needed. Therefore they are presented here:
〈 ˙x(s) ˙x(s ′ )〉 = − ld
2 √ e
2 lp
− |s−s′ |
ld sin
|s − s ′ |
ld
− π
4
(3.30)
⇒ ˙x 2 = ld
. (3.31)
4 lp

30 CHAPTER 3. POLYMERS IN CONFINED GEOMETRY
〈t(0.5) · t(s/L + 0.5)〉
1
0.9
0.8
0.7
0.6
0.5
-0.4
-0.2
exact free
ld/L = 10
ld/L = 1
ld/L = 0.5
ld/L = 0.1
ld/L = 0.05
Figure 3.4: Tangent-tangent correlation function 〈t(0.5) · t(s/L + 0.5)〉 for different confining
potentials measured in terms of ld/L. For the persistence length we have chosen lp/L = 1.
Tangent-tangent correlation function
From eq. (3.26) the tangent-tangent correlation is easily calculated. The representation
of the tangents in the weakly-bending-rod approximation is
t(s) ≈ ˙x(s), ˙y(s), 1 − 1
2 2
˙x(s) + ˙y(s) 2
. (3.32)
Inserting and expanding gives
0
s/L
〈t(s) · t(s ′ )〉 = 1 + 〈 ˙x(s) ˙x(s ′ )〉 + 〈 ˙y(s) ˙y(s ′ )〉
2 ′ 2 2 ′ 2
˙x(s) + ˙x(s ) + ˙y(s) + ˙y(s )
− 1
2
− 1 2 ′ 2 2 ′ 2 2 ′ 2 2 ′ 2
˙x(s) ˙x(s ) + ˙y(s) ˙x(s ) + ˙y(s) ˙x(s ) + ˙y(s) ˙y(s )
4
= 1 + 2 〈 ˙x(s) ˙x(s ′ )〉 − 2 ˙x 2 . (3.33)
Only terms to second order are kept, the statistical independence of the x- and
y-direction, the translational invariance of eq. (3.27) and finally the rotational
symmetry around the z-axis have been used.
Inserting the results eqs. (3.30) and (3.31) gives
〈t(s) · t(s ′ )〉 = 1 − ld
√2
exp −
2 lp
|s − s′ ′ | |s − s |
sin −
ld
ld
π
+ 1
4
(3.34)
This result is sketched in figure 3.4 for lp/L = 1 and different degrees of
confinement. Obviously, for increasing confinement, the correlations along the
whole polymer do not decay exponentially anymore (as in eq. (2.12)), but reach
0.2
0.4
.

3.3. ANALYTICAL RESULTS 31
a constant non-zero smaller one, that can be calculated, for a distance along the
polymer large compared to the deflection length |s − s ′ | ≫ ld, to
〈t(s) · t(s ′ )〉 = 1 − ld
. (3.35)
2lp
This is due to the restriction of possible orientations. For strong confinement the
polymer is only allowed to orient along the tube, which gives rise to constant
non-zero values for the tangent-tangent correlation. But there is still a short
range which is of the order ld where the correlation decay as in the free case
(exponentially) before ‘noticing’ the influence of the tube and becoming constant.
This is best seen for ld = 0.1. The short dip below the constant value is due to
the deflection (this is better seen in figure 5.4(d) on page 62).
The exact exponential line from eq. (2.12) is shown in figure 3.4 as well and
is quite different from the near free result for ld = 10. This is due to the fact,
that our result (3.34) is consistent with the exact result only up to linear order
in L/lp, which is seen upon expanding for the free limit ld → ∞
〈t(s) · t(s ′ )〉 = 1 − |s − s′ |
lp
≈ exp − |s − s′ |
lp
+ O(l −1
d )
,
(3.36)
where in the last line the expansion is resummed to give, in the same order, the
expansion of the exponential. This is the same result known for the a semi-flexible
polymer in bulk solution, eq. (2.12), up to first order L/lp, i.e. for the stiff limit.
This shows that our condition that L ≪ lp is necessary, but if lp ≫ ld we obtain
good results too, looking at eq. (3.34). For further discussion, see the section
5.2.2 about the simulation results.
Mean-square end-to-end distance
The mean-square end-to-end distance 〈R 2 (L)〉 can be calculated by a simple but
tedious integration over the tangent-tangent correlation function. The details can
be found in appendix B.2. There we obtain the result for the correlation function
of the end-to-end vectors
〈R(s) · R(s ′ )〉 =
1 − ld
ss
2 lp
′ + l3 d
2 √ 2 lp
s
− l − e d sin
s
ld
+ π
4
e − |s−s′
|
′ |s − s |
ld sin
ld
s′ ′
− s l − e d sin +
ld
π
4
+ π
4
+ 1
√ 2
,
(3.37)
which reduces for s = s ′ = L to the the mean-square end-to-end distance
2
R (L) = 1 − ld
L
2 lp
2 + l3
d
1 −
2 lp
√
L
− L
l 2 e d sin +
ld
π
. (3.38)
4

32 CHAPTER 3. POLYMERS IN CONFINED GEOMETRY
R 2 (L) /L 2
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.01
0.1
1
L/ld
10
lp/L = 0.1
lp/L = 0.5
lp/L = 1
lp/L = 5
lp/L = 10
Figure 3.5: Mean-square end-to-end distance R 2 versus the confinement strength L/ld.
Shown are curves for different stiffnesses, from the stiff lp/L = 0.1 to the flexible lp/L = 10
regime.
This depends explicitly on two ld/lp and ld/L of the three possible parameters
L, lp and ld (or combinations thereof). But we expect the main dependence only
on ld/lp due to our scaling analysis in section 3.2.2. It can be seen that this is
actually true also for result above, since it is valid only where the weakly-bending
rod approximation is applicable, which is for lp ≫ L and/or lp ≫ ld. Therefore
eq. (3.38) should only be considered in this range 4 .
The limit of large deflection length ld → ∞ corresponds to a free polymer.
Hence we should recover the results for a polymer in bulk solution. The series
expansion in the stiff regime, up to first order L/lp, of the above result, eq. (3.38),
is identical to that of the bulk solution, eq. (2.21).
Parallel mean-square end-to-end distance
100
1000
2 2
R (L) = L 1 − L
, (3.39)
3 lp
In a slightly different way a similar expressions for the end-to-end distance and
correlation function projected onto the symmetry axis can be derived. This will
be denoted by R2
|| . In our notation this is just the z-component of the end-toend
distance.
Starting with the relation eq. (2.27) and the above result for the undulation
correlation, eq. (3.26), we can use the Wick theorem to arrive at a relation for
4 E.g. for lp/L = 0.5 the part depending explicitly on L/ld changes the result for the meansquare
end-to-end distance only by 14% for L/ld = 1 and 0.03% for L/ld = 10.

3.3. ANALYTICAL RESULTS 33
the projected length correlation
˙R||(s) ˙ R||(s ′ ) = 1 − ˙r||(s) 1 − ˙r||(s ′ )
= 1 − 2 ˙x 2 + 1
2 ′ 2
˙x(s) ˙x(s )
4
+ ˙y(s) 2 ˙y(s ′ ) 2
+ ˙x(s) 2 ˙y(s ′ ) 2 + ˙y(s) 2 ˙x(s ′
)
2
= 1 − 2 ˙x 2 + ˙x 2 2 + 〈 ˙x(s) ˙x(s ′ )〉 2 . (3.40)
It has been used that the x- and y-directions are uncorrelated an that the system
is cylindrical symmetric.
Now again a tedious integration—the details are found in appendix B.3—
results in
R||(s)R||(s ′ )
= 1 − ld
ss
2 lp
′ + r||(s)r||(s ′ )
= 1 − ld
+
2 lp
l2 d
16 l2
ss
p
′ + l3 d
32 l2 min(s, s
p
′ )
2 − cos 2 |s − s′
|
+ l4 d
128 l 2 p
2s
− l + e d
− e −2 |s−s′ |
ld
2 − cos 2s
ld
ld
2s′
− l + e d 2 − cos 2s′
ld
(3.41)
− 1 .
As indicated in the first line, the first two summands are missing if the stored
length correlation function r||(s)r||(s ′ ) is examined.
For s = s ′ = L this reduces to mean-square parallel end-to-end distance
R||(L) 2 =
1 − ld
+
2 lp
l2 d
16 l2 p
+ l3 d
32 l2 L +
p
l4 d
64 l2 p
L 2
e −2L/ld
2 − cos 2L
− 1 .
ld
(3.42)
The problem with these results is, that they diverge in the limit of no confinement,
that is for ld → ∞
r||(s)r||(s ′ ) → ∞, (3.43)
which is unfortunately not a sane result.
An approximate derivation of the mean parallel end-to-end distance
A direct way to calculate the mean projected length R||(L) is the integral
L =
R||(L)
0
ds 1 + ˙r⊥(s) 2 , (3.44)

34 CHAPTER 3. POLYMERS IN CONFINED GEOMETRY
measuring the contour length of a space curve in the usual way (cf. [18]). Expanding
the square root in the weakly-bending rod limit |˙r⊥| ≪ 1
L ≈ R||(L) + 1
2
= R||(L) + 1
2
R||(L)
0
R||(L)
0
ds ˙r⊥(s) 2
ds ˙x(s) 2 + ˙y(s) 2 , (3.45)
and taking the ensemble average, where we approximate the average over the
integral by the integral of the average. Using the symmetry of the problem and
eq. (3.27) gives
The result finally is
R||(L) =
L ≈ R||(L) + 1
2
=
R||(L)
L
1 + ld
4 lp
〈R ||(L)〉
0
1 + ld
4 lp
ds 2 ˙x 2
. (3.46)
≈ L 1 − ld
+
4 lp
l2 d
16 l2 p
+ · · · , (3.47)
with an approximation for the stiff/confined limit ld/lp ≪ 1. This result is up to
first order identical to the square-root of eq. (3.42).
3.3.2 Averages by Fourier series
Instead of using Fourier integrals, which supposes the approximation that the
polymer is infinitely long, Fourier series can be used. This means fixing specific
boundary condition on the polymer5 . We are going to use “tracked ends” x(0) =
y(0) = x(L) = y(L) = 0, where the end points are fixed on but free to move
along the tube-axis, since the eigenfunctions for free ends render the calculations
to tedious. The “tracked ends” lead to a normal Fourier-sinus transformation,
with the notation
r⊥(s) = 2
hk sin(ks), (3.48a)
L
with eigenvalues k = n π
L
k
L
hk =
0
(n ∈ N).
ds r⊥(s) sin(ks), (3.48b)
5 Different boundary condition in this problem means, using a different set of eigenfunctions
of the operator ∇ 4 , which are not just sines, but combination of the sines, cosines and their
hyperbolic counterparts. The eigenfunctions are listed in [46].

3.3. ANALYTICAL RESULTS 35
x 2 (s/L)
1.5e-05
1e-05
5e-06
0
0
0.2
Integral ansatz
Series ansatz
0.4
Figure 3.6: Comparison of the mean-square undulations along the polymer x 2 (s/L) versus
the contour length s/L obtained by Fourier integral and Fourier series (‘tracked ends’)
calculation.
Inserting this transformation into the Hamiltonian eq. (3.20) and using the
equipartition theorem we arrives at (in comparison to eq. (3.25))
s/L
L
〈xkxk ′〉 = 〈ykyk ′〉 = δkk ′
2
lp
0.6
0.8
1
.
k4 −4
(3.49)
+ 4ld The back-transform—which is very tedious, and is not presented here—can
be calculated exactly. We obtain
〈x(s)x(s ′ )〉 = L2
2
ɛ
c 3
1
sinh
c cos c − cosh c sin c
cos 2c − cosh 2c
· cosh(c − sc) sin(c − sc) sinh s ′ c cos s ′ c
+ sinh(c − sc) cos(c − sc) cosh s ′ c sin s ′ c
+ sinh c cos c + cosh c sin c
· cosh(c − sc) sin(c − sc) cosh s ′ c sin s ′ c
− sinh(c − sc) cos(c − sc) sinh s ′ c cos s ′ c
1
,
(3.50)
which is valid for s > s ′ , using besides the dimensionless parameters ɛ and c,
the abbreviations sc = s/ld and s ′ c = s ′ /ld. For further calculations this is a
very inconvenient expression. To obtain more results an algebraic program like
mathematica should be used.
For s = s ′ the mean-square undulations are obtained, which is compared to
eq. (3.27) in figure 3.6. The results for the inner part of the polymer are identical.
Due to the boundary conditions, the result obtained by Fourier series has to
go to zero at the boundaries, in contrast to the Fourier integral solution, which
stays constant. In the range of validity of the weakly-bending rod approximation

36 CHAPTER 3. POLYMERS IN CONFINED GEOMETRY
the differences between the approaches will not effect the results of calculations,
besides minor boundary effects. Therefore the series ansatz is not further investigated.
The calculation become very tedious.
3.3.3 Radial distribution function
The radial distribution function (RDF) p(R) is another interesting quantity. It is
the probability distribution of the end-to-end distance vector R that incorporates
more information than only the moments. It is defined as
p(R) = 〈δ(R − |R|)〉 . (3.51)
All configurations with the specified end-to-end distance |R| = R are counted.
For the free polymer this quantity can be calculated—or at least approximated—
over the whole stiffness range. The result for the Gaussian chain was shown in
eq. (2.5). For other regions confer [7] and [47].
The calculations to obtain the RDF are quite involved for the free polymer.
For a confined polymer this becomes even more difficult. The advantage of the
polymer in bulk solution is, that the problem can be formulated by exclusively
using tangent-vectors describing the segments, independent of the embedding
space. By introducing a confining environment a non-local interaction term has
to be introduced, since the problem is now dependent on the actual positions of
the chain joints, which are given as integrals over the tangents. In Fourier space
problems arise due to higher order modes.
Therefore we do without an analytical approach and resort to simulation
results discussed in section 5.2.5.

Chapter 4
Simulation methods
This chapter starts with a review of the Monte-Carlo method in general and how
to calculate partition sums in the canonical ensemble. The Metropolis algorithm
is introduced as a method to implement the idea of Markov chains in computer
programs. In this context the problem of correlated sampling and the importance
of the quality of random number generators are discussed.
Then we turn to the discussion of these methods in the context of the wormlike
chain model in bulk solution and in confined environments. Finally we give
a list of the important observables which are obtained in simulations and where
they are used.
4.1 Motivation
As in most cases in physics there is no way of exploiting all interesting parameter
values of a given problem analytically. So one has to resort to simulation
techniques [4, 13, 22]. We saw in the chapters 2 and 3 that analytical results are
limited to either the very stiff and flexible polymer, or the strongly confined and
totally free environment.
For problems from biotechnology it is very important to obtain quantitative
results particularly in the intermediate regime—where up to now only qualitative
scaling arguments were available (cf. section 3.2). The problem of flexible DNA
confined to nano-channels with lateral dimensions in the order of the persistence
length, as introduced in section 3.1, is just in this complicated parameter regime.
Also the properties of semi-flexible F-Actin filaments, with a contour-length of
about the persistence length, are in this parameter range.
We therefore have to rely on simulation techniques. The Monte-Carlo simulation
is the method of choice, since it allows to effectively sample the important
conformations of the polymers. The problem of other methods, e.g. molecular dynamics
(MD) simulations, is the large difference in time-scales. Typical changes
on the monomer level are much faster than conformational changes of the whole
37

38 CHAPTER 4. SIMULATION METHODS
polymer with N ≫ 1 monomers. Since in MD simulation the smallest time-scale
of the model must be simulated, it cannot be effectively applied to our problem.
We therefore concentrate on the Monte-Carlo simulation in the following sections.
For a very good introduction also refer to [35].
4.2 The Monte-Carlo method
In statistical physics ensemble averages of fluctuating quantities are of special
interest. To obtain these averages, integrals of the following form have to be
evaluated (schematically for one dimension)
〈A〉 p =
+∞
−∞
dx A(x)p(x) . (4.1)
φ(x)
This example could be an average of an observable A(x) (“score function”)
weighted by the probability distribution p(x), where the integral corresponds—in
the language of statistical mechanics—to a ‘sum’ over all states. But the method
explained in the following is applicable to every integral presentable in this form.
When eq. (4.1) is not analytically solvable then one has to resort to numerical
methods. The direct way would be to divide the integration range—which can
be of any dimension—into n equidistant intervals and evaluate the sum over
{φ(xi)} on the discrete set of points {xi} (Riemann integral). In this way the
error of the approximative result in this deterministic algorithm reduces with
n −1/d depending on the dimensionality d. Therefore this method becomes very
ineffective for higher dimensions d. But in particular high-dimensional integrals
is what we are calculating in statistical mechanical problems, therefore we need
a smart way to increase the efficiency of the numerical integration.
The basic idea of Monte-Carlo 1 evaluation of an integral is to choose the n
discretization points/sample points {xi} randomly out of the probability distribution
p(x) such that the sum has to be performed only over the observable
quantity at the so generated points {A(xi)}. This algorithm has the advantage
of a statistical error proportional to n −1/2 , independent of the dimension which
outperforms the deterministic calculation for n ≥ 3. But both algorithms have
the problem, that they are very inefficient, when the problem under investigation
has a large inherent variance, due to the fact that the computational time grows
proportional to n and the variance is inversely proportional to √ n.
This problem is seen in the following way: If the probability distribution of
the random numbers is such, that the important region of the observable is poorly
sampled and therefore the variance is large, a huge number of samples may be
needed to get a reliable result. It may happen, that the interesting region is
1 The name ought to bring the casinos of Monte-Carlo to mind.

4.3. SAMPLING THE CANONICAL ENSEMBLE 39
barely hit at all during the numerical evaluation procedure. Therefore so called
variance reduction techniques are needed.
The most prominent idea is that of importance sampling. By replacing the
initial probability distribution p(x) for the random numbers by an importance
density q(x)—which should be chosen such that the poorly sampled regions are
hit more frequently, otherwise the problem can be made worse—one can reduce
the need for a large number of samples significantly! When using the importance
density an additional step in the evaluation has to be performed: Since the
probability distribution of the random samples has been changed to q(x) the
score function has to be adjusted to Aq(x) = A(x)f(x)/q(x). It can be shown
that this results in the same value for the desired average of the integral but the
variance can be reduced by orders of magnitudes.
The nice idea of simple Monte-Carlo integration can reduce the error in high
dimensional integrals, but without the importance sampling method still a large
number of problems would not be solvable in a reasonable amount of computing
time!
4.3 Sampling the canonical ensemble
In our problem of sampling averages for polymers we are working in a canonical
ensemble. Hence we have to deal with the probability distribution in the canonical
ensemble for a given configuration/micro state C
P (C) = 1
Z(β) exp −βH(C) , (4.2)
where H(C) is the energy of the micro state and β−1 = kBT is the temperature.
The normalization constant is the canonical partition sum
Z(β) =
exp −βH(C) , (4.3)
C
the sum over all micro states weighted by the Boltzmann weight. In the case of
continuous states the sum becomes an integral.
Typically one deals with a large number of particles in a system, but already
a system consisting of N = 100 particles with two possible states per particle
would involve a number of 2 100 configurations which is not directly summable
in a lifespan of an average physicist on a typical computer system (now imagine
a system of N ≈ 10 23 particles. . . ). Even simulating this system by the
Monte-Carlo approach of choosing n micro states by equal probability (out of a
flat/uniform probability distribution) is not effective since most micro states have
very low probability 2 . Therefore we have the problem described above, where the
2 E.g. the energy landscape of a high dimensional problem is mainly flat, but has some
narrow deep minima. In this case it is very unlikely that the minima are hit. But since they
are governing the behavior of the system, they should be sampled effectively.

40 CHAPTER 4. SIMULATION METHODS
variance will be large, since we barely hit contributing configurations.
Introducing the importance sampling method again and choosing random
states directly out of the distribution eq. (4.2) would be smart, but than we
are trapped in a vicious circle since the partition sum Z, which would be needed
for that, was what we had to calculate in first place. But we were not able to do
that. The solution to that problem is found by using Markov chains.
4.3.1 Markov chains
The method of Markov chains in connection with Monte-Carlo simulations was
introduced by N. Metropolis in [34] together with an algorithm, introduced in
the next section 4.3.2. The concept is to generate a sequence of successive configuration
Ci which sample the configuration space in a representative manner.
Each step in this chain only depends on the present configuration. This already
is the Markov property.
A specific chain of n configurations {C0, . . . , Cn} out of the whole configuration
space occurs with the joint probability P (C0, . . . , Cn). Writing this down as
a sequence of events gives a product of conditional probabilities gives
P (C0, . . . , Cn) = P (Cn|Cn−1, . . . , C0) · P (Cn−1|Cn−2, . . . , C0)
· · · P (C2|C1, C0) · P (C1|C0) · P (C0)
n
= P (C0) P (Ci|Ci−1), (4.4)
i=1
where in the last line the Markovian property has already been introduced (cf.
[45]). Markovian property means that the next step only depends on the current
state of the system, the past has no influence on future steps 3 . For the conditional
probabilities this results in a dependence only on the previous configuration
P (Ck|Ck−1, . . . , C0) = P (Ck|Ck−1). (4.5)
The advantage of this property is that we can introduce (for a time homogeneous
Markov chain) a fixed set of transition rates Wij between every two
states Cj → Ci of the configuration space irrespective of all the other states. This
rates form a transition matrix with n × n elements. The probabilities of the
configurations than follow a discrete time master equation, which should have a
equilibrium probability distribution {πi = π(Ci)} (where Ci is some configuration
of the allowed space) for our static problem. By demanding detailed balance
Wijπj = Wjiπi, (4.6)
3 A famous example is the Markovian Scopa player: This player plays his cards only on basis
of the last one played. He does not remember what has been played before. . .
(Scopa is a famous Italian card game)

4.3. SAMPLING THE CANONICAL ENSEMBLE 41
it is ensured that the system evolves towards equilibrium. So there is a direct
balance between every two states. There are other ways to ensure the evolution—
e.g. circular balance between all the states—into an equilibrium distribution, but
detailed balance is the easiest. The most important point is, that there is nothing
like an absorbing state. In the following sections we will only deal with the
detailed balance condition.
Knowing this detailed balance condition, it is easy to construct the transition
probabilities for the canonical ensemble, since we know the equilibrium distribution,
eq. (4.2),
Wij
Wji
= πi
πj
= exp (−β∆E) , (4.7)
where ∆E = H(Ci) − H(Cj) is the energy difference between the configurations.
The normalizing factor (the partition sum) Z drops out! We only need ratios
of equilibrium probabilities which we know to be the Boltzmann weights. This
finally renders the desired evaluations possible.
The question which remains is how to construct an algorithm which ‘moves’
us through configuration space with this transition probabilities, since storing the
transition matrix W as a whole is not possible for large configurations spaces.
4.3.2 Metropolis algorithm
The Metropolis algorithm is a recipe how to generate a Markov chain for an initial
configuration C0 → C1 → · · · → Ck−1 → Ck → · · · (cf. [34]). If we choose—as
will be described in more detail later—a worm-like chain as an example, then the
algorithm would consist of three main steps:
1. Choose one of the N discrete chain segments at random.
2. Change this segment into a new one by an arbitrary rotation. Be careful
to have the back-rotation possible as well, to ensure detailed balance!
3. Check if this move is accepted or rejected.
These three steps are iterated over for as long as the desired accuracy for the
value to calculate is achieved. In this context one has to bear in mind, that
successive steps may be correlated so that one should wait a specific time before
using a configuration for sampling. This problems will be addressed later.
The last step is where the above transition probability comes into play. When
starting off a configuration Ck a test configuration Ct is generated, for which we
can calculate the energy difference ∆E(Ct) = H(Ct) − H(Ck). With this the
transition is accepted with the probability—using eq. (4.7):
p = min 1, exp(−β∆E) . (4.8)

42 CHAPTER 4. SIMULATION METHODS
This means in terms of energy: If the energy is lowered the system evolves to the
preferred lower energy state. If the energy is increased, the system moves only
with a Boltzmann factor into that less preferred region. But one has to allow for
this movement “up the energy-hill” since the system is fluctuating around the
equilibrium configuration.
In an computer algorithmic form this consists of either accepting the move
right away since ∆E is lowered. Or a random number of a uniform deviate has
to be drawn which must lie below the transition probability exp(−β∆E).
So the new state is:
Cr accepted with probability: p
Ck+1 =
. (4.9)
Ck rejected with probability: 1 − p
Instead of the usual non-differential acceptance probability, eq. (4.8), other
schemes are possible—like the Glauber dynamics—but we will not go into the
details of this but stick we the equations derived above.
The condition of detailed balance is a sufficient but not necessary condition
for the Monte-Carlo scheme to work properly. It can be shown that e.g. sequential
updating of the particles—after a particle is moved, in the next trials this particle
is skipped, so detailed balance is obviously not satisfied—gives a working MCscheme
as well (see [13]).
4.3.3 Correlations and error calculation
When using the Metropolis algorithm one has to assure that the simulation is
exploring the configuration space. There are several problems which have to be
taken into account. Normally two configurations which are only separated by
some small number of MC steps are “quite close” together in configuration space
(in particular when being close to a steep energy minimum, a large number of MC
steps are rejected, so that one has to be careful to sample the fluctuation around
the minimum efficiently). When taking a sample configuration too frequently (in
the worst case every step), one has an ensemble of samples which are correlated.
This asks for careful treatment of error estimation and for a smart way of choosing
trial steps.
Choosing a trial Monte-Carlo step must be done in the following way:
• Make as large steps as possible to explore all configuration space without
having too many steps rejected, since large changes in energy ∆E are not
very probable to be accepted.
• On the other hand the steps should be small enough to have an overall
high enough acceptance rate. But choosing the step size to small—which
makes the acceptance rate large—will not drive the simulation far through
configuration space.

4.3. SAMPLING THE CANONICAL ENSEMBLE 43
Therefore for each simulation the acceptance rate must be adjusted before the
sampling is started. Acceptance rates in the range 25–50% are usually giving
good results.
Some additional points have to considered. When starting a simulation from
an arbitrary initial configuration, one has to let the system equilibrate for “some
time” (the number of MC-steps needed for equilibration, has to determined independently)
before the sampling is started. Otherwise non-equilibrium configurations
are sampled for an equilibrium average.
Furthermore ergodicity has to be ensured: The trial steps have to be chosen
such that the whole configuration space is covered, if the simulation would be
running long enough. In this context one has to be particularly sure, that the
system does not get trapped in local minima, showing quasi ergodicity.
Statistical error
Having a set of uncorrelated data points {x1, . . . , xn}—by experiment or simulation—a
measure for the error of the average over the samples
〈x〉 = 1
n
n
xi, (4.10)
is calculated through the statistical variance, being the mean-square deviation of
the data points from the mean
σ(x) 2 = 1
n − 1
i=1
n
xi − 〈x〉 2 . (4.11)
This actually represents the width of the distribution of data points. Since we
expect the average to converge to the exact value xs in the limit n → ∞ the
estimated error of the mean is given by
i=1
∆x = σ(x)
√ n . (4.12)
But eq. (4.11) only holds as long as the samples taken are uncorrelated, since
otherwise the error calculated is to small compared to the real one—the samples
look ‘similar’ as long as they are correlated. The aim is to have a measure after
how many MC-steps samples must be taken to have them uncorrelated or to
extract a real error estimate from correlated data. Three possible solutions are
presented in the following.
Correlation function to make independent samples Up to now there was
no real time evolution in Monte-Carlo since steps are generated mechanically in
fixed intervals which have a priori nothing to do with the real time evolution

44 CHAPTER 4. SIMULATION METHODS
of the problem. As described in [4] the dynamic interpretations of Monte Carlo
averaging can be introduced in terms of a master equation. Besides being able
to use this description to calculate correlations this is the starting point for MC
simulations of dynamic processes.
By this interpretation we write for the probability of having a configuration
x at step i as having it at time t: p(xi) ≡ p(x, t).
With this dynamic interpretation, the direct way to find out after how many
steps the samples are uncorrelated is to measure the time dependence of a suitable
correlation function. This correlation function normally decays as an exponential
in time
φ(t) = exp − |t|
, (4.13)
τ
where the time constant τ gives the time when the correlations are decayed to
1/e of the initial value 4 .
Approximation for the correlation time Besides measuring the, in the
above dynamic interpretation, time-dependent correlation function directly, there
is an alternative way to get an estimate for it by sampling different averages (cf.
[4, 22]). Having a large quantity n ≫ 1 of samples xi of an observable x we use
the statistical error
and take the mean of the square
(δx) 2 = 1
n 2
δx = 1
n
n xi − 〈x〉 2
i=1
= 1 2
x
n
− 〈x〉
2
1 + 2
n
xi − 〈x〉
i=1
+ 2
n 2
i=1
n
n
i=1 j=i+1
〈xixj〉 − 〈x〉 2
(4.14)
n
1 − i
n
〈x0xi〉 − 〈x〉 2
〈x2 〉 − 〈x〉 2
. (4.15)
In the second line the time invariance of the correlation in equilibrium 〈xixj〉 =
〈x0xj−i〉 and a change of the summation index j → j −i has been used. The time
invariance only holds if an initial transient regime, where the system ‘forgets’ its
initial configuration, is omitted.
Now introducing the ‘time’ ti = i δt, where δt is the time unit used, e.g.
δt = 1 for a sample every MC step or δt = N for one MC step per polymer
4For the worm-like chain model described in section 4.4 a good correlation function to
measure is the tangent-tangent correlation function
φ(t) =
〈ti(0) · ti(t)〉 − 〈ti〉 2
,
i
which shows reasonable decaying behavior in the case of an unconfined polymer, but fails to be
usable when confining the molecule, due to correlations by the tube.

4.3. SAMPLING THE CANONICAL ENSEMBLE 45
(where the first would not be very effective which will be explained later in
more detail). Transforming the sum into an integral—which is allowed when the
“correlation time” between successive states is s large compared to δt—than gives
(T = tn = n δt being the total sampling time)
2
(δx) = 1 2
x
n
− 〈x〉
2
1 + 1
δt
T
0
dt 1 − t
φx(t) , (4.16)
T
with the normalized autocorrelation function (or “linear relaxation function”)
φx(t) =
〈x(0)x(t)〉 − 〈x〉2
〈x 2 〉 − 〈x〉 2 . (4.17)
This function is normalized φx(0) = 1 and decays to zero φx(t → ∞) = 0, since
the samples of x should become uncorrelated when they are separated by a large
time interval.
Setting a time-scale
∞
τx = dt φx(t), (4.18)
0
in which we assume the correlation function φx(t) to decay essentially to zero,
with τx ≪ T which just means sampling long enough. This also means that φx(t)
is essentially nonzero only for t ≪ T , so that the term t/T in eq. (4.16) can be
neglected and the integration extended to the upper limit of infinity
2
(δx) = 1 2
x
n
− 〈x〉
2
1 + 2 τx
. (4.19)
δt
This shows that the error over correlated data must be enhanced in comparison
to the error calculated over uncorrelated data (〈x2 〉−〈x〉 2 )/n (which would mean
setting τx = 0 here). The factor by which this error is enhanced is also called
statistical inefficiency. The time τx can then be seen as the autocorrelation time.
In the case of sampling in steps much larger than the correlation time δt ≫ τx
the result for uncorrelated sampling is reobtained, since the parenthesis is unity.
On the other hand using τx ≪ δt results in a form independent of the sampling
step size
2
(δx) = 2 τx 2
x − 〈x〉 2 , (4.20)
T
because the unity term in the parenthesis can be omitted. This means that it
is not useful to increase the number of correlated samples by reducing the time
between two samples. In this regime there is no change in the error. This would
only increase the simulation time by taking more (useless) samples!
We finally have a method at hand to calculate the autocorrelation time τx
without directly sampling and integrating it. By measuring the mean 〈x〉 and
mean-square 〈x 2 〉 of x as well as the statistical error 〈(δx) 2 〉 the autocorrelation
time is obtained by rearranging eq. (4.19) or (4.20).

46 CHAPTER 4. SIMULATION METHODS
Blocking method There are situations, when a set of samples has been obtained,
but it has not been taken care of measuring the correlation time. When
the whole set of sampling points is available a different approach can be made to
extract an useful error estimate. It is a blocking method and therefore related to
real space renormalization (cf. [13, 35] and the original paper [12]).
The idea is to calculate the error over a successively smaller set of sample
points and observe how this error behaves. The first step is to calculate the error
over the whole set of points. Then the set is reduced to half its size by taking
the mean of two successive point as new points for the smaller sequence. Again
the error is calculated over this new set.
By repeating this until the set is too small, a sequence of errors is obtained,
which is first increasing and then should show a plateau (at the end fluctuations
become very large, so that the last points are unreliable). It can be shown (see
[12], also for further details and plots) that the real uncorrelated error corresponds
to the value of the plateau. If there is no plateau visible, than too few sample
points are available to uncorrelate the error by this method and one can just
get a lower border for the error, which is given by the largest error value in the
sequence.
4.3.4 Recursive averaging and error calculation
In the simulations a lot of averages and errors thereof have to be calculated. If
this is done at the end of the program run all the sample data has to be stored.
This can result in a large memory usage. The smart way is to calculate the
averages and errors recursively from the last value and the just sampled one.
The mean in the Nth step is denoted by (where xi is the sample taken in
MC-step i)
〈x〉 N = 1
N
N
xi. (4.21)
Inserting this in the same formula written down for N +1, results in the recursive
equation for the average
i=1
〈x〉 N+1 = N
N + 1 〈x〉 1
N +
N + 1 xN+1. (4.22)
To equate the relation for the mean-square deviation 〈∆x 2 〉 N = 〈x 2 〉 N − 〈x〉 2
N
the result eq. (4.22) has to be used besides the same for mean-square average
x 2
N+1
N 2
= x
N + 1
1
+ N N + 1 x2N+1. (4.23)
Inserting and after some rearranging of terms the results is
(N + 1) ∆x 2
N+1 = N ∆x 2 N 2. + 〈x〉N − xN+1 (4.24)
N N + 1

4.4. SIMULATION OF UNCONFINED WORM-LIKE CHAINS 47
So the average and the error are easily calculated with minor storage overhead.
The only drawback is, when a blocking technique has to be used to free the
samples of the ‘omnipresent’ correlation in MC simulations, then all the averages
have to be stored anyway.
4.3.5 Random number generators
The quality of Monte-Carlo simulations depends crucially on the quality of the
pseudo-random numbers used. The quality of the linear congruential generators,
as for example implemented in most standard libraries, have poor statistical
quality and a very small period. Other generators have to be used. There is a
large list of different types but most of them fail to pass statistical tests rather
spectacularly. There is no exact measure for the quality of a generator but there
exists a list of structural mathematical and statistical tests which give a ‘feeling’
for the quality.
A good overview over the important details of random number generators and
examples is given in the review [28].
A fast generator with sufficient quality is based on the linear feedback shift
register or Tausworthe generator (see [27] and [29]). The implementation from
the GNU Scientific Library 5 has been used in our simulation. The advantage of
this generator, besides its structural mathematical qualities, is its speed. The
random number generation consumes up to one forth of the computational time
of our simulations. Test with different generator have been performed, but the
Tausworthe generator seems to be good enough.
If for some reason an even better generator is needed, the RANLUX generator
described in [30] should be used, which can provide, by choice of a numerical
constant, uncorrelated numbers down to the last digit of the used floating point
variable. This has been theoretically justified by chaos theory. The quality of
this generator goes on costs of computational speed which is up to 20 times lower
than in the aforementioned generator.
4.4 Simulation of unconfined worm-like chains
Simulation of semi-flexible polymers are based on the discrete worm-like chain
model as introduced in section 2.3.2. The basic Hamiltonian is given by eq. (2.13),
which can be directly used to perform simulations
N−1
e = βH ({ti}) = −k ˆti · ˆti+1 with k = βεl 2 , (4.25)
i=1
5 http://www.gnu.org/software/gsl/manual/gsl-ref 17.html#SEC262 (17.11.2004)

48 CHAPTER 4. SIMULATION METHODS
where the prefactor k is related to the persistence length lp by eqs. (A.8) and
(A.10). As input parameter for the simulation the flexibility ɛ is taken so that
everything is measured in units of total length L (cf. eq. (3.2)). All output is
normalized by L as well.
This Hamiltonian is used to perform an off-lattice simulation where the polymer
can orient without any restriction (cf. [25]).
In a straightforward algorithm the polymer is represented by just its N segments,
where the positions of the joints can be ignored as long as the polymer is
unconfined. The basic steps are:
1. perform a random movement on a randomly chosen segment
2. calculate the change in configuration energy ∆e
3. accept/reject the move corresponding to the Metropolis algorithm
(4. take samples after a specific number of steps)
This three (four) basic steps are the essential part of the MC scheme. They will
be discussed now in more detail.
Step 1: The trial movement in the n-th MC-step is chosen by drawing a
random number m to specify a segment to change ˆt (n)
m → ˜t (n)
m . The orientation
is changed by a multiplication with a rotational matrix R l k
, which is chosen out
of a set that has been calculated in the initial phase of the simulation. The
superscript l numbers one of the d spacial dimensions and k the specific matrix
chosen from the set. It turned out to be optimal having 20 d equidistant discrete
matrices (20 for each dimension) out of a maximal rotation angle ±δα/2. This
angels must be tuned to give an overall acceptance rate of 25–50%.
Detailed balance is obviously guaranteed: choosing a rotation matrix by first
choosing one direction at random and than again randomly a specific rotation 6 .
Thus this step is subsumed by—with random numbers m = 1, . . . , N, l = 1, . . . , d
and k = 1, . . . , 20—
˜t (n)
i
=
Rl k · ˆt (n)
m for i = m
ˆt (n)
. (4.26)
i for i = m
One segment m is changed, where the orientation of all others is kept fixed as
shown in figure 4.1.
An alternative approach to the simulation in two dimension is by noting that
the polymer configuration is fully specified by knowing the angles θi between each
segment and a fixed global direction (see figure 4.1). The simulation can than
be build on trial moves changing these angles, i.e. taking a random change on
6 E.g. it would be wrong to use in each step a rotation in all d-direction, when applying the
rotation always in the same order. This is due to the non-commutativity of rotations, which
breaks detailed balance in this case. But choosing the order of application randomly, again
obeys detailed balance, but has an unnecessary calculation overhead.

4.4. SIMULATION OF UNCONFINED WORM-LIKE CHAINS 49
ˆt1
θ1
ˆt2
θ2
ˆtm
˜tm
Figure 4.1: Trial move in MC-algorithm for the WLC. The randomly chosen segment m is
changed, whereas all other tangents are kept fixed.
positing m with δθ where the angle θm → ˜ θm = θm + δθ must be changed. We
tested this algorithm but it did not show any major advantage neither in accuracy
nor speed (one has to calculate trigonometrical functions in every step). Since it
is not easily generalized into three dimensions the preference has been given to
the first algorithm, since it can supply a general code for arbitrary dimensions.
Step 2: For the chosen trial move the change in configuration energy ∆e must
be calculated. The advantage of the algorithm is, that this energy change is given
by a simple term, since the orientation on only two joints is changed (see again
figure 4.1)
∆e = −k
N
i=1
˜t (n)
i
= −k ˜t (n)
m − ˆt (n)
m ·
· ˜t (n)
i+1 − ˆt (n)
i
θm
˜θm
ˆtm+1
· ˆt (n)
i+1
ˆt (n)
m+1 + ˆt (n)
m−1
θm+1
ˆtm+1
θm+1
ˆtN
θN
ˆtN
θN
for: i = 1, . . . , N, (4.27)
which can be easily implemented by using two dummy segments with length zero
ˆt0 = ˆtN+1 = 0.
Step 3: According to the Metropolis algorithm introduced in section 4.3.2
this move is finally accepted, if the total configuration energy is reduced ∆e < 0.
In the case of an energy increase, it is accepted corresponding to the Boltzmann
weight x < exp(−∆e) by drawing another random number 0 ≤ x < 1. Otherwise
the move is rejected
ˆt (n+1)
i
=
˜t (n)
i
accepted
. (4.28)
ˆt (n)
i rejected
Step 4 is set in brackets, since it is not useful to take a samples from the
configuration during every MC-step. Two successive configurations are normally
strongly correlated. So we estimate the correlation time τ of one observable (see
page 44), which is taken as the sample step size, after which a configuration is
taken. This correlation time is also used to determine the error estimate on the
averages.

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

52 CHAPTER 4. SIMULATION METHODS
detail will be the cylindrical symmetric harmonic potential as already introduced
in eq. (3.20). The accuracy of this approximation and how to relate it to the
other potentials will be discussed in section 5.
4.5.1 Harmonic potential
The configuration energy function ∆e corresponding to eq. (4.25) is given by the
discretized version of eq. (3.20)
N−1
∆e = βH ({ti}) = −k ˆti · ˆti+1 + g
2
i=1
N 2
xi + y 2 i , (4.29)
such that we now need the tangents and the positions of the beads in the simulation.
The latter can be directly calculated from the tangents, but storing them
is faster.
We have to relate the prefactor g to the Odijk deflection length ld by discretizing
the harmonic potential part of eq. (3.20). Additionally one has to use
dimensionless units in the simulation. Hence all positions are measured in units
of the segment length l
γ
2
L
0
ds x(s) 2 + y(s) 2 = γl3
2
→ γl3
2
g/2β
Nl
0
ds
l
i=0
x(s)
l
2
2
y(s)
+
l
N 2
xi + y 2 i , (4.30)
where xi, yi are normalized by the segment length l = L/N. Upon using the
one finds
definition ld := (4κ/γ) 1
4 ⇒ γ = 4κ/l 4 d
g = βγl 3 = 4βκl3
l 4 d
i=0
(2.20)
= 4lpl 3
l 4 d
= 4c4
N 3 . (4.31)
ɛ
The actual parameter used for the simulations is the collision parameter c as
defined in eq. (3.3).
With these prerequisites the simulation can be implemented straightforwardly.
Only one additional MC move must be added since the space is not isotropic
anymore. Now global movements of the polymer perpendicular to the tube axis
must be attempted. This can be implemented by using a virtual segment ‘zero’.
If this is randomly selected, all positions are shifted by a random amount out of
±∆r⊥. The detailed balance condition is therefore obeyed. This maximum shift
is, along with the maximal rotation angle δα, adjusted in the initial simulation
phase. The z-position of the first bead can be kept fixed, since there is still the
rotational symmetry around the tube/z-axis.

4.6. SAMPLED QUANTITIES AND THEIR RELEVANCE 53
4.5.2 Hard-wall tubes
Hard-wall tubes with circular and quadratic cross section have been simulated.
With these potentials no additional potential energy has to be calculated, but in
every MC-step it has to be checked if any of the beads crosses the border of the
allowed space. This can also be stated in terms of a infinite energy barriers
∞ for |r⊥| >
Vtube(r⊥) =
d
2 , (4.32a)
Vchannel(r⊥) =
0 else
∞ for |x| > w
2
0 else
or |y| > w
2
, (4.32b)
where d is the tube diameter and w the width of the square channel.
Close to the walls approximately only half of the moves are accepted. The
adjustment of the acceptance rate has to be done with care. Therefore the initial
configuration is put fully stretched parallel to the tube-axis, very close to the
walls (in the channel close to a corner, where even less moves are accepted). The
acceptance rate is then adjusted by letting the configuration evolve freely and
putting it back to the wall from time to time, since it is moving to the center
of the tube. In this way we can be sure, that we are still accepting movements
close to the boundaries and are not sticking to the wall. For the main sampling
part of the simulation, the polymer is then set initially totally stretched along
the tube-axis. The initial configuration is decaying very fast, so that this specific
chosen configuration does not matter.
4.6 Sampled quantities and their relevance
During the simulation run a whole bunch of observables are sampled. Some of
them make only sense when confining the chain, as described in the previous
section. This is indicated at the corresponding location. It follows a list of the
observables, how they are calculated and their relevance in experiments:
mean (square) end-to-end distance 〈R〉, 〈R 2 〉:
The end-to-end distance is given by the sum over all tangents
R =
N
ˆti. (4.33)
From this formula both averages can be obtained straightforwardly.
i=1
This is the quantity of interest in our comparison to analytical results obtained
in chapter 3. In particular, we are going to investigate the scaling
properties in section 5.2.3.

54 CHAPTER 4. SIMULATION METHODS
For stiff and/or confined polymers this is also a quantity, which can be
experimentally measured on fluorescently dyed polymers (e.q. [44, 38]).
mean (square) parallel end-to-end distance
2
R|| , R|| :
The parallel end-to-end distance is the component of the end-to-end distance
along the confining ‘tube’ (here z-axis). So this only makes sense in
connection with the simulation in confined geometries. In the unconfined
case the simple relation R2
2
|| = 〈R 〉 /3 must hold.
The parallel end-to-end distance is given by
R|| =
N
i=1
ˆti . (4.34)
z
This quantity should coincide with the end-to-end distance in the very
strong confined limit, due to the orientation along the axis. It is more of
theoretical interest, since it can be used to explain some interesting global
features (cf. section 5.2.4).
apparent end-to-end distances 〈Ra〉, 〈R2 a〉,
2
Ra|| , Ra|| :
Another end-to-end distance is important, when experiments of flexible
polymers in confined environments are investigated (e.g. [44, 38]). Since on
fluorescently dyed polymers, the real ends are difficult to discern from the
rest of the chain. Only a blurry picture is seen as depicted in figure 1.2.
Therefore the apparent end-to-end distance is introduced, which is the maximal
distance between two beads along the chain. The distance between
two beads i, j at positions Ri, Rj is defined as
dij = Rj − Ri
(4.35)
so that in the same spirit as before, apparent and the apparent parallel
end-to-end distance are defined as
mean-square (parallel) radius of gyration R 2 g
Ra = max |dij| , (4.36a)
i,j
Ra|| = max
i,j (dij)z. (4.36b)
A scaling analysis and comparison to experimental data is provided in section
5.2.3.
2 , Rg|| :
The radius of gyration can be calculated directly as given in eq. (2.7).
Where the form with the center of mass Rcm is preferred since it only
involves order 2N calculations not order N 2 .

4.6. SAMPLED QUANTITIES AND THEIR RELEVANCE 55
This value has importance for experiments on conformations of unconfined
polymers. When e.g. a fluorescently dyed polymer shows spots of strong
intensity (which are not resolved by the camera taking the picture), one
can weighten the configuration with the intensity, leading to the radius of
gyration obtained through the fluorescent “mass” (cf. [32]).
mean-square undulations 〈x 2 (s)〉, 〈x 2 〉, 〈y 2 〉:
This values for the undulation makes sense only in simulations of confined
polymers, since it relies on having a finite system in the lateral direction to
give finite averages. The sampling is straightforward by the perpendicular
distance of the beads from the axis.
The function 〈x 2 (s)〉 gives the mean-square undulation for each bead whereas
the values 〈x 2 〉 and 〈y 2 〉 are additionally averaged over all beads.
These results can also be compared with our results from chapter 3. This
will be done in section 5.2.1.
tangent-tangent correlation function ˆt(0.5) · ˆt(s + 0.5) :
The tangent-tangent correlation function is sampled using a fixed reference
point in the middle of the polymer, to reduce boundary effects as much as
possible. We want to be able to compare the results with analytical results
obtained for different boundary conditions (see section 5.2.2). The function
is obtained directly by using the simulated tangents.
radial distribution function p(R) = p(|R|):
Instead of only sampling an average, sampling a probability distribution
needs some additional steps. The range of possible values—here 0 ≤ R ≤
L = 1—has to be divided into bins. In each sampling step the value of R
hits one of these bins, where the corresponding counter is increased. Since
we do not obtain a whole distribution in one sampling step, we are forced
to find a different way for the error calculation of the distribution points.
Therefore, during each sampling step of τR MC-steps, 500 samples are taken
for the averages to get a rough distribution. This is then used to obtain an
averaged distribution with an error estimate.
A value of 100 bins gives a smooth looking probability distribution. Only
for strongly confined or very stiff polymers the number of bin should be
increased around the fully stretched regime.
The same sort of distribution function is obtained for R||. Both results are
shown in section 5.2.5.
In principle these distribution can be compared to experimental results.
This work is currently in progress.
All sampling is done using the recursive method introduced in section 4.3.4 in
order to avoid large arrays of data or in-between storing and averaging to reduce

56 CHAPTER 4. SIMULATION METHODS
the memory usage. Therefore the memory usage is only in the order of some
MBytes.

Chapter 5
Simulation results
In this chapter we are going to explore the scaling behavior of the worm-like chain
upon confinement, using the Monte-Carlo algorithm discussed in the preceding
chapter 4. After validating the simulation with exactly known results in bulk
solution (cf. chapter 2), we turn to the influence of a harmonic potential with
cylindrical symmetry on the polymer configuration. The undulations and the
tangent-tangent correlation function are compared to analytical results (cf. chapter
3). After becoming familiar with the influence of the confining environment,
we explore the scaling behavior of the mean-square end-to-end distance 〈R 2 〉 with
the ratio of the collision parameter to the persistence length c/ɛ.
We also go beyond an analysis of the moments by studying the full probability
distribution function of the end-to-end distance. It turns out that there is a
cross over from the well known unconfined distribution to the confined regime,
strongly peaked towards full stretching, with an intermediate bimodal distribution
function, as one decreases the channel width.
Finally we have also started to explore other types of confining potentials, in
particular hard-wall potentials with square and cylindrical cross section.
Before starting the investigation, some preliminary remarks are in order:
• In this chapter all values are measured in units of the total length L.
• In the plots the statistical error is smaller than the symbols if not stated
otherwise.
• All simulations were performed without using self-avoidance effects. We
have convinced ourselves that such effects are negligible for flexible filaments
up to a length of at least N = 500 segments.
• The simulations were performed on a standard computer desktop system
(Intel Pentium IV processor, various frequencies). A typical simulation for
fixed segment size N, flexibility ɛ and collision parameter c takes about one
hour for N = 50 and up to the order of a week for N = 1000 segments.
57

58 CHAPTER 5. SIMULATION RESULTS
R 2
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
ɛ
6
7
exact
simulation
Figure 5.1: Comparison of the simulation for a polymer in bulk solution c = 0 to the exact
formula eq. (2.21). The dependence of R 2 on the flexibility ɛ is shown.
5.1 The unconfined chain
Before one can start using a simulation algorithm to produce data for a situation
where no analytical comparison is available, it should be validated by comparing
the results with well known limiting cases where analytical results are available.
This is the case for a polymer in bulk solution, where the dependence of the
second moment of the end-to-end vector distribution on the persistence length is
well known (cf. eq. (2.21)). This limit can be simulated using our algorithm for
a collision parameter c = 0. The results in figure 5.1 show that there is perfect
agreement within the statistical error.
An additional check can be done with the tangent-tangent correlation function
which should show simple exponential behavior. In the logarithmic plot 5.2 this
results in a straight line. The simulation data for e.g. lp = 10 again fits eq. (2.12)
perfectly within the errorbars.
In addition, the radial distribution function has also been checked against
known results, such that the algorithm—at least concerning the bending energy—
is fully validated.
5.2 The harmonic potential
A convenient starting point for the discussion of confinement on the conformation
of a polymer is a harmonic potential with cylindrical symmetry. This has the
advantage that we can compare with analytical results at least in the stiff limit
(see section 3.3).
8
9
10

5.2. THE HARMONIC POTENTIAL 59
ˆt(0.5) · ˆt(s + 0.5)
1
0.1
0.01
0.001
-0.5
-0.4
-0.3
-0.2
exact
simulation
-0.1 0
s − 0.5
0.1
Figure 5.2: Comparison of the tangent-tangent correlation function from eq. (2.12) to simulation
data for c = 0. The reference point of the correlation function is fixed in the middle of
the polymer, where the zero of the plot is set.
5.2.1 Undulations
We start our discussion with the effect of the confining environment on the meansquare
transverse displacements.
Upon confining we observed that undulations are more and more suppressed,
as one would also expect intuitively. In particular we have derived a result for the
mean-square undulations, eq. (3.27), that was already shown in figure 3.6 along
with the result obtained by a Fourier series ansatz.
The relative deviation of the simulation data for the mean distance between
the chain joints and the tube axis 〈x 2 s(s)〉 from the analytical result 〈x 2 a〉, obtained
in eq. (3.27), is shown in figure 5.3. For stiff confined polymers the value obtained
in the simulations is identical to the undulations as parametrized in the weaklybending
rod limit. The points are taken from simulations with N = 50 segments 1
for a value of ɛ = 0.1 and confinement c = 1 and c = 10.
In the case of strong confinement, c = 10, the simulation coincides with the
analytical value of 〈x 2 a〉 = 1.25 · 10 −5 over most of the polymer length. The
influence of the tube is seen as the overall small value of the average undulation
along the tube. The polymer segments are not allowed to deviate much from the
axis because the energy penalty becomes to high.
Since the error is smaller than the visible point size, the deviation from the
constant analytical result is due to the effect of the boundary conditions used
in the analytical calculations. In the simulation the ends are free, such that
there is a section of the chain at its ends—a length of the order of the Odijk
deflection length 1/c—which is free to explore a larger distance from the tube
axis. Increasing the confinement reduces the effect of the boundary conditions,
1 therefore 50 data point for the discrete undulation function
0.2
0.3
0.4
0.5

60 CHAPTER 5. SIMULATION RESULTS
− 1
x 2 s (s) / x 2 a
6
5
4
3
2
1
0
0
0.2
0.4
c = 1
c = 10
Figure 5.3: Mean-square undulation x2 (s) along the polymer parametrized by the contourlength
s. Shown is the relative deviation of the simulation data x2 s(s) from the analytical
results x2
a (cf. eq. (3.27)). The simulation data was obtained for a chain of N = 50 segments,
flexibility ɛ = 0.1 and collision parameter c = 1 and c = 10, respectively.
since the chain is deflected earlier along its contour.
In contrast, for weak confinement (c 1) the tube does not constrain the
undulations anymore but only the overall center of mass motion, which is also
seen in figure 5.3 for the collision parameter c = 1. Then the shape of the
mean-square displacement 〈x 2 (s)〉 is parabolic and independent of c. The overall
deviation from the analytical result starts to increase, due to the center of mass
motion.
Finally for c ≪ 1, 〈x 2 (s)〉 is not a proper measure of the polymer configuration
since the center of mass is free to diffuse in space. In this case of unconfined
polymers the measured value in the simulation is not identical to the undulations
2 . Hence we need to study another set of quantities to explore the difference
in behavior between confinement and chain-stiffness. This is done in the next
sections.
In summary we can classify the confinement into three regimes:
1. weak confinement c ≪ 1: The polymer is completely free.
2. intermediate confinement c ≈ 1: The center of mass diffusion and global
rotations are restricted by the tube.
3. strong confinement c ≫ 1: Additionally internal bending modes are restricted.
2 The parametrization of stiff free polymers by the weakly-bending rod approximation is
anyway possible, since for free polymers global rotations can be taken into account by the
trivial phase-space factor 4πR 2 .
s
0.6
0.8
1

5.2. THE HARMONIC POTENTIAL 61
5.2.2 Tangent-tangent correlation
Now we turn to investigate the tangent-tangent correlation function defined as
φtt(s, s ′ ) = 〈t(s) · t(s ′ )〉 . (5.1)
This allows us to extract some more information on the universal properties of
our polymer system.
The influence of the tube on the tangent-tangent correlation can be divided
into three regimes, when the important length scale is again the Odijk deflection
length 1/c:
1. For segments which are close enough along the chain the influence of the
tube is negligible. Then, the tangent-tangent correlation decays exponentially
as if for a free polymer until the segments get deflected by the ‘tube’.
The free behavior is seen within the range 0 ≤ |s − s ′ | 1/c.
2. In the range of |s−s ′ | ≈ 1/c the first deflection of the segments takes place.
This is seen in a intermediate flattening of the correlation function, or, for
strong confinement, in a local minimum (see figs. 5.4).
This feature indicates that the fluctuations of the tangent-tangent correlations
around the mean 〈δt(s) · δt(s ′ )〉 = 〈t(s) · t(s ′ )〉 − 〈t〉 2 are anticorrelated
in this range, due to the deflections.
3. For large distances along the chain |s − s ′ | ≫ 1/c the correlation function
depends only on the ratio c/ɛ as expected from eq. (3.35) and becomes
constant except for boundary effects. This is the universal behavior we
expect for strong confinement.
The three regimes can be identified by inspecting the plots in figure 5.4.
The behavior of the tangent-tangent correlation for fixed stiffness ɛ = 0.1 is
shown in figure 5.4(a) where the purely exponential behavior is seen for quasifree
chains with c 1, i.e. on average less than one collision with the tube. Upon
increasing the collision parameter, the correlation function develops a plateau.
Figure 5.4(d) clearly displays all three regimes. The analytical results capture
the exponential decay as well as the saturation of the correlation function but
not the final decay at the ends. The deviation at the boundary is due to the
free ends of the simulated chain. The overview plot figure 5.4(a) shows that
boundary effects become less pronounced as the confinement becomes stronger.
Then the behavior is dominated by the constant value of strong confinement,
which depends only on the scaling parameter c/ɛ as expected from eq. (3.35).
The behavior of the correlation function for intermediate confinement with
c ≈ 3 is shown in 5.4(c). Here we observe a strong deviation from the analytical
result, which is due to the boundary conditions. In this regime the specific
choice of boundary conditions for analytical calculations is essential for the correct

62 CHAPTER 5. SIMULATION RESULTS
1
0.99
0.98
0.97
0.96
-0.5
φtt(0.5, s + 0.5)/ t 2
1
0.99
0.98
0.97
-0.5
-0.4
-0.3
-0.2
c = 0.1
c = 1
c = 3
c = 5
c = 10
c = 20
c = 100
-0.1 0
s − 0.5
0.1
(a) Normalized φtt for ɛ = 0.1 and a set of increasing collision
parameters c indicated in the graph.
-0.25
1
0.99
0.98
0.97
0.96
0.95
-0.5
0
-0.25
0
0.25
0.2
(b) same as in (a) for c = 5 · 10 −4
0.25
(c) same as in (a) for c = 3
0.5
1
0.99
-0.5
-0.25
0.3
0.5
0
0.4
0.5
0.25
(d) same as in (a) for c = 10
Figure 5.4: Normalized tangent-tangent correlation function along a chain with flexibility
ɛ = 0.1. The reference point is fixed in the middle of the chain, therefore the graphs must
be symmetric. N = 50 segments were used in the simulations. Additionally, in (b)–(d) the
analytical result eq. (3.34) is shown.
0.5

5.2. THE HARMONIC POTENTIAL 63
behavior of the tangent-tangent correlation function. The actual polymer in the
simulation has free ends and is therefore much less constrained in comparison to
the analytical calculation where the fluctuation of the boundaries are suppressed.
The influence of the free ends was already seen on the undulations in figure 5.3
as a large increase of the deviation from the analytical result. But also in figure
5.4(c) a reminiscence of anticorrelations of the segments can be seen as a small
shoulder at |s − 0.5| ≈ 1/c = 1/3.
Figure 5.4(b) shows, for a collision parameter c = 5 · 10 −4 , an effectively free
polymer. The analytical result fits the data perfectly over the whole polymer
length, except for a very small deviation at the ends. This behavior is expected
since the analytical calculation is performed in the weakly-bending rod limit,
which gives the correct result for unconfined stiff polymers up to first order in ɛ,
as shown in the discussion leading to eq. (3.36). Therefore the small deviation
are only due to the limits of the weakly-bending rod approximation (terms of
order ɛ 2 ) and not due to boundary effects.
Finite-size effects have only a very small influence on the results obtained for
the tangent-tangent correlation. The overall shape is not affected by increasing
the number of segments N. Only small global shifts of the whole correlation
function can be observed. This has only an effect on integral quantities as the
end-to-end distance, which will be discussed in the following section.
After these preliminary investigations towards the validity of the dependence
of the observables on the scaling parameter c/ɛ, we now turn to the systematic
investigation using a scaling plot.
5.2.3 Scaling plot
We have already discussed the influence of confinement on the undulations and
the tangent-tangent correlation. Now we will turn to quantities, which are of
particular interest in experiments, where it is easier to measure the end-to-end
distance and related quantities than the very small undulation or a correlation
function of the tangents.
We want to present the data in a compact form, which works out the scaling
properties expected by our investigation up to this point. In a suitable plot
we expect a data collapse. Looking back at eqs. (3.12) and (3.38) we see that
plotting the mean-square end-to-end distance 〈R 2 〉 versus c/ɛ is expected to show
the data collapse. But before turning to the plot we first give an overview over
the various parameter regimes and than try to find out in which range the scaling
is expected to start, since it is not universal for all parameters as noted in the
preceding section.

64 CHAPTER 5. SIMULATION RESULTS
Comparison of analytic and simulation results
Even if one measures all length in units of the total polymer length L, the meansquare
end-to-end distance is still a function of two length scales, the tube diameter
d and the persistence length lp. As discussed in section 3.2 this suggests to
introduce two dimensionless parameters, ɛ = L/lp as a measure of the polymer
stiffness and c = L/ld counting the number of collision of the polymer with the
tube and being related to the tube diameter through the relation in eq. (3.1).
In the discussion we have to distinguish the limiting cases of stiff (ɛ ≪ 1) and
flexible polymers (ɛ ≫ 1). In the stiff limit the main parameter characterizing
the influence of the confinement on the polymer conformation is the collision
parameter c. For c ≪ 1 the polymer is unconfined and behaves as a free polymer
with 〈R 2 〉 ∼ L 2 (cf. eq. (2.22)). For c ≈ 1 undulations for the filament are not
yet restricted but the overall rotation of the filament is hindered by the walls.
For c ≫ 1 there is strong confinement with many collisions of the polymer with
the wall.
In the flexible limit the critical scales to compare are the radius of gyration
Rg and the tube diameter. As noted by de Gennes the scaling behavior of the
unconfined chain, 〈R 2 〉 ∼ lpL, starts to change once the mean-square end-to-end
distance becomes comparable to the tube diameter d. Then the polymer starts
to stretch due to the excluded volume interaction of distant segments along the
chain, such that 〈R 2 〉 ∼ Lc/ɛ. Here we consider phantom chains where selfavoidance
effects are neglected. Hence, as discussed in section 3.2.1 one expects
no effect at all of confinement on the polymer configuration. A stretching for a
chain consisting of segments is then only due to the finite segment length.
Even for flexible chains stiffness effects will become important once the tube
diameter becomes comparable with the persistence length. Then the same arguments
as in the stiff case apply.
From the foregoing discussion we can expect to find universal behavior for
stiff chains, when the chain starts to be deflected, c 1. Since we expect to have
a dependence on the scaling parameter c/ɛ this means
c
ɛ ɛ−1 . (5.2)
On the other hand for flexible chains the relevant condition is that the tube
diameter is of the order of the segment length or less d lp which leads through
eq. (3.1) to
c
1 (5.3)
ɛ
for the scaling parameter. We can summarize this conditions for the universal
scaling regime in
c
max(1, ɛ
ɛ univ
−1 ). (5.4)
For flexible polymers we therefore expect to have a longer scaling regime.

5.2. THE HARMONIC POTENTIAL 65
R 2
1
0.99
0.98
0.97
0.96
1
10
100
c/ɛ
1000
ɛ = 0.01
ɛ = 0.05
ɛ = 0.1
ɛ = 0.18
10000
Figure 5.5: Mean-square end-to-end distance R 2 versus the scaling variable c/ɛ for different
flexibilities ɛ in the stiff regime. Along with the simulation data for chains with N = 50
segments, the analytical result from eq. (3.38) and the free result eq. (2.21) is shown.
With these consideration at hand we start the investigation of the simulation
result in the stiff limit as shown in figure 5.5. Here the mean-square end-to-end
distance 〈R 2 〉 is plotted as a function of the scaling parameter c/ɛ for different
flexibilities ɛ. In addition to the simulation data the analytical results obtained
in eq. (3.38) are shown (non-constant lines) as well as corresponding results for
a free polymer c = 0 (constant lines) from eq. (2.21).
We observe that the simulation data starts to deviate at c/ɛ ≈ ɛ −1 from the
free result represented by the horizontal lines. This is the behavior expected
by our scaling arguments (see eq. (5.2)). Both in the unconfined and strongly
confined regime the simulation data are well described by our analytical results.
This is due to the nature of the weakly-bending rod approximation which is valid
for strong confinement as well as for very stiff chains.
For the intermediate regime, e.g. for a flexibility ɛ = 0.1 in the range 1
c/ɛ 100, there is a strong deviation from the the analytical results, which is
not expected from the assumptions leading to the weakly-bending rod approximation.
But in the context of the discussion of the tangent-tangent correlation
function in the preceding section we understand this deviation. Remembering the
discussion of figure 5.4(c) for the parameter c = 3 (corresponding to the scaling
parameter c/ɛ = 30), these deviations are again due to the free boundaries in
the simulation. The mean-square end-to-end distance is an integral quantity of
the tangent correlation function (cf. section 3.3.1), therefore these deviations are
most prominent in the end-to-end distance. Since the tangent correlation function
for the simulation decays stronger than the analytical result the end-to-end
distance shows values which are too small in comparison to eq. (3.38).
Now let us turn to the flexible limit, L ≫ lp. Then we expect from eq. (5.3)
the scaling range to be larger, which can be seen in figure 5.6. Here the mean-

66 CHAPTER 5. SIMULATION RESULTS
R 2
1
0.8
0.6
0.4
0.2
0
0.1
1
10
c/ɛ
100
ɛ = 0.1
ɛ = 0.5
ɛ = 1
ɛ = 5
ɛ = 10
ɛ = 50
ɛ = 300
Figure 5.6: Scaling plot for the mean-square end-to-end distance R 2 versus the scaling
parameter c/ɛ for various flexibilities ɛ from the stiff to the flexible polymer limit.
square end-to-end distance 〈R 2 〉 as function of the scaling parameter c/ɛ is shown
for the whole range of flexibilities ɛ, from the stiff up to the flexible limit. It can
be seen that a universal behavior is visible to start around c/ɛ 5. Taking a
lower envelope of the data reveals the expected scaling.
The deviations from a lower envelope in this range can be explained by finitesize
effects of the simulated chains. Already for ɛ ≈ 5 differences from the analytic
result eq. (3.38) can be observed. This is explicitly shown in figure 5.7 for the
mean-square end-to-end distance 〈R 2 〉 versus the scaling parameter c/ɛ for a
flexibility ɛ = 5. The simulation data is obtained by increasing the number N of
the segments up to 1000. Since a simulation for chains of length N = 1000 takes
in the order of a week per data point, only single points have been simulated.
The inset gives a zoomed view of the area where the major effect of finite-size is
seen. The data obtained for N = 1000 is seen to fit the analytic result perfectly.
The data converges, in this range, for decreasing segments size to the analytic
result.
These finite-size deviations increase for more flexible chains: The condition
that the segment length l = 1/N should be at least about the size of the persistence
length or smaller—otherwise the persistence length is artificially increased
due to finite-size effects—forces us to use more segments for flexible chains.
Therefore deviations due to finite-size are increasing even more. But using much
more segments than N = 1000 is problematic due to the simulation time needed.
The form of the mean-square end-to-end distance for very flexible chain is
given by the limit of the ideal chain without persistence, the freely jointed chain
(FJC). For c/ɛ 1 the function stays effective constant and for c/ɛ 1 the
function behaves like the approximation for strong confinement eq. (3.9), only
1000

5.2. THE HARMONIC POTENTIAL 67
R 2 /L 2
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
analytic
N = 50
N = 100
N = 500
N = 1000
1
1
Figure 5.7: Mean-square end-to-end distance R 2 versus the scaling parameter c/ɛ for different
levels of discretization N (flexibility ɛ = 5) and the analytic result eq. (3.38). The inset
shows a zoomed region, where the influence of finite-size effects is seen best.
10
10
c/ɛ
the prefactor in the term (d/lp) 2 has to be adjusted 3 .
Comparison to the experiment
Now we turn to the discussion of how the simulations compare to experiments.
The corresponding experiments were performed by Walter Reisner in Bob Austins
group at Princeton. The details of the experiment are discussed in section 3.1
and in the references [38, 44].
For the experimental measurements on flexible chains the most important
quantity to investigate is the the apparent end-to-end distance 〈Ra〉 which measures
the largest possible distance between two segments along the chain. In
experiments the polymer under investigation is marked with some fluorescent
dye, such that only a blurry line is visible, as seen for DNA in the figures 1.2.
To resolve the actual position of the end point of a polymer is experimentally
very hard to achieve. One could imagine using molecular biology technology to
specifically label certain regimes along a DNA strand. But this is not how experiments
in the Austin group are performed. They use TOTO-1 fluorescent dye
to uniformly label the whole filament. In such a situation the proper analytical
quantity to compare to the theory is the apparent end-to-end distance 〈Ra〉.
3 For the FJC stretching of the chain is only due to the finite segment length, and the fact that
we did not allow for backflips. If we would have allowed for those backflips the model would,
in the stiff limit, reduce to an “Ising chain”, where each segment may either point parallel or
anti-parallel to the tube axis. Then, of course, the average end-to-end distance would again
be zero. Since we did not allow for backflips the anti-parallel segment configurations are not
allowed making our system non ergodic. This is still an interesting and useful limit since it
allows us to study the effect of stiff chain segments which—in a real system—would neither be
allowed to backflip.
100
100

68 CHAPTER 5. SIMULATION RESULTS
〈Ra〉
1
0.8
0.6
0.4
0.2
0
0.1
1
10
c/ɛ
experimental data
Odijk scaling (fit)
ɛ = 0.1
ɛ = 0.5
ɛ = 1
ɛ = 5
ɛ = 10
ɛ = 50
ɛ = 300
Figure 5.8: Scaling plot for the apparent end-to-end distance 〈Ra〉 as a function of the scaling
parameter c/ɛ. Beside the simulation data for various flexibilities ɛ and the Odijk scaling
relation, eq. (3.12), with fitted parameter, experimental data for λ-DNA (see text and [38]) is
shown.
The experimental data obtained for λ-DNA is shown in figure 5.8 in comparison
to our simulation data in the cylindrical harmonic potential. In the
experiment the geometric average of the channel width is taken as the effective
tube diameter, which can be converted approximately—since the prefactor in the
relation is not exactly known, it can only be fitted—into a collision parameter
by taking eq. (3.29) literally. The Odijk scaling relation, eq. (3.12), is also fitted
into the plot (α ′ = 0.361).
Although the geometry of the experiment is different from the simulations
performed and the conversion of the channel width to the confining scale of the
harmonic potential is not rigorous, the data fits very well into the trend seen in
the plot. Despite the good trend, a more quantitative comparison is not possible.
But the deviation in the range c/ɛ 2 can be understood:
100
1000
• There is some effect of the specific confining geometry used in the simulation.
We used a harmonic potential and the experimental data was obtained
in a rectangular channel. But this effect is expected to be small and we
already have a relatively good relation by using eq. (3.1) to compare the
geometries. See also section 5.3 for more details.
• Using a large number of segments for flexible chains could bring us into
the regime where self-avoidance due to excluded volume effects finally becomes
important in static problems (cf. discussion on page 18). Checking
this would require a large amount of computer time. Already the simulations
for large N were slow due to the fine discretization. Introducing
self-avoidance in a straightforward way increases the simulation time to an
unacceptable length. To investigate this, a further development of a novel
and fast algorithm is needed.

5.2. THE HARMONIC POTENTIAL 69
R 2 and R 2 ||
1
0.8
0.6
0.4
0.2
0
0.1
R 2
2 R|| 1
Figure 5.9: The ‘dip’-feature in the end-to-end
distance. The mean-square and mean-square
, are shown versus the collision parameter c for the
parallel end-to-end distance, R 2 and R 2 ||
flexible regime ɛ = 10.
10
c
100
1000
• In particular for biopolymers electrostatic effects play an important role.
E.g. DNA is a highly charged polyelectrolyte, where the electrostatic repulsion
will have a stretching effect, leading to a second source of self-avoidance
(cf. [36]), beside the excluded volume effect.
In our simulations electrostatic effects are not taken into account. Therefore
an extra repulsion between the segments is expected, significantly effecting
the flexible regime. This is expected to shift our results in a way, that the
trend of the experimental data in the intermediate regime is fitted better.
Simulations of polyelectrolytes need special complex techniques. Implementation
of these effects is an interesting task for future projects.
It can be noted by investigating figure 5.8, that changing the observable from
the mean-square end-to-end distance to the apparent end-to-end distance does
not change our discussion about the scaling regime leading to eq. (5.4).
The goal of this scaling plot is to use it as a gauge plot for the experiments, to
convert the apparent length measured, into the real contour length of the polymer
strand.
5.2.4 End-to-end distances
In order to understand the conformational changes which take place upon confinement,
we are going to investigate the differences between the full end-to-end
distance and its component along the tube axis. The definition of these observables
was given in section 4.6. The investigation will explain a global feature
qualitatively, only visible for more flexible chains, which is different between these
two observables.

70 CHAPTER 5. SIMULATION RESULTS
R
R ||
Figure 5.10: Behavior of the end-to-end distances of flexible chains upon confinement.
A representative example of these observables for a flexible chain ɛ = 10 is
shown in figure 5.9, where the mean-square and mean-square parallel end-to-end
distance, 〈R2 〉 and R2
|| , are plotted as a function of the collision parameter c.
The full end-to-end distance exhibits a ‘dip’ (local minimum) for c ≈ 1, when the
chain begins to notice the tube through deflections. This dip is absent for R2
|| .
The two limits, c → 0 and c → ∞, are easily understood: In the limit of
an unconfined chain (c < 1) the average of the z-component is exactly a third
of the full average R2
2
|| = 〈R 〉 /3. In the opposite limit, when the confinement
becomes larger (c > 20), both observables, 〈R2 〉 and R2
|| , behave identically
since the polymer is oriented mainly along the tube. Therefore the end-to-end
distance is nearly parallel to the tube axis.
The dip—which has also been observed without explanation in [43]—can be
explained as illustrated in figure 5.10. Upon confinement of the flexible chain,
the initially randomly oriented end-to-end distance is hindered in the perpendicular
directions. This implies that first the end-to-end distance is changed
predominantly perpendicular to the tube axis, only decreasing the value of 〈R2 〉
but leaving R2
|| unchanged. Formation of the dip should start, as seen in figure
5.9, at a value of c 1. Upon further confinement the end-to-end distance
slowly starts giving way in the direction along the tube, finally also increasing
the parallel component.
The behavior of the end-to-end distances upon confinement can also be seen
in the shape of the radial distribution functions (see next section 5.2.5).
The dip is present in all end-to-end distances introduced in section 4.6, as it
is absent for all parallel end-to-end distances.
5.2.5 Radial distribution function
In this section we are going to investigate the probability distribution of the
end-to-end distance or radial distribution function (RDF) (see section 3.3.3 for
a discussion of the RDF in the unconfined case). Figures 5.11 show some representative
numerical results for a chain with flexibility ɛ = 10 and for increasing
collision parameter c.
The RDF for the end-to-end distance R is shown in figure 5.11(a) and for
δR ||
δR
d

5.2. THE HARMONIC POTENTIAL 71
p(R)
p(R ||)
4
2
0
0
4
2
0
0
c = 2
c = 3
c = 2
c = 3
0.2
0.2
0.4
c = 6
c = 10
R
0.6
(a) RDF for R = |R|
0.4
c = 6
c = 10
R ||
0.6
(b) RDF for R || = |R |||
c = 15
c = 20
c = 15
c = 20
0.8
0.8
c = 60
c = 60
Figure 5.11: Comparison of the probability distributions of the end-to-end distances for a
chain of flexibility ɛ = 10 and increasing collision parameter c.
1
1

72 CHAPTER 5. SIMULATION RESULTS
the parallel end-to-end distance R|| in figure 5.11(b). For an unconfined chain
(c 2), p(R) is peaked around R = 0.4 and shows Gaussian behavior, whereas
p(R||) is peaked at R|| = 0. This is simply due to a phase space factor of 4πR 2
included in p(R) resulting from sampling all directions of R. Such a phase space
factor is absent from p(R||) which samples the projection of R on the tube axis.
For intermediate values of c (6 c 20) the RDF p(R) shows an interesting
bimodal distribution. If the collision parameter c is increased from c ≈ 2 the peak
is shifted towards smaller values of R such that the distribution is significantly
skewed. Then, at about c ≈ 6 a second peak at large values of R develops, which
then grows at the expense of the peak at low values of R.
This feature is closely related to the ‘dip’ discussed in the previous section.
The initial drift of the peak towards smaller values is due to the restriction
of the polymers’ configuration perpendicular to the tube axis. Such a behavior
is expected from a linear response argument. When the confinement becomes
stronger, the end-to-end distance evades the transverse compression by
also stretching along the tube axis. As a result the RDF p(R) exhibits two
peaks, one at small and the other at large values of R. This quantifies the heuristic
arguments for the dip of 〈R 2 〉 in the previous section.
For comparison the parallel end-to-end distance distribution is shown in figure
5.11(b) for the same set of parameters. Obviously the average is growing
monotonously to larger values, not showing any bimodal behavior. We have already
eluded to the fact that p(R||) is peaked at R|| = 0 for weak confinement
(due to phase space effects). Hence, increasing the confinement can only result
in a shift of this peak to larger values.
Finally for narrow ‘tubes’ (c 30) both RDFs show exactly the same shape
since the end-to-end distance is now oriented parallel to the tube-axis.
5.3 The hard walls
In the final chapter we turn to the investigation of preliminary simulation results
concerning the conformations of polymers confined by hard walls. In particular,
we are going to analyze the scaling of the end-to-end distance similar as in section
5.2.3. We study two different cross sections of the tube, a circular and square
shaped channel, and leave other geometries for future investigations.
Since we are mainly interested in comparing with the experimental data (cf.
section 5.2.3) we concentrate on the mean apparent end-to-end distance 〈Ra〉.
The simulation data for the two different channel cross sections are shown in figures
5.12 and 5.13. In both cases one finds scaling behavior with scaling variables
(dɛ) −1 and (wɛ) −1 , respectively. Here d is the tube diameter for a circular cross
section and w the channel width for a square cross section.
Comparing both plots with each other and to figure 5.6 shows that the overall
behavior is identical and that there appears to be no major influence of the chosen

5.3. THE HARD WALLS 73
〈Ra〉
1
0.8
0.6
0.4
0.2
0
0.01
0.1
1
(dɛ) −1
10
experimental data
Odijk scaling (fit)
0.5
1
5
10
50
100
1000
10000
Figure 5.12: Mean apparent end-to-end 〈Ra〉 versus the scaling parameter (dɛ) −1 in the
circular cross section tube with diameter d. Besides simulation data for various flexibilities ɛ,
experimental data and the corresponding Odijk scaling fit are shown.
〈Ra〉
1
0.8
0.6
0.4
0.2
0
0.01
0.1
1
(wɛ) −1
10
experimental data
Odijk scaling (fit)
ɛ = 0.5
ɛ = 5
ɛ = 10
ɛ = 50
ɛ = 300
Figure 5.13: Mean apparent end-to-end 〈Ra〉 versus the scaling parameter (dɛ) −1 in the square
cross section tube with width w. Besides simulation data for various flexibilities ɛ, experimental
data and the corresponding Odijk scaling fit are shown.
100
1000

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

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)

78 APPENDIX A. DISCRETE WORM-LIKE CHAIN CORRELATIONS
The persistence length lp was defined through the spatial tangent-tangent
correlation function in eq. (2.12), which is for the discrete model
ˆtm · ˆtn = exp −|m − n| l
. (A.5)
To derive the relation between k and lp the correlation function is calculated
using eq. (A.4)
1 N
ˆtm · ˆtn =
dˆti ˆtm · ˆtn
−βH({ki})
e
Z({ki})
i=1
{ki=k}
1 N
=
dˆti ˆtm · ˆtm+1
ˆtm+1 · ˆtm+2 · · ·
Z({ki})
i=1
· · ·
ˆtn−1 · ˆtn
−βH({ki})
e
{ki=k}
1 ∂
=
Z({ki})
|m−n|
Z({ki})
∂km · · · ∂kn−1 {ki=k}
(A.3)
(A.4) 1
=
I0(k) N−1
∂ |m−n|
N−1
I0(ki) , (A.6)
∂km · · · ∂kn−1
i=1 {ki=k}
where in the second line
ˆti · ˆti+2 = ˆti · ˆti+1
ˆti+1 · ˆti+2 has been used, which
can be seen by writing the product of tangents as the cosines, using trigonometric
theorems and the properties of the averages. For the 3D case below it is shown
in [24, p. 377].
Now using [1, (9.6.28)] in the form I ′ 0(k) = I1(k) gives
|m−n|
I1(k)
ˆtm · ˆtn =
I0(k)
which results in the desired relation in 2D
−1 I1(k)
2D: lp = −l ln
I0(k)
lp
!
= (A.5), (A.7)
. (A.8)
For a given value of lp, k has to be determined numerically, since this formula is
not invertible in a closed form.
A similar calculation, following the above lines, can be done for three dimension.
By substituting dˆti = dφi dcos θi, we obtain for the partition function
Z3D({ki}) = (4π) N
N−1
i=1
sinh ki
, (A.9)
ki

which, after taking all the derivatives of the partition sum and solving for the lp,
results in
3D:
lp = −l ln coth(k) − 1
−1 ,
k
(A.10)
for the desired relation. Some parts of the calculations are presented in [11] for
the equivalent problem of the Heisenberg model for ferromagnets. This result
must be inverted numerically as well.
79

80 APPENDIX A. DISCRETE WORM-LIKE CHAIN CORRELATIONS

Appendix B
Calculation for the confined
polymer
B.1 Undulation correlations
The Fourier back transform of the undulation correlation function, eq. (3.25),
into real space is the integral
〈x(s)x(s ′
dk dk′
)〉 = eiks
2π 2π e−ik′ s ′
〈˜x(k)˜x ∗ (k ′ )〉
dk kBT e
=
2π κ
ik(s−s′ )
k4 + 4l −4 , (B.1)
d
integrating out the δ-function in eq. (3.25).
For evaluation of the remaining integral, we can use the complex analysis. We
have to sum over the residues of the integral kernel, hence we need to find the
poles of ˜ f(k). Expanding the denominator, the zeros are found
˜f(k)
k 4 + 4l −4
d = k 2 + i2l −2
2 −2
d k − i2ld
= k − (1 − i)l −1
−1
d k + (1 − i)ld · k + (1 + i)l −1
−1
d k − (1 + i)ld . (B.2)
These poles, as well as the path of integration, are shown in figure B.1. The
integration over the real axis is done, for s > s ′ , by closing the contour in the
upper half complex plane, where the exponential suppresses any contribution of
the closed path.
The Fourier transformation is then given by
〈x(s)x(s ′ )〉 = kBT
2πκ 2πi
81
Res ˜ f(k) + Res
k=−1+i
˜ f(k)
k=1+i
. (B.3)

82 APPENDIX B. CALCULATION FOR THE CONFINED POLYMER
(−1 + i)l −1
d
(−1 − i)l −1
d
Im k
(1 + i)l −1
d
(1 − i)l −1
d
path of integration
for s > s ′
Re k
path of integration
for s < s ′
Figure B.1: Complex plane of the function ˜ f(k) with indicated poles.
Inserting the residues, we obtain for s > s ′
〈x(s)x(s ′ )〉 = kBT l3 ⎧ ⎫
s − s′
s − s′
⎪⎨ exp i exp −i ⎪⎬
d ld
ld
s − s′
+
exp −
8κ ⎪⎩
1 + i
1 − i ⎪⎭
ld
= kBT l3 d
4 √ 2κ exp
′
s − s′ s − s
− cos −
ld
ld
π
(B.4)
4
which—by additionally closing the contour for s < s ′ in the lower half plane—
results in
〈x(s)x(s ′ )〉 = kBT l3 d
4 √ 2κ exp
− |s − s′ ′ | |s − s |
cos −
ld
ld
π
. (B.5)
4
This is the result in eq. (3.26) used for further calculations.
B.2 End-to-end distance correlation
Here we present the details of the calculation for the mean-square end-to-end
distance 〈R2 (L)〉. We start by calculating the distance between a point r(s) on
the contour and the origin of the polymer r(0), given by the integral
R(s) = r(s) − r(0) =
s
0
dτ t(τ). (B.6)

B.2. END-TO-END DISTANCE CORRELATION 83
s ′
τ ′
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Figure B.2: Integration ranges for the integral eq. (B.8).
Using this, the end-to-end distance correlation function can be calculated by an
integral over tangent-tangent correlation function, eq. (3.34),
〈R(s) · R(s ′ s
)〉 = dτ
=
=
=
s
0
s
0
0
dτ
dτ
1 − ld
2 lp
3
s ′
dτ
0
′ t(τ) · t(τ ′ )
s ′
dτ
0
′ 〈t(τ) · t(τ ′ )〉
s ′
dτ
0
′
1 − ld
√2 −
e
2 lp
|τ−τ′
|
′ |τ − τ |
ld sin −
ld
π
+ 1
4
ss ′ − ld
s s ′
√2 dτ dτ
lp 0 0
′ K(|τ − τ ′ |). (B.7)
The integral kernel has been abbreviated to K(x) = e −x/ld sin (x/ld − π/4). The
integral over the absolute value in the last line has to be split. For s ≥ s ′ the
integration range is shown in figure B.2 and is therefore calculated as
s
0
dτ
1
s ′
dτ
0
′ K(|τ − τ ′ s ′ τ
|) = dτ dτ
0 0
′ K(τ − τ ′ s ′
) +
s
0
+
s ′
s ′
dτ dτ
0
′ K(τ − τ ′ )
s
τ
2
dτ
s ′
dτ
τ
′ K(τ ′ − τ)
, (B.8)
3
where the numbers denote the corresponding areas in the figure. As an example

84 APPENDIX B. CALCULATION FOR THE CONFINED POLYMER
the first integral is calculated here, the others are evaluated in the same way,
s s ′
s (τ−s ′ )/ld
dτ
dτ dx e −x
sin x − π
4
s ′
0
dτ ′ K(τ − τ ′ ) = −ld
s ′
s
= ld
√2
= l2 d √2
= l2 d
2
dτ
e
τ/ld
s ′
(s−s ′ )/ld
τ−s′
− ld sin
0
1
√2 − e s−s′
ld sin
s
− l + e d sin
s
ld
τ − s′
ld
dy e −y sin y −
τ
− l − e d sin τ
ld
s ′ /ld
s/ld
′ s − s
+
ld
π
4
′ s
− π
s′
− l − e d sin
4
ld
dz e −z sin z
− π
. (B.9)
4
The trigonometric relations used can be found in [1]. Calculating the remaining
integrals and combining the results gives
〈R(s) · R(s ′
)〉 = 1 − ld
ss
2 lp
′ + l3 d
2 √
e
2 lp
− |s−s′
|
′ |s − s |
ld sin +
ld
π
4
s
− s
l − e d sin +
ld
π
s′ ′
− s l − e d sin +
4
ld
π
+
4
1
(B.10)
√
2
valid for all s, s ′ .
From this result the mean-square end-to-end distance is obtained in the lines
following eq. (3.37).
B.3 Parallel end-to-end distance correlation
The mean-square end-to-end distance is calculated in detail, by inserting the
derivatives of the average undulations eqs. (3.30) and (3.31) in eq. (3.40)
˙R||(s) ˙ R||(s ′ ) = d2
ds ds ′
R||(s)R||(s ′ )
= 1 − ld
+
2 lp
l2 d
16 l2 p
+ l2 d
8 l2 e
p
−2 |s−s′ |
ld sin 2
′ |s − s |
−
ld
π
4
=〈 ˙r ||(s) ˙r ||(s ′ )〉
. (B.11)
An integration for s ≥ s ′ yields the desired result for the projected length correlation
R||(s)R||(s ′ ) s s ′
= dτ dτ
0 0
′ R||(s) ˙ ˙ R||(s ′ )
= 1 − ld
+
2 lp
l2 d
16 l2
ss
p
′ + l2 d
8 l2 s s ′
dt dt
p 0 0
′ K(|τ − τ ′ |), (B.12)

B.3. PARALLEL END-TO-END DISTANCE CORRELATION 85
with K(x) = e −2x/ld sin 2 (x/ld − π/4)). The integration over K(|τ − τ ′ |) has to
be treated as in eq. (B.8).
Evaluating only the first integral explicitly
τ
0
dτ ′ K(τ − τ ′ ) = 1
2
τ
dτ
0
′ τ−τ′
−2 l e d
0
and performing the second integration gives
s ′
0
dτ
τ
0
dτ ′ K(τ − τ ′ ) = ld
8
1 − cos
ld
′ τ − τ
2 −
ld
π
4
= − ld
dx e
4 2τ/ld
−x (1 − sin x)
= ld
2τ
− l 1 + e d −2 + sin
8
2τ
+ cos 2τ
, (B.13)
s ′
2τ
− l dτ 1 + e d −2 + sin
0
2τ
+ cos
ld
2τ
ld
= l2 2s ′ /ld
d
dτ
16 0
1 − 2e −x + e −x (sin x + cos x)
= l2 d
16
2s ′
ld
2s′
− l + e d
ld
2 − cos 2s′
ld
ld
2s′
− l + e d 2 − cos 2s′
ld
− 1 . (B.14)
The remaining integrals are treated equivalently. Finally we obtain the following
result valid for all s and s ′
R||(s)R||(s ′ )
= 1 − ld
ss
2 lp
′ + r||(s)r||(s ′ )
= 1 − ld
+
2 lp
l2 d
16 l2
ss
p
′ + l3 d
32 l2 min(s, s
p
′ )
+ l4 d
128 l2
− e
p
−2 |s−s′
|
ld 2 − cos 2 |s − s′
|
(B.15)
ld
2s
− l + e d 2 − cos 2s
− 1 .
This result is used in the lines following eq. (3.41).

86 APPENDIX B. CALCULATION FOR THE CONFINED POLYMER

Glossary
L filament length
lp
persistence length
ɛ = L/lp
flexibility
ld
Odijk deflection length
c = L/ld
collision parameter
s arc length
r(s) trajectory
r⊥(s) = (x(s), y(s)) undulations
r||(s) stored length
ti, t(s)
ˆti
tangent, segment
normalized tangent
l = |ti| segment length
φtt(s, s ′ ) = 〈t(s) · t(s ′ )〉 tangent-tangent correlation function
b Kuhn segment length
N number of segments
v monomer volume
β = (kBT ) −1 inverse temperature
H Hamiltonian
Z partition sum
κ bending modulus
ε interaction energy
d tube diameter (or dimension)
w tube width
γ harmonic potential strength
R(s) distance to current position
R(L) end-to-end distance
p(R) radial distribution function (RDF)
Rg(L) radius of gyrations
R||(L), Ra(L) parallel, apparent end-to-end distance
〈R〉,
R|| , 〈Ra〉 mean (parallel, apparent) end-to-end distance
〈R2 〉, R2
2
|| , 〈Ra〉 mean-square (parallel, apparent) end-to-end dist.
2 Rg mean-square radius of gyration
〈x2 (s)〉 mean-square undulation
87

88 APPENDIX B. CALCULATION FOR THE CONFINED POLYMER

Bibliography
[1] M. Abramowitz and I. A. Stegun, editors. Pocketbook of Mathematical Functions.
Verlag Harri Deutsch, 1984.
[2] S. R. Aragón and R. Pecora. Dynamics of wormlike chains. Macromolecules,
18:1868–1875, 1985.
[3] R. Austin. Nanopores: The art of sucking spaghetti. Nature Materials,
2:567–568, 2003.
[4] K. Binder and D. W. Heermann. Monte Carlo Simulation in Statistical
Physics: An Introduction. Springer-Verlag Berlin Heidelberg, 2002.
[5] C. Bustamante, Z. Bryant, and S. B. Smith. Ten years of tension: singlemolecule
DNA mechanics. Nature, 421:423–427, 2003.
[6] C. Bustamante, J. C. Macosko, and G. J. Wuite. Grabbing the cat by the
tail: manipulating molecules one by one. Nature Reviews Molecular Cell
Biology, 1:130–136, 2000.
[7] H. E. Daniels. The statistical theory of stiff chains. Proceedings of the Royal
Society of Edinburgh, 63A:290–311, 1952.
[8] P.-G. de Gennes. Scaling Concepts in Polymer Physics. Cornell University
Press, 1979.
[9] M. Doi. Introduction to Polymer Physics. Clarendon Press, Oxford, 1996.
[10] M. Doi and S. F. Edwards. The Theory of Polymer Dynamics. Clarendon
Press, Oxford, 1986.
[11] M. E. Fisher. Magnetism in one-dimensional systems — the Heisenberg
model for infinite spin. American Journal of Physics, 32:343–346, 1963.
[12] H. Flyvbjerg and H. G. Petersen. Error estimates on averages of correlated
data. The Journal of Chemical Physics, 91:461–466, 1989.
[13] D. Frenkel and B. Smit. Understanding Molecular Simulation, From Algorithms
to Applications. Academic Press, 2002.
89

90 BIBLIOGRAPHY
[14] E. Frey. Physics in cell biology: On the physics of biopolymers and molecular
motors. ChemPhysChem, 3:270–275, 2002.
[15] E. Frey and T. Franosch. Statistische Physik — Biologische Physik und Soft
Matter. Lecture notes (Freie Universität Berlin), WS 2002/03.
[16] E. Frey and K. Kroy. Brownian motion: paradigm of soft matter & biological
physics. Review submitted to Annalen der Physik, 2004.
[17] E. Frey, K. Kroy, and J. Wilhelm. Viscoelasticity of biopolymer networks and
statistical mechanics of semiflexible polymers. In S. Malhotra and J. Tuszynski,
editors, Advances in Structural Biology, volume 5, pages 135–168. Jai
Press, 1998.
[18] R. Granek. From semi-flexible polymers to membranes: Anomalous diffusion
and reptation. Journal de Physique II, 7:1761–1788, 1997.
[19] H. Kleinert. Path Integrals in Quantum Mechanics, Statistics, Polymer
Physics and Financial Markets. World Scientific, third edition, 2004.
[20] O. Kratky and G. Porod. Röntgenuntersuchung gelöster Fadenmoleküle.
Recueil Travaux Chimiques (Pays Bas), 68:1106–1122, 1949.
[21] K.-D. R. Kroy. Viskoelastizität von Lösungen halbsteifer Polymere. PhD
thesis, Technische Universität München, 1998.
[22] D. P. Landau and K. Binder. A Guide to Monte Carlo Simulations in Statistical
Physics. Cambridge University Press, 2000.
[23] L. D. Landau and E. M. Lifschitz. Lehrbuch der theoretischen Physik,
Band VII, Elastizitätstheorie. Akademie Verlag Berlin, 1965.
[24] L. D. Landau and E. M. Lifschitz. Lehrbuch der theoretischen Physik,
Band V, Statistische Physik Teil 1. Akademie Verlag Berlin, 1979.
[25] G. Lattanzi, T. Munk, and E. Frey. Transverse fluctuations of grafted polymers.
Physical Review E, 69:021801, 2004.
[26] M. le Bellac. Quantum and Statistical Field Theory. Oxford University Press,
1991.
[27] P. L’Ecuyer. Maximally equidistributed combined Tausworthe generators.
Mathematics of Computation, 65:203–213, 1996.
[28] P. L’Ecuyer. Random Number Generation, chapter 4 of the Handbook on
Simulation, Jerry Banks Ed., pages 93–137. Wiley, 1998.

BIBLIOGRAPHY 91
[29] P. L’Ecuyer. Tables of maximally-equidistributed combined LSFR generators.
Mathematics of Computation, 68:261–269, 1999.
[30] M. Lüscher. A portable high-quality random number generator for lattice
field theory simulations. Computer Physics Communications, 79:100–110,
1994.
[31] F. C. MacKintosh and C. F. Schmidt. Microrheology. Current Opinions in
Colloid & Interface Science, 4:300–307, 1999.
[32] B. Maier and J. O. Rädler. Conformation and self-diffusion of single DNA
molecules confined to two dimensions. Physical Review Letters, 82:1911–
1914, 1999.
[33] J. F. Marko and E. D. Siggia. Stretching DNA. Macromolecules, 28:8759–
8770, 1995.
[34] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and
E. Teller. Equation of state calculations by fast computing machines. The
Journal of Chemical Physics, 21:1087–1092, 1953.
[35] K. P. N. Murthy. Monte Carlo Methods in Statistical Physics. Universities
Press (India) Private Ltd, 2004.
[36] P. Nelson. Biological Physics: Energy, Information, Life. W. H. Freeman &
Co, 2003.
[37] T. Odijk. On the statistics and dynamics of confined or entangled stiff
polymers. Macromolecules, 16:1340–1344, 1983.
[38] W. Reisner et al. Statics and dynamics of single DNA molecules confined in
nanochannels. Unpublished.
[39] M. Rubinstein and R. H. Colby. Polymer Physics. Oxford University Press,
2003.
[40] N. Saitô, K. Takahashi, and Y. Yunoki. The statistical mechanical theory of
stiff chains. Journal of the Physical Society of Japan, 22:219–226, 1967.
[41] D. W. Schaefer, J. F. Joanny, and P. Pincus. Dynamics of semiflexible
polymers in solution. Macromolecules, 13:1280–1289, 1980.
[42] P. J. Schneider and D. H. Eberly. Geometric Tools for Computer Graphics.
Morgan Kaufmann Publishers, 2002.
[43] C.-Y. Shew. Conformational behavior of a single polymer chain confined
by a two-dimensional harmonic potential in good solvents. The Journal of
Chemical Physics, 119:10428–10437, 2003.

92 BIBLIOGRAPHY
[44] J. O. Tegenfeldt et al. The dynamics of genomic-length DNA molecules
in 100-nm channels. Proceedings of the National Academy of Sciences,
101:10979–10983, 2004.
[45] N. van Kampen. Stochastic Processes in Physics and Chemistry. North-
Holland Personal Library, Elsevier Science B.V., 2001.
[46] C. H. Wiggins et al. Trapping and wiggling: Elastohydrodynamics of driven
microfilaments. Biophysical Journal, 74:1043–1060, 1998.
[47] J. Wilhelm and E. Frey. Radial distribution function of semiflexible polymers.
Physical Review Letters, 77:2581–2584, 1996.
[48] H. Yamakawa and M. Fujii. Wormlike chains near the rod limit: Path integral
in the WKB approximation. The Journal of Chemical Physics, 59:6641–6644,
1973.

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