Diffusion of fluorescent molecules in micro-
and nanostructured environments
DISSERTATION
zur Erlangung des akademischen Grades
doctor rerum naturalium
(Dr. rer. nat.)
im Fach Physik
eingereicht am
Fachbereich Physik der
Universität Dortmund
im Juni 2007
von
Herrn Dipl.-Phys. Thorsten Pieper
geboren am 20.7.1975 in Münster
Gutachter:
1. Professor Dr. Dieter Suter
2. Dr. Harald P. Mathis
Zusammenfassung
Thema der Arbeit ist die experimentelle Untersuchung von molekularer Diffusion in flüssigen
Umgebungen. Die Untersuchung von diffundierenden Molekülen liefert Informationen über die
Moleküle selbst, als auch über die Struktur des umgebenden Mediums. Diese Arbeit befasst
sich mit Diffusionsphänomenen auf Größenskalen unterhalb von 100µm. Die flüssige Phase
besteht aus Wasser oder binären Wasser/Tensid-Mischungen. Die Mikrostrukturen sind Flüs-
sigkeitskanäle, die in Rohlinge aus kristallinem Silizium geätzt werden. In diesen Kanälen wird
die diffusionsbeschränkte Durchmischung an einer Flüssigkeitsgrenzfläche untersucht. Diese Ex-
perimente fanden im stationären Fluss statt. Umgebungen in der Größenordnung von Nanometern
werden aus Wasser/Tensid-Mischungen hergestellt, die aufgrund von Selbstorganisation supra-
molekulare Ordnungen ausbilden. Von den möglichen Ordnungen werden speziell die mizellare
Phase und die lamellare Phase untersucht. Der Einfluss von Cholesterin auf die Mobilität von
Fluoreszenzmolekülen in der lamellaren Phase steht im Mittelpunkt des Interesses. Als experi-
mentelle Methode wird in erster Linie die Fluoreszenzkorrelationsspektroskopie (FCS) eingesetzt.
Desweiteren werden Grundlagen zur Verfolgung von Molekülen gelegt, indem Videoaufnahmen
von Fluoreszenzpartikeln ausgewertet werden. In dieser Arbeit werden tiefergehende Erkennt-
nisse in die Natur der Brownschen Bewegung in nahezu idealen Systemen erarbeitet. Aus den
Experimenten lassen sich Schlussfolgerungen für Untersuchungen in natürlichen Umgebungen
ziehen, wie sie zum Beispiel in eukaryotischen Zellen vorliegen.
Abstract
The thesis deals with phenomena of molecular diffusion in liquid environments. The examination
of diffusing molecules provides information about the molecules themselves, and also about the
structure of the surrounding media. This work deals with diffusion phenomena on length scales
below 100µm. The liquid phase consists of water or water/surfactant emulsions. The microstruc-
tures are fluidic channels etched in wafers of crystalline silicon. In these channels, diffusion-
-limited mixing at a liquid boundary is examined. These experiments took place in a stationary
liquid flow. Environments on the nanoscale are made of binary liquids forming supramolecular
orders by self-assembly. From these structures, the micellar and the lamellar structure are exam-
ined in particular. The focus is on the influence of cholesterol on the mobility of probe molecules
in the lamellar phase. The main experimental method is Fluorescence Correlation Spectroscopy
(FCS). Additionally, fundamentals for the tracking of single molecules are established by analyz-
ing video recordings of fluorescing particles. The thesis provides advanced insight into the nature
of molecular motion in nearly ideal systems. Further conclusions for investigations in natural
environments, as they exist in eukaryotic cells, can be drawn from the experiments.
i
Contents
1 Diffusion and fluorescence 3
1.1 Molecular motion in liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.1 Concentration fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.2 Diffusion and Brownian motion . . . . . . . . . . . . . . . . . . . . . . 6
1.1.3 Concentration gradients . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.4 Flow and diffusion in microchannels . . . . . . . . . . . . . . . . . . . 11
1.2 Fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.1 Fluorescence dyes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.2 Photostability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Confocal optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.1 Focus calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4 Fluorescence Correlation Spectroscopy . . . . . . . . . . . . . . . . . . . . . . 19
1.4.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4.2 Variance of the autocorrelation . . . . . . . . . . . . . . . . . . . . . . 22
1.4.3 Triplet state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.4 Directed flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 Simulation of FCS 25
2.1 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Photon emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 Tests of the simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4 Diffusion in oriented planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3 Fluorescence detection 37
3.1 Open microscope setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2 Fluorescence Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.1 Camera properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.2 Single particle measurements . . . . . . . . . . . . . . . . . . . . . . . 43
3.3 ConfoCor 2 setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
i
4 Diffusion in microchannels 46
4.1 Flow profiles in microchannels . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.1.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.1.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1.3 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1.4 Analysis and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2 Diffusion limited mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2.2 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2.3 Analysis and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3 Particle tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3.1 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3.2 Analysis and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.4.1 Outlook: Real time image processing and 3D molecule tracking . . . . . 62
5 Diffusion in surfactant bilayers 64
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2 Membrane system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.2.1 Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.2.2 Preparation of Surfactant Emulsions . . . . . . . . . . . . . . . . . . . 66
5.2.3 Micellar phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.2.4 Lamellar phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.3 FCS measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.3.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.3.2 Autocorrelation model functions . . . . . . . . . . . . . . . . . . . . . . 69
5.3.3 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6 Conclusions 77
A List of manufacturers 80
B Publications and conference contributions 82
ii
List of Figures
1-1 Macroscopic and local concentration . . . . . . . . . . . . . . . . . . . . . . . . 5
1-2 Simulated Poissonian distributions in subvolumes . . . . . . . . . . . . . . . . . 6
1-3 Schematic of a concentration gradient . . . . . . . . . . . . . . . . . . . . . . . 8
1-4 Diffusion with reflecting walls . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1-5 Probability distributions of a 1D Random Walk . . . . . . . . . . . . . . . . . . 10
1-6 Jablonski diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1-7 Chemical structures of Rhodamine 6G and octadecyl Rhodamine B . . . . . . . . 14
1-8 Fluorescence emission vs. excitation rate . . . . . . . . . . . . . . . . . . . . . 15
1-9 Fluorescence excitation and emission spectrum of Rhodamine 6G . . . . . . . . 15
1-10 Inverted confocal microscope for FCS and definition of the focus parameters . . . 18
2-1 Fluorescence intensity signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2-2 Theoretical vs. simulated diffusion coefficient . . . . . . . . . . . . . . . . . . . 32
2-3 Average autocorrelation and standard deviation from 12 simulations . . . . . . . 33
2-4 Standard deviation compared to theoretical approximations . . . . . . . . . . . . 33
2-5 Concept of the orientation model . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2-6 Simulated correlation curves for two-dimensional diffusion under different angles 36
2-7 Simulated correlation curves of the orientation average . . . . . . . . . . . . . . 36
3-1 Upright setup of the open microscope . . . . . . . . . . . . . . . . . . . . . . . 38
3-2 Intensity profile of the Argon ion laser . . . . . . . . . . . . . . . . . . . . . . . 39
3-3 Nanopositioner P-734 and confocal microscope setup . . . . . . . . . . . . . . . 40
3-4 Measurement setup for image detection and camera . . . . . . . . . . . . . . . . 40
3-5 Schematic drawing of the imaging setup. . . . . . . . . . . . . . . . . . . . . . . 41
3-6 Maximum exposure time vs. ROI . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4-1 Photograph of Y-mixer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4-2 Setup for the flow measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4-3 Sketch of the junction between the inlets with trapezoidal cross sections . . . . . 48
4-4 CFD simulation of the velocity profile . . . . . . . . . . . . . . . . . . . . . . . 49
iii
4-5 Correlation curves for fast and slow running fluid . . . . . . . . . . . . . . . . . 50
4-6 Flow velocity vs. y and z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4-7 CFD simulation of diffusive mixing . . . . . . . . . . . . . . . . . . . . . . . . 54
4-8 CFD results for the concentration distribution . . . . . . . . . . . . . . . . . . . 54
4-9 Correlation data from the mixing experiment . . . . . . . . . . . . . . . . . . . 55
4-10 Velocity profiles in the outlet channel . . . . . . . . . . . . . . . . . . . . . . . 56
4-11 Captured frame of diffusing particle . . . . . . . . . . . . . . . . . . . . . . . . 59
4-12 Mean square distance for the particle . . . . . . . . . . . . . . . . . . . . . . . . 60
5-1 Phase diagram of C12E5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5-2 Chemical structure of cholesterol. . . . . . . . . . . . . . . . . . . . . . . . . . 66
5-3 Spherical and rod-like micelle . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5-4 Lamellar bilayer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5-5 FCS data for the micellar phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5-6 Example for the FCS data from the surfactant/cholesterol/water emulsion . . . . 72
5-7 Diffusion coefficient D in the lamellar phase with 60 wt % surfactant . . . . . . . 73
5-8 Diffusion coefficient D in the lamellar phase with 80 wt % surfactant . . . . . . . 73
5-9 Counts/molecule value vs. cholesterol content . . . . . . . . . . . . . . . . . . . 74
iv
Acknowledgments
The main part of this work was done in the Fraunhofer BioMOS research group in Sankt Augustin.
As my scientific advisor, I especially thank Dr. Harald P. Mathis for his continuous mentoring. He
introduced me into a new and interesting field and contributed greatly to successful collaborations
with other institutions.
Special thanks goes to Professor Dieter Suter from the University of Dortmund who supervised the
progress of my work during three years. The chapter about bilayer systems is based on his idea. I
thank him for giving me the chance to present my results in several seminars and on conferences.
From the same group, I thank Svetlana Markova for her engagement in our common project.
I express my gratitude towards Professor Masataka Kinjo from the Hokkaido University for his
generous invitation to his laboratory. I am grateful that I was allowed to gain insight into research
and life in Japan. The time in Sapporo was highly instructional for me. From the laboratory, I
especially thank Dr. Goro Nishimura who gave me important advice for the experiments with
microemulsions. For all situations when life became difficult I thank Yu Ohsugi who has become
a good friend. He, and all other lab members, have left constant memories. I look back with joy
and hope to meet you again in the future. For the financial support, I thank the Japan Society for
the Promotion of Science (grant PE05567).
I also thank Professor Günter Wagner for his supervision of our students and his insights into
microengineering. He always kept clear sight even in times of trouble.
Dr. Jörg Ackermann was and still is helping with difficult theoretical questions. He had a main
part in developing the computer simulation.
From Dr. Thomas Kirner I received good advice about how to improve microfluidic devices.
The CFD simulations were carried out in close cooperation with Dr. Barbara Steckel at Fraunhofer
SCAI. I thank Dipl.-Math. Andrea Wagner for her support in questions of hydrodynamics. She
modeled the simulation grid and managed the computations using the FLUENT software package.
Benjamin Greiner is my friend. All questions concerning our common work were discussed with
him in a close and trustful atmosphere. I also thank him for proofreading.
The following students contributed to this work with all their time and energy. Volker Inden
characterized the nanopositioning unit. Kai Lueghausen did a great job in setting up the real time
image processing devices. Embarek Belhadj constructed a temperature control unit with high
v
precision. Christian Müller offered his help and experience for the experiments with enzymatic
reactions.
Erwin Rambow is the number one person to ask when electrical and mechanical setups have to be
realized quickly.
I would also like to thank some former members of the BioMOS group. These are Dipl.-Ing. Mar-
tina Jünger and Dipl.-Ing. Thomas Palutke who were responsible for fabricating the microfluidic
devices.
Last not least, I thank Dr. Antonia Delago for her encouragement and proofreading.
vi
Introduction
To date the principles of molecular organization during the life cycle of a cell are widely unknown.
Understanding cellular mechanisms contributes to the development of new drugs as well as med-
ical devices for diagnosis and therapy. Recent progress in microscope technology now enables us
to access critical questions concerning these applications. Particularly fluorescence methods have
developed quickly and have become an important tool for visualizing processes in cells. This has
also influenced some related fields, such as Biophysics, Microengineering, and Material science.
Present activities aim at improving the optical resolution, and extending the information yield
from measurements [1, 2, 3].
While physics has succeeded in describing observations on the macroscopic and on the atomistic
level, intermediate systems represent still a borderline between microscopic and nanoscopic world.
Because of the large number of molecules involved, new approaches are necessary to describe the
observed phenomena adequately.
Biological cells combine a vast number of functional molecules, structures and interaction pro-
cesses. For a principle understanding of cellular functions, it is thus useful to simplify the regarded
system. This can be achieved by transferring the system of interest to an in vitro environment. In
vitro experiments are essential to reduce the complexity and the number of unknown parameters.
Many biomedical applications require the implementation of biochemical assays in small devices.
For diagnostic applications, for example, in vitro assays are usually the preferred choice. They
can be standardized easily, so that the time for each test can be minimized. Moreover, practical
reasons demand devices that are hygienic, disposable, and small. Because the majority of appli-
cations is based on testing human body fluids (blood, urine, sweat, etc.), there are often liquid
specimen involved. For certain purposes, e. g. for blood examinations, it is also desirable that
these devices work continuously. This is one reason, why it is necessary to design continuous
flow reaction chambers.
The handling of liquids on the micrometer scale gives rise to many unsolved problems. The
current work focuses on certain issues dealing with molecular diffusion in artificial environments.
These environments include silicon based microchannels and lipid/water microemulsions showing
supramolecular order. These structures have length scales reaching from 100µm down to the
1
nanometer range. The transition from the micrometer to the nanometer scale is interesting in so
far as fundamental changes in the diffusion behavior can be observed.
The first objective of this thesis is to characterize a particular microfluidic channel geometry in
its laminar flow behavior. As a next step, the properties of the channel for diffusive mixing are
examined using fluorescent molecules in very low concentrations. Numerical simulations and
experiments using Fluorescence Correlation Spectroscopy (FCS) as the method of analysis are
combined to give a detailed description of the mixing device. The channels are test environments
for realizing enzymatic reactions inside microfluidic channels. The necessity for continuous flow
systems in assay technology has been noted by a number of researchers [4].
FCS is a microscopy method basically used to determine low concentrations and diffusion coeffi-
cients. In its simplest form, FCS does not provide image resolution.
A different set of experiment was performed using a newly developed image detection microscope.
By analyzing images of fluorescing particles, it is shown, how specific information about a single
particle can be extracted from such images. The experiment and instrument setup are part of a
bigger project on a new fluorescence detection system for tracking single molecule motion. By
characterizing the optical properties of the system, first steps towards Single Molecule Detection
(SMD) are undertaken successfully.
As a second objective, diffusion in lamellar bilayers is studied using a surfactant/water model
system, which exhibits some fundamental properties of biological membranes. These systems
show characteristic features of nanostructured environments. Again, FCS was used as the method
of choice. A computer simulation, which is developed in chapter 2, is applied for testing a new
FCS model used to describe the measurement data. The experiments were carried out entirely in
vitro. No biological cells were examined.
The thesis is structured as follows: Chapter 1 gives a short overview about the mathematical
background of diffusion and fluorescence. Chapter 2 explaines the development of the computer
simulation for FCS. In chapter 3, the techniqual equipment which was used for the experiments is
described. Three different microscope setups were used. In chapter 4, experimental measurements
in microfluidic channels are presented. The results are analyzed and interpreted for the desired
application. In the second part, the trajectory of a single fluorescing particle is recorded and ana-
lyzed. Chapter 5 is a separate study dealing with diffusion in model membranes. The simulation
from chapter 2 is applied to evaluate a new model equation for FCS. In the conclusion (chapter 6),
the results are reviewed and evaluated in the present context of research. Possible consequences
for investigating diffusion in natural environments are given.
2
Chapter 1
Diffusion and fluorescence
Molecular motion is often based on stochastic behavior [5]. The observation that thermal fluctua-
tions in liquids exist, were first described by Brown in 1827 [6]. He discovered the effect in a light
microscope when he saw a characteristic tremor on swimming pollen. Hence, this phenomenon is
generally referred to as Brownian motion.
In this chapter, the microscopic description of molecular motion is outlined in section 1.1. The
general concept of Brownian motion is valid for liquids, gas phases, and also solids. Here, only the
liquid phase is of importance. In the presented stochastic theory, the molecules in a solution are
represented by discrete objects on an imaginary lattice. Transition probabilites between the lattice
sites are defined according to the mobility properties of the molecules. The observed macroscopic
phenomena can be derived from the microscopic theory. The time course of a concentration
gradient of a dissolved substance is described by Fick’s laws. They are discussed in section 1.1.3.
With respect to the conducted experiments, the special case of diffusion in a confined vessel is
pointed out.
Light emission due to fluorescence is the essential attribute to make the visual observation of
molecules possible. The term fluorescence, which subsumes the physical effects of electronic
excitation and photon emission, is explained in section 1.2. The three-state model of electronic
states in a fluorescent molecule is cited. It allows to give a relation between the local excitation
intensity and the photon emission rate of a single molecule.
As the main experimental technique, the principle of confocal microscopy is illustrated in section
1.3. In confocal microscopy, a beam of laser light is focused into the liquid specimen. The size
of the observation volume is given by the diffraction limited spot of the incident light. The fluo-
rescence intensity trace as the outcome of a confocal fluorescence experiment is utilized. There
are different ways to analyze the intensity traces and extract relevant information from it. Here,
Fluorescence Correlation Spectroscopy (FCS) is mainly used throughout chapters 2-5. Important
equations for analyzing the correlation curves are derived from basic assumptions.
3
1.1. MOLECULAR MOTION IN LIQUIDS
For small observation volumes, as they appear in FCS, the local fluctuations of the solute molecules
become significant. It is therefore suggestive to begin with the definition of the concentration of a
substance in solution.
1.1 Molecular motion in liquids
The fluid state of aggregation is characterized by the fact, that the force driven by thermal energy is
almost equal to the intermolecular forces which can be described by the Lennard-Jones potential.
In contrast to the solid state, the molecules are able to exchange their position with their neighbors.
During a melting transition from solid to liquid state, the density is usually slightly reduced. Water
is a well-known exception from this rule. A correct description of liquids requires a combined
approach of classical and quantum mechanical physics [7]. Gases and liquids can be regarded as
continuous media in many experiments. This means, on macroscopic length scales all properties
like density, pressure or velocity can be treated as continuous variables. These quantities are
a result from averaging over a large ensemble of particles [8]. But, because matter consists of
molecules and atoms, a lower limit for the continuum assumption exists. The smaller the observed
structures are, the higher the natural fluctuations of macroscopically defined quantities become.
For the mathematical treatment of diffusion the microscopic and the macroscopic description are
distinguished.
1.1.1 Concentration fluctuations
The concentration C of a substance in a volume (of a gas or solvent) is defined by the ratio of the
total particle number Ntotal and the total volume Vtotal
C = NtotalVtotal
(1.1)
Usually, the true concentration C is not determined by counting all particles in the total volume,
but by measuring the average number of particles N in a smaller subvolume V . Implicitly, it is
assumed that this concentration C˜ is equal in each subvolume, which is equivalent to C˜ = C.
This is true, if the system is in equilibrium, and the subvolume still contains a number of particles
n ≫ 1. When going to very small subvolumes, this assumption is not valid anymore. Due to
spatial fluctuations, statistical fluctuations arise (Fig. 1-1). The distribution of the real particle
number N in different subvolumes V then follows a Poissonian distribution with a parameter
µ = N , which is equal to the average number of particles [9]
P (n | µ = N) = N
n
n! e
−N . (1.2)
4
1.1. MOLECULAR MOTION IN LIQUIDS
Figure 1-1. Illustration of the difference between macroscopic concentration and local concentration.
While there are almost equal numbers of particles in the four squares of side length L/2, in those with
length L/4 fluctuations become evident.
If the number of particles is represented by a random variable X, the fluctuations are given by
the variance V ar(X) of X. Poisson distributions are characterized by the fact that the variance is
equal to the mean number of particles
Var(X) = N. (1.3)
The relative fluctuation of a number counting experiment is given by
∆N
N =
√
Var(X)
N =
1√
N
. (1.4)
In Fig. 1-2 three Poisson distributions with different parameters are shown.
Remarks
The Poisson distribution becomes a Gaussian distribution for large N according to the central
limit theorem (see e. g., [10]).
Concentrations are given in units of 1 mol. The molar concentration c is the number of particles
expressed by multiples of Avogadro’s number NA
c = NV NA
. (1.5)
The determination of a concentration requires that a number of molecules is related to a volume.
If the volume is small, the measurement error is governed by the statistical number fluctuation. To
reach a relative error of 10% in the concentration measurement an average of 100 molecules must
be present in the volume. A higher precision can be reached when the measurement is extended
5
1.1. MOLECULAR MOTION IN LIQUIDS
µ = 10.0
µ = 1.0
µ = 0.1
number n
P(
n
)
2520151050
1
0.1
0.01
0.001
1e-04
Figure 1-2. Simulated Poissonian distributions of particles in 104 subvolumes. Uniform distributions
of 103, 104 and 105 particles in the total volume underly the data points. The theoretical functions
(solid lines) result from Eq. 1.2, with the factorial being replaced by the gamma function Γ(n). µ is
the average number of particles in a subvolume.
over time. Then, temporal fluctuations (e. g., due to Brownian motion) contribute to an increased
precision. This principle is availed in Fluorescence Correlation Spectroscopy.
1.1.2 Diffusion and Brownian motion
In correlation spectroscopy, the fluctuations in a small observation volume are utilized to deter-
mine the concentration and the diffusion coefficient. When resting particles are observed over a
period of time, number fluctuations do not exist. Particle motion, on the other hand, leads to a
varying particle number inside the observation volume and in the adjacent volumes. This fluctu-
ation implicates a temporal and spatial correlation of the particle number. The correlation time is
the characteristic resident time for a particle inside the observation volume. If the number of par-
ticles is determined in time intervals much larger than the correlation time, then the measurements
are uncorrelated and statistically independent.
The term diffusion is ambigious. It is used for the microscopic process of Brownian motion, and
also for the way to equilibrium in a concentration gradient. In the first case, the microscopic mo-
tion is meant. The second case implies the macroscopic consequences of the stochastic molecular
motion. It is often visualized as a vessel of two different substances which are separated by a wall
(Fig. 1-3). After removing the wall, the system tends to equalize the concentration gradient. This
effect can be traced back to the molecular Brownian motion. When pollen, smaller particles, or
molecules frequently collide momentum is transferred between them. This leads to a change in
direction of motion which stays linear until the following collision. The frequency of collisions
6
1.1. MOLECULAR MOTION IN LIQUIDS
can be calculated from the mean free pathlength λ¯ of the molecules and their average velocity v¯
[11]
fcoll =
v¯
λ¯ . (1.6)
When the term Brownian motion is used in the sense of migration of the same molecule type within
the molecular order, the process is called self-diffusion. Water molecules at normal conditions1
with v¯ ≃ 1m/s and λ¯ ≃ 1 pm [7] collide at a frequency of fcoll = 1012/second. Brownian motion
cannot be observed on bigger particles in a liquid because the mass of the particle grows faster
than the total momentum due to collisions with the surrounding molecules.
As a result, there is no net force on the particle. A classical solution for the problem of molecular
motion is therefore impossible to obtain. Assuming discrete collisions, the trajectory of a molecule
is continuous, but not differentiable. Of course, strictly, the idea of molecules as rigid bodies is
not justified.
The motion of a diffusing molecule can be formally discretized by dividing the motion path into
intervals. This gives rise to the concept of the Random Walk.
Random Walk
A Random Walk (RW) is a path between discrete sites. At each end point, the next step is chosen
randomly from a set of sites. The Random Walk can be generalized leading to the formalism of
Markov chains. In the limit of infinitely small steps, the path becomes a continuous curve (Wiener
process). Some restrictions can be imposed on a Random Walk to simplify the computational
realization, e. g.
• The sites are represented by the cross points of a rectangular lattice.
• The distance ǫ between steps and the length of the time interval ∆t are constant. This
implies that the following position is chosen from the neighboring sites.
• The probability for staying at the present position is zero.
The simple RW is appropriate to simulate Brownian motion with finite precision given by the
lattice parameter ǫ. The RW is symmetric, which means that the transition probabilities P+ = P−
are independent from the direction of movement. Fig. 1-5 shows three particle distributions
after a 1D Random Walk of duration q∆t. If the origin is the starting point of all molecules, the
resulting distribution follows a Gaussian normal distribution. An asymmetric RW with transition
1using the definition: 101.325 kPa, 25◦C
7
1.1. MOLECULAR MOTION IN LIQUIDS
Figure 1-3. Schematic display of a concentration gradient in a vessel before and after removing
the separating wall (solid line). The concentration is shown in a gray scale and as a function of x
(solid-dotted line).
probabilities P+ > P− generates a net drift with velocity v in the positive x-direction
P+ =
1
2 +
v∆t
2 ǫ
P− =
1
2 −
v∆t
2 ǫ . (1.7)
1.1.3 Concentration gradients
The macroscopic view of diffusion is based on the observation that the concentration of a sub-
stance tends to equalize. Fig. 1-3 shows schematically how a concentration gradient levels off
when the separating wall is removed. The effect is a result of particle fluctuations. Mathemati-
cally, it is described by the flow rate F which is defined as the particle number or the mass/(time
× area). The flow rate of a diffusing substance through the unit area is proportional to the concen-
tration gradient perpendicular to this area [12]
~F = −D ~∇C(~r, t). (1.8)
The proportionality constant D is the diffusion coefficient. D has the unit area/time. Fick’s second
law [5] can be derived from Eq. 1.8 by multiplying it with the divergence operator. Subsequent
inserting of the continuity equation leads to
∂C(~r, t)
∂t = D∆C(~r, t). (1.9)
The diffusion equation describes the temporal and spatial propagation of a concentration distribu-
tion. The integration of Eq. 1.9 gives the concentration as a function of position and time. Assum-
ing an initial point-like distribution of molecules, which is equivalent to C(x, t0) = δ(x−x0), the
8
1.1. MOLECULAR MOTION IN LIQUIDS
solution of the diffusion [12] equation is given by
C = N
2
√
πDt
e− x
2
4Dt . (1.10)
where N is the total number of molecules. In two and three dimensions, the following identities
hold
C = N4πDt e
− r
2
4Dt
C = N8(πDt)3/2 e
− r
2
4Dt . (1.11)
With different initial and boundary conditions, the analytic solutions to Eq. 1.9 can become com-
plex. In the special case of a step-like initial condition in one dimension, the solution of the
diffusion equation is [12]
c(x, t) = 12 c0
[
Erf
( x√
4Dt
)]
. (1.12)
Erf denotes the errorfunction [10]. The following function specifies diffusion in a vessel whose
walls are in a distance of −l and +l from the origin. The walls are reflecting. The initial distri-
bution of particles c(x) is 1 for 0 ≤ x < h and 0 for −h ≤ x < 0 and the solution becomes
[12]
c(x, t) = 12 c0
[
∞∑
n=−∞
(
Erf
(n2l + x√
4Dt
)
+ Erf
(n2l − x− 2h√
4Dt
))]
. (1.13)
This relation is visualized in Fig. 1-4 and compared to results from a RW simulation. The com-
puter program simulates the Brownian motion of 1.28×106 particles on a one-dimensional lattice
with length 256. The influence of reflecting walls becomes significant as soon as the travelled
distance is in range of the walls.
Because diffusion is a stochastic process, there is no linear relationship between time and the
distance from the starting point of a molecule. No direction is preferred and the average distance
〈x(t) − x(t0)〉 is always zero. Rather a linear relationship for the mean square distance holds
〈x2(t)〉 = 2D t. (1.14)
This relation is known as Einstein-Smoluchowski equation. The symbol 〈. . . 〉 is meant as a sum-
mation over the position of all molecules. Accordingly, in two and three dimensions it is
〈r2(t)〉 = 4D t,
〈r2(t)〉 = 6D t. (1.15)
9
1.1. MOLECULAR MOTION IN LIQUIDS
simulation
without reflexion
with reflexion
c(t)
c(t0)
x
c(
x)
/c
0
10.50-0.5-1
1
0.8
0.6
0.4
0.2
0
Figure 1-4. Plot of Eq. 1.13 and Eq. 1.12 compared to simulation data. c(t0) marks the step-like
initial distribution of particles (all particles on the right side). The crosses show their positions after
performing a Random Walk of 10000 steps.
after 900 steps
after 400 steps
after 100 steps
normal distribution
x
P(
x
)
100500-50-100
0.04
0.03
0.02
0.01
0
Figure 1-5. The probability distribution of end points for a 1D Random Walk after q steps follows a
normal distribution with width σ = √2Dq∆t. The parameters are D = 1 and ∆t = 1. Theoretical
normal distribution and simulation data for q = 100, 400 and 900 steps.
10
1.1. MOLECULAR MOTION IN LIQUIDS
The diffusion coefficient D and the size of a particle, or molecule, are correlated by the Stokes-
Einstein equation [13]
D = kB T6 π η Rh
. (1.16)
which is valid for spherical objects. The relation is frequently used to express the size of molecules
by their hydrodynamic radius Rh. Rh is not necessarily equal to the real radius because the molec-
ular shape is rarely spherical. According to Stoke’s laws for friction in fluids, the relation depends
on the Boltzmann constant kB , the temperature T , and the viscosity η of the medium. In general,
η is a function of the temperature, too. A molecule’s mass M is proportional to its volume, and
thus M ∝ R3h. Because D is inversely proportional to Rh, it is difficult to distinguish molecules
with different weight by their diffusion coefficients. In practice, the diffusion coefficients of two
molecules only differ significantly if their mass ratio is greater than ≈ 6. Table 1.1 lists the coeffi-
cients of a number of species diffusing in water. For flexible molecules and other geometries Eq.
1.16 must be corrected. The correction for rod-like objects is [14]
D = kB T ln (2r)6 π η a . (1.17)
Here, it is assumed that the diffusing molecule is a prolate ellipsoid with a long axis a and an axis
ratio r ≫ 1.
Another useful relation is the average time between the random arrival of two molecules in a
volume of radius a. It can be derived from the diffusion equation 1.9 [15]
t¯ = (4 πD aC)−1 (1.18)
For a concentration of one molecule in a femtoliter and a diffusion coefficient of D = 10−10 m2/s,
the time to observe 100 molecule transitions is about T = 128ms. Slower molecules with D =
10−12 m2/s need an accordingly longer time, T = 12.8 s. Here, it is assumed that each molecule
is counted. In a real fluorescence measurement, the detection efficiency and photonic fluctuations
must be considered as well.
1.1.4 Flow and diffusion in microchannels
Liquid flow on the micrometer scale shows some special properties compared to normal scale sys-
tems. For the description of liquid flows in microchannels some basic laws from hydrodynamics
are recalled. The volume V of a fluid running through a pipe of radius R and length l is given by
Hagen-Poiseuille’s law [11]
dV
dt =
π
8 η l (p1 − p2) R
4, (1.19)
11
1.2. FLUORESCENCE
substance & solvent T [K] D [m2/s]
H+ in H2O [16] 298 9.31× 10−9
Na+ in H2O [11] 298 1.25× 10−9
Cl− in H2O [11] 298 1.78× 10−9
ethanol in H2O [11] 298 1.08× 10−9
R6G in H2O [17] 298 2.8× 10−10
hemoglobin in H2O [18] 293 6.3× 10−11
insulin in H2O [18] 293 8.2× 10−11
spherical particle (polystyrol, ∅ = 1µm) in H2O 295 4.45× 10−13
Table 1.1. Experimental diffusion coefficients of some substances in water.
where the pressure difference between inlet and outlet is p1 −p2. The value η names the viscosity
of the fluid. In fluid dynamics, it is important to distinguish laminar from turbulent flow. Laminar
flow is characterized by the fact that the stream lines follow a predictable direction. Vortices can
exist, but in general the flow shows a stationary behavior. Turbulence, on the other hand, occurs
when the flow velocity exceeds the limit where the friction force at the boundary layer becomes
strong. The flow changes from laminar to unstable and chaotic behavior. A local perturbation
grows exponentially in contrast to laminar flow where it decays exponentially. The transition
from laminar to turbulent flow can be characterized by the dimensionless Reynolds number [5]
Re = v¯ ρ dη . (1.20)
For applications in microchannels with diameters d smaller than 100µm and in aqueous solution,
the flow velocity has to be greater than 20m/s before, at a Reynolds number between 2000 and
3000, a transition to turbulent flow occurs. Hence, the flow in microstructures is laminar.
1.2 Fluorescence
Fluorescence occurs when a molecule emits a photon after it has been electronically excited by
light absorption. The excitation of electrons and the emission of photons are stochastic processes.
The spectrum of the emitted light is shifted to longer wavelengths (Stokes shift) [19]. The energy
difference is transferred to neighboring molecules through collisions. This transmission occurs
radiationless by exciting molecular oscillations (vibrational relaxation). Other processes, such as
internal conversion, also play a role in the relaxation process. The diagram of electronic states for
a fluorescent molecule is depicted in Fig. 1-6. The photon emission probability follows an expo-
nential distribution. The average time between absorption and emission of a photon is typically in
the range of nanoseconds. This time is referred to as lifetime τ10 = k−110 of the S1 electronic state.
The reason for the existence of a continuous emission spectrum is, that the density of oscillation
12
1.2. FLUORESCENCE
Figure 1-6. The Jablonski diagram shows possible electronic transitions of a molecule (left). The
abbreviations are IC = internal conversion, F = fluorescence, VR = vibrational relaxation, ISC =
intersystem crossing, P = phosphorescence. It is often simplified to a ground state S0, an excited state
S1, and a triplet state T1 (right). The image was redrawn from [17].
states is much higher than the one for electronic excitation. There are fluorescence molecules with
varying spectra from the UV to the IR. The excitation and emission spectra of a typical dye is
shown in Fig. 1-9. The intensity of the excitation light Iexc = P/A depends on the photon flux
Φexc and the photon energy h c/λ
Iexc = Φexc
h c
λ =
ν
A t
h c
λ , (1.21)
where h names Planck’s constant, c the speed of light and ν the number of photons with wave-
length λ. A stands for the illuminated area. The energy diagram is often simplified to a three state
system of a ground state S0, an excited state S1 and a triplet state T1. By setting up the differential
equations for the transition rates, the fluorescence emission rate kf can be related to the excitation
intensity Iexc [17],
kf =
φf σexc Iexc
1 + τ10 σexcIexc (1 + k12/k20)
(1.22)
The fluorescence quantum yield φf , the cross section for excitation σexc, and the transition rates
k12, k20 vary among different fluorophores (cp. Tab. 1.2). In case of small intensity the linear
approximation holds
kf = φf σexc Iexc (1.23)
Fig. 1-8 shows the emission rate of one Rhodamine 6G molecule plotted against the excitation
rate. The emission rate reaches a saturation value. The number of emitted photons ν in a time
interval follows a Poisson distribution with a parameter µ = kf∆t
P (ν|µ) = µν e
−µ
ν! . (1.24)
13
1.2. FLUORESCENCE
In general, µ depends on the local light intensity Iexc(~r) via Eq. 1.22.
The Poisson distribution has the feature that the sum of m stochastically independent processes
again follows a Poisson distribution with a parameter µ =
∑m
k=1 µk. The resulting distribution
is given by their convolution. In the following, this relation is referred to as recovering property
[20].
P (ν|µ) = P (ν|µ1)⊗ · · · ⊗ P (ν|µm). (1.25)
The fluorescence excitation and emission cycle can be interrupted by an electronic transition from
the singlet S1 to the triplet T1 state (intersystem crossing). This transition corresponds to a (for-
bidden) inversion of the electron’s spin. The inversion is forbidden unless a second molecule
participates in the reaction and mirrors the spin difference. For the same reason, the probability
for the back transition from T1 → S0 is small. Thus, the triplet state lifetime is much longer than
the singlet state lifetime. While a molecule is in the triplet state, it appears dark. The emission of
a photon from the triplet state is called phosphorescence to set it apart from fluorescence.
1.2.1 Fluorescence dyes
In the experimental section, two fluorophores are used: Rhodamine 6G (R6G) and octadecyl
Rhodamine B (R18). The chemical structures are shown in Fig. 1-7. The molecular properties
of Rhodamine 6G and R18 are summarized in Tab. 1.2. R18 and R6G differ in the chemical
structure of their side chains. This has an influence on the excitation and emission spectra. R18 is
an amphiphilic molecule with a polar head and an unpolar tail. It is useful in lipid environments.
Figure 1-7. Chemical structures of Rhodamine 6G (left) and octadecyl Rhodamine B (right).
1.2.2 Photostability
Fluorescent molecules have a limited stability, especially while in the excited state. The loss
of the fluorescence property is also called bleaching and describes the photo degradation of the
14
1.2. FLUORESCENCE
kmax
kf(Iexc)
100µW
[kWcm−2]
Iexc [photons s−1m−2]
k f
[p
ho
to
ns
/s
]
1000010001001010.10.01
1e+301e+291e+281e+271e+261e+251e+241e+23
1e+08
1e+07
1e+06
100000
10000
1000
Figure 1-8. Fluorescence emission rate kf vs. excitation rate for a molecule with properties from
Tab. 1.2. On the top abscissa the excitation rate is converted into an intensity value. The saturation
value is kmax = φfk20 τ−110 / (k20 + k12). The arrow marks a typical experimental value at a laser
power of 100µW.
emission
excitation
wavelength λ [µm]
in
te
n
sit
y
[a.
u
.
]
700650600550500450400
1
0.8
0.6
0.4
0.2
0
Figure 1-9. Fluorescence excitation and emission spectrum of Rhodamine 6G (in arbitrary units).
15
1.3. CONFOCAL OPTICS
symbol unit R6G R18
molecular weight M 479.02 731.5
excitation cross section σexc m2 1.7× 10−20 n. d.
rate S1 → T1 k12 s−1 1.1× 106 n. d.
rate T1 → S0 k20 s−1 4.9× 105 n. d.
lifetime τ10 ns 4 1.5
quantum yield φf 0.936 n. d.
photo bleaching (at P < 150 kW/cm2) φbl 5e− 6 n. d.
excitation maximum λ nm 525 555
emission maximum λem nm 547 578
Table 1.2. Some properties of Rhodamine 6G (from [17, 21]) and R18 in water. No data (n. d.) was
available for some properties of R18.
molecule. The time until a fluorophore has been bleached is exponentially distributed with a
bleaching lifetime of τbl = k−1bl . The rate kbl can be expressed by the photo bleaching quantum
yield [22]
φbl = kbl τ10. (1.26)
It depends on specific properties of the fluorophore and on external influences, such as the oxygen
and ion content of the solution [23]. The photostability is an important parameter for single
molecule experiments when fluorophores are observed for a longer time.
1.3 Confocal optics
Fluorescence Correlation Spectroscopy is a method basically used to determine low concentrations
and molecular diffusion coefficients in liquid environments. The sensitivity in FCS is on the
single molecule level and the measurement data directly result from the observation of a small
ensemble of molecules. Ensemble averaging is necessary for the standard correlation function
approach that was proposed by Magde and Elson [24, 25] in 1974. The key in FCS is the reduction
of the observation volume by using a confocal microscope (Fig. 3-1). The confocal principle
is a simple way to co-locate the excitation and observation volume of a strongly focused light
spot. In the focus center, fluorescent molecules emit photons shifted to longer wavelengths in
response to electronic excitation. This light is separated from the excitation light by an optical
filter. Background light is suppressed by an aperture (pinhole) placed in the back focal plane.
The fluorescence time series I(t) is detected with a photo diode. An example for I(t) is shown
in chapter 2, Fig. 2-1. From I(t) the autocorrelation, Eq. 1.30, is computed. The correlation
curve contains information about temporally correlated processes in the fluctuating signal. This
information can be extracted from the correlation function although the underlying processes are
stochastic. The basic parameters of interest are usually the diffusion coefficient and the particle
16
1.3. CONFOCAL OPTICS
number in the observation volume. To determine the diffusion coefficient, knowledge about the
size of the observation volume is required.
1.3.1 Focus calculation
For the analysis of the fluorescence intensity time series, it is necessary to introduce some optical
functions which describe the light distribution in the beam path. The molecule detection efficiency
MDE(~r, z) is defined as the product of excitation intensity Iexc(~r, z) and the collection efficiency
function CEF (~r, z) [26]
MDE(~r, z) := CEF (~r, z) Iexc(~r, z) (1.27)
The shape of this function can be appoximated by an effectively Gaussian profile along all three
axis [27]
MDE(~r, z) ≈ W (~r, z) = W0 exp
(
−2x
2 + y2
w20
− 2z
2
z20
)
(1.28)
This approximation considerably simplifies the further calculation. The Gaussian parameters w0
and z0 correspond to the beam waists, where the effective intensity has decreased to 1/e2. A
theoretical calculation for them was given by Rigler and Mets [26]
wth ≥
λ
nπ tan (α/2)
zth ≈
e s0
tanα × 0.72. (1.29)
s0 is the pinhole radius divided by the magnification. The pinhole size is important with respect to
the collection efficiency function. The optimal size is a compromise between collection efficiency
and suppression of background light from planes above and below the focal plane. zth must
be corrected by a factor depending on sinα. Here, it is 0.72 for NA = 1.2. In practice, the
theoretical values are hardly reached. The values hold under ideal conditions of a Gaussian beam
profile and an optimal aperture position. The finite size of the objective’s illuminated rear side puts
an upper limit on the focusing half angle α = arcsin (NA/n). The theoretical focus parameters
are given for two chosen configurations in Tab. 1.3. The refractive index n of the specimen is
ideal if it matches the one of the immersion medium between the objective and the cover glass.
Differences in the refractive index lead to small changes of the focus waist. Microscope objectives
are calibrated for a cover glass thickness of 170µm. Even slight variations of ±10µm result in
significant errors and increase the focus waist [28]. Numerical calculations have revealed more
optical effects arising in a confocal microscope [27]. Hess’s study shows that it is important if
the back aperture of the microscope objective is completely, or only partially, illuminated with
17
1.3. CONFOCAL OPTICS
Figure 1-10. Inverted confocal microscope for FCS and definition of the focus parameters w0 and z0
(magnified). See also Fig. 3-1.
18
1.4. FLUORESCENCE CORRELATION SPECTROSCOPY
config. MO λ pinhole wth zth V0
nm µm µm µm fl
ConfoCor 2 40×, 1.2 W 543 78 0.206 1.8 0.43
open microscope 63×, 1.2 W 514 50 0.195 0.74 0.16
Table 1.3. Theoretical focus parameters for two configurations of microscope objectives (MO) ob-
tained by Eq. 1.29. The refractive index was set to 1.33.
the laser beam. These effects become relevant only at distances > 1/e2 from the beam axis.
They should be accounted for in the case that absolute concentrations are determined but can be
neglected otherwise.
1.4 Fluorescence Correlation Spectroscopy
From the intensity time series I(t) the autocorrelation function is computed in order to extract the
temporal information from the signal. This can be done after the entire signal has been recorded,
but usually, it is achieved in real time by special correlation board hardware. The correlation
function was first applied to fluorescence measurements by Magde and Elson [24, 25] in 1974,
while the mathematical concept of the autocorrelation goes back to Wiener in 1930 [29].
1.4.1 Theory
The theory of FCS is explained following the representation in [17] and [26]. The goal is to find
an analytical expression for the correlation curve of diffusing fluorescent molecules in solution.
The general approach combines the solution of the diffusion equation with the shape of an effec-
tive excitation profile, and subsequently performs an integration over space. Starting point is the
definition of the correlation function
G(τ) := 〈I(t)I(t+ τ)〉〈I〉
with 〈I〉 = 1T
∫ T
0
I(t). (1.30)
T represents the total measurement time. By setting δI(t) = I(t) − 〈I〉, the correlation function
can also be written as the fluctuation around the mean
G(τ) = 1 + 〈δI(t)δI(t+ τ)〉〈I〉 . (1.31)
19
1.4. FLUORESCENCE CORRELATION SPECTROSCOPY
The fluorescence intensity can be written as a sum of all fluorescing molecules in the sample
weighted by the excitation profile and the collection efficiency
I(t) = α
∫
R3
d3r C(~r, t)
Eq.1.28
︷ ︸︸ ︷
CEF (~r) Iexc(~r) ≈ α
∫
R3
d3r C(~r, t)W (~r). (1.32)
The constant α is the number ratio of emitted and detected photons. Absorption of the emitted light
can be neglected because the regarded concentrations are low. The effective measurement volume
W (~r) is independent of time, and thus, the fluorescence intensity fluctuations δI(t) originate
mainly from the particle fluctuation δc(~r, t). With Eq. 1.32, Eq. 1.31 can be rewritten as
G(τ) = 1 + α
2 ∫
R3 d3r
∫
R′3 d3r′W (~r)W (~r′) Φ(~r, ~r′, τ)
(
αC
∫
R3 d3rW (~r)
)2 . (1.33)
C = 〈C(~r, t)〉 is the concentration average in the thermodynamic equilibrium. The term
Φ(~r, ~r′, τ) = 〈δC(~r, t)δC(~r′, t+ τ)〉 (1.34)
subsumes the particle fluctuation. The integral in Eq. 1.33 can be evaluated if an expression
for Φ(~r, ~r′, τ) can be found. The fluctuations due to Brownian motion are essential for the FCS
method. The time course of a concentration distribution is described by Fick’s laws, Eq. 1.8 - 1.9.
But in this case, the solution cannot be obtained by direct integration of δC(~r, t). Unfortunately,
the initial conditions for δC(~r, t) are unknown because the microscopic distribution of molecules
is random. To solve the problem, the microscopic view from section 1.1.1 is helpful. It is possible
to state a relation for the fluctuation between two arbitrary molecules at time lag τ = 0,
Φ(~r, ~r′, τ = 0) ≡ C δ(~r − ~r′, 0). (1.35)
Taking this as an initial condition for the diffusion equation, the general solution for Φ is found
by integration
Φ(~r, ~r′, τ) = C
(4πDτ)3/2
exp


−
(
~r − ~r′
)2
4Dτ


. (1.36)
20
1.4. FLUORESCENCE CORRELATION SPECTROSCOPY
Inserting Φ(~r, ~r′, τ = 0) = Cδ(~r − ~r′, 0) into Eq. 1.33, the ~r′ integration can be evaluated
immediately. In the limit τ → 0, the correlation function becomes
lim
τ→0
G(τ) = 1 + α
2 C
∫
R3 d3r
∫
R′3 d3r′W (~r)W (~r′) δ(~r − ~r′)
(
αC
∫
R3 d3rW (~r)
)2
= 1 + 1C
∫
R3 d3rW 2(~r)
(∫
R3 d3rW (~r)
)2 . (1.37)
For τ → 0, G(τ) − 1 is inversely proportional to the particle concentration C. Identifying the
second term with the observation volume, it is justified to write
lim
τ→0
G(τ) = 1 + 1C
∫
R3 d3rW (~r)
(∫
R3 d3rW (~r)
)2 = 1 +
1
C V0
= 1 + 1N . (1.38)
N is the average number of molecules in the focus and V0 is the redefined observation volume of
FCS. It can be calculated analytically for the Gaussian approximation in Eq. 1.28
V0 :=
(∫
R3 d3rW (~r)
)2
∫
R3 d3rW (~r)
= π3/2 w20 z0. (1.39)
The geometry of the focus volume has the shape of an ellipsoid with semi-axes w0 and z0.
The time dependence of the correlation function must be calculated by inserting Eq. 1.36 into
1.33. The integration can be done analytically, leading to the final result [17]
G(τ) = 1 + 1N
(
1 + ττd
)−1 (
1 + τK2 τd
)−0.5
. (1.40)
The ratio of the long and short axis of the focus is the so-called structure parameter
K := z0w0
. (1.41)
If diffusion is the dominant process, the value where G(τ)−1 has decreased to 0.5N−1
√
1 + K−2
is the diffusion time
τd :=
w20
4D. (1.42)
When τd and w0 are known from an experiment, D can be identified with the coefficient of the
diffusing species. The diffusion time is proportional to the average time that a molecule spends
inside the focal volume. Examples for correlation functions are given in chapter 2.
Focus calibration
In real experiments, the exact shape of the excitation intensity and the collection efficiency func-
21
1.4. FLUORESCENCE CORRELATION SPECTROSCOPY
tion are hardly known. They depend strongly on the optical adjustment of the beam path. The
theoretical calculation of the focus parameters is not valid exactly. In practice, it is often better
to calibrate the values with a standard solution with a well-known diffusion coefficient. Here,
Rhodamine 6G solutions in concentrations of ≈ 10−8 M are used throughout.
After calibrating the focus size, the diffusion coefficient of a different species can be compared to
the calibration standard. Thereby, molecular motion can be characterized applying the accordant
model equations, for example for free diffusion, or directed motion. Diffusion coefficients can
be taken to distinguish molecules by the size, weight, and shape, and also to characterize the
viscosity and structure of the surrounding material. Chemical reactions and binding processes can
be observed if concentrations or diffusion coefficients are determined repeatedly with respect to
the reaction time. Additionally, phenomena of electronic excitation and emission can be studied
in detail (e. g., bleaching, fluorescence anisotropy, triplet state processes, molecule rotation, etc.).
Resolution
In Fluorescence Correlation Spectroscopy a microscopic volume is illuminated. The size of this
observation volume is governed by the diffraction limit of the excitation light resulting in a volume
of approximately 0.5 fl. The extension in axial direction also depends on the pinhole diameter.
The time resolution depends on the dead time of the photon detector and the sampling time of
the hardware correlator. The diffusion time/molecule is typically between 0.2 and 3ms. The
measurement result for the diffusion time is an ensemble average over time and the number of
molecules that have crossed the focal volume on a random path.
1.4.2 Variance of the autocorrelation
In an autocorrelation measurement, variations occur due to the statistical nature of the observed
processes. To determine the error of the diffusion coefficient and the concentration value, the
variance of the autocorrelation values is needed. It is useful to calculate the variance from known
experimental parameters. This also provides insight into the inherent dependencies of the error.
The first approximation for the variance of G(τ) was given by Koppel in 1974 [30].
σ2Koppel(m∆τi) =
1
M N2
[(1 + g2(∆τi)) (1 + g2(m∆τi))
(1− g2(∆τi))
+ 2mg2(m∆τi)
]
+ 1M
[2 (1 + g2(m∆τi))
N〈n〉 +
1
〈n〉2
(
1 + g(m∆τi)N
)]
. (1.43)
For the function g(τ) the normalized and reduced autocorrelation function for three-dimensional
diffusion is inserted
g(τ) =
(
1 + ττd
)−1 (
1 + τK2 τd
)−0.5
. (1.44)
22
1.4. FLUORESCENCE CORRELATION SPECTROSCOPY
The parameters are 〈n〉, the average count rate/correlator channel, and M = T/∆τ , the total
number of measurement intervals and N the average number of molecules as defined in Eq. 1.38.
m∆τi = τ are multiples of the correlation time interval.
The formula was originally derived for a Gaussian fluctuation model with a large number of
molecules in the observation volume. Furthermore, a uniform illumination profile, negligible
background, and sampling times much smaller than the observed diffusion times were assumed.
The function gives an estimate for the error, but systematic deviations occur for longer correlation
times [31]. Among other ways to calculate the standard deviation [31, 32, 33], the most useful
were given by Saffarian (Eq. 24 in [34]) and Qian [35]. In conclusion, the variance depends on the
fluorescence quantum yield, the measurement time and the number fluctuations of particles in the
observation volume. In the high concentration range, the signal-to-noise ratio G(m∆τi)/σ(m∆τi)
of the correlation function is independent of the concentration [30]. Otherwise, the method im-
proves in signal-to-noise ratio with higher particle fluctuation [24, 36]. It also has to be considered
that hardware correlators work with a logarithmic time scale and the interval time is not constant
[37].
The easiest way to obtain the variance experimentally is to record a number of correlation curves
G1...n, and then, calculate the mean 〈G(m∆τi)〉 and the variance for each correlation time accord-
ing to
σ2(m∆τi) =
1
n− 1
n∑
k=1
(Gk(m∆τi)− 〈G(m∆τi)〉)2 . (1.45)
1.4.3 Triplet state
When a molecule undergoes a transition to the triplet state it appears dark until it returns to the
excited state. Non-fluorescing molecules reduce the apparent number of molecules in the detection
volume. The initial part of the correlation curve rises by a fraction which strongly depends on the
excitation intensity. In the correlation function, the triplet fraction p appears as an exponential
term in Eq. 1.40 [17]
Gtrip(τ) = 1 +
1
N
(
1 + ττd
)−1 (
1 + τK2 τd
)−0.5
×
[
1 + p1− p exp
(
− ττtrip
)]
(1.46)
The triplet life time τtrip = k−121 typically has a value of a few µs. The saturation value in flu-
orescence emission (Fig. 1-8) is influenced by the triplet lifetime. Systematic errors can occur
resulting from this saturation but are neglected in the following chapters [38].
23
1.4. FLUORESCENCE CORRELATION SPECTROSCOPY
1.4.4 Directed flow
When the observed liquid is not at rest, but drifting in a defined direction, diffusion and stationary
flow are concurrent processes. The differential equation 1.9 for this case must be generalized by a
term depending on the flow velocity
∂C(~r, t)
∂t = D∆C(~r, t) − ~v(~r)
~∇C(~r, t). (1.47)
If the liquid moves in x-direction only, the diffusion equation simplifies to
∂C(~r, t)
∂t = D
∂2C(~r, t)
∂x2 − vx(~r)
∂C(~r, t)
∂x . (1.48)
The solution is obtained by Fourier transformation [39]. With Eq. 1.48, the correlation function
has the result [40]
G(τ) = 1 + 1N
(
1 + ττd
)−1 (
1 + τK2 τd
)−0.5
× exp
[
−
( τ
τflow
)2 (
1 + ττd
)−1
]
. (1.49)
The average time for a molecule to cross the focus induced by the flow is τflow. The local flow
velocity in x-direction is obtained by
vx(~r) =
w0
τflow(~r)
. (1.50)
24
Chapter 2
Simulation of FCS
The conditions for an experiment should usually be chosen as simple as possible to focus on
the most relevant question. However, in Fluorescence Correlation Spectroscopy (FCS) practical
uncertainties cannot be excluded completely. For example, background fluorescence and noise
from detection electronics always lower the signal-to-noise ratio. Another problem is that the
exact shape and dimension of the focus are unknown. There is also no possibility to suppress
photobleaching of the fluorophores completely. Altogether, the FCS method is good for measuring
concentrations and diffusion coefficients relative to a known standard. The necessary effort to
improve the setup and analysis to enable absolute measurements is high.
It is generally useful to construct an idealized experiment avoiding these difficulties. For this
purpose, computer simulations can be used to simulate experiments and exclude unknown param-
eters. Several approaches are in use, each of them with advantages and disadvantages. Recently,
Molecular Dynamics (MD) simulations gain popularity, especially when details about molecular
interaction have to be studied. Still, they require high computational power. Regarding FCS, the
Monte Carlo method is well suited for the relevant processes and time scales. This chapter presents
an algorithm that is used to simulate FCS experiments on a computer. It starts with the first section
2.1, describing the Brownian motion of molecules. The simple Random Walk algorithm which
was introduced in section 1.1.2 is applied to simulate their stochastic behavior. Section 2.2 deals
with how the random process of photon emission is implemented. In section 2.3, the simulation
results are verfied by comparing them to the theoretical expectations. At the end of the chapter,
the special case of two-dimensional diffusion is regarded in particular. The simulation is used to
find an FCS model which describes diffusion in bilayer systems (see also chapter 5).
2.1 Brownian motion
The algorithm is based on simulating the two fundamental processes, the Brownian motion of the
molecules and the photon emission caused by fluorescence excitation. Both are random processes.
25
2.1. BROWNIAN MOTION
The molecule diffusion is realized by a simple Random Walk (RW) on a three-dimensional lattice
with a side length (L− 1) ǫ. ǫ denotes the distance between two lattice points. The coordinates of
the lattice points are ~r(x, y, z), with x, y, z = {0 . . . L − 1} ∈ N. The simulation volume then
has a total size of
Vsim = (L− 1)3 ǫ3. (2.1)
The time axis is divided into a series of discrete points in time {ti ∈ R, i ∈ N} with ti+1 = ti+∆t.
The interval ∆t is given by the Einstein-Smoluchowski relation in three dimensions, Eq. 1.15,
∆t = ǫ
2
6D. (2.2)
Hence, the smallest time unit is defined by the diffusion coefficient D and the grid constant ǫ. At
t0, a number of N0 molecules is placed on arbitrary positions of the lattice. Because the number
of molecules in a subvolume is Poissonian distributed, the number N0 is drawn from a Poisson
distribution with the parameter µ = N¯ . N¯ is the average number of molecules in the simulation
and determines the average concentration c = N¯/Vsim NA. After each time step the molecules
move along the unit vectors to one of the six neighboring lattice sites with equal probability 1/6.
This means, the new position of the nth molecule is determined from the previous one by
~rn(ti) = ~rn(ti−1)± ǫ~ej. (2.3)
with the unit vectors ~ej , j ∈ {1, 2, 3}. The probability of a molecule to stay at its present position is
0. The boundary conditions of the simulation follow from the demand that the simulation volume
contains a Poissonian distribution of diffusing particles (see section 1.1.2). If a molecules crosses
the simulation boundary, which is equal to
~rn(ti) < 0
~rn(ti) > L− 1, (2.4)
the molecule is deleted from the simulation. To maintain the concentration on the time average,
new molecules have to appear at the same rate at which they are deleted. The starting point of a
new molecule is on a random position on one of the six faces of the lattice cube. For the rates to
enter or leave the volume,
kout = kin (2.5)
26
2.2. PHOTON EMISSION
must be fulfilled. To find an expression for this rate, the individual probabilities for a molecule to
exceed the boundary from a) a face b) an edge c) a corner must be summed up
Pin =
N¯
L3
[
6 16(L− 2)
2 + 12 26(L− 2) + 8
3
6
]
= N¯L . (2.6)
The generating probability determines the rate at which new molecules enter the simulation vol-
ume. The relations above are implemented in the simulation as follows. The RW is realized by a
program that generates a random number once every time step ∆t. For each molecule a number
between one and six is chosen randomly to state its next position. Afterwards, the implemented
molecule generating function is called once every time step. Another random number is taken to
decide if a new molecule enters the simulation volume. The (pseudo-) random number generator
ran1 suggested by Press is used [41].
2.2 Photon emission
Inside the simulation volume, a local light intensity W (~r) with wavelength λ is allocated to the
lattice points. To simulate a focused laser beam, as it appears in FCS applications, the intensity
profile is chosen as a three-dimensional Gaussian function (cp. Eq. 1.28),
W (~r) = P02π w20
exp
[
−2ǫ2
(x2 + y2
w20
+ z
2
z20
)]
. (2.7)
Here, the variable P0 is the laser’s optical power, w0 and z0 are the focus semi-axes in radial
and axial direction. The z-axis is defined as the laser beam axis. The focus semi-axes should be
small in comparison to the side length of the lattice cube and large in comparison to the lattice
parameter ǫ. In an experiment, the effective and the real excitation profiles are not identical.
The probability to detect a photon depends in the molecules’ positions. Here, for simplicity, the
collection efficiency function CEF (~r) = 1 is chosen as unity. The simplification is justified as
argued in section 1.3. The general validity of the FCS theory is not affected thereby.
The probability P (n)i that the nth molecule emits ν photons in the time interval ∆t between ti and
ti−1 is given by the Poisson expression
P (n)i (ν|µ) = µν
e−µ
ν! . (2.8)
The average value µ is assumed to be proportional to the effective excitation profile W (~r) of the
molecule (Eq. 1.23, 1.28)
µ(~r) = W (~r) λhc σexc. (2.9)
27
2.2. PHOTON EMISSION
The cross section σexc(λ) for fluorescence excitation is chosen constant. h and c name Planck’s
constant and the speed of light. Eq. 1.23 implies that only 2 electronic states exist, S0 and S1.
The triplet state is neglected. For the purpose of simulating diffusive correlation functions, this
assumption is justified because triplet state dynamics, bleaching, or emission saturation are irrel-
evant, here. In this sense, the simulation is intentionally limited to correlation functions governed
by the motion of particles.
The photon distribution from N independent molecules is given by Eq. 1.25. The recovering
property of the Poisson process allows to calculate the number of photons emitted by N molecules
at time ti directly:
µN =
N∑
n=1
µn. (2.10)
In an experiment, usually a detector (avalanche photo diode or photo multiplier) in combination
with a computer board (multichannel scaler) records the photon events within a minimal mea-
surement interval ∆T . This bin time is the interval within that photons are collected. Because
Poissonian emission processes from different molecules are independent from each other, the re-
covering property of the Poisson distribution can be applied again (Eq. 1.25). The sum of all
photons within a bin interval also follows a Poisson distribution with the parameter
µ∆T =
m∑
i=1
µN(ti). (2.11)
m = ∆T/∆t is the number of time steps in one measurement interval. µ∆t can be calculated
from the molecules’ positions and the local intensity. Hence, the number of emitted photons from
all molecules in ∆T can be simulated by drawing only one random number. µ∆T is inserted into
Eq. 2.8 instead of µ, thus returning the Poisson distributed photon number from N molecules.
Thereby, the number of calls of the random number generator can be reduced efficiently.
Remark on the period of the random number generator
The random number period of ran1 is about 109. For certain applications of random numbers the
period exhaust turns out to be a problem. This limit is exceeded in the simulation. But, as Woh-
land pointed out, the period length can be neglected because the simulated particles are deleted
before the number sequence repeats [31].
The number of detected photons is smaller than the number of emitted photons. The overall loss
results from the optical setup and the detection efficiency of the detector. The constant factor γ
subsumes the transmission coefficients θtrans of all optical components along the emission beam
28
2.3. TESTS OF THE SIMULATION
path. The fluorescence quantum yield φf can also be included in this value, resulting in
γ = Ω(NA) θtrans φf . (2.12)
A microscope objective covers the solid angle Ω which depends in the numerical aperture NA and
the refractive index n of the medium
Ω(NA) = 0.5 [1− cos (arcsin (NA/n))] . (2.13)
The refractive index is 1.33 for water. Otherwise, the photon emission is assumed to be isotropic
in every direction. To account for these effects, the number of emitted photons is multiplied by γ.
Addition of dark counts
Dark counts from noise of the detector electronic, which is anavoidable in most applications, can
be included easily. Because photon events from noise are an independent Poissonian process, it is
sufficient to add the dark count rate ζ to k∆t
µ∆T (ζ) = µ∆T + ζ. (2.14)
2.3 Tests of the simulation
The result of a simulation run is a series of natural numbers I(t), representing the number of
detected photons ν in a measurement interval ∆T . From the simulated intensity signal (example
in Fig. 2-1), the correlation function is calculated according to the defining Eq. 1.30.
The computer simulation requires to chose a set of parameters (Tab. 2.1 and Tab. 2.2). In contrast
to an experiment, all parameters, including the focus shape and size, are known. These values
are classified either as experimental parameters or simulation specific parameters. The values
are further distinguished into optical parameters of the setup, and values which describe the spe-
cific properties of the fluorophore. In the simulations, some values remained constant throughout,
whereas others had to be adjusted. Constant values were the wavelength, the objective’s numer-
ical aperture, the lateral focus size, the excitation cross section, the transmission factor, and the
quantum yield. The optical parameters were set to values demanding that the overall detection
efficiency γ was 4 % (a typical value for an FCS setup is 3 % [42]). The laser power, the vertical
focus radius, the diffusion coefficient, and the concentration were altered for different tasks. The
lattice parameter and the bin interval were constant while the lattice size and the total simulation
time were adapted.
29
2.3. TESTS OF THE SIMULATION
The performance of the simulation was verified by applying a series of tests. Necessary conditions
are (A) that the given concentration, and (B) that the given diffusion coefficient, are correctly
reproduced. The third test (C) checks for the variance according to Koppel’s formula 1.43. The
significance of (C) is limited because the calculated variance of the correlation function is only
an approximation. Nevertheless, it is important to see if the range of fluctuation is reasonably
simulated by the proposed algorithm. The influence of noise was not examined, and the value was
set to zero. The simulation parameters are listed in Tab. 2.1 and Tab. 2.2.
A simulated intensity trace is depicted in Fig. 2-1 with the number of photon counts plotted as
bars. 1000 intervals are shown. The intervals were re-binned to 0.1ms intervals to visualize the
photon fluctuation. The resulting correlation functions are analyzed by comparing them to the
theoretical functions given in section 1.4.1. The deviations from the theoretical curves are shown
as residuals in a separate graph below the actual diagram. The residuals are given in multiples of
the standard deviation σ(m∆τi)
res(m∆τi) =
〈Gsim(m∆τi)〉 −G(m∆τi)
σ(m∆τi)
(2.15)
A correlation curve from the intensity data of 12 simulation runs is displayed in Fig. 2-3. The
average correlation from the simulations is plotted as crosses with error bars representing their
standard deviation. The expected theoretical function is drawn dashed. The standard deviation
was calculated from a number of repeated realizations with Eq. 1.45. The results are:
1. Concentration
12 realizations with parameters from Tab. 2.2 give an average number of particles of 1.01±
0.05. Theoretically, a value of 1.0096 was expected. Theory and simulation are in excellent
agreement.
2. Diffusion coefficient
A range of diffusion coefficients from 1 × 10−11 to 9 × 10−11 m2/s was simulated. From
six realizations with a given diffusion coefficient the correlation functions were calculated
and fitted to the standard FCS model with free parameters N and D. Each data point with
error bars represents the mean and standard deviation from these six simulation runs. The
dashed line marks the identical relation between the theoretical and simulated/fitted diffu-
sion coefficients. Fig. 2-2 shows the correct reproduction of the diffusion coefficient for
D > 2×10−11 m/s. The standard error given by the size of the error bars is about 3 %. The
simulations for Dth = 10−11 m2/s show a systematic deviation towards a slower diffusion
coefficient. This error is due to the lattice discretization. It can be avoided by chosing a
different value for the step size ǫ. In general, the choice of ǫ is connected to the range of
diffusion coefficients which shall be simulated.
30
2.4. DIFFUSION IN ORIENTED PLANES
parameter symbol unit value
setup
laser power P mW 0.2
wavelength λ nm 514.5
numerical aperture NA 1.25
transmission factor θtrans 0.1355
noise level η 0
focus axis, lateral w0 µm 0.3
focus axis, axial z0 µm 0.6
fluorophore
diffusion coefficient D m2/s 10−11..9× 10−10
exc. cross section σexc m2 2× 10−20
em. quantum yield φf 0.9
av. no. of mol. on lattice N¯ 56
simulation
lattice size L 512
lattice parameter ǫ nm 5
simulation time T s 1.68
bin interval ∆T µs 1
internal
time step ∆t ns 4.63 − 417
av. no. of mol. in focus N 1.0096
concentration c M 5.58× 10−9
Table 2.1. Parameters for the simulation series (A) and (B), Dsim vs. Dth.
3. Variance of the correlation
The variance was determined from 12 realizations of simulated correlation curves with equal
parameter settings. The chosen settings are listed in Tab. 2.2. Unlike in (A) and (B), the
focus axes are equal forming a spherical profile. The diffusion coefficient was set to a
value that is equivalent to six simulation time steps/bin interval. The variance is calculated
according to Eq. 1.45 and compared to Koppel’s, Qian’s, and Saffarian’s formulas (see
section 1.4.2). Instead of the variance, the standard deviation is displayed in Fig. 2-4.
The simulated data and the theoretical approximations agree within the accuracy of their
derivation. However, systematic differences are obvious.
The simulation fulfills the tests. In the following section, it will serve as a tool to evaluate an FCS
model for diffusion in oriented planes.
2.4 Diffusion in oriented planes
There are environments that show a supramolecular order of two-dimensional layers. In these
surroundings, Brownian motion can be restricted to planes. This type of diffusion prevails, for ex-
ample, in lamellar bilayers and membranes. When the molecules diffuse on a plane, its orientation
in relation to the elongated observation volume is important (Fig. 2-5). The resulting correlation
31
2.4. DIFFUSION IN ORIENTED PLANES
parameter symbol unit value
laser power P mW 0.1
focus axis, lateral w0 µm 0.2
focus axis, axial z0 µm 0.2
diffusion coefficient D m2/s 2.5× 10−11
simulation time T s 4.19
time step ∆t ns 166.7
av. no. of mol. in focus N 0.1496
Table 2.2. Simulation series (C), variance test, differs from (A) and (B) in these parameters.
simulated intensity
interval no.
ph
o
to
n
co
u
n
ts
10008006004002000
80
70
60
50
40
30
20
10
0
Figure 2-1. Simulated intensity signal from molecules with the properties in Tab. 2.1. The time
intervals were re-binned to 0.1 ms. Average number of molecules in the focus N = 1.01.
theory
average ± sdev
Dth[m2/s]
D
si
m
[m
2 /
s]
1e-101e-11
1e-10
1e-11
Figure 2-2. Theoretical diffusion coefficient vs. Dsim from the simulation. Each point is the result of
6 simulation runs. Parameters were set according to Tab. 2.1.
32
2.4. DIFFUSION IN ORIENTED PLANES
theory
12 simulations
G
(τ
)−
1
1
0.75
0.5
0.25
0
τ [s]
re
sid
u
al
s
/σ
10.10.010.0011e-041e-051e-06
1
0
-1
Figure 2-3. Average autocorrelation and standard deviation from 12 normalized simulations of 4.19
seconds each. The parameters were set according to Tab. 2.2. The residuals below show the difference
between the simulated curve and the theoretical function in multiples of the standard deviation.
Koppel
QianSaffarian
simulation data
τ [s]
σ(
G
(τ
))
10.10.010.0011e-041e-051e-06
0.03
0.025
0.02
0.015
0.01
0.005
0
Figure 2-4. Standard deviation of 12 simulations compared to Saffarian’s, Qian’s and Koppel’s
theoretical approximations.
33
2.4. DIFFUSION IN ORIENTED PLANES
Figure 2-5. In the orientation model, molecular diffusion is restricted to two-dimensional planes
intersecting the focus intensity ellipsoid at arbitrary angles θ and vertical intercepts. Integrating con-
tributions from a number of orientations results in an effective angle θm.
function is expected to depend on the plane’s normal angle [43]. Though it is possible to orientate
a membrane (in a magnetic field or on a surface), and then perform an FCS measurement, it is ex-
perimentally easier to examine diffusion in non-oriented samples. In this case, there are a number
of non-oriented layers contributing to the signal simultaneously. For the following, it is assumed
that the orientation of these layers in space is random. The contributions from different orienta-
tions are integrated resulting in an effective angle. For lamellar systems showing two-dimensional
diffusion and random orientation, the next equation will be used as a correlation function
Go(τ) = 1 +
1
N
(
1 + ττd
)−0.5(1+cos θm) (
1 + τK2τd
)−0.5 sin θm
. (2.16)
The effective angle θm is the magic angle θm = arccos
(
1/
√
3
)
≈ 54.7◦. This model is referred
to as the orientation model. It is based on a single diffusion coefficient. The simulation from the
previous chapter is used to find out if Eq. 2.16 is an allowed model for orientation averages. To
adapt it to the problem of diffusion in bilayers, the algorithm was modified in three points
1. Two-dimensional diffusion
The molecules were still distributed randomly on a three-dimensional lattice. But diffusion
was restricted to planes only. Lattice steps in vertical direction were excluded. The time
step was set according to two-dimensional diffusion, ∆t = ǫ2/4D.
2. The excitation profile is given an axis ratio of six to ensure a significant asymmetry.
3. The excitation intensity ellipsoid can be tilted by a user-defined angle.
Because of the elongation, it was necessary to increase the simulation volume to a lattice size of
10243. The simulation parameters are listed in Tab. 2.3. A series of 26 correlation functions was
34
2.4. DIFFUSION IN ORIENTED PLANES
parameter symbol unit value
setup
laser power P mW 2
wavelength λ nm 514.5
numerical aperture NA 1.25
transmission factor θtrans 0.1355
focus axis, lateral w0 µm 0.3
focus axis, axial z0 µm 1.8
fluorophore
diffusion coefficient D m2/s 2.5× 10−11
exc. cross section σexc m2 2× 10−20
em. quantum yield φf 1.0
av. no. of mol. on lattice N¯ 64
simulation
lattice size L 1024
lattice parameter ǫ nm 5
simulation time T s 4.19
bin interval ∆T µs 1
angles θ ◦ 5− 85
internal
time step ∆t ns 250
av. no. of mol. in focus N 0.43140
concentration c M 7.94× 10−10
diffusion time for θ = 0◦ τ0 ms 0.9
Table 2.3. Parameters for the simulation series to determine the orientation average of two-
dimensional diffusion.
simulated. 20 different angles of the excitation profile reaching from 5, 10, 15, . . . , 85◦ were cho-
sen in the computations. The resulting intensity traces were summed up with equal weights. After
computing the correlation function from this data, the result represents the orientation averaged
correlation curve. Additionally, the separate correlation functions for each angle were normalized
and then averaged, resulting in a smoother curve. Example correlation for some discrete orien-
tations are given in Fig. 2-6. As expected, the apparent diffusion time increases with the angle
of inclination. Values of up to 2.6-fold the definition τ0 = w20/4D are reached (see inset of Fig.
2-6). This coincides with the value determined by Weiss [43]. The correlation average from 20
different angles is shown in Fig. 2-7 (plus symbols). The average curve resulting from the in-
tensity traces are plotted as cross symbols. For comparison, the simulation data are compared to
the expected function Eq. 2.16. The simulation data differs slighty from the model. While the
(orientation averaged) diffusion time appears to be correctly matched by the model, the shape of
the correlation curve shows systematic deviations. In Fig. 2-7, σ was estimated using the result
of six simulation runs with equal parameters. The error originates from an inherent dependency
K = K(θ). Further analysis of Eq. 2.16 is necessary and is still being developed. Nevertheless,
the model provides to be a reasonable function to describe diffusion in bilayers.
35
2.4. DIFFUSION IN ORIENTED PLANES
angle [◦]
τ d
/τ
0
9060300
2.6
2.2
1.8
1.4
1
85◦
70◦
50◦
30◦
10◦
τ [s]
G
(τ
)-1
10.10.010.0011e-041e-051e-06
1
0.75
0.5
0.25
0
Figure 2-6. Simulated correlation curves for two-dimensional diffusion under different angles of
inclination towards the excitation volume. The inset shows the corresponding diffusion times relative
to τ0.
orientation model
av intens trace
av corr curve
G
(τ
)-1
1
0.75
0.5
0.25
0
τ [s]
re
sid
u
al
s/σ
10.10.010.0011e-041e-051e-06
4
2
0
-2
-4re
sid
u
al
s/σ
Figure 2-7. Average correlation curves obtained from simulations with angles between 5◦ and 85◦.
+ symbols represent the normalized and averaged correlation curves while × symbols are derived
from the superimposed intensity data. The residuals are displayed in units of σ which were estimated
from separate simulations.
36
Chapter 3
Fluorescence detection
The history of Fluorescence Correlation Spectroscopy (FCS) begins in 1955 when the confocal
microscope was invented by Minsky [44]. The idea to use the fluctuations of fluorescent molecules
transitting through a small observation volume was born in 1974 when Magde and Elson imple-
mented the first concept of FCS [25]. Its application and related methods have quickly progressed
since then. Today, there are a variety of microscopes commercially available. Some of them com-
bine Fluorescence Imaging and FCS capability, so that biological specimens can be analyzed at
distinct positions.
The experimental part of this work begins with a specification of the laboratory equipment. Three
different setups were used. Two of the instruments were assembled in the laboratory. They are
part of an ongoing project of building a fluorescence detection system for tracking single molecule
motion. The first one, the open microscope setup, is introduced in section 3.1. It is applied for
recording fluorescent intensity traces and correlation functions. The main advantage is a position-
ing unit with a specified accuracy in the nanometer range. The imaging setup, as the second one,
consists of a microscope stage which can be equipped with a camera (section 3.2). This system is
able to record fluorescent images and save them for subsequent image analysis. The sensor of the
camera offers high frame rates and good sensitivity at the same time. The equipment is used to
observe the Brownian motion of a single particle in chapter 4. The performances of these systems
regarding their optical characteristics are equivalent to commercial systems. The experiments on
surfactant bilayer systems (chapter 5), on the other hand, were carried out on a commercial micro-
scope. The setup is described shortly in section 3.3. In this chapter, only techniqual specifications
that are necessary for the experiments are discussed. Information about companies can be found
in the Appendix A, List of manufacturers.
37
3.1. OPEN MICROSCOPE SETUP
Figure 3-1. Upright setup of the open microscope. The light from the green laser is reflected by
a dichroic mirror and coupled into a high-aperture objective. The beam is focused into the sample.
The emission light from fluorescent molecules is collected by the same objective and projected on the
detector’s sensor by the tubus lens (epifluorescent setup). Excitation light is absorbed by a transmission
filter with appropriate filter characteristics. The pinhole reduces the focus extension in the vertical
direction. Exact positioning of the sample is possible with two nanopositioners for xy- and z-direction.
3.1 Open microscope setup
Confocal detection in chapter 4 was performed using an open microscope. The construction of the
microscope follows the description given in [42]. In contrast to Fig. 1-10, the setup is in upright
position with the microscope objective viewing from top to bottom (Fig. 3-1). The laser is an argon
ion gas laser. The output laser intensity is reduced by a dichroic mirror with a transmission of 10%
and neutral wedge filters. The experiments were carried out at a laser wavelength of 514.5 nm.
The laser worked in the modelocked operation mode at a pulse frequency of 84MHz. This time
scale is smaller than the correlator’s sampling time. Thus, the pulse frequency was not observed
as an artefact in the correlation curves. The beam profile was not Gaussian because the laser tube
suffered from aging effects (Fig. 3-2). The deviation from the optimal profile is a problem for
38
3.1. OPEN MICROSCOPE SETUP
Figure 3-2. Intensity profile of the Argon ion laser measured with a beam diagnostics system. In
modelocked operation, the profile was not Gaussian. The intensity is displayed in a miscolored scale.
The graphs at the side of the image represent the projected intensities in two directions.
absolute concentration determination, but does not perturb the relative measurements presented
in this work. There is a dichroic mirror in the beam path that reflects the excitation light and
transmits the emission light. The emission light is projected on a pinhole with a tubus lens (focal
length 165mm). The pinhole diameter is 50µm. As a microscope objective either an Apochromat
63× (NA 1.2 Water immersion) or a Plan Neofluar 63× (NA 1.4 Oil immersion) was used. Some
properties are summarized in Tab. 1.3. The optional beam expander appearing in Fig. 1-10 was
not installed in the open microscope setup. Before a measurement series, the pinhole position
was adjusted in horizontal and vertical direction to find the maximum fluorescence intensity. The
detection unit consists of an avalanche photo diode with a dead time of 50ns and a dark count rate
of < 100 counts/s. The photo diode is connected with the TTL input of a hardware correlator.
The correlator has a minimum sampling time of 200 ns. A list of manufacturers for all devices is
given in the appendix.
Nanopositioning
The position of the focus was controlled by a positioning unit with 3 nm precision (Fig. 3-3).
The unit consists of a table to carry the sample. The positioner is fixed under the objective. The
inner frame of the table can be moved horizontally by piezoelectric forces in response to the
applied voltage. The maximum amplitude in both directions is 100µm. The vertical position of
the objective is controlled by a focus positioner (PIFOC) with similiar properties. It is attached
between the objective and the mount.
The overall response time of the positioner including all electronics and interfaces is an important
measure to evaluate the suitability for particle tracking. The system’s response time was deter-
mined in a dynamic measurement [45]. The response time is about 7.1ms while performing a
Random Walk.
39
3.2. FLUORESCENCE IMAGING
Figure 3-3. Nanopositioner P-734 (left). Confocal microscope setup with position controller PI-E500
in the background (right).
3.2 Fluorescence Imaging
The term Fluorescence Imaging subsumes different methods like FLIM (Fluorescence Lifetime
Imaging), LSM (Laser Scanning Microscope), and derivates of them. Here, we concentrate on
direct detection of fluorescence images with a camera mounted on top of a microscope. Image
detection is carried out with the newly built demonstrator microscope (Fig. 3-4 & 3-5). It allows
to switch between three different wavelengths and white light illumination. The viewport on top
can be equipped with a camera, or a confocal detection unit consisting of a pinhole holder and an
optical fiber.
Figure 3-4. Measurement setup for image detection (left). The pinhole holder on top can be replaced
by the camera (right).
The goal is to develop a detector which is capable of observing single fluorescing molecules over
a long period of time. Single molecule detection with spatial resolution puts high demands on
the sensitivity of the detector. The basic method of recording images with a camera has suffered
40
3.2. FLUORESCENCE IMAGING
Figure 3-5. Schematic drawing of the imaging setup.
from slow readout rates and insufficient sensitivity for a long time. Today, with improvements in
CCD and CMOS technology, these problems have been overcome. Recently developed cameras
achieve high sensitivity and frame rates at the same time [46].
3.2.1 Camera properties
The camera in Fig. 3-4 is based on CMOS technology. It combines the advantages of high frame
rates and good sensitivity. The sensor is a monochrome sensor with a maximum gray scale depth
of 10 bit. The readout is controlled by a frame grabber hardware which was installed into a PC.
The interface is described in detail in [47].
Resolution
The resolution of a digital camera in real space is given by the pixel resolution divided by the
overall magnification of objectives and lenses along the optical detection path. The camera res-
olution is limited by the pixel size and distance. The CMOS sensor has a sensitive area A of
8.576× 6.861mm2 at a maximum resolution of 1280× 1024 pixel. The pixel size is given by the
41
3.2. FLUORESCENCE IMAGING
250fps 200fps 150fps
100fps
50fps
length of square ROI
ex
po
su
re
tim
e
[m
s]
9008007006005004003002001000
20
15
10
5
0
Figure 3-6. The diagram shows the maximum exposure time which depends on the chosen frame rate,
the readout time, and the size of the ROI [47].
sensor dimensions divided by the number of pixels,
Apix =
(8.576mm
1280
)2
= (6.7µm)2 (3.1)
Including the scale factor M = 63 of the microscope objective (Plan Neofluar 63× 1.4 Oil), the
theoretical resolution of the camera in real space is 106 nm. This is below the limit of optical
diffraction.
Sensitivity
The exposure time/frame can be set according to Fig. 3-6. It depends on the size of the ROI
and the chosen frame rate. The manufacturer (Kamerawerk Dresden) specifies a sensitivity of
S = 3.29V lx−1 s−1. For practical purposes it is useful to convert to photonic units. The lux is a
photometric unit which takes into account the eye’s sensitivity function for different wavelengths.
The camera manufacturer defines 1 lux as
1 lux = 5.56× 10−3 Wm−2 (3.2)
integrating the luminosity function over the visual band. In fluorescence measurements, a smaller
wavelength band is detected, and thus,
1 lux = 1.4641× 10−3 Wm−2 (3.3)
42
3.2. FLUORESCENCE IMAGING
at λem = 555 nm is used alternatively. The light intensity on the sensor can be expressed as
I = WλApix t
= µApix t
hc
λem
(3.4)
where Wλ = h c/λem represents the photon energy. h names Planck’s constant, and µ the number
of photons. I is inserted into the definition of the sensitivity S
S = UI t =
U Apix λem
µ h c , (3.5)
where U stands for the sensor’s voltage output. If the output voltage range of the sensor amplifier
is Umax − Umin is distributed on a 8 bit gray scale (256 values), the smallest voltage difference is
∆U = Umax − Umin256 . (3.6)
This holds for an amplifier gain of one. The gain can be raised to higher values taking a loss in the
signal-to-noise ratio. Inserting the minimum voltage difference ∆U = (2.70mV−1.59mV)/256 =
4.34µV into Eq. 3.5 and solving for µ, gives the number of photons that are necessary to increase
the pixel value by 1:
µ = ∆U Apix λemS h c ≈ 242. (3.7)
With this value it is possible to estimate the requirements for single molecule detection. Taking
the emission rate of a Rhodamine molecule of k10 = 108 s−1 from Fig. 1-8, the number of emitted
photons in a 5ms frame interval is 5 × 105. A detected pixel value of 60 is assumed to show
contrast to the background of ≈ 40. The number of photons which have to be detected on a pixel
is Nmin = 60 × 242 = 14520. Thus, the total transmission factor of the optical setup must be
higher than Nmin/5×105 = 0.03. This is a reasonable value, which is provided by this equipment.
3.2.2 Single particle measurements
Imaging systems in microscopy offer the possibility to observe many different molecules at the
same time. Using different types of fluorophores in different colors, other molecules can be labeled
and visualized. This can provide useful information about conglomerates of molecules in natural
environments, such as a cell. If the goal is to localize a single molecule in an assembly, the
microscope’s diffraction limit does not prevent a molecule from being detected. If the sensor
sensitivity is good enough, single molecules can be observed, yet their position is only known
down to half of the emission light’s wavelength. The goal of single molecule detection is to
distinguish molecules from each other on the basis of their observable properties. It is interesting
to determine as many physical properties as possible. Aside from their color, molecules can be
43
3.2. FLUORESCENCE IMAGING
identified by other parameters. As an example, the diffusion coefficient can be measured directly
by analyzing a molecule’s trajectory. This is done by plotting the root mean square distance of
the particle against time (Eq. 1.15). By observing a Brownian object over a period of time, the
relation can be computed directly from subsequent images. The position of the particle has to be
determined from each image. Deviations from the linear relation indicate types of motion other
than free diffusion.
It is desirable to observe molecules over long periods of time. Problems occur if the molecule
wanders out of the focus plane. This motivates to develop optical or mechanical tracking systems
which follow a trajectory in real time. Mechanical tracking can be realized by placing the sample
on a positioner which is controlled electronically and coupled to an imaging software. Using
a differential approach, the particle’s present distance from the central position is determined,
and then translated into a control command for the positioner. The imaging setup, the software,
and the position controller form a feedback loop. Single molecule detection is possible under
certain conditions. The fluorescence quantum yield, exposure time, speed of motion, and the
light collection efficiency of the objective must be optimized. It is one goal to find out, if the
Fluorescence Imaging setup is suitable for single molecule detection and tracking.
44
3.3. CONFOCOR 2 SETUP
3.3 ConfoCor 2 setup
The setup uses a Zeiss ConfoCor 2 inverted microscope with a motorized stage. The objective is
a C-Apochromat 40× with a numerical aperture of 1.2, water immersion type. It is equipped with
a correction ring for varying cover glass thicknesses. The HeNe laser works at a wavelength of
543 nm. The laser power is reduced to 100µW by an acousto-optic tunable filter. The detection
channel consists of a variable pinhole, which is set to a diameter of 78µm. The fluorescence
signal is detected by a built-in avalanche photodiode (SPCM-AQR 13). The emission light passes
a long pass emission filter with a cut-off wavelength of 560 nm. A constant temperature in the
sample chamber is maintained by an electronically controlled holder. The schematic setup was
shown earlier in Fig. 1-10. Further details can be taken from the company’s brochure [48].
45
Chapter 4
Diffusion in microchannels
Liquids in microchannels show laminar flow behavior. This hinders a uniform mixing of reagents
because mixing is limited by the laws of diffusion. As pointed out in the introduction, adequate
mixing of liquids is an important precondition for chemical reactions in microfluidic systems. In
microfluidic flows, the kinetics of chemical reactions are limited because the reactant molecules
need some time to find a reaction partner. A quantitative analysis of chemical reactions inside a
liquid channel requires characterization of the mixing behavior inside the channel.
For the purpose of establishing enzymatic reactions in microchannels, a Y-shaped mixing channel
with two inlets and one outlet was designed and produced. The simple concept of the Y-mixer
[49] has the advantage that well-defined conditions exist. There is only one interface between
the liquids. The mixing process is predictable. In certain applications this channel geometry is
preferable. The fluid channel has a size of only 100µm which is about the size of a hair. Due to
the small dimensions of the channel, the placement of the laser focus is difficult.
The velocity profile of the flow channel is simulated using the Computational Fluid Dynamics
software FLUENT, and validated by FCS. The results described by Goesch [40] are reproduced.
After that, the experiment is carried on further to determine a concentration gradient inside the
microchannel. The purpose is to quantify the effect of diffusion limited mixing on the micrometer
scale.
4.1 Flow profiles in microchannels
The microchannels are straight channels covered with Borosilicate glass. The glass was attached
on the surface by anodic bonding. The channels were etched into silicon wafers with a thickness
of 300µm. Pictures of the microfluidic channels are shown in Fig. 4-1. To enable FCS as a
method, the water running through the channels contains the fluorescent dye Rhodamine 6G in
low concentration. The microchannel is first characterized by determing the velocity profile in the
horizontal and vertical cross section.
46
4.1. FLOW PROFILES IN MICROCHANNELS
4.1.1 Setup
The measurement setup was the open microscope shown in section 3.1. A photos of the experi-
ment is depicted in Fig. 4-2. The laser power was 400µW at first, and later reduced to 100µW
at the rear side of the objective. The Apochromat 63× 1.2 W was used as a microscopy objective.
The space between the objective and the microfluidic glass surface is filled by a drop of water
acting as an immersion medium. The cross sections are trapezoidal (Fig. 4-3). The width on
Figure 4-1. Microscopy image of the Y-shaped mixing channel. The total length of the outlet is
l = 3cm. When liquids containing different species are fed into the inlets, a liquid boundary separating
a concentration gradient is formed (right).
the upside is 64µm for the inlets and 100µm for the outlet. The angle between the walls and
surface is 54.7◦. This is a consequence of the silicon crystal’s anisotropic (110-) structure. The
channels have a constant depth of 40µm. The floor widths of inlet and outlet channels are 7.4µm
and 43µm, respectively. The summed-up cross sections of the two inlets match the outlet cross
section A▽ = 2867µm2 to minimize acceleration of the liquid at the junction.
A constant flow rate F is guaranteed by a syringe pump which maintains a constant pressure on
the gastight syringes. The fluid channels and the syringes are connected with flexible polyethylene
hoses. The flow rate of the syringe pump can be adjusted in units of volume/time.
Focus positioning
The laser focus can be positioned inside the microchannel by two nanopositioners, one for hori-
zontal, and the other for vertical alignment (Fig. 3-3). The horizontal position can be adjusted by
moving the xy-positioner on which the microstructure is attached (see Fig. 4-2). The vertical po-
sition is set by moving the microscope objective in relation to the microstructure. The coordinate
axes are defined in Fig. 4-1 and Fig. 4-3.
Visual observation of the channel was not possible with this setup. Instead, the position of the
walls and the maximum flow velocity can be determined by inspecting the fluorescence intensity
trace and the correlation function.
47
4.1. FLOW PROFILES IN MICROCHANNELS
Figure 4-2. Setup for the flow measurements. Polyethylene hoses are attached to the microfluidic
chip and connected to each channel. A drop of water as an immersion medium was applied between
the surface and the objective.
Figure 4-3. Sketch of the junction between the inlets with trapezoidal cross sections. The top of the
structure is drawn in a hatching pattern, while the bottom floor is white. The walls are blue. Direction
of view from top left.
CFD Simulation
The three-dimensional geometry of the microfluidic channel was reconstructed as a grid model in
a software specialized on Computational Fluid Dynamics (CFD). Fig. 4-3 schematically shows
a clipped drawing based on the grid model of the channel. Subsequently, the velocity profile of
running water inside the channels was simulated using a numerical simulation that applies the
Finite Element Method (FEM). It is based on a forward integration of the equations of motion for
small volume elements. In this iterative calculation the momentum and mass conservation law
must be obeyed. The simulation starts with a uniform velocity distribution of vinit = 1mm/s
over the area of the inlets. The calculation shows that the stream becomes a stationary flow profile
after a distance of 0.145mm. The velocity profile is color coded and visualized in Fig. 4-4. An
analysis of the simulation data showed that the cross sectional profiles in horizontal and vertical
direction are compliant with the assumption of a parabola. In the strict mathematical sense, the
profiles are not exactly parabolic. The influence of the side and bottom walls on the profile cannot
be neglected for certain scenarios. Furthermore, a vertical acceleration of the fluid in negative
48
4.1. FLOW PROFILES IN MICROCHANNELS
Figure 4-4. CFD simulation of the velocity profile in the outlet channel. In the stationary profile the
central maximum is ≈ 2 vinit = 2mm/s (by courtesy of Fraunhofer SCAI).
z-direction occurs at the junction. The reason is the sudden increase in floor diameter. This gives
additional room for the liquid at the bottom of the outlet channel. This component is neglected at
the point of measurement which is far from the junction (≈ 1 cm).
4.1.2 Calibration
The focus parameters were determined before the actual experiment. The focus calibration re-
sulted in focus parameters w0 = 0.32 ± 0.01µm and z0 = 3.2 ± 0.3µm. These values differ
significantly from the theoretical calculation in Tab. 1.3. This difference is partially attributed
to variations in the glass thickness. The glass layer which covers the microfluidic channels has
a thickness of 200µm which differs from the standard cover glass thickness 170µm. This has a
strong influence on the diameter of the laser focus [28]. For comparison, calibrations with cover
glasses lead to a smaller value for z0 of about 2.2±0.3µm. The non-Gaussian beam profile of the
laser (Fig. 3-2) is also expected to contribute to the aberration.
4.1.3 Experimental procedure
The syringes were filled with the same Rhodamine 6G solution in a concentration < 10−8 M. The
flow rate was set to F = 0.5µl/min for both inlets. Following from this, the average flow velocity
in the outlet is theoretically given by v¯th = F/A▽ = 5.8mm/s.
The laser focus was placed at a distance of ≈ 10mm from the junction. After the channel bound-
aries had been found, the focus was vertically placed in the center, and horizontally, at a side wall
of the outlet channel.
49
4.1. FLOW PROFILES IN MICROCHANNELS
fit
exp. data, τflow = 0.040 ms
exp. data, τflow = 0.119 ms
G
(τ
)-1
0.15
0.1
0.05
0
τ [ms]
re
sid
u
al
s
1010.10.010.001
0.005
0
-0.005
-0.01
Figure 4-5. Correlation curves for fast and slow running fluid fitted with model function 1.49. The
number of molecules are N = 5.3 and N = 5.9, respectively.
After the microchannel and the laser focus had been aligned, a series of 20− 23 consecutive cor-
relation measurements of 30 s each were carried out. The focus position was moved in horizontal
or vertical direction from one side to the opposite with a step size of 4µm (2µm, resp.) between
single measurements.
4.1.4 Analysis and results
The recorded correlation functions were fitted with the FCS equation for combined diffusion and
flow, Eq. 1.49. Free fit parameters were τflow and N . In the analysis, the diffusion constant was
kept constant at the literature value of D = 280µm2/s. Example correlation functions are shown
in Fig. 4-5. The local flow velocity is obtained by inserting τflow into Eq. 1.50. The measurement
error is determined from Gaussian error propagation. The error of w0 can be neglected compared
to the error of τflow. Two points of measurement at the sides were excluded because the laser
focus overlapped with the wall, and the fluorescence signal was unusable for analysis. The flow
profiles were fitted with the parabolic equation
vx(y, z) =
3
2 v¯
(
1− y
2
d2 −
z2
h2
)
(4.1)
The origin of the yz-coordinate system is the center of the channel. v¯ names the average velocity
along the central y0- or z0-plane. d and h are the channel’s half width, resp. height, in these
planes. Eq. 4.1 does not decribe the full velocity profile because the geometry specific boundary
conditions are not kept. The FCS measurements show a very good agreement between the mea-
50
4.2. DIFFUSION LIMITED MIXING
sured flow velocity and a parabola fit (Fig. 4-6). The average flow velocity in the z0-plane section
could be determined to 5.60 ± 0.03mm/s, and 5.59 ± 0.03mm/s in the y0-plane. Compared to
the theoretical estimate v¯th = 5.8mm/s, which was integrated over the total channel area, the
values coincide within an error margin of 5%. The channel dimensions resulting from the fits are
87.3µm for the width (in the z0-plane) and 41.9µm for the height. Pretending that the profile
was rectangular, and the flow velocity was constant over the cross section, the flow rate through
the channel would be F = 4 h d v¯ = 1.23µl/min exceeding the expected value by 23 %. Due to
slight misalignment the width comes out too big compared to the true value of d = 71.5µm in
the central plane. The height agrees well with the given value from the production process. The
results confirm that flow profiles can be determined with high precision inside microchannels.
4.2 Diffusion limited mixing
Besides dynamic properties, like the average time of molecules flowing through the laser focus,
FCS also provides information about the concentration of fluorescent molecules in the solution.
This experiment is an extension of the previous. We expect to see the equalization of a substance
along a concentration gradient in streaming water. Fluid dynamics predict laminar behavior of
the fluid. The absence of any convection or turbulence allows to observe undisturbed diffusion.
Again, the measurement was set up according to Fig. 4-1 and Fig. 4-2. The concentration gradient
∂c(x, y, z)/∂y levels off along the flow direction.
In the following, the dependence of c(y | x0 = 1 cm; z0 = 0 cm) on the y-coordinate in the center
of the channel is determined. It is difficult to chose two distant measurement positions in the
outlet channel without misaligning the chosen vertical plane of measurement. Instead of moving
the channel along the x-coordinates, the flow velocity was changed. This simplifies the experiment
greatly.
The flow in x-direction and the diffusion process are perpendicular to each other. They are com-
petitive in the sense that one process can be dominant depending on the average flow velocity. The
lower limit of the flow velocity, below that diffusion is the dominant process, can be estimated by
the channel dimension and the diffusion coefficient of the dye molecules. With Eq. 1.15, the
average time for a Rhodamine molecule to propagate across the half width d of the channel be-
comes t¯ ≈ d2/6D = 3 s. Dividing the length of the channel behind the junction by this value, the
minimum flow velocity for observing diffusive mixing is vmin > x0/t¯ = 3.3mm/s.
CFD Simulation
The mathematical treatment of the mixing problem, as it appears in this experiment, cannot
be solved analytically. The calculation presented in section 1.1.4, Eq. 1-4, is valid only for
51
4.2. DIFFUSION LIMITED MIXING
fit
FCS measurement
y [µm]
v
x
[m
m
/s]
40200-20-40
10
8
6
4
2
0
fit
FCS measurement
z [µm]
v
x
[m
m
/s]
40200-20-40
10
8
6
4
2
0
Figure 4-6. Flow velocity along the y-direction (top) and the z-direction (bottom) across the central
planes z0 and y0 of the microchannel. The error bars are given by the error of the fitted parameter
τflow.
52
4.2. DIFFUSION LIMITED MIXING
liquids at rest. It can serve as an analytical approximation at most. Here, a concentration bound-
ary and a flow profile appear simultaneously. The problem is three-dimensional (instead of one-
dimensional) and additionally, reflecting walls restrict the channel on four sides.
This problem can again be solved with the help of the numerical CFD simulation. Diffusion
processes are not automatically accounted for. The calculation of diffusing molecules in high
dilution cannot be realized by the FEM alone. It is realized by an additional simulation step using
a species transport model. In this model, the fluorophore solution is defined as an equivalent liquid
with properties partially matching those of the Rhodamine molecule. The dynamic viscosity of
this equivalent liquid is chosen eight times higher than the viscosity of water. This is according
to the diffusion coefficient ratio of water and Rhodamine molecules. The mass fraction of the
equivalent liquid was chosen as 1/1000. Thus, the overall viscosity of the simulated liquid does
not differ from water. The details about the simulation are explained in the project report [50].
The result of the CFD simulation including the species transport model is depicted in Fig. 4-7. Fig.
4-8 shows the according cross sectional distributions from the central plane. At a flow velocity of
v ≈ 2mm/s in the center of the channel, the Rhodamine is evenly distributed after ≈ 9− 10mm.
Theoretically, a value of about v × t¯ = 6mm was expected (cp. introduction to this section). To
observe the concentration step in an experiment, a higher flow velocity should be chosen than in
the simulation.
4.2.1 Calibration
Again, the focus parameters were determined before the experiment. Values of w0 = 0.317 ±
0.003µm and z0 = 3.2± 1µm were obtained.
4.2.2 Experimental procedure
One syringe was filled with pure water, the other one with the Rhodamine 6G solution (c <
10−8 M). The flow rate was adjusted to F = 0.5µl/min in both inlets. The microchannels were
then carefully filled with both solutions. At the junction, a boundary of two adjacent liquids is
formed and molecular exchange is enabled. The system was left alone for several minutes until
diffusion and flow have became stationary at constant rates. The laser power was 100µW in this
experiment.
Then, 28 correlation measurements were carried out across the channel center. As a compromise
between total measurement time and data quality, the duration for one curve was 45 s. In between
measurements the laser focus was moved 3µm further along the y-axis. For comparison, the flow
rate was increased to F = 2.0µl/min in both channels. The measurement was repeated at the
53
4.2. DIFFUSION LIMITED MIXING
Figure 4-7. CFD simulation of diffusive mixing in the Y-shaped microchannel. The images are cross
sections taken at distances 0, 0.1, 0.5, 1, 2, 3, 4, and 5 mm from the junction (image by courtesy of
Fraunhofer SCAI).
20
10
5
4
3
2
1.5
1
0.5
0.1
0
in mm
y [µm]
m
as
s
fra
ct
io
n
o
fw
/R
h6
G
0-20-40-60-80-100
1e-03
9e-04
8e-04
7e-04
6e-04
5e-04
4e-04
3e-04
2e-04
1e-04
0e+00
Figure 4-8. CFD results for the central concentration distribution across the outlet cross sections.
The curves represent concentration distributions along the y-axis in the center of the channel. The
distance from the junction is given in the legend (image by courtesy of Fraunhofer SCAI).
54
4.2. DIFFUSION LIMITED MIXING
τflow =0.0095N =1.3
τflow =0.0093N =1
τflow =0.011N =0.81
τflow =0.012N =0.62
τflow =0.016N =0.4
τflow =0.051N =0.34
τ [ms]
G
(τ
)−
1
10.10.010.001
3.5
3
2.5
2
1.5
1
0.5
0
Figure 4-9. Correlation data from the mixing experiment. Straight lines are best fits. The arrow
points in direction of decreasing concentration.
same positions. It is expected that the dye concentration in one half of the channel is lower at the
higher flow rate.
4.2.3 Analysis and results
As in the previous experiment, the correlation functions were fitted to Eq. 1.49 with free parame-
ters N and τflow. Some correlation curves are shown in Fig. 4-9. The data from three measurement
positions at the sides were excluded from the analysis because the measurement was disturbed by
unpredictable optical effects originating from focus overlaps with the side walls. The local fluo-
rophore concentration at the chosen positions was determined for the low and for the high flow
rate. The uncertainty in concentration is given primarily by the standard deviation of the correla-
tion function. Koppel’s theoretical expression, Eq. 1.43, was applied resulting in a value of about
10 %. Only for absolute concentration measurements the error in the observation volume must be
taken into account.
Figure 4-10 shows the measured Rhodamine concentration gradient along the profile of the mi-
crochannel. Additionally, the velocity profiles are depicted in the top diagram. The concentration
values are given in arbitrary units (a. u.) on the left, and in mol/l on the right axis. The latter fol-
low from the calibrated focus parameters. They should not be regarded as absolute values because
the open microscope setup does not provide true concentration determination without information
about the function MDE(r, z).
The fall-off in concentration along the y-axis is significant at the higher flow rate. At the low flow
rate, the concentration is almost constant across the channel. Subsequently, Eq. 1.13 was fitted
55
4.2. DIFFUSION LIMITED MIXING
fit
v¯ = 22.8 mm/s
v¯ = 6.09 mm/s
y [µm]
v
x
[m
m
/s]
403020100-10-20-30-40
40
35
30
25
20
15
10
5
0
fitted Eq.1.13
F=2 µl/min
F=0.5 µl/min
y [µm]
c
[m
o
l/l
]
c
[a.
u
.
]
1e-09
1e-10
403020100-10-20-30-40
1
0.1
Figure 4-10. Velocity profiles in the outlet channel for high and low flow rate (top). Distribution of
Rhodamine 6G fluorescent dye in the microchannel at two different flow rates (bottom). The concen-
tration data for the higher flow rate was fitted to Eq. 1.13 (fit parameters c0 = 2.3, t = 1.4 s).
56
4.2. DIFFUSION LIMITED MIXING
to the values for the higher flow rate including terms of fourth order. This approximate equation
describes the data surprisingly well. However, the fitted time parameter t = 1.4 s is much larger
than the average flow time from the junction to the point of measurement. Ignoring the fact that
the model is not strictly valid, this result indicates that the mixing is more efficient than expected.
The experimental data is not precise enough to allow for a better quantitative comparison with the
simulated concentration distribution. Qualitatively, the theoretical approximation by Eq. 1.13 and
the CFD simulation are both confirmed.
To improve the accuracy of the data, several correlation functions can be averaged at each position.
This requires a considerably longer measurement time. As an additional measure of the local con-
centration, the average fluorescence intensity can be taken into account. In the low concentration
range, the mean fluorescence intensity is proportional to the dye concentration. Further improve-
ment can be reached by establishing well-defined conditions of the laser focus. The glass layer
thickness has to be changed to 170µm. The beam profile can be optimized by using a different
laser.
57
4.3. PARTICLE TRACKING
4.3 Particle tracking
As was shown in the previous sections, the detection of fluorescent molecules in solution gives
information about local and dynamic properties like the molecule concentration, the flow veloc-
ity, or the diffusion coefficient. However, sometimes the relevant information is hidden due to
averaging over a large number of molecules. Despite the single molecule sensitivity of a confocal
microscope, the FCS method requires to observe a molecule ensemble to determine properties of a
molecule species from the intensity trace. In contrast to that, it is advantageous to determine prop-
erties of single molecules, e. g. in intracellular measurements. If fluorescence labeled molecules
can be distinguished from each other, certain cell functions could be revealed easier.
One goal of single molecule detection (SMD) techniques is to track a single molecule trajectory
and to determine specific parameters during observation [46, 51]. Thus, new information can
be obtained considering that the results are subject to fluctuations. For a detailed discussion of
the significance of single molecule measurements see e. g. [52]. In this section, the principle of
single molecule measurements is shown in vitro using a micrometer sized particle. This has the
advantage of a high signal-to-noise ratio and slow particle motion. The setup for video microscopy
is introduced for this purpose.
4.3.1 Experimental procedure
Image detection was realized using the microscope setup described in section 3.2. The solution
was prepared by diluting a stem solution of fluorescent microspheres by a factor of 100. A hang-
ing drop was applied to the surface of a cover glass and put under the objective. The microscope
objective was a Plan Neofluar 63×, 1.4 Oil immersion. White light was used for illumination
because the parallely installed (blue) laser did not match the excitation spectrum of the fluores-
cence dye. The particle was observed for 20 s at a frame rate of 50 fps. The exposure time was
8ms/frame. The digital film was recorded in the AVI file format.
4.3.2 Analysis and results
After recording, 1000 images were captured from the film and converted into ASCII files (using
the conversion utility of the MPlayer software). A short program was written to find the pixel
coordinates with maximum gray value in each frame of the film. The background pixel value
was ≈ 40 on the average, while the particle intensity reached a maximum of 123. The pixel with
highest intensity in each frame was assigned to the particle’s center position. The time interval was
given by the inverse of the frame rate 50 fps. The trajectory in two dimensions was reconstructed
from the 1000 coordinates (Fig. 4-11).
58
4.3. PARTICLE TRACKING
 0
 50
 100
 150
 200
 250
ROI x [pixel]
R
O
I y
 [p
ixe
l]
frame
path
 0  50  100  150  200
 0
 50
 100
 150
 200
 30
 40
 50
 60
 70
 80
 90
 100
 20  30  40  50  60  70  80  90
y
x
trajectory
start
end
Figure 4-11. Captured frame from the film of a diffusing particle. Black dots in the image are dirt
on the objective or optical filter. The microsphere’s trajectory is drawn in the image plane (top) and
separately as a graph (bottom).
59
4.3. PARTICLE TRACKING
time [s]
〈∆
r2
〉[µ
m
2 ]
1001010.1
100
10
1
0.1
Figure 4-12. The mean square distance for the path of the particle shown in Fig. 4-11. The diffusion
coefficient is determined from the slope of the linear fit between 0.1 and 6 s.
In case of a Brownian motion, the mean square distance from an arbitrary starting point is pro-
portional to time (Eq. 1.15). When the mean square distances are computed from the coordinates
of the particle, its diffusion coefficient can be determined. The microsphere motion in Fig. 4-11
supposedly conforms to a free particle. Fig. 4-12 shows the linear relation between 〈r2〉 as it was
determined from the particle path. D is obtained from the slope of the linear fit between 0.1 and
6 s,
D = 〈∆r
2〉
4∆t = 0.445µm
2/s.
The expected diffusion coefficient D1µ = 0.429µm2/sec for a particle with a diameter of 1µm is
known from Eq. 1.16. Theoretical and experimental value agree very well within an error margin
of 5 %. The hypothesis that the particle motion is Brownian can be accepted. Deviations from the
linear relation occur after 10 sec. Apparently, the particle performed a path returning close to its
starting point.
Conclusion for mechanical tracking
The experiment shows that the diffusion coefficient for a single particle can be determined with
good precision. For this particle and framerate the time that is necessary to determine D can be
reduced theoretically to about half a second. This is considerably shorter than in FCS. To follow
the trajectory of the particle mechanically in real time, the response time of the nanopositioner has
to be considered. The response time of 7.1ms is known from a dynamic measurement. The step
size can be assumed to be at the limit of the optical resolution. The highest diffusion coefficient
which could be followed is then Dmax ≈ 2µm2/s. In this case, the camera and image processing
60
4.4. DISCUSSION
software must work at a frame rate of 150 s−1. Mechanical tracking of small molecules in aqueous
solution is still out of reach in this concept. However, tracking of particles bigger than 100 nm, or
tracking of single molecules in media with higher viscosity, seems to be a realistic goal.
4.4 Discussion
In section 4.1 and 4.2, the laminar flow profile and the properties of a microfluidic mixing chan-
nel were examined. The experimental velocity profiles agree well with the classic equations for
laminar flow in such devices.
Mixing of an initially uneven concentration could be observed. Differences in local concentrations
inside a microchannel were successfully detected. The concentration gradient can be adjusted by
changing the flow rate. Diffusion due to Brownian motion appears as the only source of mixing in-
side the channel. There is no evidence, that other effects contribute to better or worse mixing. The
measurement data agree well with predictions from the CFD simulation. Analytical expressions
for complex geometries in three dimensions do not exist. Simple analytical calculations give qual-
itatively good estimates. Numerical simulations help solving the problem for special cases with a
precision only limited by computational power. For a quantitative comparison, the measurement
precision must be improved. Repeated measurements at the same positions reduce the stochastic
error as mentioned in section 4.2.3. The optical properties of the cover glass have to be adapted
to the standard of 170µm in microsopy. This type of glass requires very careful handling during
the bonding process and is rarely used. Theoretical focus calculations which include the refractive
indices of the material along the beam path can also contribute to higher precision.
The experiment shows that uniform mixing is quickly reached in the 100µm channel at low flow
rates. Nevertheless, for the purpose of observing biochemical reactions in a continuous flow, the
Y-channel mixer has a disadvantage. At high flow rates, the medium has left the microchannel
before uniform mixing has been reached. In low concentrations, the time for the initiation of enzy-
matic reactions in the outlet channel might be too short. However, as the sensitivity for detecting
single molecule transitions is given, there is still a chance that the Y-channel mixer can be used
as a reaction chamber for continuous flow systems. In this case, fluorescent switches as sensor
molecules can be applied. A four component system consisting of Amplex UltraRed (Invitrogen),
glucose oxidase, glucose, and horseradish peroxidase (HRP) was tested in small sample drops
and in microfluidic channels. Amplex UltraRed separates the fluorescent dye resorufin upon the
reaction with H2O2. The latter is a reaction product of the enzymatic reaction between glucose
oxidase and glucose in the presence of HRP. The reaction kinetics were analyzed according to
the Michaelis-Menten theory for enzymatic reactions. Confocal intensity measurements showed
that the reaction is initiated even below micromolar concentrations of glucose. For even lower
61
4.4. DISCUSSION
concentrations, single molecule events have to be counted and discriminated from the background
using one of the SMD techniques. For the analysis of single molecule reaction data, the fluores-
cence intensity signal has to be interpreted in an appropriate way. Stochastic methods based on
information from the photon counting histograms (PCH) of the fluorescence signal are promising
tools which are presently being developed [53].
The problem of mixing on the microscale has been approached earlier. Most suggestions are
based on the idea to multiply the area of contact between the two liquids. This can be achieved by
intersecting and recombining the stream [54, 55]. Other methods rely on chaotic mixing [56]. In
both cases, the experimental realization is not easy because complicated flow geometries have to
be produced. Especially for the chaotic mixer, another disadvantage is the unpredictable molecule
distribution after the mixing stage. Through the mixing process chemical reactions are accelerated,
but the internal conditions do not reach a completely homogeneous distribution. Nevertheless,
in very low concentrations, a ’homogeneous’ destribution of a molecule species does not exist.
Statistical fluctuations of the local concentration dominate the molecular distribution as pointed
out in section 1.1.1. These remain an important factor and must be taken into account in theoretical
calculations as well as in experiments. The necessary measurement time and the significance of
the result is affected thereby [52]. New numerical simulations that combine reaction kinetics,
diffusion, and liquid flow would contribute to the development of microreaction devices with
predictable properties.
For the purpose of measuring flow profiles, other methods are available. Two methods called
double-focus fluorescence cross correlation, and Particle Image Velocimetry (PIV) have been suc-
cessfully applied to fluidic channels [57, 58].
4.4.1 Outlook: Real time image processing and 3D molecule tracking
Real time image processing of a single molecule observation demands the experimental condi-
tions to be improved. Because of the low fluorescence yield from a single molecule, especially
the background noise has to be reduced. Laser irradiation instead of white light is recommended.
Furthermore, the exposure time must be optimized to gain a sufficient number of photons/pixel,
and, at the same time, the frame rate has to be high enough to keep track of the molecule posi-
tion. Regarding the sensitivity of the CMOS camera, which was calulated in section 3.2, single
molecule detection should be a reachable goal. To realize a tracking system, the position detection
has to be processed in real time [42, 59]. There are several algorithms for detecting the center of
a particle’s position to be tracked in fluorescence images data. An efficient approach which can
be realized in FPGA hardware (Field Programmable Gate Array) was suggested by Lueghausen
[47]. Position control has to be connected to the software to receive the coordinates with a shortest
possible delay. Generally, three points have to be considered:
62
4.4. DISCUSSION
1. Constraints have to be given for the maximum amplitude of the positioner in every direction
to prevent it from damage.
2. The mechanical precision is higher than the optical resolution of a light microscope. In
biological cells, some relevant structures can be much smaller. Single molecule tracking
does not improve resolution.
3. Photobleaching limits the time for observation in the case of molecules.
Tracking in three dimensions suffers from another problem, that is the molecule’s vertical direction
of motion. Due to the optical symmetry along the beam axis, the up- and down-direction cannot
be distinguished from each other. Different solutions to this problem were proposed. One is
to operate the objective nanopositioner in a scanning mode. The PIFOC performs an oscillation
around the molecules present position. The zero point has to be adjusted as soon as a change in the
molecules vertical position has been detected. This requires a rather complicated computational
and piezomechanical effort and slows down the tracking performance. There is a method available
for nanoparticles which is based on analyzing the fraction of forward scattered light [60]. A
different approach is a wedge prism located at the back focal plane. The idea was presented by
Mizutani [61]. The prism separates the fluorescence image into two halfs. Hereby, an optical
asymmetry is introduced into the detection path. The molecule’s vertical direction of motion can
be distinguished from each other by the asymmetric intensity distribution on the detector.
63
Chapter 5
Diffusion in surfactant bilayers
The preceding chapter 4 focussed on molecular motion in structures with micrometer expansions.
It was shown that, on this scale, fluids and molecules act consistently with the classic theories.
When diffusion is observed on a nanoscale, additional phenomena emerge. In this chapter, diffu-
sion experiments are carried out to measure the dynamics inside supramolecular structures. The
size of the observation volume in FCS, as it is used here, is fixed. Instead, scaling to the nanome-
ter range can be achieved by reducing the size of the observed physical system. The examination
of nanostructured environments can bring out new information about molecular interaction. In
this chapter, the diffusion coefficient – as a measure of mobility – is determined for binary sys-
tems composed of surfactant/water emulsions in different mixing ratios. The mixtures of lipid and
water form a micellar and a lamellar phase. Additionally, the influence of a third substance is in-
vestigated (cholesterol). In a ternary systems, structural changes can occur. Fluorescent molecules
are added to the lipid bilayers to probe these changes. The presence of cholesterol is expected to
have an influence on the molecular mobility. The obtained results affect the related field of Cell
Biology, as a major part of it deals with diffusion in the cellular membrane.
5.1 Introduction
In eukaryotic cells, the plasma membrane is mainly composed of glycerophospholipids, sphin-
golipids, and a sterol. The amphiphilic structure of the lipid molecules results in the formation
of a bilayer membrane. For their function, cells need an exchange of molecules and ions with
their surrounding through the cell membrane. The knowledge of how these transport processes
are controlled by the cell are important, e. g., for the development of pharmaceuticals. The fact
that the composition of the cell membrane is maintained by intracellular functions suggests that
it is a key factor for the cell’s life cycle and communication. Especially the cholesterol and lipid
content of the membrane play an important role for the regulation of protein transport inside and
outside the cell [62]. Cholesterol within the membrane has an influence on the molecular order by
64
5.2. MEMBRANE SYSTEM
hydrophobic and steric interaction [63, 64]. The purpose is to quantify the effect that cholesterol
has on the properties of surfactant membranes.
Mechanical properties of cell membranes as well as the exchange of matter through the mem-
branes depend strongly on the diffusion of lipid molecules inside the membrane. To study dif-
fusion in membranes, several methods have been used. Light scattering and NMR methods pro-
vide high precision due to averaging over large ensembles of particles. On the other hand, when
diffusion on small scales with few or even single molecules shall be observed, fluorescence mi-
croscopy methods provide a better way to determine more interesting properties. For this study,
Fluorescence Correlation Spectroscopy (FCS) was chosen as an experimental technique offering
the advantages of small sample volumes and short measurement times.
Because of the similarity between the lamellar phase of surfactant/water emulsions and the bi-
ological cell membrane they have been considered as simplified model systems for membranes.
Adding other substances to the system can also affect the phase transitions and the properties of
the individual phases. Cholesterol is an ingredient in natural membranes. Because its fraction in
the total lipid content varies among different cell types, its function is not completely clear. Here
the effect of cholesterol on a surfactant/water system, in particular the lamellar phase, is studied.
By using a surfactant/water model, the fundamental influence of cholesterol incorporated in bi-
layers is examined. In contrast to lipids of biological membranes, the chosen surfactant consists
of a headgroup and a single saturated dodecyl chain. The molecular order of the system is sim-
plified compared to biological membranes composed of a variety of complexer lipids. It is shown
that cholesterol does not reduce diffusional mobility in complex structures only, but in basic lipid
assemblies as well.
5.2 Membrane system
5.2.1 Components
C12E5 As a model membrane, the system C12E5/water was chosen. The lipid pentaethylene
glycol monododecyl ether (C12E5) is a non-ionic surfactant with a molecular weight of M =
406.6 gmol−1. For the molecule length values from 22 Å to 26 Å [66, 67] were reported. The
critical micelle concentration for C12E5 is 64µM at 25◦C [68]. As can be seen in the diagram
(Fig. 5-1), the emulsion of C12E5 and H2O shows different phases depending on the surfactant
concentration and temperature.
65
5.2. MEMBRANE SYSTEM
Figure 5-1. Phase diagram of C12E5 and H2O. L1 and Lα denote the micellar and the lamellar phase,
respectively. H1 is the hexagonal phase, L2 and S the inverse micellar and the solid phase. V1 is the
cubic phase. L3 is a sponge phase. Unnamed areas are bicontinuous. The gray dots mark the systems
that were measured. The image was redrawn from [65].
Figure 5-2. Chemical structure of cholesterol.
R18 The use of a fluorescent marker molecule is required to detect light from the transparent
system. For this purpose, the fluorophore octadecyl Rhodamine B (R18) was chosen (D = 420×
µm2 s−1 in water, M = 731.5 gmol−1). R18 is an amphiphilic molecule with a polar head and an
unpolar tail (see Fig. 1-7 and Tab. 1.2.). The dye molecules enter into the bilayer system and probe
its mobility. Here, it is generally assumed that their diffusion is related to the lipid molecules, but
an independent verification would be appropriate. There is no information available if R18 prefers
the lipid-ordered or disordered phase.
Cholesterol has a molecular weight of 386.65 gmol−1 and a density of 1.07 g cm−3 at 20◦C.
The molecule length is 16 Å. The structure of cholesterol is depicted in Fig. 5-2.
5.2.2 Preparation of Surfactant Emulsions
Lipid and cholesterol were weighed with a micro balance, vortexed, heated and sonicated until
the system became completely homogeneous. Then, milliQ water containing fluorescent dye was
66
5.2. MEMBRANE SYSTEM
Figure 5-3. Spherical and rod-like micelle
added. The 20 wt % C12E5 and the 60 % solutions contained 10−8 M R18, while the concentration
in the 80 % solution was 10−7 M. The samples were then again heated to 60-70◦C, sonificated,
vortexed, and frozen (-80◦C). These steps were repeated until the samples had become homo-
geneous gels. For the nominal 60 wt % emulsions, the fraction of C12E5 and cholesterol varied
between 0.596 and 0.643. In the subsequent measurements of the 80 wt % emulsion the lipid
fraction was kept constant at 0.80. The ratios of substances are listed in Table 5.3.4.
5.2.3 Micellar phase
In the micellar phase, the dye molecules enter the micelle and the observed diffusion will be that
of the micelle. From standard diffusion theory, the expected diffusion coefficient D is related
to the apparent hydrodynamic radius Rh of a sphere by Eq. 1.16. The diffusion coefficient of a
typical micelle with a diameter of 5.0 nm is Dmic = 86µm2 s−1 with T = 293K, and the viscosity
of water η = 1mPa s. Corrections can be applied for elliptical or flexible geometries, but will be
neglected for the further analysis.
5.2.4 Lamellar phase
In the lamellar phase, the lipid molecules form a bilayer structure. The membrane layer thickness
is about double the length of a surfactant molecule δ = 2 ls. The repeat distance of the lamellar
structure is
Xlam = 2 ls/φs (5.1)
where φs is the volume fraction of the surfactant in the emulsion. Assuming φs = 0.6 (0.8) and
ls = 26 Å, the repeat distance has a value of Xlam = 8.7 nm (6.50 nm). The molecular order of
the lamellar phase is maintained by van der Waals forces between the alkyl chains and hydrogen
bonding forces between water molecules and the lipid headgroups [69].
The FCS measurements probe the bilayer dynamics in a volume of ≈ 0.5µm3, at a distance of
60µm from the cover glass surface. Because this distance leaves room for about 7000 − 9000
67
5.3. FCS MEASUREMENTS
Figure 5-4. Lamellar bilayer
bilayers, it is assumed that surface effects are no longer important and the bilayers are randomly
oriented. In addition, the horizontal alignment of the measurement position is varied across the
sample drop. Remaining orientations of bilayers are averaged out by repeated measurements.
The molecular diffusion of each molecule is restricted to a single two-dimensional bilayer. The
resulting propagation differs from that of free diffusion [12], but the observed signal represents
an average over all possible orientations. Within the laser focus, which represents the sensitive
volume, there is room for about 50 bilayers when they are arranged horizontally and 300 when
they are arranged vertically.
Following the procedure of Galla and Sackmann for proteins and lipids in biological membranes
[70], the diffusion of dye and lipid molecules are related to their molecular weights as
Ddye = Dlipid
(Mlipid
Mdye
)1/2
. (5.2)
When C12E5 is compared to R18, the correction factor is 0.75.
Experimental data show that the shear viscosity in such binary or ternary systems can easily span
orders of magnitude from 1mPas to 105 Pas depending on concentration, pressure and tempera-
ture [71, 72, 73], but the effect of cholesterol on these parameters has not been studied so far.
5.3 FCS measurements
The experiments were carried out on the ConfoCor 2 microscope described in chapter 3.
5.3.1 Calibration
The focus parameters were calibrated with a 10−8 M aqueous solution of Rhodamine 6G. The
waist parameter determined in this way fell between w0 = 0.22 − 0.25µm, with an average of
w¯0 = 0.241µm. The structure parameter K was set to a fixed value of 6.0 corresponding to a
68
5.3. FCS MEASUREMENTS
vertical focus radius of z0 = 1.3 − 1.5µm. The theoretical calculation using Rigler’s and Mets’
formula resulted in a slightly smaller value for wth (cp. Tab. 1.3).
Adding C12E5 to pure water changes the refractive index of the sample solution. Hamano gives a
linear equation to calculate the refractive index of a C12E5/water mixture [74] for a wavelength of
633 nm and a temperature of 20◦C. According to this, the mixtures consisting of 20, 60, and 80
wt % C12E5 have refractive indices of n′ = 1.357, 1.407, and 1.432.
The difference in optical refraction changes the size of the focused laser spot. Taking this into
account, the focus waist is estimated in these three media to be wˆ0 = 0.236, 0.228, and 0.224,
respectively. These values were used in the evaluation of the correlation functions.
Two additional experimental adjustments were done to take the different refractive index into
account. The microscope objective has a correction ring which can be used to adjust the refraction
properties to varying cover glass thicknesses. Measurement series with 20 wt % and 50 wt %
C12E5 emulsions showed the highest fluorescence count rates if the correction ring was in the
maximum position of 0.18µm. In all following experiments the correction ring was kept in this
position. Secondly, the length of the optical light path was reduced by setting the focus to a
position 60± 1µm above the cover glass surface.
5.3.2 Autocorrelation model functions
The standard model proved to be unsatisfactory for the analysis of the diffusion measurements in
the lamellar phase. This is related to the fact that the molecules are not free to diffuse in three
dimensions, but are confined to the two-dimensional bilayers. The orientation of these bilayers
with respect to the laser beam is random, so the measurements detect an orientational average.
The orientation model from section 2.4, Eq. 2.16, is used as an alternative. The triplet term in Eq.
1.46 is added.
5.3.3 Experimental procedure
During the FCS experiments, the samples were placed in Lab-Tek 8-NUNC sample chambers. The
bottom glass type is Borosilicate #1 with a thickness between 0.13 and 0.16µm. The micellar
emulsions are liquids and 4 − 5 drops can be poured into the chambers. For the highly viscous
gel emulsions (60% surfactant and more) a spatula must be used. The vessel was placed into the
holder and covered with a non-transparent lid. The temperature control unit was set to 30◦C. Table
5.3.4 lists the different samples that were used for this study, showing the different Molar ratios
of cholesterol and C12E5.
Five subsequent correlation functions of 30 s each were averaged for one measurement. For each
sample, 8 to 20 measurement runs were performed. The correlation functions were fitted to the
69
5.3. FCS MEASUREMENTS
theoretical expressions of Eq. 1.40, Eq. 2.16 to determine the diffusion time and, using Eq. 1.42,
the diffusion coefficients. Additionally, the cpm value was determined.
5.3.4 Results
Figure 5-5 shows a typical measurement result from the micellar phase, for a sample of 20 wt
% C12E5. The experimental correlation data were fitted with the standard FCS function for free
diffusion, Eq. 1.46. The best fit was obtained for the average parameters τd = 2800 ± 800,
N = 0.32± 0.03, τtrip = 3µs, p = 0.14. According to Eq. 1.42, this corresponds to a diffusion
coefficient Dmic = 5.4 ± 2µm2/s. This value agrees with the expectation that it arises from the
diffusion of prolate micelles. Their average size can be estimated using the corrected Stokes-
Einstein expression for rod-like objects (Eq. 1.17). An axis ratio of 40:1 is presumed which was
given in [14] resulting in a = 180 nm for the long axis. For the given sample, this amounts to
about 5000 micelles in the focal volume, of which on average N = 0.32±0.03 carry a fluorescent
molecule.
In the lamellar phase two different ratios of C12E5 and water were examined, 60 wt % and 80 wt %
surfactant. In this case, fluorophores diffuse independently within the lipid bilayers. The average
number of molecules in the detection volume was 1.1± 0.3 for the low concentrated lamellar and
11±6 for the high concentrated lamellar phase. The higher value for the 80 % lamellar phase was
expected from the sample preparation.
Fitting these data with the standard model of unrestricted (3D) diffusion (Eq. 1.46), a significant
deviation is found, as shown in Fig. 5-6. However, there is no systematic deviation if the experi-
mental data is fit to the orientation model, Eq. 2.16. For the example shown in Fig. 5-6, the two
models yield diffusion times of τd = 1.32 ± 0.02 (standard model) and τd = 0.901 ± 0.008 ms
(orientation model), corresponding to diffusion constants D of 12.4 and 18.2µm2/s. While this
analysis results in different values for the diffusion constants, the dependence on the cholesterol
content does not change. The values of the measured diffusion constants are consistent with the
assumption that the fluorescent R18 molecules were confined to the lipid bilayers. To check for
a fraction of R18 in the water phase, the data was fit to a two-component model function. The
results show no significant component with a high diffusion constant.
Figure 5-7 summarizes the change of the diffusion constant in the 60 % lamellar phase upon the
addition of cholesterol. At low concentrations, no substantial effect is visible, but at a cholesterol
content of about 8 %, a sudden reduction by ≈ 20 % occurs. A further increase of the choles-
terol concentration yields an additional continuous reduction. The total decrease in the molecular
mobility is ≈ 30 % for a cholesterol Molar ratio of 0.2.
The same measurements were also performed for the samples with the higher lipid contents. As
shown in Fig. 5-8, the diffusion constant for low cholesterol content is approximately half of the
70
5.3. FCS MEASUREMENTS
phase C12E5 chol. H2O ratio lipid frac
wt % mg mg mg
mic 20 80 0 320 0 0.20
lam 60 243.5 48.8 162.6 0.211 0.643
233.7 38.1 154.8 0.171 0.637
255.3 38.3 170.7 0.158 0.632
230.5 33.9 156.8 0.155 0.628
243.2 31.3 163.0 0.135 0.627
267.7 31.5 177.7 0.124 0.627
255.6 25.3 168.9 0.104 0.624
199.3 19.6 132.9 0.103 0.622
250.0 21.4 167.5 0.090 0.618
227.1 16.8 151.6 0.078 0.617
219.7 12.1 145.8 0.058 0.614
253.6 9.7 173.6 0.040 0.603
229.4 8.2 153.6 0.038 0.607
227.0 4.1 156.8 0.019 0.596
lam 80 153.8 3.05 40.7 0.021 0.794
159.5 10.75 43.4 0.071 0.797
159.0 15.7 44.6 0.104 0.797
159.4 17.2 44.4 0.113 0.799
155.8 20.5 45.0 0.138 0.797
159.5 26.8 48.2 0.177 0.794
Table 5.1. The table lists the amount of substances used in the microemulsions. The ratio value gives
the Molar ratio of cholesterol versus C12E5. The lipid frac value denotes the fraction of C12E5 and
cholesterol.
corresponding data for the 60 % system. Again, a sudden decrease of the mobility is observed at
a cholesterol content of ≈ 8 %, and a continuous decrease at higher concentrations.
The triplet fraction in the correlation amplitude is at 16 % on the average. The cpm value is be-
tween 11− 14 kHz for low concentrations of cholesterol. Increasing the cholesterol concentration
reduces the count rate by up to 50 % (Fig. 5-9). This may be a consequence of increased scattering
in these samples. Visually observable was a small turbidity in the sample with a cholesterol ratio
of 0.2 at room temperature. Alternatively, the count rate reduction may occur due to fluorescence
quenching by the cholesterol.
In addition to the graphical representation of the diffusion data, Table 5.2 also gives the numerical
values for the extreme cases. The last column of this table also shows the inferred diffusion
constants of the lipid molecules, which were calculated from the measured diffusion constants by
correcting for the different molecular weights using Eq. 5.2.
71
5.3. FCS MEASUREMENTS
correlation data
one component fit
G
(τ
)-1
5
4
3
2
1
0
τ [ms]
re
sid
u
al
s
1000010001001010.10.010.001
0.20.10
-0.1
-0.2
-0.3
Figure 5-5. FCS data for the micellar phase. The diffusion time in this example is τd = 1.99 ± 0.05
ms. The residuals show the absolute difference between fit and correlation data.
correlation data
orientation fit
standard fit
G
(τ
)-1
1
0.8
0.6
0.4
0.2
0
τ [ms]
re
sid
u
al
s
1000010001001010.10.010.001
0.02
0
-0.02
Figure 5-6. Example for the FCS data from the surfactant/cholesterol/water emulsion. The exper-
imental data (crosses) are compared to best fits using the standard (single-component) model (dash-
dotted line) and to the orientation model (full line). The cholesterol to surfactant ratio was 0.172. The
diffusion times are τd = 1.32 ± 0.02 ms (standard model) and τd = 0.901 ± 0.008 ms (orientation
model).
72
5.3. FCS MEASUREMENTS
 0
 5
 10
 15
 20
 25
 30
0.10.020
di
ffu
sio
n 
co
ef
f. 
[µm
2 /s
]
ratio chol:C12E5
experimental data
Figure 5-7. Result for the diffusion coefficient D of R18 in the lamellar phase with 60 wt % surfactant.
The Molar ratio of cholesterol to surfactant increases to the right. A reduction in molecular mobility
is apparent at a ratio of 0.08. Error bars denote the standard deviation. Guidelines for the eyes are
drawn dashed.
 0
 5
 10
 15
 20
0.10.020
di
ffu
sio
n 
co
ef
f. 
[µm
2 /s
]
ratio chol:C12E5
experimental data
Figure 5-8. Diffusion coefficients for the 80 wt % surfactant emulsion as a function of the cholesterol
content.
73
5.4. DISCUSSION
C12E5 phase chol:C12E5 D(probe) D(lipid)
wt % ratio [ µm2/s]
20 micellar – 5.4 –
60 lamellar 0 26 34
60 lamellar ≈ 0.2 16 21
80 lamellar 0 14 19
80 lamellar ≈ 0.2 9 12
Table 5.2. Overview of diffusion coefficients D in C12E5/water/cholesterol emulsions determined
from the orientation model.
 0
 2
 4
 6
 8
 10
 12
 14
 16
 0.01  0.1
cp
m
 [k
Hz
]
ratio chol:C12E5
day 0
day 9
Figure 5-9. Counts/molecule value (cpm) as a function of cholesterol content from the same sample
on two different days. The brightness is reduced by increasing cholesterol content.
5.4 Discussion
In the micellar phase (at 20 wt % lipid concentration), the observed correlation function is con-
sistent with a model of freely diffusing micelles with a length of the order of 180 nm and a given
axis ratio of 40:1. This size might be an overestimate, since the interaction between the micelles
should result in a smaller diffusion constant than for micelles at low concentrations. Using this
value, an upper limit of 0.14 as the total volume fraction of micelles in the solution can be given.
Here, the volume reduction of the micellar solution of about 7 % was already taken into account.
The lamellar system does not show free, three-dimensional diffusion. The correlation curves rather
suggest that the Brownian motion of the probe molecules is restricted to the lamellar bilayers. The
orientation model used for the analysis of the correlation function takes into account an average
over many bilayer orientations and yields plausible fitting results.
74
5.4. DISCUSSION
The measured diffusion coefficients are generally higher than in DMPC membranes at equal tem-
perature [75]. This is attributed to a higher rigidity of DMPC membranes. Regarding the influence
of cholesterol, we can conclude that a cholesterol addition above a fraction of 0.08 in the C12E5
bilayer reduces the molecular mobility in the chosen surfactant/water emulsion. Because of the
small observation volume in FCS, it is justified to claim that the structural changes of the molecu-
lar order take place on a scale below the focus diameter of 0.5µm. This reduction is a consequence
of stronger molecular interaction induced by the presence of cholesterol molecules. Comparing
this effect with observations in DLPC, the reduction of D by 30 % is slightly smaller than in
DLPC where reductions by ≈ 50 % were observed at the same cholesterol fraction [76]. The
reason might be a tighter packing of cholesterol molecules in double chain amphiphiles.
It is known that cholesterol interacts with alkyl chains. There are reports that cholesterol increases
the fluidity in the center of DMPC membranes while decreasing it near the polar headgroup region
[77]. A dependency on the length and asymmetry of the alkyl chains was also found [78]. In the
present case, cholesterol has about 2/3 the length of a C12E5 molecule. Its rigid steroid ring
is thought to restrain the free rotational motion of about 5-6 segments of the surrounding lipid
molecules. Compared to biomembranes with lipids being asymmetric and longer, cholesterol
may be incorporated differently. Nevertheless, the measurements show a similar influence of
cholesterol on diffusion. Apparently, the fluorescent probes do not detect a higher fluidity of the
inner alkyl groups, but rather the reduced mobility at the polar/unpolar interface.
Comparing the results to earlier measurements, it is striking, that in a majority of experimental
studies cholesterol was found to decrease the diffusion coefficient in phosphatidylcholine mem-
branes. Molecular Dynamics simulations support these findings. The structural influence of
cholesterol seems to be independent of a possible phase transition though it might induce one
in higher concentration. Thus, the structural influence of cholesterol on the molecular order in
membranes seems to be a rather universal feature.
There are other effects in artificial membranes that possibly play a role in the correct description
of molecular motion. It is conceivable that a wobbling or convective motion of the bilayers as a
whole is important. As the lamellae are larger units – yet not immobile – this possibility must be
taken into account. The time scale that these effects appear on should be much smaller than the one
for diffusion. It was also reported that molecules diffusing in nanostructured liquids show effects
of anomalous diffusion [79, 80, 81]. This can appear, for example, in the presence of coexisting
gel and fluid phases. It can also be found in binary solutions of DMPC/DSPC, as recent studies
using computer simulations and FCS experiments have indicated [82].
In conclusion, the results show a reduction of molecular mobility in a simple surfactant/water
system induced by cholesterol at concentrations above 8 %. The data helps to generalize the
75
5.4. DISCUSSION
understanding of chemical interaction between cholesterol and lipids, and thus, the principle of
the membrane order.
76
Chapter 6
Conclusions
In the foregoing chapters, the microscopic method Fluorescence Correlation Spectroscopy (FCS)
was used to examine liquid flow and molecular diffusion in environments with different geome-
tries. The small observation volume of the microscope was placed at defined positions inside
a microchannel. The velocity of the water stream could be examined by observing fluorescent
molecules in low concentration. The flow profile was successfully measured in a cross section
of the channel. Beforehand, the three-dimensional velocity distribution had been computed with
a simulation software based on the Finite Element Method (FEM). The experimental results are
consistent with the expected (quasi-)parabolic velocity profiles.
Then, the problem of diffusion-limited mixing in microdevices was approached. A Y-shaped mix-
ing channel was designed and produced. The FCS method proved to be capable of detecting rel-
ative concentration differences with good precision inside the channels. Again, CFD simulations
and the experimental measurements provided consistent results. The accuracy of the concentration
measurements was not ideal but can be enhanced by several means. The goal of characterizing the
mixing properties of this specific channel was reached.
Despite the intended biochemical application, the experiments in microchannels proceed on a
decidedly physical starting point. The microfluidic simulations and experiments contribute to
develop accommodated reaction channels for future use. Further experiments on realizing an
enzymatic reaction inside the channel have been taken up. It is challenging to find the right
conditions for reproducible reaction kinetics. Besides the physical point of view, questions of
surface chemistry and other disciplines have to be taken into account as well.
In the biological sciences, there is a demand to examine processes in eukaryotic cells. These
processes are often based on mechanisms with only few molecules involved. This requires to
build an optical system with a high spatial resolution and the sensitivity to observe single molecule
motion in a natural surrounding. By means of image analysis it could be demonstrated how the
diffusion coefficient of a microsphere can be determined from fluorescent images. The principle
can be transferred to single molecule detection. Provided that the sensitivity and frame rates are
77
chosen adequately, the diffusion coefficient can be determined precisely in a shorter time than
in FCS. It is desirable to extract the maximum information content from the fluorescent images.
Different techniques can be realized for the purpose of tracking a molecule trajectory. There is a
fundamental difference between the software and the hardware approach. The software solution
is based on image recognition of single fluorescent molecules. As long as the real time data are
two-dimensional, this method is restricted to the plane. It has the advantage though, that many
molecules can be tracked at the same time.
From the hardware side, there are two main ways to establish a tracking system. The first approach
relies on mechanical tracking by moving the support of the sample. The second method needs
control of the laser beam by fast re-adjustment of two mirrors. The principle is availed in laser
scanning microscopy, yet with a fixed scanning scheme. Both ways share the fact that the molecule
remains in the region of interest for as long as possible. The advantage is that the molecule
is almost constantly excited during observation. This enables to measure real single molecules
parameters. The disadvantage of a tracking system is, that it must be realized in three dimensions.
This is a challenge for the optical and the mechanical detection system, which was pointed out in
section 4.4.1. A mechanical tracking system can only track one molecule independently at a time.
The use of quantum dots instead of fluorescent molecules offers a promising way to address the
problem of photo degradation.
The chapter about surfactant/water emulsions can be regarded separately from the experiments in
microstructures because it was an independent project. Yet, the experimental method and the inter-
pretation of the data are based on the preceding fundamental facts. In the context of molecular mo-
tion, it is regarded as a straight forward way to focus on a physical property when going to smaller
scales. The chapter deals with diffusion in systems forming nanostructures by self-assembly. It is
shown that cholesterol as a third component has a structural influence on the molecular order. The
bilayer structure leads to a behavior that can be described by two-dimensional diffusion in planes
rather than free diffusion in solution. Additionally, a modified model equation for FCS was found
that leads to better results for the corresponding diffusion coefficients. It will be useful in further
studies of randomly oriented lamellar media.
In environments that exhibit an intrinsic molecular structure, additional phenomena come into ef-
fect. Perturbed motion of molecules is likely to be observed in material that contain an anisotropic
molecular order. Especially, the effects of anomalous diffusion have gained interest recently. It
is largely unknown to what extend these effects play a role in intracellular mechanisms. Fluores-
cence Recovery After Photobleaching (FRAP) has often been used to examine these phenomena.
With FCS, anomalous diffusion can be observed e. g., by variation of the focus diameter [2].
Recently, new developments have lowered the resolution of optical microscopes down to 20 nm
[1, 83]. The microscope makes use of an method named STED (STimulated Emission Depletion).
78
Higher resolution offers a way to depict cellular structures which were unaccessible otherwise.
Other researchers have taken an effort to increase the focal volume. This allows to measure even
lower concentration down to the pM range [84]. Depending on the application, both concepts
expand the possibilites of fluorescence techniques. Apart from these advancements, it is still
rewarding to continue working on methods for Single Molecule Detection and analysis.
79
Appendix A
List of manufacturers
• Chemicals
octadecyl Rhodamine B (R18) – Invitrogen (Carlsbad, California)
C12E5 – SigmaAldrich (St. Louis, Missouri)
• Detectors and photon counting
beam analyzer LBA-7xxPC – Spiricon (Logan, Utah)
CCD camera Cohu 4812 – Cohu (San Diego, California)
correlator ALV-5000 – ALV GmbH (Langen, Germany)
CMOS camera LogLux i5 CL – Kamerawerk Dresden (Dresden, Germany)
multi channel scaler PMS 300 – Becker und Hickl GmbH (Berlin, Germany)
photo diode SPCM-AQR 14 – PerkinElmer (Kirkland, Canada)
photo diode SPCM-AQR 13 – PerkinElmer (Kirkland, Canada)
frame grabber mvTITAN-CL – MatrixVision (Oppenweiler, Germany)
• Fluorimeter
FluoroMax 2 – Jobin Yvon Instruments S.A. Inc. (New Jersey)
• Lasers
Ar-Ion BeamLok 2080 – Spectra Physics (Mountain View, California)
HeNe LGK 7786 P – Lasos (Jena, Germany)
• Liquid handling
Gastight 1750 syringes – Hamilton (Reno, Nevada)
syringe pump SP260p – World Precision Instruments (Sarasota, Florida)
• Mechanical stages and plates
tubus system parts – Linos (Göttingen, Germany)
linear stages – Newport (Irvine, California)
80
• Microscopes
ConfoCor 2 – Carl Zeiss (Jena, Germany)
• Microfluidics
photo masks – GeSIM (Großerkmannsdorf, Germany)
etching, bonding and microfluidic connection – Fraunhofer BioMOS (Sankt Augustin, Ger-
many)
• Objectives
C-Apochromat 63× 1.2 W – Carl Zeiss (Jena, Germany)
Plan Neofluar 63× 1.4 Oil – Carl Zeiss (Jena, Germany)
Achromat 40× 1.2 W – Carl Zeiss (Jena, Germany)
• Optical filters, lenses, dichroic mirrors
dichroic mirror 540DRLPO2 – Omega Optical (Brattleboro, Vermont)
filters – AHF Analysetechnik (Tübingen , Germany)
neutral wedges – Linos (Göttingen, Germany)
tubus lens f = 164.5mm – Carl Zeiss (Jena, Germany)
• Oscilloscope
Wavemaster 6000 – LeCroy (Chestnut Ridge, New York)
• Positioning
PZT sensor/servo module controller E-509.C3A – PI Physik Instrumente GmbH & Co. KG
(Karlsruhe , Germany)
piezo amplifier module E-503.00 – PI Physik Instrumente GmbH & Co. KG (Karlsruhe,
Germany)
display module E-516.i3 – PI Physik Instrumente GmbH & Co. KG (Karlsruhe, Germany)
PIFOC P-721 focus nanopositioner – PI Physik Instrumente GmbH & Co. KG (Karlsruhe,
Germany)
• Software
GAMBIT (grid model) – Fluent Inc. (Lebanon, New Hampshire)
FLUENT (CFD simulation) – Fluent Inc. (Lebanon, New Hampshire)
MPlayer version 1.0rc1-4.0.2 – MPlayer Team, http://www.mplayerhq.hu
81
Appendix B
Publications and conference contributions
1. Pieper, Thorsten ; Svetlana Markova ; Masataka Kinjo ; Dieter Suter ; Effect of cholesterol
on diffusion in surfactant bilayers ; submitted ; 2007
2. Pieper, Thorsten ; Svetlana Markova ; Masataka Kinjo ; Dieter Suter ; Comparative study
on the stabilizing effect of cholesterol on lamellar bilayers ; poster award ; EABS & BSJ
2006 Naha (Japan) ; 2006
3. Pieper, Thorsten ; Benjamin Greiner ; Harald P. Mathis ; From 2D to 3D Molecule Tracking ;
poster presentation ; 4th International Workshop on Scanning Probe Microscopy Berlin ;
2003
4. Pieper, Thorsten ; Benjamin Greiner ; Harald P. Mathis ; 3D Molecule Tracking ; poster
presentation ; Bunsen Tagung Kiel ; 2003
5. Pieper, Thorsten ; Benjamin Greiner ; Harald P. Mathis ; 2D Molecule Tracking ; short talk ;
annual DPG conference Dresden ; 2003
6. Mathis, Harald P. ; Benjamin Greiner ; Thorsten Pieper ; Spatially Resolved Realtime Single
Molecule Detection for Protein Dynamic Investigations ; poster presentation ; 295. WE
Heraeus Seminar Bad Honnef ; 2003
82
Selbständigkeitserklärung
Hiermit erkläre ich, die vorliegende Dissertation selbstständig und ohne unerlaubte fremde Hilfe
angefertigt zu haben und mich dabei keiner anderen als der von mir ausdrücklich bezeichneten
Quellen und Hilfen bedient habe. Ebenfalls sind alle von anderen Personen bereitgestellten Mate-
rialien oder erbrachten Dienstleistungen als solche gekennzeichnet. Die Dissertation wurde in der
jetzigen oder einer ähnlichen Form noch an keiner anderen Hochschule eingereicht und hat noch
keinen sonstigen Prüfungszwecken gedient.
Bonn, im Juni 2007
Thorsten Pieper
83
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