Quantitative Microscopic Techniques for Monitoring Dynamic

Transcription

Quantitative Microscopic Techniques
for Monitoring
Dynamic Processes in Microarrays
Proefschrift
ter verkrijging van de graad van doctor
aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof. dr. ir. J.T. Fokkema,
voorzitter van het College voor Promoties,
in het openbaar te verdedigen op 12 maart 2002 om 13:30 uur
door
Lennert Richard VAN DEN DOEL
natuurkundig ingenieur
geboren te Vlissingen.
Dit proefschrift is goedgekeurd door de promotoren
Prof. dr. ir. L.J. van Vliet,
Prof. dr. I.T. Young
Samenstelling promotiecommissie:
Rector Magnificus,
Prof. dr. ir. L.J. van Vliet
Prof. dr. I.T. Young
Prof. dr. ir. J.J.M. Braat
Prof. dr. G.W.K. van Dedem
Prof. dr. T.W.J. Gadella
Prof. dr. J. Greve
Prof. dr. J.G. Korvink
voorzitter
Technische Universiteit Delft, promotor
Technische Universiteit Delft, promotor
Technische Universiteit Delft
Technische Universiteit Delft
Universiteit van Amsterdam
Universiteit Twente
Albert Ludwig University Freiburg, Duitsland
Advanced School for Computing and Imaging
This work was carried out in the ASCI graduate school.
ASCI dissertation series number 75.
ISBN 90-77017-48-8
c 2002, L.R. van den Doel, all rights reserved.
You play, you win.
You play, you lose.
YOU PLAY.
—Jeanette Winterson, The Passion
Voor Thomas & Jolanda
Voor Rosanne,
fonkelende ster, mijn kleine engel
—R
Contents
1
Labs-on-a-chip
1.1 Combinatorial Chemistry
1.2 Human Genome Project
1.3 Intelligent Molecular Diagnostic Systems
1.4 Development of a Microarray reader
1.5 Outline of this thesis
Part I
1
1
2
2
4
5
Quantitative Reading of Microarrays
2
Yeast as a Cell Factory: the Glycolytic Pathway
2.1 Yeast: Saccharomyces cerevisiae
2.2 The Glycolytic Pathway
2.3 Summary
9
9
11
13
3
Conventional Microscope-based Microarray Reader
3.1 Microarray Fabrication
3.2 Nanoliter Liquid Handling
3.3 Conventional Microscopy Concepts
3.3.1 Infinity-corrected Optical System
15
15
17
19
20
vii
viii
CONTENTS
3.4
3.5
4
5
3.3.2 Köhler Illumination
Characterization of the Princeton Versarray CCD camera
3.4.1 Readout noise
3.4.2 Dark current
3.4.3 Linearity and Sensitivity
Summary
22
24
25
27
29
33
Quantitative Reading of Microarrays
4.1 Limiting Factors of the Instrument
4.2 Detection Limit of the Microarray Reader
4.2.1 Improving the Detection Limit (for 1 and N
measurements!)
4.3 Readout Noise Limited Detection
4.3.1 Ideal Detection System
4.3.2 Single Element Detection System with Readout Noise
4.3.3 Many Pixels Detection System with Readout Noise
4.4 Stray Light Limited Detection
4.5 Minimum Detectable Signal for Different Configurations
4.6 Modifications to the Microscope System
4.7 Conclusions and Discussion
35
35
36
Monitoring Enzyme-catalyzed Reactions in Microarrays
5.1 Imaging with the modified optical system.
5.2 NADH Calibration
5.3 Monitoring Enzyme-catalyzed reactions
5.4 Conclusions
49
49
50
54
64
38
39
39
40
42
43
44
45
48
Part II Monitoring Dynamic Liquid Behavior in Micromachined Vials
6
Dynamic Liquid Behavior in Micromachined Vials
6.1 Introduction
6.1.1 Liquid pinning: ring stains from coffee droplets
6.1.2 Modeling dynamic liquid behavior
6.1.3 Impedance-based liquid volume sensor
6.2 Survey of Microscopic Techniques
6.2.1 Conventional Fluorescence Microscopy
6.2.2 Confocal Fluorescence Microscopy
6.2.3 Interference-Contrast Microscopy
71
71
71
74
76
78
78
80
82
CONTENTS
6.3
Conclusions
ix
86
7
Electromagnetic Theory of Interference-Constrast Microscopy
7.1 Introduction
7.2 Phase Difference
7.3 Intensity of Electric Field
7.4 Comparison between Simulation and Experimental Results
7.5 Alternative Description for Intensity of Electric Field
7.6 Summary
87
87
88
91
95
97
99
8
Temporal Phase Unwrapping Algorithm
8.1 Phase Unwrapping: Spatial vs. Temporal
8.2 Temporal Phase Unwrapping Algorithm
8.3 Computation of Reference Frame
8.4 Precision and Accuracy of Phase Estimation
8.5 Simultaneously Sampling in Space and in Time
8.6 Summary
101
101
103
107
108
110
115
9
Experimental Results
9.1 Introduction
9.2 Experimental Setup
9.3 Monitoring Evaporation in Square Micromachined Vials
9.3.1 Lateral Scaling of Height Profiles
9.3.2 Evaporation Rate in Square Vials
9.4 Monitoring Evaporation in Circular Micromachined Vials
9.4.1 Radial Scaling of Height Profiles
9.4.2 Evaporation Rate in Circular Vials
9.5 Calibration of the Electronic Volume Sensor
9.6 Nanometer-Scale Height Measurements
9.7 Conclusions and discussion
Appendix Deegan’s Theory on Diffusion-Limited Evaporation
117
117
118
118
120
125
127
127
133
135
137
140
140
References
145
Summary
149
Samenvatting
151
Dankwoord
155
1
Labs-on-a-chip
One of the most widely exploited fields of research during the last decade is that of combinatorial chemistry. The stated goal of this approach is to drastically reduce the time and
cost that is required for obtaining new lead compounds. Currently, many samples are tested
against a large number of targets in microarrays for high-throughput screening. In the future,
this technology will lead to a miniaturized lab-on-a-chip. The Delft University of Technology
is financing an interdisciplinary research program, ’Intelligent Molecular Diagnostic Systems’, in which analytical labs-on-chips are being developed in order to gain insight into the
metabolic regulation of the glycolytic pathway in yeast cells.
1.1
COMBINATORIAL CHEMISTRY
As early as the 1950s and 1960s a technique to synthesize different chains of peptides 1 on
a solid phase was developed. Before that time peptide chains could be synthesized only in
solution. The main advantage of solid phase peptide synthesis over traditional solution phase
chemistry is that after each cycle of adding a peptide to the chain, the solid carrier beads
remain insoluble and excess reagents and unreacted materials can be rinsed away. This allows
for step-by-step purification through simple washing steps. This technique was developed by
Merrifield and published in 1963 [1]. In 1984 he won the Nobel Prize in Chemistry for this
research.
1a
peptide yields two or more amino acids on hydrolysis. Peptides are formed by the loss of water from the NH2
and COOH groups of adjacent amino acids. Peptides form the constituent parts of proteins. There are 20 naturally
occurring amino acids from which proteins are synthesized.
1
2
LABS-ON-A-CHIP
Combinatorial chemistry is a technique that adapts this technique of peptide synthesis for
the construction of large libraries of structurally related molecules. These libraries can be
constructed as a mixture of all these related molecules in the same reaction vessel or all these
related molecules can be constructed in separate vessels individually by parallel synthesis.
After construction, the library is screened for receptor-inhibitor activity. The active compounds are possible candidates for drug development. High-throughput screening (HTS) is
the screening and testing of hundreds to thousands of compounds in a relatively short period
of time: construction of the library and screening should take approximately the same amount
of time. Combinatorial chemistry requires adaptation of classical synthesizing techniques,
automization of liquid handling, and miniaturization of reaction vessels for the construction
of large compound libraries. High-throughput screening requires sensitive detectors to monitor the chemical activities in all vessels. This screening generates enormous amounts of data.
These screening results must be integrated with biological data and structural information of
the library.
1.2 HUMAN GENOME PROJECT
Similar developments can be seen in the field of genetics. In October 1990 the United States
Department of Energy and the National Institutes of Health joined forces in the Human Genome
Project. Two of the goals of this project are to discover all the 30 to 35 thousand genes of
the human genome and to determine the 3 billion nucleotides of the complete human genome
sequence. In February 2001 the working draft versions of the human genome were published
in Nature [2] and Science [3]
Much effort in the Human Genome Project is put into miniaturization, e.g. the fabrication of
high-density combinatorial arrays of oligomers (short chains of nucleotides). The oligomers
are fabricated on an array by photolithographic techniques: in N processing steps 4N different
oligomers (4 different nucleotides) with a length of N nucleotides can be fabricated. These
arrays allow for large-scale hybridization assays.
Now that the draft version of the human genome is available, the new challenges are to find
out what the function is of all genes, how all the genes are regulated, to gain understanding of
dynamic living systems, and to model complex biological systems.
1.3
INTELLIGENT MOLECULAR DIAGNOSTIC SYSTEMS
In the context of these developments, the Delft University of Technology is financing an
interdisciplinary research program for development of miniaturized analytical labs-on-chips
for the quantification of a variety of chemical compounds in a short period of time using a
minimum amount of reagents. Furthermore, the system should allow for interpretation of the
acquired quantitative data in combination with expert knowledge and historical data. One can
think of various applications for such a system:
INTELLIGENT MOLECULAR DIAGNOSTIC SYSTEMS
3
• Quality control in the food industry: the presence of residues from pesticides, antibiotics,
or hormones is unwanted, but the detection poses enormous problems due to the required
sensitivity and also to the enormous variety of chemical compounds present.
• Forensic research: a maximum amount of information must be obtained out of a minimum amount of sample in a short period of time. This requires not only efficient
screening for a large number of compounds, but also interpretation of the gathered data.
• Medical diagnostics: more and more tests are carried out to obtain as complete as
possible a picture of every patient who is hospitalized or treated poly-clinically. A
system which is capable of carrying out a multitude of assays and of interpreting the
results is likely to lead to significant cost-reductions as well as to improvement in the
quality of medical care.
• Epidemiology: this requires fast and massive screening using small sample volumes
from many individuals.
• Bioprocess industry: controlling or improving fermentation processes requires rapid
tests on the activities of several biochemical compounds, such as enzymes. Based on
the interpretation of the results from the tests the fermentation processes can be tuned
or further optimized.
The last application is being used in a demonstrator system within the framework of this
interdisciplinary research program. This demonstrator system will be built around silicon
based microarrays containing reaction vials with volumes on the order of a few nanoliter.
Each vial will be equipped with sensors and actuators to monitor and control a number of
important parameters. In this demonstrator all substrates involved in the twelve reactions of
the glycolytic pathway, which is the conversion of glucose to ethanol and carbon-dioxide,
will be put into small reaction vessels. All these reactions are enzyme-catalyzed reactions.
The twelve enzymes are acquired from cell-free extracts of yeast cells taken from the wellconditioned environment of a chemo-stat. Yeast is a unicellular eukaryotic organism, which
is frequently used as a model system for other higher eukaryotes, including man. One of the
fermentation processes taking place in eukaryotic organisms is the fermentation of glucose.
Differences in the environment of the yeast and or genetic modifications of the yeast will
result in differences in enzyme activity levels along this pathway. Enzyme activity levels are
monitored by measuring fluorescence levels of NADH. This fluorescent molecule is involved in
all the reactions of the glycolytic pathway. The aim is to gain insight in the metabolic regulation
of this pathway by correlating the experimental data with the environmental conditions and
or genetic modifications.
There are a number of challenges involved in this research program:
• Chemistry must be adapted to nanoliter scale. Substrates and co-factors have to be
deposited in the vials. After drying of the solvent, the solute must be kept stable in the
vials during storage. When using these prepared microarrays for measurements, the
solutes must be dissolved again. The sample solution with the enzymes must be put
into the vials without any degeneration of the enzymes. Fast evaporation of the solution
must be avoided to guarantee that the desired reactions are actually taking place.
4
LABS-ON-A-CHIP
• Liquid handling needs to be modified to the nanoliter range. Ultra small volumes must
be fast and automatically dispensed in (sub)-nanoliter vials in an accurate, precise, and
reproducible manner. A system must be developed that allows for pipetting solutions
containing various substrates at different concentrations. This system must be automated since it will be used to prepare hundreds of identical microarrays. Furthermore,
techniques must be developed to deposit the sample solution in all vials in a massively
parallel manner.
• A lab-on-a-chip is more than just a miniaturized micro-titer plate: microarrays will
contain vials with built-in intelligence, i.e. integrated actuators and sensors: every vial
will be equipped with a temperature sensor and a heater to control the temperature, a
sensor to measure the pH-value of the solution, a detector to measure the volume of the
injected solution, etc.
• The biochemical reactions will produce many fluorescent signals simultaneously. Therefore, sensitive detectors and detection methods are necessary in order to perform quantitative measurements during monitoring of all the biochemical reactions.
• High-throughput screening generates enormous amounts of data. In order to extract useful information, it is essential to develop dedicated data analysis systems. Furthermore,
data interpretation systems must be developed to derive new insight into the underlying
biochemical principles.
1.4
DEVELOPMENT OF A MICROARRAY READER
The research that will be presented in this thesis focuses on the development of a microarray
reader: a detection system to monitor all the biochemical reactions in all vials of the microarray.
This microarray reader will be built around a wide-field microscope system equipped with
a scientific grade CCD camera and a motorized stage for scanning the microarray. With
low power optics, i.e. low magnification, low Numerical Aperture, and therefor a low lightgathering power, a microarray with approximately a hundred vials can be imaged entirely
onto the CCD camera. This allows for parallel reading of the microarray. Wide field-of-view
imaging, however, is at the expense of absolute sensitivity. With high power optics only a
single vial per image can be acquired. This implies a better sensitivity, but requires sequential
scanning of the microarray. The research questions we will try answer in this research are
• What are the limiting factors of our microarray reader? What determines the lowest
concentration of fluorescent molecules that can be detected with our system? It will
turn out that stray light, originating in the (excitation) optical path of the microscope is
the limiting factor.
• Given this limiting factor, what are the possibilities to improve the detection limit of our
system? If stray light is the limiting factor, a better detector does not help, since it will
increase the signal level as well as the noise level, leaving the Signal-to-Noise Ratio
unchanged.
OUTLINE OF THIS THESIS
5
• What is the dynamic range of concentrations of fluorescent NADH molecules that can
be detected with our system under limitation of stray light?
• Given this stray light limitation, what modifications can we apply to our system to
improve the scan rate of the microarray?
• The enzyme activity levels will be derived from the production or consumption rate of
NADH. Given the scan rate of our microarray reader, what is the dynamic range of
enzyme activity levels that can be monitored with our system?
1.5
OUTLINE OF THIS THESIS
This thesis consists of two parts. The first part of this thesis presents the research into the
development of a microscope-based microarray reader. This research fits completely within
the framework of the interdisciplinary research program to develop analytical labs-on-chips.
The second part of this thesis presents research that has been performed to monitor dynamic
liquid behavior in the vials of our microarrays. Evaporation of liquid samples is a key problem
in the development of microarray technology. The research in the second part of this thesis
presents a type of microscopy, interference-contrast microscopy, that allows for quantitative
monitoring of the air-liquid interface during drying of a liquid sample in a vial. One of the
most striking results of this research is that the evaporation rate is linearly proportional to the
perimeter of the vial and not to the area of the air-liquid interface, as one would intuitively
expect. This research also presents calibration results of an electronic liquid volume sensor,
that will be integrated in our lab-on-a-chip.
Quantitative Reading of Microarrays Chapter 2 presents in more detail the biochemical
challenge for which microarray technology can provide the right analytical tools: exploring
the metabolic regulation of the glycolytic pathway in yeast cells.
Chapter 3 describes the fabrication process of the microarrays and the different liquid handling
strategies. Furthermore, some important concepts of microscopy will be introduced. Finally,
this chapter presents the characteristics of the CCD camera used in the experiments. This
chapter tries to provide sufficient background knowledge in order to fully understand the
modifications we will apply to our microscope system to improve its performance.
Chapter 4 describes in detail what the limiting factors are of measuring low levels of fluorescence in our microarrays. The limiting factors are either readout noise in the detector or stray
light originating in the (excitation) optical path. Given these limitations, we have modeled
various detection systems to investigate what the strength is of the minimum detectable signals. These models amount to modifications to our microscope system that we have applied
to improve its performance.
Finally, Chapter 5 presents the experimental results of reading microarrays with our modified conventional microscope based microarray reader. The functionality of our microarray
reader to monitor reactions of the glycolytic pathway is demonstrated by monitoring NADH
producing and consuming enzymatic reactions in our microarrays at various conversion rates.
6
LABS-ON-A-CHIP
Monitoring Dynamic Liquid Behavior in Micromachined Vials Chapter 6 looks from
different points of view at the process of evaporation to motivate this research and explains
the spatial and temporal sampling requirements for quantitative monitoring of the evaporation process in the vials of our microarrays. Interference-contrast microscopy is a type of
microscopy that satisfies these sampling requirements.
Chapter 7 presents a classical electromagnetic theory to describe the generation of the dynamic
fringe patterns, which are observed with interference-contrast microscopy.
Chapter 8 introduces a temporal phase unwrapping algorithm that computes the dynamic height
profiles of the meniscus from these fringe patterns. This algorithm analyzes the interferograms
point-by-point in time without using any spatial information.
Chapter 9 presents experimental results of monitoring the evaporation process in micromachined vials with the technique of interference-contrast microscopy. We have monitored the
evaporation process in square vials as well as in circular shaped vials of different sizes. Furthermore, this chapter will show experimental results of the calibration of a dedicated electronic
volume sensor.
Part I
Quantitative Reading
of
Microarrays
2
Yeast as a Cell Factory:
the Glycolytic Pathway
This chapter presents a biochemical challenge for which microarray technology can provide
the right analytical tools: exploring the metabolic regulation of the glycolytic pathway in
yeast cells. Yeast, a eukaryotic cell, is a model system for other higher eukaryotes, including
man. The glycolytic pathway converts glucose to ethanol and carbon dioxide. This metabolic
pathway, consisting of 12 enzyme-catalyzed reactions, is part of a larger compartmented
metabolic network, which is interlinked at various levels. Studying this basic process in yeast
is required not only to optimize this fermentation process. The insights gained can also be
used for other applications, such as the production of pharmaceutical (human) proteins by
yeast. One century of research aimed at increasing the glycolytic capacity (rate of alcoholic
fermentation) has been large unsuccessful. Microarray technology allows for doing quantitative measurements, cheap and fast, in a massive parallel manner: instead of monitoring a
single enzymatic reaction, as performed in the past century, it is possible to monitor all the
different reactions simultaneously in microarrays and extract large amounts of information
from all these measurements. The central theme in this research program is the development
of analytical tools for an integral analysis of the metabolic regulation of this pathway, rather
than applying metabolic engineering to the yeast cell itself.
2.1
YEAST: Saccharomyces cerevisiae
The yeast Saccharomyces cerevisiae is probably better known as brewers or bakers yeast. In
the past 8000 years, this fungus has played an important role in the production and conservation
of food due to its ability to ferment glucose to ethanol and carbon-dioxide. Nowadays, this
yeast is used as a ’multi-purpose cell factory’; it is not only used for the production of ethanol
9
10
YEAST AS A CELL FACTORY: THE GLYCOLYTIC PATHWAY
Fig. 2.1
An image of a budding Saccharomyces cerevisiae cell acquired with electron microscopy.
and carbon-dioxide, beer and wine, but also for glycerol, various flavors, different enzymes
and vitamins, pharmaceutical (human) proteins, etc.
Yeast, which is a unicellular eukaryote1 , has become a unique and powerful model system for
biological research, because it has some attractive features:
• yeast is cheap and easy to cultivate,
• yeast has short generation times,
• detailed genetic and biochemical knowledge of yeast has accumulated over the past 100
years, and,
• molecular techniques for genetic manipulation can be easily applied to the yeast genome.
This organism, therefore, provides a highly suitable system to study basic biological processes that are relevant for many other higher eukaryotes, including man. The complete
genome sequence of Saccharomyces cerevisiae [4] is known: the complete genome consists
of 16 chromosomes with approximately 12 million nucleotides and about 6400 genes. The
function of 40% of the genes of S. cerevisiae is still unknown. Figure 2.1 shows an image of
Saccharomyces cerevisiae acquired with electron microscopy. In this image the organism is
in the state of budding (cell reproduction).
In the past century many researchers have tried to improve the yeast strain, producing relatively
more desired product and less (unwanted) by-products, to make it stronger, to let it remain
productive under many different conditions, and to make it faster, thereby increasing the rate
of production of the product of interest. The most important technique to accomplish this is
1A
eukaryotic organism has its genetic information contained in a separate cellular compartment: the nucleus of
the cell. In addition to their nuclear genomes, all eukaryotic cells contain small additional extranuclear genomes,
which are contained in the mitochondria (and in plastids in photosynthesis performing eukaryotes). The mitochondria
are self-autonomous, self reproducing organelles within the cytoplasm of eukaryotic cells that are bounded by two
membranes. These organelles are responsible for the energy conversion of most of the cellular energy metabolites
into adenosine triphosphate (ATP).
THE GLYCOLYTIC PATHWAY
11
metabolic engineering: improving cellular activities by manipulation of enzymatic, transport
and regulatory functions of the cell with the use of recombinant-DNA technology. Simply
speaking, one modifies some nucleotides or even a complete gene in the genome of the yeast
cell. These modifications at the DNA level are transcribed onto RNA. The modified RNA is
translated into a modified protein. The modified protein has a different structure, a different
function and a different regulation with respect to the unmodified protein.
Due to these modifications it will alter the entire metabolic process of interest: metabolic
pathways are not simple linear pathways, but interlinked metabolic networks. There is regulation of enzyme activity at various levels: at the transcription level from DNA to RNA, at
the translation level from RNA to the protein, covalent modifications of proteins might occur,
and finally, there is activity regulation by substrates, products, and effectors. Furthermore, the
control of flux through pathways is not confined to a single bottleneck enzyme, but is shared
by multiple enzymes. Finally, metabolism is compartmented in eukaryotic cells; metabolic
processes take place in different areas within the yeast cell.
In this research program we want to provide analytical tools based on microarrays to monitor
the effects of modifications to the yeast genome and the changes in the environment of the yeast
in a systematic way. We will focus, in particular, on the metabolic regulation of the glycolytic
pathway. One of the approaches to study this metabolic regulation is to vary environmental
parameters and / or introduce mutations in the yeast genome. After that, one tries to relate
these measured parameters (RNA transcript level, protein activity, metabolite concentration)
to the quantity to be optimized, e.g. the glycolytic capacity. The central research themes of
this program are focusing on the biochemical challenge to perform the enzymatic reactions of
the glycolytic pathway in microarrays, the technological challenge to construct labs-on-a-chip
in which the reactions can take place, to monitor the enzymatic reactions in these microarrays,
and the ’bioinformatic’ challenge to derive new insights on the regulation of this metabolic
process from the acquired data. The research described in this part of this thesis focuses
on developing a quantitative microarray reader for monitoring the different reactions of this
pathway in microarrays.
2.2
THE GLYCOLYTIC PATHWAY
The yeast Saccharomyces cerevisiae is well-known for its alcoholic fermentation. The biochemical process of the fermentation from glucose to ethanol and carbon-dioxide is known as
the glycolytic pathway, which is shown in Figure 2.2. This pathway consists of 12 enzymecatalyzed reactions. Table 2.1 lists all twelve enzymes involved in the glycolytic pathway.
This table shows in what type of assay these enzymes are involved and if the enzyme activity
is measured via a direct reaction or via a coupled reaction. An example of a direct reaction is
the following:
HXK
glucose + ATP → G6P + ADP + photon,
(2.1)
where G6P is glucose-6-phosphate, and HXK is the enzyme hexokinase. In this reaction the bioluminescent substrate ATP (adenosine triphosphate, bioluminescence wavelength: 580nm)
is converted to ADP (adenosine diphosphate). The rate of bioluminescence of the ATP is a
measure for the enzyme activity. It is also possible to measure the activity of hexokinase via
12
glucose
ATP
ADP
hexokinase
glucose-6-phosphate
phosphoglucoseisomerase
fructose-6-phosphate
ATP
ADP
phosphofructokinase
fructose-1,6-bisphosphate
triose-pisomerase
fructose-bis-Paldolase
dihydroxyglyceraldehyde-3-phosphate
NAD
glyceraldehyde-3Pacetonphosphate
dehydrogenase
NADH
1,3-diphosphoglycerate
phosphoglyceratekinase
ADP
ATP
3-phosphoglycerate
phosphoglyceratemutase
2-phosphoglycerate
enolase
phosphoenolpyruvate
ADP
ATP
pyruvatekinase
pyruvate
lactatedehydrogenase
CO2
pyruvatedecarboxylase
aceetaldehyde
NADH
NAD
lactate
NADH
NAD
alcoholdehydrogenase
ethanol
Fig. 2.2 This figure shows the Glycolytic pathway: the fermentation of glucose to ethanol. This
pathway consists of 12 enzyme-catalyzed reactions.
two coupled reactions (the procduct of the first reaction is a substrate in the second reaction):
glucose + ATP
G6P + NADP
HXK
→
G6P−DH
→
G6P + ADP
gluconolactone + NADPH,
(2.2)
where NADP is Nicotinamide Adenine Dinucleotide Phosphate. In these two reactions, the
generation of the fluorescent product NADPH (excitation wavelength 360nm, emission wavelength 450nm) is monitored. Note that the first reaction must be the conversion rate limiting
reaction. Otherwise, the activity of the enzyme glucose-6-phosphatedehydrogenase (G6PDH)
is measured. The last column in Table 2.1 gives the specific activity of the enzymes in a well
conditioned environment of a chemostat, where the reactions are aerobic, i.e. in the presence
of oxygen, and where the reactions are glucose limited, i.e. the conversion of glucose to
ethanol is fully determined by the amount of glucose and not by other factors. The enzyme
activity is expressed in units per milliliter (U/ml ): an enzyme activity of 1U/ml means that
1µMol of substrate in 1ml solution is converted in 1 minute of time.
SUMMARY
13
Table 2.1 List of enzymes involved in the Glycolytic pathway. The third column gives the type of
assay. The fourth column shows if the enzyme activity is measured via a direct reaction or via a coupled
reaction. The last column gives the specific enzyme activities measured in a chemostat, where the culture
in under aerobic glucose limited conditions [5]. The unit U is defined in the text.
Enzyme
Abbrev.
Assay
Reaction
Specific
activity
U/ml
hexokinase
phosphoglucose-isomerase
phosphofructokinase
fructose-bis-P-adolase
triose-P-isomerase
glyceraldehyde-3P-dehydrogenase
3-phosphoglycerate-kinase
phosphoglycerate-mutase
enolase
pyruvate-kinase
pyruvate-decarboxylase
alcohol-dehydrogenase
HXK
PGI
PFK
FBA
TPI
GAPDH
PGK
PGM
ENO
PYK
PDC
ADH
ATP
NADPH
ATP
NADH
NADH
NADH
ATP
ATP
ATP
ATP
NADH
NADH
direct
coupled
direct
coupled
coupled
coupled
direct
coupled
coupled
direct
coupled
direct
1.7
2.8
0.3
1.0
52.5
5.6
7.4
6.8
0.7
2.9
0.6
9.1
The experimental procedure to monitor the effects of an adjustment to the stable and wellcontrolled environment of a chemostat to the glycolytic pathway consists of the following
steps. Two different samples are taken from the fermentor. From both samples the cellfree extract is extracted. These cell-free extracts are added to microarrays containing all
required substrates for the enzymatic reactions. The substrates are available in two different
concentrations to check whether or not the reactions are enzyme-limited. The enzyme activity
levels of all reactions taking place in the microarray are observed by monitoring the generation
or decrease in signal level from either ATP or NAD(P)H. Bioluminescence has the great
advantage, that is does not require any excitation source. It produces, however, a very weak
signal, even weaker than fluorescence. Therefore, all the assays of the glycolytic pathway
will be designed in such a way that the enzyme activity can be measured via the NAD(P)H
production or consumption. The measured enzyme activity levels indicate the effects of the
modified environmental conditions or modifications to the yeast genome.
2.3
SUMMARY
Microarray technology provides useful tools for a study of the regulation of the glycolytic
pathway in yeast cells. The glycolytic pathway consists of 12 enzyme-catalyzed reactions.
These reactions take place on microarrays where all required substrates are already available.
The enzymes are acquired from two different cell-free extracts from yeast cells from the
14
stable and well-conditioned environment of a chemostat. The enzyme activities are related
to the conversion rate of either the bioluminescent molecule ATP or the fluorescent molecule
NADPH. The generation of these products or the consumption of these substrates will be
monitored for each reaction in time.
The next chapter will present the fabrication process of the microarrays and the different liquid
handling strategies. The microarray reader for acquisition of these signals will be build around
a wide-field microscope system. The next chapter will also present some basic concepts of
quantitative microscopy. A good understanding of these concepts is necessary to understand
the modifications we will apply to our microscope system to improve its performance. Finally,
we will present results of a number of experiments that will bring us some steps closer to the
desired goal.
3
Conventional
Microscope-based
Microarray Reader
In the previous chapter the biochemical challenge of this research was introduced: development of a microarray reader to monitor enzyme-catalyzed reactions in microarrays. This
microarray reader will be built around a conventional microscope system equipped with a
scientific grade CCD camera. This chapter will present the fabrication process of the microarrays and different liquid handling strategies. Furthermore, we will introduce two concepts
of modern microscopy: the infinity-corrected optical system of a microscope and the concept
of Köhler illumination. Finally, we will present the characteristics of the CCD camera used
in the experiments. This chapter tries to provide sufficient background knowledge in order
to fully understand the modifications we will apply to our microscope system to improve its
performance.
3.1
MICROARRAY FABRICATION
The microarrays that will be used in this research are manufactured in silicon at DIMES (Delft
Institute for Microelectronics and Submicron Technology, Delft University of Technology,
NL-2628 CD Delft, the Netherlands). With conventional silicon fabrication techniques a
variety of different geometries can be manufactured. In our experiments we used either
square of circularly shaped vials with different sizes and depths. Table 3.1 gives an overview
of the volumes for typical dimensions of the different vials.
Fabrication process. The silicon wafers are first covered with a thin layer of silicon nitride.
On top of this layer a thin film of photo resist is deposited. This film is exposed through the
masking pattern of the microarrays. After removal of the exposed photo resist, the wafer is
15
16
CONVENTIONAL MICROSCOPE-BASED MICROARRAY READER
Table 3.1 This table gives an overview of the volumes in nl of the different vials used in the experiments.
depth
µm
20
30
50
square vial
circular vial (diameter)
300 × 300µm2 400 × 400µm2 300µm
400µm
1.8
2.7
4.5
3.2
4.8
8.0
1.4
2.1
3.5
2.5
3.8
6.3
etched using plasma etching. During this process, the silicon is removed at the locations of
the vials. The remaining photo resist is removed by rinsing the surface with acetone. Both
wet and dry etching techniques can be employed on the silicon wafers for the realization of
the vials.
Wet etching is performed in potassium hydroxide (KOH). Because of the crystal structure of
silicon, this technique results in an anisotropic etching of the silicon. The silicon wafers have
a < 100 > surface plane orientation. The etching rate for the < 111 > crystal planes is
much slower in KOH solution than for the other crystal planes (typically by a factor of 50).
Therefore, wet etching of the silicon results in vials with the shape of a truncated pyramid,
which has < 111 > planes as side walls for circular as well as for square etching masks.
These two types of vials are shown in Figure 3.1(a) and 3.1(b) respectively. In the case of a
circular window, underetching of the silicon nitride will take place resulting in free hanging
silicon nitride in the corners of the vial, as can be seen in Figure 3.1(a). The angle between
the < 111 > planes and the < 100 > plane is about 54.7 degree.
Dry etching is conducted by Reactive Ion Etching (RIE), which results in an anisotropic
etching. The RIE etched vials have a cylindrical or cubic shape after etching with a circular
or square masking pattern respectively. These vials are shown in Figure 3.1(c) and 3.1(d).
These two types of vials have been used in most of our experiments.
Microarrays can also be manufactured in glass. Glass wafers are then covered with a thin layer
of polysilicon, which is then processed to form the etching mask. Wet etching is conducted
in a hydrofluoric acid (HF) solution. Glass is an amorphous material, and thus isotropic for
etching. This process results in a cilindrical or cubic shape with rounded edges and corners.
These shapes are shown in Figure 3.1(e) and 3.1(f). In the initial phase of this research glass
microarrays have been used in experiments, but, from a "lab-on-a-chip" point-of-view, silicon
microarrays are to be preferred, since they allow for easy integration of electronic sensors.
The microarrays used in the experiments typically contain 5 × 5 vials. The center-to-center
distance ranges from 800µm to 1200µm. The dimensions of the microarray itself are identical
for all microarrays 2 × 1cm2 . In Figure 3.2 one of the microarrays is shown.
From these dimensions follows that the vial density is in the range from 70 to 150 vials per
square centimeter. The relatively large center-to-center distance is needed for a dedicated
filling method (not described in this thesis). With this manual filling method even smaller
vials could be filled. In the next section, however, an automated liquid handling procedure
will be described for which the minimal vial width is about 300µm.
NANOLITER LIQUID HANDLING
(a) Silicon, circular, KOH
(b) Silicon, square, KOH etched
(c) Silicon, circular, RIE etched
(d) Silicon, square, RIE etched
(e) Glass, circular, HF etched
(f) Glass, square, HF etched
17
Fig. 3.1 This figure shows six different types of vials that can be fabricated with standard silicon
fabrication techniques.
3.2
NANOLITER LIQUID HANDLING
One of the difficulties involved in the miniaturization of high-throughput screening technologies is liquid handling. Fluid volumes on the order of a nanoliter need to be injected into the
vials on the microarray. Within the framework of this research program, different nanoliter
liquid handling strategies have been developed. In this section we will briefly discuss three
techniques.
Manual liquid handling with Eppendorf Transjector An Eppendorf Transjector 5246
(Eppendorf, Netheler-Hinz GmbH, 22331 Hamburg, GE) is used to manually inject liquid
samples into the vials. This machine is primarily used for In Vitro Fertilization (IVF), in
which sperm cells are injected through the cell membrane into an oval cell. In order to inject
liquid volumes into a vial, a special capillary the Femtotip II is used. The Femtotip II is
about 5cm long and the outlet of the capillary has an inner diameter of less than 0.5µm.
The capillary tapers off to the outlet. In order to inject a small liquid sample into a vial, the
Transjector builds up a sudden injection pressure. This pressure is held during the injection
time. After injection the pressure is exponentially released to the compensation pressure. It
18
1 cm
4
4
DIOC-IMDS
2 cm
Fig. 3.2 A 2 × 1cm2 microarray etched in silicon with two arrays of 5 × 5 vials. The center-to-center
distance is 800µm.
Fig. 3.3 The experimental setup for filling microarrays manually with the Eppendorf Transjector. The
Femtotip II is placed at an angle under a low magnification objective (working distance ≈ 1cm). On
the right a "Dutch Dime" is placed (diameter is 15mm).
is necessary that the tip of the capillary touches the surface of the vial in order to release the
liquid drop from the tip and to get the liquid in the vial.
This kind of liquid handling obviously does not meet our demands. With this device it is
only possible to fill one vial at a time with just one liquid sample. Furthermore, to avoid fast
evaporation of the liquid sample, a mixture of glycerol and water (1 : 1, v/v) must be used.
Figure 3.3 shows the experimental setup for manually filling the vials with a solution.
Automated liquid handling: Electro Spray Dispensing Besides this manual liquid handling technique an automated liquid handling technique has been developed in this research
program: Electro Spray Dispensing [6]. The piston of a syringe is driven by a precision
displacement pump. This guarantees a well-controlled liquid flow at the outlet of a capillary
(inner diameter ≈ 60µm) connected to the syringe. A microarray is placed on a motorized
xyz-stage at a distance of about 250µm under the outlet of the capillary. The metal capillary
CONVENTIONAL MICROSCOPY CONCEPTS
19
Fig. 3.4 A liquid droplet is formed by an aerosol of small liquid droplets from the capillary, induced
by a strong electric field.
is connected to a high voltage, where the microarray is grounded. This potential difference
induces a large electric field. This electric field produces an aerosol of small droplets of the
conductive liquid, which are accelerated towards the microarray. This forms uniform spots
with a diameter on the order of 200µm. Figure 3.4 shows the outlet of the capillary and
the formation of a droplet in the microarray. The spraying can be stopped and resumed instanteneously by increasing and decreasing the distance between the array and the outlet of
the capillary. The increased distance lowers the electric field strength. This liquid handling
technique allows for noncontact, accurate and reproducible dispensing of ultrasmall liquid
volumes [6].
Coarse massive parallel filling In the case of manual filling as well as in the case of
electrospraying fast evaporation of the liquid sample occurs. To avoid fast evaporation, a
third technique has been used. A droplet of about ≈ 20µl is pipetted on the microarray and
spread over it, such that all wells are filled. Then a microscope slide is placed on top of the
microarray. The microarray and the microscope slide are pressed firmly together to remove
the excess liquid. The Van der Waals forces exerted on the microarray and microscope slide
prevent evaporation of the liquid in the vials. This technique allows for massive parallel filling
of practically all vials of a microarray.
3.3
The previous sections described how the microarrays can be fabricated and how they can be
filled. The next sections will describe how the fluorescent signals generated in the vials of the
microarrays can be detected.
Our microarray reader is based upon a Zeiss Axioskop microscope system equipped with a
scientific grade CCD camera. In this section we will present two concepts of modern quantita-
20
Intermediate
image plane
Objective
Object plane
d=45 mm
Fig. 3.5
ftube=150 mm
This figure shows the image forming beam path in a finite-corrected optical system.
tive microscopy: the infinity-corrected optics and Köhler illumination. These concepts form
the basis of the modifications we will introduce in the next chapter to improve the performance
of our microarray reader.
3.3.1
Infinity-corrected Optical System
Nowadays, all major microscope manufacturers provide infinity-corrected optical systems for
their microscopes. In this section we will explain the concept of infinity-corrected optics and
its advantages with respect to a finite-corrected optical system.
Finite-corrected optical system. In a finite-corrected optical system the objective forms
an image at the primary image plane at a finite distance from the objective. Figure 3.5 shows a
beam in the image forming path of a finite-corrected optical system. The well-known standards
for microscopes with a finite-corrected optical system are the following:
• The parfocal distance d equals 45mm, as indicated in Figure 3.5. The parfocal distance
is a pure mechanical dimension. It is the distance between the object plane and the
objective shoulder. When the objective is changed, the image focus is more or less
retained (parfocality).
• The tube length ftube equals 150mm, as indicated in Figure 3.5. The tube length is the
distance between the objective shoulder and the primary or intermediate image plane.
The mechanical tube length is 160mm. This is the distance between the objective
shoulder and the shoulder of the body tube that supports the ocular.
• The objective thread size equals 0.8” = 20.32mm, (W 0.8 × 1/36”).
With these standard dimensions the objectives from different microscope manufacturers could
be exchanged between different microscopes. One of the major shortcomings of this approach
is that the design for intermediate accessories, such as fluorescence filters is rather difficult,
since they have to be placed in the image forming bundle between the objective and the
21
Objective
Intermediate
image plane
Tube lens
Object plane
fobj
Fig. 3.6
parallel optical
path
ftube
This figure shows the image forming beam path in a infinity-corrected optical system.
image plane. These intermediate accessories may introduce magnification errors. An infinitycorrected optical system overcomes this difficulty.
Infinity-corrected optical system. In an infinity-corrected optical system the objective
forms an image at infinite distance from the objective. This means that the exit bundle of the
objective is a parallel bundle of rays. A second (fixed) lens must be placed after the objective to
form an image at the primary image plane. This lens is called the tube lens. Figure 3.6 shows
a beam in the image forming path of an infinity-corrected optical system. It is a common
misconception that the tube lens can be placed at an infinite distance from the objective. This
can be easily seen in Figure 3.6: after passing through the objective, light from an object on
the optical axis propagates parallel to this axis towards the tube lens. Light coming from the
periphery of the object forms also a parallel bundle of rays, but propagates at an angle with
respect to the optical axis towards the tube lens. Because of this angle there are instances
where these rays of light can no longer be captured by the tube lens. This occurs if the tube
lens is placed too far from the objective. As a conclusion, the term infinity implies that light
after passing through the objective forms a bundle of parallel rays. In an infinity-corrected
optical system the intermediate accessories can be placed in the parallel optical path without
affecting the overall magnification. The overall magnification M for an infinity-corrected
optical system, as shown in Figure 3.6. is given by
M=
ftube
,
fobj
(3.1)
where ftube and fobj are the focal lengths of the tube lens and the objective.
Table 3.2 shows the dimensions of four major microscope manufacturers that they have applied
in their infinity-corrected optical system. Zeiss called its infinity-corrected optical system Infinity Color-corrected System, (ICS), Olympus Universal Infinity System, (UIS), Nikon Chrome
Free Infinity (CFI60), where 60 refers to the adapted parfocal distance of 60mm, and Leica
Harmonic Component System (HCS), which is the successor of the DELTA infinity-corrected
optical system.
22
Table 3.2 Dimensions of the infinity-corrected optical systems of four major microscope manufacturers
Leica
Nikon
Olympus
Zeiss
∗
3.3.2
parfocal distance
mm
tube focal length
mm
objective thread
45
60
45
45
200
200
180
165
M 25 × 0.75∗
M 25 × 0.75∗
W 0.8 × 1/36”
W 0.8 × 1/36”
HCS
CFI60
UIS
ICS
) This objective thread has a diameter of 25mm.
Köhler Illumination
The illumination scheme of Köhler [7] for a conventional microscope was published in 1893,
and is still used in modern microscopy. Figure 3.7(a) shows the excitation and emission paths in
an unfolded infinity-corrected epi-illumination fluorescence microscope system. This means
that the objective and condenser as indicated in Figure 3.7(a) are the same optical component,
and that the condenser aperture has the same size as the back focal plane of the objective,
i.e. the exit pupil of the objective. The optical path of the excitation light is drawn from the
light source (Hg-lamp) to the image plane. Right before the second tube lens, at the position
of the emission filter, the excitation light is blocked. This is the reason why the rays of the
excitation light are drawn as dashed lines. The emission light originates at the object plane,
because in that plane fluorescence is generated. Only the cone of light that is captured by the
objective and imaged onto the image plane is drawn. By reversal of rays, the propagation of
the emission light from the object plane towards the light source can also be drawn. These
rays are drawn as dashed lines. In Figure 3.7(a) both tube lenses are identical, but there are
two separate tube lenses in the microscope. The size of the condenser aperture determines the
extent of the light source that is being used for excitation. The size of the field stop determines
the size of the field-of-view. The field-of-view is homogeneously illuminated.
Effects of closing the field stop. Figure 3.7(b) shows a microscope adjusted to Köhler
illumination, identical to Figure 3.7(a), but here the aperture of the field stop is smaller. The
following can be seen in Figure 3.7(b):
1. Closing the field stop results in a smaller field-of-view. Compare the image planes in
Figure 3.7(a) and Figure 3.7(b).
2. The contribution of the light originating from a single point of the light source to the
intensity of the excitation light in the object plane (in terms of excitation power per unit
area) does not change. Only the total illuminated area gets smaller. The total extent of
the light source is used for illumination of the field-of-view.
3. Since the total excitation power per unit area remains constant, the probability to generate fluorescence in the field-of-view remains constant as well.
Collector
Tubelens
Condenser
Objective
Tubelens
Emission
filter
Excitation
filter
Hg lamp
Fieldstop
Condenser
Aperture
Object
plane
Backfocal
plane
Image
plane
(a) This figure shows the optical paths of the excitation light and the emission light in a conventional epi-illumination fluorescence microscope with an infinity-corrected optical system. This
microscope system is adjusted to Köhler illumination.
Collector
Tubelens
Condenser
Objective
Tubelens
Emission
filter
Excitation
filter
Hg lamp
Fieldstop
Condenser
Aperture
Object
plane
Backfocal
plane
Image
plane
(b) This figure shows a microscope adjusted to Köhler illumination. The difference with Figure 3.7(a) is that the aperture of the field stop is smaller.
Collector
Tubelens
Condenser
Objective
Tubelens
Emission
filter
Excitation
filter
Hg lamp
Fieldstop
Condenser
Aperture
Object
plane
Backfocal
plane
Image
plane
(c) This figure shows a microscope adjusted to Köhler illumination. The difference with Figure 3.7(a) is that the aperture of the condenser is smaller.
Fig. 3.7
23
24
4. The collection efficiency of the objective, in terms of the maximum half opening angle
of the cone of light that can be collected, is unaltered. The intensity of the collected
emission light (in terms of power per unit area) remains constant.
Effect of closing the condenser aperture. Figure 3.7(c) shows a microscope adjusted
to Köhler illumination, identical to Figure 3.7(a), but here the aperture of the condenser is
smaller. In practice, this is not possible, since an objective has a fixed Numerical Aperture, and
thus the size of the back focal plane is fixed. As explained before, in fluorescence microscopy
the condenser is the objective. This implies that condenser aperture equals the back focal
plane of the objective and cannot be changed. Figure 3.7(c), however, clearly illustrates what
happens:
1. The illuminated field-of-view does not change.
2. The contribution of the excitation light, that passes the condenser aperture, to the intensity of the excitation light in the object plane does not change. A smaller fraction of the
extent of the light source, however, passes the condenser aperture. As a result the total
excitation power in the object plane is smaller.
3. Since the total excitation power per unit area is smaller, the probability to generate
fluorescence in the field-of-view is smaller.
4. Since the condenser aperture is related to the size of the back focal plane of the objective,
the maximum half angle of the cone of light that can be collected gets also smaller. This
implies that the collection efficiency of the objective gets smaller and fewer emission
photons are captured.
As a conclusion, in order to have the highest collection efficiency, high power objectives are
needed. This implies a field-of-view on the order of the size of a single vial. As a consequence,
the microarray should be readout sequentially, i.e. well per well. As described in the previous
chapter, the fluorescence signals are in the shorter wavelength part of the spectrum. This
will generate a large amount of stray light. Stray light originates in the excitation path. This
noise contribution can be minimized by closing the field stop as much as possible. Köhler
illumination gives rise to the maximum lateral resolution. We are, however, not interested
in spatial resolution. With critical (nonuniform) illumination an image of the light source is
focussed on to the object plane. This gives a relatively small area of the field-of-view that is
intensely illuminated. Critical illumination increases the excitation power in a small region.
3.4 CHARACTERIZATION OF THE PRINCETON VERSARRAY CCD CAMERA
The second component of our conventional microscope based microarray reader is a CCD
camera. Our CCD camera is the Princeton Versarray 512B Back-illuminated CCD camera
with a Tektronic 512 × 512 DB CCD chip. The CCD array consists of 512 × 512 pixels. Each
pixel is 24 × 24µm2 . The CCD array has a 100% fill factor. The physical dimensions of the
CCD chip are 12.3 × 12.3mm2 . This camera has three different gain settings: low ( 21 ×),
medium (1×), and high (2×). This CCD camera offers the possibility for readout at 100kHz
CHARACTERIZATION OF THE PRINCETON VERSARRAY CCD CAMERA
25
or at 1MHz and has a 16 bit analog-to-digital converter. The operating temperature can be
set from room temperature down to −50◦ C. The CCD camera is controlled via an ST133
Controller from Roper Scientific. Images are collected and analyzed using Matlab. Matlab
communicates with the CCD camera via the VinView software of Roper Scientific.
In this section we will present the most important characteristics of this CCD camera. For
these experiments we have used a number of well-known procedures [8].
3.4.1
Readout noise
Readout noise is the lower noise bound of the CCD camera. It has the following properties:
readout noise
• originates in the preamplifier of the CCD camera,
• is independent of the signal and the integration time,
• is additive Gaussian noise, defined in terms of the RMS error,
• depends on the readout rate, and
• is assumed to be stationary, and ergodic, i.e. the noise of all pixels is independent,
identical and constant in time, such that the variances of the readout noise per pixel can
be added.
The Princeton Versarray offers the possibility to collect the data at two readout rates: 100kHz
and 1MHz . Reading out the camera means that the shutter was kept closed during data
collection, and that an integration time of 0 seconds was used. The electronic gain was set to
low. The CCD camera was cooled to −40◦ C during this experiment.
Readout rate = 100kHz . In this experiment, each pixel of the CCD array is read out 1000
times. This implies, that for each pixel a histogram of the readout levels can be constructed.
Figure 3.8 shows four of these histograms for four different pixels of the CCD array. The
histogram in Figure 3.8(a) corresponds to pixel (8,8) on the CCD array, the histogram in
Figure 3.8(b) to pixel (504,8), the histogram in Figure 3.8(c) to pixel (8,504), and the histogram
in Figure 3.8(c) to pixel (504,504). Gaussian distributions defined by the sample mean and
the sample variance are overlaid to each of the histograms. These histograms indicate that
the readout noise of the pixels is normally distributed. Each and every pixel, however, has
a different average readout level (µ) and a slightly different readout noise level (σ): the
readout levels vary from 102.9#ADU (Analog-to-Digital Unit) to 109.1#ADU (the first
column of the CCD has an average readout level of 1493.8#ADU ). The readout noise levels
have an average value of σ = 1.14#ADU (the lowest readout noise level is 0.98#ADU ).
Figure 3.9(b) shows the readout noise levels of all pixels. The lowest value in this image
corresponds to the value of the 5% percentile (σ = 1.06) and the highest value in this image
corresponds to the value of the 95% percentile (σ = 1.208). This image shows that the
readout noise has a positive gradient towards the lower half of the CCD array. This gradient
is introduced by dark current. The total readout time of the full chip is approximately 2.5
µ=104.851
σ=1.071
0.35
0.3
relative frequency
relative frequency
26
0.25
0.2
0.15
0.35
0.3
0.25
0.2
0.15
0.1
0.1
0.05
0.05
99
100 101 102 103 104 105 106 107 108 109
110
µ=103.536
σ=1.065
111
99
µ=107.261
σ=1.142
0.3
0.25
0.2
0.15
(b) (504,8)
relative frequency
relative frequency
(a) (8,8)
0.35
100 101 102 103 104 105 106 107 108 109 110 111
readout level of Versarray [#ADU]
0.3
µ=107.424
σ=1.198
0.25
0.2
0.15
0.1
0.1
0.05
0.05
99
100 101 102 103 104 105 106 107 108 109 110 111
(c) (8,504)
99
100 101 102 103 104 105 106 107 108 109 110 111
(d) (504,504)
Fig. 3.8 Histograms of readout levels (@100kHz ) of four different pixels of the CCD array. The
curves are Gaussian distributions defined by the mean µ and standard deviation σ.
seconds. During this readout time dark current is collected as well. This dark current accounts
fully for this gradient1 .
Readout rate = 1MHz . The same experiment is repeated at the high readout rate. Figure 3.10 shows the histograms corresponding to the same pixels as in Figure 3.8. Again,
each and every pixel has a different average readout level and a different readout noise level:
the readout levels vary from 19.0#ADU to 30.2#ADU (the first column of the CCD has
an average readout level of 0#ADU ). The readout noise levels have an average value of
σ = 2.19#ADU (the lowest readout noise level is 1.96#ADU ). Figure 3.11(a) shows
the readout level of all pixels. The lowest value in this image corresponds to the value of
the 5% percentile (26.0#ADU ), and the highest value to the 95% percentile (27.4#ADU ).
next section will show that the dark current at −40◦ C is on average 1.5#ADU /s/pixel (low gain). During
the readout time of 2.5s 3.8#ADU of dark current is collected at the lowest rows of the CCD array. The extra
variance due to this dark current equals the electronic gain G times the variance of the dark current. The former
equals 0.1#ADU /e− as will be shown in Section 3.4.3, the latter equals the average value of the dark current,
since this a Poisson process. Thus,
√this extra variance is 0.1 × 3.8 = 0.38. The overall standard deviation σ due to
readout noise and dark current is 1.072 + 0.38 = 1.14#ADU .
1 The
(a)
27
(b)
Fig. 3.9 These images show the readout levels (left) and the readout noise level (right) (@100kHz )
of all pixels. The range in the left image is (103.6 − 107.3ADU ). The range in the right image is
(1.06 − 1.21ADU ). The readout noise has a positive gradient towards the lower, right-hand corner of
the image.
Figure 3.11(b) shows the readout noise levels of all pixels. The lowest value in this image
corresponds to the value of the 5% percentile (σ = 2.11) and the highest value in this image
corresponds to the value of the 95% percentile (σ = 2.27). This image shows the spatial distribution of the readout noise level per pixel. This spatial distribution is a Gaussian distribution
with a standard deviation of 0.09#ADU . The spatial distribution of the readout noise in a
single read out image has a Gaussian distribution with a standard deviation of 2.17#ADU .
The temporal distribution equals the spatial distribution. This implies that for a high readout
level the readout noise is ergodic; reading out the CCD array at 1MHz introduces no spatially
varying noise due to dark current.
3.4.2
Dark current
Dark current consists of electrons induced by (thermal) vibrations, which cannot be distinguished from photo-electrons. The production of dark current yields a Poisson distribution
and is dependent on the operating temperature
of the CCD and the integration time. The dark
current is proportional to exp − ∆E
,
with
k
the Boltzmann constant (1.38 × 10−23 J/K),
kT
and T the absolute temperature in K: as a rule of thumb, the dark current doubles with a
temperature increase of 6K. This implies that dark current can be drastically reduced by
cooling.
To measure the dark current flux in e− /s/µm2 the Princeton Versarray was set to the low
readout rate (100kHz ). With forced air and built-in Peltier cooling the Versarray can be cooled
down to about −50◦ C. In this experiment the shutter was kept closed and series of 100 images
0.15
relative frequency
relative frequency
28
µ=28.127
σ=2.11
0.1
0.15
µ=26.976
σ=2.158
0.1
0.05
0.05
18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
0.15
(b) (504,8)
relative frequency
relative frequency
(a) (8,8)
µ=25.399
σ=2.181
0.1
0.05
0.2
µ=25.733
σ=2.193
0.15
0.1
0.05
18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
(c) (8,504)
18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
(d) (504,504)
Fig. 3.10 Histograms of readout levels (@1MHz ) of four different pixels of the CCD array. The
curves are Gaussian distributions defined by the mean µ and standard deviation σ.
were taken at different temperatures in the range from −20◦ C to −40◦ C with steps of 2◦ C.
For each operating temperature two integration times were used: 0.0s, and 16.0s. The series
of images acquired with an exposure time of 0.0s corresponds to reading out the CCD array.
Images of the sample mean per pixel of each series were computed for each series. For each
temperature, images of the dark current flux in #ADU /s were computed from the sample
mean acquired with an exposure time of 0.0s and 16.0s. Figure 3.12 shows the image of the
dark current rate at −40.0◦ C. The lowest value in this image corresponds to the 5% percentile
(1.04#ADU /s) and the highest value to the 95% percentile (1.83#ADU /s). This implies
that the dark current is larger than the readout noise for integration times longer than a second.
The gradient of the dark current rate in Figure 3.12 indicates that the temperature of the CCD
array is not uniform: the temperature on the left side is lower than the temperature on the
right side of the CCD array. This is due to nonuniform performance of the Peltier cooling
element(s).
For all different temperatures in this experiment, images of the dark current rate as shown in
Figure 3.12 are computed. The average of the dark current rate and the standard deviation of
the dark current rate for these temperatures are shown in Figure 3.13.
(a)
29
(b)
Fig. 3.11 These images show the readout levels (left) and the readout noise level (right) (@1MHz )
of all pixels. The range in the left image is (26.0 − 27.4ADU ). The range in the right image is
(2.11 − 2.27ADU ).
Fig. 3.12 This image shows the spatial distribution of the dark current rate in #ADU /s. The lowest
value corresponds to the value of the 5% percentile (1.04#ADU /s), the highest value to the value of
the 95% percentile (1.83#ADU /s). The dark current rate on the right side of the CCD array is higher
than on the left side
3.4.3
Linearity and Sensitivity
Linearity. To measure the linearity of the photometric response of the Princeton Versarray,
the CCD camera was mounted on a bright-field microscope system. The microscope was
adjusted to Köhler illumination. The field-of-view was empty. After the light level in the
Dark current flux (#ADUs/pixel/s)
30
15
10
7
5
3
2
1.5
-37.5
-35
-32.5
-30
-27.5
-25
-22.5
-20
Temperature (degrees Celsius)
Fig. 3.13 This graph shows the average dark current rate in #ADU /s in the temperature range from
−40◦ C to −20◦ C.
Table 3.3
Mean and standard deviation of the regression coefficient R2 over the CCD array.
Gain
low
medium
high
µ
σ
0.9998
0.9998
1.0000
1.8 × 10−5
2.0 × 10−5
2.3 × 10−5
object plane was adjusted to comfortable viewing by the human eye (light intensity not too
bright, not too dim), a neutral density filter with a transmission of 1% was placed right after
the halogen lamp. Series of images were acquired at the three different gain settings of
the camera with different exposure times ranging from 0.1s to 10.0s. Each series consisted
of 25 images. The average photometric response at a certain integration time and gain was
computed. Figure 3.14 shows the photometric response for a single pixel for the three different
gain settings. This figure shows that in the low gain mode just half of the dynamic range of
the camera can be used. This is the reason why the low gain is given as 0.5×. For each
pixel on the CCD array the linearity of the photometric reponse is computed as the regression
coefficient R2 to the data, i.e. the photometric response as a function of the exposure time.
Table 3.3 shows the average regression coefficient of the CCD array and the standard deviation
of the regression coefficient for the three diffent gain settings. Images showing the regression
coefficient for all pixels suggest artefacts in the respons of the CCD camera due to the nonuniform illumination. These artefacts are not present: the values of the regression coefficient
differ only in the fifth decimal.
mean grey value [104 ADUs]
31
6
5
HIGH
4
MEDIUM
3
LOW
2
1
2
4
6
8
10
exposure time [s]
Fig. 3.14 This graph shows the average photometric response as a function of the integration time for
the three different gain settings of the CCD camera for a single pixel.
Sensitivity. The relative sensitivity or electronic gain G in grey level per photo-electron
(#ADU /e− ) can be measured using statistical methods under the assumption that the CCD
camera is photon limited. That is, Poisson noise, due to the quantum nature of light, is the
dominating noise source.
The grey level I of a pixel is proportional to a number of collected photoelectrons Ne according
to
I = G · Ne ,
(3.2)
where G is the electronic gain. The number of photo-electrons Ne is drawn from a Poisson
distribution: the variance of the number of photo-electrons Var [Ne ] acquired in an integration
time T equals the expected value of the number of photo-electrons E[Ne ] acquired in an
integration time T . From this follows:
E[I] = G · E[Ne ]
(3.3)
Var [I] = G2 · Var [Ne ] = G2 · E[Ne ] = G · E[I]
The sensitivity G follows now as:
G=
Var [I]
,
E[I]
(3.4)
where E[I] can be computed as the average value of a pair of images and Var [I] can be
computed as
1
Var [I] = Var [I1 − I2 ].
(3.5)
2
A series of pairs of images is acquired at different light intensities. The light intensity is varied
by placing different neutral density filters in the optical path. In each image the same uniform
variance [104 ADUs2]
32
2
1.5
HIGH: 0.36 ADU/e-
1
MEDIUM: 0.18 ADU/e0.5
LOW: 0.09 ADU/e1
2
3
4
5
6
Fig. 3.15 This graph shows the measured pixel variance Var [I] as a function of the average value
E[I]. The slope of these curves is the electronic gain or relative sensitivity.
region was selected for computation of the pixel variance Var [I] and average value E[I]. This
experiment is repeated for the three different gain settings. The results of these measurements
are shown in Figure 3.15. Table 3.4 gives the value of the electronic gain for three settings of
the CCD camera.
Table 3.4
The electronic gain G in #ADU /e− for the different gains.
gain setting
electronic gain
#ADU /e−
LOW
MEDIUM
HIGH
0.09
0.18
0.36
The same analysis can be used to compute the maximum signal-to-noise ratio SNR:
SNR = 10 log
E 2 [I]
Var [I]
= 10 log (G · E[Ne ]) .
(3.6)
Figure 3.16 shows the SNR as a function of the mean grey value, which is based on the same
images as in the previous experiment. The curves in this graph show the theoretical SNR for
the three different electronic gains G. The image based SNR is computed using the image
SNR [dB]
SUMMARY
60
55
33
LOW
pixel based SNRs
(Photon limited)
MEDIUM
HIGH
50
45
image based SNR
40
35
30
1
2
3
4
5
6
Fig. 3.16 This graph shows the SNRs as a function of the average intensity computed in a uniform
region of an image for the three different gains. The curves are the Photon limited SNRs for these gains.
The image based SNR is independent of the electronic gain.
variance, instead of the pixel variance. This computation is independent of the electronic gain
G. Here the variance is limited by the variation in pixel gain.
3.5 SUMMARY
This chapter described the how the microarrays are fabricated and what methods have been
developed in order to fill these microarray with solutions containing fluorescent molecules.
The microarrays can be read out with a conventional micoscope system equipped with a
scientific grade CCD camera. High power objectives amount to the maximum excitation
and collection efficiency. This implies sequential scanning of the microarray. Closing the
field stop reduces the contribution of stray light. A back-illuminated CCD with an average
Quantum Efficiency of approximately 70% in the visible part of the spectrum guarantees an
efficient collection of the low wavelength emission photons of NADH. In the next chapter, we
will look in detail to fluorescence measurements with this experimental set-up.
4
Quantitative Reading
of Microarrays
Chapter 2 introduced the biological challenge of this research program: development of
an analytical tool based on microarrays for integral analysis of metabolic or regulatory
networks. The demonstrator application for our developed microarray technology is the
glycolytic pathway, i.e. the production of ethanol and carbon dioxide from glucose by yeast.
This analytical tool is based on microarrays fabricated in silicon. The fabrication process
as well as the different nanoliter liquid handling procedures are described in the previous
chapter. A conventional microscope system equipped with a scientific grade CCD camera is
used to measure the fluorescent signals generated in the biochemical reactions in all vials of
the microarray. The most important concepts of quantitative microscopy are also described
in the previous chapter. This chapter will describe in detail what the limiting factors are
for measuring low levels of fluorescence in these microarrays, and what modifications to our
microscope system we have implemented in order to overcome this limitation.
4.1
LIMITING FACTORS OF THE INSTRUMENT
Reading the microarray, i.e. measuring fluorescence levels from all the vials, is hampered
by many unwanted factors from: (a) the sample (biological / biochemical variations), (b) the
optical system, and (c) the detector. The latter two characterize the instrument and are independent of the sample. These variations should be smaller than the biological / biochemical
variations. There are two possible limiting factors (noise sources) that determine the detection
limit of the system, i.e. the lowest fluorescence concentration that can be distinguished from
a blank (control) solution. The first limiting factor of the detection limit is stray light. Stray
light originates in the optical (excitation) system by reflections (flare) at air-lens interfaces and
35
36
QUANTITATIVE READING OF MICROARRAYS
scattering at diffuse surfaces within the optical system (glare). Other contributions to stray
light are (Raman) scattering in the solvent and autofluorescence in the objective. Furthermore, there are contributions to the fluorescent signal from non-specific (auto-)fluorescence
of the solvent, or ingredients. Not only the deterministic part of stray light is important, but
in particular the stochastic part of the stray light. This contribution of stray light is not only
due to Poisson statistics, but more likely to variations of the interfaces of the microscope
slide, the bottom of the microarray, and air-bubbles in the solvent. The second limiting factor
is at the detection side of the microscope system: readout noise. Readout noise originates
in the preamplifier of the CCD camera. The readout noise is a lower bound of the camera
noise. Minimizing the largest noise contributions will result in a lower detection limit of the
microscope system.
4.2 DETECTION LIMIT OF THE MICROARRAY READER
We will take a closer look at fluorescence measurements to determine the detection limit. We
assume that the concentration of fluorescent molecules is low to avoid serious attenuation of
the excitation light deeper in the sample and to avoid reabsorption of the emitted fluorescence:
in other words a linear relation between the excitation power and the emission power. Figure 4.1(a) shows the response of a detection system as a function of the concentration of the
fluorescent particles. The dashed lines indicate the distribution of the blank solution. The
solid lines indicate the distributions of the different fluorescent solutions. These distributions
imply that the blank solution as well as the fluorescent solutions are measured with a certain
precision and accuracy. Both precision and accuracy are defined by the system, i.e. we do not
consider here the biological variation. The accuracy of the fluorescent solutions equals the
accuracy of the blank solution. This means that a possible bias (background signal) can be
removed by subtracting the average blank signal from the fluorescent signals. This does not
correct for the stochastic variation. We assume in this figure that the probability density functions of the blank solution and the fluorescent solutions can be estimated extrinsically during
a calibration procedure, i.e. each concentration can be independently measured many times
on the same microarray (built-in redundancy). The results of this calibration procedure are a
probability density function of the instrument’s response to a blank solution and probability
density functions of the instrument’s responses to various fluorescent solutions.
Figure 4.1(b) shows the definition of the detection limit. We define the minimum detectable
signal such that 97.5% of all blank solutions will be classified as blank and only 2.5% will be
erroneously treated as originating from the fluorescence process to be measured. This means
that the strength of the minimum detectable signal equals the mean signal of the blank solution
plus two times the standard deviation of the signal of the blank solution. If we choose the
minimum detectable signal as the mean of the blank plus three times the standard deviation
of the blank, then in 99.5% of the measurements of a blank solution, these measurements will
be classified as a blank solution. As a consequence, the probability that a fluorescent solution
with a concentration corresponding to the detection limit will produce a signal larger than the
minimum detectable signal is 50%.
When we measure the fluorescent signal of a solution in a vial, we need to estimate the
underlying concentration of fluorescent particles. In other words, which concentration (and
n
mea
of fl
kσblank
mean of blank (C=0) solution
distribution of
fluorescent solution
ion
olut
nt s
sce
uore
distribution of
blanco solution
Detector response [#ADUs]
DETECTION LIMIT OF THE MICROARRAY READER
ConcentrationC of dye molecule [M]
lutio
e
ons
t so
cen
ores
n
u
to fl
esp
ed r
ct
e
exp
minimum detectable signal
kσblank
mean of blank (C=0) solution
distribution of
blanco solution
Detector response [#ADUs]
(a) Due to instrumental variations the response to a concentration of fluorescent particles has a certain distribution.
ConcentrationC of dye molecule [M]
detection limit
(b) The minimum detectable signal equals average response of the blank plus k times
the standard deviation of measuring the blank.
Fig. 4.1
Fluorescence response model.
37
38
corresponding expected flux λ) has the highest probability of producing this measurement. For
a single measurement this amounts to assigning the concentration according to the estimated
calibration curve. In statistical terms, for one measurement as well as for many measurements, we select the molecular concentration that maximizes the a posteriori probability that
a fluorescent solution λ generates this response:
~ =
argmax λ P (λ|X)
N
Y
pλ (xi ),
(4.1)
i=1
assuming equal a priori probabilities. Note that averaging measurements results in a probability density function with a variance (of the sample mean) that is inversely proportional
to the number of measurements. For an infinite number of measurements, the sample mean
equals the expectation value of the probability density function. As a conclusion, repeating
the number of measurements will result in a lower detection limit. In theory, it is possible to
distinguish between a blank solution and a solution with a single fluorescent particle under the
assumption that a very large number (→ ∞) of measurements can be performed. Note that
an exact measurement does not contribute to an exact interpretation of the underlying biology
/ biochemistry, since they will still be limited by the biological variation.
4.2.1
Improving the Detection Limit (for 1 and N measurements!)
From Figure 4.1(a) the following possibilities to improve the detection limit can be derived.
1. Reduce the variation of measuring the blank solution. The variation of the blank has
contributions of stray light (i.e. scatter, autofluorescence, reflections, and non-specific
fluorescence) and detector noise (i.e. dark current, and readout noise). Contributions
of stray-light can be minimized by closing the field stop as far as possible and only
illuminate (a fraction of) a single vial. The noise contributions of the detector can be
minimized by cooling the CCD and imaging the emission photons onto as few pixels
as possible. e.g. using on-chip binning.
2. Improve the sensitivity of the detection system. This is achieved by using high power
objectives. The area of the field of view is proportional to 1/M 2 , where M is the
magnification of the objective. As a result, the excitation light intensity per unit area in
the field-of-view is proportional to M 2 . The detection (emission) efficiency is mostly
influenced by the Numerical Aperture of the objective. The light gathering power of an
objective in Köhler illumination is proportional to NA2 /M 2 [9]. Under the assumption
that there is a linear relation between excitation and emission energy, we can say that
the intensity of the emission light is proportional to M 2 . Combining these results, it
follows that the light gathering power of an objective (with epi-illumination adjusted
to Köhler illumination) is proportional to NA2 . In the case of readout noise limited
detection, the Quantum Efficiency of the detector is of equal importance to the readout
noise. In the case of stray light limited detection, a better detector does not help, since
it will increase the noise level as well as the signal level, leaving the Signal-to-Noise
Ratio unchanged. This means that the graph in Figure 4.1(a) will only be scaled in the
vertical direction.
READOUT NOISE LIMITED DETECTION
39
Reducing the variation of measuring the fluorescent solutions implies a better accuracy. The
measured variation of the fluorescent solutions contains contributions of stray light and detector
noise. A third contribution to the variation of the fluorescent solutions is photon noise. Photon
noise obeys a Poisson distribution due to the quantum nature of light. Repeating identical
biochemical processes in different vials will generate the same Poisson process. A fourth
contribution is caused by differences in injected sample volume. Differences in volume, and
therefore different amounts of ingredients, will result in a different observed emission rate λ. If
the differences of liquid volumes could be taken into account, e.g. with a dedicated electronic
liquid volume sensor as presented in the second part of this thesis, or could be neglected, then
at high fluorescent concentrations the limiting factor of the variation is photon shot noise. For
low concentrations, the same argument holds for both the fluorescence solutions and the blank
solution.
In the following sections we will present possible modifications to overcome the limiting
factors of our conventional microscope system and to improve the performance of the detection
system in terms of the detection limit.
4.3
In Figure 4.1 it is assumed that the response of the detection system is linearly proportional
to the emission power, and that fluorescence is a linear process, i.e. the emission power
depends linearly on the excitation power. This assumption is true for low excitation power
and low concentrations of fluorescing particles [10]. Furthermore, this figure suggests that
the variation of the fluorescence measurements grows linearly with increasing concentrations.
This linear behavior is not true, and this figure is only drawn this way for clarity.
In the following sections we will estimate the strength of the minimum detectable signal in
the case that readout noise is the limiting factor. In the previous section a definition for the
minimum detectable signal was derived: the mean of the blank µ plus k times the standard
deviation σ of the blank distribution (Imin = µblank + kσblank ). Another statistic of interest
is the coefficient of variation CV = σ/µ. We will show that the CV can be improved either
by repeating the measurement n times in the same vial with individual measurement time
T (n × T ), or by increasing the measurement time T to nT (1 × nT ). The result of this
improvement in CV , however, has no biological / biochemical meaning, since it only implies
a better estimate of the biochemical process in that single vial, characterized by the emission
rate λvial in that vial. It does not say anything about the biological variation of the process
involved, characterized by the probability density function of λ.
4.3.1
Ideal Detection System
First, we will consider the case that we have an ideal imaging and detection system: there
are no external noise sources, the system consists of perfect components, and each emitted
photon that enters the detection system is detected. This means true photon counting. The
quantum nature of light shows that the number of photons emitted in a fixed time interval
T varies. This number obeys a Poisson distribution where σ 2 = µ. Therefore, the exact
40
concentration of fluorescent molecules cannot be measured: the coefficient of variation for a
single measurement with this ideal detection system yields:
√
σ
λT
1
single
CV ideal = =
(4.2)
=√ ,
µ
λT
λT
where λ is the emission rate of the fluorescent solution in photons per second.
If we want to discriminate a fluorescent solution from a blank solution (λ = 0) with this ideal
system within a measurement time of T = 1s, then only a single photon must be detected.
If we want to detect this single photon with a probability of 99%, then the probability of not
detecting a single photon is 1%:
P (n = 0; λT ) =
e−λT (λT )n
= e−λT = 0.01.
n!
(4.3)
In terms of Poisson statistics this translates to an emission rate λ = 4.61photons/s. This
single measurement has a CV = 47%, which is much larger than the biological variation.
Repeating this measurement n times (n × T ) or increasing the measurement time T to 1 × nT ,
changes the coefficient of variation into:
1
1
single
1×nT
CV n×T
,
ideal = CV ideal = √ CV ideal = √
n
nλT
(4.4)
the CV improves with the square root of the number of measurements. With n = 87 the CV
of these measurements becomes 5%. Note that either repeating the measurement n times or
increasing the measurement time n times amounts to the same improvement of CV due to
the absence of detector noise.
4.3.2
Single Element Detection System with Readout Noise
Next, we will replace the ideal detector with a more realistic detector: a detector with readout
noise. Readout noise originates in the preamplifier of the sensor (photodiode or CCD camera).
The readout noise is independent of the signal and the integration time, but it depends on
the readout rate. Readout noise is additive Gaussian noise with zero mean. The readout
noise of the sensor, in units of electrons, is in terms of the RMS error at a certain readout
frequency, e.g. the Photometrics Series 200 CCD camera has a readout noise level σr =
11.7e− RMS @500kHz [11] and the Princeton Versarray 512B back-illuminated CCD camera
has a readout noise level of σr = 12.4e− RMS @100kHz as shown in the previous chapter.
This translates to the same number of photon equivalents, assuming a Quantum Efficiency
of 100%. To avoid that reading out the camera results in a negative number of photons, a
constant bias nbias is added to all signals: reading a blank solution will not result in zero
photons, but in a number of photons related to the normal distribution of the readout noise
with a constant bias added. The probability density function related to the blank solution is a
gaussian distribution with mean nbias and standard deviation σr : N (µ = nbias , σ = σr ).
In the case of reading a fluorescent solution, there is first the probability that the fluorescent
solution generates nfl photons given an emission rate λ, and secondly the probability that nd
"photons" are detected, given that nfl photons were emitted. To be more precise: each value
Blank:
0
nbias
0
+
Readout noise
N(0,σr)
N(nbias,σr)
41
Output:
n=N(nbias,σr)
nbias
SINGLE
MEASUREMENT
!!!
Fluorescence:
P(nfl;λt)
Bias:
nbias
nbias
nfl
+
Readout noise
N(0,σr)
nbias+nfl
N(nbias+nfl,σr)
Output:
n=N(nbias+nfl,σr)
Fig. 4.2 Reading a blank solution with a detector with readout noise results in drawing from the
distribution N (nbias , σr ). Reading a fluorescent solution is first drawing from a Poisson distribution
defined by the emission rate λ and the measurement time T , and then drawing from the readout noise
distribution N (nbias + nfl ).
of nfl has a probability of occurring equal to P (nfl ; λT ). The distribution N (nbias + nfl ; σr )
of the readout noise is centered around n = nbias + nfl and weighted with this probability.
Figure 4.2 tries to make this clear for a single measurement. Since the readout noise does not
depend on the signal itself, this implies that the overall probability density function of reading
a fluorescent solution is the convolution of a Poisson distribution given by the emission rate
λ and the measurement time T and a normal distribution defined by the RMS error of the
readout noise σr :
p(n; λT, σr ) = p(nfl ; λT ) ∗ N (nbias ; σ = σr ),
(4.5)
where ∗ is the convolution operator. The probability P (nd |nfl ) is a drawing from this distribution. For λT large enough, this convolution results in
pan approximately normal distribution
with
mean
n
+
λT
and
standard
deviation
σ
=
σr2 + λT : N (µ = nbias + λT, σ =
bias
f
p
2
σr + λT ).
In this case, the absolute minimum detectable signal equals nbias + kσr . We are, however,
only interested in the relative minimum detectable signal kσr , i.e. we subtract the average
blank level. This translates to an emission rate λ = 36photons/s, given k = 3, T = 1s, and
σr = 12e− . The CV of a single measurement with this non-ideal single-element detector is
p
σr2 + λT
CV single
=
,
(4.6)
single
λT
which is 37.3%. Repeating this experiment n times (n × T ) yields
p
σr2 + λT
1
1
single
n×T
CV single = √ CV single = √
.
λT
n
n
(4.7)
With n = 56 measurements it follows that CV = 5%. Increasing the measurement time T
to 1 × nT yields
p
σr2 + nλT
CV 1×nT
=
.
(4.8)
single
nλT
42
With n = 14.2 follows that CV = 5%. A longer measurement time is superior to repeated
measurements in terms of CV .
4.3.3
Many Pixels Detection System with Readout Noise
Finally, the point sensor is replaced by a CCD camera, which is built as a 2-D array of photosensitive elements. Each element can be characterized as a point sensor. In this case, the
photon detector is a CCD camera consisting of a large number of pixels, M . Each and every
pixel suffers from readout noise. We assume here that the readout noise is
• stationary: statistical properties do not change over time, and therefore do not change
from measurement to measurement, and
• ergodic: statistical properties derived from an ensembe of measurements (= #pixels)
are identical to the properties derived from a single element in a series of measurements.
i.e. the noise of all pixels is independent, identical and constant in time, such that the variances
of the readout noise per pixel can be added:
2
σr,M
=
M
X
σr2 = M σr2 .
(4.9)
m=1
Reading a√
blank solution will result in a normal distribution
with mean M ×nbias and standard
√
deviation M × σr : N (µ = M × nbias , σ = M × σr ). The fluorescent signal is spread
over all pixels. This results in a spatio-temporal Poisson distribution. The variance of the
2
fluorescent signal summed over all pixels σf,M
is given by
2
σf,M
=
M
X
i=1
σf2
=
M X
i=1
σr2
λT
+
M
= M σr2 +
M
X
λT
i=1
M
= M σr2 + λT,
(4.10)
where λT /M means that the total emission photon flux λT is spread over M pixels. Reading
a fluorescent
p solution many times results in a normal distribution N (µ = M × nbias +
λT,
σ
=
M σr2 + λT ). For this detector the relative minimum detectable signal equals
√
k M σ = λT = 3.6 × 103 photons with k = 3, and M = 104 and σr = 12e− . The CV of
this single measurement with this CCD camera yields:
p
M σr2 + λT
single
CV M
=
,
(4.11)
λT
which is 33.4%. Repeating this experiment n times yields
p
M σr2 + λT
1
1
single
n×T
CV M = √ CV M
=√
.
λT
n
n
With n = 45 it follows that CV = 5%. Increasing T to 1 × nT yields
p
M σr2 + nλT
CV 1×nT
=
.
M
nλT
(4.12)
(4.13)
CV [%]
STRAY LIGHT LIMITED DETECTION
50
43
Ideal system (Poisson limited)
λ=4.6 photons / s
Single element detector
λ = 36 photons / s
20
Many (104) element detector
λ = 36x 103 photons / s
10
increased T
#repeats
5
T=1s
σr = 12eM = 104 pixels
2
2
5
10
20
50
100
#repeats, c.q. increase T
Fig. 4.3 The different CV for different configurations as a function of the number of repeated measurements or increased measurement time.
With n = 6.7 it follows that CV = 5%.
As a conclusion, measuring low levels of fluorescence requires multiple measurements in
order to achieve a sufficient CV . Furthermore, the minimum detectable signal can be lowered
by imaging the fluorescent signal from the vial onto as few pixels as possible. Thus, measuring
very low levels of fluorescence is at the expense of spatial resolution. Further on in this chapter
we will show through modifications of the emission path of a fluorescence microscope, how this
can be achieved. Furthermore, we will estimate the minimum detectable signal in photons for
different configurations based on the model presented in this section. Figure 4.3 summarizes
these conclusions: with an ideal detector very low levels of fluorescence can be detected at
the expense of precision. A CCD camera, on the other hand, can detect fluorescence much
more precise, but with a much lower sensitivity.
4.4 STRAY LIGHT LIMITED DETECTION
In the case of stray light limited detection a better detector (higher QE, lower readout noise)
does not result in a better detection limit: the detector yields not only a better response to the
signal, but also to the noise, i.e. the Signal-to-Noise Ratio will not improve. This means that
Fig. 4.1(a) will only be scaled in the vertical direction and the detection limit will not shift to
lower concentrations.
If stray light (with a photon generation rate λstray photons/s) is the limiting factor, then
measuring a blank solution with a measurement time T will result in an average response
of λstray T photons, neglecting the bias nbias or the number of sensitive elements M . This
is again a Poisson process: the variance of the blank measurements equals λstray T . The
44
√
strength of the absolute minimum detectable signal becomes Imin = λstray T + k λstray T .
Since the detection method is only based on counting photons, it is not possible to distinguish
between a photon from a stray light source or a photon coming from the fluorescence process
of interest. Measuring a fluorescent solution will result in an absolute average response of
(λ + λstray )T photons with a standard deviation equal to the square root of that number. The
coefficient of variation for a single measurement under limitation of stray light is
p
(λ + λstray )T
single
CV stray =
.
(4.14)
λT
In this expression the average signal is corrected for the average blank response. Repeating
the measurement or increasing the measurement time results in the same improvement of the
coefficient-of-variation as in the case of an ideal detector:
p
(λ + λstray )T
1
1
single
1×nT
√
√
,
(4.15)
CV
=
CV n×T
=
CV
=
stray
stray
stray
λT
n
n
This expression for the CV is independent of any detector parameters. Note that we only
include the deterministic component of the stray light.
The amount of stray light is already minimized by using high power objectives, dedicated
filter blocks and closing the field stop as far as possible. Another option to gain signal with
respect to stray light is to use dyes with a much higher quantum yield. This can also be
achieved by using other solvents. To avoid fast evaporation from open nanoliter reactors we
have used a mixture of glycerol / water in the past. Practically all dyes, however, have a
higher quantum yield in an aqueous solution. The massive parallel coarse filling procedure
covers the vials and allows for using the appropriate buffer (aqueous solution) as solvent. This
will result in a better quantum yield. Stray light could be avoided completely by detecting
luminescence instead of fluorescence, where luminescence light is emitted in the absence of
excitation light. Furthermore, the limiting factor of stray light can be overcome by exploiting
other techniques, such as Fluorescence (Cross-)Correlation Spectroscopy [12]. With this
technique the fluorescence signal from a confocal volume is detected as a function of time
and the correlation spectrum is computed. This technique is limited by Raman scattering of
the solvent. The dynamic range of these techniques is from 10−9 M down to 10−12 M [12].
In principle, lower concentrations can be detected with this technique, but the measurement
time is inversely proportional to the concentration of the fluorescent dye.
4.5 MINIMUM DETECTABLE SIGNAL FOR DIFFERENT CONFIGURATIONS
We will now compute the strength of the minimum detectable signal for different configurations
of uor system in the case of readout noise limited detection. These configurations differ in the
optical emission path and in the type of detector. The excitation part of all configurations is the
same. This implies that the generated emission photon flux is the same for all configurations.
The area of each vial on the microarray is 200×200µm2 . From these examples we can predict
the improvement of the detection limit for our new configuration, if readout noise limits the
detection.
MODIFICATIONS TO THE MICROSCOPE SYSTEM
45
1. The configuration used for the first experiments consisted of a Zeiss 20 × /0.75 FLUAR
objective, a 1× camera mount, and the Photometrics Series 200 CCD camera (no binning). The vial of a microarray is imaged onto M = 588×588 ≈ 3.5×105 pixels (pixel
size = 6.8 × 6.8µm2 ). The readout noise√per pixel is σr = 11.7e− . This translates to a
minimum detectable signal of Imin = k M σr = 2.1 × 104 e− with k = 3. NAD(P)H
emits around 450nm. The Quantum Efficiency of this CCD camera is approximately
15% at this wavelength. The minimum detectable signal converts then to 1.4 × 105
photons. The CV of this measurement is 5%.
2. The new configuration consists of a 20×/0.75 objective. The tube lens is replaced by an
inverted 5× objective. The overall magnification is 4×. The Photometrics (no binning)
is replaced by the Princeton Versarray back illuminated CCD camera (no binning). The
vial will be imaged onto M = 33×33 ≈ 1.1×103 pixels. The readout
noise per pixel is
√
σr = 12.4e− . This translates to a minimum detectable signal k M σr = 1.2 × 103 e− .
That is equivalent to 1.8 × 103 photons (QE ≈ 70%@450nm) without binning. In
theory this will translate to a detection limit which is a factor of 80 lower. The CV of
this measurement is 24%
3. Another interesting configuration is the following: the optical configuration remains
identical, but instead of the Princeton Versarray CCD camera, a photo multiplier tube
is mounted on the microscope. The limiting factor for a PMT in photon counting mode
is the dark current. The dark current is expressed in Equivalent Noise Input (ENI). A
typical range for the ENI is 10−15 − 10−16 W . This translates to 0.3 to 3 thousand
photons per second [13]. For short integration times the minimum detectable signal for
a PMT is on the same order of magnitude as for the new configuration.
Experiments to be presented in the next chapter will show that our new configuration is,
unfortunately, still operating under the limitation of stray light. The new configuration does not
result in an improved detection limit, but due to the better collection efficiency, the fluorescence
signals can be acquired much faster. This translates to a higher readout rate of the microarray.
4.6
Figure 4.4 shows the modifications to our microscope system in terms of the excitation and
emission light beams. Figure 4.5 shows photographs of the new configuration. The modifications consist of the following:
1. The amount of stray light is already minimized by closing the field stop as far as
possible, such that only (a fraction of) a single vial is illuminated. Note, that this has
no effect on the excitation power: only that excitation light is blocked, that would not
lead to excitation anyway. We have chosen a (relatively) high magnification objective
(20 × /0.75), such that the excitation power per unit area is large. The field of view is
just a single vial. This implies a sequential scanning of the microarray.
2. The high NA objective implies a high collection efficiency. Furthermore, the trinocular
tube with the eyepieces and the relay optics (M = 1×) has been replaced by an inverted
46
Excitation part
Collector
lens
Tube lens
Emission part
20x/0.75
20x/0.75
Condenser Fluorescing
sample
Detection part
Inverted
5x/0.25
on-chip
binning
200µm
Readout
Hg-lamp
WD
Field stop Excitation
filter
Object
plane
Emission
filter
CCD-chip
Shutter
(WD=9.3mm)
Fig. 4.4 The excitation and emission beams of our modified conventional microscope system. The
closed field stop minimizes the influence of stray light. The 20×/0.75 objectives ensures large excitation
power per unit area, and high collection efficiency. The inverted 5 × /0.25 objective enlarges the photon
flux by a factor of 25. The CCD camera with large pixels (and on-chip binning) implies a high sensitivity
at the expense of spatial resolution.
5×/0.25 Zeiss FLUAR objective. This enlarges the emission photon flux in photons per
unit area by a factor of 25: we are not interested in spatial resolution, only in sensitivity.
• Both objectives are infinity-corrected objectives. This means that the exit bundle
of the objectives is a parallel bundle. Placing the infinity-corrected objectives
back-to-back creates a proper imaging system.
• The size of the back focal plane of the 5× objective is a little smaller than the
extent of the tube lens. The consequence is that the effective NA of the system is
a little smaller.
• Another consequence of the smaller extent of the 5× objective is vignetting [14].
This can be avoided by closing the field stop far enough.
• In principal, instead of a 5× objective, an identical 20× objective could be used.
Because of the distance between the mechanical shutter and the CCD array, ≈
1cm, a very large working distance is necessary to make a sharp image. The 5×
objective has a working distance WD = 9.3mm.
3. The scientific grade CCD camera with small pixels is replaced by a CCD camera with
large pixels. This implies that only one tenth, the ratio of the pixel sizes, of the photon
flux is necessary to get the same increase in signal level per pixel, given that both cameras
have the same overall conversion factor from photons to gray level. This gain can be
further increased by using on-chip binning. This gain in signal at reduced readout noise
level implies a better SNR.
47
Princeton Versarray 512B
Back Illuminated CCD
INVERTED Zeiss FLUAR
5x / 0.25 WD = 9.3 mm
Zeiss Axioskop
Zeiss FLUAR
20x / 0.75
(a)
Nikon F to C mount
adapter
Tube for INVERTED
Zeiss FLUAR 5x / 0.25
with Nikon F-mount
(Replaces trinocular tube)
Zeiss FLUAR 20x / 0.75
(b)
Fig. 4.5 The upper photo shows the microscope: the trinocular tube is removed and replaced by an
inverted 5 × /0.25 Zeiss FLUAR objective. A dedicated mount connects the Princeton Versarray CCD
camera to the microscope. The lower photo shows the camera mount and inverted objective in detail.
48
4.7 CONCLUSIONS AND DISCUSSION
This chapter discusses the measurement of absolute fluorescence signals in a conventional microscope system equipped with a scientific grade CCD camera. Two limiting factors for the
detection limit can be distinguished: stray light originating in the excitation part of the microscope system or readout noise in the photon detection part of the system. This chapter showed
possible modifications to a conventional microscope system to improve the sensitivity in terms
of the detection limit. This chapter showed that measuring very low levels of fluorescence is
at the expense of spatial resolution. Measuring low levels of fluorescence requires multiple
measurements in order to achieve a sufficient CV . The modifications to our microscope system consisted of a lower overall magnification by replacing the tube lens in the microscope
with an inverted 5× objective and replacing the detector with a new CCD camera with large
pixels and a higher Quantum Efficiency. In the case of readout noise limited detection, these
modifications should result in a detection limit which is about two orders of magnitude better
than our unmodified configuration. In the case of stray light limited detection, these modifications result in a faster scanning repetition rate of the microarray. The next chapter will show
the performance of our modified microarray reader, and will demonstrate the functionality of
this system by monitoring NADH consuming and producing enzyme-catalyzed reactions in
our microarrays.
5
Monitoring Enzyme-catalyzed
Reactions in Microarrays
This chapter presents results of a number of experiments that have been performed with
the microarray reader described in the previous chapter. First, we show that the modified
optical system produces sharp images. Secondly, we determine the dynamic range of our
detection system for the fluorescent molecule NADH. Furthermore, we have monitored one
NADH producing and one NADH consuming enzyme-catalyzed reaction in our microarrays.
The experimental results demonstrate that our microarray reader can be used to monitor all
the reactions in the glycolytic pathway. In the end, this system can be used as an analytical
tool to gain insight in the metabolic regulation of the glycolytic pathway.
5.1
IMAGING WITH THE MODIFIED OPTICAL SYSTEM.
First, we will show that our modified microscope system produces sharp images. Note that
we are not interested in spatial resolution: replacing the tube lens with an inverted 5 × /0.25
FLUAR objective introduces aberrations. The inverted 5× objective demagnifies the image.
Therefore, ordinary test patterns for microscopy are not suitable. As a simple test pattern a
series of lines each with a thickness of 0.1mm and 5 divisions per millimeter (line-to-line
spacing is 0.2mm) was printed on plain paper (80gr /m2 ) by a HP laser jet 5m at 600dpi .
This test pattern was placed under the microscope and observed with trans-illumination. Figure 5.1(a) shows an image of this test pattern acquired with the modified microscope system.
As can be seen in Figure 5.1(a) the field-of-view of the camera is much larger than the fieldof-view of the optical system. Figure 5.1(b) shows the intensities along a line in the image of
Figure 5.1(a). From this figure we conclude the following:
49
50
MONITORING ENZYME-CATALYZED REACTIONS IN MICROARRAYS
• Practically no photons reach the detector in the area outside the field-of-view of the
optical system.
• The field-of-view is not homogeneously illuminated. This non-homogeneity is also
present in an unmodified microscope system adjusted to Köhler illumination [15].
• A rough estimation of the sampling density follows from the distance between the two
outer minima in Figure 5.1(b). The physical distance is 1.2mm. In the image this corresponds to a distance of 109 pixels. The sampling density is approximately 91mm −1 .
Given the pitch of 24µm, it follows that the overall magnification of the modified optical
system is approximately 2.2×. One would expect an overall magnification of 4×. The
working distance of the 5× objective is "only" 9.3mm. The distance between the mechanical shutter of the CCD camera and the CCD array itself is larger than this working
distance. According to Equation 3.1, the overall magnification in an infinity-corrected
optical system was defined as the ratio of the focal lengths of the two lenses in the
system. Our system effectively uses longer focal lengths. This amounts to a smaller
ratio, and thus a lower overall magnification.
• The diameter of the field-of-view of the optical system corresponds to 181 pixels or
2.1mm.
5.2
NADH CALIBRATION
As mentioned in Chapter 2 all enzymatic reactions of the glycolytic pathway can be monitored
via the production or consumption of the fluorescent molecule NADH. In some reactions the
enzyme activity is measured directly via NADH, in other reactions it is measured via two
coupled reactions. An example of a direct reaction is the reaction in Expression 2.1, the
reactions in Expression 2.2 are an example of coupled reactions. First, we have performed
experiments to determine the dynamic range of NADH concentrations that we can detect with
our microarray reader. The detection system we have used in the experiments is the microarray
reader described in Section 4.6 with a 20 × /0.75 FLUAR objective and a 5 × /0.25 FLUAR
objective placed back-to-back, both from Zeiss. Filterset 01 from Zeiss (excitation: BP 365
/ 12, beam splitter: FT 395, and emission: LP 397). The Princeton Versarray 512B CCD
camera is mounted on the microscope with an in-house built camera mount. Solutions of
NADH were prepared by serial dilution in Millipore demineralized water. A microarray with
vials with dimensions 200 × 200 × 20µm3 is filled with one of the NADH solutions using the
massive parallel filling technique described in Section 3.2. With this technique a microscope
slide is shifted over the microarray to avoid fast evaporation; this slide serves as a cover slip.
For the experiments we used a microscope slide cut from a Pyrex wafer. The thickness of
this microscope slide is 0.5mm. This material has a transmittance of almost 95% for light
with a wavelength of 360nm. As a result, reflections of the excitation light are much less on
this material than on an ordinary microscope slide. All 25 vials of the microarray are read
sequentially. In order to minimize the contribution of stray light the field stop is closed as
far as possible. As a result only a fraction of the vial is illuminated. Figure 5.2 shows an
NADH CALIBRATION
51
Intensity in %
(a)
100
80
60
40
20
100
200
300
400
500
line position in pixels
(b)
Fig. 5.1 The upper image shows an image of a test pattern with lines (line-to-line distance is 0.2mm).
The field-of-view of the camera is much larger than the field-of-view of the optical system. The bottom
graph shows the intensities along a line in the upper image.
52
400µm
Fig. 5.2 This image shows a vial of a microarray filled with an NADH solution. The spot of the
emission light is smaller than the extend of the vial. Trans-illumination is used to make the edges of the
vial visible.
image of a single vial etched in silicon with dimensions of 400 × 400µm2 . For acquisition
of this image epi-illumination as well as trans-illumination is used. The silicon wafer is not
completely opaque, but a little transparent. The light from the halogen lamp makes the edges
of the vial visible and also causes the bright spot in the left region. The bright spot in the vial
is the emission light generated by the NADH molecules in the vial. It is clear that the size of
the illumination spot is smaller than the size of the vial.
Figure 5.3(a) shows the results of these measurements. The data points are the average
intensities of the 25 vials, the error bars correspond to a 3σ-interval. From this graph, we
conclude that there is a linear response for NADH concentrations up to a concentration of
1mMolar NADH. The gain setting of the CCD camera in this experiments was set to low ( 21 ×).
The dynamic range of the CCD camera with this gain setting is approximately 4×104 #ADU .
From the graph in Figure 5.3(a) it is not possible to determine the detection limit for NADH.
Figure 5.3(b) shows the same data on a logarithmic scale. The average response of the blank
solution is subtracted from the other data points. The minimum detectable signal is drawn in
this graph as three times the standard deviation of the blank solution. The minimum detectable
signal crosses the calibration curve at a concentration of 5µMolar . This NADH concentration
is the detection limit of our microarray reader. The dynamic range of NADH ranges from
5µMolar to 1mMolar . As expected, the minimum detectable signal is far above the readout
noise of the detector. We conclude that the microarray reader is operating under limitation of
stray light.
Detector response [1000 #ADUs]
NADH CALIBRATION
30
53
Volume vial = 200 x 200 x 20 µm3
Exposuretime = 0.1 s (no Binning, 1x gain)
Cover slide = Pyrex
±3σ−interval
25
20
15
10
5
200
400
600
800
Concentration NADH [µMolar]
Detector response [1000 #ADUs]
(a)
10
Volume vial = 200 x 200 x 20 µm3
Exposuretime = 0.1 s (no Binning, 1x gain)
Cover slide = Pyrex
Datapoints: averaged over 25 different vials
1
±3σ- interval
minimum detectable signal
0.1
3σ-blank
5
10
Detection
limit = 5 µMolar
50
100
500
1000
Concentration NADH [µMolar]
(b)
Fig. 5.3 The upper graph shows the calibration for NADH measured with our microarray reader. The
lower graph shows the same calibration curve on a logarithmic scale. The detection limit for NADH is
5µMolar .
54
5.3 MONITORING ENZYME-CATALYZED REACTIONS
Glucose-6-phosphate-dehydrogenase (G6P-DH). The following enzyme-catalyzed reaction has been monitored in our microarrays:
G6P + NAD
G6P−DH
→
Gluconolactone + NADH,
(5.1)
where G6P is glucose-6-phosphate, and G6P-DH is the enzyme glucose-6-phosphate-dehydrogenase.
In this reaction the fluorescent molecule NADH is produced.
• A cocktail of G6P, NAD, and activator MgCl2 , containing equal concentrations of G6P
and NAD is prepared. The initial concentration is 2mM . The concentration MgCl2 in
the cocktail is 5mM .
• Solutions with different enzyme concentrations of G6P-DH are prepared: 0.4U/ml,
0.2U/ml, 0.1U/ml, and 0.05U/ml. An enzyme concentration of 1U/ml equals an
amount of enzyme molecules that convert 1µMol of substrate in 1ml solution per
minute of time.
• The reaction will be initialized by mixing two equal volumes of the cocktail and the
enzyme solution. After mixing, the substrate concentrations as well as the enzyme
concentrations are reduced by a factor of 2. After bulk mixing, ≈ 5µl will be spread
over the microarray. The microarray is shifted under a Pyrex microscope slide. After
attachment, the microarray and the microscope slide are pressed firmly together for
about half a minute.
• Reading out the microarray will start approximately two minutes after initialization of
the reaction. A single scan of the 25 vial array takes on the order of half a minute. The
exposure time / integration time per vial is 0.1s. The microarray will be continuously
readout for a chosen number of scans.
• The raw results consists of an average intensity in #ADU measured in an image In out
of an image sequence n = 1, 2, · · · , N . The image sequence, as well as the time are
stored.
Figure 5.4 shows the acquired image sequence of a single vial during the reaction. This image
sequence consists of 100 images. Figure 5.5 shows an acquired image sequence of a single
vial in which evaporation occurs during the reaction. The latter measurement sequence is not
used in the analysis.
We assume that the enzyme-catalyzed reaction is performed under pseudo-first-order conditions. This means that the following equation describes the NADH concentration:
t
[NADH(t)] = [NAD(0)] 1 − exp −
.
(5.2)
τ
At t = 0 no NADH has been produced and as t → ∞ all NAD is converted to NADH and
the concentration NADH at the end-point of the reaction equals the initial concentration of
MONITORING ENZYME-CATALYZED REACTIONS
Fig. 5.4
reaction.
55
This figure shows an acquired image sequence of a single vial during an enzyme-catalyzed
Fig. 5.5 This figure shows an acquired image sequence of a single vial in which evaporation occurs
during an enzyme-catalyzed reaction.
40
40
40
40
40
20
20
20
20
20
intensity in 1000 #ADUs
56
0
40
20
0
40
20
0
40
20
0
1484
0
5000 0
40
0
0
5000 0
40
1474
0
5000 0
40
0
1529
0
5000 0
40
0
5000 0
40
0
-
0
5000 0
40
0
5000 0
40
0
20
0
0
5000 0
1490
0
5000 0
40
0
5000 0
40
0
5000 0
40
0
5000 0
0
5000 0
5000
20
1628
1643
0
5000 0
40
5000
20
1580
1553
0
5000 0
40
20
1522
1642
0
5000 0
40
20
1554
20
1511
1646
20
20
5000
20
1501
20
1297
0
5000 0
40
20
1718
1548
1594
0
5000 0
40
20
20
1590
20
1542
0
5000 0
40
20
1633
20
40
0
1614
5000
5000
20
-
1607
0
5000 0
time in s
Fig. 5.6 This figure shows for 5 × 5 vials the acquired signals of the above mentioned reaction with
an enzyme concentration of 0.025U/ml and the fitted pseudo-first-order reaction model. The italic
numbers in the graphs are the values of the characteristic time τ in seconds.
NAD: [NADH(∞)] = [NAD(0)]. Figure 5.6 shows for 5 × 5 vials the acquired signals
of the above mentioned reaction with an enzyme concentration of 0.025U/ml and the fitted
first-order reaction model. A nonlinear fit procedure (Matlab) using the Levenberg-Marquardt
optimization method has been used to fit this model to the data. Two extra parameters have
been added to the model in Equation 5.2: a vertical offset and a time shift t0 . The italic numbers
in the graphs are the estimated characteristic times τ for this reaction. This characteristic time
τ indicates the amount of time in which the concentration NAD is reduced by a factor e: in
a time τ after initialization of the reaction the NAD concentration is only 37% of the initial
NAD concentration, and after 3τ , the NAD concentration is 5% of the initial concentration,
and the NADH concentration has reached a level of 95% of the end-point of the reaction.
The following four tables (Table 5.1-5.4) give the estimated characteristic times τ in 5 × 5
vials of a microarray. For all four experiments the same microarray has been used. After
each experiment the microarray is cleaned to avoid cross-contamination. The final enzyme
concentrations in the reactions are 0.025U/ml, 0.05U/ml, 0.10U/ml, and 0.20U/ml. The
numbers between brackets are the values of the χ2 -merit function of the nonlinear fit procedure.
Table 5.5 summarizes the experimental results from the previous four tables. The graph in
Figure 5.7 shows the values of the reciprocal of the measured characteristic times, τ1 as a
57
function of the enzyme concentration [G6P-DH]. This graph shows that our measurements
correspond to the expected behavior of this reaction.
Table 5.1 This table shows the measured characteristic times τ in s for the reaction in Equation 5.1.
The enzyme concentration is 0.025U/ml. The numbers betweens between brackets are the values of
the χ2 -merit function of the nonlinear fit.
1484 (39.9)
1474 (42.2)
1542 (45.1)
1718 (65.1)
1297 (39.3)
1614 (56.2)
1529 (71.2)
1501 (33.8)
1511 (49.7)
1633 (70.1)
1548 (46.1)
1490 (66.1)
1554 (59.9)
1521 (33.6)
1590 (34.3)
1646 (91.6)
1628 (41.8)
1553 (32.5)
1606 (65.5)
1594 (45.2)
1642 (79.3)
1643 (63.0)
1580 (53.2)
-
The enzyme concentration is 0.05U/ml. The numbers betweens between brackets are the values of the
χ2 -merit function of the nonlinear fit.
1133 (6.8)
1181 (5.4)
1169 (7.2)
1223 (9.7)
1146 (8.6)
1149 (9.9)
1265 (8.1)
1140 (11.4)
1231 (8.1)
1242 (5.7)
1090 (10.5)
1234 (7.0)
1204 (10.9)
1033 (8.7)
1196 (14.3)
1133 (9.2)
1263 (9.3)
1226 (12.9)
1241 (13.9)
1147 (13.1)
1229 (13.8)
1203 (8.2)
1203 (12.0)
1171 (12.7)
1136 (11.2)
χ2 -merit function of the nonlinear fit. Measurements are shown in Figure 5.8.
473 (15.8)
481 (13.4)
525 (8.5)
-
693 (7.0)
723 (9.3)
708 (8.4)
578 (11.5)
641 (7.1)
1045 (6.8)
947 (13.23)
803 (7.5)
744 (7.7)
1166 (5.8)
814 (7.3)
1328 (7.9)
1257 (6.2)
789 (10.5)
58
351 (5.8)
363 (11.2)
458 (40.5)
-
346 (7.0)
355 (10.6)
369 (8.3)
333 (34.8)
375 (6.5)
362 (6.0)
311 (9.7)
342 (5.0)
337 (12.1)
485 (49.4)
378 (9.6)
358 (9.8)
359 (13.9)
334 (13.4)
183 (23.7)
374 (6.3)
351 (5.8)
396 (12.5)
267 (19.7)
209 (23.8)
Table 5.5 This table summarizes the results from the tables above: the first column gives the enzyme
concentration, the second column gives the number of monitored reactions in a 5 × 5 microarray, the
third column gives the mean characteristic time µτ , the fourth column gives the standard deviation of the
measured characteristic times στ , and the last column gives the coefficient of variation of the experiment
CV τ
1/τ [s-1]
Enzyme conc.
U/ml
0.025
0.05
0.1
0.2
reactions
(out of 25)
23
25
17
23
µτ
s
1561
1183
807
348
στ
s
85
56
261
65
CV τ
%
5.5
4.8
32
19
0.003
0.0025
0.002
0.0015
0.001
0.0005
0.05
0.1
0.15
0.2
0.25
G6P-DH enzyme concentration [U/ml]
Fig. 5.7 This graph shows the reciprocal of the measured characteristic times,
enzyme concentration [G6P-DH].
1
τ
as a function of the
40
40
40
20
20
20
0
40
30
20
40
30
20
473
0
0
4000 0
40
2000
0
0
2000
0
4000 0
40
2000
0
2000
0
4000 0
2000
20
2000
0
4000 0
0
4000 0
4000
2000
4000
2000
4000
1257
0
4000 0
40
2000
2000
20
0
4000 0
40
20
744
2000
2000
20
20
641
4000
20
0
4000 0
40
20
0
4000 0
40
20
0
578
0
4000 0
40
2000
2000
1166
0
2000 4000 0
40
803
20
10
20
0
4000 0
40
2000
5
1328
0
4000 0
40
2000
0
4000 0
20
2000
708
0
4000 0
40
2000
20
947
20
10
20
525
20
0
20
0
4000 0
40
2000
40
723
0
4000 0
40
2000
0
4000 0
40
481
0
2000
20
1045
693
0
4000 0
40
20
5
2000
59
814
2000
0
4000 0
2000
789
4000
time in s
Fig. 5.8 This figure shows for 5 × 5 vials the acquired signals of the above mentioned reaction with an
enzyme concentration of 0.10U/ml and the fitted pseudo-first-order reaction model. The italic numbers
in the graphs are the values of the characteristic time τ in seconds. Evaporation corrupted the experiment
in 8 out of 25 vials.
60
Lactate-dehydrogenase (LDH). The following enzyme-catalyzed reaction has also been
monitored in our microarrays:
LDH
pyruvate + NADH → lactate + NAD,
(5.3)
where LDH is the enzyme lactate-dehydrogenase. In this reaction the fluorescent molecule
NADH is consumed.
• A cocktail of pyruvate, and NADH, containing equal concentrations of pyruvate and
NADH is prepared. The initial concentration is 2mM .
• A stock solution with an enzyme concentration of 0.6U /ml is used to prepare solutions
with different enzyme concentrations of LDH. The table below shows the amounts of
cocktail solution, enzyme solution and H2 O that are added together to initialize the
above reaction.
Table 5.6 This table shows the amounts of cocktail solution, enzyme solution, and H2 O that are added
together to initialize the enzyme-catalyzed reaction. The enzyme concentration in the stock solution is
0.6U /ml .
cocktail
µl
50
50
100
100
H2 O
µl
16.7
33.3
83.3
91.2
enzyme
µl
33.3
16.7
16.7
8.4
final enzyme conc.
U/ml
0.2
0.1
0.05
0.025
• After bulk mixing, ≈ 20µl will be spread over a 2cm long region along the long edge
of a Pyrex microscope slide. The microarray is shifted under the wetted part of the
Pyrex microscope slide. After attachment, the microarray and the microscope slide are
pressed firmly together for about half a minute.
• Reading out the microarray will start approximately three minutes after initialization
of the reaction. A single scan of the 25 vials of the array takes on the order of half a
minute. The exposure time / integration time per vial is 0.1s.
• The raw results consists of an average intensity in #ADU measured in an image In out
of an image sequence n = 1, 2, · · · , N . The image sequence, as well as the time are
stored.
We again assume that the enzyme-catalyzed reaction is performed under pseudo-first-order
conditions. This means that the following equation yields for the NADH concentration:
t
[NADH(t)] = [NADH(0)] exp −
.
(5.4)
τ
50
0
0
638
0
0
0
2000 0
50
1000
681
0
0
0
0
0
2000 0
50
1000
517
0
2000 0
50
1000
546
0
2000 0
50
1000
0
1000
0
2000 0
1000
0
2000 0
1000
1000
0
2000 0
2000
1000
641
0
2000 0
50
1000
0
2000 0
40
1000
2000
536
500
1000
572
2000
707
2000
469
0
1000
682
739
627
0
2000 0
50
1000
50
0
2000 0
50
1000
498
0
2000 0
50
1000
50
0
2000 0
50
1000
503
749
589
0
2000 0
50
1000
495
0
2000 0
50
1000
611
0
2000 0
50
1000
50
638
0
2000 0
50
1000
558
50
758
50
0
2000 0
50
1000
50
50
764
0
2000 0
50
1000
50
718
61
1000
588
20
0
2000 0
1000
2000
time in s
Fig. 5.9 This figure shows for 5 × 5 vials the acquired signals of the above mentioned reaction with an
enzyme concentration of 0.1U /ml and the fitted pseudo-first-order reaction model. The italic numbers
in the graphs are the values of the characteristic time τ in seconds.
At t = 0 no NADH has been consumed and as t → ∞ all NADH is consumed: [N ADH(∞)] =
0. Figure 5.9 shows for 5 × 5 vials the acquired signals of the above mentioned reaction with
an enzyme concentration of 0.1U/ml and the fitted first-order reaction model. One extra
parameter to incorporate a vertical offset has been added to the model in Equation 5.4. The
italic numbers in the graphs are the estimated characteristic times τ for this reaction.
The following four tables (Table 5.7-5.10) give the estimated characteristic times τ in 5 × 5
vials of a microarray. For all four experiments the same microarray has been used. After each
experiment the microarray has been cleaned. The final enzyme concentrations in the reactions
are 0.025U/ml, 0.05U/ml, 0.10U/ml, and 0.20U/ml. The numbers between brackets are
the values of the χ2 -merit function of the nonlinear fit procedure.
Table 5.11 summarizes the experimental results from the previous four tables.
The graph in Figure 5.10 shows the values of the reciprocal of the measured characteristic
times, τ1 as a function of the enzyme concentration [LDH].
62
The enzyme concentration is 0.025U/ml. The numbers betweens between brackets are the values of
the χ2 -merit function of the nonlinear fit.
1466 (78.7)
1484 (80.7)
1154 (91.9)
1192 (80.8)
1210 (85.1)
1732 (140)
1363 (77.0)
1081 (124)
1169 (135)
1240 (102)
1383 (95.1)
1235 (61.2)
1069 (84.6)
1132 (66.4)
1381 (89.4)
1381 (105)
1248 (69.9)
1215 (63.4)
924 (81.5)
1133 (81.8)
1403 (69.6)
1491 (66.0)
1488 (53.3)
993 (32.5)
926 (27.6)
953 (26.8)
936 (33.1)
1073 (26.6)
915 (33.6)
941 (23.9)
889 (27.8)
941 (30.3)
1069 (23.8)
990 (26.9)
956 (31.4)
822 (30.1)
942 (28.1)
1040 (22.8)
951 (35.8)
958 (23.6)
950 (33.1)
925 (29.7)
1114 (30.1)
1070 (27.4)
862 (36.6)
894 (36.6)
1024 (25.6)
1362 (57.6)
718 (8.1)
638 (9.8)
558 (19.0)
495 (19.9)
517 (22.8)
764 (10.4)
681 (11.1)
611 (11.5)
503 (23.5)
546 (17.0)
758 (11.9)
638 (23.5)
589 (15.0)
498 (22.1)
572 (20.0)
749 (10.2)
682 (8.7)
627 (9.3)
500 (23.1)
469 (37.5)
739 (12.4)
707 (8.6)
641 (14.7)
536 (22.2)
588 (13.7)
245 (11.3)
247 (18.8)
250 (19.0)
217 (15.7)
212 (19.0)
201 (33.7)
-
193 (25.9)
183 (30.9)
172 (13.0)
175 (17.3)
178 (21.4)
-
63
Table 5.11 This table summarizes the results from the above tables: the first column gives the enzyme
concentration, the second column gives the number of monitored reactions in a 5 × 5 microarray, the
third column gives the mean characteristic time µτ , the fourth column gives the standard deviation of the
measured characteristic times στ , and the last column gives the coefficient of variation of the experiment
CV τ
1/τ [s-1]
Enzyme conc.
U/ml
0.025
0.05
0.1
0.2
reactions
(out of 25)
23
25
25
10
µτ
s
1286
980
613
206
στ
s
181
106
93
31
CV τ
%
14.1
10.8
15.2
15.2
0.006
0.005
0.004
0.003
0.002
0.001
0.05
0.1
0.15
0.2
0.25
LDH enzyme concentration [U/ml]
Fig. 5.10 This graph shows the reciprocal of the measured characteristic times,
enzyme concentration [LDH].
1
τ
as a function of the
64
5.4 CONCLUSIONS
We have shown that our modified microscope system produces sharp images. The overall
magnification is approximately two times. By closing the field stop as far as possible, the
amount of stray light is minimized and only a fraction of the vial is illuminated. The dynamic
range of our system for NADH ranges from 5µM to 1mM . Our system has a linear response
in this range. To demonstrate the functionality of our system, we monitored two enzymecatalyzed reactions at different conversion rates. One reaction produces NADH, and one
reaction consumes NADH. The enzyme concentrations are given as enzyme activity levels:
identical enzyme concentrations for two different enzymes amount to identical conversion
rates, regardless if the reaction consumes or produces NADH. For this reason it was expected
that the curves in Figure 5.7 and Figure 5.10 would have the same slope. Inspection of these
curves shows that this is not the case. Possible causes for this deviation are that (a) the stock
solution of one of the enzymes contained a different concentration, or (b) the difference in
temperature in which the reactions were monitored. The reaction with G6P-DH was monitored
at ≈ 30◦ C, and the reaction with LDH at ≈ 26◦ C. We have used a pseudo-first-order kinetic
model to describe the reaction. Although this model describes our experimental data very well,
it is, chemical speaking, perhaps not the right model. Our model assumes that the reactions
reach an end-point, where all NADH is either produced or consumed. Other chemical models
do not assume a "one-way" reaction, but a reversible ("two-way") enzyme-catalyzed reaction:
in that case the reaction reaches an equilibrium. A study to other chemical models falls outside
the scope of this thesis.
Part II
Monitoring Dynamic
Liquid Behavior in
Micromachined Vials
6
Dynamic Liquid Behavior
in Micromachined Vials
Evaporation is a key problem in the development of microarray technology, especially in
the case of microarrays with open reactors. In order to establish completed (bio)-chemical
reactions and to be assured of sufficient readout time for acquiring all fluorescent signals from
all vials on a microarray, the evaporation process should be extended as long as possible.
In previous chapters successful solutions to overcome the problem of evaporation have been
demonstrated. In this part of this thesis we will look in close detail at the phenomenon
of evaporation in micromachined subnanoliter vials. We will not only describe the process
of evaporation qualitatively in terms of some underlying physical principles, we will also
introduce a microscopic technique, interference-contrast microscopy, that allows quantitative
monitoring of the evaporation process. First, we will look from different points of view to the
process of evaporation to motivate this research.
6.1
6.1.1
INTRODUCTION
Liquid pinning: ring stains from coffee droplets
One would expect that the evaporation process of liquid droplets was described many years ago.
On the contrary, just recently some underlying principles of the dynamic process of evaporation
have been published, e.g. by Deegan et al in Nature [16] and in Physical Review [17].
Figure 6.1 shows the drying of a coffee droplet. Initially, the coffee particles were homogeneously distributed in the droplet, but during evaporation they become more and more
concentrated at the edge, leaving a ring like stain. These rings are also found after drying
of spotted DNA materials on microarray structures for Comparative Genomic Hybridization
71
72
DYNAMIC LIQUID BEHAVIOR IN MICROMACHINED VIALS
"The thrill of the spill"
(New Scientist) [18]
(a) t = 0min
(b) t = 10min
(c) t = 20min
(d) t = 30min
(e) t = 40min
(f) t = 50min
Fig. 6.1 This figure shows six images of a drying coffee droplet. It can be seen that the droplet is
pinned to its initial position, and that it leaves a ring-like stain after drying.
(CGH). According to Deegan, ring formation as shown in Figure 6.1 occurs if and only if the
following two conditions are satisfied:
1. the contact line of the drop is pinned to its initial position, and
2. evaporation takes place at the edge of the drop.
On smooth teflon, no pinning of the contact line of the droplet occurs, and the drying droplet
will contract: no ring will be formed after drying. Covering the drop with a lit with a small hole
over the center, reduces the evaporation at the edge. Although there is contact line pinning,
the deposit will be uniformly distributed, due to the reduced evaporation at the edge.
Contact line pinning. In a static situation, without evaporation, the contact angle between
the solvent and the surface on which the droplet lies, is determined by the balance between the
surface tension of the solvent, which tries to pull the droplet into a sphere, and the adhesive
forces, which try to attract the liquid to the surface. If the droplet lies on a perfectly flat and
smooth surface, the edges of an evaporating droplet have to move inwards to maintain the
required contact angle. This is shown in Figure 6.2(a). But real surfaces are not that smooth.
INTRODUCTION
73
(a) A contracting droplet on smooth surface.
(b) A pinned droplet.
Fig. 6.2 The top figure shows an evaporating droplet on a flat surface, which contracts to maintain the
required contact angle. The bottom figure shows a drying droplet pinned to its initial position. The loss
of liquid at the edge is replenished by liquid from the bulk of the droplet.
If the surface is just a little bit rough, when the edge loses some liquid, it only has to move a
little bit inwards to find a position, where the roughness of the surface gives it the right contact
angle again. This way, it moves so slowly inwards that it is effectively pinned in place. This
pinning occurs for many different solvents and many different surfaces.
Pinning on the underlying surface prevents the droplet from shrinking. This implies that the
footprint of the droplet remains constant. The evaporated liquid at the edge is replenished by
liquid from the bulk of the droplet. This means that there is a flow of liquid moving outwards
to the edge of the droplet. This is shown in Figure 6.2(b). The driving force of the outward
liquid flow is the elasticity of the surface, which tries to minimize the surface area. As the
solvent is evaporating at the edge, the outward flow of the solvent with particles, like in a
droplet of coffee, will deposit the particles at the edge, which make the surface even rougher.
As a result the edge is even more firmly pinned. Finally, when all the liquid has evaporated,
the (coffee) particles will form a ring of stain.
Deegan’s model of diffusion limited evaporation. Deegan states that the functional form
of the local evaporation rate along the air-liquid interface depends on the rate limiting step
of the evaporation process. If the rate-limiting step is the transfer rate across the liquidvapor boundary, then the local evaporation rate ~j(r) is constant. This implies that the total
evaporation rate is proportional to the area of the air-liquid interface. If the rate-limiting step
is the diffusive relaxation of the saturated vapor layer immediately above the drop, then ~j(r)
is strongly enhanced towards the edge of the drop. In this case the total evaporation rate is
proportional to a characteristic length of the droplet. Deegan concludes that the evaporation
process in his experiments is diffusion limited.
74
The diffusion goes from the saturated vapor layer (us ) immediately above the droplet to the
ambient vapor pressure (u∞ ). The evaporation process will rapidly attain this steady state.
The diffusion equation reduces then to Laplace’s equation:
∇2 u = D
∂u
~
= 0, ~j = −D∇u,
∂t
(6.1)
where u is the mass of vapor per unit volume, and D is the diffusion constant for vapor in air.
This boundary value problem has an electric analogon, if u is replaced by the electric potential
~ Futhermore, Deegan assumes that the pinned droplets
V and ~j(r) with the electric field E.
have the shape of a spherical cap during evaporation (with a constant footprint of radius R).
The solution to this boundary problem is well approximated by [17]
~j(r) = j0 f (λ) n̂, λ = π − 2Θc ≈ 1 ,
2 λ
2π − 2Θc
2
1 − Rr
(6.2)
where Θc is the contact angle of the droplet with the underlying surface, and n̂ the unit normal
vector of the air-liquid interface. From this expression follows that the rate of mass loss ~j(r)
diverges towards the edge of the droplet (r → R).
Section 6.2.1 will show observations of an evaporating Rhodamine solution in a micromachined vial. These observations show that the liquid is pinned in place to the edge of the vial
and that the fluorescent particles first move towards the edges of the vial and finally towards the
four corners of the square vial during the evaporation. This implies that there is evaporation
at the edge of the vial. These observations in micromachined vials are in agreement with the
observations of Deegan.
6.1.2
Modeling dynamic liquid behavior
In a collaborative research project with the Institute for Microsystem Technology of the Albert Ludwigs University Freiburg and the Electronic Instrumentation Laboratory of the Delft
University of Technology, Korvink et al have developed a physical model that describes the
evaporation process in micromachined vials in terms of the shape of the air-liquid interface
during evaporation, the local evaporation rate, and the flow field in the liquid. In this section
we will only summarize the approach to compute this phenomenon.
Qualitatively speaking, the following processes take place after injection of a liquid sample in
a vial. It is assumed that after injection the liquid "instantaneously" assumes its equilibrium
shape s(t). In the next instant a fraction of the liquid evaporates. According to Korvink, the
driving force of evaporation is the non-equilibrium that arises between the liquid phase and
the gaseous environment: if the partial pressure of the gas molecules is not at its equilibrium
value, it will lead to either evaporation or condensation. Korvink assumes that the current
density of molecules through the gas-liquid interface ~j is fully determined by the difference
of the equilibrium partial pressure ps in the saturated gas phase and the actual value of the
vapor pressure pd [19, 20]:
r
~j = √1 αc (ps − pd ) M n̂,
(6.3)
RT
2π
INTRODUCTION
75
where αc is the condensation coefficient of the material, M is the molar mass of the material,
R is the gas constant, and T is the absolute temperature. 1 The above equation is only valid for
a flat boundary separating two infinite half-spaces of the liquid phase and the gas phase. For a
convex, respectively a concave gas-liquid interface the vapour pressure increases respectively
decreases. The vapour pressure ps of the saturated gas above a curved surface is a function
of the mean curvature κ [19, 20]:
2mγlg
ps (κ) = ps (0) × exp
κ ,
(6.4)
kB T ρ
where ps (0) is the vapour pressure for a flat surface with curvature κ = 0, m is the molecular
mass, γlg is the liquid-gas surface tension, kB is Boltzmann’s constant, and ρ is the mass
density. This implies that the shape s(t) of the gas-liquid interface defines the mass flux
~j from the liquid across the meniscus to the gaseous environment. The mass flux ~j varies
with position, because of the space-varying shape s(t). In the same instant of evaporation,
the remaining liquid volume must compensate for its content loss by adjusting its surface
shape s(t)2 . This means that there is a liquid flow inside the liquid phase to replenish the
non-uniform loss of mass over the droplet surface.
In order to avoid solving for the fully-coupled evaporation, surface evolution, and fluid flow,
it is assumed that the surface evolution is sufficiently slow compared to the evaporation and
the fluid-flow. Choosing a time-step small enough effectively decouples the surface evolution
from the fluid flow. This leads to the following (simplified) algorithm:
1. Given the mass M (t) of the liquid in the vial, compute the fluid shape s(t) at instant t.
2. Given the fluid shape s(t), compute the mean curvature κ(t) of the surface elements.
The mean curvature is related the vapour pressure ps (κ) of the saturated gas above the
surface according to Equation 6.4. The vapour pressure ps (κ) in turn is related to the
mass flux density ~j according to Equation 6.33 .
3. Compute the space varying mass flux density ~j(r) on the liquid surface. Integrate ~j(r)
over the total surface S to compute the total mass change in the interval ∆t.
R
4. Compute the remaining mass of the liquid M (t + ∆t) = M (t) + ∆t ~jd~s.
5. Return to step 1, as long as boundary conditions are met.
The algorithm has two computationally intensive operations: the computation of the instantaneous surface shape s(t) from the Navier-Stokes equations, and the computation of the space
1 In
other words, Korvink assumes that the rate-limiting step of the evaporation is the transfer of molecules through
the interface. As explained in the previous section, Deegan assumes that the rate-limiting step of the evaporation is
the diffusion from the saturated vapor layer immediately above the air-liquid interface. Our experiments will prove
that Deegan’s assumption is right.
2 The air-liquid interface of a liquid sample in a circular shaped vial is a spherical cap as will be shown in Section9.4.
This means that the curvature κ is constant over the air-liquid interface. This implies a constant local evaporation
rate ~j(r) and the total evaporation rate will be proportional to R2 and varies as a function of time, when the meniscus
changes from convex to concave. Our experiments will show that this is not the case.
3 Since this model describes the interface transfer limited evaporation, the actual value of the vapor pressure p = 0
d
76
Fig. 6.3 This figure shows the initial surface s(0) of the liquid volume. This computation did not
include the boundary condition that the liquid was pinned to the edge of the vial.
varying mass flux ~j(s). Figure 6.3 shows the initial surface s(0) after injection of the liquid.
The computation of the surface s(0) did not use the boundary condition that the liquid was
pinned in place at the boundary of the vial.
Verification of the computational results based on the model above, especially the shape s(t)
of the air-liquid interface, requires quantitative measurements of the physical quantities of
interest. These quantities can be derived from microscopic image sequences acquired during
evaporation of the liquid in the vials.
6.1.3
Impedance-based liquid volume sensor
In the first part of this thesis, it has become clear that an electronic volume sensor is an
important sensor for a lab-on-a-chip. Such a sensor does not only indicate the presence of
liquid in a vial, moreover it measures the amount of liquid in the vial.
Silicon integrated microarrays make it possible to use an electrical method for detecting the
presence and amount of liquid in a vial. Conventional silicon fabrication techniques enable the
use of planar electrodes. This opens up the possibility for complete integration of the sensor
electrodes and the electronics for a dedicated electronic volume sensor. Figure 6.4 shows a
first generation of a microarray with integrated electrodes on the bottom of each of the twelve
different vials with sizes ranging from 100 × 100 × 6.0µm3 up to 500 × 500 × 6.0µm3
and volumes from 60pl to 1.5nl . When liquid is dispensed in the vials, the liquid alters the
impedance between the electrodes. The impedance can be correlated to the volume of the
liquid.
A schematic cross-section of such a reactor is shown in Figure 6.5. A silicon wafer is covered
with a 2µm thick layer of SiO2 . Aluminium electrodes with a height of 300nm are patterned
onto this layer. A 500nm thick cover of Six Ny is fabricated for electrical insulation. Finally,
the vials are formed in a 6.0µm thick layer of SiO2 . An aluminium layer was sputtered
on the backside of the wafer to form an Ohmic contact. Grounding of the Si wafer reduces
the effect of external electrical interferences, and minimizes parasitic current between the
electrodes through the silicon substrate. The electrical equivalent circuit of Figure 6.5 is
shown in Figure 6.6. In this figure Cf and Rf are the capacitance and the resistance of the
liquid. The current through the Si substrate is negligible, because of the small capacitance
and the grounding of this substrate. From Figure 6.6 follows that the capacitances CSix Ny ,
INTRODUCTION
77
Fig. 6.4 This photograph shows a first generation of a microarray with integrated electrodes at the
bottom of each of the twelve vials. The area of the array is 5 × 5mm 2 , the area of the different wells
varies from 100 × 100µm2 to 500 × 500µm2 . The bright areas within the dark rectangles and circle
are the aluminium electrodes.
the capacitance Cf , and the resistance Rf of the liquid contribute to the impedance Zx :
s
2
Rf2
2
|Zx | =
+
,
(6.5)
1 + ω 2 Cf2 Rf2
ωCSix Ny
where ω = 2πf , and f is the frequency of the AC current. With a floating impedance
method the impedance Zx can be measured. Such a method minimizes the contribution of
parasitic components. The coaxial cables used for connection are the main causes for parasitic
capacitances. An AC voltage source operating at 20kHz in combination with a current-tovoltage converter is used to measure the current.
Optical measurements of the remaining amount of liquid in the vial during evaporation can
be used for calibration of the impedance measurements of this electronic volume sensor. In
Section 9.5 we will show experimental results of this electronic volume sensor.
The previous sections illustrate the need for quantitative measurements especially of the shape
of the air-liquid interface during evaporation. These measurements should allow verification
78
Cf
SiO2polymer
liquid
CSixNy
Rf
left
electrode
CSiO2
SixNy
SiO2
CSi N
x y
right
electrode
CSiO2
RSi
Si
RSi
RSi
Al
Al-electrodes
Fig. 6.5 This figure shows a cross-section
through a vial with two aluminium electrodes patterned on the bottom.
Fig. 6.6 This figure shows the electric equivalent
of a vial equipped with two electrodes filled with
liquid.
of the underlying physical model. A straightforward measurement that follows from this
is the remaining liquid volume in the vial. The measurements of the liquid volume enable
calibration of an electronic liquid sensor. First, we will give a survey of microscope techniques
that could be used to perform the required measurements. In the following chapters we will
focus on a specific microscope technique in combination with a signal processing algorithm
that produces the required quantities of interest.
6.2
SURVEY OF MICROSCOPIC TECHNIQUES
Optical monitoring of the evaporation process in sub-nanoliter vials requires a type of microscopy that allows reconstruction of the shape of the liquid volume, especially from the
air-liquid interface, during evaporation. This implies that the three-dimensional volume needs
to be properly sampled in space (both lateral and axial) as well as in time.
6.2.1
Conventional Fluorescence Microscopy
Consider the case that a vial is filled with a homogeneous fluorescent solution. The fluorescent
signal of the particles indicate the presence of the liquid. The total fluorescent intensity is a
measure for the initial liquid volume in the vial. The total fluorescent intensity does not change,
however, during evaporation, and is therefore not a good measure. A second drawback is that
with a conventional wide-field microscope a three-dimensional volume is imaged onto a twodimensional image sensor and depth information is lost. It is not straightforward to retrieve the
three-dimensional shape of the liquid from these intensity differences. As a conclusion, video
microscopy with a conventional microscope fulfills only the temporal sampling requirement.
Figure 6.7 shows five instants of the evaporation proces of a Rhodamine solution in a square
vial. The left parts in Figure 6.7 show the recorded images, whereas the right parts show the
intensity distribution of the recorded images. At t = 0, just after injection of the liquid, the
signal from the vial is uniform, besides some geometrical effects along the sidewalls and in
(a) t = 0min
(b) t = 5min
(c) t = 10min
(d) t = 15min
79
(e) t = 20min
Fig. 6.7 A series of images showing the evaporation process of ethyleneglycol-water (90%/10%, v/v)
in a vial at different times. Note that the images show the fluorescent intensity profile, not the shape of
the meniscus.
the corners of the vial. This implies that the meniscus of the liquid is virtually flat and the
vial is completely filled. After filling of the vial, the liquid is pinned to the edge of the vial.
During evaporation, the pinning of the liquid ensures that the evaporation from the edge is
replenished by liquid from the center part of the liquid in the vial [17]. This can be seen
qualitatively in Figure 6.7: the fluorescent signal gets weaker in the center, until the bottom
of the vial is reached. The signal disappears from that area at some instant between 10 and
15min. While the evaporation continues, the liquid is stuck in the four corners of the vial. At
these spots the concentration of fluorescing particles is furhter increasing. Finally, when the
solvent is completely evaporated, the particles loose their fluorescent behavior and the signal
disappears. In Figure 6.8 the average fluorescence signal from the entire vial and the signals
from the center, from one of the sidewalls and from one of the corners of the vial are shown
intensity in #ADUs
80
800
600
Intensity in corner
400
Average vial intensity
200
Intensity at
sidewall
Intensity in centre
5
10
15
20
time in minutes
Fig. 6.8 The average fluorescence signal from the vial, the signals from the center of the vial, the
sidewalls of the vial and the corner of the vial as a function of time. It can be seen that, besides some
geometrical effects, the average signal of the vial does not decrease during the evaporation process.
as a function of time. It can be seen in this graph that the average fluorescence signal from
the vial remains almost constant during evaporation. The slight increase of the signal in the
beginning is caused by geometrical effects, due to the shape of the vial. These effects are less
than 10% of the signal. Furthermore, it can be seen that the signal in the corners of the vial
increases steadily, because of the accumulation of the fluorescent particles at these spots.
Again, conventional fluorescence microscopy allows a qualitative description of the evaporation process, but it is not a suitable technique to do quantitative measurements of the shape of
the meniscus.
6.2.2
Confocal Fluorescence Microscopy
A second well-established method to analyze three-dimensional microvolumes is confocal
fluorescence microscopy. This kind of microscopy, however, fulfills only the spatial sampling
requirement. In a confocal microscope system a pinhole in the excitation path causes point
illumination4 , whereas a second pinhole in front of the detector prevents out-of-focus light
from reaching the detector. This way only light from a small volume in the sample is collected.
As a result an image of a three-dimensional volume can be acquired by scanning a stack of
slices at different depths. Due to the time-consuming scanning the temporal resolution of
confocal microscopy is poor. The left part of Figure 6.9 shows different cross-sections of
three-dimensional confocal images of a vial filled with an evaporating fluorescent solution.
These figures clearly show the shape of the vial. The right part of Figure 6.9 shows the
4 In
practice, a confocal microscope is equipped with a focused laser beam, which is focused sufficiently small for
point illumination
81
height in µm
54o
n
m
lu
co
n
itio
os
lp
xe
pi
row pixel position
height in µm
(a) t = 0min
n
m
lu
co
n
itio
os
lp
xe
pi
row pixel position
height in µm
(b) t = 3min
n
m
lu
co
os
lp
xe
pi
n
itio
row pixel position
(c) t = 6min
Fig. 6.9 The left figures show cross-sections through confocal images of an evaporating liquid sample
in a square vial. The shape of the meniscus is clearly visible. The vial has the shape of a truncated
pyramid. The right figures show the shape of the meniscus computed from the confocal images.
three-dimensional shapes of the meniscus at the same instants as the left part. These shapes
of the meniscus are computed by measuring at each lateral position the axial position where
the fluorescent signal vanishes, since this indicates the position of the air-liquid interface.
82
6.2.3
Interference-Contrast Microscopy
A third type of microscopy that complies with the temporal sampling requirement as well as
with the spatial sampling requirement is interference-contrast microscopy. Unlike the first two
types of microscopy, in which the observed intensity is proportional to the local concentration
of the fluorescent molecules in the sample, the measured contrast in interference-contrast
microscopy is an extrinsic property of the sample under observation. The evaporation process
of the liquid is observed with epi-illuminated imaging with a narrow band light source. Part
of the incident light reflects at the air liquid interface. Another part reflects at the bottom of
the vial and interferes with the first reflected part at the air liquid interface. This results in
constructive and destructive interference as a function of the distance to the bottom of the vial.
Evaporation of the liquid changes the profile of the meniscus. As a result of this the interference
patterns are dynamic. Imaging of the meniscus profile yields a fringe pattern in which the
isophotes correspond to isoheight curves of the meniscus. Figure 6.10 shows three images of
an acquired image sequence of the evaporation process in one of the vials. The instant t = 0s
corresponds to an image with a perfectly flat meniscus. In Section 8.3 we will discuss how
this instant is measured. Furthermore, Figure 6.10 shows the height profiles of the meniscus
along one half of a diagonal of the vial, which resulted in the observed interferograms. These
profiles are computed with the algorithm to be described in Section 8.2.
Description of dynamic fringe patterns. The following observations are made from the
interferogram sequence acquired during evaporation of the liquid. These observations can
also be seen in Figure 6.11. This figure shows the time evolution of the fringe pattern along a
line through the center of the vial.
1. First, the meniscus is convex and the fringes propagate towards the center of the vial
and disappear. This indicates that the meniscus becomes flat. After the final fringe has
disappeared, i.e. when the meniscus is perfectly flat, the fringes reappear in the center
of the vial and propagate towards the sidewalls of the vial. During evaporation more
and more fringes are generated. This implies that the meniscus becomes more concave.
This is shown in the right part of Figure 6.10. This can also be seen in Figure 9.3.
2. Starting from the instant when the meniscus is flat and no fringes are present, until the
moment that the liquid reaches the bottom and the liquid film breaks, 29 bright fringes
appear in the center of the vial. We will prove that the number of observed fringes
depends on the wavelength of the incident light and the refractive index of the liquid
(and weakly on the angle of incidence). The left part of Figure 6.12 shows the time
evolution of the fringes during evaporation in the center of the vial, recorded with a
wavelength of 500nm and 29 bright fringes can be counted, whereas the right part of
of Figure 6.12 shows the time evolution of the fringes in the same vial, now recorded
with a wavelength of 600nm and 34 bright fringes can be counted.
3. Just before the instant at which the liquid film breaks at the bottom of the vial, a dark
fringe originates in the center of the vial. As will be explained in Chapter 7 this is
related to the difference of the refractive index of the liquid and the refractive index of
the bottom.
83
8
6
4
2
t =-53
0
(a) t = −53s
8
6
4
2
t =97
0
(b) t = 97s
8
6
4
2
t =164
0
(c) t = 164s
Fig. 6.10 The left figures show the dynamic interference patterns as recorded in a 6.0µm deep vial.
The right graphs show the one-dimensional height profiles (computed as explained in Section 8.2) along
a diagonal.
sidewall of vial
time
center of vial
sidewall of vial
t=0s
84
bottom of vial
Fig. 6.11 This figure shows the time evolution of the fringes along a line through the center of the vial.
In the center of the vial most modulations occur, whereas along the sidewalls of the vial no modulations
occur at all.
85
number of bright fringes counted from t=0
t=0: meniscus is flat
5
5
10
10
15
15
20
20
25
25
30
bottom of the vial
500 nm
600 nm
Fig. 6.12 The left figure shows the time evolution of the fringe pattern in the center of the vial recorded
with a wavelength of 500nm, whereas the right figure shows the time evolution of the fringe pattern in
the center of the same vial, now recorded with 600nm.
86
4. The contrast between the dark and bright fringes increases in time, i.e. the contrast
increases with decreasing height. This can be seen in Figure 6.10. On the other hand,
this implies that for larger heights of the liquid, the fringe patterns will not be observable,
because of too little contrast. This will be further explained in Section 7.5.
5. The spatial frequency of the fringes increases during evaporation. At a certain moment
this results in spatial undersampling. Spatial phase unwrapping algorithms that analyze
the fringe patterns will not be successful in these regions due to aliasing. As will be
discussed later, our temporal phase unwrapping algorithm will unwrap most of these
regions successfully.
6.3
CONCLUSIONS
In previous chapters we have shown successful solutions to overcome the problem of evaporation. There are, however, a number of motivations to study the process of evaporation
in detail. Monitoring the evaporation process requires sufficiently high spatial sampling as
well as temporal sampling. Conventional fluorescence video microscopy complies only with
the temporal sampling requirement, and confocal fluorescence microscopy only with the spatial sampling requirement. The technique of interference-contrast microscopy complies with
both sampling requirements. In the next chapter we will present an electromagnetic theory
to describe the generation of the fringe pattern at the meniscus of the thin liquid sample. In
Chapter 8 we will describe a signal processing algorithm to analyze these fringe patterns and
to derive the height profiles of the meniscus. In Chapter 9 we will present results of a number
of experiments we have performed with this technique.
7
Electromagnetic Theory of
Interference-Constrast
Microscopy
This chapter presents a classical electromagnetic theory to describe the generation of the
dynamic fringe patterns, which are observed during evaporation of a liquid sample in a 6µm
deep micromachined vial. The fringe patterns will be described as modulations of the intensity
of the electric field as a function of the height of the air-liquid interface. This model explains
most of our observations as described in Chapter 6. This model is based on the classical
theory of the generation of fringes of equal thickness in thin films [21] and it shows analogies
to the model of fluorescence interference-contrast microscopy as presented by Lambacher and
Fromherz [22].
7.1 INTRODUCTION
The optical model of interference-contrast microscopy is based on the classical theory of
the generation of fringes of equal thickness in thin films [21]. According to this theory
interference patterns are observed due to an optical path difference between two parts of an
incident plane wave. One part of the incident plane wave is reflected from the upper surface
of the thin film, whereas the other part is reflected from the lower surface of the thin film.
Lambacher and Fromherz [22] expanded this model to describe microscopic observations
of interference patterns generated by a fluorescent cyanine dye embedded in a LangmuirBlodgett film. The dye is spaced from a reflecting silicon surface by a layer of silicon dioxide.
Their optical model takes the distance dependent excitation of the fluorescent dye as well as
the distance dependent emission of the dye into account. Lambacher and Fromherz call this
type of microscopy fluorescence interference-contrast microscopy. This chapter presents an
electromagnetic model to describe the observations of interference patterns in sub-nanoliter
87
88
ELECTROMAGNETIC THEORY OF INTERFERENCE-CONSTRAST MICROSCOPY
vials etched in silicon dioxide with a typical depth of 6µm as shown in Chapter 6. This model
is an extension of the classical theory and shows analogies to the model of Lambacher and
Fromherz.
The observed interference patterns show modulations of the intensity of the electric field as a
function of the height d of the air-liquid interface (the meniscus) of the liquid in the vial. In
this chapter an expression for the intensity of the electric field will be derived. This derivation
takes into account that the interference patterns are observed via epi-illumination of the liquid
sample in the vial with an incoherent unpolarized narrow-band light source. Furthermore, the
incident light has a limited angle of incidence. The derivation of an expression for the intensity
of the electric field consists of the following steps. The first step derives an expression for the
phase difference between the part of the incident plane wave that is reflected at the air-liquid
interface and the part of the incident plane wave that is reflected at the bottom of the vial.
The second step is to derive an expression for the sum of the electric field vectors of these
parts. In the third step the electric field vector will be integrated analytically over all angles
of polarization. Then, two weight functions are introduced to take the different wavenumbers
and angles of incidence into account. This results in an expression for the intensity of the
electric field, which can only be evaluated numerically.
7.2
PHASE DIFFERENCE
The optical model of interference-contrast microscopy is shown in Figure 7.1. The microscope
is adjusted to Köhler illumination: this implies incoherent illumination, and the incident light
consists of plane waves. This plane wave is incident at an angle Θ0 with respect to the normal of
the air-liquid interface. After refraction at the air-liquid interface, this plane wave propagates
towards the reflecting bottom of the vial at an angle Θin . The direct part of the incident plane
wave interferes with the part of the incident plane wave that is reflected at the bottom of the
vial. Whether this interference is constructive or destructive depends on the height d with
respect to the bottom of the vial and the angle of incidence Θ0 . Thus, interference occurs at
every height d, but, as will be explained further on, an image sensor will only observe the
interference at the air-liquid interface. First, the air-liquid interface is considered to be parallel
to the silicon dioxide bottom of the vial, as shown in Figure 7.1.
The optical path difference OPD between the direct part and the reflected part of the incident
plane wave at a height d above the reflecting bottom of the vial is given by
OPD = nliq (ABCD − FD) − nair (EF ),
(7.1)
where nliq and nair are the refractive indices of the liquid (ethylene-glycol, nliq = 1.432),
respectively the air. In Figure 7.1 it can be seen that the optical path length AB equals FD.
The OPD reduces then to
OPD = nliq (BCD) − nair (EF ).
(7.2)
d
The optical path length BCD equals twice the optical path length BC . With cos(Θin ) = BC
the OPD can be rewritten as
2 nliq d
OPD =
− nair (EF ),
(7.3)
cos(Θin )
PHASE DIFFERENCE
Eout,1
Eout,2
Eout,3
in
c
pl ide
w ane nt
av
e
Eout,4
E
F
plane of incidence
Θo
E
A
γ
air
E
Ethyleneglycol
E
D
B
d
Θin
reflec
ting in
89
Z
terfac
e-m
irror
C
X
ide)
n diox
Y
well (silico
bottom of
Fig. 7.1 A plane wave, incident at an angle Θ0 , is refracted at the air-liquid interface of the liquid
sample in the vial. After refraction, the direct part of this plane wave interferes with the part of this plane
wave that is reflected from the bottom of the vial. Whether this interference is constructive or destructive
depends on the height d with respect to the bottom of the vial. The arrows indicate the different parts of
the electric field: Ek is the part of the electric field in the plane of incidence (xz -plane), and E⊥ is the
part of the electric field perpendicular to the plane of incidence (y-axis).
where Θin is the angle of incidence in the liquid sample. Furthermore, from Figure 7.1 follows
EF
, where DB equals 2 d tan(Θin ). The angle Θ0 is the angle of incidence in
sin(Θ0 ) = DB
the air. Inserting this in Equation 7.3 yields
OPD =
2nliq d
− 2nair d sin(Θ0 ) tan(Θin ).
cos(Θin )
(7.4)
Applying Snellius’ law nair sin(Θ0 ) = nliq sin(Θin ) yields
OPD =
2 nliq d
2 nliq d sin2 (Θin )
−
= 2 nliq d cos(Θin ).
cos(Θin )
cos(Θin )
(7.5)
The assumption in this derivation is that the air-liquid interface is parallel to the bottom of the
vial. We will now prove that Formula 7.5 is also valid, when the air-liquid interface is inclined
at an angle α with respect to the bottom of the vial. This situation is shown in Figure 7.2.
Again, the optical path difference is given by Equation 7.1. Furthermore, AB equals FD and
with Snellius’ law follows nliq (BB 0 ) = nair (EF ). The optical path difference reduces then
to
OPD = nliq (B 0 CD).
(7.6)
in
c
pl ide
w ane nt
av
e
90
E
θ0
air
F
A
Ethyleneglycol
D
B
α
B' θin
d
C
reflecting bottom -silicon dioxide
Fig. 7.2 An incident plane wave is refracted at the air-liquid interface, which is inclined at an angle α
with respect to the bottom of the vial. The optical path difference between both parts of this plane wave
is given by Equation 7.5.
From Figure 7.2 follows cos(2Θin ) = cos2 (Θin ) − sin2 (Θin ) =
Equation 7.6 yields
OPD = nliq (1 + cos(2Θin ))(CD),
where CD follows from cos(Θin ) =
OPD =
d
CD .
B 0C
CD
. Inserting this in
(7.7)
This leads to
nliq d
(cos(2Θin ) + 1),
cos(Θin )
(7.8)
which reduces to the optical path difference given by Equation 7.5. This proves that the optical
path difference between the direct part and the reflected part of the incident plane wave given
by Equation 7.5 does not depend on the angle α between the air-liquid interface and the bottom
of the vial.
Again, the optical path difference causes interference between the direct part and the reflected
part of the incident plane wave as a function of the height d above the bottom of the vial.
The modulations at a height d in the liquid, however, cannot be observed. The reason is that
in a microscope system the outgoing rays, indicated in Figure 7.1 as Eout,i (i = 1, 2, 3, 4),
are observed in different uncorrelated points on an image sensor. An image sensor mounted
on a microscope can only observe interference, if two rays combine at a single point on the
image sensor, which seem to originate from the same point, but have different optical paths.
According to Figure 7.1, the rays Eout,2 and Eout,4 originate from the same point A, but these
rays are observed at different points on an image sensor. The rays Eout,2 and Eout,3 have
different optical paths, but they do not originate from the same point, nor are they observed at
the same position on an image sensor. When d is extended to the air-liquid interface, however,
these two rays shift towards each other and end up in the same position. The image sensor
mounted on the microscope will observe this as interference. This implies that the observed
91
INTENSITY OF ELECTRIC FIELD
interference patterns originate at the air-liquid interface and that the modulations inside the
liquid cannot be observed at all.
The phase difference Φin corresponding to the optical path difference given by Equation 7.5
is
4π nliq d cos(Θin )
Φin =
,
(7.9)
λ
where λ is the wavelength of the incident light. In terms of the wavenumber k =
phase difference becomes
Φin = 2 nliq k d cos(Θin ).
7.3
2π
λ ,
this
(7.10)
Electric Field Vectors. As indicated in Figure 7.1, the electric field vectors of the incident
~ ⊥ ), and vectors parallel
plane wave are split into vectors normal to the plane of incidence(E
~ k ). Using the notation from Figure 7.1, the electric field vector
to the plane of incidence (E
~ i and the electric field vector E
~ r after reflection at the air-liquid
of the incident plane wave E
interface are given in Cartesian coordinates by


cos(γ0 ) cos(Θ0 )
~i = E 
,
sin(γ0 )
E
cos(γ0 ) sin(Θ0 )
(7.11)


k
cos(γ0 ) cos(Θ0 )r01

⊥
~r = E 
sin(γ0 )r01
E

,
k
cos(γ0 ) sin(Θ0 )r01
(7.12)
where E is the amplitude of the electric field of the incident plane wave. The angle γ0 is
the angle of polarization with respect to the plane of incidence. The angle Θ0 is the angle of
k
incidence of the incident plane wave. The coefficients rij and rij⊥ are the Fresnel coefficients
for reflection at the interface between media i and j for respectively the parallel component
and the normal component with respect to the plane of incidence. The index 0 is for the air, 1
for the liquid in the vial and 2 for the silicon dioxide under the bottom of the vial. The electric
~ trt after refraction at the air-liquid interface, reflection at the bottom of the vial
field vector E
and again refraction at the liquid-air interface is given in Cartesian coordinates by

~ trt = E eiΦin
E

k
k k
cos(γ0 ) cos(Θ0 )t01 r12 t10


⊥ ⊥
sin(γ0 )t⊥

,
01 r12 t10
k
k k
cos(γ0 ) sin(Θ0 )t01 r12 t10
(7.13)
k
where Φin is the phase difference as defined in Equation 7.10. The coefficients tij and t⊥
ij are
the Fresnel coefficients for transmission through the interface between media i and j for the
parallel component and the normal component with respect to the plane of incidence.
92
For completeness, we give here the definitions of the Fresnel coefficients [23] based on the
notations from Figure 7.1:
rij⊥
=
k
rij
=
t⊥
ij
=
k
tij
=
ni cos(Θi ) − nj cos(Θj )
,
ni cos(Θi ) + nj cos(Θj )
−nj cos(Θi ) + ni cos(Θj )
,
nj cos(Θi ) + ni cos(Θj )
2ni cos(Θi )
,
ni cos(Θi ) + ni cos(Θj )
2ni cos(Θi )
,
nj cos(Θi ) + ni cos(Θj )
(7.14)
(7.15)
(7.16)
(7.17)
where ni and nj are the refractive indices of the two media, and Θi and Θj are the angle of
incidence, and the angle of refraction. These two angles are related by Snellius’ law.
The sum of the two parts of the reflected plane waves follows as


k
k
k k
cos(γ0 ) cos(Θ0 )(r01 + t01 r12 t10 eiΦin )

⊥
⊥ ⊥ iΦin
~ = E
sin(γ0 )(r01
+ t⊥
)
E
(7.18)

.
01 r12 t10 e
k
k
k k iΦin
cos(γ0 ) sin(Θ0 )(r01 + t01 r12 t10 e
)
~ 2 describes the observed interference as a function of
The intensity of the electric field |E|
the height d for a single incident plane wave with a certain angle of polarization, angle of
incidence, and wavenumber. The sum of all these single-quantum modulations gives rise to
the observed interference patterns [22].
Angle of Polarization. The first step to compute the intensity of the electric field is to
~ 2 analytically over all angles of polarization. The result of this computation can
integrate |E|
be split into an offset, which is not important, and a modulating part:
OFFSET
}|
{
2
k 2
k 2 k 2 k 2
⊥ 2
⊥ 2 ⊥ 2 ⊥ 2
~
h|E| iγ0 = π(r01 + r01 + t01 r12 t10 + t01 r12 t10 ) +
z
⊥ ⊥ ⊥ ⊥
2π(r01
t01 r12 t10
|
+
(7.19)
k k
k k
r01 t01 r12 t10 ) cos(Φin ) .
{z
MODULATING PART
}
The amplitude term preceding the cosine function shows very little dependency on the angle
of incidence Θin in the interval of interest [0, 0.2]rad .
Transmission Wavenumber Spectrum. The second step to evaluate the intensity of the
electric field is performed by averaging over all wavenumbers. For the experiments, a narrow
bandpass filter (600FS10-50, Andover Corporation, Salem, NH, USA) with a central wavelength λc = 602.3nm and a full width at half maximum (FWHM) of 9.7nm is placed in the
illumination path. The transmission spectrum of this filter S(k, σk ) (normalized in amplitude)
in terms of the wavenumber k can be approximated by a flattened Gaussian function:
(k − kc )2
(k − kc )2
S(k, σk ) = 1 +
exp −
,
(7.20)
2σk2
2σk2
relative transmission
93
1
0.8
0.6
FWHM
0.4
0.2
10.3
10.4
10.5
10.6
wavenumber k[µm-1]
Fig. 7.3 The filled curve shows the relative transmission of the measured spectrum of the narrow
bandpass filter. The black curve shows the relative transmission of the approximated spectrum as
defined by Equation 7.20.
with σk = 0.0443µm−1 corresponding to the FWHM of the filter. Figure 7.3 shows the
relative transmission of the measured spectrum and of the approximated spectrum.
Angle of Incidence. Finally, we will introduce a weight function for the angle of incidence.
Assume that the microscope system used for the experiments is adjusted to Köhler illumination [9]. This implies that the backfocal plane of the objective is a congruent plane of the light
source. Furthermore, we assume that the backfocal plane is uniformly filled. See Figure 7.4.
Each point in the back focal plane corresponds to a certain angle of incidence Θ0 , according
to the relation r0 ∝ tan(Θ0 ), with r0 the distance to the optical axis in the back focal plane,
and Θ0 the angle of incidence. All points with the same distance to the optical axis correspond
to the same angle of incidence. The probability that an incident plane wave has an angle of
2
incidence smaller than the angle Θ0 is proportional to πr0 , which is a circular fraction of the
back focal plane. The probability that the angle of incidence of a plane wave is smaller than
the maximum angle of incidence Θmax is, of course one with r0 = R the radius of the back
0
focal plane. The cumulative probability density function equals then ( rR )2 . The probability
density function that a certain incident plane wave has an angle of incidence Θ0 is then equal
0
to 2r
R2 , the derivative of the cumulative probability density function, which is proportional to
tan(Θ0 ). This defines the weight function for the different angles of incidence.
The halogen lamp used for the experiments, however, does not fill the back focal plane of
the objective (Zeiss FLUAR 20 × / 0.75) completely1 . Given the definition of the Numerical
1 Note
that in interference-contrast microscopy, this does not affect the resolving power of the optical system: the
period length of the modulations, which is reciprocal to the resolution is only determined by the angle between the
94
R
R
r'
Θ'
r'
optical axis
back focal plane
(front view)
back focal plane
(side view)
object
plane
objective
Fig. 7.4 Each point in the back focal plane corresponds to a certain angle of incidence Θ0 , according
to the relation r0 ∝ tan(Θ0 ), with r0 the distance to the optical axis in the back focal plane, and Θ0 the
angle of incidence.
Aperture (NA = nair sin(Θ0,max )) and a light source large enough to fill the back focal plane,
the following relation yields the maximum angle of incidence:
NA
tan(Θ0,max ) = q
.
n2air − NA2
(7.21)
The halogen lamp fills only one-fifth of the diameter of the back focal plane. In that case,
tan(Θ0,max ) equals one-fifth of the right hand side of Formula 7.21. As said before, the angles
of incidence Θ0 and Θin are related by Snellius’ law. The maximum angle of incidence in
the liquid Θin,max follows then as




n
1
NA
air
 .
Θin,max = arcsin 
sin arctan  q
(7.22)
nliq
5 n2 − NA2
air
With these two weight functions for the wavenumber and the angle of incidence of the incident
light, the intensity of the electric field at a height d above the bottom of the vial can be computed
as:
Z Θin,max Z ∞
2
~
~ 2 i dΘin dk,
|E(d)|
=
tan(Θin )S(k, σk )h|E|
(7.23)
Θin =0
k=0
γ0
air-liquid interface and the liquid-SiO2 interface. Even with just normal incidence the same high frequency pattern
can be generated. The highest frequency, that an objective can image onto a sensor is determined by the illumination
wavelength and the Numerical Aperture of the objective and is defined by fc = 2NA
. In transmission microscopy,
λ
an incompletely filled back focal plane would, of course, seriously affect the resolution.
COMPARISON BETWEEN SIMULATION AND EXPERIMENTAL RESULTS
95
2
~ iγ in Formula 7.19. The maximum angle
where S(k, σk ) is defined in Formula 7.20 and h|E|
0
of incidence Θin,max follows from Formula 7.22.
7.4
COMPARISON BETWEEN SIMULATION AND EXPERIMENTAL RESULTS
Expression 7.23 is evaluated numerically as a function of the height d above the bottom of the
vial with the following set of parameters:
1. central wavenumber of the transmission filter: kc = 10.4µm−1 ,
2. width of the transmission spectrum of the filter: σk = 0.0443µm−1 ,
3. refractive index of the liquid in the vial: nliq = 1.4319,
4. refractive index of the silicon dioxide bottom: nsil = 1.46, and
5. maximum angle of incidence: Θin,max = 0.155rad.
The result of this computation is shown in Figure 7.5.2 This computation is compared to
measurements in a digital recording of an evaporating ethylene-glycol sample. In each frame
of this recording the intensity in the center of the vial is measured. The modulating intensity,
measured as a function of time, is unwrapped with the algorithm that will be presented in
Chapter 8. This temporal phase unwrapping algorithm gives the height of the liquid as a
function of time. A parametric plot of the measured modulating intensity against the computed
height of the liquid is also shown in Figure 7.5. Recall the observations mentioned in Chapter6.
The second observation was that 29 bright fringes appeared in the center of the vial from the
moment that the meniscus was perfectly flat (t = 0s). This can be checked in Figure 7.5. The
third observation was that the final fringe just above the bottom of the vial is a dark fringe. Our
model agrees with this. Furthermore, the fourth observation was that the contrast increases
with decreasing height. This can also be seen in Figure 7.5. Finally, as can be seen in Figure 7.5
the period of the measured modulations equals the period of the simulated modulations from
Formula 7.23. The amplitude of the measured modulations, however, drops much faster than
that of the simulated modulations. We think that the poor agreement between our model
and the experiments is due to out-of-focus light. The focal plane for the experiments is the
bottom of the vial. This implies that in the beginning of the recording, when the meniscus is
convex, the meniscus is not focused and contains large out-of-focus contributions. During the
evaporation, the meniscus gets more and more in focus and the contributions of the out-offocus light get smaller and finally disappear. The contributions of the out-of-focus light result
in a reduced contrast for large values of d.
2 The bottom of the vial is covered by a layer of uniform thickness of Si N . The incident light is reflected at the
x y
Six Ny SiO2 interface. The refractive index of Six Ny is approximately 2.1, which is larger than the refractive index
of the SiO2 bottom of the vial (nsil = 1.46). The Fresnel coefficients in Equations 7.17 dictate that no phase shift
of πrad occurs at reflection on this interface. Of course, this layer introduces an additional constant optical path
difference. This extra OPD almost equals an integer number of periods for the wavelengths used in the experiments.
Therefore it is not taken into account and we subtract a phase shift of πrad from the simulation results to remove the
extra phase shift introduced by reflection on the liquid SiO2 interface used in the model.
Height d in µm
t=60 s. t=30 s.
t=90
s.
t=120 s.
t=210 s. t=180 s.t=150 s.
t=0 s. t=-30 s.
8
t=-60 s.
7
6
5
4
1
2
3
Simulated modulations
Measured modulations
Time t in s
96
Intensity
Fig. 7.5 This graph shows the intensity of the electric field as a function of the height d above the
reflecting bottom of the vial. One curve is based on evaluation of Expression 7.23, whereas the other
curve has been experimentally acquired. The moment t = 0s corresponds to the moment in the recording
of the interference pattern where the meniscus is perfectly flat.
ALTERNATIVE DESCRIPTION FOR INTENSITY OF ELECTRIC FIELD
7.5
97
ALTERNATIVE DESCRIPTION FOR INTENSITY OF ELECTRIC FIELD
The simulation results in Figure 7.5 indicate that the modulations can be written as a single cosine function with varying amplitude A(d) cos(Φ(d)), where the argument Φ(d) is a function
that defines the period of the modulations, and the amplitude A(d) is a function that defines
the envelope of the modulations.
Length of Modulations. The argument of the cosine function Φ(d) is, of course, related to
the phase difference as defined in Equation 7.10. The different wavenumbers can be replaced
by a single effective wavenumber keff and the different angles of incidence can be replaced
by a single effective angle of incidence Θeff
in . This yields
Φ(d) = 2 nliq keff d cos(Θeff
in ),
(7.24)
From this equation follows, that the length of the modulations ∆d, i.e. the change in the height
d corresponding to a phase difference Φ(d) = 2πrad equals
∆d =
2π
2nliq keff cos(Θeff
in )
.
(7.25)
Because of the symmetry of the transmission spectrum S(k, σk ) around k = kc , we conclude that the period of the modulations is proportional to kc . This implies that the effective
wavenumber keff = kc . The effective angle of incidence Θeff
in corresponds to the effective
√
√
eff
1
1
radius of the backfocal plane 2 2R, thus Θin = 2 2Θin,max . For small values of d, the
length of the modulations depends weakly on the angle of incidence [21]. For large values
of d, however, both the angular distribution and the spectral bandwidth will have a significant
effect on the fringe visibility.
Finally, we compared the positions of the zero-crossings in our measured modulations with the
predicted positions of the zero-crossings 2k+1
4 ∆d, k ∈ {0, 1, 2, 3, . . .}. The error between the
measured positions of the zero-crossings and the predicted positions has a standard deviation of
2.5nm, which equals approximately one percent of the length of the modulations. Remember
that there is a bias as explained in footnote 2. Here we are only interested in the precision of
the measurements.
Envelope of Modulations. In order to find an expression for the envelope or the amplitude
factor A(d) of the modulations, we will look in closer detail at Equation 7.23. First, we will
only consider the integral dependent on the wavenumber k and omit the dependency on the
angle of incidence Θin . Insert Equation 7.10 and the modulating part of Equation 7.19 into
Equation 7.23. For clarity, we replace the cosine function by a complex exponential function.
This yields
Z ∞ k2
k2
X(d) = <
1 + 2 exp − 2 exp (j2nliq k d cos(Θin )) dk ,
(7.26)
2σk
2σk
k=0
where X(·) just indicates a function of the height d, and <{·} is the real part. From a
mathematical point of view the modulations are caused by the fact that kc 6= 0. In order to
98
compute the envelope, we must set kc = 0. The following example illustrates this point of
view:
Consider the following function fmod (d):
Z ∞
(x − x0 )2
exp −
exp (jxd) dx,
fmod (d) =
2σx2
−∞
where the subscript mod indicates that this is an modulating function. Apply the variable
transformation x0 = x − x0 . This yields:
Z ∞
x02
fmod (d) =
exp − 2 exp (j(x0 + x0 )d) dx0
2σx
−∞
Z ∞
x02
= exp (jx0 d)
exp − 2 exp (jx0 d) dx0
2σx
−∞
= exp (jx0 d) fenv (d),
where the subscript env indicates that this function is the envelope of the function
fmod (d). This means that the modulating function can be described by a complex
exponential function with period 2π
x0 and amplitude function fenv (d). Figure 7.6 shows
the real part of the modulating function fmod (d) and its corresponding envelope fenv (d)
π
in the interval d = [0, 5] with x0 = 2π and σx = 10
.
1
fenv(d)
fmod(d)
0.5
1
-0.5
2
3
4
5
d
-1
Fig. 7.6 This figure shows the real part of the modulating function fmod (d) and its corresponding
π
envelope fenv (d) in the interval d = [0, 5] with x0 = 2π and σx = 10
.
Setting kc = 0 yields the following:
Z ∞ k2
k2
X(d) = <
1 + 2 exp − 2 exp (j2 nliq cos(Θin ) d k) dk . (7.27)
2σk
2σk
k=−∞
Note that we have replaced the integral range from (0, ∞) to (−∞, ∞), and omitted the
correcting factor of 21 . We recognize the function X(d) as the Fourier transform of the transmission spectrum. Note that k and d are physical quantities with reciprocal units. Recalling
the omitted dependency on the angle of incidence Θin , results in the envelope A(d) corre2
~
sponding to the modulations |E(d)|
from Equation 7.23. This shows that the envelope of the
modulations is a Fourier transform of the wavenumber spectrum (weighted with the angle of
99
SUMMARY
incidence Θin ): in a similar manner as in a Fourier Transform Spectrometer [24] the produced
interferogram encodes the spectrum of the light. Neglecting constant factors, the envelope
A(d) is given by,
Z
Θin,max
A(d) =
Θin =0
2
(4 nliq d2 σk2
F (Θin ) tan(Θin ) σk ×
(7.28)
√
cos2 (Θin ) − 3) exp(−2 3 n2liq d2 σk2 cos2 (Θin ))dΘin ,
where F (·)
√ stands for the amplitude term of all Fresnel coefficients in Equation 7.19. An extra
factor of 3 has been added to the exponential term for proper scaling.
This analysis shows that the amplitude of the modulations decreases as a function of the
height d. As a consequence, the modulations will not be observable above some height3 .
Furthermore, from Fourier analysis we know that the Fourier transform of S(k, σk ) can be
broadend by narrowing this function. Ultimately, with monochromatic light, the fringe patterns
would be visible up to large heights d.
7.6
SUMMARY
This chapter presents a classical electromagnetic theory to describe the generation of fringe
patterns in the vials during the evaporation of liquid in the vial. The main concept of this
theory is that one part of an incident plane wave, that is reflected on the air-liquid interface,
interferes with another part of the same incident plane wave, that is reflected on the bottom of
the vial. The phase difference between both parts of this plane wave is given by
Φin = 2 nliq k d cos(Θin ).
(7.29)
The fringe patterns are observed as modulations of the intensity of the electric field. The
computation of the intensity of the electric field takes into account the angle of polarization,
the transmission spectrum, the angle of incidence, and the maximum angle of incidence.
The intensity of the electric field can be computed by numerical evaluation of the following
equation:
Z Θin,max Z ∞
2
~
~ 2 i dΘin dk,
(7.30)
|E(d)| =
tan(Θin )S(k, σk )h|E|
γ0
Θin =0
k=0
~ 2 iγ in Formula 7.19. The maximum
where S(k, σk ) is defined in Formula 7.20 and h|E|
0
angle of incidence Θin,max follows from Formula 7.22. The electromagnetic model agrees
with experimental results. The period of the modulations is given by:
∆d =
2π
√
.
2nliq kc cos( 12 2 × Θin,max )
(7.31)
With kc = 10.43µm−1 , nliq = 1.432 and Θin,max = 0.155rad follows ∆d = 211.6nm.
3 Under
the assumption that the optical system can still spatially resolve the modulations.
8
Temporal Phase
Unwrapping Algorithm
The dynamic interferograms as shown in Chapter 6 can be regarded as dynamic contour maps
of the air-liquid interface: each isophote curve in an interferogram is an isoheight curve.
In this chapter we will present a temporal phase unwrapping algorithm that computes the
dynamic height profiles of the meniscus from these contour maps. This algorithm analyzes the
interferogram point-by-point in time without using any spatial information. Analysis of this
algorithm will show that this approach has a less severe spatial sampling requirement than a
spatial phase unwrapping algorithm.
8.1
PHASE UNWRAPPING: SPATIAL VS. TEMPORAL
The acquired interferograms I[m, n; t] as shown in Figure 6.10 and Figure 6.11 can be described by:
I[m, n; t] = A[m, n; t] cos(Φ[m, n; t]) + B[m, n; t] + N [m, n; t],
(8.1)
where A[·] and B[·] are the space and time varying amplitude and background. The signal
N [m, n; t] is an additive noise signal. The argument Φ[m, n; t] is the phase map of the
interferogram. The absolute or unwrapped phase map is related to the height d of the liquid
via
√
4πnliq cos( 21 2 × Θin,max )
Φ[m, n; t] =
d[m, n; t].
λc
(8.2)
101
102
TEMPORAL PHASE UNWRAPPING ALGORITHM
This equation is identical to Equation 7.24 in Chapter 7. The length of a single period ∆d
follows from this equation as
∆d =
2nliq cos( 12
λ
√c
.
2 × Θin,max )
(8.3)
Due to the cosine operator in the interferograms, the unwrapped phase map must be retrieved
from the wrapped or relative phase map Φ̂[m, n; t]:
Φ̂[m, n; t] = Φ[m, n; t] mod 2π.
(8.4)
The goal of an unwrapping algorithm is first to derive the wrapped phase map Φ̂[m, n; t]
from the observed interferogram I[m, n; t] by measuring the relative phase at every time. The
second step is to retrieve the absolute phase map Φ[m, n; t] with respect to a reference point
by adding 2π to the relative phase at every located 2π discontinuity. Section 8.3 will describe
how this reference point is determined.
Several algorithms exists that unwrap 2D phase maps derived from a static interference pattern, e.g. [25, 26]. These algorithms compare phase values at neighboring pixels. Spatial
unwrapping algorithms might suffer from the following three problems.
1. Phase errors originating in regions with a high noise level propagate outside these regions
and corrupt the remaining part of the image.
2. The fringe pattern must be sampled at a spatial frequency that is at least twice the
highest frequency in the interferogram to meet the requirements of the Nyquist sampling
theorem. One of our observations in Chapter 6 is that the images are undersampled in
regions with a steep meniscus. In Section 8.5 we will treat the aspects of sampling in
detail.
3. A spatial unwrapping algorithm will fail when true discontinuities are present in the
phase map. These discontinuities are observed as segmented fringes and are caused by
height differences. Experimental results to be shown in Chapter 9 will prove that our
temporal phase unwrapping algorithm can easily deal with these discontinuities.
To avoid the problems mentioned above we propose to analyze the acquired interferograms
in time, point-by-point as will be described in Section 8.2. Temporal phase unwrapping was
described earlier, e.g. by Huntley and Saldner[27].
Temporal phase unwrapping with a four-step interferometer. Huntley and Saldner used a four-step interferometer to generate four intensity images at phase steps of
π
2:
Ik [m, n; t] = A[m, n; t] cos(Φ[m, n; t] + Φk ) + B[m, n; t] + N [m, n; t],
where
Φk =
(k − 1)π
, k = 1, 2, 3, 4.
2
103
A phase map is directly computed from these four images according to the following
formula:
∆I42 [m, n; t]
Φ[m, n; t] = arctan
,
∆I13 [m, n; t]
where
∆Iij [m, n; t] = ∆Iij [t] = Ii [m, n; t] − Ij [m, n; t].
Thus,
∆I42 [t]
π
3π
) − cos(Φ[t] + ))
2
2
= A[t](sin(Φ[t]) + sin(Φ[t])) = 2A[t] sin(Φ[t])
= A[t](cos(Φ[t] +
With this method the phase difference between two successive phase maps is straightforward to compute:
∆Φ[m, n; t]
= Φ[m, n; t] − Φ[m, n; t − 1]
∆I42 [t]∆I13 [t − 1] − ∆I13 [t]∆I42 [t − 1]
.
= arctan
∆I13 [t]∆I13 [t − 1] + ∆I42 [t]∆I42 [t − 1]
Thus,
∆Φ[m, n; t]
cos(Φ[t]) sin(Φ[t − 1]) − sin(Φ[t]) cos(Φ[t − 1])
= arctan
sin(Φ[t]) sin(Φ[t − 1]) + cos(Φ[t]) cos(Φ[t − 1])
sin(Φ[t] − Φ[t − 1])
= arctan
cos(Φ[t] − Φ[t − 1])
With the method of Huntley and Saldner the phase map is computed directly from a series
of interferograms with known phase steps. However, in our case, the interferograms are not
recorded with known phase steps between successive images and it is not possible to derive
the phase map directly from successive images. The phase difference between two successive
images measured at different position is a continuum due to the space varying evaporation
of the liquid during the experiments. As can be seen in Figure 6.10 and Figure 9.3, the total
height change in the center of the vial is maximal. This leads to a maximal phase difference
at that position in two successive images. On the other hand, at the border of the vial there is
practically no height change: the phase difference is minimal.
8.2
In the remainder of this section the spatial dependency of the physical quantities in Equation 8.1
will be dropped: I[t] = I[m, n; t] and so forth. The square brackets indicate that the signals
are discrete signals.
104
In order to compute the phase map, we have to estimate the phase at every point in our signal.
The phase of a discrete signal can be estimated by applying a discrete Fourier transform. To
avoid discontinuities in the periodic representation of the signal, a window function must be
used before applying the Fourier transform. In our algorithm, this window function has a fixed
width. The bandwidth of our signal varies spatially. Subsampling the signal with a spatially
varying subsampling factor results in an approximately uniform bandwidth. The signal will
be subsampled in such a way that approximately two periods of the subsampled signal fall
within the width of the window function. This avoids that the low frequency component of
interest gets mixed with the DC component of the signal.
Figures 8.1(a)–8.2(b) show the successive steps of the temporal phase unwrapping algorithm.
The following recipe gives a short overview of the different steps.
1. Subtract the background B[t] from the interferogram I[t]. The result of this computation
is denoted as I 0 [t], which equals A[t] cos(Φ[t]) + N [t]. See figure 8.1(a).
2. Compute a subsampling factor fsub in order to subsample I 0 [t] in such a way that
0
approximately two periods of the subsampled interferogram, denoted as Isub
[t] fall into
a Kw number of data points wide interval.
0
3. Multiply a Kw number of data points wide region of Isub
[t] with a Kw number of data
points wide window function (see Figure 8.1(b)) and get the wrapped phase from the
frequency spectrum. The wrapped phase is denoted as Φ̂sub [t]. See Figure 8.2(a).
4. Retrieve the absolute phase, Φsub [t], by unwrapping the wrapped phases. Interpolate
the absolute phase to its original length. This is denoted as Φ[t]. Finally, convert the
absolute phase to the height of the meniscus as a function of time. The height of the
meniscus is denoted as d[t]. See figure 8.2(b).
Background removal. The first step in the computation of the phase map Φ[t] from the
interferogram I[t] is to subtract the background B[t] from the interferogram. The time dependent background is estimated by low-pass filtering I[t] with a Gaussian kernel with a very
large standard deviation. The result of this operation is denoted as I 0 [t] and is shown in Figure 8.1(a). The signal I 0 [t] consists of the cosine function with varying amplitude and the
additive noise term.
Subsampling. The next step is to subsample the low frequency signal I 0 [t] with a factor
fsub . The temporal sampling density is constant, but the temporal bandwidth varies spatially
as can be seen in Fig. 6.11. To satisfy the Nyquist criterion in the time dimension over the
entire image, the signal in time is oversampled: the oversampling rate increases towards the
sidewalls of the vial. We propose to use a window function with a fixed width of 128 points
and to subsample the low frequency signal I0 [t]. This subsampling is performed in such a
0
way that approximately two periods of Isub
[t] fall within the width of a window W [t]. The
window function is defined by

!2
!4 
!
Kw
Kw
t
−
t
−
(t − K2w )2
1
1
2
2
 exp −
,
(8.5)
W [t] = 1 +
+
2
σt
8
σt
2σt2
Intensity in #ADU's
105
10
5
-5
-10
1000
1250
1500
Frame number
Intensity in #ADU's
(a) This graph shows the background corrected data I 0 [t] in the center point of the
vial in the frame series 750 – 1500.
10
Window
function
7.5
5
2.5
20
40
60
80
100
120
-2.5
-5
-7.5
-10
(b) This graph shows the window function W [t] and the
0 [t]W [t].
product Isub
Fig. 8.1
The first two steps of the temporal phase unwrapping algorithm.
where σt = 16.0, and Kw is the fixed width of the window. The subsampling factor fsub is
defined as the ratio between the average length of the modulations in I 0 [t] and the required
average length of the modulations within the width Kw of the window. The former equals the
number of data points N in I 0 [t] divided by the number of periods in I 0 [t], where each period
has two zero-crossings ZC , and the latter simply equals Kw /2. Thus, the subsampling factor
fsub yields:
2 2 NI 0 [t]
fsub =
,
(8.6)
Kw #ZC
where [·] is the round operator.
In order to estimate the number of zero-crossings #ZC in I 0 [t] we apply a low-pass filter in
the time dimension with a variable width σ that suppresses the noise sufficiently, but does
not corrupt the signal. This implies that the width of the filter must be tuned in such a way
that it removes all zero-crossings introduced by the noise, but not the zero-crossings of the
signal. First, the number of zero-crossings #ZC is counted in I 0 [t]. Then I 0 [t] is filtered
flat meniscus
π
Phase
106
π
2
1000
1250
π
2
−
1500
Frame number
−π
-2
-1
0
1
2
4
3
calibrated phase wraps
Height in µm
(a) This graph shows the estimated phases in the frame series 750 – 1500.
8
6
4
2
1000
1250
1500
Frame number
(b) This graph shows the height profile, which follows by unwrapping of the estimated wrapped phases.
Fig. 8.2
The last two steps of the temporal phase unwrapping algorithm.
√
with a Gaussian kernel G[t; σ = 2] and the number of zero-crossings is counted again. This
operation removes zero-crossings due to the high frequency components
√ of the noise. This
filtering is repeated with the size of G[t; σ] increased by a factor of 2 until the number of
zero-crossings does not change after two successive filterings. In this situation the noise is
suppressed sufficiently and only true zero-crossings are present in the signal. The number of
zero-crossings can vary from 1 at the sidewall of the vial to 84 in the center of the vial. Under
assumption that the evaporation rate remains constant1 , the average number of periods within
Kw can be computed with Formula 8.6. As a result of the resampling the average number of
modulations within Kw of the subsampled interferogram varies from 1.6 to 2.4 modulations.
0
[t]. This way of subsampling is necessary to avoid
The subsampled signal is denoted as Isub
0
that F{Isub [t]W [t]} is mixed with the DC component of the frequency spectrum. The operator
1 In
Chapter 9 we will show that this assumption is true. Furthermore, this can al ready be seen in Figure 8.2(b),
which shows the height of the liquid as a function of time.
COMPUTATION OF REFERENCE FRAME
107
0
F{·} denotes the Fourier transform. The window function W [t] and the product Isub
[t]W [t]
are shown in Figure 8.1(b).
Note that the discrete Fourier transform (DFT) is a least squares algorithm for a number
of uniformly spaced frequencies k · 2π
N , k = {0, 1, . . . , N − 1} [28]. The aforementioned
subsampling strategy guarantees a proper match between the actual frequency and two times
the fundamental frequency 2π/N of the discrete frequencies of the DFT.
Phase estimation. The third step in the algorithm is to compute the 1D wrapped phase
map Φ̂sub [t]. One way to do this is to apply an FFT algorithm to get a Fourier transform
0
of a windowed part of Isub
[t]. The window function avoids discontinuities in the periodic
0
representation of Isub [t] computed with the discrete Fourier transform. The phase to be
estimated equals the phase of the bin with the maximum amplitude in the discrete frequency
0
spectrum. For each point Isub
[t0 ] the wrapped phase Φ̂sub [t0 ] is computed by multiplying a
0
128 wide window W [t] with Isub
[t + t0 ] around the instant t0 and computing the Fast Fourier
Transform of this product. The wrapped phase at the instant t0 follows from the discrete
0
frequency spectrum: Φ̂sub [t0 ] = Phase{W [t]Isub
[t0 ]}. The wrapped phase map, denoted as
Φ̂sub [t] is shown in Figure 8.2(a).
Unwrapping and scaling. After these three steps the 1D wrapped phase map is computed.
Next, the wrapped phase map is calibrated with respect to a reference point as described in
Section 8.3. In this calibration step the number of observed phase wraps is set to zero at the
instant where the meniscus is flat, i.e. we start counting phase wraps from this instant on.
The final step of the algorithm is to unwrap the wrapped phase map. The unwrapping is done
with an unwrapping operator found in literature [26]. Finally, the subsampled unwrapped
phase map Φsub [t] is interpolated by a factor fsub to the original length of I[t]. The phase
map Φ[t] is scaled with a factor ∆d
2π , with ∆d given in Equation 8.3, to an absolute height in
micrometers with respect to the flat meniscus. The result of this unwrapping algorithm is the
height of the liquid d[m, n; t] in a single data point [m, n] as a function of time. The height
d[t] with respect to the bottom of the vial is shown in Figure 8.2(b).
8.3
COMPUTATION OF REFERENCE FRAME
It is not possible to calibrate all wrapped phase maps at the final image of the image sequence,
because the liquid does not reach the bottom of the vial at every point at the same time. The
wrapped phase maps can be calibrated, however, when the meniscus is flat. At that instant
an interference pattern is absent, because the meniscus is parallel to the bottom of the vial.
In this time interval the OPD is approximately independent of the position. Although no
fringe pattern is being observed, the measured intensity over the entire vial is influenced by
the actual phase difference. This means that the intensity over the entire vial shows the same
modulation in time due to the time-varying OPD when the meniscus changes from convex to
concave. This has been monitored and is shown in Figure 8.3. The bottom graph shows the
modulation of the average intensity as a function of time, whereas the top graph shows the
variation of the average intensity (measured after background correction). The frame with the
smallest variation in average intensity corresponds to the one with the flat meniscus. In this
Average (#ADU's) StDev(#ADU's)
108
2
1.5
1
0.5
946
981
1010
1040
1071
1099
1000
1100
1000
1100
1125
60
55
Frame number
Fig. 8.3 The frame, where the meniscus is perfectly flat, is determined by a minimum variation in
intensity. This frame is used to trigger all relative height profiles by setting the number of counted
2π-discontinuities to zero in this frame.
figure, the standard deviation σframe of the intensity distribution of frame i is computed as
σframe
v
u
u
=t
M X
N
X
1
(Ii [m, n] − hI[m, n]it=1010...1070 )2 ,
N M − 1 m=1 n=1
(8.7)
where hI[m, n]it is the time-average over 60 frames in the time window where the meniscus is
almost flat. From Figure 8.3 follows that the meniscus becomes perfectly flat in frame 1040.
The time t = 0 in Figure 7.5 corresponds to this frame number. All wrapped phase maps are
calibrated by setting the number of observed phase wraps at that instant to zero. The labels
of the calibrated phase wraps are indicated in Figure 8.2(a).
The instant of the flat meniscus defines the absolute height of the vial. The absolute height of
the vial follows from Figure 7.5: the height of the vial is 6.13µm.
8.4
PRECISION AND ACCURACY OF PHASE ESTIMATION
Accuracy. The phase estimation is biased. Figure 8.4 shows the bias of the phase estimation
of the temporal phase unwrapping algorithm. The bias of the phase estimate is computed by
unwrapping cos(Φ[t]) with unit amplitude and no noise. The argument of the cosine function
Φ[t] is given by α K2πw (t − K2w ), where α defines the number of periods of Φ[t] within Kw .
The number of periods varies from 1.5 periods at the bottom of this graph to 2.5 periods at
the top. Recall that the number of periods within Kw varies approximately from 1.6 to 2.4
periods due to the subsampling. The bias is less than 0.01rad . It can be seen in Figure 8.4
that the bias of the phase estimation gets very large for windows smaller than 1.58 periods.
The bias in these regions is at least one order of magnitude larger than in the remaining part
of this figure. From simulations follows that
PRECISION AND ACCURACY OF PHASE ESTIMATION
0
64
96
128
2.5
2.0
2π
=
phase
phase = π
phase = 0
unbiased phase
phase = -π
= -2π
0.05
32
2.5
phas
e
0.1
number of periods α in Kw
Bias in rad
0
109
2.0
-0.05
-0.1
1.5
0
32
64
96
1.5
128
Fig. 8.4 This figure shows the bias of the phase estimation for a cosine function with a varying number
of periods α in Kw . The cosine function has unit amplitude and no noise. The bias is less than 0.01rad .
In the region below α = 1.58 the bias is at least one order of magnitude larger.
1. varying the amplitude A[t] as a function of the time t (A[t] cos(Φ[t])) results in a larger
bias.
2. The bias increases when the number of periods within Kw deviates more from two.
3. The bias of the phase estimation with exactly 2 periods within Kw is minimal.
4. The phase estimation Φ = 0 is only unbiased when the amplitude A[t] is constant.
The shape of the regions below 1.58 periods is maintained.
Precision. So far, we have not taken the additive noise term of Equation 8.1 into account.
Figure 8.1(a) shows a part of the interferogram in the center of the vial. The amplitude A[t]
is on the order of 7ADUs (Anolog-to-Digital Units). The standard deviation of the noise σN
is estimated by
v
u
T
u 1 X
σN = t
(I[t] − I[t] ∗ G[t; σ = 2.0])2 ,
(8.8)
T − 1 t=1
where G[·; σ] is a Gaussian kernel with standard deviation σ and ∗ is the convolution operator.
A typical noise level for the measurements presented in this part of this thesis is σN = 0.77.
The Signal-to-Noise Ratio (SNR) is on the order of 10. The variation of the phase estimation
is measured by applying the algorithm to a series of 1000 artificially generated interferograms
with A = 1, B = 0, and N [t] drawn from a Gaussian distribution with a standard deviation
RMS error of phase estimation (rad)
110
1.5
1
Standard Deviation
of additive Gaussian noise
SNR = 1
0.5
SNR < 1
0.1
0.2
0.3
SNR > 1
0.4
0.5
0.6
0.7
0.8
0.9
1
Amplitude A of interferogram (#ADU)
Fig. 8.5 This graph shows the RMS error of the phase estimation as a function of the amplitude of the
interferogram with a constant noise level. Even when the amplitude equals the standard deviation of the
noise, the phase can still be estimated with an RMS error of 0.16. If the SNR is below 1, the RMS error
increases rapidly.
of 0.1 (SNR=10). The RMS error of the phase estimation is 0.015rad . We conclude that for
this SNR our phase estimation is very precise.
Our observation that the contrast increases during evaporation (See also Figure 8.1(a)) suggests, under the assumption that the noise level remains constant in time, that the SNR increases
during evaporation. For most regions this statement holds. For some regions, however, the
gain in amplitude is reduced by spatial undersampling. Note that the optical spatial resolution
( 2NA
λ ) is not the limiting factor, but the sampling density related to the video camera and
the acquisition software. It is important to determine the lowest required SNR for a proper
estimation of the phase map. To investigate this empirically, a series of 1000 interferograms
were artificially generated with A = 0.01 · · · 1.0, B = 0, and N [t] drawn from a Gaussian
distribution with a standard deviation of 0.5. For each series the RMS error of the phase
estimation is measured. The results of these simulations are shown in Figure 8.5. The RMS
error of the phase estimation is below 0.16rad , when the SNR is higher than 1. Below this
SNR the RMS error increases rapidly.
8.5
SIMULTANEOUSLY SAMPLING IN SPACE AND IN TIME
In the first part of this chapter three possible problems from which spatial unwrapping algorithms might suffer were mentioned. One of these problems is sampling. In this section we
will discuss in detail the aspects of sampling. From a signal processing point of view there is
a fundamental advantage to do the unwrapping in time.
111
The Nyquist sampling criterion dictates that the fringe pattern needs to be sampled at least at
twice the highest frequency in the interferogram to enable reconstruction of the analog fringes.
Here, however, we are not interested in reconstruction of the fringes, but in reconstruction of
the meniscus that produced this fringe pattern. Image acquisition with a CCD camera with a
100% fill factor for the photo-sensitive sites, however, is not just sampling the signal with a
2D impulse train. Image acquisition with a CCD camera can be described in two steps: first,
the continuous spatial signal is convolved with the transfer function of the pixel. In the ideal
case, each pixel on the CCD element has a uniform sensitivity. This implies that the transfer
function of a pixel is a block function. Convolution of the signal with the pixel transfer
function is equal to a multiplication of the Fourier transform of the signal and the Fourier
transform of the pixel transfer function in the frequency domain. The Fourier transform of the
pixel transfer function is a sinc-function. This sinc function has its first zero-crossing at the
sampling frequency. The second step in image acquisition is to sample the convolved signal
with a 2D impulse train. As long as the fringe pattern is spatially sampled at a frequency
above the Nyquist frequency, then spatial reconstruction of the fringe pattern is possible with
this kind of acquisition.
In our experiments, the fringe patterns are sampled in space as well as in time. For this
moment, we assume that our temporal sampling satisfies the Nyquist criterion. The left image
of Fig. 8.6 shows a one-dimensional fringe pattern generated at the meniscus interface as a
function of the contact angle of the meniscus as indicated in Fig. 9.3: from the bottom of
this figure upwards the contact angle increases. This results in a steeper meniscus, which is
assumed in this example to be straight, and therefore a spatially denser fringe pattern. We
assume in this example that the fringe pattern is generated with monochromatic light, such
that the amplitude of the fringe pattern is constant for different heights. For three different
contact angles of the meniscus the amplitude of the fringe pattern is shown. The right image
shows the effects of imaging the left image onto a CCD camera. First, the left figure is
convolved with a uniform filter, the pixel transfer function. As a result of this convolution,
the amplitude decreases with increasing frequency, and at a certain frequency (due to a larger
contact angle), the amplitude of the fringe pattern becomes negative. This phase jump of π
radians is visualized in region 3 in the right image. Secondly, the filtered fringe pattern is
sampled with a 1D impulse train, indicated by the dots.
height d in µm
1
-1
-0.5
1
1
-1
-0.5
0.5
-1
1
-0.5
0.5
0.5
αmax
contact
angle
contact angle
2
3
4
5
6
7
8
lateral position
region 1
region 2
region 3
contact angle
tan(α)=∆d SD
0.1
0.05
-0.8
-0.4
0.4
0.8
-0.1
-0.2
0.2
0.1
-0.05
-0.1
tan(αmax)=∆d SD/2
lateral position
-0.2
region 1
region 2
region 3
height
0.2 0.4 0.6 0.8 1
Amplitude
region with
low SNR
112
113
Fig. 8.6 The left graph shows a fraction of Fig.9.3, indicating the contact angle. The left image shows
a one-dimensional fringe pattern generated at the meniscus interface as a function of the contact angle
of the meniscus as indicated in the left figure: from the bottom upwards the contact angle increases,
which results in a steeper (straight) meniscus and therefore a spatially denser fringe pattern. We assume
in this example that the fringe pattern is generated with monochromatic light, such that the amplitude
of the fringe pattern is constant for different heights. The right image shows the effects of imaging the
left image onto a CCD camera. First, the left image is convolved with a uniform filter, the pixel transfer
function. As a result of this convolution, the amplitude decreases with increasing frequency, and at a
certain frequency, the amplitude of the fringe pattern becomes negative. This phase jump of π radians is
visualized in region 3. Secondly, the filtered fringe pattern is sampled with a 1D impulse train, indicated
by the dots. In region 1, the spatial sampling density SD is larger than the Nyquist frequency fN ,
defined by the contact angle: spatial as well as temporal unwrapping is possible in this region. In region
2, the spatial sampling density satisfies fN /2 ≤ SD ≤ fN : aliasing occurs spatially, only temporal
unwrapping is possible. In region 3: the spatial sampling density is below fN /2: no unwrapping is
possible, due to irrecoverable phase jumps.
In region 1, the spatial sampling density SD is larger than the spatial Nyquist frequency,
defined by the contact angle α. Spatial as well as temporal reconstruction of the analogue
fringe pattern is possible in this region.
In region 2, the spatial sampling density gets below the spatial Nyquist frequency, but remains
above half the Nyquist frequency. The sinc function corresponding to the pixel transfer function is still positive, as shown in the most right part of Fig. 8.6. This implies that convolution of
the signal with the pixel transfer function does not introduce any unrecoverable phase jumps.
Each point in the interferogram is spatially undersampled, but still sufficiently sampled in time,
although with reduced amplitude. This means that the meniscus profile can be reconstructed
in time point-by-point. Since the meniscus has the smoothest shape possible, this temporal
reconstruction also allows spatial reconstruction of the meniscus. It is, of course, not possible
to reconstruct the analog spatial fringe pattern, necessary for spatial unwrapping.
In region 3: the spatial sampling density gets below half the Nyquist frequency. The first
negative lobe of the sinc-function starts to coincide with the Fourier spectrum of the signal
and a phase jump occurs, as shown in the right part of Fig. 8.6. From this moment on even
temporal reconstruction is not possible.
To summarize, spatial reconstruction of the meniscus requires spatial sampling at the Nyquist
frequency, whereas temporal reconstruction requires spatial sampling at half the Nyquist
frequency. For this reason, we choose temporal reconstruction.
Temporal sampling requirement. The evaporation speed of the liquid, i.e. the change
in height per unit of time, requires a minimal temporal sampling density. Obviously, the
evaporation speed is maximal in the center of the vial. The evaporation speed is defined as
the height change of a single fringe ∆d in a time window τ∆d . The analysis requires that
approximately two periods of the temporal fringe pattern correspond to the fixed window
width Kw . This requirements yields
Kw ≤ 2τ∆d Rsampling ,
(8.9)
Intensity in #ADU's
114
∆d (µm)
10
time window τ∆d (s)
5
-5
-10
τ∆d (s).Rsampling (s-1)>=Kw/2
Frame number
#data points
Fig. 8.7 In the time window τ∆d the change in height of the liquid is ∆d. The number of data points
in this time window τ∆d Rsampling should be larger than K2w
where Rsampling is the sampling rate (15 frames per second). Figure 8.7 visualizes this
requirement. According to the Nyquist criterion, this implies 32 times oversampling of the
fringe pattern. Note that a wider window implies a better resolution of the discretized frequency
spectrum, but requires a higher sampling density. Formula 8.9 can be rewritten as follows:
max
vevap.
≤
2
∆d Rsampling .
Kw
(8.10)
max
With Kw = 128 and ∆d = 0.2116µm follows vevap.
= 0.05µm/s. The evaporation speed
in the center of the vial follows from Figure 7.5: the average evaporation speed is 6.13µm in
305s, which equals 0.02µm/s.
Spatial sampling requirement. On the other hand, the spatial sampling density is limited
by the angle α between the meniscus and the bottom of the vial: the projection of the fringes
on the image sensor gets denser when the meniscus gets steeper. Figure 8.8 shows a fraction
of the meniscus at an angle α with respect to the bottom of the vial. The right part of this
~ 2 measured at different continuous positions
figure shows the intensity of the electric field |E|
r. The bottom part shows four pixels of the camera, on which the fraction of the meniscus
is projected. The Nyquist criterion dictates that one period of a periodic signal is sampled
with at least two samples. This is visualized in this figure. This requirement translates to the
following:
2 tan(α)
SD ≥
,
(8.11)
∆d
with SD the sampling density in µm−1 . For the figures shown in Figure 6.10, the region
of interest (i.e. half the width of the vial) is 136pixels wide. Given the width of that vial
136
(300µm), the spatial sampling density SD = 150
= 0.91µm −1 . With ∆d = 0.2116µm
from Equation 8.3 follows that α must be smaller than 5.5 degrees. This criterion will not be
met in the regions near the sidewalls of the vial towards the end of the evaporation process.
For this reason, spatial unwrapping is not possible in these regions. Temporal unwrapping,
however, is easy, because the overall change in height in these regions is minimal!
SUMMARY
115
Relative Intensity |E(r,d)|2
bright fringe
en
dark fringe
α
pixel
d in µm
m
pixel
pixel
∆d (µm)
s
u
isc
pixel
1/SD (µm)
position in vial r
Fig. 8.8 The meniscus is at an angle α with respect to the bottom of the vial. The right part shows the
~ 2 measured at different continuous positions r. The bottom part shows
intensity of the electric field |E|
two pixels of the detector on which one period of the fringe pattern is projected. This is the minimal
sampling density for recovery of the analogue fringe pattern as dictated by the Nyquist criterion.
8.6 SUMMARY
This chapter presented a temporal phase unwrapping algorithm. We propose to do the phase
unwrapping temporally, instead of spatially, because temporal phase unwrapping requires only
spatial sampling at half the Nyquist frequency, whereas spatial unwrapping requires spatial
sampling at the Nyquist frequency. Furthermore, spatial unwrapping algorithms fail when
the fringe patterns contain segmented fringes. In the next chapter we will show experimental
results that prove that our temporal phase unwrapping algorithm can easily deal with these
discontinuities.
The algorithm consists of the following steps: after background correction, the interferogram
is subsampled in such a way that approximately two periods of the fringe pattern fall within
the width of a window function. The Fourier transform of the product of the window function
and the subsampled fringe pattern is computed. The phase to be estimated simply equals the
phase of the bin with the maximum amplitude in the frequency spectrum. Finally, the wrapped
phases are unwrapped and the absolute phase is converted to the height of the liquid at that
position. All heights are triggered by setting the counted number of phase wraps to 0 at the
instant that the meniscus is flat.
Analysis of our algorithm shows that the accuracy of our algorithm is less than 0.01rad , and
the precision in terms of the RMS error is better than 0.16rad , when the SNR is higher than
1. With ∆d = 211.6nm/2πrad phase shift, this translates to an accuracy of 0.3nm and a
precision better than 5.4nm.
9
Experimental Results
This chapter presents experimental results of monitoring the evaporation process in micromachined vials with the technique of interference-contrast microscopy. We have monitored
the evaporation process in square vials as well as in circular shaped vials of different sizes.
This chapter will show that the evaporation rate of the liquid is linearly proportional to the
width of the vial, and not, as one would intuitively expect to the area of the air-liquid interface.
Furthermore, this chapter will show experimental results of electronic volume measurements
and optical volume measurements for calibration of the dedicated electronic volume sensor.
Finally, we will show that with this technique it is possible to measure height differences of
objects patterned on the bottom of a vial.
9.1
INTRODUCTION
In Chapter 7 we introduced an electromagnetic theory to describe the generation of dynamic
interference patterns, which are observed during evaporation of a liquid sample in a 6µm deep
micromachined vial. Chapter 8 presented a temporal phase unwrapping algorithm to analyze
these fringe patterns. With this algorithm the fringe patterns are analyzed in time point-bypoint. In this chapter we will show the results of various experiments we performed with the
technique of interference-contrast microscopy. With this technique we have monitored the
evaporation process in square vials with a width in the range from 75µm to 300µm, and in
circular vials with a diameter in the range from 100µm to 300µm. For all experiments we
retrieved sequences of height profiles of the meniscus. Korvink has used these height profiles
for comparison with the simulation results of his physical model as described in Chapter 6.
From these height profile sequences it is straightforward to compute the remaining liquid
117
118
EXPERIMENTAL RESULTS
volume in the vial as a function of time and the evaporation rate of the liquid. We will present
a counterintuitive result: the evaporation rate is linearly proportional to the width of the vial and
not to the area of the air-liquid interface. This means that the evaporation process is diffusion
limited. In an appendix at the end of this chapter we will treat briefly the theory of Deegan
of diffusion-limited evaporation in pinned droplets [17]. Similar experiments have been
performed in vials with electrodes patterned on the bottom of the vial. These measurements
are used to calibrate the dedicated electronic volume sensor as described in Chapter 6. We
will present electronic volume measurements as well as optical volume measurements. We
noted in Chapter 8 that true discontinuities result in segmented fringe patterns and that spatial
phase unwrapping algorithms are not capable of analyzing segmented fringe patterns. We will
show experimental results that our algorithm can easily deal with these discontinuities. We
can measure the height of these discontinuities on nanometer-scale.
9.2 EXPERIMENTAL SETUP
The measurements are performed on a Zeiss Axioskop with a 20 × /0.75 FLUAR or a 5 ×
/0.25 FLUAR objective. A Sony DXC-960MD 3CCD color video camera is mounted on
the microscope via a 2.5× camera mount from Zeiss. The CCD camera is attached to a
Matrox Marval G200 AGP framegrabber. An adjustable cold light illuminator was used for
epi-illumination. A 600FS10-50 AM-23831 narrow bandpass interference filter from Andover
Corporation was placed right in front of the illuminator. To direct the light downwards in the
microscope a beamsplitter was placed in the illumination path. Actually, this beamsplitter
was just the dichroic mirror of a fluorescence filterset from Zeiss (filterset #15, excitation:
BP 546 / 12, beamsplitter: FT 580, emission: LP 590), where we removed the excitation
filter as well as the emission filter from the filterblock. Microarrays with different sized vials
were etched in silicon dioxide at DIMES (Delft Institute for Microelectronics and Submicron
Technology). The vials were manually filled with pure ethylene-glycol using an Eppendorf
Transjector 5246. The image processing package SCIL Image (TNO Institute of Applied
Physics (TPD), Delft, The Netherlands) was used to process the video files.
9.3
MONITORING EVAPORATION IN SQUARE MICROMACHINED VIALS
Five digital video files of the evaporation process in five different sized square vials were
acquired with the experimental setup as described above. Table 9.1 shows some details of
the acquisition. Because of symmetry, the region of interest is only one quarter of the vial.
Recall that the video files are analyzed in time point-by-point and that no spatial correlating
functions whatsoever have been used to compute the height profiles. First, we will verify
whether or not the condition for the temporal sampling rate is satisfied. This condition is
given by Equation 8.10:
2
max
vevap.
≤
∆d Rsampling ,
(9.1)
Kw
where Kw is the width of the window function, ∆d = 0.2116µm the period of a single modulation, and Rsampling the temporal sampling rate given in Table 9.1. The temporal sampling
119
Table 9.1 Details of the acquisition. The first column gives the dimensions of the different vials. All
vials have a depth of 6.1µm. The second column gives the objective used for acquisition. For all but
one vial, a 2.5× camera mount was used. The third column gives the region of interest of each vial.
For reasons of symmetry, this region is only a quarter of the vial. The fourth column gives the temporal
sampling rate. The last column gives the length of the recording and the frame where the meniscus is
flat.
vial dimensions
µm2
objective
region of interest
pixels
Rsampling
frames/s
frames
flat/total
75 × 75
100 × 100
150 × 150
200 × 200
300 × 300
20 × /0.75
20 × /0.75
20 × /0.75
5 × /0.25
20 × /0.75∗
175 × 155
113 × 206
173 × 159
118 × 106
136 × 258
15
15
30
15
15
193 / 1055
368 / 1544
362 / 3906
528 / 2776
1040 / 4572
∗
) 1.0× camera mount used instead of 2.5× camera mount.
Table 9.2 Details of the analysis. The first column gives the dimensions of the different vials. All
vials have a depth of 6.1µm. The second column gives the time interval between the instant that the
meniscus is flat and the instant that the meniscus reaches the bottom. These values follow from Table 9.1.
The third column gives the measured evaporation rate. The fourth column gives the width of the window
function used for the analysis, and the final column gives the maximum evaporation rate as defined by
Equation 9.1.
vial dimensions
µm2
∆T = Tbottom − Tflat
s
vevap
µm/s
Kw
max
vevap.
µm/s
75 × 75
100 × 100
150 × 150
200 × 200
300 × 300
57.5
78.4
118.1
149.9
235.5
0.11
0.078
0.052
0.041
0.026
32
64
128
128
128
0.20
0.099
0.099
0.050
0.050
requirement is met, if the average evaporation rate is less than the maximum evaporation rate.
The average evaporation speed is simply defined as the height of the vial divided by the time
interval ∆T between the instant that the meniscus is flat, Tflat and the instant that the meniscus
reaches the bottom in the center of the vial, Tbottom . Table 9.2 gives details of the analysis of
each acquired video and shows that the condition for the temporal sampling rate is satisfied.
120
Figure 9.1 shows six height profiles (out of a sequence of 4572 height profiles!) of the meniscus
of a liquid sample in a 300 × 300µm2 square vial1 . These height profiles are equally spaced in
time. The instant t = 0.0s corresponds to the moment that the meniscus is perfectly flat. The
instant t = 234.0s corresponds to the moment right before the meniscus reaches the bottom of
the vial and erupts. Note that the computation of the height profiles is spatially uncorrelated,
but correlated in time. This correlation in time is inversely proportional to the change of the
meniscus level during evaporation.
Fringe pattern - contour map. As mentioned before, the fringe pattern can also be regarded
as a contour map of the meniscus. Figure 9.2 shows one quarter of the vial as shown in
Figure 6.10 at t = 97s. From the retrieved height profiles of the meniscus at this instant
we calculated a contour map with the contour lines of the dark (dashed lines) and the bright
fringes. In Figure 9.2 these contour lines are laid over the fringe pattern.
9.3.1
Lateral Scaling of Height Profiles
Figure 9.1(f) shows the profile of the meniscus just before the liquid film breaks at the bottom
of the vial. In the upper left corner of this figure errors are visible (possibly) due to spatial
undersampling of the fringe pattern. Note that these unwrapping errors do not occur in the
bottom part of this figure, because the sampling density in the y-direction is larger than the
sampling density in the x-direction given in Table 9.1. Further on in this section we will address
the cause of these errors in more detail. In order to do that, we will show in this section that
the height profiles are laterally scalable, i.e. the shape of the meniscus is the same for the
different vials, when they are rescaled to the same dimensions. From this observation we can
easily demonstrate that the spatial sampling requirement as defined in Equation 8.11 is not
always met for the different vials. Figure 9.3 illustrates this observation by showing the time
evolution of a single line of the meniscus during evaporation. The time difference between
two successive lines is 10s. This figure shows that the angle α between the meniscus and the
bottom of the vial approximates the maximum angle αmax as defined by Equation 8.11 and
can get larger than αmax , which implies that the spatial sampling requirement is not fulfilled.
The height profiles in Figure 9.3 correspond to the time evolution of the fringes as shown in
Figure 6.11.
In order to demonstrate the lateral scalability of the meniscus we will show three onedimensional height profiles for each of the different vials at five different instants in time.
This results in fifteen sets of five height profiles. Two of these one-dimensional height profiles
are perpendicular to the sidewall of the vial and go through the center of the vial, the third onedimensional height profile is a diagonal of the vial. We assume that the entire two-dimensional
height profile is scalable at this instant in time, if these one-dimensional height profiles for
the different vials are scaled versions of each other. Furthermore, if the height profiles at two
successive instants in time are scaled versions, then it is assumed that all height profiles in
between are scaled versions of each other. The instants in time t for this analysis are chosen
1 The only way to visualize the performance of our algorithm properly is, of course, with a video showing the evolution
of the 3D height profiles of the meniscus.
150
150
100
100
50
50
0
0
6
6
4
4
2
0
150
2
0
150
100
50
0
100
t =-59.0
50
0
(a)
t =0.0
(b)
150
150
100
100
50
50
0
0
6
6
4
4
2
0
150
2
0
150
100
50
0
100
t =58.0
50
0
(c)
t =117.0
(d)
150
150
100
100
50
50
0
0
6
6
4
4
2
0
150
2
0
150
100
50
0
(e)
121
t =175.0
100
50
0
t =234.0
(f)
Fig. 9.1 This figure shows six height profiles of the meniscus of a liquid sample in a 300 × 300µm2
vial at equally spaced instants in time. Because of symmetry only a quarter of the vial is shown.
122
height d in µm
Fig. 9.2 This figure shows the contour lines of the dark (dashed lines) and the bright fringes, calculated
from the retrieved height profile laid over the fringe pattern in a 300 × 300µm2 vial.
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
αmax
150
75
0
distance to center of vial in µm
75
150
W*
0.42 W*
Fig. 9.3 This figure shows the time evolution of a single line of the meniscus through the center of a
300 × 300µm2 vial. The time difference between two successive lines in 10s. The angle αmax is the
maximum angle between meniscus and bottom for spatial analysis.
123
line segment in vial:
k=0
d in µm
8
6
4
2
σd in nm
100
50
k=1
d in µm
8
6
4
2
100
σd in nm 50
k=2
d in µm
σd in nm
100
50
d in µm
σd in nm
100
50
d in µm
8
6
4
2
σd in nm
100
50
k=4
100
50
W
0.5W
W
0.5W
W
8
6
4
2
100
50
W
0.5W
8
6
4
2
100
50
W
0.5W
8
6
4
2
100
50
W
0.5W
W
0.5W
8
6
4
2
100
50
W
0.5W
W
0.5W
8
6
4
2
100
50
W
0.5W
8
6
4
2
100
50
W
0.5W
8
6
4
2
100
50
0.5W
W
0.5W
8
6
4
2
100
50
8
6
4
2
8
6
4
2
k=3
W
0.5W
8
6
4
2
W
0.5W
8
6
4
2
100
50
W
0.5W
Fig. 9.4 This figure shows sets of height profiles of the five different square vials, "horizontal",
"vertical", and "diagonal", for different instances in time, defined by k from Equation 9.2. For each set
of height profiles an error estimate σd is computed. Since all sets of graphs lie on top of each other, we
conclude that the height profiles are laterally scaled versions of each other.
by:
t = Tflat + k
Tbottom − Tflat
4
, k = 0, 1, 2, 3, 4.
(9.2)
Figure 9.4 shows for each value of k three pairs of graphs. The upper graph of each pair
shows a set of five height profiles of five different vials. The lower graph of each pair shows
an error estimate. The left set of graphs shows "horizontal" (x-direction) height profiles, the
mid set of graphs "vertical" (y-direction) height profiles, and the right set of graphs "diagonal"
height profiles. All height profiles have been rescaled to 100 × 100 data points. In Figure 9.4
the horizontal axis of each graph is defined by W , with W = 100. The error estimate σd is
124
Table 9.3 Required vs actual spatial sampling density. The first column gives the size of the vial.
The second column defines the maximum angle of the meniscus as defined by Equation 9.4. The third
column gives required spatial sampling density as defined by Equation 9.3. The fourth column gives
the image size of one quarter of the vial, and the last column gives the actual sampling density in both
directions.
vial dimensions
µm2
tan(α)
SD required
µm−1
region of interest
pixels
SD
µm−1
75 × 75
100 × 100
150 × 150
200 × 200
300 × 300
0.39
0.29
0.19
0.15
0.097
3.68
2.76
1.84
1.38
0.92
175/155
113/206
173/159
118/106
136/258
4.67/4.13
2.26/4.12
2.31/2.12
1.18/1.06
0.91/1.72
measured as follows: at each position there are five data points, since there are five different
sized vials. The error σd is then defined as half the difference of the data point above the
median value and the data point below the median value. As can be seen in the graphs in
Figure 9.4 this error remains below 100nm, except for the regions where there is spatial
undersampling. From the graphs in Figure 9.4 we conclude that the height profiles have the
same shape and that they are scaled version of each other.
Finally, we can verify whether or not the condition for the spatial sampling density is met.
This condition is given by Equation 8.11:
2 tan(α)
,
(9.3)
∆d
where SD is the sampling density, α is the angle of the meniscus as shown in Figure 9.3. This
figure shows that, given that the height profiles are scalable,
SD ≥
6.1µm
,
(9.4)
0.42W ∗
where W ∗ is half the width of the vial. Table 9.3 shows the required spatial sampling density
to satisfy Equation 9.3 and the actual spatial sampling density for the different vials. From
this table we conclude the following:
tan(α) =
1. The required spatial sampling density as defined by Equation 9.3 given in the third column of Table 9.3 equals the Nyquist frequency, since it is the condition to sample at two
samples per period. The highest spatial frequencies of the fringe patterns correspond to
half SD required . These frequencies are lower than the cutoff frequency of the objectives
used for the experiments. The cutoff frequency fc is defined as fc = 2NA
λ . For all
measurements fc = 2.5µm−1 , except for the 200 × 200µm2 vial fc = 0.83µm−1 .
This implies that even the most dense fringe patterns are passed through the objective.
2. In four cases the spatial sampling requirement is not satisfied. Section 8.5, however,
explained that the temporal phase unwrapping algorithm required spatial sampling at
125
half the Nyquist frequency, i.e. SD
2 . From this point of view, the requirement for the
spatial sampling density is satisfied for all cases. This implies that undersampling is
not the cause of the unwrapping errors shown in Figure 9.1(f). Still, in these cases the
spatial sampling is so low, that they are imaged with a reduced amplitude. This results
in a Signal-to-Noise Ratio, which is too low for proper analysis.
9.3.2
Evaporation Rate in Square Vials
Figure 9.1 show six profiles of the meniscus during evaporation. A straightforward measurement is the remaining liquid volume in the vials as a function of time. Of course, directly
related to that is the evaporation rate of the liquid. The results of these measurements are
shown in Figure 9.5(a). The different curves in Figure 9.5(a) show how the remaining liquid
volumes in the different sized vials, all with a depth of 6.1µm, change as a function of time.
The slope of each of these curves is the evaporation rate. The straight curves in Figure 9.5(a)
indicate that the evaporation rate is constant during evaporation. The evaporation rate as a
function of the width of the vial is shown in Figure 9.5(b). This graph shows that the evaporation rate is proportional to the width of the vial! This is a counterintuitive result: one
would expect that the evaporation rate is proportional to the area of the air-liquid interface,
which is roughly proportional to the square of the width of the vial. According to Deegan [17]
this result implies that the evaporation is diffusion-limited. In an appendix at the end of this
chapter we will briefly discuss the theory of Deegan on diffusion limited evaporation.
126
remaining liquid
volume in pl.
700
volume of well
552 pl
size of well
600
300x300µm2
500
1.32
400
300
pl/s
200x200µm2
246 pl
200
0.83
100
0.62 pl/
s
pl/s
150x150µm2
139 pl
61 pl
35 pl
-50
0.41 pl/
2
100x100µm2
75x75µm
s
0
0.31 pl/s
50
100
150
200
250
time in s.
evaporation rate in pl/s
(a) This graph shows the remaining liquid volume as a function of time for square
wells of various size. The depth of the wells is 6.1µm. The slope of the curves is
the evaporation rate.
1.5
1.25
1
0.75
0.5
0.25
50
100
150
200
250
300
width of wells in µm
(b) The evaporation rate as a function of the width of the wells.
Fig. 9.5 The upper graph shows the remaining liquid volume as a function of time and the bottom
graph shows the evaporation rate as a function of the width of the vial.
MONITORING EVAPORATION IN CIRCULAR MICROMACHINED VIALS
9.4
127
The evaporation process has been monitored in four different sized circularly shaped vials.
Table 9.4 shows some details of the acquisition. First, we will verify whether or not the
condition for the temporal sampling rate of Equation 9.1 is satisfied. Table 9.5 gives details
of the analysis of each acquired video and shows that the condition for the temporal sampling
rate is satisfied.
Figure 9.6 shows six height profiles (out of a sequence of 5148 height profiles!) of the meniscus
of a liquid sample in a circular vial with a diameter of 200µm . Except for the first one, these
height profiles are equally spaced in time. The instant t = 0.0 corresponds to the moment that
the meniscus is perfectly flat. The instant t = 142.0 corresponds to the moment just before
the meniscus reaches the bottom of the vial and erupts.
Fringe pattern - contour map. In analogy to the square vials, Figure 9.7 shows an image
of the fringe pattern in a circular shaped vial with a diameter of 300µm on which the contour
lines of the corresponding height profile are overlaid.
9.4.1
Radial Scaling of Height Profiles
In this Section we will show that the profiles of the meniscus are radially scaled profiles of each
other. Deegan [17] assumes that an evaporating pinned droplet has the shape of a spherical
cap: the droplet is so small that the gravitational force is negligible with respect to the surface
tension, which tries to minimize the surface area of the droplet, i.e. which tries to pull the
droplet into a sphere. We will make the same assumption. Figure 9.8 shows a cross-section
through a circular vial with diameter R filled with a liquid sample. The meniscus has the
shape of a spherical cap with radius ρ. The height of the meniscus in the center of the vial
with respect to the border of the vial is H and at a distance r from the center of the vial, the
height of the meniscus is h. In triangle 4OAB and triangle 4OA0 B 0 the following equations
Table 9.4 Details of the acquisition. The first column gives the diameter of the different vials. All
vials have a depth of 6.1µm. The second column gives the objective used for acquisition. For all vials,
a 2.5× camera mount was used. The third column gives the region of interest of each vial. For reasons
of symmetry, this region is only a quarter of the vial. The fourth column gives the temporal sampling
rate. The last column gives the length of the recording and the frame where the meniscus is flat.
vial diameter
µm
objective
region of interest
pixels
Rsampling
frames/s
frames
flat/total
100
150
200
300
20 × /0.75
20 × /0.75
20 × /0.75
5 × /0.25
234 × 424∗
164 × 156
234 × 203
349 × 325∗
30
30
30
15
240 / 2399
668 / 3870
875 / 5148
325 / 3427
∗
) region of interest covers entire vial.
128
Table 9.5 Details of the analysis. The first column gives the diameter of the different vials. All vials
have a depth of 6.1µm. The second column gives the time interval between the instant that the meniscus
is flat and the instant that the meniscus reaches the bottom. These values follow from Table 9.4. The
third column gives the measured evaporation rate. For all vials Kw = 128. The final column gives the
maximum evaporation rate as defined by Equation 9.1.
vial dimensions
µm
∆T = Tbottom − Tflat
s
vevap
µm/s
max
vevap.
µm/s
100
150
200
300
72.0
106.7
142.4
206.8
0.085
0.057
0.043
0.030
0.099
0.099
0.099
0.050
yield:
4OA0 B 0 : ρ2 = (ρ − H + h)2 + r2 ,
4OAB : ρ2 = (ρ − H)2 + R2 .
(9.5)
Solving this set of equations for h yields:
h = h(r) =
H 2 − R2 +
√
H 4 − 4H 2 r2 + 2H 2 R2 + R4
2H
(9.6)
First, we have fit this model to the height profiles of the meniscus in the vial with a diameter
of 300µm. Note that the region of interest covers the entire vial. Before fitting the model, the
profiles were resampled to a square grid of 100 × 100 data points. The result of the nonlinear
fit will be a radius R in pixels defined on the resampled grid. Of these 104 data points, 7645
data points correspond to the meniscus, the remaining 2355 data points to the surrounding
area of the vial. The domain of the data points is chosen symmetrically around the origin:
{x =p(−50, 50), y = (−50, 50)}. The variable r of the model of Equation 9.6 is replaced with
r = (x − x0 )2 + (y − y0 )2 . The model of the spherical cap is fit to the sequence of profiles
every second, i.e. one fit per 15 frames. The optimization method used for the non-linear fit
is the Levenberg-Marquardt method (Mathematica 3.0, Wolfram Research Inc., 1996). The
initial values of the parameters were chosen as follows: R = 50.0, H = 2.5, x0 = 0.0, y0 =
0.0. In most cases the iterative minimization of the χ2 merit function took six iterations. The
χ2 merit function is given by the sum of the squared residuals. Figure 9.9 shows the results
of this experiment. Figure 9.9(a) shows that the value of the χ2 -merit function increases in
time. This can be explained by the fact that the unwrapping errors propagate in time. The
unwrapping errors are the largest contributions to the χ2 -merit function. The model of the
spherical cap does not fit the shape of the meniscus, when it becomes flat. Around this instant
the optimization method fails to find the correct values for the parameters H and R. This can
also be seen in the graphs for the radius R and the position (x0 , y0 ). The optimization method
finds the correct values for the height H. Note that in Figure 9.9(c) H is given with respect
100
100
75
75
50
50
25
50
25
0
100
75
25
0
0
6
6
4
4
2
0
2
0
t=-28.0
50
25
0
(a)
100
75
t=0.0
(b)
100
100
75
75
50
50
25
25
50
25
0
100
75
08
0
6
6
4
4
2
0
2
0
t=35.5
50
25
0
(c)
t=71.0
100
75
75
50
50
25
50
25
(e)
100
75
(d)
100
0
129
75
100
t=106.5
25
0
0
6
6
4
4
2
0
2
0
50
25
0
75
100
t=142.0
(f)
Fig. 9.6 This figure shows six height profiles of the meniscus of a liquid sample in a 300 × 300µm2
vial at equally spaced instants in time. Because of symmetry only a quarter of the vial is shown.
130
Fig. 9.7 This figure shows the contour lines of the dark (dashed lines) and the bright fringes, calculated
from the retrieved height profile laid over the fringe pattern.
Spherical cap
A'
H
B'
r
A
Liquid
in vial
ρ
ρ
h
R
B
ρ
O
Fig. 9.8 This figure shows a vial with a diameter R filled with a liquid sample. The meniscus has the
shape of a spherical cap with radius ρ. In the center of the vial the height of the meniscus with respect
to the border of the vial is H, at a distance r from the center of the vial the height of the meniscus is h.
to the bottom of the vial. Due to the fact that the height profiles of the meniscus have been
resampled to 100 × 100 pixels, the value of the radius R is not exactly correct. Figure 9.10
shows a spherical cap fitted through the (subsampled) data of a height profile of the meniscus.
From these results we conclude that the model of a spherical cap is in agreement with our
Radius R in pixels
value of χ2 merit function
200
150
131
60
57.5
55
52.5
50
100
47.5
45
50
42.5
50
100
150
200
time in s
50
6
5
4
3
150
200
time in s
(b) Radius R in pixels as a function of
time
Shift from origin in pixels
height H in µm
(a) Value of χ2 -merit function as a function of time
100
2.5
2
1.5
1
0.5
x0(t)
2
50
1
y0(t)
100
150
200
time in s
-0.5
50
100
150
200
time in s
(c) Height H with respect to the bottom
of the vial in µm as a function of time
-1
(d) Position (x0 , y0 ) as a function of
time
Fig. 9.9 The graphs above show the value of the χ2 -merit function, the radius R, the height H with
respect to the bottom of the vial, and the position (x0 , y0 ), all as a function of time. The instant t = 0
corresponds to the moment that the meniscus is flat. Around this moment, the optimization method
failed, because the model does not fit to the flat shape of the meniscus. These nonlinear fit results were
obtained from the height profiles computed in a vial with a diameter of 300µm.
measurements. Finally, in order to prove our assumption that the height profiles are radial
scaled version of each other, we have fit the model of a spherical cap to every circular vial at
the instant T = Tflat + 21 (Tbottom − Tflat ), i.e. halfway during the evaporation. The results
of these measurements are listed in Table 9.6. Again, all height profiles have been resampled
to 100 × 100 pixels.
These results prove that the height profiles are radially scaled versions of each other. With
this in mind, it is staightforward to check whether or not the condition of the spatial sampling
density has been fulfilled. This condition is stated in Equation 9.3. Figure 9.11 shows a
cross-section through a circular vial, at the moment that the liquid reaches the bottom of the
vial. The shape of the meniscus is a spherical cap. The aspect ratio in this graph is strongly
out of proportion!
-40
-20
0
y-po
sitio
n in
pixe
20
ls
40
6
5
height H in µm
132
4
3
20
40
0
pixels
x-position in
-20
-40
Fig. 9.10 This figure shows a spherical cap fitted through the (subsampled) data of a height profile of
the meniscus.
Table 9.6 Parameters of spherical cap fitted to meniscus in different sized circular vials. The first
column gives the diamter of the vial. The second column gives the frame number of the height profile,
which the model was fitted to. The third column gives the value of the χ2 -merit function. The fourth
column gives the position (x0 , y0 ), and the final two columns give the radius R and the height H of the
model with respect to the bottom of the vial.
vial dimensions
µm
frame
value of χ2
(x0 , y0 )
pixels
R
pixels
H
µm
100∗
150
200
300∗
1320
2269
3012
1876
310.6
16.5
2.58
27.4
(-0.52,-0.21)
(0.73,0.69)
(-1.91,-0.03)
(0.50,0.91)
51.57
102.7
103.5
50.6
3.17
3.10
3.10
3.09
∗
From this figure the following equation for αmax follows:
tan(αmax ) =
R
2RH
R2 + H 2
= 2
,
ρ
=
,
ρ−H
R − H2
2H
(9.7)
where the expression for ρ follows from the second equation in Formula 9.5. Table 9.7 shows
the required spatial sampling density to satisfy Equation 9.3 and the actual spatial sampling
density for the different vials. From this table we conclude that in all cases the requirement
for the spatial sampling density is fulfilled.
αmax
133
ρ
ρ
R
Spherical cap
H
αmax
Fig. 9.11 This figure shows the profile of the meniscus at the instant that it reaches the bottom of the
vial. The meniscus has the shape of a spherical cap. The aspect ratio in this figure is strongly out of
proportion!
Table 9.7 Required vs actual spatial sampling density. The first column gives the diameter of the vial.
The second column defines the maximum angle of the meniscus as defined by Equation 9.7. The third
column gives required spatial sampling density as defined by Equation 9.3. The fourth column gives the
region of interest, and the last column gives the actual sampling density in both directions.
vial dimensions
µm2
tan(α)
SD required
µm−1
region of interest
pixels
SD
µm−1
100∗
150
200
300∗
0.249
0.165
0.123
0.082
2.35
1.56
1.16
0.77
234/424
164/156
234/203
349/325
2.34/4.24
2.19/2.08
2.34/2.03
1.16/1.08
∗
9.4.2 Evaporation Rate in Circular Vials
As with the square vials, we have measured the remaining liquid volume in the circular vials
as a function of time and the evaporation rate of the liquid. The results of these measurements
are shown in Figure 9.12(a). The different curves in Figure 9.12(a) show how the remaining
liquid volumes in the different sized vials, all with a depth of 6.1µm, change as a function
of time. The slope of each of these curves is the evaporation rate. The straight curves in
Figure 9.12(a) indicate again that the evaporation rate is constant during evaporation. The
evaporation rate as a function of the width (diameter) of the vial is shown in Figure 9.12(b).
This graph shows that the evaporation rate is proportional to the diameter of the vial.
134
700
volume of well
remaining liquid
volume in pl.
diameter of vial
600
500
300 µm
433 pl
400
1.13
p
l/s
300
193 pl
200 µm
200
0.71 pl/s
108 pl
100
150 µm
0.55 pl/s
48 pl
100 µm
0.35 pl/s
-50
50
100
150
200
time in s.
evaporation rate in pl/s
(a) This graph shows the remaining liquid volume as a function of time for circular
vials of various diameter. The depth of the vials is 6.1µm. The slope of the curves
is the evaporation rate.
1.2
1
0.8
0.6
0.4
0.2
50
100
150
200
250
300
width of wells in µm
(b) The evaporation rate as a function of the diameter of the wells.
Fig. 9.12 The upper graph shows the remaining liquid volume as a function of time and the bottom
graph shows the evaporation rate as a function of the diameter of the vial.
CALIBRATION OF THE ELECTRONIC VOLUME SENSOR
9.5
135
CALIBRATION OF THE ELECTRONIC VOLUME SENSOR
Figure 6.4 shows a first generation of a microarray with integrated electrodes at the bottom of
each of the twelve vials. When liquid is dispensed in the vials, the liquid alters the impedance
between the electrodes. The impedance is related to the volume of the liquid and can be
measured with a floating impedance method. An AC voltage source operating at 20kHz is
used to measure the impedance. In this chapter we will only show results of two experiments
performed in the lower left vial of the microarray shown in Figure 6.4. A detailed description
of all experiments falls outside the scope of this thesis. This vial has an area of 300×300µm2 .
The electrodes have an area of 300 × 75µm2 .
In these experiments a vial is filled with pure ethylene-glycol. The evaporation process is
monitored with the technique of interference-contrast microscopy and with electric impedance
measurements. Figure 9.13 shows the results of these experiments. Figure 9.13(a) shows the
remaining liquid volume in the vial as a function of time for two (!) successive measurements.
These curves have been measured exactly like the curves in Figure 9.5(a) and in Figure 9.12(a).
The slopes of these curves, the evaporation rates, are 1.55pl /s and 1.59pl /s. This graph shows
that optical volume measurements can be reproduced accurately and precisely. Figure 9.13(b)
shows the output voltage of the electric impedance measurement as a function of time. The
shape of both curves is the same. There is, however, a bias present in the electric impedance
measurements. The instant that the liquid reaches the bottom of the vial is clearly visible.
When the liquid hits the bottom of the vial, the liquid film breaks, and the remaining liquid
is drawn to the sidewalls of the vial. This leads to a sudden increase in impedance, and,
consequently, to a decrease in current through the vial. Figure 9.13(c) shows the output of the
electric impedance measurements as a function of the liquid volume in the vial. Again, one
measurement is biased with respect to the other measurement. This bias is due to impurities
in the solution left in the first experiment.
From these experimental results we conclude that it is possible to electronically measure the
volume with good precision, but with poor accuracy.
136
5
700
Voltage in V
Volume in pl
600
500
400
300
4
3
2
200
1
100
-100
-50
50
100
150
-100
200
250
time in s
Voltage in V
(a) Optical volume measurements: liquid
volume as a function of time.
-50
50
100
150
200
250
time in s
(b) results of electric impedance measurements: voltage as a function of time.
5
t=-50
t=0
t=50
t=-50
t=100
4
t=0
t=150
t=50
t=100
3
t=200
t=150
t=200
2
1
100
200
300
400
500
600
700
Volume in pl
(c) Output of electronic volume sensor as a function of the liquid volume.
Fig. 9.13 These graphs show the results of optical and electronic volume measurements. These
measurements are performed in a square vial of 300 × 300µm2 .
NANOMETER-SCALE HEIGHT MEASUREMENTS
9.6
137
The temporal phase unwrapping algorithm is applied to measure height differences in other
micromachined picoliter vials. In these specific vials two aluminium electrodes have been
patterned on the bottom of the vial. The electrodes introduce a height difference in the vial,
which is observed as a phase jump in the interference pattern. Figure 9.14 shows the onedimensional time evolution of the fringe pattern along a line through the center of the vial
including the electrodes. It is, of course, only possible to compute the wrapped phase jump.
Therefore, the measured absolute height difference limits itself to a maximum of ∆d. The
phase jump is computed as follows. First, the interference pattern is recorded and the frame
where the meniscus is flat, is computed as described in Section 8.3. This computation is
performed in the region of the vial where the bottom is present. With the temporal phase
unwrapping algorithm the wrapped phase in each point of this frame is computed. This
computation makes the wrapped phase jump ∆Φ visible. Due to the different optical behavior
of aluminium (phase shift π on reflection) and silicon dioxide (no phase shift on reflection)
an additional phase difference of size π must be added. Since we measure only the wrapped
phase jump, an integer number of 2π might be added to get the true phase difference ∆Φ0 . In
formula:
∆Φ0 = (∆Φ + π) + k 2π.
(9.8)
We measured the average phase in the computed wrapped phase map in three regions of
25 × 25pixels: the left electrode, the true bottom of the vial and the right electrode. The
average phase of the left electrode is 2.32rad , the average phase of the right electrode is
2.10rad and the average phase of the bottom is 2.82rad . Given that the electrodes have a
defined height of 0.3µm, we conclude that the height of the left electrode is 300.6nm with a
standard deviation of 4.1nm. The height of the right electrode is 293.2nm with a standard
deviation of 4.5nm. The precision on the bottom of the vial is 7.6nm. The final result of this
computation is shown in Figure 9.15.
t=0
region of electrode
time
region of electrode
138
300 nm
height profile of bottom
Fig. 9.14 This figure shows the time evolution of the fringes along a line through the center of the
vial including the electrodes. The phase jump due to the height difference at the bottom of the vial is
clearly visible.
139
(a) This image shows the electrodes on the bottom of the vial
300 nm
(b) This figure shows the retrieved height profile of the electrodes.
Fig. 9.15 The top image shows the frame where the meniscus is flat. The electrodes are clearly visible
on the left and right side of the image. The bottom figure shows the measured height difference between
the bottom of the vial and the electrodes on both sides of the vial.
140
9.7 CONCLUSIONS AND DISCUSSION
In this chapter we have shown a number of experimental results:
1. All digital videos of the evaporation process have been sufficiently sampled in time as
well as in space. In some cases, however, the spatial sampling density, was so low, that
the fringe pattern was recorded with reduced amplitude. As a result the Signal-to-Noise
Ratio was too low for proper unwrapping.
2. Comparison of the height profiles in different vials showed that the shape of the height
profiles is scalable. The meniscus in the circularly shaped vial can be very well modeled
by a spherical cap.
3. From the fact that the evaporation speed is linearly proportional to the width of the vial,
we conclude that the evaporation process is diffusion limited.
4. Experiments with the electronic volume sensor showed that the volume of the liquid can
be measured in a precise manner. The measurements, however, lack accuracy, because
of a bias in the output signal of the electronic volume sensor.
5. With the technique of interference contrast microscopy and our temporal phase unwrapping algorithm it is possible to measure height differences of objects patterned on the
bottom of a vial with a precision of about 5nm.
Appendix Deegan’s Theory on Diffusion-Limited Evaporation
The theory presented in this appendix is based on a publication of Deegan [17] in Physical
Review. Deegan observed the drying of liquid droplets pinned to the surface. Deegan considers
the local evaporation rate ~j, the rate of mass loss per unit surface area per unit of time from
the drop by evaporation. He states that the functional form of ~j depends on the rate limiting
step of the evaporation process:
• if the evaporation process is interface-transfer limited, i.e. the rate limiting step is the
transfer rate across the liquid-vapor boundary, then ~j is a constant, which implies that
the total evaporation rate is proportional to the area of the liquid-vapor interface.
• if the evaporation process is diffusion limited, i.e. the rate limiting step is the diffusive
relaxation of the saturated vapor layer immediately above the droplet, then ~j is not a
constant, but ~j is strongly enhanced towards the pinned edge of the droplet. In general,
the total evaporation rate in the diffusion limited regime is proportional to the radius
of the drop (a characteristic length). In this article, Deegan builds a physical model to
describe this behavior of ~j.
Deegan performed more or less the same measurements as presented in this chapter: he
monitored the evaporation process of small droplets and measured the evaporation rate as a
function of the radius of the droplet. As in our experiments, he found that the evaporation
rate of the droplets is proportional to the radius of the droplets. From this result he concludes
APPENDIX
141
Shperical cap
H
h
v
R
r
Fig. A.1 This figure shows a droplet with the shape of a spherical cap. The droplet has radius R and
height H. The annular element is at a distance r from the center and has a height h. It is assumed that
there is an outward flow field v.
that the evaporation is diffusion limited. Furthermore, he showed that the evaporation rate is
constant, which implies that the evaporation process is a steady-state process.
Deegan’s physical model of the drying of droplets shows that there is an outward flow of the
liquid inside the droplet and that the probability of escape of a molecule is greater towards
the edge of the droplet. Deegan solves a mass balance for an annular element of a spherical
shaped droplet at a distance r from the center of the droplet, as shown in Figure A.1:
∂
(ρ2πrdr h) =
∂t
ρ2π [rh(r)v(r) − (r + dr )h(r + dr )v(r + dr )] −
p
~j(r)2πr dr2 + dh2 ,
(A.1)
where the left-hand side of the equation is the change of mass in an annular element as a
function of time. The first term of the right-hand side of the equation is the difference between
the mass that flows in and out of the annular ring, given an outward and vertically averaged
flow-field v(r). The second term of the right-hand side of the equation is the mass loss due to
evaporation through the vapor-liquid interface. Division by 2πrdr yields
s
2
∂h
∂h
1 rh(r)v(r) − (r + dr )h(r + dr )v(r + dr ) ~
ρ
= ρ
−j 1+
∂r
∂t
r
dr
s
2
∂h
1 ∂
.
(A.2)
= −ρ
(rhv) − ~j(r) 1 +
∂r
r ∂r
Solving for the flow-field v results in:
v(r) = −
1
ρrh
Z
0
r


s
2

∂h
∂h 
+ρ
dr.
r ~j(r) 1 +

∂r
∂t 
(A.3)
142
Fig. A.2 This image shows the maximum projection of an image sequence acquired from a 200 ×
200 × 6.1µm3 vial filled with an evaporating ethylene-glycol sample containing fluorescent beads with
a diameter of 0.5µm.
From this equation it is clear that a nonzero v arises when there is a mismatch between the
local evaporation rate and the rate of change of the height profile.
In order to demonstrate the presence of the flow-field in our vials, an ethylene-glycol sample
containing fluorescent beads with a diameter of 0.5µm was injected in a 200 × 200 × 6µm3
vial and monitored during drying. Right after injection, the fluorescent beads were homogeneously distributed in the liquid sample and the beads showed Brownian motion. During the
evaporation process all the beads moved towards the sidewalls of the vial. This demonstrates
the presence of the flow-field v in our vials. Figure A.2 shows the maximum projection of an
image sequence, like a photograph of a dynamic scene acquired with a very long exposuretime,
acquired from a 200 × 200 × 6.1µm3 vial filled with an evaporating ethylene-glycol sample
containing fluorescent beads with a diameter of 0.5µm.
(Numerical) computation of v(r) requires specification of h = h(r) and ~j(r). Deegan assumes
that the droplet has the shape of a spherical cap, as defined in Equation 9.6:
h = h(r) =
H 2 − R2 +
√
H 4 − 4H 2 r2 + 2H 2 R2 + R4
2H
(A.4)
143
APPENDIX
This corresponds physically to assuming that R is small and that the gravitational force is
negligible with respect to the surface tension, which tries to pull the droplet into a sphere.
As said before, the evaporation is diffusion limited, which means that the evaporation is a
steady-state process. This means that the diffusion equation reduces to Laplace’s equation:
∇2 u = D
∂u
' 0, ~j = −D∇u,
∂t
(A.5)
where u is the mass density of the vapor in the air, and D is the diffusion constant for the
vapor in air. The boundary conditions for this problem are the following:
1. Along the surface of the droplet the air is saturated with vapor: u is a constant us .
2. The mass density u converges to the ambient vapor pressure, u∞ far away from the
droplet.
This boundary value has an electric analog of a charged conductor, when u is replaced by the
~ Deegan gives an approximation for the
electric potential V and ~j(r) by the electric field E.
analytical solution to this problem:
1
~j(r) ∝ q
1−
r 2
R
n̂.
(A.6)
This solution for ~j(r) shows that the evaporation diverges near the edge of the droplet (r → R).
This edge enhancement arises from the greater probability of an evaporating molecule’s escape
when leaving from the edge than when leaving from the center of the droplet.
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Summary
Quantitative Reading of Microarrays. The interfaculty research center Intelligent Molecular Diagnostic Systems employs chemistry in microarrays, fluorescence reading, and bioinformatics to gain insight into the metabolic regulation of the glycolytic pathway. In this research
program silicon-based labs-on-chips are being developed; each vial on the microarray will
be equipped with various sensors and actuators. With this kind of microarray technology
enormous amounts of data will be generated. Development of dedicated data analysis and
data interpretation systems is necessary to extract useful information from the experimental
data and to combine that with present data and knowledge.
Currently, the enzyme-catalyzed reactions of this pathway can be monitored in microarrays via
the production or consumption of NADH. The activity levels of the enzymes involved in the
reactions can be derived from the production or consumption rates of this fluorescent molecule.
The biochemical reactions are monitored with a modified conventional microscope system
equipped with a scientific grade CCD camera. With low power optics fluorescent signals from
an entire microarray can be acquired in a single image. This allows for a parallel readout of
the microarray. Wide field-of-view imaging, however, is at the expense of absolute sensitivity.
High power optics guarantees a high collection efficiency, but implies a sequential readout of
the microarray. With this kind of microarray reading system, there are two possible limiting
factors in the instrument that determine the lowest detectable concentration of fluorescent
particles. These limiting factors are stray light and readout noise. If readout noise limits
the detection, then the detection limit can be lowered by imaging the fluorescent signal onto
as few pixels as possible. This can be accomplished by choosing a CCD camera with large
pixels, where each pixel has a low readout noise level. A second modification is to demagnify
the image by placing a second objective back-to-back to the primary objective lens in the
149
150
SUMMARY
emission path. If stray light limits the detection, then a better detector will not result in a
lower detection limit, since it will not improve the signal-to-noise ratio of the measurements.
Applying the same modifications to the microscope system will result in a faster scanning rate
of the microarray. We have demonstrated the functionality of our conventional microscopebased microarray reader for this research program by monitoring an NADH producing and
consuming reaction at different enzyme activity levels in the microarrays.
Monitoring Dynamic Liquid Behavior in Micromachined Vials. Evaporation is a key
problem in the development of micrarray technology, especially in the case of open reactors.
Within the framework of this research program methods to avoid fast evaporation of solvents
have been developed. Still, the process of evaporation at a nanoliter scale is an interesting
phenomenon to study, and a requirement for calibration of a dedicated impedance based volume sensor. Quantitative measurements of the dynamic process of evaporation in the vials
of our microarrays requires sufficient spatial (lateral and axial) as well as temporal sampling.
Interference-contrast microscopy is a technique that satisfies both sampling requirements.
With this technique an interference pattern is generated at the air-liquid interface of the evaporating liquid sample. This interference pattern shows fringes of equal height. With a temporal
phase unwrapping algorithm these fringe patterns are analyzed in time point-by-point. With
this algorithm the height profile of the air-liquid interface can be reconstructed without using
any spatial correlating function. The instant that the air-liquid interface is perfectly flat is
used to calibrate all one-dimensional height maps. Applying this technique to reconstruct
the shape of the interface that generated this fringe pattern requires only spatial sampling at
half the Nyquist frequency, whereas a spatial phase unwrapping technique requires spatial
sampling at the Nyquist frequency. A straightforward measurement from these results is the
remaining liquid volume as a function of time, and from that the evaporation rate. A counterintuitive result is that the evaporation rate is proportional to the perimeter of the vial, and not
to the area of the air-liquid interface. This can be explained by the fact that the liquid is pinned
to the edge of the vial and that diffusion is the rate limiting step of the evaporation process.
Furthermore, when there are objects present on the bottom of the vials, these objects introduce phase differences in the interference pattern, which are observed as segmented fringes.
Spatial phase unwrapping algorithms will fail analyzing patterns with segmented fringes. A
temporal phase unwrapping algorithm, however, is insensitive for these kind of segmented
fringes. With this technique it is possible to measure the height of objects on a nanometer
scale.
Samenvatting
Kwantitatief lezen van microarrays. Het interfacultair onderzoekscentrum Intelligente
Moleculaire Diagnostische Systemen past chemie toe in microarrays, detecteert fluorescentie
in deze microarrays, en gebruikt bioinformatica om inzicht te verwerven in de metabolische
regulatie van de glycolyse. In dit onderzoeksprogramma worden op silicium gebaseerde
laboratoria-op-chips ontwikkeld; iedere reactor in de microarray zal worden voorzien van
verscheidene sensoren en actuatoren. Met dit soort microarray technologie zullen enorme
hoeveelheden gegevens worden gegenereerd. Ontwikkeling van toegesneden systemen om
deze experimentele gegevens te kunnen analyseren en interpreteren is noodzakelijk om nuttige
informatie te verkrijgen en deze te combineren met reeds beschikbare gegevens en kennis.
Op dit moment kunnen de reacties van de glycolyse, gekatalyzeerd door enzymen, worden
gevolgd in microarrays door middel van de productie of consumptie van NADH. De activiteit
van de enzymen, die in die reacties een rol spelen, kunnen worden afgeleid van de productie
of consumptie snelheden van dit fluorescerend molekuul. De biochemische reacties worden
gevolgd met een aangepast helderveld microscoop systeem, dat is uitgerust met een CCD
camera geschikt voor wetenschappelijke doeleinden. Met een optisch systeem met lage vergroting en numerieke apertuur kunnen de fluorescentie signalen van een gehele microarray
worden verkregen in een enkel beeld. Dit biedt de mogelijkheid om de microarray op een
parallele manier uit te lezen. Het afbeelden van een groot gezichtsveld gaat echter ten koste
van absolute gevoeligheid. Een optisch system met een hoge vergroting en numerieke apertuur garandeert een hoge en efficiënte lichtopbrengst, maar impliceert het sequentieel uitlezen
van een microarray. Met een dergelijk systeem om microarrays uit te lezen, zijn er twee
mogelijke beperkende factoren in het instrument die de laagst detecteerbare concentratie van
fluorescerende deeltjes bepalen. Deze beperkende factoren zijn strooi licht en uitlees ruis.
151
152
SAMENVATTING
Als uitlees ruis de detectie beperkt, dan kan de detectie grens worden verlaagd door het fluorescentie signaal af te beelden op zo weinig mogelijk beeld elementen als mogelijk. Met
een CCD camera met grote beeld elementen, waarbij ieder beeld element een laag niveau van
de uitleesruis heeft, kan dit worden bereikt. Een tweede aanpassing is om de afbeelding te
verkleinen door een tweede objectief rug-tegen-rug ten opzichte van het eerste objectief te
plaatsen in het emissie lichtpad. Als strooi licht de detectie beperkt, dan zal een betere detector
niet resulteren in een lagere detectie grens, omdat dat de signaal-ruis verhouding van de metingen niet zal verbeteren. Het toepassen van dezelfde aanpassingen aan het microscoop systeem
zal in een snellere uitlees snelheid van de microarray resulteren. Wij hebben de functionaliteit
van onze microarray lezer, gebaseerd op een helderveld microscoop, voor dit onderzoeksprogramma aangetoond door een NADH producerende en een NADH consumerende reactie te
volgen met verschillende ezym activiteiten.
Observeren van Dynamisch Gedrag van Vloeistof in Subnanoliter Reactoren Verdamping is een sleutel probleem in de ontwikkeling van microarray technologie, zeker in
het geval van open reactoren. In het kader van dit onderzoeksprogramma zijn er methodes
ontwikkeld om snelle verdamping van oplossingen te voorkomen. Dat neemt niet weg dat
het verdampingsproces op de schaal van een nanoliter een interessant fenomeen is om te
bestuderen. Bovendien is een dergelijke studie noodzakelijk om een toegesneden volume
sensor gebaseerd op het meten van impedantie te kunnen kalibreren. Kwantitatieve metingen
van het dynamische verdampingsproces in de reactoren van onze microarrays eisen voldoende
ruimtelijke (laterale en axiale) en temporele bemonstering. Interferentie-contrast microscopie
is een techniek die voldoet aan beide eisen voor de bemonstering. Met deze techniek wordt
een interferentie patroon gegenereerd op het lucht-vloeistof oppervlak van de verdampende
vloeistof in de reactor. Dit interferentie patroon toont ringen van gelijke hoogte. Met een
temporeel fase-reconstructie algoritme worden deze ringen patronen punt-voor-punt in de tijd
geanalyzeerd. Met dit algoritme kan het hoogte profiel van het lucht-vloeistof oppervlak
worden gereconstrueerd zonder gebruik te maken van functies die ruimtelijke verbanden aanleggen. Het moment dat het lucht-vloeistof oppervlak volkomen vlak is, wordt gebruikt om
alle een-dimensionale hoogte profielen te kalibreren. Het toepassen van deze techniek om de
vorm van het oppervlak te reconstrueren dat dit ringen patroon genereerde, vereist slechts een
ruimtelijke bemonstering op de halve Nyquist frequentie. Een ruimtelijk fase-reconstructie
algoritme daarentegen vereist ruimtelijke bemonstering op de Nyquist frequentie. Een voordehandliggende meting met deze resultaten is het overgebleven vloeistof volume als functie
van de tijd, en daaruit de verdampingsnelheid. Het resultaat is, tegen alle verwachtingen in,
dat de verdampingssnelheid evenredig is met de omtrek van de reactor, en niet met het oppervlak van het lucht-vloeistof oppervlak. De verklaring hiervoor is het feit dat de vloeistof
vast zit aan de rand van de reactor en dat diffusie de snelheidsbeperkende stap is van het verdampingsproces. Verder zullen objecten die aanwezig zijn op de bodem van de reactor, een
fase verschil introduceren, dat zich laat zien door een patroon van onderbroken ringen. Een
ruimtelijk fase-reconstructie algoritme zal er niet in slagen om patronen met onderbroken ringen te analyseren. Een temporeel fase-reconstructie algoritme daarentegen is ongevoelig voor
dit soort onderbroken ringen. Met deze techniek is het mogelijk om de hoogte van objecten
te meten op de schaal van een nanometer.
Curriculum Vitae
Richard van den Doel was born on January 15th 1974 in Vlissingen. He obtained his VWO
gymnasium diploma at the Stedelijke Scholengemeenschap Middelburg. In 1992 he started
his study Applied Physics at the Delft University of Technology. He obtained his M.Sc. in
1997. He did his graduation project at the Pattern Recognition Group under supervision of
prof. dr. I.T. Young. The goal of his graduation project was to develop semi-automated
procedures to calibrate microscope systems equipped with a scientific grade CCD camera and
an automated xyz-stage. In 1997, he continued working in the Pattern Recognition Group
of the Faculty of Applied Sciences and started his Ph.D. project within the framework of
an interfaculty research program ’Intelligent Molecular Diagnostic Systems’. The central
theme of this research center is to develop analytical systems that are able to analyze liquid
samples that contain a variety of many different biochemical compounds. Furthermore, these
systems should allow for dedicated data analysis and interpretation in combination with expert
knowledge and historical data. Prof. dr. I.T. Young is leader of this program and was initially
his direct supervisor. After one year in this project, prof. dr. ir. L.J. van Vliet became
his supervisor. The focus of his Ph.D. project was the development of a microarray reader
to collect all fluorescent signals from the microarray. A second subject in his thesis is the
study of the dynamic process of evaporation in subnanoliter wells. Currently, he is a post-doc
in a research project focusing on the analysis of MRI images of low water food products.
Furthermore, he is a teacher of two first-year courses.
153
Dankwoord
You play, you win. Ik wil mijn promotoren, Lucas van Vliet en Ted Young, bedanken voor
hun begeleiding tijdens mijn promotie onderzoek. Lucas, je bent altijd zeer kritisch en je neemt
pas iets aan als je het helemaal begrijpt. Ik moet eerlijk toegeven, voor mij is dat vaak een stap
verder gedacht dan ik zelf doe. Ted, sinds ik mijn eerste college Systemen & Signalen bij je
volgde, heb ik je bewonderd om je manier van doceren. Ik hoop de komende jaren nog veel van
je te leren op dit gebied. Veel dank ben ik verschuldigd aan Kari Hjelt. Aanvankelijk moest
ik erg wennen aan de manier waarop je wetenschap bedrijft, maar in de loop van de tijd ben
ik je steeds meer gaan waarderen voor je onuitputtelijke hoeveelheid (on)wetenschapppelijke
ideeën. Ik vind het nog steeds jammer dat je halverwege het DIOC programma terug bent
gegaan naar Finland. Dr. Geld, ik zal je niet vergeten! Verder wil ik iedereen van het DIOC
programma ’Intelligent Molecular Diagnostic Systems’ bedanken: Rob Moerman, Sylvia
Picioreanu, Hans Frank, Gijs van Dedem, Heidi Dietrich, Christina Apetrei (Analytische
Biotechnologie), Lina Sarro, Michiel Vellekoop, Bonnie Gray, Ventzeslav Iordanov, Peter
Szczaurski en Dilaila Craido (Electronische Instrumentatie Laboratorium en DIMES), Piet
Verbeek (Patroonherkennen), Marcel Reinders, Lodewyk Wessels, Henk Verbruggen, Eugène
van Someren, en Peter van der Veen (Informatie en communicatietheorie / Control, Risk,
Optimization, Sytems & Stochastics). Veel dank voor Wim van Oel voor het maken van
allerlei onderdelen voor de microscoop. Dank aan Michael Young voor het uitvoeren van
de analyse van een aantal opnames van verdampingsexperimenten. Verder wil ik bedanken:
Bernd Rieger, David Tax, Elżbieta Pekalska,
Frank de Jong, Marjolein van der Glas, Michael
,
van Ginkel, Michiel de Bakker, Ronald Ligteringen, Stephanie Ellenberger, en alle anderen
van Patroonherkennen.
155
156
DANKWOORD
You play, you lose. Na het overlijden van Rosanne heb ik heel veel steun gekregen van
Ela, Bob, en Lucas. Lieve Ela, bedankt voor alles, wat je tegen me gezegd hebt, en wat je
me gegegeven hebt. Bedankt voor je arm om mij heen. Bedankt voor alles! Bob, bedankt
voor de wijze woorden, die je tegen me gesproken hebt. Je hebt me laten zien dat je nooit
en nergens de enige bent. Lucas, ik heb je van het eerste begin tot het verdrietige eind op de
hoogte gehouden. Het zal voor jou ook niet de meest makkelijke periode zijn geweest. Je
hebt op een later moment op me in moeten praten, opdat ik mijn onderzoek niet zou opgeven.
Bedankt daarvoor, ik had dat echt nodig. Bedankt voor alle steun.
YOU PLAY! Lieve Jolanda, lieve Thomas, we hebben al zoveel meegemaakt, zoveel gelukkige
gebeurtenissen, maar ook hele verdrietige. Mijn promotie kwam als het ware tussendoor. Nu
is dit ook een afgerond hoofdstuk. Wie weet wat we met z’n drieën nog allemaal zullen
beleven. Het maakt niet uit, zolang wij maar samen zijn. Ik hou van jullie... van hier tot aan
de maan (en terug)! En dan ben ik nog maar net begonnen!

Quantitative Microscopic Techniques for Monitoring Dynamic

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