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Doctoral Thesis

Micromachined viscosity sensors for the characterization of DNAsolutions

Author(s): Rüst, Philipp Lukas

Publication Date: 2013

Permanent Link: https://doi.org/10.3929/ethz-a-010031569

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Diss. ETH No. 21299

Micromachined Viscosity Sensors

for the Characterization of DNA

Solutions

A dissertation submitted to

ETH ZURICH

for the degree of

Doctor of Sciences

presented by

PHILIPP LUKAS RUST

Master of Science ETH in Mikro- und Nanosystemen

September 17th, 1983

citizen of Thal (SG)

accepted on the recommendation of

Prof. Dr. Jurg Dual, examiner

Prof. Dr. Christofer Hierold, co-examiner

2013

I have always believed that scientific research is another domain where a form of

optimism is essential to success: I have yet to meet a successful scientist who lacks the

ability to exaggerate the importance of what he or she is doing, and I believe that

someone who lacks a delusional sense of significance will wilt in the face of repeated

experiences of multiple small failures and rare successes, the fate of most researchers.

Daniel Kahneman in ”Thinking, fast and slow”

Acknowledgments

First of all, I would like to thank Prof. Dr. Jurg Dual for giving me the opportunity

to work on a very interdisciplinary project during my time as as PhD candidate. He

gave me the freedom and the time to explore new fields on my own, also by attending

conferences on various topics around the world. I am especially thankful for giving

me the opportunity to work at Viscoteers GmbH, where I enjoyed to work not only as

an engineer, but where I also could experience the various pleasures and challenges an

industrial environment poses.

Prof. Dr. Christofer Hierold for accepting to co-supervise my thesis.

A big thank you goes to Dr. Damiano Cereghetti for performing the molecular biology

experiments presented in this thesis and for providing critical and inspiring comments

on my work. Thank you also for reading chapter 4 of the manuscript. I also liked the

scientific and non-scientific discussions we had when we were waiting for reactions to

finish.

Dr. Joe Goodbread and Dr. Klaus Hausler for their input the regarding the wireloop,

viscometry and the gated PLL.

Dr. Lukas Bestmann, who laid the ground for the project and gave a lot of input from

the perspective of diagnostics.

Dr. Stefan Lakamper for his critical comments on my work.

The various colleagues who contributed directly or indirectly to this project. I would

like to mention especially Simon Muntwyler and Felix Beyeler for very helpful tips on

micro-fabrication; Juho Pokki and Muhammad Arif Zeeshan for the time they spent at

the SEM together with me; Bengt Wunderlich for tips using the evaporator; Thomas

Liebrich for the tips on error analysis.

Raoul Hopf, Ivo Leibacher, Wiebke Jager and David Hasler for their contribution to the

project with their theses.

v

Thomas Wattinger for accepting to share the office with me. I know, I have a very dy-

namic manner of typing on the keyboard and commenting the reaction of the computer.

Thank you very much for the helpful and motivating discussions in the scientific area.

A big thank you to my colleagues from the Center of Mechanics. I very much enjoyed

the warm atmosphere at the institute, the Christmas dinners and summer parties, the

Panetonatas, the seminar weeks, the discussions during the lunches and coffee breaks,

the Super-Kondi, etc.

Dr. Stephan Kaufmann for the IT support. I also enjoyed the time working as a teaching

assistant for his lectures.

Gabi Squindo, the administrative assistant of the institute. Jean-Claude Tomasina for

the fabrication of the large scale mechanical parts.

Dr. Stefan Blunier and Donat Schweiwiller for running the clean room in the CLA

building where I spent many hours.

Last but not least a very, very big thank you goes to Ueli Marti. He fabricated the gated

PLL which was a big part of the project. If electronic equipment did not work, he could

fix it immediately. He also supported me when the project was in its challenging phases

with valuable hands-on ideas and motivating discussions.

vi

Contents

Abstract ix

Zusammenfassung xi

Nomenclature xiii

List of Figures xviii

List of Tables xxi

1. Introduction 1

1.1. Chip based nucleic acid testing . . . . . . . . . . . . . . . . . . . . . . . 2

1.2. Dynamic viscometry for nucleic acid testing . . . . . . . . . . . . . . . . 11

1.3. Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2. Modeling of the structural mechanics and the fluid structure interaction 17

2.1. State-of-the-art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2. Mechanical model of the cantilever . . . . . . . . . . . . . . . . . . . . . 19

2.3. Fluid structure interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4. Combined model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.5. Calculation of viscosity and density . . . . . . . . . . . . . . . . . . . . . 42

3. Cantilever system for viscosity and density sensing 45

3.1. State-of-the-art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2. System description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.3. Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4. Characterization of DNA solutions 81

4.1. Rheology of DNA solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.2. Titration experiments with DNA solutions . . . . . . . . . . . . . . . . . 92

vii

Contents

4.3. Polymerase chain reaction – PCR . . . . . . . . . . . . . . . . . . . . . . 102

4.4. Rolling circle amplification – RCA . . . . . . . . . . . . . . . . . . . . . . 108

5. Conclusion and outlook 119

A. Fabrication 123

A.1. Assembly and bonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

A.2. PDMS lid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

A.3. Heater chip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

A.4. Resonator chip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

B. Correction factor for gated measurement 127

C. Error analysis 129

Curriculum vitae 145

viii

Abstract

The miniaturization of diagnostic technology has been drawing great attention during

the last few years. Advantages of the miniaturization apart from less space require-

ments are the reduced need of sample volume and reagents as well as a smaller power

consumption. Due to their small size, the time required for analysis can often be re-

duced, resulting in a faster diagnosis and medical response. Other objectives in the

development of these lab-on-a-chip devices are increased simplicity to reduce the tech-

nical skills needed for their use and increased robustness. This allows the use of such

devices in harsh environments found for example in developing countries. Diagnostic

methods which directly probe the existence of a certain sequence in DNA can be used

for a variety of tests, ranging from the detection of pathogens to the identification of

a hereditary disease. The technical equipment required for such tests may however be

extensive, as many steps from preparation to the sample evaluation are needed. Many

of the recently developed miniaturized devices are designed for one or two steps only.

Especially the evaluation of the sample after an amplification reaction is often done

with bulky optical equipment needed for fluorescence measurements. These reactions

are routinely employed in the analysis of DNA. The main goal of the project described

in the thesis was the development of a novel method for the evaluation of products of

amplification reactions.

The general idea is, that the fluid mechanical properties, i.e. the viscosity and the density

of a sample change over the course of the reaction. This change shall be probed with

a viscosity or density sensor, fulfilling the requirements of size and sensitivity needed

for a miniaturized device. The advantages compared to fluorescence measurements are

the reduced level of complexity and the fact, that the measurement can take place

in the reaction chamber itself during the progress of the reaction. A viscosity and

density sensor based on a U-shaped, vibrating cantilever was developed for this task.

Depending on the properties of the surrounding fluid, the cantilever’s damping and

resonance frequency change. In order to facilitate the design of the cantilever, a model

was developed. It takes geometry parameters and fluid properties as inputs and gives

ix

Abstract

the resonance frequency and the damping as output. The structural model is based on

Euler-Bernoulli beam theory. The influence of the fluid is modeled in a semi-analytical,

two-dimensional approach, where an analytical model serves as a basis and correction

factors for the actual geometry are calculated from finite element simulations.

The cantilever sensors were fabricated using standard, silicon based micro-fabrication

technology. In a first step, the sensors were then characterized with respect to sensitivity

and accuracy. Parasitic effects, such as damping effects, self-heating and temperature

influences were investigated. Based on the measurement of the sensitivities of the reso-

nance frequency and the damping with respect to viscosity and density, a linear scheme

for the calculation of the fluid properties was developed.

In a second step, different solutions which contained 110 bp and 10 kbp long strands of

DNA were used to determine the minimal amount of DNA the sensor is able to detect.

It could be shown, that the sensor can also be used to measure the intrinsic viscosity of

the 110 bp strands. The measurements with the 10 kbp solutions showed the limitations

of the method for measuring the intrinsic viscosity. Due to the high strain rates and

frequencies involved, non-Newtonian effects prevent the measurement of the intrinsic

viscosity.

The third step was the assessment of the sensor for the observation of an amplification

reaction. The measurement of PCR products prepared off-chip showed promising results,

where an amplified product could successfully be distinguished from an unamplified

sample. However, the reproducibility of these experiments was poor. For this reason,

the rolling circle amplification was tested as an alternative. It was possible to distinguish

between an amplified and an unamplified product with off-chip reaction products with

much better confidence compared to the PCR experiments. The reaction was also carried

out on the chip itself, while the resonance frequency and the damping of the cantilever

sensor were measured in real-time. A very good distinction between amplification and

negative control could be shown.

x

Zusammenfassung

Die Miniaturisierung von diagnostischen Methoden hat in den letzten Jahren grosse

Aufmerksamkeit bekommen. Die Vorteile einer Miniaturisierung ist nicht nur der gerin-

gere Platzbedarf, sondern auch weniger benotigtes Probevolumen, kleinere Mengen an

Reagenzien und ein reduzierter Energiebedarf. Dank der geringen Grosse kann oft auch

die Analysezeit verkurzt werden, was zu einer schnelleren Diagnose und dadurch einer

schnelleren medizinischen Intervention verhelfen kann. Weitere Ziele bei der Entwick-

lung dieser so genannten Lab-on-a-Chip Devices sind eine erhohte Robustheit und eine

Reduktion der benotigten technischen Fahigkeiten fur die Bedienung des Gerats. Dies

ermoglicht die Benutzung in einer rauen Umgebung wie sie etwa in Entwicklungslandern

vorgefunden wird.

Diagnostische Methoden welche das Vorhandensein einer bestimmten DNA-Sequenz

prufen, konnen fur vielfaltige Tests verwendet werden, von der Detektion von Krank-

heitserregern bis hin zur Diagnose einer Erbkrankheit. Der Umfang der benotigten tech-

nischen Gerate kann aber relativ gross sein, da von der Probenvorbereitung bis zur

eigentlichen Auswertung oft viele Schritte notig sind. Viele der neu entwickelten, minia-

turisierten Instrumente sind aber nur fur einen oder zwei dieser Schritte gedacht. Speziell

die optische Auswertung mittels Fluoreszenzfarbstoffen wird noch haufig mit unhandli-

chen Geraten gemacht. Das Hauptziel dieses Projektes war die Entwicklung einer neu-

artigen Methode zur Auswertung von Produkten aus Amplifikationsreaktionen. Solche

Reaktionen werden bei der Analyse von DNA-Proben routinemassig durchgefuhrt.

Das Detektionskonzept beruht darauf, dass sich die fluiddynamischen Eigenschaften ei-

ner Probe wahrend der Reaktion andern. Diese Anderung soll mittels eines Sensors, der

die Viskositat und Dichte einer Flussigkeit messen kann, ermittelt werden, wobei die

Anforderungen an Grosse und Sensitivitat eines miniaturisierten Gerates erfullt werden

sollen. Die Vorteile gegenuber einer Fluoreszenzmessung ist die reduzierte Komplexitat

und die Tatsache, dass die Messung im Probengefass bei laufender Reaktion durchgefuhrt

werden kann. Fur diese Aufgabe wurde ein Viskositats- und Dichtesensor basierend auf

einem schwingenden Biegebalken entwickelt. Abhangig von den Eigenschaften der ihn

xi

Zusammenfassung

umgebenden Flussigkeit andern sich Resonanzfrequenz und Dampfung des Balkens. Um

die Dimensionierung zu vereinfachen wurde ein Modell entwickelt. Es benotigt die Geo-

metrie des Balkens und die Eigenschaften der Flussigkeit als Eingangsparameter und

gibt die Resonanzfrequenz und Dampfung aus. Das mechanische Modell basiert auf der

Euler-Bernoulli Theorie. Der Einfluss der Flussigkeit ist auf semi-analytischem, zwei-

dimensionalem Weg modelliert. Dabei dient ein analytisches Modell als Basis und der

Einfluss der eigentlichen Geometrie wird mittels einer Finite-Elemente-Simulation be-

rechnet.

Die Sensoren wurden mittels Silizium-basierter Mikrofabrikationstechnologien herge-

stellt. In einem ersten Schritt wurden diese Chips dann hinsichtlich Sensitivitat und

Genauigkeit charakterisiert. Parasitare Effekte wie Dampfung, Eigenerwarmung und

Temperatureinflusse wurden zudem untersucht. Basierend auf der Messung der Sensi-

tivitaten der Resonanzfrequenz und Dampfung bezuglich Dichte und Viskositat wurde

ein lineares Schema fur die Berechnung der Flussigkeitseigenschaften entwickelt.

In einem zweiten Schritt wurden Losungen, welche 110 Basenpaare und 10’000 Basen-

paare lange DNA-Strange enthielten, benutzt, um die minimal benotigte Konzentration,

welche der Sensor messen kann zu ermitteln. Es konnte gezeigt werden, dass sich der

Sensor eignet, um die intrinsische Viskositat der 110 bp-Losungen zu messen. Wegen

der hohen Scherraten und Frequenzen sind die gemessenen Viskositaten und Dichten

der Losungen mit den langen DNA-Ketten schwieriger zu interpretieren. Die Verstri-

ckung der einzelnen Ketten und das daraus resultierende nicht-Newtonsche Verhalten

sind der Grund dafur.

Der dritte Schritt war die Evaluation der Sensoren fur die Uberwachung von Ampli-

fikationsreaktionen. Die Messungen von PCR-Produkten, welche nicht auf dem Chip

produziert wurden, zeigten vielversprechende Resultate, wobei ein amplifiziertes Pro-

dukt von einem nicht amplifizierten unterschieden werden konnte. Die Reproduzierbar-

keit war jedoch schlecht. Aus diesem Grund wurden Versuche mit der Rolling Circle

Amplification gemacht. Dabei unterschieden sich amplifiziertes und nicht amplifiziertes

Produkt viel starker als bei der PCR. Die Reaktion wurde auch auf dem Chip selber

durchgefuhrt, wobei der gleichzeitig die Resonanzfrequenz und die Dampfung des Bie-

gebalkens gemessen wurde. Auch hier konnte sehr gut zwischen amplifiziertem Produkt

und Negativkontrolle differenziert werden.

xii

Nomenclature

Acronyms

µPIV Micro particle image velocimetry

AFM Atomic force microscope

bp, kbp (kilo) base pairs

CE Capillary electrophoresis

cssDNA Circular ssDNA

DNA Deoxyribonucleic acid

FEA Finite element analysis

FRET Forster resonance energy transfer

FSI Fluid structure interaction

gDNA Genomic DNA

gPLL Gated phase locked loop

ICP Inductively coupled plasma

LOC Lab-on-a-Chip

LOD Limit of detection

MEMS Micro Electro Mechanical System

MRSA Methicillin resistant Staphylococcus aureus

NTP Nucleoside triphosphate

PCB Printed circuit board

xiii

Nomenclature

PCR Polymerase chain reaction

PDMS Polydimethyl siloxane

PECVD Plasma enhanced chemical vapor deposition

QCM Quartz crystal microbalance

qPCR Quantitative PCR

RCA Rolling circle amplification

RNA Ribonucleic acid

RT-PCR Reverse transcriptase PCR

SDOF Single degree of freedom system

SOI Silicon on insulator

ssDNA Single stranded DNA

VCO Voltage controlled oscillator

WLC Worm like chain

Symbols

[η] Intrinsic viscosity

α Mark-Houwink exponent

x Mean value of x

β Dimensionless number

δ Ratio between different spring constants, depth of penetration, model

parameter of the WLC model

∆α Phase difference

ε Model parameter of the WLC model

η Dynamic viscosity, displacement function

xiv

Nomenclature

η∗, η′, η′′ Complex viscosity with real and imaginary part

ηS Viscosity of the solvent

ηred Reduced viscosity

Γ Hydrodynamic function

κ Argument of the transfer function H

u Velocity field

µ Line distributed mass

∇ Nabla operator

Ω Correction function

ω Angular frequency

Φ Magnetic flux

φ Phase

φref Reference phase

ρ Density of a fluid

ρS Density of silicon

σ Standard deviation

τ Parameter

θl Torsional displacement of the longitudinal beam

θtr Torsional displacement of the transversal beam

ϕ Displacement function

~µ Vector of material properties

~f Frequency vector

A,B,C,D,E, F Constants

xv

Nomenclature

a0, a1, a2 Fitting constants

B Magnetic field strength

b Width of a single beam

b0, b1, b2 Fitting constants

c Concentration

c∗ Critical concentration

clin, ctor, cbend Linear, torsional and bending spring constant

d Width of the cantilever

D0 Intrinsic damping

d1, d2 Distances to the lid and to the back plate

df Frequency difference (damping)

E Young’s modulus

e Efficiency

F General, unknown force

F (t) Arbitrary force

f(x, t), f(x, ω) Line distribute force

fpeak Peak frequency

fres Resonance frequency

G Shear modulus

G(ω), G′, G′′ Shear modulus with its real and imaginary part

G1, G2 Amplification factors

H Transfer function

h Thickness of the cantilever

xvi

Nomenclature

I, I∗ Area moment of inertia, unity matrix

I0 Amplitude of the excitation current

Iex Excitation current in bit value

IS Ionic strength

K Inverse of the sensitivity matrix

K Mark-Houwink constant

k, k∗ Wave number

K1, K0 Modified Bessel functions of the second kind, constant

KH Huggins constant

l Length of the cantilever

LC Contour length

LP Persistence length

M Moment, Molarity

MW Molecular weight

n Number of conductor loops

Nn, N0 Number of species at cycle n and 0

p Pressure

Qi Quality factor regarding the damping component i

Rloop, R1 Resistances

S Sensitivity matrix

Si,j Sensitivity of measurand i with respect to parameter j

T Torsional moment, period

t Time

xvii

Nomenclature

Uind, Uamp (Amplified) induced voltage

V Volume

v(t) Velocity

v0 Velocity amplitude

w Linear displacement

wl Displacement of longitudinal beam

wtr Displacement of the transversal beam

x, y, z Coordinates

Z Impedance

xviii

List of Figures

1.1. The concept of the PCR . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2. PCR chip designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3. The concept of the RCA . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4. Detection methods based on fluorescence . . . . . . . . . . . . . . . . . . 8

1.5. Alternative detection methods . . . . . . . . . . . . . . . . . . . . . . . . 10

1.6. Measurement principle in dynamic viscometry . . . . . . . . . . . . . . . 13

1.7. Generic amplitude and phase curve of an SDOF system . . . . . . . . . . 14

2.1. Dimensions of the cantilever . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2. Impedance model of the cantilever . . . . . . . . . . . . . . . . . . . . . . 21

2.3. Simulation of the clamping . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4. Calculation of the induced voltage . . . . . . . . . . . . . . . . . . . . . . 28

2.5. Procedure to calculate the fluid force Γ . . . . . . . . . . . . . . . . . . . 32

2.6. Influence of the velocity amplitude on Γ . . . . . . . . . . . . . . . . . . 34

2.7. Influence of the density, viscosity and frequency on Γ . . . . . . . . . . . 35

2.8. Influence of the beam width on Γ . . . . . . . . . . . . . . . . . . . . . . 35

2.9. Influence of the wall distance on Γ . . . . . . . . . . . . . . . . . . . . . . 37

2.10. Correction for a rectangular cross section . . . . . . . . . . . . . . . . . . 38

2.11. Combined model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.12. Calculated dependency on viscosity and density . . . . . . . . . . . . . . 43

3.1. Exploded view of the sensor . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2. Fabrication of the resonator chip . . . . . . . . . . . . . . . . . . . . . . 53

3.3. Fabrication of the lid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.4. Chamber filling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.5. Heater chip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.6. Increase in hydrophility with annealing . . . . . . . . . . . . . . . . . . . 59

3.7. Flow diagram of the gated PLL . . . . . . . . . . . . . . . . . . . . . . . 60

3.8. Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

xix

List of Figures

3.9. Small bubbles at the cantilever . . . . . . . . . . . . . . . . . . . . . . . 63

3.10. Resonance frequency and damping in vacuum . . . . . . . . . . . . . . . 64

3.11. Resonance frequency in air . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.12. SEM image of the clamping . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.13. Tuning of the impedance model . . . . . . . . . . . . . . . . . . . . . . . 67

3.14. Difference between water and glycerol solutions . . . . . . . . . . . . . . 69

3.15. Measured and simulated induced voltage . . . . . . . . . . . . . . . . . . 71

3.16. Measured and simulated sensitivities . . . . . . . . . . . . . . . . . . . . 73

3.17. Viscosity and density calculation . . . . . . . . . . . . . . . . . . . . . . 75

3.18. Influence of temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.1. Intrinsic viscosity of DNA solutions . . . . . . . . . . . . . . . . . . . . . 84

4.2. Models for DNA solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.3. Mark-Houwink exponent from WLC simulations . . . . . . . . . . . . . . 89

4.4. Resonance frequency and damping for different concentrations of DNA . 96

4.5. Measured viscosity and density of DNA solutions . . . . . . . . . . . . . 99

4.6. Reduced viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.7. Measured resonance frequency and damping for PCR samples . . . . . . 104

4.8. PCR with alternative negative control . . . . . . . . . . . . . . . . . . . 106

4.9. Bad reproducibility of PCR experiments . . . . . . . . . . . . . . . . . . 107

4.10. RCA off-chip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.11. Microscope images after filling the RCA samples . . . . . . . . . . . . . . 111

4.12. Gels of on-chip RCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.13. RCA on chip for 90 minutes . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.14. Comparison of RCA and non-reacting solution on chip . . . . . . . . . . 115

4.15. Evaluation of the slope after 30 minutes . . . . . . . . . . . . . . . . . . 116

B.1. Correction for the gated measurement . . . . . . . . . . . . . . . . . . . . 128

xx

List of Tables

2.1. Standard values for the fluid mechanic simulations . . . . . . . . . . . . . 33

2.2. Parameter study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.3. Implemented designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.1. Calibration liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.2. Measured and calculated sensitivities . . . . . . . . . . . . . . . . . . . . 73

3.3. Temperature dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.1. 10 kbp and 110 bp solutions used for titration experiments . . . . . . . . 94

4.2. Chip cleaning procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.3. LOD and nmin in titration experiments . . . . . . . . . . . . . . . . . . . 97

4.4. Values for the intrinsic viscosity . . . . . . . . . . . . . . . . . . . . . . . 98

4.5. PCR reaction composition . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.6. PCR with alternative negative control . . . . . . . . . . . . . . . . . . . 105

4.7. RCA composition off-chip . . . . . . . . . . . . . . . . . . . . . . . . . . 109

A.1. Assembly of the components . . . . . . . . . . . . . . . . . . . . . . . . . 123

A.2. Fabrication steps PDMS lid . . . . . . . . . . . . . . . . . . . . . . . . . 123

A.3. Fabrication steps heater chip . . . . . . . . . . . . . . . . . . . . . . . . . 124

A.4. Fabrication steps resonator chip . . . . . . . . . . . . . . . . . . . . . . . 125

B.1. Measured and calculated correction for the gPLL . . . . . . . . . . . . . 128

xxi

1. Introduction

Miniaturized diagnostic devices have been drawing great attention during the last few

years. They are called lab-on-a-chip (LOC) devices or micro total analysis systems (mi-

croTAS). In an ideal design they are small, fast, need less sample or reagent volume,

less power and are cheaper compared to their large scale counterparts. These prop-

erties make them a very promising tool for various applications. With the advent of

microfluidics and the possibilities of MEMS principles the way to chip based biological

and medical tools was opened.

There is a large range of devices and methods developed in the last decades, from the

very simple paper lateral flow strip tests (e.g. pregnancy tests) to chips with a very broad

variety of functionality such as micro mixers, valves reservoirs, etc. built in. Given the

wide range of microfluidic principles [1], such as pressure driven flow, electro-kinetic flow

or acoustic waves and the wide range of biochemical processes, ranging from nucleic acid

based reactions, immunoassays to cellular assays, the possibilities for new inventions are

immense. Areas of application include drug development, (point-of-care) diagnostics,

agriculture and ecology just to mention a few.

LOC devices have a great potential to improve global health [2]. In developing countries

with almost no resources for sophisticated laboratory equipment, they may improve the

fast and cheap detection of infectious diseases and efficient allocation of medication. The

advantages of LOC devices in such an application are their cheapness, small size and

low power consumption. But the focus lies not only in the cheapness of the diagnostic

tool but also in the simplicity of the handling and robustness. With a generally lower

amount of trained people available, the tools must be easy to use. Also storage and

transportation is a challenge. It may happen that temperatures go up to 40 C and no

refrigeration is available.

The advantages of LOC devices for pathogen detection are useful as well in developed

countries. One example is the detection of methicillin resistant Staphylococcus aureus

(MRSA), which is an antibiotic resistant bacterial species. It causes infections in an

endemic degree [3]. This makes it sometimes necessary to isolate patients when they

1

1. Introduction

enter the hospital. There is a controversy, whether a screening for MRSA at hospital

entries is useful or not [4]. The main issue is the fact that ”rapid” tests take up to half

a day or more when including all the steps needed for the analysis and not only the two

to four hours the actual test takes. This makes it therefore difficult to take action when

necessary. A device that can be used on entry of the patient with a sample-in-answer-out

time of a few minutes would make screening much easier and also more effective.

The analysis of clinical samples can be divided into three steps: Sample preparation,

reaction and analysis. In a LOC, ideally, all three steps are carried out on the same

system and a sample-in-answer-out device is pursued. Sample preparation includes

taking the sample from the patient (blood, saliva, urine), purification and concentration

of the substance needed for the reaction. The actual reaction is enzymatic in many

cases. The analysis of the reaction outcome is strongly linked to the reaction itself. The

following section 1.1 is a short review about chip based nucleic acid testing, since the

goal of this work is such a system.

This work is embedded an a larger project at the institute, where sample preparation

based on ultrasonics and sample evaluation based on viscometry are developed. The aim

of this work is to develop a system for nucleic acid testing based on viscometry where

the reaction and the sample evaluation take place on the same device.

Work in this field has started a few years ago with the thesis of Lukas Bestmann [5].

He developed a cartridge for sample preparation of whole blood. Together with Daniel

Bachi, a microchip with a reaction chamber for the polymerase chain reaction (PCR)

was built. The reaction outcome was evaluated with a melting curve analysis based

on commercially available fluorophores. The system could detect factor V Leiden point

mutations and discriminate between Gram-positive and Gram-negative bacteria.

1.1. Chip based nucleic acid testing

Chip based nucleic acid testing is a very promising field, due to its general nature (testing

a sequence of nucleotides). It can be used for the detection of a high variety of pathogens,

the diagnosis of many different hereditary diseases, water testing, testing for genetically

modified organisms and more. Many standard tests used in centralized laboratories are

based on nucleic acid testing, therefore, many protocols and test sequences exist already.

The main challenge is to adapt them to work in a miniaturized environment or, in other

words, to build a device that is compatible with the existing protocols.

2


dsDNA

Denaturation

Annealing

Elongation

50 C°

50 °C

96 °C

72 °C

ssDNA

Primer

Nucleotide

Polymerase

Figure 1.1: Polymerase chain reaction: In a first step, the mix is heated to approximately96 C to separate the two strands (denaturation). After cooling to 50 C theprimers attach to the single strands according to their sequence (annealing).The polymerase then elongates the hybrids using the nucleotides present inthe solution.

Two reactions are used in this work. The first one, the polymerase chain reaction, is a

widespread reaction used in diagnostics and many other fields in biology. The second

reaction is the rolling circle amplification (RCA). It is an isothermal reaction which

produces very long strands of DNA and therefore is well suited to increase the viscosity

of the product.

1.1.1. PCR

The PCR is a technique to copy a small number of DNA strands in order to get a very

high number of double stranded DNA (dsDNA). It was developed in the 1980s by Kary

Mullis [6]. The whole reaction consists of temperature cycles which are repeated 20

to 40 times. One cycle is illustrated in figure 1.1. In each cycle, the number of DNA

strands is doubled if the reaction works ideally. By using a clever design of the primers,

specificity for a certain base sequence (target) can be achieved.

Different variations of the PCR exist [7]: In multiplexed reactions multiple sets of primers

are used to amplify different sequences in one sample. Reverse transcriptase PCR (RT-

PCR) is used to detect RNA (ribonucleic acid) in a sample. In this reaction the enzyme

reverse transcriptase is used to convert the RNA into its complementary DNA. The

DNA sample can then be amplified using conventional PCR.

3

1. Introduction

Another variation is real-time PCR or quantitative PCR (qPCR), where the amount

of DNA is estimated. After each cycle the fluorescence level is measured and is an

indication of the amount of DNA present in the mix. This can be done either using

fluorescent dyes such as SYBR green or ethidium bromide or by using fluorophores

and Forster resonance energy transfer (FRET). The advantage of this method is the

possibility to quantify the amount of DNA present in a sample. This is especially useful

in combination with RT-PCR, where the amount of expressed RNA is of interest.

The advantage of using PCR instead of classical cell culture is that it is very specific,

independent of antibiotic treatments and faster as well.

Northrup et al. [8] were the first ones who carried out the PCR on a miniaturized

silicon chip and developed a portable system. During the following years, the field was

investigated intensely. The main advantage of the miniaturization, additional to the

advantages of LOC devices in general such as portability, low reagent consumption is the

reduced thermal inertia. This results in much shorter cycle times, which are ultimately

limited by the bio-molecular processes [9]. However there are also disadvantages. The

main issue is the high surface-to-volume ratio. All ingredients, namely the DNA and the

enzymes, tend to stick to surfaces, which reduces the efficiency of the reaction. Another

disadvantage is the fact, that for very low molar concentrations of target molecules a

minimum volume is necessary. Statistically, at least one molecule has to be present in a

diluted sample with a high enough confidence level [10].

Design concepts for miniaturized PCR chips

There are two design concepts for chip PCR which are widely used: Stationary chamber

PCR and continuous flow PCR. The first concept is shown in figure 1.2 a). In the first

design, the reaction takes place in the same chamber. It is basically the miniaturization

of a PCR tube with an integrated heater in the simplest case. It is appealing due to

its rather straight forward design. However, since usually the whole chamber has to be

heated, the thermal inertia is still quite large. In contrast to the continuous flow design

no mechanism to generate flow is necessary.

The second group are continuous flow type chips as conceptually shown in figure 1.2

b). In this design the liquid is moved to different areas with different temperatures

on the chip. The challenges in this design is to engineer the temperature distribution

on the chip. The temperature isolation between different areas of the chip has to be

good. A general problem in microfluidics is the absorption of reaction components on

the surface due to the large surface-to-volume ratio. Since the liquid is moved through

4


a)

b)

c)

d)

Figure 1.2: PCR chip designs (reprinted from Park et al. [7], with permission from El-sevier). a) Stationary, closed chamber with one heater; b) Continuous flowwith one heater for each temperature; c) Continuous flow with droplet gen-eration; d) Droplet based stationary PCR in an open system.

5

1. Introduction

a relatively long channel in the case of a continuous flow type design, this effect is even

amplified. Another challenge is the flow profile. It is not uniform across the cross section

in general. This means that the temperature cycling is not uniform which may cause

problems for the PCR. Means for the generation of flow have to be provided, which is

not the case for stationary chamber designs. This could be a problem for a portable

device. Variations of this design are oscillatory and circular devices, where the chip has

three different temperature regions. In the first group, the sample is moved forth and

back, in the second group the sample is moved in a closed circular channel.

The third group are droplet based systems shown in figure 1.2 c) and d). The reaction

takes place in a droplet which is either generated in a flow system using flow focusing,

fluidic junctions or electric fields as in c) or in an open system as in d). The advantage

of droplet based systems is their very low thermal inertia especially when using an

open system [9]. Other advantages are the possibility to produce a very high number

of droplets which serve as a single reaction chamber. This reduces the risk for cross

contamination. However, up to now, these systems have limited portability due to the

necessity of external valves, tubes and instruments.

1.1.2. RCA

The reason why heating up to 95 C is necessary in the PCR is the separation of the two

strands into single strands in order to free the sites where the primers can attach. Since

this step is time consuming and energy intensive, isothermal methods are an interesting

alternative. There exists a variety of isothermal methods for nucleic acid amplification

[11]. In many cases, the strand displacement ability of an enzyme is used to separate

the two strands instead of heating the mixture up.

The RCA is particularly interesting for a viscosity based sensor, since very long strands

can be produced. The principle of the reaction is sketched in figure 1.3. Usually, circular

single stranded DNA (cssDNA) serves as the target. The DNA polymerase of the Φ29

bacteriophage is used for amplification. It has strand displacement ability, meaning

that the enzyme displaces (but does not degrade) DNA paired downstream from its

synthesis direction. The primers bind to the target and the enzyme starts replication. If

amplification of the whole DNA present is needed, random primers are used. They bind

to various positions on the cssDNA. This way, a very high number of DNA molecules

can be achieved in a decently short time. Additionally the primers bind also to the

sequences which were polymerized. The fact that long strands are produced is expected

to increase the viscosity drastically.

6


ssDNAPrimer Polymerase

Strand displacement

Figure 1.3: Principle of the rolling circle amplification: The primer binds to the targetDNA. Then the polymerase starts amplification. Due to its strand displace-ment ability, very long strands are produced, when a circular target is used.

There have also been efforts to run the RCA on a chip. E.g. Sato et al. [12] used micro

beads. The detection was fluorescence based. In a PhD thesis by Koster [13], the RCA

was run on a 1024 well chip with a reaction volume of 150 nl. The detection was based

on different fluorescent methods.

1.1.3. Post-reaction analysis - detection of DNA

Fluorescence based detection

A complete chip based diagnostic device has to include the analysis of the reaction

product. On one side, the DNA content has to be made ”visible” to the eye or to an

instrument. In most of the cases this is done optically using fluorescence. On the other

side, it is not enough to only know if there is DNA, often the strand length distribution

is of interest.

An overview of some common fluorescence methods is illustrated in figure 1.4. A very

common method is capillary electrophoresis (CE). A long channel in which the DNA

strands are separated in a flow according to their length is used. At the end of the

channel, an optical detector registers the fluorescence and a graph with fluorescence

peaks is produced. Another method is the melting curve analysis, where the change

in fluorescence over temperature is plotted. A strong decrease is observed around the

temperature where the dsDNA is separated into two strands when increasing the tem-

perature. This method is particularly interesting for single nucleotide polymorphism

7

1. Introduction

Threshold

Flu

ore

scnce

Cycle number

a)

c) d)

b)

t

Flu

ore

scen

ce

Fluorescent dye

Figure 1.4: Methods used together with fluorescent detection. All of them are based onfluorescent dyes, which re-emit light if they bind to double stranded DNA. a)Capillary electrophoresis is used to separate DNA fragments with differentlength; b) The melting curve analysis (reprinted from [14]) shows at whichtemperature two strands are separated, indicated by a decrease of fluores-cence; c) For qPCR the fluorescence is monitored for each cycle. By usingdifferent concentrations of the target, it can be quantified; d) Hybridizationarrays have different locations with immobilized oligomers. DNA specificallybinds to these locations.

8


analysis as shown by Bestmann [5], since the melting temperature depends on the exact

sequence.

Figure 1.4 c) shows an illustration of the qPCR with four different concentrations. It is

worth noting, that a plot of the number of dsDNA molecules in the mix has usually an

S-like shape, which is an intrinsic property of the PCR. At the beginning, the reaction

is limited by the number of targets or the limit of detection of the detection system.

Then the reaction has an exponential phase, where the number of dsDNA molecules is

doubled with each cycle. At the end, the reaction is limited by the number of primers,

nucleotides and the health of the polymerase.

Pipper et al. [15] show a droplet based system where a RT-PCR is carried out on. In this

method, a dilution series of the sample is made. The cycle number where the fluorescence

crosses a certain threshold is then plotted versus the logarithm of the concentration. The

method facilitates the quantification and validation of the reaction.

Another group of chips for fluorescent DNA analysis are hybridization arrays. A chip

is coated with a large number of oligo nucleotides in a predefined pattern with different

sequences. DNA in the sample then hybridizes and produces a fluorescence pattern.

There exists a variety and different combinations of the methods mentioned above.

However, the disadvantage of using fluorescence is the need for bulky equipment such

as laser sources, filters and detectors. Although this can partly be overcome with LEDs

or photo diodes it adds another degree of complexity to the instrument. Nevertheless,

these methods are probably the most common ones found in current lab-on-chip devices.

Electrochemical detection

Electrochemical methods are based on electrodes placed in the reaction chamber. Dif-

ferent mechanisms influence the electrical properties near the electrode surface, which

can be measured [16]. The methods can be classified into reagent-less and reagent-based

methods. In the first group the electrical signal change is induced by hybridization of

surface immobilized DNA with the sample itself. In the second group the concentra-

tion of a reagent near the surface is influenced by hybridization or presence of DNA

molecules. For example Defever et al. [17] show a system based on redox probes in the

mixture. Similar to ethidium bromide, these probes bind to dsDNA. Therefore the elec-

trochemical properties of the fluid change, which can be detected with the electrodes.

The method is illustrated in figure 1.5 a).

9

1. Introduction

a) b)

Figure 1.5: Alternative detection methods: a) Electrochemical detection of DNA(reprinted with permission, from Defever et al. [17]. Copyright 2011 Ameri-can Chemical Society). The redox probes (red) bind to DNA. As the concen-tration is increased, the conductivity of the liquid is reduced; b) Cantileverscan be used to detect molecules in a solution in the static mode, where themolecule binds to a (functionalized) surface and induces mechanical stressor in dynamic mode, where the added mass of the molecule changes theresonance frequency (after Alvarez et al. [18]).

10

1.2. Dynamic viscometry for nucleic acid testing

Mechanically based detection

Figure 1.5 b) shows two different detection principles which can be applied using a

cantilever. In general, one can distinguish between static and dynamic mode. In the

first case the principle is based on a reaction that takes place on the surface of the

cantilever and induces a mechanical stress. Due to the stress, the cantilever is bent. In

the dynamic mode, the resonance frequency or the damping of the cantilever is measured.

The resonance frequency is very much dependent on the (added) mass of the cantilever.

For both methods, the cantilever has to be coated, which may be especially difficult when

using the static mode, where only one side has to be functionalized. The advantage of

functionalization, however, is the high specificity that can be achieved. Typical coatings

are (swelling) polymers, self assembled mono layers, hydro gels or brush macromolecules.

These principles have been applied for the detection of DNA [19], but usually a signal

amplifying mechanism, such as gold beads or functionalized polymers is used. Also

many applications with antibody detection or the detection of whole cells have been

shown [20, 18].

Since the (mass) sensitivity in the dynamic mode depends on the Q-factor it may get

very difficult to measure when the cantilever is immersed in aqueous solutions, therefore

mostly the static mode has been used recently.

Another mechanical detection method is the use of piezoelectric transducers [21]. Modes

of operation are the thickness extensional mode, thickness shear mode, the lateral ex-

tensional mode and the flexural mode. Detection of DNA has been shown with these

types of sensors using coatings that hybridize with the target DNA [22]. Surface acous-

tic wave transducers, which work in a similar way, have also been shown to be able to

detect DNA [23, 24] .


The approach which is presented in this thesis is based on the measurement of the

fluidic properties of the sample. Assuming that the density and the viscosity change

during the reaction, a device measuring one or more of these quantities can be used

to monitor the progress of the reaction or to analyze the reaction outcome at the end.

Curtin and coworkers [25] tried to apply this principle to PCR using micro particle

image velocimetry (µPIV). However, they were not able to see any viscosity change.

11

1. Introduction

Compared to the fluorescent methods described above the viscometric approach has

some advantages. The viscosity measurement is in general less complex, since it needs

only a detecting unit. This is in contrast to the fluorescent methods (FRET, ethidium

bromide), where an excitation source (laser diode, UV lamp) and a read-out unit are

needed. If measuring the viscosity or density directly, the detection is label-free. This

means on one side, that the DNA does not have to be labeled with fluorescent dyes

or similar means. And on the other side, the sensing element does not have to be

functionalized either. This makes the fabrication of the sensors simpler and also more

stable over time, which may be important for applications in developing countries. All

in all, the viscometric approach should make it possible to yield much simpler devices.

Another advantage of this approach is the fact, that the actual measurement can take

place in the reaction chamber itself. This means, that a real-time measurement could

be possible, given that the sensitivity of the measurement is high enough.

However, the simplicity comes with the disadvantage of less information. If detecting

viscosity and density, only one or two values can be read out. This is in contrast to CE

devices, where information about the length of DNA strands is available. Multiplexed

assays, which are possible with fluorescent dyes of different color, are not possible as

well. In other words, a device based on a viscosity measurement could only detect one

pathogen at once. However, if one aims for a sample-in-answer-out device, a yes or no

answer may be enough.

One advantage of using fluorescence is its very low limit of detection. SYBR green for

example can detect 20 pg of dsDNA when used in an agarose gel [26]. A pure viscosity

measurement will never be sensitive enough to detect such low concentrations (compare

section 4.2). An approach to overcome this problem is to use a reaction that amplifies

the viscosity, namely the RCA.

Selectivity for a particular sequence comes from the selectivity of the polymerization

process via the primers. A second level of selectivity can be achieved by using fluorescent

markers which bind to a specific sequence in the amplified product. This method is not

feasible with viscometry. However, an approach to selectively increase the viscosity

of the product is to link the strands of the amplification product using specific DNA

sequences and a linking agent such as ligase.

12


I

B

FL

Visco

sity

Density

Measurement of resonancefrequency and damping

Calibration curve

Density and viscosityDirect analysis of reaction

Res

onance

freq

uen

cy/ D

am

pin

g

Figure 1.6: Observation of the reaction with a vibrating cantilever. Left: the U-shapedresonator is immersed in the liquid, which influences the resonance frequencyand the damping depending on the amount and length of DNA strands. Thecantilever is driven with the Lorentz force FL generated by an alternatingcurrent I and a permanent magnetic field B. Right: for the calculation ofthe viscosity and density a calibration is needed.

1.2.1. Working principle

The working principle of the system discussed in this thesis is illustrated in figure 1.6.

A U-shaped cantilever vibrates in the liquid 1. It is driven by the Lorentz force, which

is generated by an alternating current flowing through the cantilever and a permanent

magnetic field from a magnet. The liquid influences the resonance frequency and the

damping of the system. These two values can be used directly to observe or analyze a

reaction. Intuitively, an increase in damping and a decrease in the resonance frequency

is expected as more and more DNA strands are polymerized.

If the explicit values for viscosity or density are sought (e.g. in SI units), a scheme for

the calculation of these values out of the resonance frequency and damping has to be

found. One approach is to use an accurate model to calculate calibration curves, which

1The concept of a U-shaped cantilever is based on an idea by Joe Goodbread, Portland, USA andJurg Dual, Center of Mechanics, ETH Zurich, where a bent metal wire of several millimeters in sizewould have been used to measure viscosity and density. The device shown in this thesis is basicallythe miniaturization of this concept.

13

1. Introduction

Frequency

Am

plitu

de

Phase

Á

Frequency

df

fres

f fres peak/

df

¢®

¢®

Figure 1.7: Generic amplitude and phase curve of the transfer function of an SDOFsystem. The resonance frequency fres and the peak frequency fpeak areindicated on the amplitude spectrum. The value df is the difference betweena phase difference of ±∆α with respect to the driving force and is a measurefor damping. The resonance frequency is indicated by zero phase difference.

relate the resonance frequency and damping to the properties of the fluid. A simpler,

linear approach based on sensitivities is presented in section 2.5.

In figure 1.7, the two important parameters resonance frequency fres and the damping df

are illustrated. If the modes of vibration of the cantilever are well separated, one mode

can be treated as a single degree of freedom system (SDOF). The resonance frequency

and the frequency of the velocity peak fpeak coincide for such a system, whereas the

frequency of the displacement peak does not coincide with the resonance frequency for

a damped system. The resonance frequency is defined as the frequency at which the

phase difference of the velocity signal crosses zero, as shown on the right of figure 1.7.

Damping can be characterized in many different ways. For this thesis, the notion of the

frequency difference df shall be used. It is defined as the difference between the two

frequencies where the phase difference φ between excitation and readout is ±∆α. In the

case where ∆α is 45, df is identical to the commonly used term bandwidth.

These definitions make sense in view of the fact, that a phase locked loop (PLL) will

be employed to measure fres and df . This kind of electronics sets the phase difference

φ between excitation and readout to a predetermined value by adjusting the frequency

of the excitation. Setting φ to zero and ±∆α, directly reveals fres and df . Throughout

the thesis, ∆α = 22.5 is used.

The Q-factor can be calculated from

Q = 2πStored energy

Energy dissipated per cycle=fresdf

tan(∆α) (1.1)

14

1.3. Outline of the thesis

It is inversely proportional to the damping of the system. This is not the case for df ,

which increases, as the resonance frequency of a system increases while its damping is

constant. The Q-factor has several components [27]:

1

Qtot

=1

Qvisc

+1

Qac

+1

Qtherm

+1

Qsupport

+1

Qmaterial

(1.2)

Qvisc represents viscous losses. Qac represents acoustic losses, which arise through ra-

diation of energy in a compressible fluid. Qtherm represents thermoelastic losses, which

are caused by the energy flow due to strain-induced temperature gradients as the can-

tilever vibrates. The energy that flows into the support due to the stresses that arise at

the clamping causes Qsupport to decrease. The energy loss due to material damping is

indicated by Qmaterial.

An important property of sensors is their sensitivity. In the case of a vibrating cantilever,

the resonance frequency and the damping are sensitive to changes in viscosity and density

amongst other influences. The sensitivities are defined as

Sfres,ρ =∆fres∆ρ

Sfres,η =∆fres∆η

(1.3)

Sdf,ρ =∆df

∆ρSdf,η =

∆df

∆η(1.4)

where ∆ indicates a difference in the corresponding value. So, e.g. Sfres,ρ indicates the

sensitivity of the resonance frequency with respect to density changes. The dependen-

cies of the resonance frequency and damping are not linear in general. Therefore, the

sensitivities are not constant for an arbitrary range of fluids.

1.3. Outline of the thesis

The model of the vibrating cantilever is described in chapter 2. After introducing an

impedance based mechanical model along with an expression for the induced voltage, the

influence of the surrounding fluid is investigated numerically. Based on these simulations,

the geometry for the cantilever is derived and a procedure for the calculation of the

viscosity and density out of the resonance frequency and damping is given.

Chapter 3 covers the experimental part treating the cantilever as a viscosity and density

sensor. After presenting the details about fabrication and the experimental setup, the

15

1. Introduction

sensor is characterized in view of sensitivity and accuracy. Parasitic effects such as

temperature effects and losses are investigated as well.

Chapter 4 covers experiments with DNA solutions. In a first step the cantilever’s re-

sponse to solutions of different concentrations of 100 bp and 10 kbp long DNA strands

in buffer is studied. Afterwards, the sensor is tested for the use as a diagnostic device

using the PCR. Due to the difficulties encountered with this reaction, the sensors were

also tested using the RCA with very good results.

16

2. Modeling of the structural

mechanics and the fluid structure

interaction

A model of the cantilever immersed in the fluid shall be used for the design of the

sensor, i.e. for the optimization with respect to sensitivity and to check the fulfillment of

technical limitations. In order to relate the measured resonance frequency and damping

to the properties of the fluid, an accurate model of the resonator is of great help.

After a short discussion of the state-of-the art of modeling immersed cantilevers in sec-

tion 2.1, the model is introduced. It has to include the structural mechanics of the

resonator itself and the forces which arise due to the surrounding fluid. The combina-

tion of these two domains is called fluid-structure-interaction (FSI). Accordingly, the

modeling is divided into two parts. The mechanical model will be first introduced in

section 2.2. It reflects the sensor’s mechanical behavior in vacuum and is built by con-

necting sub-elements which are based on Euler-Bernoulli beam theory. It is therefore

one-dimensional. For predictions of the sensor’s behavior in a fluid, a model for the

fluid forces on the cantilever is introduced in section 2.3. The model is two-dimensional.

Therefore, the complete model including structural mechanics and fluid flow shown in

section 2.4 is a 1D-2D-model.

The model itself will give the resonance frequency and damping as an output for a given

geometry, density and viscosity. For the calculation of the properties of the surrounding

liquid the problem has to be inverted. Namely a mathematical expression for the vis-

cosity and density for given resonance frequency and damping is sought. An approach

for the inverted problem will be presented in section 2.5.

The modeling is experimentally validated in the next chapter in section 3.3, after the

introduction of the experimental setup.

17

2. Modeling of the structural mechanics and the fluid structure interaction

2.1. State-of-the-art

Ideally, the model describing the behavior of the cantilever is three-dimensional and

includes the mechanical structure and a fully coupled viscous, compressible fluid in an

infinite or finite volume. Although some of these demands are built into commercial finite

element (FEA) software nowadays, the computational requirements are still tremendous

when not applying any simplifications. Such simulations are extremely time consuming

or even not possible at all. Since the frequency domain is of interest, the governing

equations have to be formulated accordingly. Numerical simulations tend to be time

consuming and have to be run on a powerful computer. After all, they work only in one

direction, meaning that the properties of the fluid, i.e. viscosity and density, are given.

However, what is sought is an inverse formulation in order to be able to calculate the

fluid’s properties. Although some groups have tried modeling with 3D FEA including

several physical domains (e.g. Basak et al. [28]), the general goal is to find an analytical

or at least semi-analytical expression for the cantilever’s vibration.

Elmer and Dreier [29] derived expressions for the fluid forces on a cantilever for an in-

finitely thin cross section. In order to calculate the forces, the cantilever is assumed to

be much longer than the largest dimension of the cross section. If this is the case, the

derivatives in longitudinal direction are much smaller compared to the ones in lateral di-

rection, at least for the low order modes. Therefore, solving the Navier-Stokes equations

reduces to a two-dimensional problem. The third important assumption in their model

is an inviscid fluid along with small vibration amplitudes compared to the cantilevers

width.

In 1998 Sader derived expressions making similar assumptions, namely the cantilever

is still assumed to be infinitely thin. However, the fluid is assumed to be viscous in

contrast to the Elmer-Dreier model. To model a rectangular cross section, Sader derived

a correction factor which is applied to the expression for the force on a circular cylinder.

This expression was derived by Stokes [30]. The Sader model has been further simplified

by Maali et al. [31]. They reduced the values for the fluid force acting on the cross

section calculated by Sader to expressions depending on the width of the cantilever

and the boundary layer thickness and four constants. Various groups applied either

of these models to compare their experimental results with theory (see section 3.1).

The expressions of Maali found an application in a paper by Youssry et al. [32], where

they used them for the direct calculation of viscosity and density out of the resonance

frequency and damping.

18

2.2. Mechanical model of the cantilever

b

h

A A-

Frame

Vibrating part

Fluid d

b

l

b

A AVibrating part

Figure 2.1: Dimensions of the cantilever. The longitudinal beam has the length l andwidth b. The transversal beam has the length d and the same width. Thebeam’s thickness is h.

The work of Sader was extended by himself and his coworkers to include torsional

vibrations [33] and compressible fluids [34] for the application with gases. Recently

they extended their model to work with rectangular cross sections with arbitrary ratios

between thickness and width [35].

A completely different approach was chosen by Etchart et al. [36]. They compared

their measurements with the expression for the FSI of a vibrating sphere and a simple

harmonic oscillator.

Although there are expressions for the problem with a circular cross section surrounded

by a boundary (see e.g. Retsina et al. [37]), there is no literature on the problem where

the cross section is rectangular and has a finite width/thickness ratio and is bounded

by a wall.

The structural vibration is calculated from the Euler-Bernoulli beam equations in most of

the cases found in literature. If only one mode of vibration is of interest, the equations

can be simplified. Boskovic et al. [38] used the approximation of a simple harmonic

oscillator to describe the first mode and used Saders expressions for the fluid load.


The dimensions used in the mechanical model are shown in figure 2.1. Since the model

is one-dimensional in nature, the lateral dimensions l and d are given with respect to the

center line of the beam. The transversal and longitudinal beams are introduced with

the same width b.

19


The expressions yielding resonance frequency and damping for a cantilever or a clamped-

clamped beam are well known. However, in this case, the geometry is a bit more

complicated. For this reason, a model which can predict the resonance frequency of

the first mode of vibration shall be derived in this chapter, following the impedance

modeling theory shown by Dual [39].

The impedance of a mechanical element is defined as

Z =F (t)

v(t)(2.1)

where F is the force at a given point and v the point’s velocity. Using this concept,

the response of different mechanical elements, such as springs, point masses or beams,

can be calculated. The approach is much easier to implement than e.g. a finite element

analysis, but on the other side limited to rather simple geometries.

2.2.1. Impedance model

The model which is used to describe the mechanical behavior of the resonator is sketched

in figure 2.2, where also the variables are defined.

The following assumptions and restrictions apply.

• The compliance of the mechanical support of the cantilever can be modeled with

linear and torsional spring elements.

• The layers of silicon oxide, silicon nitride and gold can be modeled as added mass

on the cross section of the beam but do not add to the cantilever’s stiffness.

• Temperature dependency is described by a temperature dependent Young’s mod-

ulus and the linear expansion coefficient.

• There is symmetry with respect to the x-y-plane at the center of the transversal

beam. The model is therefore restricted to symmetric modes and can e.g. not

describe the second mode of vibration which is antisymmetric.

• Acoustic effects, i.e. compressibility and acoustic streaming are neglected.

• The influence of the corners is neglected.

• The excitation can be described by a single force at the center of the transversal

beam.

20


F tL( )

µl( , )x t

wtr( , )x t1

wl( , )x tx

y zZ10

Z20

Z30 Z12

Z31

Z11

x1

y1

z1

Figure 2.2: Impedance model of the cantilever. The first index i in the impedance Zijrepresents the type of load (1: force, 2: bending moment, 3: torsion) andthe second index j indicates the position (0: at the support, 1: at thecorner, 2: at the center of the transversal beam). The linear and torsionaldisplacements of the longitudinal and transversal beams are indicated by wl,wtr and θl, respectively. The excitation is introduced as a point force FL(t)at the center of the transversal beam.

Spring elements

The model consists of two types of elements. The first type are springs, which are

introduced to represent the non-rigid support. The impedance of a spring element can

be calculated from the Hookean law as in equation 2.2, where clin, wl and F are the

linear spring constant, the deflection in z-direction and an unknown force F , respectively.

With

F = clinwl(0, t) (2.2)

F = F0eiωt (2.3)

wl,t(0, t) = iωwl(0, t) (2.4)

assuming a harmonic motion (equations 2.3 and 2.4) and some manipulation yield an

expression for the impedance Z10 of the spring:

Z10 =F

wl,t(0, t)=−iclinω

(2.5)

The expressions for the impedances for bending Z20 and torsion Z30 can be derived in

an analogous manner. For this the linear spring constant clin is replaced by the spring

21


Variableboundary condition

Load

Figure 2.3: A three dimensional FEA model of a single clamped beam on a large blockwas used to calculate the spring constants clin, ctor and cbend. The area onwhich the boundary condition was set to either fixed, free in z-direction orcompletely free to calculate wbeam, wlin + wbeam and wtot is marked with apink rectangle.

constants cbend and ctor relating moments to angles of rotation and wl,t is replaced by

wl,xt and θl. This yields

Z20 =−icbendω

Z30 =−ictorω

(2.6)

The numerical values for the spring constants are deduced from three dimensional finite

element simulations of a single cantilever which is attached to a large, solid block. The

total deflection of the cantilever’s tip caused by a force applied at the tip is given by

wtot = wbeam + wlin + wbend (2.7)

where wbeam is the deflection caused by the compliance of the cantilever itself and wlin

and wbend are the deflections caused by the linear and bending compliance of the clamp-

ing. The deflections wi are calculated with Comsol Multiphysics R© by setting the bound-

ary conditions at the clamping to either fixed (wbeam), free in z-direction (wlin+wbeam) or

free (wtot) as shown in figure 2.3. The spring constants of each element can be calculated

as

cbeam =F

wbeamclin =

F

wlincbend =

Fl2

wbend(2.8)

where cbeam is the spring constant of the beam only. The total spring constant ctot is

given by1

ctot=

1

cbeam+

1

clin+

l2

cbend(2.9)

22


Introducing a parameter δ with wlin = δ(wlin + wbend) and using Hooke’s law yields

clin =1

δ

(ctotcbeamcbeam − ctot

)(2.10)

cbend =l2

1− δ

(ctotcbeamcbeam − ctot

)(2.11)

The torsional spring constant ctor is calculated analogously by applying a bending mo-

ment to the clamped beam in the simulation. More details regarding the calculation of

the spring constants are outlined in the semester thesis of Hasler [40].

Beam elements

The second type of elements are beams. The expressions for the impedance of these

parts of the cantilever are derived based on Euler-Bernoulli beam theory. The partial

differential equation describing the movement of a beam is given by

EIw,xxxx (x, t) + ρsAw,tt (x, t) = f(x, t) (2.12)

where EI is Young’s modulus times the second moment of area, ρs the density of the

beam’s material and A its cross section. The line distributed force f(x, t) acts on the

cross section of the beam. This equation is of general nature. Therefore, w can be wl or

wtr. Assuming time harmonic behavior with angular frequency ω and using a separation

ansatz yields

w(x, t) = ϕ(x)eiωt (2.13)

f(x, t) = f(x)eiωt (2.14)

With this we get

ϕ′′′′ − k4ϕ = f/EI (2.15)

where k4 = ω2 ρAEI

is the wave number.

The influence of the surrounding liquid is introduced via the line distributed force f . In

literature, this is usually done in the following form

f(x) =π

4b2ρω2Γ(ω)ϕ(x) (2.16)

23


where b and ρ are the width of the beam and the density of the fluid, respectively. Note

that this expression is a complex function of ω and involves contributions related to the

acceleration, velocity and displacement of the cross section. The hydrodynamic function

Γ is complex valued with Γ = γ′ + iγ′′. The real and the imaginary part of Γ can be

interpreted as added mass and added damping due to the fluid. The exact form of the

function Γ(ω) will be introduced in section 2.3. The fluid forces can be included in the

wave number, yielding a new expression for k

k?4 =ω2

EI

(bhρs + b2

π

4ρ(γ′(ω) + iγ′′(ω))− iD0

). (2.17)

Here, a value D0 to include damping effects discussed in section 3.3.6 is already intro-

duced. Solving the ordinary differential equation 2.15 yields

ϕ(x) = A cos(k?x) +B sin(k?x) + C cosh(k?x) +D sinh(k?x) (2.18)

This equation has four constants A, B, C and D. They can be replaced by introducing

boundary conditions and the definition of the impedances at the corresponding points.

In order to link the movement of the transversal beam to the longitudinal one, an

expression for the torsion in the longitudinal beam has to be found. The derivation

starts with the partial differential equation for torsion

θ,tt =GI?

ρsIpθ,xx (2.19)

where G is the shear modulus, I? is the torsion moment of inertia and Ip is the polar

moment of inertia. For a rectangular cross section I? is given by Timoshenko and

Goodier [41]

I? =1

3bh3

(1− 192

π5

h

b

∞∑n=1,3,5,...

1

n5tanh

nπb

h

)(2.20)

In the model, only the first term of the sum is used due to the fast convergence of

n−5. Again, equation 2.19 is of general nature, therefore θ can be either θl or θtr.

Assuming harmonic behavior, an ordinary differential equation with the wave number

p = (ω2ρsIp/GI?)1/2 is found

η′′(x) + p2η(x) = 0 (2.21)

24


Solving this equation yields

η(x) = E sin(px) + F cos(px) (2.22)

with two unknown constants E and F . They will be replaced later by introducing

boundary conditions and impedances.

Longitudinal beam

By introducing the impedances Z10, Z20 and Z11, three of the constants A-D can be

eliminated (equations 2.23a, 2.23b and 2.23c). The fourth one can be eliminated with

the boundary condition 2.23d which is based on the assumption of zero bending moment

at the end of the longitudinal beam. Therefore the angular momentum of the transversal

beam as well as any corresponding effects are neglected.

Z10 =Q(0, t)

wl,t(0, t)= i

EI

ω

ϕ′′′(0)

ϕ(0)(2.23a)

Z20 =M(0, t)

−wl,xt(0, t)= −iEI

ω

ϕ′′(0)

ϕ′(0)(2.23b)

Z11 =Q(l, t)

wl,t(l, t)= i

EI

ω

ϕ′′′(l)

ϕ(l)(2.23c)

M(l, t) = 0 = −EIϕ′′(l, t)eiωt (2.23d)

Here, the expressions for the bending moment M(x, t) = −EIw,xx and the shear force

Q = −EIw,xxx have been used. Note that the index in ϕl(x) has been omitted for

the sake of brevity. With these definitions, an expression for the impedance Z11 at the

corner can be found:

Z11 =iEI

ωk?3

(α1ζ + α2) sin(k?l)− cos(k?l) + (−α1 − α2ζ) sinh(k?l) + ζ cosh(k?l)

(α2 + α1ζ) cos(k?l) + sin(k?l) + (−α1 − α2ζ) cosh(k?l) + ζ sinh(k?l)

(2.24)

where

ζ =α2 cos(k?l) + sin(k?l) + α1 cosh(k?l)

sinh(k?l)− α1 cos(k?l)− α2 cosh(k?l)(2.25)

25


and

α1 =1

2(ξ10k

?3 − 1/(ξ20k?)) ξ10 = i

EI

ωZ10

α2 = −1

2(ξ10k

?3 + 1/(ξ20k?)) ξ20 = −i EI

ωZ20

are introduced for better readability.

Equation 2.22 has two unknown constants E and F . These can be replaced by intro-

ducing the following impedances

Z30 =T (0, t)

θl,t(0, t)= −iGI

?

ω

η′(0)

η(0)(2.26a)

Z31 =T (l, t)

θl,t(l, t)= −iGI

?

ω

η′(l)

η(l)(2.26b)

where the definition of the torsional moment T (x, t) = GI?θl,x has been used. The index

in ηl is again dropped for the sake of brevity.

This yields an expression for the impedance Z31 at the corner due to torsion

Z31 = −ipGI?

ω

cos(pl) + i GI?

ωZ30p sin(pl)

sin(pl)− i GI?ωZ30

p cos(pl)(2.27)

Transversal beam

The impedance Z12 at the center of the transversal beam is given by

Z12 =Q(d/2, t)

wtr,t(d/2, t)= i

EI

ω

ϕ′′′(d/2)

ϕ(d/2)(2.28)

With this definition for Z12 and the following boundary conditions, the constants A-D

are eliminated from equation 2.18. The boundary conditions read

Z31 =M(0, t)

−wtr,x1t(0, t)= −iEI

ω

ϕ′′(0)

ϕ(0)(2.29a)

Z11 =Q(0, t)

wtr,t(0, t)= i

EI

ω

ϕ′′′(0)

ϕ(0)(2.29b)

ϕ′(d/2) = 0 (2.29c)

26


Equation 2.29a links the torsional component of the longitudinal beam to the bending

of the transversal beam. The link between the shear forces in the two beams is made

with equation 2.29b. The last boundary condition 2.29c enforces symmetry. The form

of Z12 is identical to equation 2.24. However ζ, k?, α1 and α2 take different values.

Finally, all unknown constants have been replaced with impedances and the springs

and beams are connected with appropriate boundary conditions. This allows calculat-

ing 1/Z12 for a given ω which yields a complex value for the inverted impedance at

this frequency. In order to calculate the resonance frequency and the damping df , the

frequencies fulfilling the following conditions have to be found numerically

Arg(1

Z12

)!

= 0 =⇒ fres (2.30)

Arg(1

Z12

)!

= ±∆α =⇒ df (2.31)

2.2.2. Temperature dependency

The model includes a temperature dependent Young’s modulus in the form of

E(T ) = E(T0)−BTe−T0/T (2.32)

where B is a constant taken from literature [42]. All the dimensions (i.e. l, d, b and h)

were scaled according to

l(T ) = l(T0)(1 + α(T )(T − T0)) (2.33)

where α (values from Virginia Semiconductor Inc. [43]) is the temperature dependent

thermal expansion coefficient in the 110 direction of the silicon crystal.

2.2.3. Induced voltage

The control system which is employed to measure resonance frequency and the damping

is based on an excitation with the Lorentz force and an inductive readout of the cantilever

vibration. For this reason, the induced voltage Uind, which is generated in the moving

beam in the permanent magnetic field, is an important design parameter. A first order

model describing the most important effects for the first mode is sought to give guidelines

27


B

A

Uind

di

li

Figure 2.4: Model used to calculate the induced voltage in the cantilever. The inducedvoltage Uind is proportional to the time derivative of the magnetic flux Φ.The magnetic flux is proportional to the area A spanned by the deflectionamplitude and the widths di. The magnetic field strength B is assumed tobe uniform.

for the design. The restrictions that apply are mentioned as the equations are introduced.

The model which is used for the estimation of this quantity is sketched in figure 2.4.

The force is reduced to a point force acting on the center of the transversal beam and

the magnetic field is assumed to be homogeneous. Thus, equation 2.28 can be used

straightforward. The amplitude of the Lorentz force is given by

FL = I0BK1

n∑i

di (2.34)

where I0 is the amplitude of the driving current, B the field strength of the magnetic

field, di is the length of one conductor line in transversal direction and n is the number

of conductor loops. The equation is simplified assuming that the mean of all di is equal

to the length d of the transversal beam. In this case, the sum degenerates to nd. The

factor K1 reflects the fact, that the conductor loops are spread over the surface of the

cantilever. It is estimated from

K1 =n∑i

diwl,t(li)

dwl,t(l)(2.35)

where l and li are the nominal length of the transversal beam and the positions of the

transversal conductor lines, respectively. The velocities wl,t(li) have to be calculated at

the positions li as shown in figure 2.4.

The induced voltage is then given by

Uind = −∂Φ

∂t= −nK1dBwtr,t(d/2, t) (2.36)

28

2.3. Fluid structure interaction

where Φ is the magnetic flux produced by the permanent magnet. The velocity can be

calculated from the impedance model using

wtr,t(d/2, t) =FL

2Z12

= I0dBK1abs

(1

2Z12

)(2.37)

The factor 2 comes from the fact, that only half of the beam is modeled and therefore

the impedance at the center of the transversal beam has to be doubled to calculate the

force. From these relations, the induced voltage for a given current amplitude can be

calculated:UindI0

= K21n

2d2B2 1

2Z12

(2.38)

It is important to note that this expression contains the impedance 1/Z12 and therefore

depends on the accuracy with which this value can be predicted.

With the theory presented in section 2.2, it is now possible to calculate fres and df of

the cantilever in vacuum, because the fluid forces represented by Γ are zero for vacuum.

The next step is now, to find an expression for Γ with a fluid.


In this section, an expression for Γ(ω) in equation 2.16 shall be derived. This term

reflects the forces acting on the cantilever due to the surrounding fluid. There is an

analytical expression for the case, where the cross section of the beam is circular. It

was derived by Stokes [30]. However, in the case of the system presented here, the

cross section is rectangular. Sader [44] proposed a complex valued, frequency dependent

correction factor for thin beams with a rectangular cross section in the form of

Γrect(ω) = Ω(ω)Γcirc(ω) (2.39)

where Γcirc is the analytical solution of the circular cross section from Stokes and Ω(ω)

is the correction factor.

The analytical solution for the circular cross section is given by

Γcirc = 1 +4iK1(−i

√iβ)√

iβK0(−i√iβ)

(2.40)

29


where K0 and K1 are modified Bessel functions of the second kind and

β = ρωb2/(4η) (2.41)

is a dimensionless number. It is sometimes called Reynolds number (e.g. in [44]) al-

though it is not the same as the commonly known definition of the Reynolds number.

The numbers for the correction factor Ω(ω) given by Sader are based on the assumption

of a very high ratio b/h between width of the beam and its thickness. In the present case,

however, this ratio is around 3. An additional assumption of Sader is, that there is no

nearby boundary. In the case of the viscometer, however, the boundary is 200 µm away

from the cantilever surface. Assuming that its width b is 200 µm and is the characteristic

length scale, the assumption of a well separated boundary is no longer valid.

In order to get an expression which is also valid for a nearby wall and an almost quadratic

cross section, a procedure to calculate Γ using finite element analysis will be presented in

this section. It is assumed, that the general form of the physically based hydrodynamic

function 2.40 along with equation 2.39 is still valid for this geometry. However, a new

formulation for the complex valued correction function Ω(ω) has to be found.

The general idea is to use finite element simulations for this task, where different ge-

ometry cases are employed to derive Ω(ω) based on these simulations. In the FEA

simulations the Navier-Stokes equations are solved. They read

ρ∂u

∂t︸︷︷︸1

+ ρ(u · ∇)u︸︷︷︸2

= ∇

−pI + η(∇u + (∇u)T )︸︷︷︸3

(2.42)

where u is the velocity field, I is the identity matrix, ∇ is the Nabla operator and p

the pressure. If the velocity amplitude v0 of the beam’s vibration is small enough, the

nonlinear term (2) can be neglected. This case is called Stokes flow, where second order

effects, such as streaming are neglected. For an estimation of the order of magnitude

of the different terms it is assumed, that a Stokes boundary layer is built up by the

oscillation. The boundary layer thickness δ is given by

δ =

√2η

ρω(2.43)

where η and ρ are the viscosity and the density of the fluid and ω is the angular frequency.

Using the relations ∇ ≈ 1/δ, |u| ≈ v0 and ∂/∂t ≈ ω and the properties of water for

30


ρ and η and a frequency of 12 kHz leads to the conclusion, that Stokes flow can be

assumed for a vibration amplitude v0 of up to 0.1 m/s.

2.3.1. Finite element model

The half model of a rigid, rectangular cross section moving translationally in an incom-

pressible, linearly viscous fluid was built in Comsol Multiphysics R©. All the simulations

presented here were made with version 4.2a. The velocity distribution around a cross

section of the cantilever was simulated in time-domain with the Creeping flow (Stokes

flow) interface of the CFD module. A sketch showing the geometry parameters and

an image of the calculated velocity magnitude is shown in figure 2.5. The velocity

vy = v0 sin(ωt) with v0 = 10−5 m/s was prescribed on the boundary of the cross section.

This is smaller than in the experiment (see chapter 3), which is, however not a problem

as long as the assumption of Stokes flow is valid. The outer boundaries were set to

either open boundary or wall for different cases. A symmetry boundary condition was

set on the symmetry line. The mesh size was set to 0.1 · 10−6 m. This allows to resolve

the boundary layer (2 µm according to equation 2.43) with at least 10 nodes even at

100 kHz. In order to further increase the accuracy, the mesh growing rate hgrad was

set to 1.025. Three periods T = 2π/ω were simulated and 20 time steps per period

were saved. The BDF solver was used for time integration, together with the direct

solver Pardiso. Preset values were used regarding solver settings, except for the toler-

ance atol, which was set to 0.05 · 10−5. The total force f(t) acting on the cross section

was calculated by integration of the total stress over the boundary at each time step.

The model was first built with the graphical user interface of Comsol Multiphysics R© and

then saved as an .m-file. This allows to run the simulations in a batch mode (e.g. for

different frequencies) via the Matlab R© interface.

The simulated force values were fitted to f(t, ωi) = f0 sin(ωit + κ), where f0 and κ are

the force amplitude and phase, respectively. From these values

H0 = f0/v0 (2.44)

H = H0eiκ (2.45)

can be calculated. The quantity H is therefore complex and depends on ω. It relates

the results from the time domain simulations to the frequency domain, hence it shall be

called transfer function. From equation 2.16 it follows that

31


d2

h b

d1

v v !ty= sin( )0

symmetry

0

0.2

0.4

0.6

0.8

1x10

-5

wall/open

wall

Figure 2.5: Left: Parameters used in the fluid mechanic model of the two dimensionalflow around the cantilever, with the gray area indicating the simulated part.Right: FEA result showing a close up view of the velocity magnitude in thetime domain in m/s.

H =f

v= −iπ

4b2ρωΓsim (2.46)

With this equality, theoretical values for the fluid forces, namely the one obtained from

the Sader theory, and simulated values can be compared.

2.3.2. Influence of different parameters on Γ

Sader gives a numerical approximation for the value of Ω(ω) as a function of τ =

log10(β) (see equation 2.41). It seems therefore natural to use a similar expression here.

According to equation 2.41, the hydrodynamic interaction function Γ should depend on

ρ, η, b, and ω (see figure 2.5 for the naming of the dimensions). Obviously, for the case

at hand, the dependency on b is more complex. Additionally, there is a dependency on

the thickness h of the beam and the distances d1 and d2 from the wall. The influence of

all these parameters on Γ shall be investigated. The following plots show values for the

real and imaginary part of Γ. For a cantilever with dimensions of the final design (see

section 2.4.1), a 1% increase in the real part of Γ changes the resonance frequency and

df by -0.3% and -0.7% respectively. The same change in the imaginary part changes

the value for df by 1% but does hardly affect the resonance frequency. The standard

32


Variable Value

Simulation interface Creeping flowb 200 µmh 70 µmd1 200 µmd2 ∞ (open boundary)v0 10 µm/sρ 1000 kg/m3

η 0.001 Pas

Table 2.1: Standard parameters used in the fluid mechanic simulations.

values used in the simulations are given in table 2.1. They were used, where not stated

differently.

Amplitude

In principle, the amplitude of the vibration should not have an influence on Γ as long as

the forces from the fluid scale linearly with the amplitude. In order to verify, that the

amplitudes are in the linear range and no second order effects are present, simulations

with the non-linearized Navier-Stokes equations were made. For this, the Laminar Flow

interface of Comsol Multiphysics R© was used in which the full Navier-Stokes equations

are solved instead of using the Creeping flow interface. The geometrical parameters were

as in the standard configuration, except that d2 was set to 450 µm. Five amplitudes

v0 from 10−5 to 10 m/s were tested as shown in the legend in figure 2.6. The figure

shows the difference of the real and imaginary part of Γ for different amplitudes and

frequencies with respect to the simulations with v0 = 10−5 m/s.

According to the results shown in figure 2.6, the amplitude has almost no influence

below values of 1 m/s. Below 0.1 m/s the change compared to the simulations with

v0 = 1 ·10−5 m/s is smaller than 0.1%. The typical amplitude in the experiments shown

in chapters 3 and 4 is v0=10 mm/s. It can therefore be concluded, that firstly the

assumption of Stokes flow is valid and secondly no non-linear fluid mechanic effects have

to be expected at the vibration amplitudes present in the experiments.

Density, viscosity and frequency

In order to see the influence of the viscosity, density and frequency, these three param-

eters were varied in a simulation. The frequency range was 5 to 100 kHz, the viscosity

range was 1 to 10 mPas and the density was varied between 600 and 1400 kg/m3. The

33


-(¡

)+(

)Im

Im

¡0

Re

Re

(¡)-

()

¡0

0.8 1 1.2 1.4 1.6 1.8 2

x 104

-0.2

0

0.2

0.4

0.6

0.8 1 1.2 1.4 1.6 1.8 2

x 104

-0.5

0

0.5

1

1.5

Frequency [Hz]

1e-5 m/s

1e-1 m/s

1 m/s

5 m/s

10 m/s

Figure 2.6: Simulations of the real and imaginary part of Γ for different frequencies andamplitudes v0. The differences to Γ0 related to v0=1 · 10−5 m/s are shown.Below v0=1 m/s no difference can be observed. The forces due to the fluidscale therefore linearly below this value and Stokes flow can be assumed.

width and thickness of the cross section were held constant. A graph showing the real

and imaginary part of Γ for different ρ, η, and ω is shown in figure 2.7.

From the plot it can be concluded, that there is indeed only a dependency on ρω/η,

since all points fall onto the same line. The black line indicates the theoretical value

Γcirc for a circular cross section. The offset comes from the fact, that the expression is

not valid for a rectangular cross section. However, the shape of the line looks similar to

the simulated values, justifying the approach of using Γcirc together with a correction

factor Ω(ω).

Width of the beam

The next step is to investigate the influence of the beam’s width b . Simulations with

different widths and different frequencies were done for this. The results are plotted in

figure 2.8 for three different frequencies with respect to the ratio b/h. The numbers are

indicated as differences between the theoretical value for an infinitely thin beam [44]

and the simulated value.

34


¯ ½b ! ´= /42

¡circ

Simulation varying , and½ ! ´

-(¡

)Im

Re(¡

)

102

103

1

1.5

2

102

103

0

0.2

0.4

0.6

0.8

Figure 2.7: Influence of density and viscosity on Γ. The circles indicate simulated values,where ω, η and ρ were varied keeping all other parameters constant. Ifplotted with respect to β, they all fall on one line. The solid line indicatesthe theoretical value for Γcirc.

0 1 2 3 4 5 6 7 80.1

0.2

0.3

0.4

0.5

0.6

0 1 2 3 4 5 6 7 8-0.05

0

0.05

0.1

0.15

-(¡

)+Im

sim

Im

(¡)

sader

Re(¡

)-si

mRe(¡

)sa

der

b h/

25 kHz55 kHz95 kHz

Figure 2.8: Differences between simulated values Γsim with a finite ratio b/h and thetheory for an infinitely thin cross section Γsader.

35


As the ratio increases, the difference between the value for a thin beam and the simulated

value converges to zero. Notably, this is already the case at ratios slightly above one for

the imaginary part. For the real part, however, the difference between the two values

is still around 10% even when the ratio is above 7. An alternative would be to derive

a new formulation for the fluid forces, e.g. by replacing b2 by bh in equation 2.46 as

well as in equation 2.41. This would also be meaningful in the sense, that the real part

of Γ can be interpreted as added mass, whereas the imaginary part can be interpreted

as added damping, which intuitively are proportional to the area of the cross section.

However, since b and h are geometry parameters which will not change for the same

sensor, the correction can directly be included in Γ here instead of reformulating the

whole approach.

Wall distance

The last geometry parameter to analyze is the distance between the cross section and

the walls. For this, the boundary conditions on all outer boundaries were set to wall.

Figure 2.9 shows simulations for different distances d1 between the upper wall and the

top side of the cross section with 5 kHz. The different lines indicate different values for

the distance d2 to the bottom wall. The tabulated values from Brumley et al. [35] are

given for reference.

The values show a distinct decrease until around 300-400 µm. This justifies the assump-

tion, that the related length scale for d1 and d2, where boundary effects are important,

is approximately twice the beam’s width. Since the chamber of the sensor should be as

small as possible, it will be necessary to build the lid as close as possible to the cantilever.

Another aspect is that due to processing restrictions, the distance between the lower side

of the cantilever will always be approximately 450 µm. At a distance d1=200 µm the

real part of Γ is around 6% within the asymptotic value, whereas at 400 µm it is within

1.5%. It is therefore important to include both walls in the modeling.

2.3.3. Calculation of the correction factor Ω(ω)

It has been shown that the general form for the hydrodynamic function Γ is a reasonable

basis. The correction factor Ω(ω) shall be in a similar form as the one proposed by Sader

[44]. It will be a function of ρ, η and ω. The correction factor Ω(ω) will be different for

different wall distances and different ratios b/h.

36


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.4

1.6

1.8

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.15

0.2

0.25

d1 [mm]

d ¹2=300 m

d ¹2=450 m

d ¹2=600 m

-(¡

)Im

Re(¡

)

Brumley, 2010

Figure 2.9: Influence on Γ of different distances to the walls at the top side d1 and thebottom side d2 at 5 kHz. The solid line indicates the value obtained byBrumley et al. [35], where no boundary is considered.

Comparing the simulations with the prediction for the circular cross section, the real

and imaginary part of the correction Ω(ω) can be calculated from equations 2.39 and

2.46:

Ω(ω) = Ω0eiϕ =

4i

πρωb2H0

Γ0

ei(κ−γ) (2.47)

using Γcirc = Γ0eiγ and H = H0e

iκ.

To implement the results of the FEA simulations in the impedance model, a second

order polynomial was fitted to the amplitude Ω0(τ) and phase ϕ(τ) with τ = log(β).

Ω0(τ) = a2τ2 + a1τ + a0 (2.48)

ϕ(τ) = b2τ2 + b1τ + b0 (2.49)

In figure 2.10 Γ is plotted for four different cases. Γcirc, Γsim and Γsader indicate the

theoretical values for a circular cross section, the simulated value and the function value

from the correction of Sader [44], respectively, for a rectangular cross section with width

b = 200 µm. The function Γrect = Ω(ω)Γcirc is calculated with the simulated values

for Ω(ω) from equations 2.48 and 2.49. The values for Γrect and Γsim fit nicely, which

37


Re¡()

0.5 1 1.5 2

x 104

1

1.2

1.4

1.6

1.8

Frequency [Hz]

0.5 1 1.5 2

x 104

0.05

0.1

0.15

0.2

0.25

Frequency [Hz]

-(

)Im

¡

¡rect

¡circ

¡sim

¡sader

Figure 2.10: Real and imaginary part of the hydrodynamic function for a circular crosssection Γcirc, the corrected function according to Sader Γsader, the simulatedvalues Γsim and the corrected values Γrect using the fitted Ω(ω).

indicates, that the second order polynomial fit worked well. For both, the real and the

imaginary part, the correction of Sader is, as expected from figure 2.8, not high enough

for the rectangular cross section.

The hydrodynamic function Γ goes into the value for the force (equation 2.16) and

finally into the wave number k?. The resonance frequency and the damping are then

numerically calculated from the impedance model. The numerical calculation involves

a search for the value of the frequency where the phase of 1/Z12 is zero assuming all

other parameters are given. The other parameters of interest are the properties of the

fluid, namely η and ρ. These two parameters do not affect the coefficients ai and bi of

the polynomial of Ω(ω). It is therefore possible to calculate Ω(ω) for one geometry (i.e.

wall distances, b, h) and using the same expression for various frequencies, viscosities

and densities because these values are parametrized.

2.4. Combined model

In the preceding sections, the theory to describe the behavior of the cantilever was

developed. In this section, the combination of the beam theory with the FSI and the

implementation thereof are presented, summarizing the theory shown above. Much of

38

2.4. Combined model

the implementation in Matlab R© was done by David Hasler [40] during his semester

thesis, where he also included the temperature influence on Young’s modulus E and on

the geometry.

The strategy for combining the results from the FSI simulations and the impedance

model is outlined in figure 2.11. The calculation of the fluid force in time domain for

a given v0 and a set of frequencies ωi is done with Comsol Multiphysics R©. The output

of this simulation are values for the force per length f(t) for different frequencies over

a range of three periods T . With these data, the transfer function H is constructed

in Matlab R© (build H.m) according to equation 2.45. The correction factor Ω(ω) is

constructed subsequently. For this purpose a second order polynomial is fitted to the

simulated values of H0 and κ using a set of frequencies ωi implementing equation 2.47

(fit omega.m). The spring constants required for the impedance model are calculated in

Comsol Multiphysics R©. The impedance model described in section 2.2.1 is programmed

in Matlab R© (wireloop.m). The force f on the cross section exerted by the fluid is

calculated according to equation 2.39 requiring the correction factor Ω(ω) and the hy-

drodynamic function Γcirc for the circular cross section. The outputs of the model are

the resonance frequency, df and the magnitude of 1/Z12. The latter is used to calculate

the induced current according to equation 2.38.

2.4.1. Influence of design parameters

Up to now, the influence of geometry parameters on fluid forces have been investigated.

Here, the influence of the geometry parameters on the final measurement shall be illu-

minated. The following design goals and restrictions apply

• Due to the implementation of the readout system (see section 3.2.4), the resonance

frequency has to be below approximately 25 kHz. There is a trade-off between

precision in the measurement of the frequencies and the maximum frequency the

instrument can handle since the frequency is measured with a counter.

• The Q-factor should be as high as possible. A value of 15 is approximately the

minimum that is acceptable for the readout system. A high Q-factor makes it also

easier to achieve high velocity amplitudes which increases the readout signal. For

this reason, silicon is the material of choice, because it is possible to reach fairly

high Q-factors in contrast to polymers, where material damping may be dominant.

39


ComsolFE simulations

Ò Calculations in

MatlabÒ

Analytical models

build_H.mrect_sweep.m

fit_omega.m

wireloop.m

Calculation ofthe springconstant

Theory for circularcross section:analytical expressionfor ¡circ

Impedance model

f t !( , )iCalculate H

Calculatecorrection factor

H( )!i

c cclin tor

bend

, arg Z(1/ )12 y

F t( )

µl( , )x t

wtr( , )y twl( , )x t

x

zZ12

Z31

Z11

f

t

Figure 2.11: Implementation of the combined 1D-2D model. The fluid structure interac-tion and the spring constants come from Comsol Multiphysics R©. All inputsare combined in a Matlab R© file wireloop.m.

40

2.4. Combined model

Changed parameter fres [Hz] df [Hz] Q factor Uind [µV] Sη,df Sρ,fres

- 12’802 215 24.7 2.4 100 -3.7h= 50 µm 8’319 193 17.9 2.9 90 -2.7b= 100 µm 15’148 331 19.0 4.2 156 -3.2l= 1300 µm 17’489 253 28.6 2.2 115 -5d= 1300 µm 13’757 223 25.6 1.7 103 -4

Table 2.2: Parameter study for different geometries done with the impedance modelaround η =1 mPas and ρ =1000 kg/m3. The standard configuration wasd1=200 µm, d2 = ∞, b=200 µm, l=1600 µm, h=70 µm and d=1600 µm.Each row represents one configuration, whereas the parameter indicated inthe first row was changed with respect to the standard configuration.

• The sensitivities of fres and df with respect to the viscosity and the density have

to be maximized.

• The readout is done inductively. In order to have a high signal to noise ratio, the

induced voltage Uind should be as high as possible.

• The size of the overall sensor should be as small as possible. Advantages of a small

system are reduced thermal inertia and less required sample volume.

The results of a parameter study are outlined in table 2.2. In summary, the width b has

a high influence on the Q-factor (the wider the higher) as well as on the sensitivities

(the wider the lower). Due to this influence, there is a trade-off between getting a high

Q-factor and a high sensitivity when setting the width of the cantilever.

There is another trade-off between induced voltage and sensitivity. The higher the

induced voltage, the lower is the sensitivity for different geometries. The main factor

influencing the induced voltage is the length of the transversal beam. A high signal can

be achieved by increasing this parameter. This comes, however, with the disadvantage

of a larger sensor and therefore a larger chamber.

The lengths l and d should be as small as possible for space reasons. However, a very

small l drastically increases the resonance frequency. This effect can be annihilated by

making the thickness smaller. A smaller thickness means, on the other side, a smaller

Q-factor.

Based on these considerations the dimensions of the cantilever were set to l=1600 µm,

b=200 µm, h=70 µm and d=1600 µm. These dimensions were used in all experiments

presented in this thesis except for the ones shown in section 3.2.4. Micro machining has

the advantage, that a variety of geometries can be implemented on one wafer, as long as

41


Parameter Value range

Geometry U-shape and plate 1’600×1’600 µmLength l 1’300 and 1’600 µmLength d 1’300 and 1’600 µmWidth b 100, 200 and 400 µmThickness h 70 µm

Table 2.3: Fabricated geometries (see chapter 3) based on the design considerationsshown here.

the vertical dimensions are not varied. For this reason a variety of other geometries was

also drawn on the different generations of masks. Geometry variations from the ones

mentioned were: Plate geometry (1600×1600 µm) instead of a cantilever, a different

length l (1300 µm) and beam widths b (100, 400 µm). The geometries are summarized

in table 2.3.

2.5. Calculation of viscosity and density

In the actual measurement, the resonance frequency and the damping are measured by

the electronics. These data might be enough information to make a statement about

the progress of a bio-chemical reaction. However, if the instrument is used as a viscosity

and density sensor, these values have to be converted to units of viscosity and density.

This is in principle possible, since there are two input quantities fres and df and two

output quantities ρ and η. The difficulty is that both output quantities depend on both

input quantities.

Youssry et al. [32] propose a model-based method to calculate ρ and η. They use a

cantilever clamped on one side. Therefore Euler-Bernoulli beam theory and the FSI

model proposed by Sader [44] or Maali [31] can be used. After applying some simplifica-

tions they get expressions for ρ and η which depend only on the damping and resonance

frequency in vacuum and in the liquid. In our case, the model describing the cantilever

is more complicated, therefore this procedure is not applicable.

The expected densities and viscosities are close to the ones of water. In this small

range, the resonance frequency and damping will change almost linearly with the fluid’s

properties. Therefore it will be possible to use a linear scheme for the calculation of ρ

and η out of fres and df . Figure 2.12 shows two contour plots derived from the model.

The plot on the left shows the dependency of the resonance frequency on viscosity and

density, respectively, in the range of interest. The plot on the right shows the dependency

42

2.5. Calculation of viscosity and density

12558.9

12605.4

12652.0

12698.5

1 1.2 1.4 1.6 1.8990

1000

1010

1020

1030

1040

1050

223.1

236.0

248.7

261.6

274.4

287.2

1 1.2 1.4 1.6 1.8990

1000

1010

1020

1030

1040

1050

Viscosity [mPas] Viscosity [mPas]

Den

sity

[kg/m

]3

Figure 2.12: Influence of density and viscosity changes on the resonance frequency (left)and df (right) in a close range around the properties of water.

of df on the two fluid parameters. The resonance frequency depends on both quantities

by approximately the same amount. The damping however mainly depends on the

viscosity. This means, that the calculation of the fluid properties is possible using the

measured resonance frequency and damping of the first mode only. The behavior of the

damping is almost linear. The contour lines of the resonance frequency, however, are

slightly non-linear. Nevertheless, a linear approach is used for back-calculation here.

Assuming linearity, the change in resonance frequency ∆fres and damping ∆df with

respect to a reference fluid can be calculated from the sensitivities Si,j. The approach

is formulated in matrix notation as[∆fres

∆df

]︸︷︷︸

~f

=

[Sfres,ρ Sfres,η

Sdf,ρ Sdf,η

]︸︷︷︸

S

·

[∆ρ

∆η

]︸︷︷︸

~µ

(2.50)

where ∆ρ and ∆η are the changes in density and viscosity respectively. The values for

viscosity and density can then be calculated as

~µ = S−1 ~f = K ~f (2.51)

where K is the inverse of S. This gives the changes in viscosity and density with respect

to a reference fluid. Since linearity is assumed, the properties of the reference fluid have

to be close to the fluid which should be measured.

43


There are two possibilities to get the values for Si,j. The first is to use the impedance

model. This requires the model to almost perfectly represent the reality. If this is not

the case, calibration factors have to be introduced. The values of the calibration factors

can be found via a calibration measurement. The second option to derive Si,j is to make

calibration measurements with different fluids and to calculate the sensitivities directly

from these.

An approach to increase the range of the calculation scheme 2.50 is to replace η and ρ by

functions of one or both of these, based on a linearization of equation 2.40. As pointed

out by different researchers [44, 31], the following relations for the viscous part of the

Q-factor and fres for a slender beam hold, if the modes of vibration are well separated

and dissipative effects are small

Q =

4µπρb2

+ <(Γ)

=(Γ)fres = fres,vacuum

(1 +

πρb2

4µ<(Γ)

)−1/2(2.52)

where µ = b2ρS. Therefore, both, the imaginary and the real part of Γ influence the

damping whereas the resonance frequency is only influenced by the real part of Γ. Over

a wider range of β, these exhibit an approximately logarithmic behavior. It would

therefore be reasonable to use the logarithm of η and ρ in equation 2.50. In the inviscid

limit, we have =(Γ) → 0 and <(Γ) → 1. For this case, the resonance frequency only

depends on the square root of the density. However, the values of the sensitivity matrix

prove, that the inviscid limit can not be assumed here.

44

3. Cantilever system for viscosity and

density sensing

After a brief literature review about miniaturized viscosity sensors with a focus on

cantilever sensors the fabrication of the chip and the experimental setup are presented.

The main goal of this chapter is the characterization of the chip as a viscosity and density

sensor. The sensor shall be characterized with respect to sensitivity and accuracy. In

doing so, measurements with the chip are also compared to the model shown in chapter

2 in order to assess the model’s accuracy. Finally, parasitic effects, namely self heating,

damping and temperature influences are investigated.


The methods to determine the density and the viscosity of a liquid can be divided into

two groups. In the first group, the measurement is based on a continuous flow. Classic

devices such as capillary viscometers and rotational rheometers work according to this

principle. The second group are devices which make use of an oscillating structure.

The frequency can range from several hundred Hertz as for medium sized resonators to

several Megahertz as it is the case for quartz crystal sensors.

In general, it has to be distinguished whether the fluid under investigation exhibits

non-Newtonian behavior or not. The behavior of Newtonian fluids is not shear rate

dependent nor frequency dependent. However, the viscosity of complex fluids such as

dispersions and emulsions is shear rate and frequency dependent in general. Accord-

ingly, the frequency range of the device used to characterize such liquids is of high

importance. The results for large scale instruments, such as oscillatory rheometers are

usually not comparable to the results obtained with miniaturized instruments when

testing non-Newtonian fluids [45]. Large scale instruments operate at frequencies below

approximately 100 rad/s. Above this frequency, inertial effects of the instrument itself

45

3. Cantilever system for viscosity and density sensing

start to play a dominant role and make measurements impossible. The frequency range

of smaller devices is in the kHz to MHz range.

The resonance frequency and damping of oscillating devices are often influenced by the

density as well. For this reason, these devices can partly also be used to measure the

density and the viscosity at the same time. The following overview focuses on small

scale devices designed for the measurement of viscosity.

3.1.1. Miniaturized viscosity and density sensors

Stationary flow devices

Miniaturized stationary flow devices are mostly based on the assumption that the flow

pattern behaves according to the theory of Hagen-Poiseuille. Lee and coworkers [46]

present a microfluidic device with channels and reservoirs for a reference fluid. Srivastava

et al. [47] use a capillary pressure driven flow, with the advantages of a very small sample

volume and the fact that no pump is necessary. However the precision of their device is

limited mostly due to the fact, that the flow is dependent on surface properties, which are

hard to control. The device of Tang et al. [48] is based on a pressure drop generated by

degassed PDMS. It was used for assaying endoglucanase activity. These types of sensors

usually require a reference fluid to improve the accuracy, which makes the system and

the actual measurement more complex.

Curtin et al. [25] used µPIV to determine the viscosity of PCR solutions. With µPIV

the whole flow field is recorded and analyzed. This gives much more insight into the

physics but complicates the system.

Surface based devices

Two groups of devices based on piezoelectric materials shall shortly be introduced here.

The first group are piezo crystals, where the whole crystal oscillates in a certain mode.

They are known as quartz crystal microbalance (QCM) and have a very high sensitivity

to mass. Since the added mass is also depending on the viscosity of a fluid, they are also

sensitive to this parameter. The second group are surface acoustic wave based sensors

[49]. Here mainly the surface of the crystal is in motion.

When a surface in contact with a fluid oscillates in tangential direction, a decaying wave

is generated. The decay length of this wave is equivalent to the boundary layer thickness

δ as defined in equation 2.43. The boundary layer thickness decreases with increasing

frequency. It is a measure of the length scales in which an oscillatory sensor is influenced

46


by the liquid. Accordingly, when measuring a dispersion with a sensor working in the

MHz range, the sensor may not be able to ”see” the dispersed phase but only the

dispersion medium if the particles are larger than the boundary layer thickness [45].

This is for example the case, if whole blood should be characterized. Quartz resonators

and acoustic wave sensors work in the MHz range, whereas MEMS based sensors work

at intermediate frequencies in the kHz range, which partly solves the problem.

3.1.2. Resonant cantilever sensors

There has been a lot of research focusing on the use of miniature-sized cantilevers for

viscosity and density sensors. Advantageous is the simplicity of a cantilever. It can, in

general, be well described from a modeling point of view. There are many fabrication

technologies, which allow various materials and sizes to be used. A big disadvantage

of using small cantilevers is the fact, that damping by the fluidic environment can be

tremendous. This reduces the signal amplitudes and the Q-factor, which makes the

measurement less accurate.

For this reason, most of the cantilever sensors are rather large compared to other MEMS

devices. The range goes from millimeter sized cantilevers [50] to cantilevers with a

length of a few hundred microns [51]. The complexity of a single chip can reach from

the implementation of a single cantilever to a chip including the readout circuitry [52].

Geometry

There is a variety of geometries discussed in literature. More plate-like geometries where

e.g. employed by Goodwin et al. [53] or Ghatkesar and coworkers [54]. Etchart and

coworkers [36] used a doubly clamped thin silicon beam geometry, Riesch et al. [55] used

a similar setup. The cantilever geometry of a U-shaped beam discussed in this thesis has

been employed by Agoston et al. [56] and Requa and Turner [57]. Some researchers, e.g.

McLoughlin et al. [58] used commercially available AFM tips. Even more complicated

geometries, e.g. plates with holes, were discussed in a paper by Herrera-May et al. [59].

A very clever construction has been presented by Linden et al. [60]: In their device, only

one side of the cantilever is in contact with the fluid. This reduces the sensitivity on

one side, but on the other side, the Q-factor and the signal amplitude can be increased.

47


Mode

In many cases, people use the first mode of vibration, however also higher order modes

can be employed. Wilson et al. [50] used the second mode of their cantilever. The

combined use of many higher order modes was described by Ghatkesar [54, 51]. The

advantage of this concept is, that the most sensitive ones with respect to either viscosity

or density can be chosen. Also torsional [61] and longitudinal as well as transversal

modes [62] have been discussed.

Excitation and readout methods

A straightforward way to excite the cantilever is to mount it on a piezo crystal [54]. In

this thesis, the Lorentz force is used for actuation. This approach has been reported by

several groups [52, 56, 57, 53]. Excitation methods by means of an alternating magnetic

field and a magnetic coating of the cantilevers was shown by Zhao et al. [63] and Vidic

et al. [64]. Magnetic actuation of a nanocomposite cantilever (Fe3O4 particles in SU-8)

was recently presented by Suter and coworkers [65]. Piezoelectric [50] and thermal [66]

actuation are discussed as well. Some researchers do not use an active excitation, but

use the thermal noise spectrum of the cantilever [58] instead.

The readout is very often made via optical means [63, 58, 54, 56, 64, 55]. Another

very common method is a piezo-resistive readout [53], which can be enhanced using

a Wheatstone bridge [52]. An inductive readout, similar to the one presented in this

thesis, is presented e.g. by Requa and coworkers [57]. Alternatives are discussed by

Boisen et al. [67], including hard contact digital or a capacitive (mainly not for liquids)

readout.

Application

Many researchers present proof-of-concept experiments, where glycerol solutions are

common liquids for the investigation of the influence of viscosity and density on the

cantilever. However, these sensors find applications in various fields, including medical

science (e.g. glucose monitoring [63]), car industry [56], and oil and gas industry [36, 53].

Performance

The performance of the sensors presented in literature is partly hard to compare. Very

often, the goal is rather a proof-of-concept, than the full characterization of a sensor.

In order to compare different designs in a general aspect, the performance of the sensor

48


has to be expressed with respect to density and viscosity. These data are however

rarely given explicitly, which makes the comparison between different research groups

challenging.

The properties of test fluids reach from very low viscosity and density shown with

pressurized hydrocarbons [53, 36] to values of the viscosity which are not higher than a

few hundred mPas. Accuracy is usually expressed as an error with respect to a reference

measurement or literature values. Etchart et al. [36] state an error in viscosity and

density of 30-40% with various hydrocarbons exhibiting viscosities between 0.22 and

104 mPas based on an empirical calibration. Wilson et al. [50] state an error of 3.5% in

density. In their well-founded analysis, Ghatkesar et al. [54] state a resolution of 1.5%

in viscosity and 0.06% in density by using different modes and evaluating the Q-factor

and the resonance frequency. McLoughlin et al. [58] state a maximum deviation of 6.8%

in viscosity from literature values using an AFM cantilever.

3.1.3. Micro rheometers

The devices discussed up to now are intended for Newtonian fluids only. There is also

research on the development of miniature devices, which can be used for the characteri-

zation of non-Newtonian fluids. Except for the cantilever sensors, they are not employed

in a resonant mode.

The first group of devices intended for rheological measurements are inspired by the

large scale rheometers. Christopher et al. [68] presented a device with a horizontally

moving plate. Through the measurement of amplitude and phase difference between

driving force and plate position, they could measure the loss and storage modulus of a

PDMS film up to 1000 rad/s. Cheneler et al. [69] used two plates, whereas one plate

is moved in the direction perpendicular to the surface. This causes a squeezing of the

fluid which is between the two plates. The main difficulties seem to be evaporation and

to include the meniscus in the modeling.

Han et al. [70] presented a flow based device made of PDMS, where they successfully

measured properties of poly(ethyleneoxide).

Some groups try also to employ cantilevers for the measurement of non-Newtonian

fluids. In their publication, Belmiloud et al. [71] describe measurements with silicon gels.

Moatameti et al. [72] showed good agreement between measured and predicted phase

and amplitude of a cantilever for the non-Newtonian polystyrene in diethyl phtalate

solutions. Mather et al. [73] presented measurements with different non-Newtonian

49


PCB

PDMS Lid

Resonator Chip

Heater Chip

10mm

Figure 3.1: Exploded view of the sensor.

fluids using a cantilever. Their measurements show good agreement with extrapolated

reference measurements done with a rheometer.

3.2. System description

The sensor presented here is assembled out of three parts: the resonator chip with the

cantilever, the lid and the heater chip on the backside. Figure 3.1 shows the whole

setup. The chamber which contains the fluid is formed by a cavity which goes through

the resonator chip and by a cavity in the lid. The liquid to be measured is filled with a

pipette via holes and channels in the lid.

The lid is attached to the silicon chip with oxygen plasma assisted bonding and the

heater chip is glued to the resonator chip. This assembly is glued onto a PCB in order

to facilitate handling and electrical connections. The gold loops on the cantilever and

the resistive heater which is on the heater chips as well as the temperature sensor are

electrically connected by wire bonding from the top and the bottom of the chip.

The design boundaries were mostly discussed in section 2.4.1. Additional limitations

come from the readout electronics. It limits the resonance frequency to values below

25 kHz. The induced voltage Uind is proportional to the area A indicated in figure 2.4.

Because Uind should be as large as possible, the lateral dimensions have a lower limit.

The damping of the system should not be too large. A Q-factor significantly larger than

1/2, which is the limit at which an oscillation is possible, is required. The sensor is

designed for the measurement with DNA solutions. The viscosity and density of these

50


liquids are close to the values of water, which means that the viscosity will be around

1 mPas and the density will be around 1000 kg/m3.

Reaction volumes in molecular biology are usually between 10 and 100 µl. As already

mentioned in the introduction, a smaller volume is advantageous with respect to speed

but has also disadvantages. Because the sensor should first be tested with reaction

products which come from a standard sized PCR tube, the volume is around 10 µl.

In the following sections each part is described in detail. The assembly procedure of the

parts and detailed fabrication recipes can be found in Appendix A.

3.2.1. Resonator chip

There were several development steps during the time of the project. The first generation

of chips was made without the gold loops on top of the cantilever. The silicon of the

device layer was highly doped. This renders it conductive and the cantilever can be

used as conductor loop directly. In a second generation, the gold loops were introduced

in order to increase the induced voltage Uind. Various process development steps and

device improvements were made with each subsequent generation. Different geometries

were fabricated, and an insulating layer of silicon oxide between the conductor loops

and the device layer, as well as a protection layer made of silicon nitride on top of the

gold were introduced. Both layers serve as electrical insulation from the environment.

The description of the fabrication steps that follows refers to the final design which was

used for almost all measurements described here.

Fabrication

The fabrication of the silicon chip is shown in figure 3.2. Three foil masks I-III for

photolithography were used (JD phototools, Oldham, UK). The first one contains the

structure of the gold loops, the second one the geometry of the cantilever and the

last one, which is used for the backside etching, contains only the geometry of the

chamber. The fabrication is based on a double side polished silicon-on-insulator wafer

(IceMOS Technology, Belfast, UK). The thickness of the handle layer, buried oxide

layer and device layer were 450 µm, 1 µm and 70 µm, respectively. This means, that the

cantilever has a thickness of approximately 70 µm and the part of the chamber formed

by the silicon chip is 521 µm deep. The resistivity of the device layer was 1-10 Ω cm.

In order to insulate the conducting gold loop, a layer of silicon oxide with a thickness of

approximately 150 nm was deposited in a plasma enhanced chemical vapor deposition

51


(PECVD) process (step 1 in figure 3.2). This thickness is estimated from the deposition

rate. The gold loops which are used for driving the cantilever and reading out the

induced voltage were deposited in the next step. A 7 µm thick layer of negative photo

resist (AZ nLOF2070, Microchemicals GmbH, Ulm, Germany) was spin coated, exposed

with mask I and developed. The recipe of this step is based on a recipe by Shih-Wei

Lee (Group of Micro and Nanosystems, ETH Zurich). Afterwards a 12 nm thick layer

of titanium and a gold layer with a thickness of 200 nm were deposited by evaporation.

The excessive gold was then removed in a lift-off process in n-methyl-2-pyrrolidone (step

2 in figure 3.2).

The cantilevers were then structured by depositing a 10 µm thick layer of photo resist

(AZ4562, Microchemicals, Ulm, Germany) and exposing with mask II. Afterwards dry

etching steps through the silicon oxide layer with reactive ion etching (RIE) and the

device layer with an inductive coupled plasma (ICP) (step 3 in figure 3.1) followed. In

order to ensure a homogeneous etch rate and to avoid over-etching of large unused areas

(hatched in figure 3.2), these areas were also covered with resist while leaving trenches

of 100 µm. The backside of the wafer was then structured with AZ4562 and mask III.

After this step the wafer was glued upside down onto a 500 µm thick support wafer with

white wax (Crystalbond R© Aremco 555, Electron Microscopy Sciences, USA). The lower

part of the fluidic chamber was formed by ICP etching vertically through the wafer.

At the same time the chips were separated. This was achieved by leaving small lines

around the chip uncovered during the back-side etching step. The unused parts of the

device layer around the cantilevers were removed with a tweezer after etching. By RIE

etching the buried oxide from the back side, the cantilevers were then released (step 4

in figure 3.2). For this, the chips were attached to a support wafer with Kapton R© tape.

The last step was the deposition of a silicon nitride with a PECVD process serving as

a protection layer. The contact pads were protected with a microscope slide during

deposition.

Process and device development

In the first generation of chips, the silicon of the cantilever was highly doped. This

makes it possible to use the cantilever itself as a conductor, which greatly simplifies the

fabrication because the gold loop on top of the cantilever is not needed. However, with

this design the electrical insulation from the environment is difficult to achieve with

micro fabrication techniques. The gold loops introduced in the second generation solve

this problem, because PECVD can be used to add a protection layer at the end of the

52


1

2

3

4

5

SiliconSilicon Oxide

Silicon NitrideGold

Cutting linefor side view

not to scale

Figure 3.2: Fabrication of the resonator chip: 1: A 150 nm thick layer of silicon oxide isdeposited in a PECVD process on an SOI wafer. 2: the 200 nm thick metallicconductor loops are evaporated and structured with a lift-off process 3: thesilicon oxide and the device layer are dry etched 4: the handle layer andthe buried oxide are dry etched from the back side 5: a protection layer ofsilicon nitride is deposited.

53


process. It is important to completely insulate the gold loops from the environment,

but also from the silicon. Without the oxide layer between the silicon and the gold,

the signal can be distorted by applying a potential to the liquid that surrounds the

cantilever but also by incident light.

The insulation on top of the gold loops has two functions. The first one is the avoidance

of an ohmic contact between the fluid and the conductor. If there is a contact, the signal

can be distorted, especially if the fluid has dissolved salts in it, which is the case for

buffered solutions. Without an ohmic contact, no current can flow through the liquid

between the conductor lines, which would e.g. allow electrolysis. The second function

of the nitride layer is to facilitate the plasma bonding with the lid.

3.2.2. PDMS Lid

Probably the most common material for microfluidic devices is polydimethylsiloxane

(PDMS). It can be easily bonded to glass by oxygen plasma assisted bonding. Also

other surfaces like silicon oxide and silicon nitride can be used. Other advantages are

its transparency and bio-compatibility. For these reasons and due to the fact, that its

handling and fabrication are simple, PDMS was chosen as material for the lid. PDMS

has also its disadvantages. One of them being the fact that its surface is hydrophobic.

This is a disadvantage for microfluidic applications. With a plasma treatment it can be

rendered hydrophilic. However, this effect vanishes with time.

Fabrication

The fabrication steps for the PDMS lid are shown in figure 3.3. An ICP etched silicon

wafer was used as mold. It was fabricated by applying a 10 µm thick photo resist etch

mask (AZ4562) to a 500 µm thick silicon wafer. The mask contains the features for the

upper part of the fluid chamber and the fluidic channels. As these features are rather

large, a foil mask can be used (ordered from Fotosatz Salinger AG, Zurich). Afterwards,

it was etched in an ICP process to a depth of 200 µm and the resist was removed

subsequently.

In order to avoid sticking of PDMS to the silicon mold, the wafer was put into an

atmosphere of chlortrimethylsilane (Sigma Aldrich, Buchs, Switzerland) for a 30 minutes

before using it (Bengt Wunderlich, Biochemisches Institut, University of Zurich, oral

communication). Sylgard 184 silicone base and the curing agent (Dow Corning, Midland,

USA) were mixed in a ratio of 10:1 with a plastic fork and degassed in vacuum at a

54


A-ABase

Spacer

Lid

a) c)b)

Screws

PMMA

Photo resist

Silicon

POM

PDMS

PC A A

Boundary ofthe cavity inthe silicon chip

Figure 3.3: Fabrication of the lid. a) The mold is made by ICP etching a silicon waferto a depth of 200 µm. b) The PDMS mixture is poured onto the wafer,where a plastic fixture assures a uniform thickness of the lid. c) Finally, theholes for the fluidic connections are punched and the lids are separated witha scalpel.

pressure of approximately 50 mbar to remove bubbles. In order to ensure that the

thickness of the lid is uniform, the silicon mold was put into a holder (fabricated by

J.-C. Tomasina, Center of Mechanics) consisting of a base, a spacer and a lid as shown

in figure 3.3 b). The 1 mm thick spacer made of poly methyl methacrylate (PMMA)

defines the thickness of the PDMS. The spacer was fabricated by laser cutting (VLS3.5,

Universal laser systems, USA), the base is made of polyoxymethylen (POM), the lid is

made of polycarbonate (PC). After pouring the mixture onto the mold, the assembly

was put at 70 C for 4 hours. The lids were then separated with a scalpel and the holes

were punched with a sharpened metal tube.

The lid was attached to the resonator chip via oxygen plasma bonding. For this, the

lid and the resonator chip were treated in an oxygen plasma asher (Diener electronics,

type nano) for 30 s at a pressure of 0.4 mbar and a power of 50 W. The time is critical

because the bond is based on the creation of Si-OH groups at the surface of the PDMS.

If the exposure to the plasma is too short, there are not enough Si-OH sites, if it is too

long, there will be a non-sticking silica layer. By laying the lid onto a glass slide, it can

then be easily aligned when assembling it with the resonator chip.

Chamber design

The design of the chamber containing the fluid is very critical to reduce the formation of

bubbles when the chamber is filled. In contrast to many other microfluidic devices, where

only channels or very thin cavities are present, the cantilever protruding the chamber

55


43

21

1mm

Side view

PDMS

Silicon

Heater chip

Figure 3.4: Filling of the chamber. The overhanging PDMS shown in insert view ensuresthat the bottom of the chamber is filled first (1 and 2). The liquid stops atthe support of the cantilever (red circle in 2) until the bottom is completelyfilled (3).

makes the situation much more complex. Another order of complexity is added by the

fact that multiple types of surfaces (PDMS, silicon nitride, silicon oxide, silicon side

walls) are present. The plasma assisted bonding step, which is used to attach the lid

to the silicon chip makes the PDMS hydrophilic. This property is, however, lost after a

few hours. There are ways to simulate two phase flow, e.g. with the level set method.

The main problem with simulations is that accurate predictions depend on the validity

of the boundary conditions such as contact angles. As these are hard to predict (e.g. the

sidewall may have some residues of passivation material from the etching process) and

also change over time and with each liquid, a robust design had to be found empirically.

Several iterations of different lid designs led to the one shown in figure 3.4.

The feature leading to a complete filling of the chamber is the overhanging part of

PDMS shown in the insert. It prevents the liquid from flowing directly to the outlet and

thus allows to fill the bottom of the chamber reliably. The fluid stops where it reaches

56


the support (red circle in frame 2) until the bottom of the chamber is completely filled

(frame 3).

3.2.3. Heater chip

The first function of the heater chip is to keep the temperature inside the chamber

at a constant value. Due to the high dependency of the viscosity of many fluids on

temperature, this functionality is of importance. The second function of the heater chip

is to run temperature cycles. As many biochemical reactions, most notably the PCR,

play with different enzyme activities and DNA binding temperatures, a heater chip shall

make it possible to run such reactions on the chip. A third functionality which showed

to be useful when reusing the chips is the evaporation of solvent residues after cleaning

the chips. The chip was designed and fabricated by Ivo Leibacher during his master

thesis [74], where more details about the process development are given.

The heater chip is glued to the bottom side of the resonator chip with the resistive heater

looking downward from the chamber in order to be able to contact the resistors from the

bottom side of the PCB. Because the functional part of the heater/sensor is outside the

chamber, good heat conduction from the resistors to the inside of the chamber has to

be guaranteed. Therefore the substrate has to be as thin as possible without affecting

the manufacturing. For this reason a 300 µm thin silicon wafer serves as substrate.

The heater chip has a resistive meander-like platinum heater and a resistive sensor loop

incorporated as illustrated in figure 3.5.

The first fabrication step is an application of a 200 nm thick layer of silicon oxide in a

PECVD process to the silicon wafer. Both resistors were then fabricated during the same

step with a lift-off process based on the negative resist nLOF2070 done after evaporation

of a 180 nm thick layer of platinum on 20 nm titanium. The final step, after dicing and

cleaning, is the application of a protective layer and annealing of the platinum layer.

For this a droplet of polyimide (593052-250ML, Aldrich Chemistry) was applied to the

surface and the chip was heat treated with a temperature ramp going up to 350 C. This

anneals the platinum and cures the polyimide layer.

Process development

It was observed that the resistance of the gold loops as well as the temperature coefficient

of the resistivity of the platinum film were different compared to the expected values

57


Heater

Temperaturesensor

2 mm

Figure 3.5: The backside lid has a heater and temperature sensor incorporated.

from literature. It is known, that the properties of thin films can be different compared

to values for bulk material (see e.g. [75]).

The resistance of the gold loops on top of the cantilever was calculated with Comsol

Multiphysics R©. For this the electric currents, DC module was used. The two dimen-

sional geometry of the conductor was drawn and an electric potential of 10 V was applied

as a boundary condition to one contact pad, while the other pad was hold at 0 V. The

conductivity was set to 4.55 · 107 S/m, which is the value for bulk material [76]. The

measured resistance was 30-40% higher than the calculated value.

The temperature coefficient of the resistivity of the platinum temperature sensor is

0.84 ·10−3 1/K at 20 C, if it is measured before the protective polyimide layer is applied.

After the application of the protection layer it increases to 2.3 · 10−3 1/K. According to

Zhang et al. [75], the bulk value is 3.9 ·10−3 1/K. The same group reports 1.4 ·10−3 1/K

for a 28 nm thick film, other groups [77, 78] report values of 2.3·10−3 (thickness: 300 nm)

and 3.2 · 10−3 1/K (thickness: 100 nm). The first conclusion is, that there is a spread in

these properties which may come from the employed fabrication technique but also an

influence of the thickness can be observed. The second conclusion is, that the application

of the protective layer has an influence on the properties of the conductor. The main

reason is most probably, that the material is annealed. This effect was also observed by

Schmid and Seidel [79].

The second property of the heater chip which changes after the application of the poly-

imide layer is the hydrophobicity. Figure 3.6 shows the back side of two heater chips

after the application of a droplet of water. The first one was annealed at 350 C with

the same temperature ramp as it is used for the curing of the polyimide, the second one

was not heat treated.

58


Annealed Untreated

2.5 mm

Figure 3.6: Increase in hydrophility of the silicon oxide layer on the inner side of theheater chip with annealing at 350 C.

With this qualitative experiment it is demonstrated, that the heat treatment makes the

surface of the back side of the chip hydrophilic. This is important because this side

of the heater chip forms the lower part of the fluidic chamber and altering the surface

properties may also change the behavior when filling the chamber. However, due to the

design of the chamber (see figure 3.4) the influence of this effect is reduced.

3.2.4. Readout

A common method to characterize the spectral behavior of a system is the measurement

of the transfer function. With this method, the amplitude and phase response of a

defined frequency range is recorded. If the resonance modes are well separated, the

resonance frequency and damping of one or several distinct modes can be treated as an

SDOF. The resonance frequency and the damping can be found by fitting a Lorentzian

to the amplitude response. In this work, a phase locked loop (PLL) is used instead. The

system is shown in figure 3.7. It is a slight variation of the gated PLL as described by

Goodbread et al. [80]. The PLL keeps the phase difference φ between the excitation and

the signal to be measured at a given value φref .

The phase of the sensor signal is detected by synchronous demodulation. For this, the

induced voltage is amplified in two inverting amplifier stages first. This signal is mixed

with a signal having the same frequency as the excitation but a phase shift of φref . After

integration the signal is fed into the VCO which is connected to the clock input pin of

the programmable logic. A gate included in the programmable logic switches between

excitation and read-out in the time domain with a waiting time between the readout

59


Measurement

Excitation

Sense gate

Read-outDrive

VCOH-Driver

Programmablelogic

G1

Resonator

Switch

microController

Phase detectorand integrator

Phase detectorand integrator

G2

Wait

Áref

fresonator

fVCO=256fdriver

fdriver

fdriver

Figure 3.7: Main components of the implementation of the gPLL. fresonator indicatesthe frequency of the induced voltage, fV CO the frequency of the voltagecontrolled oscillator and fdriver the frequency of the excitation.

and excitation phase. The programmable logic generates also a signal having the driving

frequency. An H-driver is used to generate the driving signal which is applied to the

cantilever. The amplitude of the driving signal is controlled via a second phase detector

which is 90 phase shifted with respect to the first one. The micro controller is used to

set the reference phase φref and for communication with a personal computer. With the

gating mechanism it is easier to measure a small readout signal where a large excitation

is required, because these two signals can be separated in time, thereby eliminating

electrical cross talk. Similar to a conventional lock-in amplifier, the circuit allows to

measure a relatively noisy signal.

The gated PLL has been built by Ueli Marti (Center of Mechanics, ETH Zurich). In

all measurements presented except for the ones in appendix B the number of driving

cycles was 26 and the number of measurement cycles was 3, whereas there were 2 cycles

between the end of the driving phase and the start of the measurement phase.

60


The advantage of the system is its high precision in the frequency measurement. One

measurement is in general also much faster since in the extreme case only two points at

φ = ±∆α have to be measured, compared to a transfer function measurement, where

the whole given spectral range has to be scanned or a relatively long measurement and

a subsequent FFT has to be made. Due to the fact, that the resonator’s vibration is

freely decaying, a correction has to be applied to the measurements when comparing

with simulations (see Appendix B).

3.2.5. Experimental setup

Although the sensor has a temperature control feature, most of the following measure-

ments were done in a temperature controlled Styrofoam box. The setup is shown in

figure 3.8. The temperature was held constant inside the box using a Peltier element

(not shown because it is located outside of the box). The sensor was mounted on a

sample holder which not only facilitates the measurements but also keeps the distance

to the NdFeB magnet constant. The magnet consists of two S-30-15-N magnets (su-

permagnete.ch, Switzerland) arranged in series with a diameter of 30 mm and a height

of 15 mm and an energy product of approximately 40 MGOe. The temperature was

measured with a Pt100 temperature sensor. Before using it, the system was tested with

a highly accurate and calibrated Pt100 sensor for temperature accuracy and stability.

Where not stated differently, the temperature was held at 23±0.1 in the experiments.

A phase difference ∆α = 22.5C was used in all measurements. The electronics box with

the gated PLL is connected to a personal computer with LabVIEW, where the data can

be visualized and saved. All experiments shown in the following and in chapter 4, except

the one in vacuum, were made using this box.

Bubble removal

The lid of the chip is designed in a way, that no air pockets build up when the liquid is

filled into the chamber. Nevertheless some very small bubbles can develop when the chip

is filled. They predominantly form at the inner corners of the cantilever as well as at the

side walls and are not visible by eye (see micrographs in figure 3.9). The development

of bubbles does depend on the type of liquid and on the number of times the chip has

been used. When the temperature is increased, they grow and disturb the measurement,

which is indicated by a drift in fres and df . These bubbles can be removed by putting

the filled chip at −18 C. Since the sample should not freeze, this was done in steps of

61


Pt100

Magnet

Sensor

Sampleholder

Cooling/heatingfan

to electronicsGrounding wire40 mm

Figure 3.8: Experimental setup with external temperature control.

80 s. After each step, the existence or disappearance of the bubbles was checked under

a microscope.

3.3. Characterization

Every sensor has its characteristics which are used to select the correct sensor for a

certain application. The most important ones are the sensitivity, range, calibration

constants and numbers on the expected errors such as accuracy and precision. In the first

part of this section, the behavior of the sensor in vacuum, air and liquids is investigated

experimentally and compared to the model. In the second part of this section, the

sensitivity and accuracy are quantified. In the last part, parasitic effects are discussed.

The range is not specifically treated because it was not maximized. However it shall

briefly be discussed at this place. The upper physical limit of the range of a resonant

cantilever sensor is given by the damping, which has to be below the value for critical

damping. The use of the gPLL introduces an upper limitation to the range because the

induced voltage must be high enough. This limitation depends on the amplification of

the signal as shown in section 3.2.4. With the current configuration, the upper limit is

62


After filling Cooling -18 C, 80s±

400 m¹

Figure 3.9: The small bubbles (red circles) which build up when the chamber is filledcan be removed by cooling the sample after filling.

approximately 4 mPas. The second limitation to the range is the assumption of linearity,

which has to be valid if the scheme shown in section 2.5 is used.

Along with the characterization of the cantilever as a viscosity and density sensor the

modeling presented in chapter 2 shall be verified. The model of the cantilever serves two

main purposes. The first is the facilitation of the design. The fulfillment of the require-

ments for amplitude, frequency range and sensitivity can be nicely predicted as shown

in section 2.4.1. The second purpose of the model is its use for the calculation of the

viscosity and density out of the measured values for resonance frequency and damping.

In order to fulfill this task, the model has to represent the reality very accurately.

Except for the data shown in section 3.3.3, the dimensions of the cantilever were

l = 1600 µm, d =1600 µm, b = 200 µm and h = 70 µm.

3.3.1. Resonance frequency and damping in vacuum and air

The resonance frequency and damping of three chips were measured in a low vacuum

down to 80 mbar. The data is shown in a semi-logarithmic plot in figure 3.10. The

differences with respect to ambient pressure are shown, as the absolute values of the

resonance frequency differ much more. This can be seen from the values measured in

air shown in figure 3.11. In vacuum, the added mass and added damping are reduced.

This is indicated by an increase of 18 Hz in the resonance frequency and a decrease in

df of 2.6 Hz as the pressure is lowered to 80 mbar. The simulated values give roughly

63


0

5

10

15

20

70 700Pressure [mbar]

Pressure [mbar]

dfdf

-[H

z]amb

fres-

[Hz]

fresamb

,

-3

-2

-1

0

70 700

Figure 3.10: Resonance frequency and damping with respect to ambient pressure forthree chips in low vacuum. The error is smaller than the symbols. Theblack line indicates values from the impedance model. The resonance fre-quency at ambient pressure is around 19.9 kHz and increases with decreas-ing pressure while the damping decreases by 2.6 Hz from 6.5-10 Hz in air.A two fold increase of the Q-factor is calculated from equation 1.1 with thevalues for df .

the same results. It is, however, important to note that the model does not include

compressibility. Notably, the increase in the resonance frequency is very small compared

to the differences in fres between different sensors. Extrapolating the values for df and

fres to zero pressure gives a difference of 2.9 Hz for df and a difference of 19.3 Hz for fres.

The differences between air and vacuum are very consistent as can be seen from figure

3.10. The value for df in vacuum for one chip can therefore be estimated by measuring

df in air and deducing 2.9 Hz from this measurement. Values for df in vacuum of 3.5-

7 Hz result. This is important to know, because the damping that is not caused by the

fluid has to be included in the model.

In a second step, the model shall be compared to measurements in air. Figure 3.11 shows

the measured resonance frequencies in air for 13 different chips. The measurements were

made after finishing the fabrication, but before introducing any liquid into the sensor.

The average resonance frequency is 19’815 Hz with a standard deviation of 214 Hz. The

lines indicate simulated values.

There are different reasons for the rather large spread. An important one is illustrated in

figure 3.12. Due to a small misalignment of the first mask, with respect to the back side

mask the mechanical support is slightly altered. The misalignment can be caused by a

bad alignment during the exposure of the photo resist. Another reason is the slightly

64


19

19.5

20

20.5

21

21.5

2

4

6

8

10

12

fres[k

Hz

]

df

[Hz

]

Measured

c h ¹comsol =70 m

c=1 h ¹=70 m

c h ¹comsol =69 m

Comsol 3D

Figure 3.11: Left: Measured resonance frequencies in air (diamonds) and simulated val-ues of the resonance frequency with the impedance model for air (lines) and3D simulation in Comsol for vacuum (solid line). The spring constants usedin equations 2.5 and 2.6 were set to either values calculated with FEA indi-cated by ccomsol or to infinity for a rigid clamping indicated by c =∞. Thethickness h was set to 70 µm according to the thickness of the device layerand to 69 µm to show the influence of this parameter. Right: Measureddf . The simulations are not shown, because the differences for differentconfigurations are very small, and the combined effect of the contributionsaccording to equation 1.2 need to be subtracted anyway.

negative etching profile when the wafer is etched from the backside. It can also be seen,

that there are ripples in the vertical direction on the cantilever, which result from the

mediocre quality of the film masks used. Additional factors that affect the resonance

frequency are small particles and residues on the cantilever, thickness variations of the

device layer, which can either come from supplier or from over etching of the oxide

layer, the gold layer and the protection layer made of silicon nitride. Because all these

factors influence the resonance frequency in positive or negative direction, there is no

clear dependency of the resonance frequency on the location on the wafer.

If the spread was very small (around 1-2 Hz), the chips could be probably used without

any calibration, which is however not the case. A systematic investigation of all the

factors listed above and much more process development would be necessary to reach

this goal.

The measurements of the resonance frequency shall now be compared to simulations.

Three simulations for air done with the impedance model are shown in figure 3.11.

Neglecting the compliance of the support (c = ∞) yields a resonance frequency of

almost 21 kHz. Using the stiffness from FEA simulations (ccomsol) and a thickness of

70 µm significantly reduces the resonance frequency. Reducing the thickness of the

beam to 69 µm yields a resonance frequency slightly below 20 kHz, which is in the

65


20 m¹

Device layer

Handle layer

Mask error

Overhang

Figure 3.12: Scanning electron image of the beam support viewed from the back side ofthe chip. The overhanging device layer and a small mask error causing thespread of fres in air can be seen.

region of the measured values. The black line in figure 3.11 indicates the value from a

3D finite element simulation with h =70 µm made in Comsol using the solid mechanics

interface. The material in the simulation was set to orthotropic silicon from Comsol’s

material database. The silicon oxide and the gold layer were represented by an added

mass. Except for a refinement of the mesh at the clamping of the cantilever, standard

parameters were used. The result is in good agreement with the impedance model,

where ccomsol and h = 70 µm was used. Differences result from anisotropy, the corners

and partly neglected torsional inertia of the transversal beam.

For all these reasons, it is necessary to adjust the resonance frequency in the impedance

model. In other words a calibration is necessary. The easiest way to do this, is to

introduce a scaling factor for either the thickness of the beam or the stiffness of the

support, which is shown in the next section.

3.3.2. Resonance frequency and damping in liquids

The next step is to investigate the prediction of the model including FSI. Every chip

was measured with water and a solution of 5% glycerol (Sigma-Aldrich Reagent plus

>=99%) filled into the chamber before any other experiments were made. Additionally,

for some of the chips 2% and 10% solutions were tested as well. Figure 3.13 shows

the measured fres and df for all chips and solutions along with simulated values. All

parameters (geometry, viscosity and density) were set to their nominal values in the

simulations. Data for the glycerol solutions were taken from tabulated values [81]. The

finite element correction Ω(ω) was used for the FSI, such that wall effects and the finite

66


1000 1010 1020 103012.4

12.6

12.8

13

13.2

0.9 1 1.1 1.2 1.3

180

190

200

210

220

1000 1010 1020 1030

0.9 1 1.1 1.2 1.3

fres[k

Hz]

df

[Hz]

Density [kg/m ]3

Density [kg/m ]3

Viscosity [mPas]Viscosity [mPas]

Calibration with air Calibration with water

Figure 3.13: Measured (dots) and simulated (lines) values for the resonance frequencyand damping df for water and glycerol solutions. The spring constants wereadjusted to a value where the resonance frequency fits the measured valuein air (left) and water (right) for each sensor. For this clin, cbend and ctorwere multiplied with the same factor. The black lines indicate simulationswhere the spring constants were set according to the finite element analysis.The correction for the gated measurement shown in appendix B was appliedto the simulated values of df .

width/thickness ratio are modeled according to the state of the art. The correction as

shown in Appendix B was applied to the values for df .

Let us first look at the black lines and the measurement data. The resonance frequency

is between one and four percent too high in the model. This is consistent with the

simulations for air, where the simulated frequency is higher than all measured values.

The damping calculated by the impedance model is between 7 and 12 % too high in

the model. The spread in the experiment in resonance frequency and damping which is

present in air is as well visible for measurements in liquids. When the concentration of

glycerol is increased, df goes up as predicted. As shown in section 2.5 this effect comes

mainly from the increase of viscosity. The increased damping due to the viscosity is,

however, overestimated.

67


Since the model is not representing the reality well enough to be used for the inverse

problem, there is a need to introduce one or several correction factors. The main uncer-

tainties influencing the frequencies are the thickness of the cantilever and the stiffness

of the support. It is therefore plausible to scale one of these parameters. Now let us dis-

cuss the colored lines in figure 3.13, which were generated by multiplying all the spring

constants clin, cbend and ctor (see section 2.2.1) by a common factor between 0.5 and 0.8.

Keeping in mind, that introducing this factor is actually a calibration which has to be

done for each sensor, the most efficient way is to derive this factor using measurements

made in air. The plots on the left side were generated by lowering the spring constant

until the resonance frequencies in air matched. The prediction for the resonance fre-

quency in the liquids is improved and a maximum error of 1 % results. However, the

errors are still too large to allow a back-calculation using the model.

The two plots on the right side of figure 3.13 were generated by tuning the spring

constant until the resonance frequencies in water matched. In this case, the prediction

for the resonance frequency in glycerol is much improved, because the difference between

the point of calibration and the glycerol samples is only a few Hertz in the case of the

calibration with water, whereas the difference using the air as reference value is around

7 kHz. The prediction of the damping also slightly improves by the tuning.

The same procedure was made by tuning the thickness h of the beam. The results look

very similar and are not shown for this reason.

Revising these results, the following conclusions can be drawn. The model for the fluid-

structure-interaction overestimates the influence of the liquid on the damping (df is too

high in the model) and the sensitivity with respect to viscosity changes. All in all the

model is useful for the prediction of the cantilevers behavior in a fluid and therefore a

valuable design tool. However, the model is not accurate enough to be used to solve the

inverse problem, i.e. the calculation of the fluid properties. It is worth mentioning that

Ghatkesar et al. [54] also calibrated their cantilevers in water with thickness adjustments

in the model.

An alternative to accommodate differences between sensors is to relate measurements

for each sensor to the measured value for water. This is shown in figure 3.14. The plots

show the differences in fres and df between the value measured in water and the value

measured with the liquid to be characterized.

If only the differences between water and glycerol solutions are plotted, all measure-

ments for fres coincide nicely on one point. In other words, the sensitivities of all chips

are almost identical. The same procedure has been applied for the simulated values,

68


-200

-160

-120

-80

-40

00.99 1 1.01 1.02 1.03

0

5

10

15

20

25

30

0.9 1 1.1 1.2 1.3 1.4

Viscosity [mPas]

Density [kg/cm ]3

ff

resmeasured

,-

[Hz]

reswater

,

df

df

measured

water

-[H

z]

Simulation

Average measured

Figure 3.14: Difference in resonance frequency and damping between water and glycerolsolutions measured with 9 chips. The values shown here are for water, andglycerol solutions of 1%, 5% and 10%. The error bars indicate the maximaand minima over all chips.

which also fit nicely for the resonance frequency. However, the values for damping are

consistently higher, indicating that the prediction of the influence of the fluid on the

damping is not correct. The main reason is probably the fact, that effects at the corners

of the cantilever are not considered in the model.

Summarizing the results presented in figure 3.14 the following conclusions can be drawn.

The model does nicely predict the drop in resonance frequency but does not predict the

correct values for damping, even if only differences are evaluated. On the other side,

looking only at differences should allow to use sensitivities measured for one, or only a

few chips in order to use equation 2.50. Like this, for all other chips, only one calibration

step has to be made, i.e. measuring the resonance frequency and damping in water.

3.3.3. Induced voltage

The induced voltage Uind at the conductor loops tends to be very small and has to be

maximized. It is also a direct measure for the amplitude of the cantilever’s vibration

when the field strength of the magnet is known. The induced voltage is first amplified in

two inverting amplifier stages. In order to calculate Uind, the amplifications have to be

known. The conductor loops on the cantilevers have a resistance of approximately 70 Ω.

This resistance has to be included in the calculation of the amplification. Therefore, the

amplification G1 of the first stage is given by

69


G1 =R1

Rloop

(3.1)

where R1 and Rloop are the resistances of the amplification stage and the conductor loop,

respectively. The voltage Uamp after the second amplification stage is then given as

Uamp = G1G2Uind (3.2)

where G2 is the amplification of the second stage.

The induced voltage Uind was experimentally quantified. For this, the first maximum

after the waiting phase of the gPLL of the sinusoidal sensor signal as conceptually shown

in figure 3.7 was measured. Figure 3.15 shows the induced voltage measured with water,

divided by the excitation current amplitude I0, along with the simulated values. The

current I0 was measured using a series resistor between excitation and loop and took

values around 5-20 mA. The magnetic field strength B, measured with a Hall sensor at

the sensor’s location, was approximately 250 mT. Configurations of different designs are

shown in figure 3.15. The first one is a cantilever of the first generation, where the beam

itself was conductive, therefore the number of windings is one. The second and the third

configurations are cantilevers with b =200 µm and 400 µm. Since the wider cantilever

has more space, the number of loops can be increased to eight. The last configuration is

a one side clamped plate with 1600 µm width and 1600 µm length, respectively. It has

seven conductor loops. The simulated induced voltage was calculated using equation

2.38. The spring constants of the clamping were adjusted in the model, such that the

predicted resonance frequency in water coincided with the measurements.

As expected Uind increases with an increasing number of windings. The model predicts

the increase fairly well. The differences between the model and the measured values come

from the fact, that the model has to include the whole mechanics and FSI expressions

contained in Z12 according to equation 2.38. Additional uncertainties stem from the

magnetic field strength B and the fact, that the measurement is non-stationary, whereas

in the model a stationary state is assumed.

According to this plot, the configuration with eight loops would be favored. But the

higher the number of windings the higher is the resistance of the whole loop. As shown in

equation 3.1, the amplification depends on the resistance of the loop itself. Additionally,

the thermal energy that is built up due to the excitation current is proportional to the

resistance of the loop. For these reasons, a very small resistance is favorable. The

70


Beam width [ m] / Number of loopsw ¹ n

Measurement

Model

0

5

10

15

20

25

30

35

40

200 200 400 1600 1600£

1 2 8 7

UI

ind/

[mV

/m

A]

0

Figure 3.15: Measured and simulated induced voltage Uind in water divided by the am-plitude of the excitation current I0. The error bars are smaller than thesymbols, where not visible. Cantilevers with different widths and numbersof loops were tested. The last chip had a rectangular plate instead of acantilever.

configuration with two loops showed to be a good trade-off between a high Uind and a

low resistance.

3.3.4. Sensitivity

The sensitivity is important for two reasons: Firstly, knowing the sensitivity allows the

more or less direct comparison with other resonant sensors. Secondly, the higher the

sensitivity of a sensor is, the higher is its precision. In the case of a resonant sensor,

the sensitivity is usually given in frequency per Pas. The reason for this is the fact,

that measuring with a resonant viscosity/density sensor can be divided in two steps:

the first step is the change in fres or df of the cantilever (the transducer) due to the

fluid. The second step is the measurement of the frequency. The precision of the whole

measurement depends therefore on the precision of the device used for the frequency

measurement and on the sensitivity of the transducer.

Since both sensor outputs, the resonance frequency and the damping, depend on both

material parameters, the sensitivity is a 2× 2-Matrix as shown in equation 2.50. When

experimentally characterizing the sensors, the challenge is to separate the density and

the viscosity. Based on a publication by Ghatkesar et al. [54], three different fluids were

71


Density [kg/m3] Viscosity [mPas]

Water 997 0.94Ethylene glycol 10% 1010 1.25Ethylene glycol 6% 1004.2 1.123Glycerol approx.a 4.8% 1010 1.118a) Diluted until the density was the same as in the 10%solution of ethylene glycol

Table 3.1: Properties of water and the solutions used to measure the sensitivity at 23 C.Both quantities were measured independently by commercial instruments.The maximum measurement error in the density is 0.6 kg/m3, the maximumerror in the viscosity is 0.006 mPas.

chosen for this task as lined out in table 3.1. Two solutions of ethylene glycol (99+%

extrapure, Acros Organics) in water were prepared, one having a concentration of 6%

(v/v) and the other having a concentration of 10% (v/v). The third solution contains

approximately 4.8% (v/v) glycerol (Sigma-Aldrich Reagent plus >=99%). This solution

was diluted until its density was the same as in the 10% ethylene glycol solution. The

density was independently measured with an Anton Paar DMA 35 density meter. The

viscosity was measured with an Ubbelohde capillary viscometer. The values for water

are taken from literature [81]. All values are for 23 C. This is the temperature at which

all experiments shown in this chapter and chapters 4.2 and 4.3 have been made.

The glycerol solution serves as a basis. The 10% ethylene glycol solution has the same

density, whereas the 6% solution has approximately the same viscosity. This allows the

independent measurement of the sensitivity of fres and df with respect to either the

viscosity or the density only. The sensitivities can then be calculated with equations

1.3 and 1.4. The difference between the 6% ethylene glycol solution and the glycerol

solution reveals the sensitivities with respect to density and the difference between the

10% solution of ethylene glycol and glycerol reveals the sensitivities to viscosity.

Three chips were characterized with this method. The results are plotted in figure

3.16. The dots indicate the measured values, whereas the lines indicate values from the

impedance model. The numerical values are outlined in table 3.2. The mean values for

the sensitivities fit nicely with the predictions from the model. This is true except for the

sensitivity of df with respect to the viscosity as could be expected from the discussion

in section 3.3.2.

The scatter in the measured sensitivities is rather large. For this reason, an error analysis

is made. The procedure is described in appendix C. It is based on the assumptions

of an error in df and fres of 2 Hz and errors in the fluid properties of 1 kg/m3 and

72

3.3. CharacterizationS

[Hz/

kg m

]fres½,

-3S

[Hz/

kg m

]df,

-3

½

S[H

z/m

Pas]

fres,´

S[H

z/]

df,´

mP

as

-5

-4

-3

-2

-1

0

2.7-F5 2.7-C3 2.7-G7

-200

-160

-120

-80

-40

0

2.7-F5 2.7-C3 2.7-G7

-0.4

-0.2

0

0.2

0.4

2.7-F5 2.7-C3 2.7-G7

0

20

40

60

80

100

120

2.7-F5 2.7-C3 2.7-G7

Measurement

Simulation

Figure 3.16: Measured and simulated sensitivities for three chips. The plots are arrangedin the same form as the matrix S in equation 2.50. Estimates for the errorare given in table 3.2. The abscissa indicates the chip serial number.

Mean of the measured sensitivities Sensitivities from simulations[Hz/kgm−3] [Hz/mPas] [Hz/kgm−3] [Hz/mPas]

fres -3.84 -146 -3.59 -139df 0.057 33 0.053 106

Standard deviation Error from error analysis

fres 0.6 24.8 1.2 21df 0.3 18 0.4 16

Table 3.2: Numeric data from the plots in figure 3.16 for the sensitivity matrix. Thedata is ordered as in the S-matrix in equation 2.50 and in the plots. Theerrors are in absolute values and calculated according to appendix C.

73


0.006 mPas respectively. The errors for the frequencies are based on experience, the

errors for the fluid properties are calculated from an error analysis for the reference

measurements (Ubbelohde viscometer and DMA35 density meter). The result based on

these assumptions is shown in table 3.2. They coincide well with the measured standard

deviations and explain the large scatter. Assuming, the sensitivities of the chips are the

same, the results could be improved by measuring more chips, since the uncertainty of

the mean value is reduced by increasing the number of measurements.

3.3.5. Calibration and accuracy

There are two ways to calibrate the sensors. The first one is to tune one or several of the

parameters in the model and then deriving calibration curves based on polynomials. This

would be necessary to use the calibration in a very large range, since the dependencies

of fres and df on the viscosity and density are non-linear in general. A second possibility

is to use empirically derived sensitivities. Under the assumption that the dependencies

are linear in the range of interest, the calculation of the fluid properties can be done as

shown in equation 2.50. This is expected to be the case for the low viscosity and density

DNA solutions for which the sensor is designed.

The second procedure has been applied to the data presented in figure 3.13. The vis-

cosities and densities for the glycerol solutions measured with this method are plotted

in figure 3.17 along with the tabulated values [81]. The procedure allows only the cal-

culation of differences between the fluid to be measured and a calibration fluid. In this

case water was taken as the calibration fluid. The mean measured values shown in table

3.2 were used for the sensitivity matrix.

The measurements made with the cantilever sensor fit very well with the tabulated

values. The maximum error in the density is below 0.4%, the maximum error in viscosity

is 5.5% with respect to the actual value. The results show, that the sensitivity matrix

does not have to be determined for each sensor separately. Only one calibration liquid

has to be measured, water in this case, once the sensitivities are known. The accuracy of

the measurement with one chip could be improved when calibrating the chip separately,

which was however not done here.

An important interpretation of the plot is that the values for the sensitivity are well

chosen. The error estimated from the error analysis significantly overestimates the error

between different chips. The best accuracy stated in literature is the one by Ghatkesar

et al. [54] with 0.06% in density and 1.5% in viscosity. These numbers are slightly better,

74


0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0.995

1.005

1.015

1.025

1.035

1.045

1.055

Glycerol 1% Glycerol 5% Glycerol 10%

Tabulated density

Measured density

Tabulated viscosity

Measured viscosity

Den

sity

[kg/cm

]3

Vis

cosi

ty [m

Pas]

Figure 3.17: Comparison between tabulated and measured viscosity and density. Thepoints representing the measurements (red) are the mean values of all mea-sured data over 9 chips. The error bars on the symbols indicate the maxi-mally and minimally measured value for each point. The error bars whichare set to the left of the symbols indicate the calculated maximal error ac-cording to the procedures described in appendix C, where uncertainties inthe properties of the reference fluid and sensitivities are taken into account.They are omitted for the density values, since they are smaller than thesymbols.

75


however different modes were used in by Ghatkesar et al., whereas here only the first

mode was used.

One component of the error results from the assumption of linearity of fres and df

with respect to viscosity and density changes in equation 2.50. This assumption holds

only for a small range of material properties. From figure 3.17 it follows that the error

due to non-linearity is smaller then the other measurement uncertainties. Otherwise,

an increasing difference between measurement and data sheet values would have been

observed.

3.3.6. Parasitic effects

Losses affecting the Q-factor

The measurement of the viscosity is mainly influenced by the damping of the cantilever

as can be seen in table 3.2. Damping of a mechanical system has several causes, where

only one of them is viscous damping. For this reason, any additional damping effects may

bias the viscosity measurement. The components of damping were already introduced

in section 1.2.1.

The energy loss due to material damping is negligible for single crystal silicon compared

to the other losses [82]. The effect of acoustic radiation was investigated in detail in

the semester thesis of Raoul Hopf [83]. As long as there is no coupling between the

vibration of the cantilever and the whole sensor (chip mounted on PCB), the lowest Qac

was estimated to be above 2 · 104. The expressions given by Lochon et al. [27] yield an

estimate for Qtherm = 5 · 104.

As discussed in section 3.3.1, df in vacuum is between 3.5 and 7 Hz, which results in

a Q-factor of approximately 1’200-2’400. Damping due to acoustic effects and viscous

damping are not present in vacuum. Therefore the main damping source is the me-

chanical support. Typical numbers for df in a fluid are around 200 Hz. Therefore, all

damping effects additional to viscous damping, except acoustic damping, make 2-3%

of the viscous damping. These damping effects, which are independent of the fluid are

included in the model by adjusting D0 in equation 2.17.

Self heating

The excitation of the vibration is achieved by an alternating current flowing through

the gold loops. These loops have a non-zero resistance and therefore thermal energy

is generated by the current. This energy slightly heats up the cantilever’s surface and

76


Measurement SimulationAir Water Air Water

fres -0.56±0.04 4.33±1.4 -0.95 4.55df 0.004±0.04 -0.65±0.38 -0.0003 -2.7

Table 3.3: Measured and simulated temperature dependency of fres and df in water andair. For the measurements, the standard deviation between the four chips isindicated. The values are all in Hz/ C.

hence may bias the measurement. The amplitude I0 of the current was measured to be

around 7-8 mA with water. With a resistance of the loop of 70 Ω, the ohmic power is

5 mW. Although this is much less than what is e.g. consumed by the heater chip, it is

not negligible a priori.

In order to ensure, that the thermal losses are indeed below a limit that affects the

measurement, I0 was increased experimentally by changing the excitation circuitry until

an increase in the temperature on the heater chip could be measured. This critical

limit was reached after a four fold increase of the current. Due to the fact that the

generated thermal energy is proportional to I20 it can be concluded, that self-heating

can be neglected. Another observation showing that this is indeed true is the fact,

that there was no increase in resonance frequency observed just after turning on the

excitation. When increasing I0 over the critical limit by purpose, the resonance frequency

increased.

Temperature effects

The resonance frequency and the damping of the cantilever have a temperature de-

pendency. Mainly the geometrical extension and the temperature dependency of the

stiffness cause a decrease of the resonance frequency in vacuum as the temperature is

increased.

The temperature sensitivities calculated with the impedance model are listed in table

3.3. The model for air includes the temperature dependency of the Young’s modulus

and a temperature dependent thermal expansion coefficient as lined out in section 2.2.2.

For water the temperature dependency of the liquid was also taken into account.

In order to verify the modeled values, the temperature dependency was measured in a

range of 10 C around room temperature. The measured values are also listed in table

3.3. Figure 3.18 shows the measured values of fres and df along with a linear fit for each

chip.

77


fres[H

z]df

[Hz]

1.94

1.96

1.98

2x 10

4

8

8.5

9

9.5

10

Temperature [ ]ºC

1.24

1.26

1.28

1.3x 10

4

170

180

190

200

Temperature [ºC]

Air Water

18 20 22 24 26 28

3.19

3.19

5.88

5.06

18 20 22 24 26 28

-0.57

-0.32

-0.51

-1.21

18 20 22 24 26 28

-0.61

-0.53

-0.58

-0.52

18 20 22 24 26 28

-0.02

-0.03

0.01

0.05

Figure 3.18: Resonance frequency and damping measured for four different chips over arange of approximately 10 C in air (left) and water (right). The solid linesindicate a linear fit. The legend entries indicate the slopes of the fittedcurves. The values are in Hz/ C.

78


The model predictions for the temperature dependency of the resonance frequency are

in good accordance for water. The value for air is approximately 70% lower than the

one measured in air. The predicted value for df in air is almost zero, mainly because no

temperature dependent damping was introduced. The mean value over 4 chips of the

measured temperature sensitivity is 0.004 Hz/ C, however with a fairly large variation

from chip to chip. Even the sign is different for different chips. Looking at the values in

figure 3.18, a trend can be found for each chip. The main source of loss is the mechanical

support, which is a parameter that is very hard to control. However, the absolute values

of the temperature sensitivity are very small. With the maximum measured temperature

sensitivity of 0.05 Hz/ C and a temperature accuracy of ±0.1C, the resulting error in

fres is 0.005 Hz, which is below the precision of the readout circuitry. Therefore, the

influence of the losses into the support due to an inaccuracy of the temperature can be

neglected as can be seen from equation 1.2.

The temperature sensitivity of the resonance frequency in water and in air/vacuum

differ in sign. This is due to the fact, that in air, the resonance frequency is mainly

influenced by the change of Young’s modulus, whereas the temperature dependent fluid

properties dominate with a surrounding liquid. If making experiments over a range of

temperatures, the effect of decreasing resonance frequency has to be accounted for in

the calculation of viscosity and density. Neglecting this effect would yield an error of

approximately 13%.

79

4. Characterization of DNA solutions

The device presented in this thesis was mainly developed for the detection of DNA in

an aqueous solution. The detection is enabled by the change of the liquid’s viscosity or

density when DNA is added or polymerized. At high concentrations of DNA, the fluid

can exhibit non-Newtonian behavior. The study of this behavior has been a research

topic since the discovery of DNA and its importance. The rheological treatment of DNA

solutions in literature is reviewed in section 4.1. The theory presented in this section

serves as a basis for understanding the experimental results shown in the subsequent

sections.

In section 4.2 the response of the cantilever sensor to solutions with DNA of different

lengths and concentrations is investigated experimentally. The first goal of these exper-

iments is to establish a limit of detection for different kinds of DNA solutions. For this,

the explicit values of viscosity and density need not to be known. The second goal is to

compare the results with other rheological studies of DNA solutions presented in section

4.1. This requires the calculation of viscosity and density.

The ultimate goal of the project is the development of a system that can detect the

presence of a defined sequence of base pairs in a sample with unknown DNA content.

A first attempt is made using the well established polymerase chain reaction, which is

presented in section 4.3. As will be shown, the detection seems to be possible. However

the changes in the properties of the fluid before and after the reaction are small and

partly not reproducible. For this reason, a second attempt was made using the rolling

circle amplification. The results for these experiments are presented in section 4.4.

Due to the much longer strands produced with this reaction the differences between

unamplified mix and amplified product are much larger.

4.1. Rheology of DNA solutions

Rheology of polymer solutions was and still is a means to understand the structure

and hydrodynamic properties of macromolecular polymers. From a biological point of

81


view, the size, shape and stiffness are of great importance. It is therefore not surprising,

that the fluid mechanical behavior of DNA solutions has been of great interest since

the molecule could be extracted from biological samples [84]. The rheology of DNA has

been studied even before the discovery of the double helical structure of DNA in 1953

to learn more about the structure of the molecule.

Early experiments were made with either long chains (> 10 kbp) obtained from bacterio-

phages, such as T4 or λ-phage, or with shorter chains gained from sonicated samples of

long chains. Since that time, DNA has been characterized very well. It became possible

to produce very well defined samples. Monodispersity, which is an important factor for

the rheological behavior, can easily be achieved by newer techniques such as the PCR.

Even the synthesis of a predefined sequence is feasible, at least for short chains. For

these reasons, DNA became a model molecule for the rheology of macromolecules. It is

possible to look at single molecules and their behavior in flow. For this, the molecule is

stained with a fluorescent dye [85]. Recently DNA has been used as as a sample material

to study the evolution of vortices in an abrupt planar micro-contraction [86]. DNA has

explicitly been chosen as a solute, because it is well characterized for viscoelastic flows.

The physical properties of DNA are also interesting for other reasons. For example DNA

has been employed as an engineering material for nanosciences. It has also been used

with detection methods based on its physical properties [87].

In the following, some basic rheological concepts as well as theory important for DNA

rheology will be introduced. In order to compare different approaches, a short overview

and a comparison of modeling activities found in literature as well as experimental

approaches will be shown.

4.1.1. Basic concepts

Intrinsic viscosity

The reduced viscosity is defined as [88]

ηred =η − ηsηsc

(4.1)

where η is the viscosity of the solution, ηs is the viscosity of the solvent and c is the

concentration of the solute in g/l. The intrinsic viscosity, a common quantity for the

characterization of DNA solution, is defined as

82


[η] = limc→0

ηred (4.2)

If the intrinsic viscosity is measured in the zero shear limit, it is denoted [η]0. The

viscosity of the solution can be calculated from

η = ηs(1 + [η]c+KH [η]2c2) (4.3)

where KH is known as the Huggins coefficient.

There is an empirical relation between the intrinsic viscosity of macromolecules and their

molecular weight M observed by Mark in 1938 and Houwink 1940 [88]. It is described

by

[η] = KMα (4.4)

This relation is known as the Mark-Houwink equation. K and α are solute specific

constants, whereas α is a measure for the stiffness of the polymer chain.

Tsortos and coworkers [89] recently published an extensive study of the intrinsic viscosity

over a wide range of molecular weights. In addition to their own measurements they

analyzed data for the intrinsic viscosity available in literature. Their data is shown in

figure 4.1. Two distinct regions can be identified in the log-log plot. The first region

is between 7 · 103 and 2 · 106 Da, the second region is for molecular weights of 2 · 106

to 8 · 1010 Da. This corresponds to the ranges of 10 to 3’000 bp and 3’000 to 108 bp,

respectively.

The influence of concentration

When increasing the concentration of a high molecular weight solution, the viscosity

increases almost linearly with concentration. At increasing concentrations, the molecules

start to overlap and entangle. The concentration c∗ at which this happens is estimated

by Larson [90] to be

c∗ =1

[η]0(4.5)

At concentrations above c∗ the polymer chains start to entangle. At these concentrations,

it makes more sense to look at the storage and loss moduli than measuring the intrinsic

viscosity. One molecule can then be thought to be confined in a virtual tube, when the

83


-3

-2

-1

1

2

3

0

3 4 5 6 7 8 9 10 11

log(M)

log[

] [m

l/m

g]

´

Figure 4.1: Intrinsic viscosity of DNA solutions for a wide distribution of molecularweights from Tsortos et al. [89].

fluid is relaxed [90]. After a certain time (reptation time), the molecule has completely

left this tube due to the motions of the ends.

The method of choice for the characterization of entangled DNA solutions are small

amplitude oscillatory shear experiments, where the complex shear modulus is measured.

The complex shear modulus is defined as

G(ω) = G′(ω) + iG′′(ω) (4.6)

G′(ω) being the storage modulus and G′′(ω) the loss modulus. Both depend on the

angular frequency ω.

The complex viscosity is defined as

η∗(ω) = η′(ω)− iη′′(ω) (4.7)

and is related to the shear modulus by

G′′(ω) = ωη′(ω) (4.8)

G′(ω) = ωη′′(ω) (4.9)

84


It has been shown, that the reptation time τ is connected to the crossover frequency

ωc, where G′ and G′′ cross on a plot with respect to the frequency. Typical values for

concentrated solutions are around 1 s.

Several groups have investigated entangled DNA solutions [91, 92, 93, 94] based on

micro-rheology or using cone-plate rheometers. Concentrated solutions of long DNA

(calf thymus, λ-DNA) show a distinct frequency and strain rate dependent behavior of

the shear modulus.

Salt concentration

Since DNA is a polyelectrolyte, the concentration of salt may have an influence. It is

known, that experiments in pure water are barely reproducible. At very low concen-

trations of ions, the DNA is expanded [90]. However, this effect is negligible above

approximately 10 mM salt concentration. Tsortos et al. [89] give a relation for the influ-

ence of the ionic strength IS on the intrinsic viscosity which holds between 2 mM and

1 M.

[η] = D +B√IS

(4.10)

D and B are constants depending on the length of the DNA. The smaller the molecular

mass, the lower the influence. According to Tsortos et al., there is a 6% change in

intrinsic viscosity if the salt concentration is increased from 0.1 M to 0.2 M for very long

chains. For chains below approximately 4’500 bp the effect should not be detectable

anymore.

In the experiments described in section 4.2, the concentration of KCl is 10mM plus

50mM of Tris-HCl. Therefore, the effect of ionic strength should not play a role.

4.1.2. Modeling

As stated earlier, DNA is not only of interest because of its relevance in biology. It also

serves as a model polymer due to its well known and controllable molecular properties.

Several theoretical models have been developed for DNA solutions. The following effects

are believed to be important in modeling diluted solutions of macromolecules according

to Larson [90]: Viscous drag, entropic elasticity, Brownian forces, hydrodynamic inter-

action, excluded volume interactions, internal viscosity and self-entanglement. Viscous

drag is the main factor for polymers to increase the viscosity. It is the frictional force,

that influences the flow around the molecule. Entropic elasticity comes into play as

85


soon as there are visco-elastic effects observed in the rheological behavior of the fluid.

Brownian forces arise through the random bombardment of the molecule with solvent

molecules. It can be observed for example with a bead immersed in a liquid without

any superimposed flow. Hydrodynamic interaction is an effect which is important for

long chains. It describes the effect when one end of the molecule influences the flow

around the other end of the molecule. Excluded volume effects arise through the fact

that one end of a molecule can not physically be at the same position as the other end.

This increases the mean end-to-end distance. Internal viscosity describes losses due to

the internal friction in a molecule. Self-entanglement is the effect, when two ends of the

molecule overlap and hinder each other. According to Larson, the last two effects are of

less significance.

Bead-rod and bead-spring model

The most common models are shown in figure 4.2. A straight forward model is the

bead-rod model [90]. It is defined by a number of beads, which are connected with stiff

rods. Frictional drag acts on the beads and each rod can randomly orient. The length

bk of a single element is called Kuhn segment. The mean-square end-to-end length is

< R2 >= b2kNk, where Nk is the number of Kuhn segments. The contour length of a

fully stretched polymer chain is given by LC = Nkbk. This value is a property of the

real polymer chain. It can be calculated via LC = nl, where for DNA n is the number

of base pairs and l the distance between the base pairs. The value for l is 0.34 nm. The

mean-square end-to-end distance is a property that can be measured. It is characterized

by a number C∞ in < R2 >= C∞nl2. The model parameters bk and Nk can therefore

be calculated from measurements of < R2 > and the knowledge of l and n

The length of a Kuhn step has to be around 10 times the bond length of a polymer,

since C∞ is around 10. Therefore, the longer the polymer is, the higher the number of

Kuhn steps. Since the computational efforts are rather high for this model, very long

chains are not feasible any more. This is one reason, why the bead-rod model has only

limited application for DNA.

In order to reduce the complexity of the model and to enable the modeling of larger

chains, bead-spring models can be used (figure 4.2 b)). The beads are connected by

springs instead of rigid rods. The advantage is, that the model can be much coarser.

Each spring can represent several Kuhn steps. This reduces the degree of freedom com-

pared to the bead-rod model. The interesting thing of this model is that the properties

can nicely be tuned with the spring constant. In the simplest case it is a linear Hookean

86


a) b)

c)

bk

Lc

Lp

Lc

"±=200=0.1

"±=1.5=0.01

"±=0.025=0.01

Figure 4.2: Models used to describe the rheological properties of DNA. a) Bead-rodmodel and b) Bead-spring model according to [88] c) worm-like chain ac-cording to [95].

87


spring (Rouse model). However, in reality the spring force is increased more than lin-

early when the polymer is stretched to its contour length. For this reason, nonlinear

spring forces are used. The bead-spring model can be connected to the bead-rod model

using an appropriate function for the spring force (inverse Langevin function). Even

the behavior of the worm like chain model introduced in the next section can be ap-

proximated with an appropriate spring force. One disadvantage of the Rouse model is

the fact, that it does not include hydrodynamic interaction. The Zimm model is an

extension of the Rouse model and includes this effect.

Worm-like chain model

The worm-like chain (WLC), also called Kratky-Porod model, is sketched in figure 4.2

c) for different model parameters. The main difference between the previous models is

that the chain is now described as continuous and differentiable curve in space [95]. The

characteristic equation describing the model is

< s1 · s2 >= exp

(−|s1 − s2|

Lp

)(4.11)

where s1 and s2 are unit tangent vectors at the contour points s1 and s2 respectively.

Lp is the persistence length. The persistence length is a measure of the stiffness of

the molecule. The stiffer it is, the longer its persistence length. The model is a good

option if the polymer is not stiff enough to be described by a rigid rod, but too stiff

to be treated as a random coil. The mean square end-to-end distance is < R2 >=

2LPLC [1 − LP/LC(1 − e−LC/LP )]. For a long flexible chain, where LC >> LP , we get

< R2 >= 2LPLC . Comparing to the bead-rod model above, it can be concluded that

the Kuhn segment length is twice the persistence length. One additional parameter d

is needed for the WLC model, representing the diameter of the chain. The flow around

the chain is assumed to be of Stokes type [96]. The influence of the two parameters

ε = LP/LC and δ = d/LC is illustrated in figure 4.2 c).

Comparison

Larson [90] gives numbers for the persistence length for λ phage DNA (48’502 bp) from

different experiments. For unstained DNA, a persistence length of 54 nm was found

using single molecule experiments.

Mansfield and coworkers made WLC simulations for different parameters δ and ε. One

part of their results is the prediction of the Mark-Houwink exponent α shown in a contour

88


DNA

damag

edCNTs

defect-freeCNTs1

kbp

100 bp

Figure 4.3: Contour plot of the Mark-Houwink exponent α derived from WLC sim-ulations (reprinted (adapted) with permission from Mansfield et al. [95].Copyright (2008) American Chemical Society). The meaning of ε and δis illustrated in figure 4.2. The logarithmic scales indicate the regions forDNA (with d = λl =2.4nm and Lp=50nm), damaged and defect-free carbonnanotubes.

89


plot in figure 4.3. Tsortos et al. [89] got α= 1.05 for short chains (10 to 3000 bp) and

0.69 for long chains (3000 to 108 bp). The trend in the predictions of Mansfield is

the same. For the short chains they get values for α between 0.95 and 1.3, depending

on chain length. For long chains they get values of around 0.7 to 0.8. However, the

WLC they describe is not valid anymore for very short chains. The discrepancy for low

molecular weights was as well noted by Tsortos and coworkers. They have calculated

the persistence length LP from the measured α. Their conclusion is that LP goes from

40 nm for 1000 bp down to 16 nm for 150 bp. This would mean, that the smaller

molecules are not as stiff as assumed in many other studies.

4.1.3. Experimental methods for the rheological characterization of

DNA solutions

Classical approaches

In the early phase of experimental characterization of DNA fluids, the methods of choice

have been capillary viscometers, such as Ostwald or Ubbelohde type viscometers [97].

The main purpose was to measure the viscosity of solutions at different concentrations

and to calculate [η] using equation 4.3. They give accurate results regarding viscosity.

In certain configurations it might be possible to get data for increased strain rates.

However, it is not possible to do experiments with oscillatory flow with these devices.

Because DNA solutions exhibit non-Newtonian behavior, people use conventional cone-

plate rheometers as well [98, 91]. This kind of rheometers can be run in controlled stress

or controlled strain(rate) modes and allow a much deeper insight in the properties of

the fluid. Especially at concentrations above the entanglement concentration a capillary

viscometer does not give enough information to characterize a fluid properly.

Specialized methods based on flow

There exist some improved less common methods based on flow having special features.

The main idea is usually to generate a very well defined flow. E.g. Gulati et al. [86] use a

2:1 contraction channel to study the viscoelastic flow patterns with a semi-dilute solution

of DNA. The flow patterns were visualized with particle imaging velocimetry. In order

to characterize the fluid, they used as well a rheometer in cone-plate and plate-plate

configuration, as well as an oscillatory flow viscometer. Hsieh and Liou [85] investigated

the flow pattern of λ-phage DNA in converging-diverging micro channels. The aim of

this geometry is to generate extensional flow and analyze the stretching behavior.

90


A certain disadvantage of some of these instruments is their rather large consumption of

material. Lee and Tripathi [46] present a microfluidic device which allows the measure-

ment of [η]. The device requires only 3 µl of sample volume. It is based in an optical

measurement of fluorescence and the measurement of differential pressure. For a 10 bp

long chain they get a value for intrinsic viscosity of 3.9 ml/g, however with a fairly large

error.

Microrheology

A relatively new area is microrheology, which is employed to determine the molecular

properties of polymers. With this method, a small sphere with a radius of approxi-

mately 1 µm is placed into the liquid [99, 92]. The Brownian movement of the sphere

is recorded and the mean square displacement is evaluated directly or used to calcu-

late other molecular properties. This way of measurement is called the passive mode.

One benefit of this method is its low sample consumption. In the active mode, the

bead’s position is controlled via an optical trap or electromagnetic forces and brought

to an oscillatory motion. A setup for DNA was presented in 2008, where Rajkumar and

coworkers [100] hold a micro-bead in an optical trap. The power spectral density of the

bead was analyzed in this case.

Conventional rheometers often yield rather inaccurate results for fluids with a viscosity

close to the one of water. In general, microrheology yields more accurate results for this

range.

Other methods for macromolecular polymer solutions

There are other methods to determine the properties of solutions of (long) polymers.

Some of them are used in connection with rheology. As shown above, the persistence

length LP of a polymer molecule is an important parameter describing it. Wang and

coworkers [101] used optical tweezers to hold a micro-bead. A chain of DNA is attached

to the bead and fixed at the other end. The chain is then stretched and the required

force is recorded. Using this method a force-extension curve can be derived. From this

data the contour length LC as well as an appropriate law for the spring force and out

of this the persistence length LP could be calculated. In their case, they got a value for

LP=38-47 nm for different salt concentrations.

With static light scattering it is possible to measure the square average radius of gyra-

tion. This has been done for calf thymus DNA by Godfrey and coworkers in the seventies

[102]. Dynamic light scattering can reveal additional information about the molecular

91


dynamics as shown by Sorlie and Pecor [103]. They measured the diffusion coefficient,

which is inversely proportional to the radius of gyration. In their contribution they also

compare their results to various models and get a good agreement for the Rouse model.

Godfrey carried out sedimentation experiments [104] measuring the sedimentation coef-

ficient. This value is connected to the hydrodynamic radius.

4.1.4. Conclusions

The main body of literature data on DNA solutions treats diluted solutions at low

frequencies or stationary flow. Also data for entangled DNA solutions is available only for

frequencies below 100 rad/s. This is probably due to the fact, that all the experimental

setups described above are limited to low frequencies. Except for active microrheology,

where in principle higher frequencies are achievable with the right optical equipment.

Also the principle of time temperature superposition [90] can not be applied, since

DNA denatures at elevated temperatures. This makes a direct comparison with the

experiments described below rather difficult. The use of the cantilever sensor gives

access to the properties of DNA solutions in the 10 kHz range.

The persistence length of DNA is generally be thought to be around 50 nm. Therefore

it will be interesting to test the device with chains that are much longer or shorter than

this value. The 110 bp samples (37.4 nm) used in the next section are slightly below

LP , whereas the 10 kbp (3.4 µm) samples are much longer.

4.2. Titration experiments with DNA solutions

With a first experiment with DNA solutions, the ability to distinguish between different

concentrations of DNA using the chip shall be investigated. Based on this knowledge, a

judgment whether amplification products in a PCR can be detected or not can be made.

If the sensitivity of the chip is high enough, even real time measurements (measuring fres

and df after each cycle) could be made. For this experiment the viscosity and density do

not need to be calculated explicitly. The calculation of these would introduce a certain

error. Therefore a less stringent measure of performance is the ability to distinguish two

fluids with only the measurement of fres or df .

The limit of detection xLOD, the smallest concentration that can be detected with rea-

sonable certainty, can be derived from

92


xLOD = xb + kσb (4.12)

where xb is the mean value of a blank measurement, k takes the value of 3 and σb is the

standard deviation of the blank measurement [105]. For fres and df this reads

fres,LOD = fres,buffer − kσfres,buffer (4.13a)

dfLOD =df buffer + kσdf,buffer (4.13b)

where the minus in front of k comes from the fact, that the resonance frequency decreases

for an increasing concentration. The blank measurement is in this case a measurement of

fres or df with the buffer solution. In other words, if the value of fres or df for a solution

with a certain concentration is at least k times the standard deviation different from

the measurement with the buffer solution, the chip is able to detect this concentration

of DNA.

The results from this experiment shall be compared to an ideal, fictitious PCR. For this,

solutions with different concentrations of DNA were made. The cycle number n of this

fictitious PCR can be calculated theoretically. The copy number Nn in a PCR is given

by

Nn = N0(1 + e)n (4.14)

where N0 and e are the initial copy number and the efficiency of the PCR, respectively.

Assuming an efficiency of 1, the approximate cycle number can be calculated by

n = log2

Nn

N0

(4.15)

Nn is in this case calculated from the concentration cn in cycle n using

Nn =cnMW

VNA (4.16)

where MW is the molecular weight of one strand, V is the reaction volume and NA the

Avogadro constant.

93


10 kbp solution 110 bp solution

Length [bp] 10’000 110Molecular weight [kDa] 6200 68.2Contour length [nm] 3400 37.4Concentration range [mg/ml] 0.13 - 133.4 0.0047 - 3.39Cycle number n 23 - 33 25 - 34Copy number range 3 · 1011 - 3 · 1014 1 · 1012 - 7 · 1014

Table 4.1: Properties of the 10 kbp and 110 bp solutions used for the titration experi-ments.

4.2.1. Experiment and results

Materials

In order to produce solutions with different concentrations, a titration was done. Two

types of solutions were used. The first titration series was made with solutions of dsDNA

with a length of 10 kbp, the second series had 110 bp long strands in solution. The

110 bp DNA was synthetic, bought from FRIZ Biochem (Neuried, Germany), whereas

the 10 kbp DNA was a plasmid digest purchased from Ascoprot Biotech (Zlin, Czech

Republic). In both cases the DNA was dissolved in a buffer (10 mM KCl, 50 mM Tris-

HCl, pH 7.5). The properties are outlined in table 4.1. First, stock solutions of 3.39

mg/ml for the 110 bp strands and 133.35 mg/ml for the 10 kbp strands were prepared.

The stock was then diluted by each time halving the concentration. All solutions were

prepared by Dr. Damiano Cereghetti.

The concentration range and the calculated copy number range are given in table 4.1.

The value for the copy number n in the table is based on the assumption that an

amplification of 100 ng of genomic DNA in a 25 µl reaction volume occurred under ideal

conditions (e = 1 in equation 4.14). Each sample was measured three times in order to

establish the statistics needed for the calculation of the LOD and to have a measure

for the error. The setup described in section 3.2.5 was used with the temperature set

to 23 C. The measurements presented in this section were carried out on the same

chip. This has the benefit of a lower chip consumption on one side. On the other side

there is no chip to chip variation of the resonance frequency which would have to be

taken into account when comparing the results. The cleaning protocol applied between

measurements with DNA is described in table 4.2.

94


Step 1 Remove sampleStep 2 Fill with DI water and heat for 30 s with the platinum heater at 7 VDC .

Repeat once.Step 3 Flush with DI water three times.Step 3 Dry by heating with the platinum heater on the back side while flush-

ing with nitrogen.

Table 4.2: Chip cleaning procedure applied after each sample containing DNA.

Results and discussion

Figure 4.4 a) shows the mean values of the measured fres and df for both titration

series with respect to mass concentration. The error bars indicate the minimum and

maximum values for each point. The standard deviation of fres and df is between 2 and

3 Hz. The error bars of the resonance frequency measurement are not visible since they

are too small. As expected, fres decreases with increasing concentration and df increases.

Interestingly, for the long chains, this effect is only visible at higher concentrations. In

figure 4.4 b) the same data is plotted with respect to molarity. Here, the effect is already

visible at lower values for the 10 kbp strands.

The expression in equation 4.12 gives the minimal difference between fres or df with

buffer and DNA solution which is needed to detect a minimal concentration of DNA. A

way to convert fres,LOD and dfLOD into units of concentration is to make a fit for fres

and df for the measured data. This yields a function that relates the frequencies to the

concentration. Using this function, the minimal concentration corresponding to fres,LOD

and dfLOD can be calculated .

This procedure was applied to the data shown in figure 4.4. A second order polynomial

was used for the fit. The calculated limit of detection is shown in table 4.3 in terms of

concentration, copy number and minimal cycle number. For both, the 10 kbp and the

110 bp solutions, the lowest value results when measuring the resonance frequency. The

values for the 10 kbp solutions are higher. This is somewhat counter intuitive. However

the data make sense in view of the fact, that the longer strands have a higher molecular

weight, and therefore the copy number is lower. This can also be seen if comparing the

two plots in figure 4.4. The minimal cycle number nmin is based on the assumptions

described above and equations 4.14 through 4.16. It is lower for the 10 kbp solutions.

The reason for this is the fact that these strands are longer and have therefore a higher

influence on the cantilever.

95


180

190

200

210

220

12060

12160

12260

12360

12460

12560

12660

0.001 0.01 0.1 1 10 100

fres[H

z]

df

[Hz]

10 kbp fres

110 bp fres

10 kbp df

110 bp df

Concentration [ g/¹ ¹l]

(a)

180

190

200

210

220

12060

12160

12260

12360

12460

12560

12660

5.00E-09 5.00E-08 5.00E-07 5.00E-06

fres[H

z]

df

[Hz]

10 kbp fres

110 bp fres

10 kbp df

110 bp df

Molarity [mol/ ]l

(b)

Figure 4.4: Measured fres and df for different concentrations of 10 kbp and 10 bp longDNA in buffer. The plots are with respect to mass concentration in a) inwith respect to molarity in b). The error bars indicate the minimum andmaximum values.

96


measuring fres measuring dfcmin Nmin nmin cmin Nmin nmin

110 bp 0.44 9.7 ·1013 31 2.49 5.5 ·1014 3410 kbp 2.35 5.7 ·1012 27 9.16 2.2·1013 29

Table 4.3: Limit of detection in terms of concentration cmin in mg/ml, copy numberNmin

and cycle number nmin for 110 bp and 10 kbp solutions when measuring fresand df , respectively.

It is important to point out, that the influence of the DNA is higher on fres than on df .

The LOD is lower for fres. Therefore, it will be favorable to assess the reaction outcome

via the measurement of the resonance frequency.

The minimum cycle numbers are at the upper limit of what is normally achieved. How-

ever, they are not completely out of range. For the calculation of nmin an efficiency

of 100 % over the whole reaction was assumed. This is never the case in a real PCR.

Another important fact which is not considered here is the consumption of NTPs and

primers. With higher cycle numbers the concentration of these two ingredients gets

lower and lower. This slows down or even stops the reaction at later cycles. For these

reasons, the numbers for nmin shown here represent the best possible case.

4.2.2. Comparison with existing models

In the preceding part, the values for fres and df were evaluated directly in order to

establish a limit of detection. If the chip is used to detect the progress of a reaction, this

might be enough information. As shown in section 3.3, the cantilever can also be used to

calculate the viscosity and the density of the solutions. This makes these results much

better comparable to literature values and therefore much better interpretable. The

disadvantage of calculating the viscosity and density explicitly is however, that further

uncertainties due to calibration errors are introduced.

The calculated fluid properties are plotted in figure 4.5 for both solution types. The

values have been calculated using the sensitivities from table 3.2. The error bars in the

plots are fixed percentage values of 6% for the viscosity and 0.4% for the density. The

chip that was used for the titration experiments was one of the 9 chips used in section

3.3.5. It is therefore reasonable to assume that the error will not exceed the maximum

error shown in figure 3.17. The error could further be reduced by calibrating the sensor

with the glycerol and ethylene glycol solutions separately. This has, however, not been

97


Measured [η] KMα from [89] c∗

110 bp 0.047 ±0.01 0.049 2010 kbp 0.013 ±0.008 3.8 0.26

Table 4.4: Measured intrinsic viscosity [η] in ml/mg along with 95% confidence boundsof the fit and intrinsic viscosity using the Mark-Houwink equation with α =1.05 and K = 3.5 · 10−7 ml/mg for the short strands and α = 0.69 andK = 8 · 10−5 ml/mg for the long strands (values from Tsortos et al. [89]).The last column shows the entanglement concentration in mg/ml derivedfrom the literature values for [η].

done. The error bars for the 110 bp solutions have been omitted since they would cover

the whole range of the plot.

As shown in section 4.1, the typical quantity to characterize the rheological properties

of diluted DNA solutions is the intrinsic viscosity. It can be quantified by plotting

the reduced viscosity versus the concentration. A linear least squares fit of the reduced

viscosity reveals the intrinsic viscosity [η] and the Huggins coefficient KH by comparison

with equation 4.2. Doing this for the measured data results in the plots shown in figure

4.6. For the fit, the data points were weighted with the inverse value of the errors in

order to deal with the fact, that the error is higher for the solutions with a very low

concentration. Additionally, only the data with a reasonably small error was used.

Using the parameters for the Mark-Houwink equation from Tsortos et al. [89] the liter-

ature values for [η] can be calculated. The measurements and the literature values are

summarized in table 4.4 along with the calculated entanglement concentration.

110 bp DNA

The viscosity of the 110 bp solutions increases from 0.95 to 1.7 mPas over the range of

the experiment as shown in figure 4.5 a). The increase in density of the 110 bp solutions

is almost negligible. From figure 4.6 an intrinsic viscosity of 0.047 ml/mg results. This

value for [η] agrees well with the prediction from literature and illustrates that the chip

can be used for the characterization of solutions with short DNA strands.

10 kbp DNA

According to figure 4.5 the viscosity of the 10 kbp solutions is increased from 1 mPas to

1.7 mPas, while the density increased from 1000 to 1018 kg/m3. The intrinsic viscosity

is measured to be 0.013 ml/mg for the long strands. The Huggins coefficient is negative,

98


0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1001.0

1001.5

1002.0

1002.5

1003.0

1003.5

1004.0

0.001 0.01 0.1 1 10

Den

sity

[kg/m

]3

Vis

cosi

ty [m

Pas]

Concentration [mg/ml]

(a) Measured viscosity and density of the 110 bp solutions

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

995

1000

1005

1010

1015

1020

1025

1030

0.1 1 10 100

Den

sity

[kg/m

]3

Vis

cosi

ty [m

Pas]

Density

Viscosity


(b) Measured viscosity and density of the 10 kbp solutions. Both values have to beinterpreted with care as the solutions exhibit non-Newtonian behavior for which equation2.50 may not be suitable.

Figure 4.5: Measured viscosity and density. The error bars of the viscosity values are6% and the ones for the density are 0.4% following the reasoning in section3.3.5. They are omitted for the density in plot a) due to their large size ofapproximately 4 kg/m3.

99



´re

d[m

l/m

g]

0 20 40 60 80 100 120 140-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

´red= -3.6e-6 +0.013c

(a) 110 bp Solutions


´re

d[m

l/m

g]

´red= 0.0042 +0.047c

0 0.5 1 1.5 2 2.5 3 3.5-0.05

0

0.05

0.1

0.15

(b) 10 kbp Solutions

Figure 4.6: Reduced viscosity of the 110 bp solutions in a) and the 10 kbp solutions in b).The solid lines are a linear fit for c in mg/ml, revealing the intrinsic viscosityand the Huggins coefficient. The error is larger for low concentrations sincethe difference between ηs and η tends to be smaller.

100


whereas it is positive for the short strands. The measured intrinsic viscosity is completely

out of range. It is even lower than the value of the short strands. Also the negative

Huggins coefficient seems to be unreasonable. Due to the non-Newtonian behavior, the

calculated densities may be wrong. The data has therefore be interpreted with great

care.

As shown in table 4.4 the entanglement concentration for the long strands is 0.26 mg/ml.

The solution with the lowest concentration has a concentration of 4 mg/ml, which is

already above the entanglement concentration. The fluid is therefore not expected to

behave like a linear viscous fluid anymore. In this case, various effects, which were not

directly controlled in the experiment, start to play a role. The solution is expected to

exhibit visco-elastic effects when the concentration is increased above the entanglement

concentration. This means, that the viscosity is a complex value or in other words,

the rheological behavior has to be expressed as a storage and loss modulus as shown in

section 4.1. These moduli are frequency dependent and there is also a dependency on

shear rate in general.

It is not straight-forward to calculate the influence of the storage modulus on the reso-

nance frequency. The sensors were calibrated with glycerol and ethylene glycol solutions.

These are Newtonian and therefore not suitable for a calibration for non-Newtonian flu-

ids. Belmiloud et al. [71] investigated the influence of non-Newtonian fluids on cantilever

sensors. They state that an increase in the cantilever frequency may be observed where

elasticity is dominant. According to them the storage modulus acts like an elastic foun-

dation, which would explain a higher fres than expected.

The amplitude of the cantilever is indirectly controlled by the electronics and can be

calculated, however with a limited amount of accuracy. Another possibility is to measure

the amplitude directly with a laser interferometer. These measurements are a bit tricky

however, since the light beam has to go through the PDMS lid and through the liquid,

whereas at each interface it is partly reflected.

A finite element simulation analogous to the one shown in figure 2.5 using the velocity

amplitudes from section 3.2.4 gives an estimation of the maximum strain rates produced

by the cantilever. The maximum strain rate is calculated to be 1.5 · 104 1/s. As can

be seen in figure 2.5, the flow field and therefore also the strain rates are far from

homogeneous however.

A direct comparison of the experiments to literature is not possible, because no liter-

ature data has been found on the behavior of entangled or semidilute DNA solutions

at high frequencies. The highest frequency that is usually investigated is around 100

101


rad/s. In their publications Mason et al. [91] and Bandyopadhyay and Sood [94] made

experiments with calf thymus DNA in buffer. The strands have a length of 13 kbp,

which is comparable in size to the 10 kbp solutions. They used concentrations from 1

to 10 mg/ml, which is in the same range as in the presented experiment. The com-

plex viscosity η∗ shows a drop above a strain rate of 1/s, which is far below the strain

rate estimated above. It is however important to note, that not only the strain rate

is important but also the amplitude of the strain. Large deformations may change the

conformation of the molecule by e.g. stretching or tumbling it [90]. However with the

rather small deformation present with the cantilever, this might not be the case.

In conclusion the sensor can be used for the rheological characterization of short stranded

molecules and Newtonian fluids without qualification. It is also possible to make mea-

surements with non-Newtonian fluids as shown with the 10 kbp solutions, which gives

new information on the properties of such fluids at frequencies in the kHz range. It

is, however, important to note, that there are the discussed uncertainties on how to

interpret the data.

4.3. Polymerase chain reaction – PCR

The experiments in this section were carried out to demonstrate the concept of measuring

the reaction outcome of a PCR with the cantilever. The reactions were mixed and cycled

off-chip on a commercial thermo cycler. The reaction outcome was analyzed with gel

electrophoresis in order to verify the expected result with a standard test. The part of

the samples which was not used for verification was then used for measurement with the

cantilever sensor.

Two different cases were tested. In both, there were one positive sample and several

negative controls. In the positive sample, all ingredients for a successful reaction were

present. The negative controls were designed in a way, that no reaction takes place.

All reactions including electrophoresis were prepared and carried out by Dr. Damiano

Cereghetti. In order to estimate the confidence with which the negative samples can be

distinguished from the positive ones, Welch’s t-test was made. Each sample was usually

measured three times. However, due to chip breakage some samples were measured only

twice. The statistics have therefore to be interpreted very carefully.

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4.3.1. Experimental setup

A 187 bp long region (MW=57.9 kDa) surrounding the Leiden mutation on the factor

V gene was selected as described in previous studies [5]. Genomic DNA certified for the

Factor V gene mutations was from NIBSC (National Institute for Biological Standards

and Control; South Mimms, UK). AmpliTaq Gold DNA Polymerase (5 U/µl; #4338859)

was from Applied Biosystems (Carlsbad, CA, USA). dNTPs (10 mM each; #R0192)

were from Fermentas (St. Leon-Rot, Germany). Calf thymus DNA (#D4522) was from

Sigma (Saint Louis, MO, USA)

Four mixes were prepared. Table 4.5 shows the content of the samples. The first sample

A1 contained all the ingredients necessary for a successful reaction. In A2 the polymerase

was added after cycling. As such, no amplification can take place. In A3 there were

no primers present. In A4 there was no genomic DNA, meaning that there was no

target that could be amplified. The purpose of the negative controls in the first case

is to see the influence of the different ingredients on the reaction. All samples have to

have the same composition, since the salt concentration and especially the glycerol from

the enzyme storage buffer influence the viscosity and the density of the samples. No

reaction takes place in the negative controls, which was verified with gel electrophoresis.

Accordingly, the properties of the fluids should not change before and after reaction.

4.3.2. Results

Figure 4.7 shows the measured fres and df for the solutions described in table 4.5. A1

and A2 were measured three times, A3 and A4 were measured twice. The positive

sample A1 can be clearly distinguished from the negative controls. As expected from

section 4.2.1, the distinction between positive and negative is much clearer when looking

at the resonance frequency. The last row in table 4.5 lists the p-values for the t-test

carried out for the measured resonance frequencies. The test was done for all negative

controls. For sample A2 the value is 0.005. This value is below 0.05, which is normally

chosen as the confidence level. For the two other negative controls, the values are higher

and for A4 even above 0.05. However, it is important to note, that these samples were

measured only twice. Therefore the meaning of these numbers is somewhat limited.

Alternative negative control

The negative controls in the experiment above were made to verify that a mix that is

not able to amplify DNA (because the enzyme, the primers or the target is missing)

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Common to all samplesTris, pH 8.0 15 mMKCl 50 mMMgCl2 3 mMdNTPs (in total) 0.8 mMEDTAa 4 µMDTTa 40 µMTween 20a 0.02 % (v/v)Glycerola 2% (v/v)

Specific to each sampleA1/B1 A2/B2 A3/B3 A4/B4

Polymerase 20 U (20U)b 20 U 20 UPrimers (both, 20 nt long) 2 µM 2 µM - 2 µMGenomic DNA (approx. 109 bp) 8 ng/µl 8 ng/µl 8 ng/µl -Expected Outcome pos neg neg neg

T-testp-value for T-test with respect toA1

- 0.005 (0.04) (0.08)

a) from enzyme storage buffer, b) added after cycling

Table 4.5: PCR reaction composition. The total volume is 100 µl. The cycle numberwas 40.

181

182

183

184

185

12543

12553

12563

12573

A1 A2 A3 A4

fres

[Hz]

df

[Hz]

fres

df

Figure 4.7: Measured resonance frequency and damping for PCR samples. The positivesample A1 can be distinguished from the negative ones if looking at thedifferences of fres.

104


Human gDNA Calf thymus gDNA No DNAC1/D1 C2/D2 C3 C4 C5

Primers [µM] ofeach

0.5 0.5 0.5 0.5 0.5

DNA [ng in 100 µl] 800 200 800 200 0Expected outcome pos pos neg neg neg

Table 4.6: Changed components for PCR with alternative negative control. The othercomponents can be found in table 4.5.

does not generate a positive signal. However, in an application where the aim is the

detection of pathogens or an inherited disease, such as the Factor V Leiden mutation,

the approach would be slightly different. The mix should yield a positive outcome if a

specific sequence of DNA is present. Therefore, the same target as the positive sample

should be present in the negative control, however without the sequence that would

trigger the reaction.

Table 4.6 shows the mixes used in a second experiment, which should test the chip’s

ability to detect the presence of a certain sequence. C1 and C2 are designed to yield

a positive outcome, having the sequence in the mix. C3 and C4 contain calf thymus

DNA, where the positive sequence can not be present. The difference between C1 and

C2 is the concentration of the target, as it is the case for C3 and C4. C5 has no target

DNA in it.

The measured resonance frequencies and damping values are plotted in figure 4.8. Each

point represents one measurement. D1 and D2 are duplicates of C1 and C2. They

contain the identical components, however, the reaction was carried out in parallel in a

different tube. As expected, the values for the resonance frequency of C1 and D1 are

lower than the ones for the negative controls. The values for C2 and D2 are in the same

range as the negative controls are. This shows, that the detection of a certain sequence

is possible. However, the outcome is very much dependent on the initial concentration

of the target. If the concentration is lowered by a factor of four. The result would then

be a false negative.

Repeatability

The repeatability of all the experiments showed to be a problem during further exper-

iments. The measured fres and df for four samples (A1-A4) with duplicates (B1-B4)

which were prepared according to table 4.5 are shown in figure 4.9. The data is plotted

in the order the samples were measured. The samples A1 and B1 are expected to yield

105


fres

[Hz]

df

[Hz]

fres

df

181

182

183

184

185

186

12605

12615

12625

12635

12645

C1 D1 C2 D2 C3 C4 C5

Figure 4.8: Resonance frequencies and damping values for the experiment with alterna-tive negative controls. Each point represents one measurement. A1 and B1are the positive samples.

a positive results, i.e. a lower resonance frequency and a higher df . These are indicated

by a gray background. The first measurement of A1 indeed yields a lower fres. How-

ever, the values increase with each measurement. There is a general drift visible with

an increasing number of measurements.

There are different possible reasons for these problems. First, the good results shown

in section 4.3.2 may have been positive not because the cantilever could detect the

polymerization, but another effect may have been responsible for the lower fres. One

possibility is evaporation of water altering the concentration of solutes. However, the

results in figure 4.9 indicate a problem with the measurement procedure. Thawing and

freezing the samples repeatedly and the storage of the sample may induce sedimentation

or clumping of the polymerized DNA. If this was the case, the fluid would not be

homogeneous anymore and the properties of the part of the sample transferred to the

chip would more or less randomly change.

Formulas 4.14 and 4.16 can be used to estimate the final concentration in the PCR mix.

With the data from table 4.5 and assuming an efficiency of e = 1 the final concentration

will be 1’600 µg/µl. With a more realistic efficiency of e = 0.9 and assuming that

the reaction reaches the plateau after approximately 30 cycles (n = 30) the predicted

concentration is 0.34 µg/µl. This value is in the order of magnitude of the limit of

detection. This shows, that the outcome detected by the chip is highly dependent on

the initial concentration of the target, as shown experimentally above. The second

important factor is the enzyme activity, which can decrease with storage time and slight

changes of the experimental environment.

106


179

181

183

185

187

189

12740

12760

12780

12800

12820

A1 A2 A3 A4 B1 B2 B3 B4 A1 A1 A1 B1 A2 A1 B1

fres

df

fres

[Hz]

df

[Hz]

Figure 4.9: Single measurements of PCR samples in order of time. The composition isas in 4.5, where the samples beginning with B are duplicates. The sampleswhich should yield a positive result which would be indicated by a lower fresor higher df are marked with a gray bar.

4.3.3. Conclusions

The goal of the two experiments was to show that the cantilever is able to distinguish

between a positive and a negative reaction outcome of a PCR. This has been demon-

strated by the two experiments shown above, however with bad reproducibility. The

next step would be to carry out the whole reaction on the chip, which would reduce any

uncertainties from sample handling and storage. However, various other problems arose:

The formation of bubbles could not be avoided for temperatures above approximately

40 − 50 C. From a biochemical point of view, this is not much of a problem in prin-

ciple. However, the formation of bubbles makes it impossible to draw any meaningful

conclusions from the measurement of resonance frequency and damping. The above ex-

periments show, that a reaction can be detected, however with a limited significance. In

a real application, the presence of a sequence has to be detected with a high confidence

level. Given the fact, that the reaction was not carried out on the chip and additional

uncertainties will most probably arise in an on-chip reaction the sensitivity of the chip

is too low.

There are different approaches to overcome this problem. The first approach would

be to increase the sensitivity of the cantilever. This could be done by decreasing its

dimensions. However, the read-out signal and Q-factor would also decrease and would

107


make a precise measurement more difficult. The consequence of a decreased precision

would be a higher standard deviation.

Another approach is to optimize the reaction in a way, that the viscosity or density is

changed much more. This approach is pursued in the next section with the rolling circle

amplification.

4.4. Rolling circle amplification – RCA

In the last section, it was demonstrated, that the reaction outcome of a PCR can be

analyzed by the cantilever chip. Although the differences between negative and positive

reactions are significant, the reaction outcome will be very difficult to interpret if the

whole reaction takes place on the chip as discussed above. Another challenge are the

temperature cycles which have to be done with PCR. The cycling has to be made by

either a powerful heating (and cooling) system or a rather complicated design in the

case of a continuous flow system. Additionally, the formation of bubbles is increased at

higher temperatures.

The rolling circle amplification described in this section has several advantages over a

PCR with regard to an on-chip reaction [11]. First it is an isothermal reaction which

has its efficiency optimum between 30 and 60 C, depending on the polymerase. This

means, that no temperature cycling has to be done. Due to the rather low temperatures

needed, bubble formation at higher temperatures can be avoided.

The second big advantage over the PCR is the fact, that very large molecules can be

formed. This should enhance the readout signal. Although no temperature cycling

has to be made, the reaction usually takes around one hour. However, the resonance

frequency and damping change already at an early stage of the reaction due to the

increased signal in the optimal case, thereby reducing the time needed for analysis.

As for the PCR, the mix was prepared and run out off-chip in a PCR tube in a first stage.

The products were then analyzed on the chip and with classical gel electrophoresis in

parallel. In a second step, the amplification was done on the chip and the amplification

product was analyzed with gel electrophoresis in order to verify the chip measurement.

The samples were prepared by Dr. Damiano Cereghetti.

108


Common to all samplesBuffer and enzyme patent-protected

Specific to each sampleA1 A2 B1 B2

dsDNA (48.5 kbp) 3 3a - -lambda Cl857 Sam7cssDNA (6.4 knt) - - 3 3a

M13mp18Expected outcome pos neg pos nega) added after enzyme inactivation

Table 4.7: Composition of the RCA experiment. A2 is the negative control for A1 andB2 is the negative control for B1. The amount of the targets was 30 ng ineach case. For the plasmid this corresponds to approximately 8 · 109 copies.The total volume was 60 µl.

4.4.1. Off-chip amplification

Materials

Table 4.7 shows the composition of the four mixes, which were tested. In the first

mix A1 a long strand (48.5 kbp) of dsDNA was used as target together with random

hexamer primers, both contained in the Illustra GenomiPhi V2 DNA amplification kit.

A2 is the negative control. In the third mix B1 the target was a 6.4 knt long strand

of M13mp18 cssDNA. B2 is the negative control. The manufacturer instructions were

followed except that the DNA was not heat denatured before the amplification. In both

cases the mixes of the negative control were identical with the only difference, that the

targets were added after enzyme inactivation by heating it up to 65 C for ten minutes.

The reaction was run at 30 C for 90 minutes. M13mp18 single stranded Phage DNA

(#P-107) was from Bayou Biolabs (Harahan, LA, USA). The Illustra GenomiPhi V2

DNA amplification kit (#25-6600-30) was from GE Healthcare (Buckinghamshire, UK).

The four samples were measured on the chip one after another for three times. The

temperature was set to 23 C using the temperature box shown in figure 3.8.

Results

The measured resonance frequencies and df as well as the excitation current Iex are

plotted in figure 4.10. The excitation current is expressed in a 10 bit number which is

directly proportional to the current I0 flowing through the loops. Each point represents

109


160

175

190

205

12773

12813

12853

12893

12933

A1 A2 B1 B2

700

800

900

1000

A1 A2 B1 B2

fres

[Hz]

df

[Hz]

Iex

[a.u

.]

fres

dfIex

Figure 4.10: Measured fres and df of the off-chip amplified RCA samples (left) andapplied excitation current (right). A1 contained linear DNA and B1 circularDNA. A2 and B2 are the corresponding negative controls with deactivatedenzyme. Especially in B1 the difference between positive and negativesample can nicely be seen.

the average of the three measurements. The error bars indicate the maximum and

minimum values.

As expected, the damping is increased, whereas the resonance frequency is reduced by

the DNA chains for both positive samples when comparing to its negative counterpart.

This proves, that the RCA gives the expected results when measured on the chip. The

difference between A1 and A2 is lower than the one between B1 and B2. In B1, there

is circular DNA present. This means, that the polymerase can go around the target for

a virtually infinite number of times, once the primers are paired to the target and the

polymerase has attached to this site. With only linear DNA present, as it is the case for

A1, the polymerase has to ”find” a new primer-target pair each time it has amplified

the sequence. This explains the much higher signal in B1 compared to A1.

The variation in the measurements with B1 is very high. Microscope images of the

chips after filling it with the samples are shown in figure 4.11. The sample in a) is

the negative control A2. The liquid is transparent. The liquid in b) is the positive

reaction with linear DNA and is turbid. The same is true for the positive sample with

circular DNA shown in c). The effect is very extreme and clearly visible in c), where

a cloudy phase reaching from the inlet of the chamber can be seen. Depending on how

the cloud looks the cantilever is influenced in a different way. This explains the high

variability of the measurements with B1. Mori et al. [106] observed a white precipitate

when running the loop-mediated isothermal amplification, which was identified to be

magnesium pyrophosphate. They used this by-product for the detection of the reaction

110


a) b) c)

1mmA2 A1 B1

Figure 4.11: Microscope images taken after filling the RCA samples into the chamberof the chip. Note the turbidity in b) and c), which probably comes frompyrophosphates and is more pronounced in c). The white dots are dustparticles on top of the chip.

via turbidity measurements. It is likely that the turbidity in the RCA samples is also

caused by pyrophosphates which is released by the polymerase during the reaction.

4.4.2. On-chip amplification

In the preceding section it was shown, that the changes of the fluid mechanical properties

after running an RCA are high enough to be detected by the cantilever. The next goal is

to run the reaction on the chip while recording the resonance frequency and damping at

the same time. This procedure is carried out in the experiments shown in this section.

In order to show, that the RCA runs on the chip, samples identical to B1 described

in table 4.7 were prepared. An aliquot was transfered to the chip immediately after

mixing. The chip was then heated to 30 C. The resonance frequency and damping

were recorded during 90 minutes in the box shown in figure 3.8. The rest of the sample

was amplified in a conventional thermo cycler. As for the chip the temperature was

30 C and the reaction time was 90 minutes. The enzyme of the sample on the thermo

cycler was deactivated afterwards. The reaction products from the chip and from the

thermo cycler were stained on a gel for visualization.

In figure 4.12 a) the electrophoretically stained samples are shown. Lane 2 is the product

from the thermo cycler. Since the DNA strands in the reaction product have more or

less a random length distribution and are partly very long, the sample is smeared and a

large part of the DNA is still in the pocket. For this reason, an aliquot was digested with

the restriction enzyme HpaII (R0171S, NEB, Ipswich, MA, USA). The enzyme cuts the

111


DNA at certain positions and thereby shortens the strands. Lane 3 shows the outcome

of the digested product.

The reaction product from the chip was digested as well and stained on lane 4. Compar-

ing lane 3 and 5 it can be concluded that the reaction worked on the chip with virtually

the same efficiency.

The recorded resonance frequency, damping and excitation current are plotted in figure

4.13. The following observations can be made. First, there is a temperature effect visible.

Until approximately 2 minutes after starting the measurement, the resonance frequency

is increasing and the damping and excitation current are decreasing. This is is due to

the increase in temperature to 30 C. After a short plateau, the resonance frequency

decreases and the excitation current as well as the damping increase monotonically.

After 90 minutes, the resonance frequency has decreased by approximately 5 Hz from

the initial plateau. The increase of df is 6 Hz and the increase of the excitation current

is 0.8 mA. From the experiments shown in figure 4.10, a decrease of the resonance

frequency of approximately 40 Hz would have been expected. The increase of df is also

below the expected value from the off-chip amplification. The differences can probably

be explained by the pyrophosphate visible in figure 4.11, which were observed when the

reaction was run on a thermo cycler but not when the amplification was made on the

chip. Another reason for the different behavior may come from the fact, that the off-chip

sample has to be transferred after the reaction took place. This certainly changes the

conformation of the DNA. Keeping in mind, that the molecules tend to be very long, this

may have a high influence on the behavior of the cantilever. Additionally, the cantilever

is mainly influenced by the environment which is only a few times the boundary layer

thickness away from it. The movement of the cantilever during the actual reaction may

change the reaction kinetics and the DNA conformation, which is built up near the

cantilever during the reaction. This may have an effect on the dynamic behavior as

well.

After approximately 35-40 minutes there is a kink in all the curves. The most obvious

reason for this is that the reaction ends at this point in time. In order to verify this

hypothesis, the reaction was run on the chip and stopped after 35 minutes. In parallel,

the identical assay was run on the thermo cycler, where samples in different tubes were

taken out of the thermo cycler after 20, 40, 60 and 90 minutes. The product was stained

without digestion. The result is shown in figure 4.12 (b).

The products from the thermo cycler are shown on lane 1 through 4. Even if this not

a quantitative method, it can qualitatively be seen, that the reaction indeed reaches its

112


1 2 3

Thermocycler Chip

4

10

50

150

20

30

100

40

(a) Comparison of chip and thermo cycler am-plification after 90 minutes. Lane 1: Lad-der (O’RangeRuler, ThermoScientific), lane 2:undigested RCA, lane 3 and lane 4: digestedamplification products.

20' 40' 60' 90' 35'

Thermocycler Chip

1 2 3 4 5

(b) Comparison of off-chip amplified RCA prod-ucts (lane 1 to 4) after 20, 40, 60, and 90 min-utes and on-chip amplified RCA product after35 minutes (lane 5).

Figure 4.12: RCA products electrophoretically stained on Invitrogen 10% PAGE, 1×TBE buffer run on the chip and on the thermo cycler (gels and imagesprepared by Dr. Damiano Cereghetti).

113


180

186

190

194

198

12859

12863

12867

12871

0 20 40 60 80

Time [min]

fres[H

z]

fres

df

df

[Hz]

6.6

6.8

7

7.2

7.4

7.6

0 10 20 30 40 50 60 70 80 90

I0[m

A]

Time [min]

Figure 4.13: RCA carried out on the chip for 90 minutes at 30 C. The resonance fre-quency and df are plotted on the left and the applied excitation current isplotted on the right. Note the kinks in all three curves, which indicate theend of the reaction between 30 and 40 minutes.

end before the 90 minutes time frame. Lane 5 shows the product from the chip. The

band has approximately the same intensity as lane 2. This verifies that the reaction on

the chip is close to its end as well. Therefore, the kink in the curve comes from the end

of the reaction.

At this point, two questions arise. First, where does the almost constant slope that

occurs after 40 minutes come from and does it also occur for other liquids? Second, is it

possible to (significantly) distinguish between a sample where the reaction takes place

and one where the reaction does not take place? In order to answer these questions, the

following experiment was carried out. Water, RCA buffer, a negative control (no DNA

target present) and the complete RCA mix (positive) were measured with the chip. The

measurement was run for 35 to 55 minutes at 30 C. According to the results shown

above, the reaction should almost have reached its end after this time.

The time evolution of the resonance frequency, df and the excitation current is plotted

in figure 4.14 for all four liquids. For the non-amplifying samples, the slopes of df and

excitation current are almost zero after the temperature equilibrium is reached. In the

case of the positive RCA sample, a clear increase in df and excitation current can be

seen. The decrease in resonance frequency is approximately the same as for the buffer

solution (not visible on the plot). After approximately 40 minutes the curves reach a

plateau, indicating, that the reaction has finished.

114


Slope between 27and 30 minutes

0 10 20 30 40 50 6012.7

12.8

12.9

0 10 20 30 40 50 60160

180

200

0 10 20 30 40 50 604.64

7.74

10.84

Time [s]

Water

Buffer

Negative RCA

Positive RCA

fres[k

Hz]

df

[Hz]

Iex[m

A]

Figure 4.14: fres, df and excitation current of water, buffer, a sample where the enzymewas inactivated (negative RCA) and a sample where polymerization takesplace. The positive RCA shows an increase in damping and excitationcurrent after approximately 15 minutes.

115


-4

-3

-2

-1

0

Water Buffer Negative RCA

0

0.16

0.31

Slo

pe

[mH

z/s]

fres

Slo

pe

[mH

z/s]

df

-0.5

0.5

1.5

2.5



Slo

pe

[A

/s]

I¹

ex

Figure 4.15: Average slope and standard deviation of the curves shown in figure 4.14.The distinction between a non reacting sample (water, buffer, negativecontrol) and a reacting sample (RCA) is most significant in the excitationcurrent.

The statistical evaluation of the experiment is shown in figure 4.15. The mean values

of the slopes of fres, df and Iex between 27 and 30 minutes, as shown in figure 4.14, are

plotted. The error bars indicate the standard deviation. As expected from figure 4.14

the values for the resonance frequency are almost the same for all four samples, with a

comparably high standard deviation. Although the mean value for the RCA is slightly

lower than for the other samples, it is impossible to draw any meaningful conclusions

about the course of the reaction from the resonance frequency. The situation for df and

the excitation current looks different. In both cases, the slope of the RCA samples is

higher compared to the negative control. Especially in the case of the excitation current,

the distinction between water, RCA buffer, negative control and the positive reaction is

very clear.

116


The experiments presented in this section show that the amplification of DNA with

the RCA can be detected with the cantilever. The primers used in these experiments

are non-specific. This means, that basically every sequence of DNA in the mix can be

amplified. However, there are ways to make this reaction specific and therefore useful

to detect a predefined sequence of DNA. On the other side, the system has real-time

capabilities, since the measured quantities are recorded during the whole course of the

reaction. It also benefits from the fact that the RCA is isothermal, running at 30 C

and the time of an experiment can be reduced to 20-30 minutes.

117

5. Conclusion and outlook

Modeling

The model of the U-shaped cantilever based on the concept of mechanical impedance

was implemented. Clamping was modeled via linear and torsional springs. The influence

of the liquid was modeled using a two-dimensional approach, where correction factors

for a beam with a rectangular cross section near a wall were derived using finite element

analysis. The resonance frequency in the liquid is predicted with an accuracy of 4%,

whereas df is predicted with an error of up to 15% for a 10% glycerol solution. The

main source of error seems to come from the fact, that the corners are modeled neither

in the structural part nor in the FSI. The model could further be refined taking these

effects into account. The best approach would be to derive impedances for the corners

and a correction factor for the FSI, which would be a trade off between the completeness

of a fully coupled, three dimensional approach and the present model.

A procedure for the calculation of the viscosity and density was presented. Based on

the measurement of fres and df both quantities can be calculated without the need of

another, independent measurement. The method is based on the assumption, that fres

and df change linearly with the density and the viscosity. It could be refined by using

a higher order polynomial instead of the linear approach.

Viscosity and density sensor

A cantilever-based system requiring a sample volume of less than 12 µl was fabricated

and experimentally characterized. A trade-off between sensitivity, signal level and range

had to be found. The system was optimized for measurements of fluids with properties

close to the ones of water. Due to the inductive readout, the required instrumentation

could be kept relatively simple compared to optical methods. The miniaturization of the

geometry is limited by the readout method and the fact, that viscous damping drastically

increases as the sensor’s size is reduced. For this reason, the sensor is relatively large,

compared to conventional MEMS devices. An improvement could be achieved using an

optical readout, which would on the other side complicate the setup.

119

5. Conclusion and outlook

The sensor was tested with solutions of glycerol and ethylene glycol. The measured

densities and viscosities were within 0.4% and 5.5% of the tabulated values.

A problem in microfluidics is chamber filling and the building up of bubbles. The issue

of filling has been solved by designing the chamber with an overhanging lid, making the

requirement of a hydrophilic surface unnecessary.

Towards a new diagnostic method

Solutions of short strands of DNA were successfully characterized with the sensor. The

intrinsic viscosity determined by the cantilever sensor is in good agreement with litera-

ture values. Experiments with concentrated solutions of long strands of DNA could be

used to establish a limit of detection. As these exhibit complex behavior, the cantilever’s

response could not directly be used for the rheological characterization of these solu-

tions. The rheology of DNA solutions at high frequencies has not been a research topic

up to now due to the lack of an appropriate system. Although it is not straightforward

to determine the complex viscosity directly from the resonance frequency and damping,

the system might nevertheless give some valuable insights for rheologists and molecular

biologists.

The main goal of the project was the establishment of a new analytical method for DNA

analysis. Efforts towards the combination of the PCR with the cantilever sensors showed

some promising results, however with bad reproducibility. The information that can be

gained from the cantilever is only available at the end of the reaction. The sensitivity is

too small for a real time detection of the PCR. If a yes or no answer is sought, e.g. in a

device that should detect the presence of a pathogen, this might be enough information.

However, the trend in research goes towards multiplexed and real time reactions. This

has the advantage, that multiple pathogens can be detected in one reaction, which is

especially favorable in a productive environment such as in large analysis laboratories.

Efforts to combine the cantilever sensor with the rolling circle amplification gave much

better results in contrast. Samples of reaction products amplified on a thermo cycler

as well as an amplification on the chip itself produced a very clear difference between a

”positive” and a ”negative” sample. The RCA has the advantage of being isothermal.

The measurement of the resonance frequency and the damping was done while the

reaction itself was running. The system has therefore real-time capabilities.

Future efforts in the direction of a complete lab-on-a-chip have to go into the direction

of completeness of the system. Although the reaction was shown to be running on

the chip itself, important steps, such as sample purification or pre-concentration, are

120

still open. Another way of improving the whole system has to go into the direction of

parallelization. The evaluation of multiple samples and especially negative controls are

an important feature of many diagnostic systems.

121

A. Fabrication

A.1. Assembly and bonding

# Part Process Parameters

1 Lid, Resonator Oxide plasma bonding 50 W, 0.4 mbar, 30 s2 PCB Glue with X603 Heater Glue with Vitralit 6108T Cure: 120 C, 30 min4 Wirebond5 Protection Protect front side bonds with epoxy

Table A.1: Assembly of the components. X60 was from HBM, Germany; Vitralit wasfrom Panacol, Germany. The epoxy is Loctite Double Bubble (Henkel & CieAG, Pratteln). The PCB was ordered from Eurocircuits Sarl, Switzerland.

A.2. PDMS lid

# Process Parameters Set point

1 Mold Photolitho as in A.4 step 3.1ICP etching MR C1 TE 182 cyclesClean in acetone, IPA and waterPlasma asher 600 W, 2 min

2 PDMS Mixing Ratio 10:1Degas in vacuum approx. 10 minPut mold in atmosphere of chlor-trimethylsilane

30 min

Assemble mold and holder, pour mixCure in oven 4 hours, 70 CSeparate lids and punch holes

Table A.2: Fabrication steps for the PDMS lid.

123

A. Fabrication

A.3. Heater chip


1 Insulation PECVD deposition of silicon oxide 6 min, 300 C2 Photolitho I Photolitho as in A.4 step 2.13 Evaporation Deposition of titanium 20 nm

Deposition of platinum 180 nm4 Lift-off Lift-off in NMP5 Protection Spin-coating of a photo resist layer as in

A.4 step 3.1, without exposure and devel-opment

6 Dicing Dicing of the wafer into chips7 Cleaning Bath in acetone, IPA and water8 Protection Application of a droplet of polyimide on

the heater and temperature sensorCure in oven temperature: [ C]/ 120/10/20ramp[ C/min]/time [min] 200/5/30

350/5/60

Table A.3: Fabrication steps heater chip

124

A.4. Resonator chip

A.4. Resonator chip


1 PECVD Oxida-tion

Temperature 300 C

Time 6 minutes2.1 Photolitho I AZ nLOF2070 3.5 ml

Spin coating 2000/300/3,[speed /ramp rate /time] 2500/500/45Pre-bake 110 C, 90 sExposure 152 mJ/cm2

Post bake 110 C, 90 sDevelopment AZ 826 MIF 2 min totalQDR/RD clean and dry

2.2 Evaporation Titanium 12 nmGold 200 nmno substrate rotation for both layers

2.3 Lift-off Bath in NMP Use 2 baths3.1 Photolitho II HMDS N2, HMDS, N2 300 s, 30 s, 300 s

AZ 4562 3.4 mlSpin coating 700/500/5[speed /ramp rate /time] 1700/ 1000/ 35Bake 10 minutes at 100 CExposure 700 mJ/cm2

Development AZ351 1:4 diluted 2 minutes, agitatedClean QDR, rinser dryer

4.1 RIE I Recipe Oxide1 3 min 30 s4.2 ICP I MR C1 TE 80 cycles4.3 Cleaning Acetone/IPA/Water each 5 minutes, no US5.1 Photolitho III Backside litho as in step 2.15.2 ICP II MR C1 TE 400 cycles5.3 Clean water bath 65 C

Cascade of Remover 1165 and water 65 CDry 97 C

6 RIE II Oxide1 8 min7 Nitride SiNx on PECVD 10 min

Table A.4: Fabrication steps resonator chip.

125

B. Correction factor for gated

measurement

Romoscanu [107] pointed out in his PhD thesis, that there is a systematic error in df

when measuring with the gated phase locked loop due to the free oscillation of the

resonator. He gives an equation for the correction of this effect:

dfQS = dfS − nπdf 2

S

4∆αfres(B.1)

where dfQS is the quasi stationary df which is measured with the gated PLL, dfS is the

stationary (real) df and n is the number of cycles in which the sense gate is active and

the resonator is freely decaying. This correction has been applied to the measurements

of df shown in chapter 3.

In order to verify that this correction is needed, the measured df for different solutions

of ethylene glycol (6% and 10%), glycerol (5%) and DI water are plotted in figure B.1.

A linear trend is observed as predicted by Romoscanu’s formula. The measured slope

and the predicted slope given by

πdf 2S

4∆αfres(B.2)

from equation B.1 are tabulated in table B.1. The stationary damping dfS was calcu-

lated by extrapolation to n = 0. Good agreement between measurement and theory is

observed.

127

B. Correction factor for gated measurement

dfS [Hz] Measured slope Calculated slope

Ethylene glycol 10% 226.2 -8.0 -7.9Water 207.7 -6.8 -6.7Glycerol 5% 217.1 -7.4 -7.5Ethylene glycol 6% 219.6 -7.6 -7.7

Table B.1: Sationary damping dfS along with the slope of dfQS from the measurementsshown in figure B.1. The calculated slopes according to equation B.2 aregiven in the last column.

160

170

180

190

200

210

2 3 4 5 6 7

Ethylene Glycol 10%

Glycerol 5%

Ethylene Glycol 6%

Water

Cycles of free decayn

dfQS

[HZ]

Figure B.1: Measured dfQS as a function cycles n in which the resonator is freely vibrat-ing. The solid black lines are a linear fit.

128

C. Error analysis

General procedure

The general procedures for the calculation of measurement errors are described in the

ISO guide 98-3:2008, known as GUM [108] (guide to the expression of uncertainty in

measurement). The measurement uncertainties (including error bars) shown in this

thesis have been derived with this procedure.

The combined standard uncertainty uc is given by

u2c =N∑i=1

(∂f

∂xi

)2

u2(xi) (C.1)

where f is a function that relates N measured quantities xi with uncertainties u(xi)

to the measurand y. Two examples are given in the following in order to clarify the

application of equation C.1.

Uncertainty of the sensitivities

The measured sensitivities are calculated from differences in viscosity and density of the

solutions and differences of measured fres and df . In the case of the sensitivity of the

resonance frequency with respect to density changes, the according formula is

Sfres,ρ = f(∆fres,∆ρ) =∆fres∆ρ

(C.2)

The uncertainty according to C.1 is then given as

u(Sfres,ρ)2 =

(1

∆ρu(∆fres)

)2

+

(∆fres∆ρ2

u(ρ)

)2

(C.3)

129

Error in viscosity and density

The calculation of the viscosity and density involves the inverse K of the sensitivity

matrix S. The uncertainty u2(Km,n) of each component depends on the uncertainty of

the sensitivities. It is calculated using C.1

u2(Km,n) =2∑

i,j=2

(∂Km,n

∂Si,ju(Si,j)

)2

(C.4)

where the uncertainties for S are calculated as shown in C.3. The error of the measure-

ment can then be expressed as

u(~µ)2 = (K K) ·(u(~f) u(~f)

)+ (u(K) u(K)) ·

(~f ~f

)(C.5)

where indicates element wise multiplication of the vectors and matrices defined in

equation 2.51.

130

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141

Publications and conference talks

Philipp Rust, Damiano Cereghetti and Jurg Dual, ”A Viscometric Chip for DNA Anal-

ysis”, Procedia Engineering vol. 47, pp. 136-139, 2012

Philipp Rust, Ivo Leibacher and Jurg Dual, ”Temperature Controlled Viscosity and

Density Measurements on a Microchip with High Resolution and Low Cost”, Procedia

Engineering, vol. 25, pp. 587-590, 2011

Philipp Rust and Jurg Dual, ”Novel method for gated inductive readout for highly

sensitive and low cost viscosity and density sensors”, Solid-State Sensors, Actuators and

Microsystems Conference (TRANSDUCERS), pp. 1088-1091, 2011

Philipp Rust and Jurg Dual, ”A Micromachined Device for the High Resolution Mea-

surement of Fluid Properties”, SGR Meeting, 2010

Philipp Rust, Damiano Cereghetti and Jurg Dual, ”A Micro-Liter Viscosity and Density

Sensor for the Rheological Characterization of DNA solutions in the kilo-Hertz range”,

submitted

Philipp Rust, Damiano Cereghetti and Jurg dual, ”Viscometric real-time monitoring of

the rolling circle amplification with a micromachined cantilever”, in preparation

143

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