Investigating nano-scale viscous and elastic forces with ...1209086/FULLTEXT02.pdf · Investigating visco-elastic forces at the nanometer-scale is important to the characterization

Investigating nano-scale viscous and elastic forces withintermodulation

Studies in multifrequency atomic force microscopy

PER-ANDERS THORÉN

Doctoral ThesisStockholm, Sweden 2018

TRITA-SCI-FOU 2018:17ISBN 978-91-7729-802-1

KTH School of Engineering SciencesSE-100 44 Stockholm

SWEDEN

Akademisk avhandling som med tillstånd av Kungliga Tekniska högskolan framläggestill offentlig granskning för avläggande av teknologie doktorsexamen i fysik 15 juni2018 i FB42, Albanova Universitetscentrum, Kungliga Tekniska högskolan, Roslagstulls-backen 21, Stockholm.

Opponent: Dr. Greg HaugstadHuvudhandledare: Professor David B. Haviland

Cover picture: Phase and topography of polystyerene (green) and low-density polyethylen(blue) imaged in air. The dynamic force quadrature curves for the LDPE, reveal hys-teresis: the signature of a soft viscoelastic surface.

© Per-Anders Thorén, 2018

Tryck: Universitetsservice US AB

iii

Abstract

Investigating visco-elastic forces at the nanometer-scale is important to thecharacterization of soft materials. A quantitative force measurement can be ob-tained using an atomic force microscope (AFM) with a calibrated force transducer(the AFM cantilever). In this thesis, we discuss and evaluate simple methods ofcalibration and we use these calibrations to measure dynamic force quadraturecurves for both normal and in-plane tip-surface forces using Intermodulation AFM(ImAFM).

ImAFM utilizes the nonlinearity of the tip-surface force by measuring the mix-ing between two or more drive frequencies placed close to a resonance of the AFMcantilever. The intermodulation response at many mixing frequencies provides ad-ditional observables, useful for characterization of materials. We use ImAFM nearthe first flexural resonance to measure visco-elastic materials and we show that sur-face motion plays an important role in the analysis of soft samples. To explain ourmeasurements we derive a simple model where the surface position is described byan exponential relaxation when perturbed from its equilibrium. Through numericalsimulations of this model we explain experiments for many different soft sampleswith varying properties. We further apply the intermodulation technique to softsamples in liquid.

ImAFM at the first torsional resonance frequency induces motion of the tip in-plane with the surface, enabling friction measurements between the tip and sample.Due to the high torsional resonance frequency, the tip velocity can reach severalcm/s, many orders of magnitude higher than typical AFM friction measurements.By measuring the amplitude dependence of the dynamic force quadrature curves,we can resolve the transition between the tip sticking to the surface, through stick-slip to free sliding motion.

Keywords: Atomic Force Microscopy, Nonlinear dynamics, Soft materials, High-speed friction measurements, Modeling and numerical simulations

iv

Sammanfattning

Att undersöka viskoelastiska krafter på nanometernivån är viktigt för att ka-rakterisera mjuka material. Kvantitativa kraftmätningar kan erhållas med hjälp avett atomkraftmikroskop (från engelskans atomic force microscope, eller AFM) ochen kalibrerad kraftsensor (den så kallade cantilevern). I denna avhandling disku-terar och utvärderar vi olika metoder för kalibrering och vi visar hur kalibreringenkan användas för att mäta dynamiska kraftkvadraturer för krafter riktade vinkelrättsamt parallellt mot ytan med hjälp av IntermodulationsAFM (ImAFM).

ImAFM utnyttjar att ytkraften är icke-linjär, vilket gör att två eller fler drivfre-kvenser placerade nära en av cantileverns resonanser intermodulerar när cantile-vern interagerar med ytan. Intermodulationen ger tillgång till fler observabler, vilkaanvänds för att karakterisera olika material. Vi använder denna teknik i närhetenav den första flexuralfrekvensen hos cantilevern för att mäta viskoelastiska materi-al och vi visar att ytans rörelse spelar en viktig roll i analysen av mjuka material.För att förklara våra mätningar beskriver vi en enkel modell, där ytans rörelse be-skrivs av en exponentiell avslappning till sitt jämviktsläge. Med hjälp av numeriskasimuleringar av denna modell förklarar vi mätningar på mjuka material med olikaegenskaper. Vi demonstrerar även hur ImAFM och denna enkla modell kan förklaramätningar i vätska.

ImAFM kring den första torsionella resonansfrekvensen skapar en rörelse hoscantileverspetsen i ytans plan, vilket möjliggör mätningar av friktionskrafter mel-lan spetsen och ytan. Tack vare den höga frekvensen hos den torsionella resonan-sen kommer spetsen röra sig med en hastighet motsvarande flera cm/s över ytan.Genom att mäta amplitudberoendet hos de dynamiska kraftkvadraturerna kan vistudera hur spetsens rörelsemönster ändras från att sitta fast vid ytan till hur densnabbt färdas fram och tillbaka.

Contents

Contents v

1 Introduction 11.1 Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Outline of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Atomic force microscopy 52.1 Quasi-static AFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Dynamic AFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Calibration 113.1 Flexural calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Static loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13The Sader method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Torsional calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Green’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Using flexural k to calculate torsional κt . . . . . . . . . . . . . . . . . . . . 18

3.3 Squeeze-film damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Intermodulation AFM 234.1 Frequency mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2 Dynamic force quadrature curves . . . . . . . . . . . . . . . . . . . . . . . . 254.3 Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5 The moving surface model 315.1 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.2 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.4 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.5 Additional models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

DMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

v

vi CONTENTS

Attractive PWL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Smooth PWL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Deformation-dependent PWL . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Further extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6 Intermodulation in liquid 456.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.3 Scanning in liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486.4 Hydrogels in liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

7 Intermodulation friction force microscopy 537.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567.3 ImFFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567.4 ImFFM on stiff materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

The modified Prantl-Tomlinson model . . . . . . . . . . . . . . . . . . . . . 58Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.5 ImFFM on soft materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657.6 Models which do not explain the data . . . . . . . . . . . . . . . . . . . . . 68

The modified PT model version 2 . . . . . . . . . . . . . . . . . . . . . . . . 68The side-wall model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69The MSM model for shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

8 Conclusions and outlook 75

Acknowledgements 77

Bibliography 79

A Code listings 87

B Appended papers 89

Chapter 1

Introduction

For my ally is the Force, and a powerful ally it is. Life creates it, makes it grow. Its energysurrounds us and binds us. Luminous beings are we, not this crude matter. You must feelthe Force around you; here, between you, me, the tree, the rock, everywhere, yes. Evenbetween the land and the ship. – Yoda

1.1 Force

FORCES are around us – everywhere. They determine the slow eternal motion ofcelestial bodies, the fast oscillations of atoms and the spinning of their electrons.Man has always tried to use forces for means of survival and to enhance daily

life. Some ten thousand years ago, our ancestors from the Upper Paleolithic era learnedto use the force stored in suspended plant fibers or strings of rawhide to create simplebows. The bows were used to hunt animals which fed the tribe, and to protect thetribe from hostile neighbors – both of great importance for the survival of the group.Our ancestors later harvested the power of fire by using the friction force between twopieces of wood or by hitting stones together, accelerating the evolution of mankind.

Humanity kept on evolving over thousands of years, but survival hindered any re-search or understanding of force. Not until Ancient Greece (around 12th - 9th cen-turies B.C.) had life simplified enough to allow for the rich aristocracy to spend timeon leisure activities like philosophy and natural sciences, instead of gathering food andwood. They thought of the world in terms of two fundamental forces: Love (whichbrought things together) and Hate (which caused them apart). Aristotle (384 - 322B.C.) started to observe nature and he tried to explain what he saw. He noted thatan object being pushed would remain in motion until the pushing force stopped, forexample a man pushing a cart: when his pushing stops, the cart stops [1]. He couldexplain this by his simple theory of Love and Hate, but it could not explain the trajec-tories of projectiles. Archimedes (287 - 212 B.C.) discovered his famous principle – anobject experiences a force equal to the weight of the displaced medium. This principle

1

2 CHAPTER 1. INTRODUCTION

turns out to have important consequences for material research, such as van der Waalsforces [2].

Ibn al-Haytham (also known as Alhazen, 965 - 1039 A.D.) first proposed the use-fulness of verifying theory by experiment. Galileo (1564 - 1642) also noticed the im-portance of the scientific method, which he used to e.g. test his theory of gravity bysimultaneously drop two spheres of different weight from the Tower of Pisa [3]. Yearslater, Isaac Newton (1642 - 1727) derived his law of gravity and three force laws, stillused today. Not only physics and chemistry experienced rapid evolution during the17th and 18th centuries, but also enormous advances took place in pure and appliedmathematics by Euler, LaGrange and Laplace: important discoveries which greatly as-sisted the development of theoretical models for physical phenomenon.

1.2 Friction

Understanding and controlling friction has been of great importance for thousands ofyears, resulting in many inventions to use or reduce friction. Skis and later the sledallowed for more effective hunting in the winter and opening for trade with neighboringtribes. Soon people started to put the newly developed wheel and well-greased axlesonto their sleds, leading to more effective transportation with wagons. What wouldthe modern world look like without the wheel?

Mastering friction also gave the young human race the ability to tame the fire:which has been used ever since to cook food (helping to kill bacteria and parasites)and provide heat and warmth to the tribe. This enormously increased survival chancesof the tribe and helped mankind to migrate deeper into colder regions.

Figure 1.1: Egyptian engineering. Relief found in the tomb of Djehutihotep II. A largestatue is pulled on a sled. At the front of the sled stands a man pouring water betweenthe skids and ground.

1.2. FRICTION 3

Building the pyramids was one of the greatest feats in human history. Stones weigh-ing around 500 metric tons were hauled to the top of the pyramids on very primitivesleds. The drawing in fig. 1.1 show the transportation of a colossus statue depictingDjehutihotep II (1900 BC). In the drawing there is a man standing at the front of thesled pouring water, which is a indication that the Egyptians understood the importanceof friction. It has been estimated that the number of slaves pulling the sled (approxi-mately 170) would not be enough considering the mass of the statue (roughly 600 tons)if the sled was sliding on a dry sand-wood interface [4, 5]. Their pulling (∼ 800N/man)would have to exceed the friction force Ffric. = Nµ, where µ is the friction coefficientand N the normal force. Using numbers from fig. 1.1 would give a friction coefficientof about µ = 170men·800N/man

600tons ≈ 0.23, which is close to the numbers reported for wetsand in Fall [5].

It was not until the 16th century, when Lenardo da Vinci started to perform his manyexperiments, that friction got its first scientific analysis. Da Vinci made two importantobservations while sliding blocks of different shapes over surfaces: 1) friction1 is inde-pendent of contact area and 2) friction is proportional to the weight of the block. Thesetwo observations were later rediscovered by Guillaume Amontons in the 17th century,together with a third observation: 3) sliding friction is independent of sliding velocity.These three laws are known as Amontons laws.

Friction at the nanometer-scale is of great importance for modern technology, suchas MEMS and NEMS (micro/nano-electromechanical systems): mechanical gadgetswhich are integrated in everything from our smart-phones to advanced medical sys-tems (such as micro-swimmers inserted into the blood stream). The most popularmodel to explain friction at small scales is the so called Prandtl-Tomlinson-model [6],but a very similar model was already proposed by Charles-Augustine de Coulomb inthe 18th century. Coulomb used his model to explain macroscopic friction as two cor-rugated brush-like surfaces interlocking into each other, where a force is required tobreak the locking. The corrugation model explains why some interfaces have less fric-tion than others, since the match/mismatch of their periodicities allow for differentlocking strengths.

Apart from the few scientists discussed thus far, friction has mostly been an engi-neering rather than a scientific discipline, with questions like: "How can we make thiswheel spin faster?" or "Is it possible to make this door open more easily?". One of thebiggest break-throughs, which resolved many engineering problems was the discoveryof petroleum products and how to use them as lubricants. This discovery greatly helpedthe industrial revolution by reducing the internal friction in different types of machin-ery such as steam engines and locomotives. Oil as a lubricant should be considereda technological marvel. However even better ways of reducing friction are requiredat the nanometer-scale, as energy loss due to friction increases with deceasing size,and many lubricants cannot be used due to their high viscosity. Understanding frictionat the nanometer-scale is important if we wish to keep pushing the development of

1Note that da Vinci did not talk about a "friction force" but rather just "friction" as in higher or lowerfriction. It was not until Newton finished his work on forces that scientists started to discuss friction forces.

4 CHAPTER 1. INTRODUCTION

MEMS and NEMS and scientific measurement devices. Reducing friction is also impor-tant from an environmental and financial point of view, where it has been estimatedthat about 23 % of the worlds energy consumption is lost due to friction and wear [7],and ca 30 % of the fuel consumption in a car is used to overcome internal friction [8].

1.3 Outline of this Thesis

This thesis is based on the five appended papers, labeled I-V. It begins with an intro-duction to the basic concepts used throughout the papers, such as the principles ofan atomic force microscope (chap. 2) and Intermodulation AFM (chap. 4). Chapter 3gives a brief recap of cantilever calibration methods, and summarizes the main con-cepts of Paper I. An alternative approach to the moving surface model (Papers II andIII) together with a deep discussion about further extensions of this model is givenin chapter 5. Chapter 6 provides a background for measurements in a liquid envi-ronment described in Paper IV. The introductory section of chapter 7 introduces thePrandtl-Tomlinson model, which is further extended into the model used in Paper V. Toconclude the friction chapter, a few different models are proposed which might explainshear measurements on soft materials. Although all of these models were investigatedby simulations, none of them successfully explain experiments on different polymersurfaces. The formulation of a model for in-plane tip-surface forces on soft materials isleft as an open problem for future research.

Chapter 2

Atomic force microscopy

THE BASIC CONCEPT of the atomic force microscope (AFM) [9] is depicted in fig. 2.1.The AFM followed the invention of the scanning tunneling microscope (STM) byGerd Binnig and Heinrich Rohrer in the early 1980s at IBM Research in Zurich,

the first realization of a scanning probe microscope (SPM) with a system architecturethat remains today. Todays AFM’s are sophisticated multi-purpose instruments withapplications in many fields of science, e.g. surface imaging with a resolution ≈ 1000times better than the diffraction limit of optical microscopes, material characteriza-tion at the nanometer-scale [9, 10, 11], lithography [12, 13, 14], mechanics of cancercells [15, 16] and even single atom manipulation [17], to name a few. The key elementof the AFM is the cantilever beam.

The cantilever is the force transducer, having typical dimensions: 120 µm × 30µm × 2 µm (Tap300 tapping mode cantilever from Bruker AFM probes). The beam isclamped at one end and a sharp tip is attached to its other free end. The cantileverposition is controlled with piezo-electric elements, known as the scanner. There aremany different scanner designs, but its main task is to control the probes x y-position(e.g. while scanning) and height (e.g. tracking the surface). The AFM can also be builtas a sample-scanning instrument, where the x yz-position is changed by moving thesample instead of the probe, i.e. the scanner is placed under the sample.

There are two main modes of operation: quasi-static and dynamic. Both of thesemodes will be examined in detail in the following sections 2.1 and 2.2. In the dynamicmode the cantilever is inertially excited at some frequency by oscillating (accelerat-ing) the probe base. In principle, the scanner-piezo can shake the cantilever (not forsample-scanning AFM), but it is too slow due to its high mass. A large scanner-piezois necessary to enable a greater z scan range, allowing for larger topographical fea-tures while scanning the surface. The standard scanner piezo used in the Nanowizard3 from JPK instruments has a z-range of 15 µm. Therefore, to increase the oscillationfrequency, a tiny shaker piezo is attached between the cantilever base and the scanner.

With either quasi-static or dynamic techniques, the goal of any AFM measurementis to monitor the deflection of the cantilever. The deflection can range from a few Å for

5

6 CHAPTER 2. ATOMIC FORCE MICROSCOPY

Laser

Scanner

Shaker piezo

Base Cantilever

Tip

Sample

Mirror

Photo detector

Figure 2.1: Schematic of a standard AFM setup. Not to scale. The AFM cantilever isextended from the cantilever base. At the end of the cantilever is a tip, which is used tofeel the sample surface. The deflection of the cantilever is measured with a laser, thatreflects off its free end. After reflection an adjustable mirror guides the laser beam to aphoto-detector, used for read-out. The cantilever is excited by a tiny shaker piezo, andits position is controlled by the scanner piezo.

measurements in vacuum [18], up to several 100 nm in air. Measuring such smalldeflections can be a difficult thing, but thanks to a cleverly designed optical lever, thedeflection can readily be recorded with sub-Å resolution [19]. The ultimate lower limitis determined by detector noise, which is the order of 50 fm/

pHz for a good detector1.

A detailed discussion of the optical lever is given in chapter 3.

2.1 Quasi-static AFM

The simplest and most intuitive way of operating an AFM is in a quasi-static mode,often called contact mode [9]: the cantilever tip is always2 in contact with the sampleand the scanner slowly moves the tip across the surface. While scanning, the tip-surfaceforce induces a bending of the cantilever. With a feedback circuit controlling the scan-ner, the bending is kept at a setpoint value chosen by the user. If the scanning is slow

1Calibration parameters for a Tap300 is given in table 6.1.2Assuming perfect feedback, which of course is not the case in real experiments. Non-perfect feedback

can generate artifacts such as para-shooting.

2.2. DYNAMIC AFM 7

enough, the tip-surface and cantilever bending forces are in equilibrium and the set-point controls the surface load – a crucial parameter for all AFM experiments.

If a force F is applied to the free end of the cantilever beam (i.e. to the tip), thecantilever will deflect some distance d from its unloaded position. Hooke’s law gives

F = kd (2.1)

where k is the spring constant of the cantilever. Thus, if we know k and measure d, wecan find F . There are several techniques for calibrating k. A discussion of the most pop-ular methods are given in Chapter 3. Calculating the force from eq. (2.1), we can studyhow the force changes when the probe is moved towards or away from the surface. Ex-tracting the distance dependence of the force and fitting different models, propertiesof the interaction can be found. Common models are the Derjaguin-Muller-Toporov(DMT) and Johnson-Kendall-Roberts (JKR) models, both based on Hertz contact me-chanics [20].

Quasi-static techniques are easy to use but there are unfortunately major drawbackswith contact mode AFM, of which the most serious is the enormous abrasive wear onthe tip and sample while scanning, especially for stiff and rough samples. To reducethe wear of the tip, the tip itself can be made of diamond or have a diamond-likecoating. These special type of tips are much more expensive than standard cantilevers(∼ 1700 Euro vs ∼ 30 Euro3), but are designed to have a long life-time. Unfortunately,using an "unbreakable" tip increases the wear on the surface.

2.2 Dynamic AFM

Quasi-static AFM assumes continuous force balance between the tip and sample. Whatif this assumption is violated and the dynamics of the cantilever becomes important?In the paper by Binnig et al. [9], the method of exciting the cantilever at its resonancefrequency was proposed, and later adopted by the AFM community as one of the twomain modes of operation. This mode is commonly called Tapping modeTMor AC mode,and many different dynamic methods have emerged from it [10, 21, 22, 23]. Anydynamic technique is, by nature, violating the quasi-static assumption and analysis oftip-surface force using Hooke’s law is no longer valid.

By considering tiny oscillations of the cantilever tip around its rest position, we canmodel the cantilever as a mass-loaded spring which experiences an external force Fext.Its equation of motion (EOM) is

md + kd = Fext. (2.2)

The external force consists of the drive force Fd(t), the tip-surface interaction force Ftsand a damping force Fdamp = −mγd. Equation (2.2) becomes

md +mγd + kd = Fd(t) + Fts. (2.3)

3Bruker AFM probes 24/1 2018

8 CHAPTER 2. ATOMIC FORCE MICROSCOPY

Introducing the resonance frequency ω0 =p

k/m and quality factor Q = ω0/γ theEOM becomes

1ω2

0

d +1ω0Q

d + d =1k(Fd(t) + Fts) . (2.4)

Fourier transforming each term in eq. (2.4) gives

−ω2

ω20

d +iωω0Q

d + d =1k

�

Fd(t) + Fts

�

. (2.5)

where d = d(ω) =∫∞−∞ d(t)e−iωtdt is the Fourier transform of d = d(t). Defining the

transfer function G

G(ω) =

�

1−ω2

ω20

+iωω0Q

�−1

, (2.6)

the equivalence of Hooke’s law for a dynamic system is

F(ω) = kG−1(ω)d(ω). (2.7)

Note that the quasi-static version of Hooke’s law [eq. (2.1)] is obtained when ω = 0,i.e. limω→0 G(ω) = 1.

An important property of the transfer function in eq. (2.6) is its frequency depen-dence. Plots of |G| (transfer gain) and arg(G) (phase) are shown in fig. 2.2. The mag-nitude of the transfer gain |G| can be very large near a high Q resonance, which is whydynamic techniques are more sensitive than quasi-static methods for which |G|= 1. Thesharp jump in phase at resonance depicted in fig. 2.2 (b) gives an excellent operationpoint for any feedback circuit. A tiny shift in resonance frequency (due to topographicfeatures or material changes) dramatically changes the phase of the cantilever.

2.3 Applications

There are many different AFM techniques, based on the two general modes: contactand dynamic. Depending on the goal of the measurement, an appropriate techniquemust be chosen. In bio-applications [24, 25, 26] experiments are typically done in liq-uid or buffer solutions on very soft materials such as cells. Cells are fragile and do notlike to be disturbed so care has to be taken during imaging or other measurements. Adynamic technique would therefore be preferred since it is more sensitive and requiresless interaction with the sample. Unfortunately the liquid makes it difficult to use dy-namic methods due to very strong damping force acting on the cantilever body and therisk of exciting acoustic modes in the liquid. Therefore contact mode is typically usedin liquid with soft cantilevers having spring constant less than 1 N/m.

Many types of techniques for electrical and magnetic [27, 28] characterization ofmaterials can also be performed in contact mode with conductive or magnetic can-tilevers; Kelvin probe force microscopy [29] and Intermodulation electrostatic forcemicroscopy [30] are dynamic methods for electrical characterization.

2.3. APPLICATIONS 9

0.0 0.5 1.0 1.5 2.0 2.5 3.0

1

10

100

|G|

a)Q = 500.0

Q = 10.0

0 f0 2f0 3f0

0

−π2

−π

arg(

G)

b)

Figure 2.2: Cantilever transfer function. (a) The absolute value of the transfer func-tion |G| and (b) its phase. At resonance, f0, the transfer gain peaks at |G|=Q.

Chapter 3

Calibration

THE AFM CANTILEVER is ideally suited to transduce the force F between a sharp tipand surface. A force on the tip is converted to deflection of the cantilever. Thecantilever deflection is measured while it scans the sample, but not directly in

units of meter. Although calibrated detectors exist, e.g. Laser Doppler vibrometry [31](LDV)1 and the capacitance change for a known geometry [32], they are rarely usedin commercially available AFMs. The standard setup used in most AFMs, depicted infig. 2.1, is called the optical lever technique. Details of the readout procedure are shownin fig. 3.1. In this chapter the necessary calibration procedure for the cantilever andthe optical lever will be explained carefully, giving insight into how force is measuredwith the AFM. Section 3.2 gives an introduction and summary to Paper I.

3.1 Flexural calibration

The standard optical lever measures the flexural bending of the cantilever or its tor-sional twisting, where flexural bending is by far the most common. AFM based on can-tilever flexure allows for characterization of the surface topography and the distancedependence of the interaction force normal to the surface.

Using the setup depicted in fig. 3.1, the parameter of interest is the flexural de-flection d of the cantilever from its equilibrium position. The deflection is typically afew nanometer up to some hundred nanometers2. A laser is focused at the free-endof the cantilever and the reflection is guided through a mirror and lens system to aphoto-detector. The motion of the reflected spot is large in relation to the motion ofthe tip, due to the long optical path in relation to the cantilever length. This optical’lever’ allows for very sensitive measurement of deflection. For small deflections, theoptical lever is assumed to be a linear detector with an unknown conversion factor

1The LDV measures the Doppler shift of a reflected laser beam, which is proportional to the cantilevervelocity.

2The AFM has the capability to measure sub-nm deflections as well, but in a more standard measurementthe deflection is usually larger.

11

12 CHAPTER 3. CALIBRATION

A B

DCA B

DC

τ

Fd

yx

z

y

z

x

zϕ

a)

b)

c)

Figure 3.1: The optical lever. A laser (red path) is aimed at the end of the cantileverbeam. It is reflected off the cantilever and then guided by a lens and mirror systemonto a photo-detector. The enlarged detector in (b) shows its four sub-detectors A−D,and the laser spot covering a large part of the photo-sensitive surface. In the upper leftcorner the deflection d due to an applied force F is depicted, and in the lower rightcorner the twisting φ originating from a torque τ.

3.1. FLEXURAL CALIBRATION 13

(units m/V) connecting measured detector voltage to actual tip motion in meters. Thisconversion factor is known in the literature as invOLS (inverse Optical Lever Sensitiv-ity). The word "sensitivity" in the context of the optical lever is misleading, since thevalue of invOLS does not tell us about the sensitivity of the measurement, but ratherthe responsivity, or magnitude, of the linear response function describing the opticallever. Therefore we use the term invOLR = α−1 (inverse Optical Lever Responsivity).The OLR α has units of V/m.

The photo-detector is composed of four smaller detectors addressed A to D. Theintensity on each of the four detectors are measured and used in different combinationsto find the cantilever deflection relative to its unloaded position, usually correspondingto the reflected spot on the center of the detector. For flexural deflection, the intensities(A+ B)− (C + D) are measured and the cantilever twisting is measured from (A+ C)−(B + D), see fig. 3.1 (a) and (b)3.

The calibration procedures can be split into two groups: static and dynamic tech-niques, both with their strengths and weaknesses.

Static loading

Static loading is in general destructive to the tip and/or surface, but it is very easyto perform. The simplest method is to first calibrate the optical lever α by slowly ap-proaching a stiff surface causing a bending of the cantilever. Assuming that: i) thez-scanner is calibrated (i.e. the change in base position is known) and ii) the cantilevertip is sliding freely on the surface, the conversion factor from volt to meter is obtained.The second assumption is that there is no tip force that causes longitudinal bendingmoment, also giving rise to beam flexure. Figure 3.2 (a) show such an approach mea-surement for a Tap150 cantilever on a sapphire surface. The slope of the linear regiongives α−1 = 26.6 nm/V.

Knowing the value of the inverse optical lever responsivity α−1, the stiffness k of thecantilever is found by measuring thermal noise. Thermal noise is a white noise, excitingthe cantilever with equal power at every frequency. However, in a narrow-band aroundeach resonance, the response is enhanced. The cantilever response is well-describedby the voltage power spectral density

Stot(ω) = α2Scant(ω) + Sdet, (3.1)

where Sdet is the frequency-independent detector-noise floor. The power-spectral den-sity of cantilever deflection Scant(ω) can be calculated from the fluctuation-dissipationtheorem

Scant(ω) =4kB Tω

Im�

1k

G(ω)�

=4kB T

kQ

ω40

(ω20 −ω2)2 +

�ωω0Q

�2 , (3.2)

3Please note that the illustrated laser position on the detector corresponds to a combined bending andtwisting!


−2.91 −2.90 −2.89

Scanner position [µm]

0.4

0.6

0.8

Defl

ecti

on[V

]

a)

α−1 = 27nm/V

100 150 200

Frequency [kHz]

10−11

10−9

10−7

Svv

[V2/H

z]

Stot

α2Scant + Sdet

Sdd

b)

Figure 3.2: Static calibration. (a) Approach-retract curve for a Tap150 cantilever on asapphire surface. (b) Flexural thermal noise spectrum of the same cantilever, acquiredfrom the AFM software. Numerical fits to the cantilever and detector (orange) and thecantilever only (green) are shown.

where T = (273+30) K is the ambient temperature, kB is the Boltzmann constant andG(ω) is the transfer function of the cantilever, given in eq. (2.6). Combining eq. (3.1)and eq. (3.2), an expression for the measured noise spectrum is obtained

Stot(ω) =4kB Tα2

kQ

ω40

(ω20 −ω2)2 +

�ωω0Q

�2 + Sdet. (3.3)

Using the numerical curve-fit routine scipy.optimize.curve_fit [33] (which uses the Leven-berg-Marquardt algorithm) to the noise data in fig. 3.2 (b), the unknown parameters,Q,ω0, k and Sdet of eq. (3.3) are found. For the data of fig. 3.2 (b), the fitted values are:f0 =ω0/(2π) = 154 kHz, Q = 208 and k = 3.8 N/m. Note that while ω0 and Q can befound without a value of α, we need α to determine k since α−1k enters as a productin eq. (3.3). The cantilever is now calibrated and the force on it can be calculated fromthe measured voltage on the photo-detector.

However there is a subtle problem with this approach: the static bending shapeof the cantilever when measuring α is not the same as the dynamic bending shapewhen performing noise measurements. When α is determined, a point load is appliedat the end of the cantilever, while the thermal noise is a dynamic load which excitesthe cantilever over its entire body. A conversion factor between static and dynamicstiffness can be found from finite element analysis, but this needs to be calculated foreach cantilever geometry. The conversion is approximately kdynamic = 1.101kstatic forthe AC160TS, which has similar plane view dimensions to the Tap150 [34].


In another static method, the stiffness can be calculated by pressing against a can-tilever of known stiffness [35] (so called calibration cantilever), but it requires carefulpositioning of the cantilever relative to the reference cantilever and can easily breakboth. There are also other techniques [35] which are even more involved, for examplethe loaded-mass and the pendulum methods.

It is obvious that all of the static techniques are destructive and might in many casesdestroy the tip or damage the sample. Often a hard surface is used when calibratingα against the z-scanner because it invariably causes the tip to break. Performing thecalibration at the end of the measurement might save the tip during the scan, but thecalibration is sensitive to the laser position on the cantilever, which might drift duringthe measurement. Therefore it is not certain that the laser is at the same positionduring the calibration as it was during the experiment. Surface debris can also becomeattached to the tip during calibration of α.

The Sader method

One way to solve the problems described above is to use only dynamic techniques,where one does not need to apply a static load to the tip and the tip does not need tocontact the surface. The most versatile method thus far is the so called Sader methodbased on hydrodynamic theory. It is a non-invasive technique where the cantileverdamping is calculated by finding the hydrodynamic drag force from a surrounding fluid(typically air). The hydrodynamic theory, together with a thermal noise spectrum al-lows for a complete calibration of the cantilever.

For dynamic measurements, not only the cantilever stiffness and α are needed tocalculate the force. Equation (2.7) shows that the force-deflection relationship is fre-quency dependent. Therefore the quality factor Q and resonance frequencyω0 are alsorequired to fully calculate the force. Fortunately they can be obtained directly from thenoise measurement.

A noise spectrum is measured in a narrow band around the resonance frequency, seefig. 3.3. A fit to the voltage power spectral density in eq. (3.1) is performed in a similarmanner to the invasive approach, with an unknown α-value. The quality factor andresonance frequency are obtained from the fit, together with a pre-factor in eq. (3.3)which sets the magnitude of the noise power

SDC =4kB Tα2

kQω0. (3.4)

Using the hydrodynamic theory together with the resonance frequency and qualityfactor, the cantilever stiffness is calculated [36, 37]

k = 1.101 · 0.1906ρ f b2 LQω20Γi(Re) (3.5)

where ρ f is the density of the ambient medium, L and b are the length and width of thecantilever, Q the quality factor and ω0 the resonance frequency. The prefactor 1.101is the conversion factor between static and dynamic spring constants, i.e. eq. (3.5)


0 200 400 600 800 1000

Frequency [kHz]

10−13

10−12

10−11

10−10

[V2 A

D/H

z]

Stot a)

100 120 140 160 180 200

Frequency [kHz]

10−13

10−12

10−11

10−10

Svv

[V2 A

D/H

z]

Stot

α2Scant + Sdet

α2Scant

b)

Figure 3.3: Thermal noise with the IMP software. (a) Thermal noise acquired overa large frequency range. The region marked with red is shown in (b). Fits to the can-tilever and noise (orange), and only cantilever (green) are performed for the zoomedwindow. y-axis for both (a) and (b) are in analog-digital volt units (1VAD ≈ 0.1 mV).


is actually a dynamic technique4. Γi(Re) is the imaginary part of the hydrodynamicfunction: a function of the dimensionless Reynolds number

Re=ρf b

2ω0

4µf(3.6)

with the ambient dynamic viscosity µf (units kgms ). The Reynolds number is an important

quantity in fluid mechanics. Being the ratio of inertial to viscous forces, it characterizesthe flow around the cantilever. A series expansion of the hydrodynamic function isgiven in Sader et al. [36] for a long thin cantilever beam. It was noted in [34] and latershown [38] by Sader et al. that the hydrodynamic function used in eq. (3.5) can besimplified, with one constant A, unique to each cantilever plane-view. The expressionfor the stiffness k reads

k = AQ f 1.30 . (3.7)

The Global Calibration Initiative (GCI) was launched together with a proof-of-conceptpaper [38], with the goal of a community-effort to help improving cantilever calibra-tions. The values of A for different cantilevers are found through reference-calibrationsby the community, uploaded to the GCI web page [39].

Regardless of the specific representation of the hydrodynamic function, the nextstep in the calibration procedure is to determine α. As proposed by Higgins et al. [40],we integrate eq. (3.3) over all frequencies and find the mean displacement of the can-tilever

⟨z⟩=π

2α2 f0SDCQ. (3.8)

With the equipartition theorem [41] we get

12

kBT =12

k⟨z⟩, (3.9)

which gives

α−1 =

√

√ 2kBTπk f0SDCQ

. (3.10)

For the data shown in fig. 3.3 together with eq. (3.5) and eq. (3.10) this procedureyields: f0 = 154 kHz, Q = 245, k = 4.53 N/m and α−1 =59.09 nm/V. This calibrationof α can be traced to one measurement of thermal noise which does not rely on acalibrated scanner.

It is important to point out that eq. (3.10) together with eq. (3.5) or eq. (3.7) givesthe dynamic values of k and α, whereas the approach in section 3.1 mixes static anddynamic. The non-invasive technique is also very fast: just move away a few cantileverwidths from the surface, measure a thermal noise spectrum and perform the fit. Thisprocedure can easily be done between each image to correct for drift in the AFM system.

4See table IV in Sader [34]. The value 1.101 is for an AC160TS cantilever, with very similar plane-viewdimension as the Tap150 cantilever used in this chapter.


3.2 Torsional calibration

Green’s method

If a static torque τ around the major axis is applied to the free end of the cantilever,the static twisting angle φ is

φ =1κtτ. (3.11)

To connect the measured twist to the applied torque, the torsional spring constant κt[Nm] must be known. There are many different methods to determine κt, some whichare very involved and destructive to tip and sample [35, 42, 43, 44, 45]. Althoughquasi-static techniques for measuring in-plane forces exist, such as lateral force mi-croscopy [46, 47] and particle-pushing [48], a dynamic method is favorable in manyrespects as we shall discuss in Chapter 7. Similar to the flexural eigenmodes, there isa difference between the static and dynamic torsional stiffnesses and they need to becalibrated properly. Green et al. [49, 50] derived a theoretical expression for the hy-drodynamic damping due to torsional motion. The calibration approach is very similarto the Sader method: the torsional resonance frequency ω0t and quality factor Qt arefitted after which κt and αt are calculated. The expression for the torsional stiffnessaccording to Green is

κt = 0.1592ρ f b4 LQtω20tΓt(Ret). (3.12)

The torsional twisting is well approximated by a harmonic oscillator, allowing us toreplace k→ κt and ⟨z⟩ → ⟨φ⟩ in eq. (3.9), giving the optical lever responsivity of thetorsional mode, αt in V/rad.

Using flexural k to calculate torsional κt

Green’s formula together with measured noise enables a non-invasive calibration ofthe torsional stiffness. It is a powerful and fast method, but it requires a thermal noisespectrum from which the resonance frequency and quality factor can be found. Whenmeasuring thermal noise, typically a few hundreds of spectra are averaged to improvethe fit quality. Figures 3.2 (b) and 3.3 (b) both show examples of noise peaks at theflexural resonance frequency of a soft tapping mode cantilever. The quality of thesefits is good. The stiffer torsional noise peak is different. It has been shown that stifferresonances are more difficult to calibrate with thermal noise. If the peak is too weak itgets very difficult to i) find the resonance and ii) perform an accurate numerical opti-mization to fit the model parameters. It is also easy to confuse spurious peaks with theactual resonance peak. Figure 3.4 gives an example of the torsional resonance of: (a)the same Tap150 cantilever used in fig. 3.3 and (b) a stiffer tapping cantilever (Tap525,Bruker AFM probes)5. It is clear that the fitting with the stiffer Tap525 cantilever is lesssensitive to change in the parameters. The resonance frequency can in many cases be

5Both Tap150 and Tap525 have the same plane-view dimensions, but different thickness. Their plane-view dimensions are approximately 125× 30 µm, with thicknesses of 2.1 µm and 5.75 µm respectively.

3.2. TORSIONAL CALIBRATION 19

1200 1210 1220 1230 1240 1250 1260 1270

10−3

10−2

10−1 a)

2410 2415 2420 2425 2430 2435 2440 2445 2450

Frequency [kHz]

10−6

10−5 b)Stot

α2Scant + Sdet

α2Scant

[V2 A

D/H

z]

Figure 3.4: Torsional noise. (a) Torsional noise peak for a soft Tap150 tapping modecantilever. (b) Torsional nosie peak for a stiff Tap525 tapping mode cantilever. Both(a) and (b) show numerical fits to the cantilever and detector (orange) and cantilever(green).

found with good accuracy even if the noise peak is weak in comparison to the detectornoise, but the quality factor becomes less accurate.

Stiff cantilevers are preferred for intermodulation AFM in general and intermodu-lation friction force microscopy in particular. The noise spectrum of the flexural reso-nance for a stiff cantilever is in most cases good, but the torsional calibration is challeng-ing. To address the difficulty with torsional calibration, we use a method to calculatethe torsional stiffness based on a calibrated flexural mode (Paper I).

For flexural oscillations in vacuum, the first resonance frequency of a cantileverbeam is

ω(vac)0f = 2

√

√ kf

ρc Lbh(3.13)

where kf is the dynamic flexural spring constant, ρc the cantilever density, L, b and h


are the length, width and thickness respectively. Similarily, the first torsional resonancefrequency is

ω(vac)0t =

2p

6b

√

√ κt

ρc Lbh(3.14)

with the torsional spring constant κt. Both expressions depend on the cantilever densityρc and the thickness h, where the latter has large uncertainty6. By taking the ratioof eq. (3.13) and eq. (3.14), ρc and h fall out, leaving a relation between resonancefrequencies and stiffnesses

κt = kfb2

6

�

ω(vac)0t

ω(vac)0f

�2

. (3.15)

Equation (3.15) holds in vacuum, but for a viscous environment, like air, a correctionfactor, C , needs to be applied, i.e.

κt = Ckf b2�

ω0t

ω0f

�2

, (3.16)

whereω0f andω0t are measured in air. In Paper I we showed that the correction factorfor cantilevers without or with a small tip, is C ≈ 0.15± 0.01, close to the theoreticalvalue in vacuum: C = 1

6 ≈ 0.166... For tipped cantilevers, the value of C is furthersuppressed due to disturbed fluid flow around the tip.

Equation (3.16) is an approximation for air, but it solves the problem of relying toa good torsional thermal noise fit. The torsional resonance frequency is still requiredto calculate κ, but it can be found by other means, such as frequency sweeps. Tofinally validate this approach we used finite element simulations to obtain the flexuraland torsional stiffnesses and resonance frequencies, and then calculate the stiffnessfrom eq. (3.16). We showed good agreement for both a long and a short calibrationcantilever geometry.

3.3 Squeeze-film damping

Not only a good calibration is needed to accurately determine the tip-surface force. Aswas shown by Borgani et al. [51] and others [52, 53, 54], the cantilever resonanceis also sensitive to the probe-surface distance due to so called squeezed film or lubri-cation forces. It is therefore of great importance to perform the noise calibration atan adequate distance from the surface, at least several cantilever widths away [54].In Borgani et al. it was shown that especially the quality factor drops by 30% whenclose but not making contact with the surface. Another important concept introducedby Borgani et al. is background force compensation. Especially for soft materials, thesqueezed film forces need to be removed when calculating the true tip-surface force.

6For a Tap300 cantilever the manufacturer-specified thickness is h = 3.75± 0.75µm (March 2018). Forthe Tap150 cantilever the thickness is h= 2.1± 1µm.

3.3. SQUEEZE-FILM DAMPING 21

The background compensation can also remove other long-range forces, regardless oftheir nature, so long as they give linear response.

Chapter 4

Intermodulation AFM

INTERMODULATION ATOMIC FORCE MICROSCOPY (ImAFM) was developed in 2008 byPlatz et al. [11]. ImAFM exploits a feature of driven non-linear systems called in-termodulation, or frequency mixing. Intermodulation frequently occurs in audio-

circuits, where their nonlinear nature generates distortions. It is also common in radiotransmission. In these type of circuits the intermodulation distortion (IMD) is unwantedsince it distorts the music or signal, but there are certain cases where intermodulationis desired. By monitoring the IMD from a known input signal, the nonlinearity of asystem is reconstructed. This trick is used by experimentalists to investigate nonlinearsystems, e.g. microwave resonators [55, 56] and AFM. Several ImAFM sub-techniqueshave been developed: Force curve analysis [57, 58] and fitting to an arbitrary forcemodel [59]; electrical characterization with Intermodulation Electrostatic Force Mi-croscopy (ImEFM) [30] and high-speed friction measurements with IntermodulationFriction Force Microscopy (ImFFM) [60] (see chapter 7 and paper V). This chapterintroduces the basic concepts of ImAFM and the important analysis tool: the dynamicforce quadrature curves.

4.1 Frequency mixing

Frequency mixing is a property of non-linear systems, where two or more pure drivetones generate response at new, non-drive frequencies. Intermodulation is the technicalterm for frequency mixing, and it is the essential building block of intermodulationAFM (ImAFM) [11]. Modeling the cantilever as a driven damped harmonic oscillator,its equation of motion is given by eq. (2.4). The applied forces are the drive force Fd(t)and the tip-surface interaction force Fts. Intermodulation AFM, in contrast to tappingmode AFM, uses two or more drive tones, such that

Fd(t) =N∑

i

Ai cos(2π fi t +φi), (4.1)

23

24 CHAPTER 4. INTERMODULATION AFM

0.0 0.5 1.0 1.5 2.0

Time [ms]

−20

0

20

d[n

m] a)

Free

145 150 155

Freq [kHz]

10−4

10−2

100

|d|[

nm

] c)

0.0 0.5 1.0 1.5 2.0

Time [ms]

b)

Engaged

145 150 155

Freq [kHz]

d)

Figure 4.1: Intermodulation. (a) Simulated cantilever motion, without tip-surfaceforce. Inset axis shows a zoom at the vertical dotted line. (b) Simulated cantilevermotion, with tip-surface force. Inset axis shows a zoom at the vertical dotted line.The black solid line shows the time envelopes and the black dashed line shows thesurface position. (c) Fourier transform of the free motion (a), show response only atthe two drive frequencies. (d) Fourier transform of the engaged motion (b), showsintermodulation at many frequencies.

where N is the number of drive tones, each having frequency fi , amplitude Ai and phaseφi . There is no formal constraint on the number of drive tones1 and the analysis usedthroughout this work is valid for any number of drive tones. However, two drive tonesare used for all techniques described in this thesis, where f1 and f2 are centered closeto a resonance of the cantilever such that f1+ f2

2 ≈ f0. The drive force is

Fd(t) = A1 cos(2π f1 t +φ1) + A2 cos(2π f2 t +φ2). (4.2)

The spacing between the drive frequencies is ∆ f = f2− f1, where ∆ f � f0 defines themeasurement bandwidth or the time T = (∆ f )−1 needed to give a frequency resolutionof ∆ f . A typical value of ∆ f is 500 Hz, giving a measurement time of T = 2 ms.

Figure 4.1 shows simulated data of eq. (2.4) with the drive force eq. (4.2). Pythoncode for the simulation is given in appendix A.1. Simulations of a free cantilever is

1There is an instrumental limit: lockin amplifiers are used to measure the signal at different frequencies.To get accurate measurements, the response for at least each drive frequency must be measured, requiringone lockin amplifier per drive tone.

4.2. DYNAMIC FORCE QUADRATURE CURVES 25

shown in fig. 4.1 (a) and (c) (i.e. FTS = 0) and simulations of an engaged cantilever infig. 4.1 (b) and (d). The tip-surface force used in the simulation is

Fts(z = h+ d) = (−kvz − Fad) · fstep(z), (4.3)

where kv is the material stiffness, h the equilibrium position of the tip, Fad the adhesionforce and

fstep(z) =�

1−1

e−z/z0 + 1

�

(4.4)

a smooth step function with z0 controlling the smoothing. Comparing the frequencyresponse around resonance in fig. 4.1 (c) (free cantilever) to (d) (engaged cantilever),many intermodulation peaks (IMPs) appear at linear combinations of the two drivefrequencies, such that

fIMP = nf1 +mf2, (4.5)

where n, m ∈ Z. We use a notation of addressing each IMP with its order and locationrelative to the two drives e.g. foL or foH where o = |n|+ |m| refers to the order of theintermodulation product and L/H refers to lower or higher than f1, f2. By measuringthe response of many IMPs, the nonlinearity can be characterized. The spectrum iswhat we measure and analyze.

4.2 Dynamic force quadrature curves

An important goal for AFM is to reconstruct the tip-surface force. Before the emergenceof ImAFM, the standard way of finding the force was to analyze quasi-static approachcurves [an example is given in fig. 3.2 (a)]. Assuming a model for Fts, parameters of themodel are found, for example the elastic modulus E∗ of the Derjaguin-Muller-Toporov-model (DMT) [20]. There are a plethora of models for different systems [20], e.g. theJKR-model (a "DMT-model" for soft compliant materials), Chadwick theory (spheresindenting supported membranes) and Lennard-Jones potential + van der Waals (com-monly used to describe short-range forces in non-contact AFM). In many cases thesemodels are applied to quasi-static approach curves [61].

ImAFM measures dynamic force quadrature curves at every pixel of a scan. A spec-trum is measured at each pixel at a rate of 500 pixels per second (assuming∆ f =500 Hz),and each of these spectra enables analysis of the non-linearity which generated theIMPs, i.e. the tip-surface force. Solving for Fts in eq. (2.5) and using eq. (2.6), theinteraction force is

Fts(ω) = kG−1(ω)d(ω)− Fd(ω). (4.6)

From the thermal noise calibration (see chap. 3), the values of ω0, Q and k are deter-mined, but the drive force Fd(ω) is unknown. The drive forcing can be measured fromthe response of the cantilever dfree far away from the surface, such that Fts = 0 andFd = kG−1dfree. The interaction force can be written

Fts = kG−1(ω)�

d − dfree

�

. (4.7)


If the force is expanded as a finite power series in tip position the expansion coeffi-cients of the polynomial function can be found using linear algebra [57]. Later DanielForchheimer [59] derived a method using numerical optimization to find the parame-ters of an arbitrary force model. The largest drawback with both these reconstructionmethods is the fact that they rely on a model.

A model-free method for force reconstruction used in this thesis, is the dynamicforce quadrature curves (FQC) developed by Platz [58]. The FQC do not give the in-stantaneous force at some tip position, rather they are weighted integrals of force oversingle oscillation cycles, one is in-phase with the motion (FI(A) or the conservative force)and the other quadrature to the motion or in-phase with velocity (FQ(A) or the dissipativeforce). The FQC are defined as [58]

FI(A) =1T

∫ T

0

FTS(t) cos(ωt)dt, (4.8a)

FQ(A) =1T

∫ T

0

FTS(t) sin(ωt)dt, (4.8b)

where A is the amplitude (of oscillation), T is the measurement time and ω is thecarrier frequency. They have been thoroughly explained and discussed in previouspublications [58, 61].

Using the simulated cantilever motion depicted in fig. 4.1, the FQC are recon-structed. Figure 4.2 (b) shows the conservative force FI(A) and (c) shows the dissipativeforce FQ(A) in blue. The force used in the simulation is a purely conservative force onlydepending on z, and not on velocity z. Therefore the quadrature force FQ(A) = 0. Apositive FI(A) corresponds to a net attractive force and negative FI(A) is a net repulsiveforce. Equivalently for FQ(A): a negative value corresponds to energy dissipated to thesample and a positive FQ(A) is energy obtained from the sample. Note that for eachoscillation amplitude, one value-pair for the FQC are obtained. Figure 4.3(a) shows anoscillation with amplitude A0 in orange. Integrating eqs.(4.8) for this specific oscilla-tion amplitude gives the FQC values marked with an orange dot in fig. 4.3 (b) and (c).Integrating for each amplitude gives the theoretical FQC shown with orange dashedlines in fig. 4.2 (b) and (c).

4.2. DYNAMIC FORCE QUADRATURE CURVES 27

−1.0

−0.5

0.0

FI

[nN

]

b)

5 10 15 20

Amplitude [nm]

−1.0

−0.5

0.0

FQ

[nN

]

c)

Simulated

Actual

−2 0 2 4

Tip position [nm]

−1

0

1

2

FT

S[n

N]

a)

Figure 4.2: The force. (a) The force from eq. (4.3), used for the simulation shownin fig. 4.1. (b) The force quadrature curve FI(A), in phase with motion. (c) The forcequadrature curve FQ(A) is quadrature to the motion. The force quadratures are recon-structed from the simulated data from fig. 4.1. The simulated FQCs are shown in blueand the actual [by integration of eqs. (4.8)] in orange. The parameters are: h= 15 nm,ks = 2 nN, Fad = 2 nN and z0 = 1 Å.


0 5 10 15 20 25 30

Tip position [nm]

-1.0

0.0

1.0

FT

S[n

N]

h−A0 h h+A0a)

-0.1

0.0

0.1

FI

[nN

]

A0

b)

0 5 10 15 20

Amplitude [nm]

-0.1

0.0

0.1

FQ

[nN

]

A0

c)

Figure 4.3: Integrating the force. (a) The force from eq. (4.3) (blue) and an oscil-lation with amplitude A0 around the working height h marked in orange. The forcequadrature curves FI(A) and FQ(A) are shown in (b) and (c) respectively (blue). Inte-grating eqs.(4.8) with an oscillation amplitude A0 gives one point on the FQC, markedwith an orange dot in (b) and (c).

4.3. INSTRUMENTATION 29

4.3 Instrumentation

To reconstruct the FQC, a free spectrum and an engaged spectrum are needed. Thesimulated spectra shown in fig. 4.1 (c) and (d) are obtained by calculating the DiscreteFourier Transform of the simulated motion using the Fast Fourier Transform (FFT) [62,33]. Using the FFT on exactly one beat gives a spectrum with frequency resolution∆ f ,up to the Nyquist frequency (half the sampling frequency). Only one beat is simulatedand the FFT gives the full spectrum. In experiments, the rate at which data is obtained(250 Msample/sec.) is too fast and the FFT is too slow to reduce the data in real-time.Because the cantilever filters out most of the response, only frequencies near resonancegive response above the detector noise. Therefore data acquisition is performed directlyin the frequency domain with lockin amplifiers.

Throughout this thesis a multi-frequency lockin amplifier (MLA) from Intermodu-lation products [63] is used, which enables measurement of up to 40 frequencies inparallel. The MLA and its software form an add-on compatible with AFM’s from all ofthe major AFM manufacturers. When performing multi-frequency measurements it isimportant to choose the drive frequencies carefully to avoid spectral leakage. If drivefrequencies are divisible with their frequency separation, i.e.

f1 = n1∆ f ,

f2 = n2∆ f(4.9)

where n1 and n2 are integers, all intermodulation products will occur at an integertimes∆ f . The cantilever motion will be periodic in one measurement window T = 1

∆ fand there is no spectral leakage.

Chapter 5

The moving surface model

THUS far we have only been discussing forces on stiff materials. However in a vastnumber of applications from polymers to biology and medicine research, softermaterials are of great interest. Developing new environmental friendly mate-

rials which can be recycled requires knowledge of their mechanical and rheologicalproperties. Understanding their response to different load situations and surroundingsis crucial for applications. This chapter gives an introduction to the model used in Pa-per II and III, which explains measurements on soft materials. We provide an extendedderivation of this model and discuss different formulations of the tip-surface interactionforce.

5.1 Equations of motion

The cantilever and surface are modeled as two harmonic oscillators coupled throughthe tip-surface force. The equation of motion of the cantilever is given by eq. (2.5).Equivalently, the dynamics of the surface is described by the equation of motion (EOM)

msds +ηsds + ksds = −Fts, (5.1)

with the surface stiffness ks, mass ms and dissipation ηs. The tip-surface force is Fts =Fts(s, s), where s = z− zs = (h+ d)− (z0+ ds) is the separation between the moving tipand the moving surface, see fig. 5.1.

In an earlier publication [64] a DMT force (Derjaguin-Muller-Toporov model [20])was used to model the tip-surface interaction on several soft materials. As discussedin sec. 4.2 and paper II, we use a simplified piece-wise linear (PWL) model, with onlytwo parameters: the adhesion force Fad and the stiffness kv associated with the tippenetrating the surface and displacing a volume of the material. This simplified modelstill captures the essential features of an AFM interaction model, as shown in fig. 5.2. Inaddition to this conservative part of the tip-surface force, we assume a damping force

31

32 CHAPTER 5. THE MOVING SURFACE MODEL

-d

d

h

z

z

zs

sw

0s

Figure 5.1: Coordinates. The cantilever and surface coordinates. The equilibriumposition of the tip is at some height h in the laboratory frame, and the rest position ofthe surface is at z0. The tip deflection from equilibrium is d, and that of the surface isds. The working distance is w= h− z0, and the tip-surface separation s = w+ d − ds.

−15 −10 −5 0 5 10 15

Separation s [nm]

−5

0

5

Fts

[nN

]

PWL

DMT

Figure 5.2: DMT and PWL force. A DMT-force in blue with parameters a0 = 1.5 Å,H = 8 · 10−20 Nm, R = 10 nm and E = 80 MPa. The corresponding PWL-model withkv = 0.867 N/m and Fad = 6 nN. Figure adopted from Paper II.

−ηvs associated with material flow as the tip penetrates the surface. The total force is

Fts(s, s) =

¨

0

−kvs−ηv s− Fad

if s > 0,

if s ≤ 0.(5.2)

5.2. NUMERICAL ANALYSIS 33

5.2 Numerical analysis

We simulate the cantilever and surface dynamics by coupling eqs. (2.5) and (5.1) viaa tip-surface force Fts. The system is

1ω2

0

d +1ω0Q

d + d =1k(Fd(t) + Fts(s, s)) , (5.3a)

msds +ηsds + ksds = −Fts(s, s). (5.3b)

To solve a system like (5.3) it is convenient to rewrite it as a system of first orderequations on the form

˙x = F( x), (5.4)

where x = {x i} are the state variables of the system. Introducing x1 = d, x2 = d,x3 = ds and x4 = ds, we find

x1 = x2,

x2 = ω20

k (Fd(t) + Fts(s, s))− ω0Q x2 − x1,

x3 = x4,

x4 = − 1ms

Fts(s, s)−ηs x4 −ksms

x3

(5.5)

with s = w+ x1 − x3 and s = x2 − x4. This system has in total 7 free parameters1, forthe specific tip-surface force given in eq. (5.2).

We integrate eq. (5.5) using an adaptive time-stepping solver coded in C calledCVODE [65]. Using CVODE reduces the simulation time by a factor of approximately100 compared to a higher-level language (e.g. scipy.odeint in Python). If simulationtime is not a problem, any standard ode-library can be used. Although using theCVODE-package speeds the simulation up tremendously, the simulation speed can beincreased even more by reducing the dimensionality of the system.

The harmonic motion described by eq. (5.3b) corresponds to surface oscillations ata particular frequency ωs =

p

ks/ms. As the cantilever interacts with the surface it canexcite surface waves, either capillary waves (for liquid samples) or Rayleigh waves (forsolid samples). As argued in the supplementary material of ref. [64], both capillarywaves and Rayleigh waves, with wavelengths corresponding to the tip radius, havefrequencyωs which is several orders of magnitude larger than the cantilever frequency∼ω0. The surface inertia can thus be neglected in eq. (5.3b), i.e. we discard terms oforder ω−2

s . The EOM for the surface in eq. (5.3b) becomes a first order equation:

ηsds + ksds = −Fts(s, s). (5.6)

This assumption also reduces eqs. (5.5) to a three dimensional system. The three-dimensional system is much faster to solve because it removes the fast time-scale of

1Assuming that the cantilever is calibrated, i.e. k, Q and ω0 are determined beforehand.


the surface which causes the adaptive time-stepper to take many very small steps. Weperformed comparisons of simulations with both the three and four dimensional systemand they give equivalent results for typical parameters.

There is a caveat with the three dimensional system: the tip-surface force dependson s and s = d − ds, where ds is not a state variable of the three-dimensional system.When the tip-surface force in eq. (5.2) is inserted into eq. (5.6) (for s < 0):

ηsds + ksds = kv(h+ d − z0 − ds) +ηv(d − ds) + Fad, (5.7)

ds appears on both sides of the equation. Solving eq. (5.7) for ds would allow us toformulate the full system in the form of eq. (5.4):

x3 =

¨

− ksηs

x3kv

ηs+ηv(h+ x1 − z0 − x3) +

ηvηs+ηv

x1 + Fad −ks

ηs+ηvx3

if s > 0,

if s ≤ 0,,

x1 = x2,

x2 = ω20

k (Fd(t) + Fts[(h+ x1 − z0 − x3), (x2 − x3)])−ω0Q x2 − x1.

(5.8)

It is important to first calculate the value of x3 at each time step, such that it can beused to calculate the RHS in the expression for x2.

However, when we rewrite eq. (5.7) to solve for ds, it becomes difficult to replacethe tip-surface coupling force with another model. For example a Kelvin-Voigt-like cou-pling force Fts = e− x3/v0 cannot be used because it is not linear in x3. Using the four-dimensional system eq. (5.3), any type of coupling force as a function of s and s can beused directly.

To get an intuitive feeling for the damping parameters, we introduce two relaxationtimes. The surface relaxation time τs =

ηsks

is the characteristic relaxation time of thefree surface, typically in the order of µs. Similarly, a relaxation time associated with thematerial flow is τv = ηv/kv . For numerical simulation we normalize time in terms ofthe resonance frequency of the cantilever, such that u = tω0. Reformulating eq. (5.8)in terms of u gives

ddu x3 =

¨

− 1ω0τs

x3kv/ω0

τsks+τvkv(h+ x1 − z0 − x3) +

τvkvτsks+τsks

x1 + Fad −ks/ω0

τsks+τvkvx3

if s > 0,

if s ≤ 0,,

ddu x1 = x2,ddu x2 = 1

k

�

Fd(u=ω0 t) + Fts[(h+ x1 − z0 − x3), (x2 −ddu x3)]

�

− 1Q x2 − x1.

(5.9)When analyzing the FQC (more on this in the next section) we use ω0τs/v as the pa-rameters related to the surface and interaction damping. Using the normalized timewe relate the time constant to the time between successive tip impacts given by theresonance frequency of the cantilever, ω0. Identifying two regimes for the coupledtip-surface dynamics:

i) ω0τs > 1,

ii) ω0τs < 1.(5.10)

5.3. SIMULATIONS 35

In case i) the surface relaxes slower than the frequency of tapping, i.e. if the surfaceis displaced, it has not enough time to reach its equilibrium before the cantilever tapsagain. For case ii) the surface has relaxed faster than consecutive taps, i.e. it has timeto go back to its equilibrium position.

5.3 Simulations

0.0

0.2

0.4

0.6

FI

[nm

]

c)

0 10 20 30

Amplitude [nm]

−0.2

−0.1

0.0

FQ

[nm

]

d)

0

25

50

75

[nm

]

CantileverSurface

a)

0 1 2

Time [ms]

0.0

0.6

[nm

]

b)

Figure 5.3: The moving surface model. Simulations of eq. (5.5) with a PWL-force.(a) show the envelopes of cantilever motion (green) and surface deformation (red).The rapid oscillations at the frequency ω0 are not visible on this time scale. (b) showsthe envelope of the surface motion with an expanded y-axis. (c) and (d) show theforce quadrature curves FI(A) and FQ(A). The parameters are h = 26 nm, ω0τs = 1,ω0τv = 0.2, ks = 5 N/m, kv = 0.867 N/m and Fad = 6 nN.

Simulation of either eqs. (5.5) or eqs. (5.9) gives similar results. Using eqs. (5.5)allows for modeling non-linear damping, while eqs. (5.9) gives a faster simulation.Figure 5.3 (a) and (b) show the simulations of eqs. (5.5) for the PWL interaction forcedepicted in fig. 5.2. The simulated cantilever motion is shown in green and the surfacedeformation in red. The inset shows a zoom of the surface deformation. The parame-ters of the simulation are given in the caption to fig. 5.3. When ks > kv, the cantileverpenetrates deep into the material and the surface motion is small. Yet the surface mo-tion is important for the interaction. The FQC are shown in (c) and (d). Note that, in


Haviland et al. [64] Paper II(DMT force) (PWL force)

ks 0.05 N/m 0.0469 N/mηs 2.4 · 10−7 kg/s 1.7 · 10−4 kg/skv 0.1 N/m 0.15 N/mηv 0.13 · 10−6 kg/s 3.9 · 10−6 kg/sFad 2 nN 2.86 nN

Table 5.1: Comparison of the DMT and PWL model parameters for measurements on apolycaprolactone sample.

comparison to the FQC shown in fig. 4.2, the surface and interaction damping gives aFQ, whereas FQ in fig. 4.2 (c) is zero.

An earlier publication by Haviland et al. [64] showed similar agreement betweensimulations and experiments. Instead of using our PWL model, they used a DMT forcefor the conservative part of the coupling force, with 2 additional free fit-parameters.We compare the simulated parameters from both of these models by measuring andanalyzing two similar polystyrene-polycaprolactone (PS-PCL) samples. The measure-ments where carried out a couple of years apart, with different AFM probes (both uses aTap300, but not the same cantilever). The reconstructed parameters for a PCL-domainare given in table 5.1. As ref [64] does not have a kv parameter, we used the approxi-mate linear slope of the DMT contract region. It is interesting to note that ks, kv , ηv andFad between the different measurements are reasonably close to one another. They arenot expected to be identical, as these linear force constants do depend on the tip shape.However, ηs is much larger in the latter measurements of ref [64]. This discrepancycould be explained by the fact that the sample has changed. The aged sample has adried surface that heals slower, i.e. larger τs = ηs/ks.

5.4 Optimization

Finding the best set of the 6 or 7 free parameters for the simulation to match the ex-periments can be very difficult and time consuming. By understanding how the modelparameters affect the FQC, a coarse estimate for all parameters are found by hand,giving simulated FQC with close resemblance to those measured. In practice, 5 (6)parameters are free as h can be determined by inspection. A numerical optimizationroutine is then used to fine-tune all parameters.

The optimization algorithm tries to minimize the residual, defined as

r =∑

n

Re�

F expn − F sim

n

�

+ Im�

F expn − F sim

n

�

, (5.11)

where the index n runs over frequency components of the force near resonance, for Fn

5.5. ADDITIONAL MODELS 37

above the noise. The force spectrum is

Fn = kG−1(n∆ω)dn. (5.12)

It is important to understand that the many free parameters give a very large pa-rameter space, which may have several local minima in addition to its global mini-mum. It is very difficult to know whether the current "good" set of parameters repre-sents the local or the global minimum. There exist global optimization routines, forexample the basin-hopping method (e.g. scipy.optimize.basinhopping [66, 33]), wherethe optimizer iteratively picks a new parameter set at random and solves a local min-imization problem for each parameter set. There are also brute force techniques (e.g.scipy.optimize.brute [33]) where the parameter space is split into a grid for which a localoptimization is performed at each point. As an example, our local optimization routinetakes around 60 sec for each optimization. The global optimizations techniques needthis time for each jump/grid point.

5.5 Additional models

We found that the PWL model explains experiments well for many different surfaces,both soft and stiff materials. Nevertheless, with the four-dimensional system describedby equation (5.5), it becomes very easy to replace the coupling force Fts to an arbitraryfunction. If the force is split into its conservative and dissipative parts, it can be writtenFts = Fcons. + Fdiss.. For the PWL model, the forces are

Fcons.(s) =

¨

0

−kvs− Fad

if s > 0,

if s ≤ 0,(5.13a)

Fdiss.(s) =

¨

0

−ηv s

if s > 0,

if s ≤ 0.(5.13b)

In the upcoming subsections, only the conservative force is modified while the dissipa-tive force is defined by eq. (5.13b). Plots of the different conservative forces are shownin fig. 5.4.

DMT

A popular model used in many AFM experiments is the DMT [20] model. The DMTforce is described by

Fcons.(s) = FDMT(s) =

¨

−HR6s2

−HR6s2 + 4

3 E∗p

R(a0 − s)3/2if s > a0,

if s ≤ a0,(5.14)

where H is the Hamaker constant, R the tip radius, E∗ is the elastic modulus and a0the intermolecular distance. Although the DMT model is widely used, it is not well-motivated and it requires four parameters for the conservative force alone. A plot of


0

10

a)

DMT

b)

Attractive

−10 0 10

0

10

c)

Smooth

−10 0 10

d)

Distance-dep.

Separation s [nm]

For

ce[n

N]

Figure 5.4: Different force models. The different force models discussed in section5.5.

the DMT force is shown in fig. 5.4 (a). As shown in Haviland et al. [64] the DMTmodel works equally well as the PWL model for the materials they studied. This can beexplained by a0 � |smax|, |smin|, in which case the coupling force turns on rapidly. ThePWL assumes an instantaneous turn-on of the coupling force.

Attractive PWL

It is easy to modify the PWL model to include a finite range of the adhesion force byadding an extra length scale, s0. The conservative force can be written

Fcons.(s) =

0

Fad

�

ss0− 1

�

−Fad − kvs

if s > s0,

if 0≤ s ≤ s0,

if s < 0.

(5.15)

This model is also piece-wise linear, and in the limit of s0→ 0, the standard PWL modeleq. (5.13) is regained. A plot of the force is shown in fig. 5.4 (b).

Smooth PWL

The attractive PWL model is also unphysical in that it is discontinuous in the forcegradient. The tip surface force in eq. (5.13) can be written with a step-function

Fts(s, s) = (1−H(s))(−kvs− Fad −ηv s), (5.16)


where H(s) is the Heaviside step function

H(s) =

¨

1

0

if s > 0,

if s ≤ 0.(5.17)

Replacing the discontinuous step function H(s) with a smooth step function, the tipsurface force will be continuous in s and a lot easier for the integrator to work with.There are many smooth-step functions, e.g. the logistic function, the error function,arctangent function and hyperbolic tangent. Any of these functions can be used toreplace the Heaviside function, at the expense of one extra free parameter. A plot ofthe smoothed PWL force is shown in fig. 5.4 (c) using a hyperbolic tangent tanh(s/s0)as smoothing function. In our experience, this smoothing makes little difference in theparameter values when s0� |smax|, |smin|.

Deformation-dependent PWL

In an attempt to describe very soft materials, such as hydrogels, we developed a modelwhere the adhesion force is not constant but assumed to be dependent on the surfacedeformation ds. A lifted surface (ds > 0) will form a contact with smaller area than anindented surface (ds < 0), see fig. 5.5. Assuming a linear relationship between ds andFad we have

Fad(ds) =

¨

0

Fad

�

dsd0− 1

�

if ds ≥ d0,

if ds < d0,(5.18)

where Fad is the magnitude of the adhesion force and d0 is the maximum lift of thesurface. A plot of the smoothed PWL force is shown in fig. 5.4 (d) for two differentsurface deformations ds. The solid line is for ds = 0 nm and the dotted has ds = 1 nm.

This force model gives slightly better agreement with experiments on a soft hydrogelcontact lens, but it is much more difficult to find suitable parameters and the numericaloptimizer struggles in comparison to the simple PWL model. Figure 5.6 shows a com-parison of the simulated FQC using eq. (5.13) [panels (a) - (d)] or eq. (5.18) [panels(e) - (h)] for the hydrogel. The general shape of the FQC is obtained with the simplePWL force, especially the dissipative force. However it cannot capture the observed dif-ference in adhesion force for increasing and decreasing amplitudes, as indicated by redcircles in (a) and (e). The coupling force eq. (5.18) gives better agreement to both theconservative and dissipative FQC. Although the simulated conservative force is closerto that measured, it cannot fully model the region for A> 70 nm.


contactaread > 0 d < 0s s

Figure 5.5: The adhesion force. The area of the tip covered by the surface is indicatedby the red cone. The contact area is small for a lifted surface ds > 0 (left side) andlarge for an indented surface ds < 0 (right side).


0.00

0.25

FI

[nN

] a)

0 25 50 75

Amplitude [nm]

−1

0

FQ

[nN

]

ExperimentSimulation

b)0 1 2

0

200

[nm

]

Tip Surfacec)

0 1 2

Time [ms]

0

10

[nm

] d)

Standard PWL

0.00

0.25

FI

[nN

] e)

0 25 50 75

Amplitude [nm]

−1

0

FQ

[nN

]

ExperimentSimulation

f)0 1 2

0

200

[nm

]

Tip Surfaceg)

0 1 2

Time [ms]

0

20

[nm

] h)

Deformation-dependent PWL

Figure 5.6: Simulations and experiments on a hydrogel surface. (a) and (b) showthe measured and simulated FQC on a hydrogel surface for the simple PWL forceeq. (5.13). (c) show the tip motion and surface motion, with an extended y-axis givenin (d). (e) and (f) show the measured and simulated FQC on a hydrogel surface forthe deformation-dependent PWL force eq. (5.18). (g) show the tip motion and sur-face motion, with an extended y-axis given in (h). Arrows indicate the direction of thehysteresis loops and the red circles in (a) and (e) show the amplitudes at which theadhesion is largest.


−0.4

−0.2

0.0

0.2

FI(A

)[n

N]

a)

Experiment

Simulated

Amplitude [nm]

0

20

40

60

[nm

]

b)

Cantilever

Surface

−2

0

2

[nm

]

c)

0 10 20 30

Amplitude [nm]

−0.4

−0.2

0.0

0.2

FI(A

)[n

N]

d)

0 2

Time [ms]

0

20

40

60

[nm

]

e)

0 2

Time [ms]

−2

0

2

[nm

]

f)

Figure 5.7: Hysteresis and no hysteresis. (a)-(c) show simulations without hysteresisbut with good agreement to the repulsive region of the measured conservative FQC. (d)-(f) show simulations with hysteresis but less good agreement to the repulsive region ofthe measured conservative FQC.

Further extensions

The different models given above can explain many soft materials, but not every in-teraction force observed in measurements. Especially FQCs with both hysteresis and ahard-contact repulsion are difficult to simulate. Figure 5.7 show measurements on thepolystyrene region of the PS-PCL sample described earlier, and simulations of the PWLmodel (only the conservative force quadrature is shown). The parameters used for (a)- (c) manages to capture the repulsive region of the conservative force, but cannot ex-plain the hysteresis. (d) - (f) show another set of parameters with hysteresis (not anexact match) but it does not capture the repulsive region. The parameters are given intable 5.2. The repulsive curve is obtained for a fast stiff surface (small ω0τs and largeks), while hysteresis requires a slow soft surface (large ω0τs and small ks). For the


h ω0τs ω0τv ks kv Fad

No hysteresis 26.0 nm 1.43 0.29 32.5 N/m 1.10 N/m 3.88 nNHysteresis 26.6 nm 17.8 0.29 0.15 N/m 2 N/m 4 nN

Table 5.2: Parameters of the PWL model, for simulations with and without hysteresis.The simulated conservative force quadrature are shown in fig. 5.7.

measured conservative FQC, both of these effects must occur simultaneously which isnot possible for the simple PWL model.

One possible solution to this problem is to introduce a multi-layered geometry, withseveral coupled moving-surfaces at different depths inside the material. However sucha system would increase the complexity of the model and require many additionalparameters and much longer simulation times. Another simpler solution could be tohave a deformation-dependent viscoelastic response of the surface, i.e. ks = ks(ds) andηs = ηs(ds), but we never tried this type of model.

Chapter 6

Intermodulation in liquid

MANY SAMPLES RELEVANT to biology or chemistry research require measurementsin an aqueous environment [24, 25, 67]. For bio-oriented research it is com-mon to use different types of buffer solutions since the specimen is preferably

kept in a surrounding as close as possible to its living-state. A typical measurement ona cell monitors its mechanical response to different types of drugs which are easy todeliver through the buffer solution.

In chemistry, an important research topic is corrosion. Corrosion can be studied indifferent ionic solutions where the electro-chemistry is degrading the material. Study-ing corrosion with an AFM [68] therefore requires the cantilever to be submerged inliquid. Furthermore, applying a voltage between the cantilever tip and sample, a lo-cal electro-chemical reaction can be pin-pointed with nano-meter precision. Anotherinteresting study measures the properties of highly viscous ionic liquids. It has beenshown how the ions form ordered layers close to the sample surfaces [69].

Each of the aforementioned applications requires understanding of the AFM’s be-havior in liquid. There are many difficulties regarding liquid measurements: from theset-up of the AFM probe and sample, to the cantilever calibration and finally the mea-surements. The aim of this chapter is to give some insight into the different steps ofAFM in liquid in general and ImAFM in particular – a background to Paper IV.

6.1 Setup

In our lab, three different AFM instruments are available: a Dimension Icon AFM and aFastScan AFM from Bruker Corp., and the NanoWizard from JPK instruments. Each ofthese AFM’s has the possibility to work in liquid environment, but the NanoWizard isdesigned and built for bio-applications and we use this instrument for the liquid mea-surements presented in this thesis. The cantilever holder for the NanoWizard consistsof a solid glass block, with two polished windows through which the laser is guided.The glass block directly enables liquid measurements, and no additional accessoriesare required for a basic aqueous experiment. There are two simple ways of submerg-

45

46 CHAPTER 6. INTERMODULATION IN LIQUID

Glass block

Liquid

Sample

a) b)

Shakerpiezo

Figure 6.1: Measurements in liquid. Not to scale. Two types of set-up that can be usedwhen working in liquid environments: (a) A droplet is placed between the cantileverholder (glass block) and sample. (b) shows how a large part of the cantilever holder isimmersed into e.g. a petri-dish containing the liquid. For both figures the characteristicglass block cantilever holder from JPK instruments is depicted.

ing the cantilever into the liquid: i) put a tiny µL-sized droplet covering the cantileverand parts of the sample area and ii) placing the sample in a liquid-filled petri-dish orbeaker and then carefully plunging the cantilever holder into the liquid. Both of theseoptions are depicted in figure 6.1.

The advantage with option i) is that only a small amount of liquid is required, andno container for the liquid is needed since the surface tension of the droplet is enoughto contain it. However the tiny droplet will evaporate, especially if the instrument iswarm, and might need to be refilled between measurements. Disturbances, from ei-ther external sources (i.e. 50 Hz pickup, mechanical vibrations or acoustic disturbancesfrom people walking by) or internal sources (i.e. the shaker piezo), can generate mo-tion of the droplet which couples to the cantilever and perturbs the measurement. Es-pecially periodic forces can cause serious problems since they can excite resonances ofthe droplet.

Using a petri-dish or equivalent as in ii) is beneficial since the measurements can beperformed directly in the storage vessel of the biological sample. The risk of drying thesample is negligible. Unfortunately the stiff walls imposes strong boundary conditionsgiving a high risk of resonances in the container.

The typical procedure when working in liquid is as follows: first the laser and de-tector are aligned in air and the cantilever is calibrated in air. Then the cantilever issubmerged in liquid using either i) or ii), and the laser is realigned (due to the differ-ent refractive index of air and liquid) and the cantilever is recalibrated (the damping

6.2. CALIBRATION 47

depends on the medium). It is not required to start in air before liquid, but the setupprocedure (especially aligning the detector and finding the cantilever resonance) tendsto be easier in air compared to liquid.

6.2 Calibration

260 265 270 275

10−12

10−11

10−10

[V2 A

D/H

z]

Stot

α2Scant + Sdet

Scant

a)

80 100 120 140 160 180

Frequency [kHz]

10−12

10−11

10−10

[V2 A

D/H

z]

b)

Figure 6.2: Noise measurements in air and liquid. Thermal noise measurements inair (a) and PBS buffer solution (b) for the same Tap300 cantilever. The total noiseStot( f ) is fit to eq. (3.3), where the cantilever motion Sdet( f ) is plotted with dashes.

The Sader method for cantilever calibration described in chapter 3.1 works in prin-ciple for arbitrary hydrodynamic surrounding. Equation (3.5) depends on the Reynoldsnumber, which depends on the fluids density ρ f and viscosity µ f . The density and vis-cosity for air and liquid (water at 20◦) are given in table 6.1.

To use the Sader method, the resonance frequency and quality factor of the can-tilever must be determined, where the preferred technique is to measure thermal noise.


Unit Air Liquid Descriptionρ f [kg/m3] 1.18 998.2 Fluid densityµ f [kg/(ms)] 1.86·10−5 0.001 Fluid viscosityf0 [kHz] 268 132 Cantilever resonance frequencyQ 515 7.9 Quality factork [N/m] 23.62 26.75 Spring constantα−1 [nm/V] 49.6 23.4 invOLRnf [fm/

pHz] 49 30 Detector noise floor

k·nf [fN/p

Hz] 1160 797 Detector noise equivalent forceSDC [fm/

pHz] 0.89 9.6 Thermal noise motion

k · SDC [fN/p

Hz] 20.97 257 Thermal noise force

Table 6.1: Comparison of calibration values for the same Tap300 cantilever, in air andliquid (values for water 20◦ are used).

Noise data for a Tap300 cantilever is given in fig. 6.2, for air in (a) and phosphate-buffered saline (PBS) in (b). The density and viscosity for PBS are very close to thoseof water. When the cantilever is submerged in PBS solution, the resonance frequencyis reduced by a factor of 2 and the quality factor is approximately 65 times smaller, asshown in tab. 6.1. Nevertheless the cantilever stiffness is calculated to around 25 N/mfor both environments.

As predicted by the fluctuation-dissipation theorem, the frequency-independentpower spectral density of fluctuations in force

Sthermal = 2kB Tη, (6.1)

depends on the ambient temperature T and damping η. The thermal noise force on thecantilever is therefore higher in liquid compared to in air, due to the larger damping η inliquid. The minimum detectable force is 12 times larger in liquid than in air. Increasedηmeans lower quality factor in liquid and the cantilever is less responsive. The deflectionnoise on resonance is therefore lower than that in air, such that it becomes the order ofthe detector noise, making thermal noise calibration more difficult.

6.3 Scanning in liquid

Measurements were carried out on a polymer blend sample of polystyrene–low-densitypolyethylene (PS-LDPE, see the cover art), in air and PBS buffer solution with a Tap300cantilever. In the LDPE domain, the interaction forces are analyzed with FQC, shown infigs. 6.3 (in air) and 6.4 (in liquid). Figures 6.3 and 6.4 illustrate an important featureof the liquid: there are strong background forces which have to be taken into accountwhen reconstructing the tip-surface FQC. Using the background force compensationtechnique (BG-compensation) described in Borgani et al. [51], any linear background

6.3. SCANNING IN LIQUID 49

0.0

0.6

1.2

FI

[nN

]

a)

5 10 15 20 25 30

Amplitude [nm]

−0.8

−0.4

0.0

FI

[nN

]

b)

BG comp.

No BG comp.

Figure 6.3: Force quadrature curves on LDPE in air. (a) show the conservative forcequadrature FI(A) and (b) the dissipative force quadrature FQ(A). The blue curves arebackground compensated while the orange curves are without background compensa-tion. The FQC are analyzed at the LDPE region of the sample.

force can be subtracted, regardless of its physical origin. The BG compensation in airis small for the conservative FQC, but significantly larger for the dissipative FQC as isshown in fig. 6.3. In Borgani et al. and Honig et al. [53], these additional dissipativeforces are accredited to so called squeeze film (or lubrication) effects. This can beexplained by an added mass effect on the cantilever.

In liquid (see fig. 6.4) the magnitude of the required BG compensation is muchlarger than in air, especially for the conservative force quadrature. The uncompensatedconservative FQC shows a dominantly attractive force, while the BG compensated forceis domainantly repulsive for all amplitudes.


−5

0

5

FI

[nN

]

a)

0 2 4 6 8 10 12 14

Amplitude [nm]

−10

−5

0

FI

[nN

]

b)

BG comp.

No BG comp.

Figure 6.4: Force quadrature curves on LDPE in liquid. (a) show the conservativeforce quadrature FI(A) and (b) the dissipative force quadrature FQ(A). The blue curvesare background compensated while the orange curves are without background com-pensation. The FQC are analyzed at the LDPE region of the sample.

6.4. HYDROGELS IN LIQUID 51

6.4 Hydrogels in liquid

The initial aim for developing ImAFM in liquid was for a collaboration with a researchgroup at Adolphe Merkle Institute in Fribourg, Switzerland. They are interested inthe properties of very soft hydrogels (E-modulus∼10 kPa) with magnetic iron nano-particles [26] for bio-applications. The hydrogel was supposed to be measured withImAFM and the response to be analyzed with the moving surface model described inchapter 5.

However, it turned out to be very difficult to find the liquid-hydrogel interface dur-ing approaches with ImAFM, since the change of the viscoelastic properties across theinterface is negligible and the sensitivity of the cantilever resonance in liquid is nothigh enough. One solution would be to use a colloidal probe attached to the free endof the cantilever, but colloidal probes are preferably used on soft cantilevers and softcantilevers are difficult to use with dynamic AFM techniques in liquid.

Since the liquid-hydrogel interface was difficult to find, we tried to take the hydrogelout of the buffer solution and measure the air-(still wet)hydrogel interface. Comparedto in liquid, the air-(still wet)hydrogel interface could be found and the feedback keptthe cantilever stable at this height, for some time. However after a few tens of seconds,the surface starts to wet the cantilever. After approximately one minute, the entire can-tilever and chip is covered by the hydrogel. This wetting makes even the air-hydrogelinterface too unstable to measure.

Chapter 7

Intermodulation friction force microscopy

INTERMODULATION FRICTION FORCE MICROSCOPY (ImFFM) is a new technique whichenables rapid friction measurements at the nanometer-scale, reaching tip velocitiesup to several cm/s. Using two drive tones, the response at many intermodulation

products are measured and analyzed (see chap. 4). The intermodulation response givesunique insight to the in-plane forces, such as friction or shear. Where standard mea-surement techniques only give either the average friction at some constant velocity, orresolve individual stick-slip events at very low velocities, ImAFM provides a means forcharacterization of the velocity-dependence of friction forces. In this chapter, the de-tails of ImFFM are described. Measurements on stiff materials are explained in termsof a modified Prandtl-Tomlinson model and soft materials are analyzed with a movingsurface-like model. The chapter covers Paper V.

7.1 Introduction

k η

m

x (t)

U(x)

s

x (t)t

v

Figure 7.1: The Prandtl-Tomlinson model. A mass m moves in a periodic potentialU(x). The mass is attached to a support via a spring k and damper η. The support ismoved with a velocity v.

The Prandtl-Tomlinson (PT) model [70], depicted in fig. 7.1, was proposed by Lud-

53

54 CHAPTER 7. INTERMODULATION FRICTION FORCE MICROSCOPY

wig Prandtl in 19281. It has long been used to model friction experiments [72, 73, 74].Even though it is a model derived long before the ability to measure nanometer-scaleinteractions, it is frequently used today to model AFM friction measurements. A tipwith mass m is attached to a spring k and a damper η, which are attached to a rigidsupport. The support is moved with a velocity v. The position of the support at time tis xs(t) = vt and the position of the tip is xt(t). The surface is modeled as a corrugatedpotential U(x) = U0 cos 2πx

x0, where U0 and x0 are the surface barrier and periodicity

respectively. The equation of motion for the tip is

mxt = −η xt − k(xs − xt)−dU(x)

dx. (7.1)

Introducing the resonance frequency ω0 =p

k/m and quality factor Q = k/(ω0η),eq. (7.1) becomes

1ω2

0

xt = −1ω0Q

xt − (xs − xt)−1k

dU(x)dx

. (7.2)

The PT model can explain so called stick-slip events, which are readily observed whenscanning slow across a surface. The stick-slip events occur when the tip mass becomesstuck to a minimum of the potential U(x). When the elastic energy in the spring hasincreased enough, the mass can slip out of one minimum and move to a neighboringminimum.

There are several ways to measure friction with an AFM, where the simplest is toslowly slide the AFM tip across the surface at some constant velocity and measure thecantilever twisting. If the torsional spring constant is known, the torque generatingthe measured twist can be calculated. So called friction loops are measured by firstscanning at constant velocity in one direction and then back again to the starting point.Such a measurement is illustrated in fig. 7.2 by simulation of eq. (7.2). Figure 7.2 (a)depicts the motion of the tip xt in blue and of the support xs in orange. The codeis given in appendix A.2 and the parameters in table 7.1. The support is moved at aconstant velocity for 4 ms after which it turns back to the starting position. When thesupport is moving, the tip is performing stick-slip motion due to the periodic potentialused to model the surface. The force on the support is shown in fig. 7.2 (b), wherearrows indicate the direction of the friction loop. The dashed lines indicate the meanforce on each branch, which is usually associated to the friction.

Similar to the dynamic techniques discussed in chapter 2, friction can be measuredby exciting the cantilever at one of its mechanical resonances. Dynamic techniquesgive higher sensitivity compared to quasi-static techniques [60], but three calibrationparameters are required in order to quantitatively reconstruct force. By exciting thetorsional resonance of the cantilever, the tip starts to oscillate in a twisting motion [75,76, 77]. If the twisting angle is small, only in-plane forces will be measured. It isalso possible to use the cantilever in a pendulum-like geometry, where the tip is placed

1The paper by Tomlinson [71], which historically is cited for the "Tomlinson model", does not containthe actual PT model used today [72]. It was proposed by Prandtl in 1928.

7.1. INTRODUCTION 55

0 1 2 3 4 5 6 7

Time [ms]

0

1

[nm

] a)

xt xs xt - xs

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Support position [nm]

−0.45

0.00

0.45

xt

-xs

[nm

]

b)

Figure 7.2: The Prandtl-Tomlinson model. (a) shows the motion of the support xs asa function of time (orange) and the induced stick-slip motion of the tip mass xt (blue).In (b) the force experienced by the support as a function of support position is shown,with the mean force shown as dashed lines.

Symbol Value Descriptionf0 =

ω02π 300 kHz Resonance frequency

Q 0.49 Quality factork 40 N/m Spring stiffnessv 400 nm/s Support velocityx0 0.1 nm Surface periodicity

2πU0x0

10 nN Amplitude of surface potential

Table 7.1: Parameters for the PT simulation shown in fig. 7.1.

along the principal axis of the cantilever beam [78]. Exciting the flexural mode of thecantilever will measure in-plane forces.


7.2 Calibration

To get quantitative force measurements, regardless of technique, the cantilever has tobe calibrated. Following the torsional calibration procedure described in section 3.2,the quality factor Qt and resonance frequencyω0t are determined from a thermal noisemeasurement. The invOLR α−1

t and the torsional stiffness κt are then calculated asdiscussed in sec. 3.2. The torsional stiffness κt relates an applied torque τ to the re-sulting twist angle φ according to eq. (3.11). However it is more convenient to relatethe measured photo-detector signal to the perpendicular, or in-plane force acting onthe tip. Therefore the stiffness to in-plane force

kt =κt

h2tip

(7.3)

is introduced, where htip is the tip height, typically 15-20 µm. Using this stiffness, theinvOLR value must be determined in terms of in-plane motion [nm/V].

7.3 ImFFM

Intermodulation friction force microscopy works on many commercially available AFMinstruments. Figure 7.3 show the connections required for ImFFM. For completenessfig. 7.3 shows both vertical and lateral detector signals, but for ImFFM it is enough withaccess to only the lateral signal. From the signal access module (SAM), the vertical andlateral detector signals are high-jacked and passed to a multifrequency lockin amplifier(MLA) [63]. Similar to ImAFM, the lateral detector signal is Fourier transformed anda feedback circuit keeps the amplitude of the first drive frequency at the setpoint. Thesetpoint is subtracted from the measured response amplitude to give the error signal.The error signal is fed back into the AFM controller (red arrow from OUT A (Feedback)→ Detector vertical), which is run in contact mode with setpoint = 0 V. The controllersPID feedback then adjusts the z-piezo to keep the error signal at zero. Note that we usethe vertical input port for the error signal, even though the true error signal is calculatedfrom the lateral detector signal!

The drive signal exciting the cantilever is generated from the MLA-software andpassed directly to the AFM head through the SAM. As with standard ImAFM, a two-tone drive generates a beating motion of the cantilever. With ImFFM, the drive tonesare placed close to the first torsional eigenmode instead of the flexural mode. The non-linear frictional forces generate intermodulation products when the cantilever engagesthe surface, as the two drive tones mix through the tip-surface interaction. By mea-suring several of these intermodulation products the non-linearity is reconstructed. Incontrast to ImAFM where normal forces are measured, ImFFM characterizes the forcein the plane of to the surface. The origin of these forces can be either friction or shearinteraction, depending on the material.

To excite the torsional eigenmode, it is best to use a non-standard cantilever actuatorwhich performs a rocking motion. Enhanced inertial coupling to the torsional mode

7.3. IMFFM 57

SAMAFM controllerDetector verticalDetector lateralCantilever drive

EOL triggerEOF trigger

MLAOUT 1

IN 1 (Vertical)IN 2 (Lateral)

OUT A (Feedback)TRIG IN 1 (EOL)TRIG IN 2 (EOF)

ETHERNET

AFM headDetector verticalDetector lateralCantilever drive

Intermodulation friction force microscopy

Figure 7.3: ImFFM setup. The setup schematics of ImFFM. The computer controls theAFM through the AFM software and controller. The signal access module (SAM) is usedto make connections to the multifrequency lockin amplifier (MLA).

+

-

piezo

cantilever base

Figure 7.4: The torsional cantilever holder. Schematic view of the torsional cantileverholder, not to scale. Two piezo-electric elements (brown) are driven out of sync witheach other to induce twisting of the cantilever base.


can be achieved by using two piezos side-by-side, driven in opposite direction to one-another. One such design is shown in fig. 7.4, where the left piezo expands when theright piezo retracts as indicated by the arrows. Oscillating the applied voltage inducesa twisting of the cantilever base, which inertially actuates the cantilever. Note that itusually is possible to induce torsional motion with a standard cantilever holder, due toresidual coupling. Since the high-Q torsional resonance only requires tiny base motionto get excited, very weak coupling is enough to excite the resonance. However suchexcitation typically requires a higher drive voltage compared to a torsional holder, andsome AFM controllers cannot provide sufficient voltage to generate a desired oscillationamplitude.

7.4 ImFFM on stiff materials

The modified Prantl-Tomlinson model

−1.0

−0.5

0.0

FI

[nN

]

a)

I II

0 1 2 3

Amplitude [nm]

−1.0

−0.5

0.0

FQ

[nN

]

b) I II

Figure 7.5: Measured force quadrature curves with ImFFM on graphite. Typicalforce quadrature curves on a stiff material. (a) show the conservative force quadraturecurve FI(A) and (b) show the dissipative force FQ(A). The regions of the FQC are labeledin orange (I) and green (II). In region I the tip is sticking to the surface and in regionII the tip is performing stick-slip motion.

Although the PT model works well to explain the type of measurement shown infig. 7.2, the simple PT could not explain ImFFM on graphite or mica. Examples of FQCmeasured on graphite are shown in fig. 7.5. It shows two distinct regions: a regionwhere the tip is sticking (at small amplitudes, marked in orange) which transitions intoan intermediate stick-slip phase and then finally into smooth sliding at high amplitude

7.4. IMFFM ON STIFF MATERIALS 59

(marked with green). The simple PT model can reproduce the stick-slip to smoothsliding regions of the measured FQC, but it cannot explain the sticking region at smallamplitude.

(a) (b)ti

p b

ase

Figure 7.6: Modified PT model. Drawing not to scale. Depicted in (a) is the cantilevertip with a flexible apex. (b) show the tip spring and damper together with the apexspring and damper. The coordinate of the apex is xt and the coordinate of the tip baseis xb. Figure reproduced from Paper V.

An extended PT-model has been proposed by Krylov et al. [79] and Tshiprut etal. [80], where the soft apex of the tip is modeled with an extra mass, spring anddamper. Figure 7.6 depicts the tip with its soft apex in (a) and the equivalent springand damper-system is illustrated in (b). The stiffness of the cantilever is K , Γ is itsdamping and M the effective mass. The tip base is rigidly connected to the cantileverbeam, and the soft tip apex is modeled as a single mode oscillator with stiffness, damp-ing and mass k, γ and m respectively. The tip base motion xb is measured by thedetector and the coordinate of the tip apex is xt. The tip apex is interacting with thecorrugated surface U . Note the difference between xb and xt: xb is measured, but theapex coordinate xt cannot be measured. Nevertheless its whereabouts is of great im-portance since the interaction to the surface is coupled through the apex coordinate,not the base coordinate.

The equations of motion for the system in fig. 7.6 are

M xb = −MΓ xb − K xb + Fd(t) + Fc(d, d), (7.4a)

mxt = −Fc(d, d) + Fsurf(xt, xt). (7.4b)


Unit Value Descriptionlength 1.42 Å Closest distance of neighbors in graphitemass 4.78·10−25 kg Mass of 24 carbon atoms

energy 4.85·10−20 J Thermal energy at room temperature

time 4.46·10−13 s time=r

mass·length2

energy

Table 7.2: The parameter scalings.

The drive force Fd(t) is the standard intermodulation drive at two frequencies close tothe torsional resonance. Connecting the motion of the tip base to the tip apex is donevia the coupling force

Fc(d, d) = kd +mγd (7.5)

which is a function of the tip deformation d = xt− xb and the velocity d = xt− xb. Thesurface corrugation

U(xt) = −U0 cos�

2πxt

a0

�

(7.6)

gives a non-linear conservative force

∂ U(xt)∂ xt

=2U0π

a0sin�

2πxt

a0

�

. (7.7)

Similar to the moving surface model, we add an interaction damping force −η xt. Tocomplete the model, the possibility of adding thermal noise is included, giving a totalsurface force

Fsurf(xt, xt) = −η xt −2U0π

a0sin�

2πxt

a0

�

+ Fnoise. (7.8)

For the PT simulations we found it important to use thermal noise, which changes theescape attempt rate of the tip apex out of the corrugated surface potential. The noise isa Gaussian random distributed force with a correlation time shorter than the shortestcharacteristic time scale of the system (typically the resonance frequency of the tipapex), but longer than the predicted internal time-steps taken by the integrator. If thecorrelation time is too short (shorter than the integrator time steps), the integratormight have problems converging to a solution. If the correlation time is too long, itwill not accurately model a random force. For the random number generation used toconstruct the noise, the Ziggurat method [81] is used.

Simulations

The equation system in eq. (7.4) is made dimensionless by rescaling time, length andmass. The scalings are given in table 7.2. Time is implicitly scaled through the energyscaling. Note that time is not scaled with the resonance frequency, as it was for themoving surface model.


The shortest of the two time scales of eq. (7.4), i.e. the resonance frequency of thecantileverω0 =

p

K/M and that of the tipωt =p

k/m, determines the size of the inte-grator time step. In experiments, ωt is estimated to be much larger than ω0, resultingin a very slow integration on a powerful PC. In our simulations we chooseωt ≈ 300ω0,which is a lot smaller than expected for experiments, but still large enough to success-fully reproduce measurement data. To further speed up the simulation, we simulatewith∆ f ≈ 15 kHz, whereas in experiments we use∆ f ≈ 500 Hz. In this case a typicalsimulation takes around 200 s when the equations are coded in C and integrated withCVODE [65].

−2

−1

0

FI

[nN

]

a)

I II

0.0 0.5 1.0 1.5 2.0 2.5

Amplitude [nm]

−2

−1

0

FQ

[nN

]

b) I II

Figure 7.7: Simulated force quadrature curves using the modified PT model. Sim-ulation of eq. (7.4), with parameters given in table 7.3. (a) depicts the conservativeforce quadrature FI(A) and (b) show the dissipative force FQ(A).

Simulations of eqs. (7.4) were performed in Paper V, together with a comparisonto experiments. The FQC in fig. 7.7 obtained by simulation show good agreementwith the experimental FQC in fig. 7.5. We use the simulated tip and apex motionto investigate in detail the dynamics of our friction model. The dynamics of the tipbase (cantilever) xb in fig. 7.8 (a), shows the typical beat envelope of rapid oscillatorymotion. More interesting is the dynamics of the tip apex xt, plotted in fig. 7.9, whichcannot be measured, but strongly affects the force on the cantilever. Figure 7.9 (a)shows the corresponding apex motion for a time 1/∆ f corresponding to one beat. Inthe beginning of the beat (orange region) the cantilever oscillates back and forth ina nearly perfect sine-motion, as shown in fig. 7.8 (b). At the same time, the tip apexmakes small excursions within one minimum of the corrugated potential as shown atthe beginning of fig. 7.9 (b). The tip is stuck but the cantilever is moving and deformingthe tip apex.

At t ≈ 0.12, the apex jumps to exactly a0 and then back to 0, after which it jumps


to −a0. This jumping of exactly one lattice spacing for small oscillations is the onset ofstick-slip. For larger drive amplitudes [green region in figs. 7.8 (a) and fig. 7.9(a)], thetip motion shows large jumps between many minima, making the transition to smoothsliding, as seen in fig. 7.9 (c). Small notches can be seen in fig. 7.9 (c), correspondingto the tip barely sticking during these large jumps. These notches are not seen for thecantilever motion in fig. 7.8 (c).

0.0 0.2 0.4 0.6 0.8 1.0−20

−10

0

10

20

a)

0.10 0.12 0.14

−2

0

2 b)

0.26 0.27 0.28

−5

0

5 c)

Time [1/∆f ]

Can

tile

ver

pos

itio

nx

b[a

0]

Figure 7.8: Simulations of the modified PT model. Simulation of eq. (7.4), withparameters given in table 7.3. (a) depicts the cantilever base position xb during onebeat period T = ∆ f . Two regions are marked with orange and green and they areexpanded in (b) and (c) respectively. The y-scale for each plot is in units of the latticespacing a0.

From the above discussion of the cantilever and tip motion, a stronger interpretationof the shape of the FQC measured by ImFFM can be given: for small amplitudes the tipapex is stuck in a surface minimum and the measured motion of the cantilever is due toflexing of the soft cantilever tip (region I in figs. 7.5 and 7.7). The linear slope of FI(A)for small amplitudes is therefore explained by tip flexing, and measurement of the tipstiffness can be made. Because the tip is stuck, or not moving very much (i.e. xb ≈ 0and x t ≈ 0), we may approximate the coupling force as Fc = −kAcos(ωt) in eq. (4.8).


0.0 0.2 0.4 0.6 0.8 1.0

−10

0

10a)

0.10 0.12 0.14

−1

0

1 b)

0.26 0.27 0.28

−5

0

5c)

Time [1/∆f ]

Tip

pos

itio

nx

t[a

0]

Figure 7.9: Simulations of the modified PT model. Simulation of eq. (7.4), withparameters given in table 7.3. (a) depicts the tip apex position xt during one beatperiod T =∆ f . Two regions are marked with orange and green and they are expandedin (b) and (c) respectively. The y-scale for each plot is in units of the lattice spacinga0.

Integrating this force over t = [0, 2πω ], we find

FI(A) =ω

2π

∫2πω

0

−kAcos(ωt)︸︷︷︸

Fc

cos(ωt)dt = −kA2

. (7.9)

Using equation (7.9) the simulated linear slope of FI is k =8.0 N/m or 3.32 in dimen-sionless units. This value is very close to the combined stiffness, i.e. ktot =

KkK+k ≈ 3.30

using the values given in tab. 7.3. A linear fit is shown in fig. 7.7 (a), marked with adashed line. Experiments give ktip ≈ 2 N/m, comparable to values measured by othergroups.


Symbol Value Descriptionf0 1.0×10−3 Cantilever resonance frequencyK 40 Cantilever stiffnessQ 500 Cantilever quality factor

m 2.5 Apex massk 3.6 Apex stiffnessγ 12 Apex damping

U0 2.16 PT barrier heighta0 1 PT barrier periodicityη 7.56 Interaction damping

Nnoise 50000 Number of noise samples*( fs)noise 0.249 Noise sampling frequency*σnoise 6.1 Sigma of the noise

Table 7.3: Dimensionless parameters for the modified PT simulation shown in figs. 7.8.(*An oversampling of 5 was used, i.e.. ( fs)noise = 5 · ( fs) where ( fs) is the samplingfrequency of the integrated output.)

7.5. IMFFM ON SOFT MATERIALS 65

7.5 ImFFM on soft materials

The force quadrature curves on stiff materials can be explained by the modified PTmodel, where the surface interaction is modeled by damped motion in a corrugatedpotential. This model works well for samples with an atomic lattice like graphite ormica, but breaks down when trying to explain measurements on softer viscoelasticmaterials like polymers. In this section, measurements on different polymer surfacesare shown and discussed. Figure 7.10 show examples of FQC on four soft materialsgiven in table 7.4: polystyrene, LDPE and two PDMS samples.

The FQC on polystyrene and LDPE show features similar to the FQC on graphite,depicted in fig. 7.5, but with an important difference: hysteresis between the up-beat(increasing A) and down-beat (decreasing A), as indicated by arrows in fig. 7.10. Inchapter 5 and Papers II and III, the hysteresis for standard ImAFM on polymers wasexplained by the finite relaxation time of the soft surface. The FQC shown in fig. 7.10are measured with ImFFM, where the cantilever tip is sweeping across the surface. Itis obvious that the MSM discussed earlier cannot be used directly to explain in-planeforces, but the hysteresis reveals that a finite relaxation time must be relevant. Unfor-tunately we didn’t manage to derive an interaction model which successfully describesthe ImFFM force quadrature curves on soft materials, but from our understanding ofthe basics of ImFFM measurements on stiff materials and the MSM, it is still possibleto make many interesting observations about dynamic friction measurements on softmaterials.

From simulations of the PT model, it was shown that the minimum of FI(A) cor-responds to the transition from sticking, to the onset of sliding across the sample. Infig. 7.5 (a) this transition occurs around 5 Å, which is close to the lattice parameter ofthe measured graphite (depending on the lattice orientation relative to the scan direc-tion). For the soft materials, the minimum occurs at higher amplitudes, around 2 nm(for PS) or 3 nm (for LDPE). In contrast to graphite, the sticking-region shows equallystrong dissipative force FQ(A) as conservative force FI(A). In the sticking-region ongraphite, FQ(A) was zero. This observation suggests that in the sticking region, the tipis submerged into the material and the material itself is deforming. Notice that for thePT model, the surface is assumed to be rigid and the tip apex (∼10 nm) is deforming,not the surface potential. For softer materials, the tip may penetrate into the material,as the viscoelastic material is not rigid and can flow around the tip.

Figure 7.11 visualizes how the tip sinks into the material. It is easy to realize fromfig. 7.11 that the force F needed to deform the flexible tip grows with penetration depths, which in turn gives an effectively stiffer tip. In reality the material is not infinitelystiff, giving the total "coupling stiffness" as a combination of the material and the tipstiffnesses i.e.

ktot =ktipkmat

ktip + kmat. (7.10)

For the PT model, the material is stiff and kmat� ktip⇒ ktot ≈ ktip. If the tip sinks intoa soft material, the tip becomes effectively stiffer, i.e. ktip� kmat⇒ ktot ≈ kmat.


−3

−2

−1

0

[nN

]

Polystyrene

a)

FI(A)

Polystyrene

b)

FQ(A)

0 2 4 6

−3

−2

−1

0

[nN

]

LDPE

c)

LDPE

d)

0 2 4 6

−3

−2

−1

0

[nN

]

PDMS 3.5MPa

e)

PDMS 3.5MPa

f)

0 2 4 6

Amplitude [nm]

−3

−2

−1

0

[nN

]

PDMS 2.5MPa

g)

0 2 4 6

Amplitude [nm]

PDMS 2.5MPa

h)

Figure 7.10: Measured FQC on soft materials. The left column of panels [(a), (c),(e) and (g)] show the conservative force quadrature curves for four different materials.The right column of panels [(b), (d), (f) and (h)] show the dissipative force quadraturecurves for the same four samples. Arrows indicate the direction of the hysteresis loop.

7.5. IMFFM ON SOFT MATERIALS 67

F

s Fmaterial

Figure 7.11: Tip stiffness. The force F required to deform the tip must grow withpenetration s, yielding an effectively stiffer tip.

The discussion above reveals an important difference between the MSM and themodified PT model: the MSM assumes that the tip is infinitely stiff and the hidden de-gree of freedom is in the surface, whereas the modified PT model assumes the surfaceis infinitely stiff and the hidden deformation happens to the tip apex. Since the mea-sured FQC on PS and LDPE in fig. 7.10 (a)-(d) show characteristic features from bothregimes, it is reasonable to assume that the desired model would reduce to the MSMor modified PT model in different limits.

The measurements on PDMS in fig. 7.10 (e)-(h) show a quantitatively differentshape compared to PS or LDPE. The FQC are always decreasing towards larger nega-tive force with no transition into smooth sliding. Very similar FQC were obtained anddiscussed in chapter 6, where a soft LDPE surface was submerged into buffer solu-tion and measured with standard ImAFM. From simulations of the MSM model it wasshown that the tip was always in contact with the surface. The two PDMS samples are30 respectively 40 times softer than LDPE and almost 1000 times softer than PS. It istherefore easy to suspect that something similar to the LDPE in liquid occurs, i.e. thecantilever tip sinks so deep into the PDMS, that there is never a transition into smoothsliding, even at the highest oscillation amplitude.

The low amplitude regions of fig. 7.10 (a), (c), (e) and (g) can now be used toroughly estimate the material stiffness of each sample, assuming an infinitely stiff tip.A dashed line marking the slope for each polymer is plotted and the stiffness is givenfor each sample in table 7.4. As expected, the stiffness decreases with elastic modulusbut the relative change in stiffness does not correspond to the relative change in elasticmodulus. The ratio of elastic modulus for PS to PDMS (2.5 MPa) is 1080, while theratio of stiffness between the two is just 2.3.


Material E-modulus StiffnessPolystyrene (PS) 2.7 GPa 3.9 N/mLow-density polyethylene (LDPE) 0.1 GPa 3.5 N/mPolydimethylsiloxane (PDMS) 3.5 MPa 1.9 N/mPolydimethylsiloxane (PDMS) 2.5 MPa 1.7 N/m

Table 7.4: The different polymer samples studied. E-modulus is the nominal bulk valuereported by the manufacturer (Bruker AFM Probes). The stiffness is calculated fromthe slope of the conservative force quadrature curve in fig. 7.10.

7.6 Models which do not explain the data

We tried to derive several different models, similar to the moving surface model de-scribed in chapter 5 but we never found a model which could explain the measuredFQC. As a starting point for further research, this section will give a short summary ofa few of these models and a discussion of why they didn’t work.

The modified PT model version 2

k η

m

x (t)

U(x)

s

x (t)t

F(t)

η k ss

ms

M

x (t)b

K Γ

F(t)

Figure 7.12: The modified PT model, version 2. Similar to the standard PT model,the cantilever tip interacts directly with a corrugated surface. However, the surface isnow free to move due to its viscoelastic coupling to the bulk material, through ks, ηsand ms. Components shown in red is a combination of the modified PT model and theviscoelastic corrugated surface.

7.6. MODELS WHICH DO NOT EXPLAIN THE DATA 69

As already discussed, the modified PT model was successfully used to describe fric-tion for stiff materials but it failed to model soft interactions. One possible reason towhy it didn’t work is that the extra degree of freedom is located in the tip, which theninteracts with the surface. For soft materials, a more probable design would be to movethe deformation from the tip to the surface, such that the tip is rigid and the surfacecorrugation undergoes damped spring motion, see fig. 7.12.

Although this model feels like the most promising description of a soft ImFFM mea-surement, we did not have time to fully investigate the parameter space. For the pa-rameters we tested it did not reproduce the measured data. A possible extension of thismodel would be to also include the flexible tip apex, i.e. both a flexible surface and tip,see red components in fig. 7.12. However such a model would be 6-dimensional, withat least 12 model parameters.

The side-wall model

k η

m

x (t)t

F(t)

η kss

ms

x (t)L x (t)R

ηk s s

ms

x=0

kc

Figure 7.13: The side-wall model. The cantilever is interacting with two moving side-walls, whose positions are xL(t) and xR(t). The walls are coupled to the bulk materialvia a spring ks and damper ηs, and they have an effective mass ms. The equilibriumposition of the cantilever and the two side walls is marked with a dashed line at x = 0.

The next candidate model we tried is a fundamentally different model, depicted infig. 7.13. When the cantilever tip interacts with the surface it digs out a tiny indent,which expands as the cantilever gains oscillation amplitude. The indent is modeled astwo side walls where the position of each side wall, relative to the equilibrium positionat x = 0, is free to move independently of the other. Their positions are xL(t) (leftwall) and xR(t) (right wall). If the motion of each side wall is modeled as a dampedharmonic oscillator, there are a total of 6 degrees of freedom (2 from the cantilever and


2+2 from the side walls). The equations of motion for this system are

mxt +mγ xt + kxt = Fc + Fd,

ms xL +msγs xL + ks xL = Fc,L,

ms xR+msγs xR+ ks xR = Fc,R,

(7.11)

where the side wall coupling forces are

Fc,L =

¨

−Fc

0

if xt ≤ xL,

if xt > xL,(7.12)

and

Fc,R =

¨

−Fc

0

if xt ≥ xR,

if xt < xR.(7.13)

The total coupling force is

Fc =

−kv(xt − xR)0

−kv(xt − xL)

if xt ≥ xR,

if xL < xt < xR,

if xt ≤ xL.

(7.14)

The model can be simplified by assuming a symmetric expansion of the indent, i.e.there is only one wall, which interacts with the cantilever twice per oscillation. Intro-ducing a single coordinate for the side walls, xs, such that xR = xs and xL = −xs. Theequations system (7.11) can be written

¨

mxt +mγ xt + kxt = Fc + Fd,

ms xs +msγs xs + ks xs = −sgn(xt)Fc,(7.15)

with sgn(x) being the sign operator. The coupling force is

Fc =

−kv(xt − xs)0

−kv(xt + xs)

if xt ≥ xs,

if − xs < xt < xs,

if xt ≤ xs.

(7.16)

This simplification reduces the equation system to two coupled harmonic oscillators.The full model can also be expanded by considering a coupling spring between the twoside walls, marked with red in fig. 7.13, such that when the left edge is moving, theright is pulled too.

Unfortunately this type of model does not give the right shape of FQC, as depictedin fig. 7.14, showing one simulation for a specific set of parameters. Although manyparameter combinations were tested, the model did not reproduce the shape of FQC onsoft materials. In some sense the model successfully explains the up-beat (increasingA) where the cantilever is facing a conservative elastic force when trying to widen the


0.0 0.2 0.4 0.6 0.8 1.0

Time

−20

0

20

[nm

]

a)

xt(t) xR(t) xL(t)

0 5 10 15

Amplitude [nm]

−0.1

0.0

FI

[nN

]

b)

0 5 10 15

Amplitude [nm]

−1

0

FQ

[nN

]c)

Figure 7.14: Simulations of the side-wall model. (a) depicts the motion of the can-tilever in blue, and the position of the two side-walls in orange and green. A zoomedregion at the dashed line is shown in the inset. (b) and (c) shows the reconstructedforce quadrature curves.

gap, giving the linear slope in FI(A). However during the down-beat (decreasing A)the cantilever experiences no interaction since it oscillates in the void between the twoslowly relaxing side walls – corresponding to the flat region with zero force. If therelaxation time of the side walls is decreased, such that they relax faster than the beatperiod T = 1/∆ f , a linear FI(A) is obtained for all amplitudes since the cantileveralways has to push the walls.

We also tried a slightly modified version where the side walls had non-linear springs,see fig 7.15. Smoothed versions of both forces were also tested, without success.

The MSM model for shear

In paper IV it was shown that if the surface is soft enough it always stays in contactwith the cantilever tip. The characteristic shape of these force quadrature curves aredepicted in fig. 5 (e) and (f) in paper IV, and they look similar to those measuredon the softest polymer surfaces with ImFFM. However, the standard MSM-model hasan anti-symmetric coupling force [see eq. (5.2)], i.e. different for s > 0 and s < 0.This force works well when the motion of the cantilever is perpendicular to the surface


a xL xL 0 xR a xR

Position [a.u.]

For

ce[a

.u.]

Not-saturating

Saturating

Figure 7.15: Different coupling forces. Two different versions of the coupling forcewhich were used. The blue curve corresponds to the force given in eq. (7.14), whilethe orange force saturates after some distance a · xL(R), where a is a free parameter.

and the cantilever undergoes intermittent contact with the surface. For shear motion,the tip does not break free from the surface. The force must be symmetric around theequilibrium position d = xb − xt = 0. Therefore the following coupling force is used

Fts(d, d) = −kvd −ηvd − Fad. (7.17)

Simulating the moving surface model described in chapter 5, but with the couplingforce in eq. (7.17) produces the FQC in fig. 7.16 (b) and (c). They do agree to someextent with measurements on the softest polymer surface (PDMS 2.5 MPa) – especiallythe dissipative force quadrature curve [fig. 7.16 (c)]. The conservative force [fig. 7.16(b)] agrees less well, where the simulation cannot explain the bending of the measuredFQC. The parameters used for the simulation are given in table 7.5.

For shear measurements on very soft materials, a model similar to the model de-scribed in this section is probably on the right track. But this type of model will notbe able to explain the semi-soft materials, such as PS and LDPE without further modi-fications. Comparing the measurements and simulations depicted in fig. 7.16 (b), oneinterpretation of the bending of FI is a change in surface or material stiffness as A in-creases (in this case a reduction of stiffness). A way to account for this is to introducea non-linear coupling force, or a deformation-dependent stiffness, i.e. ks/v = ks/v(d, d)and ηs/v = ks/v(d, d). Depending on the d and d dependences, many additional freeparameters might be required, and further research is needed.


0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

Time [ms]

−5

0

5

[nm

]

a)

d(t) ds(t)

0 2 4 6

Amplitude [nm]

−5.0

−2.5

0.0

FI

[nN

]

Exp.

Sim.

b)

0 2 4 6

Amplitude [nm]

−2

0

FQ

[nN

]

Exp.

Sim.

c)

Figure 7.16: Simulation of a moving surface-like model. (a) show the simulatedmotion of the cantilever (green) and surface (red). The inset is a zoom at the blackdashed line. (b) and (c) show a comparison of experimental (blue)and simulated (or-ange) force quadrature curves. The experimental force curves are measured on PDMS2.5 MPa, and the simulation is performed with a modified PWL force model.

Symbol Value Descriptionω0τs 0.2 Quality factor for surface dampingω0τv 2.7 Quality factor for interaction damping

ks 2.9 N/m Stiffness of surfacekv 3.5 N/m Stiffness of materialFad 0 nN Adhesion forceh 0 nm Working height*

Table 7.5: Value of parameters for the simulation shown in fig. 7.16. (* In this modifiedmodel, the "working height" corresponds to a lateral displacement, which is set to zero.)

Chapter 8

Conclusions and outlook

CHARACTERIZING surfaces with atomic force microscopy (AFM) is a formidableproblem, and despite extensive research during the last decades, much is stillto be done. In this thesis we use multifrequency AFM for such characterizations,

measuring both normal and in-plane forces on many different materials, and we explainthe measurements with numerical simulations of simple models. The thesis startedfrom the previous work on ImAFM by Daniel Platz [82] and Daniel Forchheimer [83].We showed how their theoretical framework could be applied to explain soft materialsand we further developed it to measure in-plane forces, such as friction and shear,on both soft and stiff surfaces. The much higher tip velocities that can be reachedwhen measuring in-plane forces (up to 5 orders of magnitude higher than standardAFM friction measurements) enables study of e.g. friction at the nanometer scale, withsliding velocities close to those relevant to applications at the meter scale.

Quantitative force measurements with an AFM requires a well-calibrated force trans-ducer. There are different techniques available for flexural and torsional cantilever cal-ibration, but many of them require a priori calibration of e.g. the AFM scanner andthey rely on several assumptions. Using only beam theory, we were able to use a flexu-ral cantilever calibration to calibrate the torsional eigenmode by a single measurementof the torsional resonance frequency. Our method enables torsional calibration of stiffcantilevers, where measuring the cantilevers response to thermal noise cannot readilybe performed.

Even though stiff cantilevers are difficult to calibrate, they are preferred for mea-suring in-plane forces with intermodulation friction force microscopy (ImFFM), dueto their higher flexural stiffness. A high flexural stiffness is necessary to keep the tipposition stable during torsional oscillation, by avoiding jump-to-contact. Using a stiffcantilever, we showed measurements of in-plane forces at tip velocities reaching cm/son mica and graphite. Their atomic periodicity shows a transition from sticking of thetip to smooth sliding. With numerical simulations of an extended Prandtl-Tomlinsonmodel, we were able to explain the measured force quadrature curves.

A similar stick-slip behavior was seen for stiff polymer surfaces, together with hys-

75

76 CHAPTER 8. CONCLUSIONS AND OUTLOOK

teresis. This hysteresis reveals that a slow relaxation of the surface is relevant. Mea-suring on even softer polymers, this slow surface relaxation becomes more importantand analysis of the interaction force suggests that the tip is always in contact with thematerial. In other words the in-plane force should be interpreted as a shear interactionfor the softest materials. Although we cannot provide a theory to explain shear mea-surements on the softest viscoelastic surfaces, we anticipate that many new interestinginsights can be drawn if such a theory is found.

Soft materials can also be measured using intermodulation AFM (ImAFM). Weshowed a simple, yet powerful, piece-wise linear model and used it to explain measure-ments on different viscoelastic surfaces. The piece-wise linear model is phenomeno-logical in the sense that it cannot be derived from first principles, but it captures theessential features of many soft surface interactions. We believe that this simple modelgives basic physical understanding of dynamic measurements on soft materials at thenanometer scale, and it emphasizes the importance of using a dynamic technique whenmeasuring viscoelastic materials.

We hope that the work presented herein will contribute to our understanding ofnanometer scale mechanical response of materials for applications in many disciplinesof science, from material research to medicine.

Acknowledgements

The journey towards my Ph. D. started 2013 with a master thesis project in the group ofProf. David Haviland. Not being an experienced researcher, Davids guidance quicklyhelped me onto the right track and I started to develop the tools necessary to finishmy master project. Towards the end of my project, David said he had an open Ph.D.position, which was a perfect continuation to my master thesis. I applied for it and I amvery happy that you accepted me as a student for the second time. I am still amazedby how much you have taught me during these 5 years!

I would also like to thank my former and current colleagues. A special mentiongoes to Riccardo Borgani – we have shared office for 4 years, and several hotel rooms atconferences around the world. I really enjoyed our many discussions and have alwayshighly valued your constructive input. I am very glad that both you and me will stay inStockholm for at least a few more years to come, and I hope our badminton-Fridays willcontinue in the fall! Hopefully we will plan and carry out a new mountain hike? DanielForchheimer, you have been an enormous knowledge bank and I appreciate that youstayed in the group. Astrid de Wijn, although you left for your position in Norway, yourhelp with nano-tribology were very valuable to me to get a good start with my project,and I am very happy how our friction paper turned out. Illia Dobryden, thank you forour many hours in the lab and I hope you will enjoy your new apartment. FedericaCrippa, your short stay here was very intense and productive, and I hope you will showme the Alps someday.

Daniel Platz, you have always given important and useful feedback and Erik Tholénhas always been very quick implementing new soft- and hardware features required forthe project. Shan Jolin and Erik Holmgren, I wish you luck with your Ph.D. studies; Ihope Thomas Weißl’s and Simo Sah’s academic carriers will be fruitful and successive;Vladislav Korenivski – thank you for your many advices, regarding both scientific andless-scientific topics. I would also like to thank Björn Koop who shares my love forleverpastej and Jan Hoh who moved back to the U.S.

This thesis would not have been possible without the help of Matthew Fieldenand Anders Liljenborg. As lab managers, they untiredly made sure all lab equipmentworked. If something broke they did their best to solve it as soon as possible.

To my many friends: thank you all for your friendship! Especially Alexander Fors-man, who has been my right-hand man since the second year during our undergraduatestudies at KTH. We share so many nice memories and I really hope the future has more

77

78 CHAPTER 8. CONCLUSIONS AND OUTLOOK

to give, where your wedding is the next thing on the agenda. My neighbor and bad-minton partner Ignacio Gallego Marcos and our other badminton partner Joshua HautDen Stein, good luck with your Ph.D.s and Ignacio: lets keep up with our Sunday walks!I would also like to thank all my new friends at SSD for a very necessary break fromscience.

Last but not least, I would like to acknowledge my family. Mamma och pappa, Malinoch Viktor, thank you for your never-ending support during my years of study. Withoutyour guidance while growing up I would not have gotten to this point in life and Iappreciate it more than I possibly can write here. I have enjoyed all of your weekendvisits to Stockholm and I am very happy about the enormous help with finding my newapartment!

Stockholm, May 2018Per-Anders Thorén

Bibliography

[1] A. Stinner, The story of force: from Aristotle to Einstein, Physics Education 29(2),77 (1994).

[2] J. N. Israelachvili, 1 - Historical Perspective, in J. N. Israelachvili (ed.), Intermolec-ular and Surface Forces (Third Edition), pp. 3 – 22, Academic Press, Boston, thirdedition ed. (2011).

[3] G. Galilei, Dialogue Concerning the Two Chief World Systems – Ptolemaic and Coper-nican, translated by Stillman Drake, University of California Press (1967).

[4] D. Dowson, History of Tribology, Longman (1979).

[5] A. Fall, B. Weber, M. Pakpour, N. Lenoir, N. Shahidzadeh, J. Fiscina, C. Wagner,and D. Bonn, Sliding friction on wet and dry sand, Physical Review Letters 112(17),3 (2014).

[6] V. L. Popov and J. A. Gray, Prandtl-Tomlinson model: History and applications infriction, plasticity, and nanotechnologies, ZAMM Zeitschrift fur Angewandte Math-ematik und Mechanik 92(9), 683 (2012).

[7] K. Holmberg and A. Erdemir, Influence of tribology on global energy consumption,costs and emissions, Friction 5(3), 263 (2017).

[8] VTT Technical Research Centre of Finland. (2012, January 12). One-third of car fuelconsumption is due to friction loss. ScienceDaily. Retrieved November 12, 2017 fromwww.sciencedaily.com/releases/2012/01/120112095853.htm.

[9] G. Binnig, F. C. Quate, and C. Gerber, Atomic Force Microscope 56(9) (1986).

[10] S. Jesse, S. V. Kalinin, R. Proksch, a. P. Baddorf, and B. J. Rodriguez, The band exci-tation method in scanning probe microscopy for rapid mapping of energy dissipationon the nanoscale, Nanotechnology 18(43), 435503 (2007).

[11] D. Platz, E. A. Tholén, D. Pesen, and D. B. Haviland, Intermodulation atomic forcemicroscopy, Applied Physics Letters 92(15), 153106 (2008).

79

80 BIBLIOGRAPHY

[12] S. C. Minne, J. D. Adams, G. Yaralioglu, S. R. Manalis, A. Atalar, and C. F. Quate,Centimeter scale atomic force microscope imaging and lithography, Applied PhysicsLetters 73(12), 1742 (1998).

[13] R. Garcia, A. W. Knoll, and E. Riedo, Advanced scanning probe lithography, NatureNanotechnology 9(8), 577 (2014).

[14] R. V. Martínez, N. S. Losilla, J. Martínez, Y. Huttel, and R. Garcia, Patterning poly-meric structures with 2 nm resolution at 3 nm half pitch in ambient conditions, NanoLetters 7(7), 1846 (2007).

[15] Q. S. Li, G. Y. H. Lee, C. N. Ong, and C. T. Lim, AFM indentation study of breastcancer cells, Biochemical and Biophysical Research Communications 374(4), 609(2008).

[16] M. Lekka, Discrimination Between Normal and Cancerous Cells Using AFM, Bio-NanoScience 6(1), 65 (2016).

[17] O. Custance, R. Perez, and S. Morita, Atomic force microscopy as a tool for atommanipulation, Nature Nanotechnology 4(12), 803 (2009).

[18] F. J. Giessibl, Atomic Resolution of the Silicon (111)-(7x7) Surface by Atomic ForceMicroscopy, Science 267(5194), 68 (1995).

[19] J. L. Hutter and J. Bechhoefer, Calibration of Atomic-Force Microscope Tips, Reviewof Scientific Instruments 64(7), 1868 (1993).

[20] J. Melcher, D. Kiracofe, S. Hu, S. Johnson, and A. Raman, Why VEDA ?, School ofMechanical Engineering and Birck Nanotechnology Center I . 47907 (2012).

[21] N. F. Martinez, S. Patil, J. R. Lozano, and R. Garcia, Enhanced compositional sen-sitivity in atomic force microscopy by the excitation of the first two flexural modes,Applied Physics Letters 89(15), 1 (2006).

[22] B. J. Rodriguez, C. Callahan, S. V. Kalinin, and R. Proksch, Dual-frequencyresonance-tracking atomic force microscopy, Nanotechnology 18(47) (2007).

[23] D. Forchheimer, S. S. Borysov, D. Platz, and D. B. Haviland, Determining surfaceproperties with bimodal and multimodal AFM., Nanotechnology 25(48), 485708(2014).

[24] P. K. Hansma, J. P. Cleveland, M. Radmacher, D. A. Walters, P. E. Hillner,M. Bezanilla, M. Fritz, D. Vie, H. G. Hansma, C. B. Prater, J. Massie, L. Fuku-naga, J. Gurley, and V. Elings, Tapping mode atomic force microscopy in liquids,Applied Physics Letters 64(13), 1738 (1994).

[25] C. A. J. Putman, K. O. Van Der Werf, B. G. De Grooth, N. F. Van Hulst, and J. Greve,Tapping mode atomic force microscopy in liquid, Applied Physics Letters 64(18),2454 (1994).

BIBLIOGRAPHY 81

[26] F. Crippa, T. L. Moore, M. Mortato, C. Geers, L. Haeni, A. M. Hirt, B. Rothen-Rutishauser, and A. Petri-Fink, Dynamic and biocompatible thermo-responsive mag-netic hydrogels that respond to an alternating magnetic field, Journal of Magnetismand Magnetic Materials 427(June 2016), 212 (2017).

[27] S. Hosaka, A. Kikukawa, Y. Honda, H. Koyanagi, and S. Tanaka, Simultaneousobservation of 3-dimensional magnetic stray field and surface structure using newforce microscope, Japanese Journal of Applied Physics 31(7), L904 (1992).

[28] D. Forchheimer, D. Platz, E. A. Tholén, and D. B. Haviland, Simultaneous imagingof surface and magnetic forces, Applied Physics Letters 103(1) (2013).

[29] M. Nonnenmacher, M. P. O’Boyle, and H. K. Wickramasinghe, Kelvin probe forcemicroscopy, Applied Physics Letters 58(25), 2921 (1991).

[30] R. Borgani, D. Forchheimer, J. Bergqvist, P.-A. Thorén, O. Inganäs, and D. B.Haviland, Intermodulation electrostatic force microscopy for imaging surface photo-voltage, Applied Physics Letters 105(14), 143113 (2014).

[31] B. Ohler, Cantilever spring constant calibration using laser Doppler vibrometry, Re-view of Scientific Instruments 78(6) (2007).

[32] J. Brugger, R. A. Buser, and N. F. De Rooij, Micromachined atomic force microprobewith integrated capacitive read-out, Journal of Micromechanics and Microengi-neering 2(3), 218 (1992).

[33] E. Jones, T. Oliphant, P. Peterson et al., SciPy: Open source scientific tools for Python(2001–).

[34] J. E. Sader, J. A. Sanelli, B. D. Adamson, J. P. Monty, X. Wei, S. a. Crawford,J. R. Friend, I. Marusic, P. Mulvaney, and E. J. Bieske, Spring constant calibrationof atomic force microscope cantilevers of arbitrary shape., The Review of scientificinstruments 83(10), 103705 (2012).

[35] M. L. B. Palacio and B. Bhushan, Normal and Lateral Force Calibration Techniquesfor AFM Cantilevers, Critical Reviews in Solid State and Materials Sciences 35(2),73 (2010).

[36] J. E. Sader, Frequency response of cantilever beams immersed in viscous fluids withapplications to the atomic force microscope, Journal of Applied Physics 84(1), 64(1998).

[37] J. E. Sader, J. W. M. Chon, and P. Mulvaney, Calibration of rectangular atomic forcemicroscope cantilevers, Review of Scientific Instruments 70(10), 3967 (1999).

[38] J. E. Sader, R. Borgani, C. T. Gibson, D. B. Haviland, M. J. Higgins, J. I. Kil-patrick, J. Lu, P. Mulvaney, C. J. Shearer, A. D. Slattery, P.-A. Thorén, J. Tran,

82 BIBLIOGRAPHY

H. Zhang, H. Zhang, and T. Zheng, A virtual instrument to standardise the calibra-tion of atomic force microscope cantilevers, Review of Scientific Instruments 87(9)(2016).

[39] J. E. Sader, SaderGCI: https://sadermethod.org/ (2015–).

[40] M. J. Higgins, R. Proksch, J. E. Sader, M. Polcik, S. Mc Endoo, J. P. Cleveland, andS. P. Jarvis, Noninvasive determination of optical lever sensitivity in atomic forcemicroscopy, Review of Scientific Instruments 77(1), 013701 (2006).

[41] R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics,Volume I, Addison-Wesley (1963).

[42] J. Stiernstedt, M. W. Rutland, and P. Attard, A novel technique for the in situ cal-ibration and measurement of friction with the atomic force microscope, Review ofScientific Instruments 76(8), 083710 (2005).

[43] D. F. Ogletree, R. W. Carpick, and M. Salmeron, Calibration of frictional forces inatomic force microscopy, Review of Scientific Instruments 67(9), 3298 (1996).

[44] M. Munz, Force calibration in lateral force microscopy: a review of the experimentalmethods, Journal of Physics D: Applied Physics 43(6), 063001 (2010).

[45] R. Alvarez-Asencio, E. Thormann, and M. W. Rutland, Note: Determination oftorsional spring constant of atomic force microscopy cantilevers: Combining normalspring constant and classical beam theory., The Review of scientific instruments84(9), 096102 (2013).

[46] C. M. Mate, G. M. McClelland, R. Erlandsson, and S. Chiang, Atomic-scale fric-tion of a tungsten tip on a graphite surface, Physical Review Letters 59(17), 1942(1987).

[47] A. Peled, J. Castro, and W. J. Weiss, Atomic force and lateral force microscopy (AFMand LFM) examinations of cement and cement hydration products, Cement andConcrete Composites 36(1), 48 (2013).

[48] D. Dietzel, U. D. Schwarz, and A. Schirmeisen, Nanotribological studies usingnanoparticle manipulation: Principles and application to structural lubricity, Fric-tion 2(2), 114 (2014).

[49] C. P. Green and J. E. Sader, Torsional frequency response of cantilever beams im-mersed in viscous fluids with applications to the atomic force microscope, Journal ofApplied Physics 92(10), 6262 (2002).

[50] C. P. Green, H. Lioe, J. P. Cleveland, R. Proksch, P. Mulvaney, and J. E. Sader, Nor-mal and torsional spring constants of atomic force microscope cantilevers, Reviewof Scientific Instruments 75(6), 1988 (2004).

BIBLIOGRAPHY 83

[51] R. Borgani, P.-A. Thorén, D. Forchheimer, I. Dobryden, S. M. Sah, P. M. Claes-son, and D. B. Haviland, Background-Force Compensation in Dynamic Atomic ForceMicroscopy, Physical Review Applied 7(6), 1 (2017).

[52] W. M. Zhang, G. Meng, J. B. Zhou, and J. Y. Chen, Nonlinear dynamics and chaosof microcantilever-based TM-AFMs with squeeze film damping effects, Sensors 9(5),3854 (2009).

[53] C. D. F. Honig, J. E. Sader, P. Mulvaney, and W. A. Ducker, Lubrication forces inair and accommodation coefficient measured by a thermal damping method usingan atomic force microscope, Physical Review E - Statistical, Nonlinear, and SoftMatter Physics 81(5), 1 (2010).

[54] J. E. Sader, R. Borgani, C. T. Gibson, D. B. Haviland, M. J. Higgins, J. I. Kil-patrick, J. Lu, P. Mulvaney, C. J. Shearer, A. D. Slattery, P.-A. Thorén, J. Tran,H. Zhang, H. Zhang, and T. Zheng, A virtual instrument to standardise the calibra-tion of atomic force microscope cantilevers, Review of Scientific Instruments 87(9)(2016).

[55] E. A. Vopilkin, A. E. Parafin, and A. N. Reznik, Intermodulation in a MicrowaveResonator with a High-Temperature Superconductor, Technical Physics 45(2), 214(2000).

[56] E. A. Tholén, A. Ergül, K. Stannigel, C. Hutter, and D. B. Haviland, Parametric am-plification with weak-link nonlinearity in superconducting microresonators, PhysicaScripta T T137 (2009).

[57] D. Platz, D. Forchheimer, E. A. Tholén, and D. B. Haviland, Polynomial force ap-proximations and multifrequency atomic force microscopy, Beilstein Journal of Nan-otechnology 4, 352 (2013).

[58] D. Platz, D. Forchheimer, E. A. Tholén, and D. B. Haviland, Interaction imagingwith amplitude-dependence force spectroscopy., Nature communications 4, 1360(2013).

[59] D. Forchheimer, D. Platz, E. A. Tholén, and D. B. Haviland, Model-based extractionof material properties in multifrequency atomic force microscopy, Physical ReviewB 85(19), 195449 (2012).

[60] P.-A. Thorén, A. S. de Wijn, R. Borgani, D. Forchheimer, and D. B. Haviland, Imag-ing high-speed friction at the nanometer scale, Nature Communications 7, 13836(2016).

[61] D. B. Haviland, Quantitative force microscopy from a dynamic point of view, CurrentOpinion in Colloid and Interface Science 27, 74 (2017).

[62] J. W. Cooley and J. W. Tukey, An algorithm for the machine calculation of complexFourier series., Mathematics of Computation 19, 297 (1965).

84 BIBLIOGRAPHY

[63] http://www.intermodulation-products.com/.

[64] D. B. Haviland, C. A. van Eysden, D. Forchheimer, D. Platz, H. G. Kassa, andP. Leclère, Probing viscoelastic response of soft material surfaces at the nanoscale.,Soft matter 12(2), 619 (2016).

[65] A. C. Hindmarsh, P. N. Brown, K. E. Grant, S. L. Lee, R. Serban, D. E. Shumaker,and C. S. Woodward, SUNDIALS: Suite of nonlinear and differential/algebraic equa-tion solvers, ACM Transactions on Mathematical Software (TOMS) 31(3), 363(2005).

[66] D. J. Wales and J. P. Doye, Global optimization by basin-hopping and the lowestenergy structures of Lennard-Jones clusters containing up to 110 atoms, Journal ofPhysical Chemistry A 101(28), 5111 (1997).

[67] M. Radmacher, M. Fritz, and P. K. Hansma, Imaging soft samples with the atomicforce microscope: gelatin in water and propanol, Biophysical Journal 69(1), 264(1995).

[68] G. Bertrand, E. Rocca, C. Savall, C. Rapin, J. C. Labrune, and P. Steinmetz, In-situelectrochemical atomic force microscopy studies of aqueous corrosion and inhibitionof copper, Journal of Electroanalytical Chemistry 489(1), 38 (2000).

[69] J. M. Black, M. Zhu, P. Zhang, R. R. Unocic, D. Guo, M. B. Okatan, S. Dai, P. T.Cummings, S. V. Kalinin, G. Feng, and N. Balke, Fundamental aspects of electricdouble layer force-distance measurements at liquid-solid interfaces using atomic forcemicroscopy, Scientific Reports 6(August), 1 (2016).

[70] L. Prandtl, Ein Gedankenmodell zur kinetischen Theorie der festen Körper, ZAMM- Journal of Applied Mathematics and Mechanics / Zeitschrift fur AngewandteMathematik und Mechanik 8(2), 85 (1928).

[71] G. T. B.Sc., CVI. A molecular theory of friction, The London, Edinburgh, and DublinPhilosophical Magazine and Journal of Science 7(46), 905 (1929).

[72] V. L. Popov, The Prandtl-Tomlinson Model for Dry Friction, pp. 173–192, SpringerBerlin Heidelberg, Berlin, Heidelberg (2017).

[73] A. Vanossi, N. Manini, M. Urbakh, S. Zapperi, and E. Tosatti, Colloquium: Model-ing friction: From nanoscale to mesoscale, Reviews of Modern Physics 85(2), 529(2013).

[74] A. Socoliuc, R. Bennewitz, E. Gnecco, and E. Meyer, Transition from stick-slip tocontinuous sliding in atomic friction: Entering a new regime of ultralow friction,Phys.˜Rev.˜Lett. 92, 134301 (2004).

[75] M. Reinstädtler, U. Rabe, V. Scherer, U. Hartmann, a. Goldade, B. Bhushan, andW. Arnold, On the nanoscale measurement of friction using atomic-force microscopecantilever torsional resonances, Applied Physics Letters 82(16), 2604 (2003).

BIBLIOGRAPHY 85

[76] B. Bhushan and T. Kasai, A surface topography-independent friction measurementtechnique using torsional resonance mode in an AFM, Nanotechnology 15(8), 923(2004).

[77] A. Yurtsever, A. M. Gigler, and R. W. Stark, Amplitude and frequency modulationtorsional resonance mode atomic force microscopy of a mineral surface., Ultrami-croscopy 109(3), 275 (2009).

[78] U. Gysin, S. Rast, M. Kisiel, C. Werle, and E. Meyer, Low temperature ultrahighvacuum noncontact atomic force microscope in the pendulum geometry., Review ofScientific Instruments 82(2), 023705 (2011).

[79] S. Y. Krylov, J. a. Dijksman, W. a. Van Loo, and J. W. M. Frenken, Stick-slip motionin spite of a slippery contact: Do we get what we see in atomic friction?, PhysicalReview Letters 97(16), 2 (2006).

[80] Z. Tshiprut, a. E. Filippov, and M. Urbakh, Effect of tip flexibility on stick–slip motionin friction force microscopy experiments, Journal of Physics: Condensed Matter20(35), 354002 (2008).

[81] G. Marsaglia and W. W. Tsang, The Ziggurat method for generating random vari-ables, Journal of Statistical Software 5(8), 1 (2000).

[82] D. Platz, Reconstructing force from harmonic motion, Ph.D. thesis, KTH, Nanos-tructure Physics (2013), qC 20130527.

[83] D. Forchheimer, Imaging materials with intermodulation : Studies in multifre-quency atomic force microscopy, Ph.D. thesis, KTH, Nanostructure Physics (2015),qC 20150209.

Appendix A

Code listings

import numpy as npfrom s c i py . i n t e g r a t e import odeintimport matp lo t l i b . pyp lo t as p l t

# De f in e the equat ion o f motiondef c a n t i l e v e r (y , t , w0, Q, k , a1 , a2 , w1, w2, p1 , p2 , ks , fad , h) :

d , dd = yFd = a1 np . cos (w1 * t + p1) + a2 * np . cos (w2 * t + p2)s = h + dFt s = np . p iecewise ( s , [ s < 0 , s >= 0] , [ lambda s : −ks s − fad , 0])dydt = [dd , w0 * w0 * (( Fd + F t s ) / k − dd / w0 / Q − d) ]return dydt

fsamp = 10e6 # Sampling f r equency , Hzdf = 500.0 # Measurement bandwidth ( d e l t a _ f ) ,Hzdt = 1.0 / fsampns = in t (np . r i n t ( fsamp / df ) ) # Number o f samples in 1 beat

f0 , Q, k = df 300.5 , 220. , 15. # C a n t i l e v e r parameterspi2 = 2 * np . p i# Dr ive f r equency , f o r c e and phasew0, w1, w2 = pi2 * f0 , pi2 * ( f0 − df / 2 . ) , pi2 * ( f0 + df / 2 . )a1 , a2 = 0.844 , 0.846p1 , p2 = 0 . , np . p i / 4.ks , fad , h = 2 . , 2 . , 10. # Model parameters

t = np . arange (0 , 2 * ns * dt , dt ) # Output t imesy0 = [0 , 1e7 ] # I n i t i a l c o n d i t i o ns o l = odeint ( c a n t i l e v e r , y0 , t , args=(w0, Q, k , a1 ,

a2 , w1, w2, p1 ,p2 , ks , fad , h) )

p l t . p l o t ( t * 1e3 , s o l [ : , 0]) # P l o t the s imu la t ed d e f l e c t i o np l t . x l a b e l ( ’ Time , s ’ ) , p l t . y l a b e l ( ’ De f l ec t ion , nm ’ )p l t . show ()

Listing A.1: A simple simulator in Python using scipy.integrate.odeint

87

88 APPENDIX A. CODE LISTINGS

import numpy as npfrom s c i py . i n t e g r a t e import odeintimport matp lo t l i b . pyp lo t as p l t

def pos_xs ( t , v0 , t_2 ) :return np . p iecewise ( t , [ t < t_2 , t >= t_2 ] ,

[ lambda t : v0 * t , lambda t : v0 * (2 * t_2 − t ) ] )

def ode (y , t , w0, Q, k , v0 , N, a0 , t_2 ) :d , v = yx1 = pos_xs ( t , v0 , t_2 )dydt = [v ,

w0 * w0 * (− N / k np . s i n (2 np . p i d / a0) − v / w0 / Q − (d− x1) ) ]

return dydt

w0 = 300e3Q = 0.49k = 40.v0 = 400.N = 10.0a0 = 0.1dt = 1e−6tend = 3. / v0t = np . arange (0 , tend , dt )y0 = [0 , 0]s o l = odeint (ode , y0 , t , args=(w0, Q, k , v0 , N, a0 , tend / 2 . ) )

f i g , [ax , ax2 ] = p l t . subp lo t s (2 , 1)ax . p l o t ( t * 1e3 , s o l [ : , 0])ax . p l o t ( t * 1e3 , pos_xs ( t , v0 , tend /2) , ’ tab : orange ’ )ax2 . p l o t ( s o l [ : , 0] , k * ( s o l [ : , 0] − pos_xs ( t , v0 , tend /2) ) )ax . s e t _ x l a b e l ( r ’ Time [ms] ’ ) , ax . s e t _ y l a b e l ( r ’ [nm] ’ )ax2 . s e t _ x l a b e l ( r ’ Support p o s i t i o n [nm] ’ ) , ax2 . s e t _ y l a b e l ( r ’ F [nN] ’ )f i g . t i g h t _ l a y o u t ()f i g . show ()

Listing A.2: Simulations of friction loops with Prantl-Tomlinson model

Appendix B

Appended papers

I Per-Anders Thorén, Riccardo Borgani, Daniel Forchheimer and David B. HavilandCalibrating torsional eigenmodes of micro cantilevers for dynamic measurement offrictional forces. (Submitted) Review of Scientific Instruments (2018)

My contribution: I developed the theory together with DBH. I performed experi-ments and analysis. I led the writing of the manuscript together with DBH.

II Per-Anders Thorén, Riccardo Borgani, Daniel Forchheimer, Illia Dobryden, Per M.Claesson, Hailu G. Kassa, Philippe Leclère, Yifan Wang, Heinrich M. Jaeger, andDavid B. Haviland On modeling and measuring viscoelasticity with dynamic AtomicForce Microscopy. (Under review) Physical Review Applied (2018)

My contribution: I developed the model with DBH. I performed experiments andanalysis. I led the writing of the manuscript together with DBH.

III Hailu G. Kassa, Johan Stuyver, Anton-Jan Bons, David B. Haviland, Per-Anders Thorén,Riccardo Borgani, Daniel Forchheimer, Philippe Leclère Nano-mechanical propertiesof interphases in dynamically vulcanized thermoplastic alloy. Polymer 135, 348-354(2018)

My contribution: I performed the numerical analysis with the moving surfacemodel.

IV Federica Crippa, Per-Anders Thorén, Daniel Forchheimer, Riccardo Borgani, Bar-bara Rothen-Rutishauser, Alke Petri-Fink, and David B. Haviland Probing nano-scaleviscoelastic response in air and in liquid with dynamic atomic force microscopy. SoftMatter, DOI: 10.1039 (2018)

My contribution: I performed experiments and analysis with FC. FC and I led thewriting of the manuscript together with DBH.

V Per-Anders Thorén, Astrid S. de Wijn, Riccardo Borgani, Daniel Forchheimer andDavid B. Haviland. Imaging high-speed friction at the nanometer scale. Nature Com-munications. 7, 13836 (2016)

89

90 APPENDIX B. APPENDED PAPERS

My contribution: I helped ASdW develop the theory. I performed all experimentsand analysis. I led the writing of the manuscript together with ASdW and DBH.

Publications not included in this thesis

• Riccardo Borgani, Daniel Forchheimer, Jonas Bergqvist, Per-Anders Thorén, OlleInganäs, and David B. Haviland Intermodulation electrostatic force microscopy forimaging surface photo-voltage. App. Phys. Lett., 14 105, 143113 (2014)

• John E Sader, Riccardo Borgani, Christopher T Gibson, David B Haviland, MichaelJ Higgins, Jason I Kilpatrick, Jianing Lu, Paul Mulvaney, Cameron J Shearer, Ash-ley D Slattery, Per-Anders Thorén, Jim Tran, Heyou Zhang, Hongrui Zhang, TianZheng A virtual instrument to standardise the calibration of atomic force microscopecantilevers. Rev. Sci Instr., 9 87, 093711 (2016)

• Per M Claesson, Illia Dobryden, Gen Li, Yunjuan He, Hui Huang, Per-Anders Thorén,David B Haviland From force curves to surface nanomechanical properties. PhysicalChemistry Chemical Physics, 35 19, 23642-23657 (2016)

• Riccardo Borgani, Per-Anders Thorén, Daniel Forchheimer, Illia Dobryden, Si Mo-hamed Sah, Per Martin Claesson, and David B. Haviland Background-Force Com-pensation in Dynamic Atomic Force Microscopy Phys. Rev. App., 7, 064018 (2017)

• Hui Huang, Illia Dobryden, Per-Anders Thorén, Lina Ejenstam, Jinshan Pan, MatthewFielden, David B Haviland, Per Martin Claesson Local surface mechanical proper-ties of PDMS-silica nanocomposite probed with Intermodulation AFM. CompositesScience And Technology, 150, 111-119 (2017)

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