DYNAMIC INVESTIGATION OF POLYMERIC MATERIALS - REPRODUCIBLE DATA ACQUISITION

Transcription

1 DYNAMIC INVESTIGATION OF POLYMERIC MATERIALS - REPRODUCIBLE DATA ACQUISITION AND PROFOUND MECHANICAL ANALYSIS Dissertation zur Erlangung des Doktorgrades Dr. rer. nat. der Fakultät für Naturwissenschaften der Universität Ulm ABTEILUNG EXPERIMENTELLE PHYSIK vorgelegt von: Alexander Michael Gigler aus Herbrechtingen 2006

2 II Amtierender Dekan: Prof. Dr. Klaus-Dieter Spindler 1. Gutachter: Prof. Dr. Othmar Marti 2. Gutachter: Prof. Dr. Paul Ziemann Tag der mündlichen Prüfung: 05. Juli 2006

3 Abstract The investigation of the mechanical properties of surfaces has attracted increasing attention over the last decades. At the same time, miniaturization of mechanical devices such as MEMS - micro-electro-mechanical-systems - has reached nanoscale dimensions. By means of the Atomic Force Microscope (AFM), the mechanical properties at these lengthscales can be accessed. Theories and simulations describing these nano-mechanics, however, are not yet conclusive. The reliable and reproducible acquisition of AFM data has often been doubted, since the calculated results of the measurements did not fully comply with the results from macroscopic experiments. The AFM, however, is able to determine the mechanical behavior of surfaces reproducibly if the apparatus is calibrated thoroughly before and after the measurements are taken. With the Digital Pulsed Force Mode (DPFM) technique, it is possible to access the entire spectrum of sample dynamics and indentation behavior. Using a commercial setup and AFM tips, the mechanical properties of a dewetting polymer mixture of styrene-butadiene-rubber (SBR) and polymethylmethacrylate (PMMA) have been investigated. From the experiments, a recipe for the reliable acquisition of AFM data has been constructed. Furthermore, an algorithmic framework based on contact mechanical models has been developed and implemented as evaluation software. This software allows the quantitative evaluation of entire data sets acquired in DPFM experiments. This amount of data has not been evaluated before, and thus adds an important statistical certainty to the results obtained from the measurements. In addition to the polymer mixture, which is a passive sample system, experiments have also been conducted on living cells. Operating the DPFM in a physiologic buffer solution allowed acquisition of maps of the mechanical response. Another field that is accessible by the AFM techniques is the determination of the lateral force between an indenter and a sample under an additional lateral relative movement. This is known as Dynamic Friction Force Microscopy (DFFM). However, the relative speeds between the tip and the sample are generally rather low and not comparable with speeds that occur in macroscopic reality. To gain more information on the lateral forces occurring at these velocities, a special add-on system to the AFM has been devised, which III

4 IV Abstract makes realistic velocities of several centimeters or even meters per second accessible. The successful operation of the apparatus is shown by exemplary measurements. Together, the reproducibility of indentation measurements and the measurement of lateral forces under realistic shear conditions might add a new quality to the experiments conducted on the nanoscale using Atomic Force Microscopy.

5 Contents Abstract Contents List of Figures List of Tables List of Abbreviations III V IX XIII XV 1 Introduction Scanning Force Microscopy Modes of SPM-Operation Static Modes Dynamic Modes Theoretical Contact Mechanics Hertzian Modeling Sneddon s Extensions to the Hertzian Model Models Incorporating Adhesion Experimental Setup for Indentation Experiments α-snom in AFM Operation Digital Pulsed Force Mode Implementation Technical Implementation of Pulsed Force Mode Analogies and Differences of PFM, JM & Force-Volume-Mode Extended DPFM - Digital CODYMode Measurements and Calibration Techniques for Digital Pulsed Force Mode Force Curves Calibration for Force Measurements Dewetting Polymer Mixtures V

6 VI Contents 4.4 DPFM on Polymers Results from Individual DPFM Force Curves Mapping Physical Quantities to the Topographical Features Reproducibility Experiments Recipe for Acquisition of Quantitative AFM-Data DPFM on Living Cells DPFM Data-Handling Data-Compression Calibration of the DPFM Apparatus Data-Evaluation Data-Preparation Forces versus Indentation Depth Zero-Assumption Data-Conversion Lateral Forces Measured by AFM High-Speed-Friction Implementation High-Bandwidth Photodetector Vibrometer Lock-In Technique Acousto-Optical-Modulator The Actual Setup Approaches for Calibration and Evaluation of Lateral Force Experiments Empirical Model for Dynamic Friction Modeling Sticking Friction Modeling Kinetic Friction Experiments Using the Novel Setup Results on Polymer Blends Optimized Samples by Microcontact Printing Results on Oligomeric SAMs Conclusions Zusammenfassung 119 References 123 A Automated DPFM Data-Evaluation 133

7 Contents VII B Composition of Inks and Etchants 137 C Recipe for µ-cp 139 D µ-cp of Fischer Projection Patterns 141 E Lateral Force Calibration 143 E.1 Friction on Slopes E.2 Friction on Flat Planes Acknowledgements 151 Issue of Statement 153 Curriculum Vitae 155 Posters - Paper - Presentations 157

8 VIII Contents

9 List of Figures 1.1 Components of an AFM setup Schematics of the AFM detection and feedback system Amplitude of a damped and driven harmonic oscillator Phase of a damped and driven harmonic oscillator Impact of two elastic spheres Indentation of a sphere into an elastic half-space Normalized plot of contact mechanical models Schematic drawing of the α-snom Force-distance curve Force vs. time Oliver-Pharr data-analysis method Reference curve on a hard substrate Transformed reference curve without phase correction Phase correction Phase correction Best fit of the reference data A zoomed view of the best fit Surface morphology of dewetting polymer mixtures Stepwise analysis of PFM curves Closeup of the force traces of PMMA and Si Results from PFM data analysis Masks used for data evaluation Relative error of the maximum force Topography at maximum force Cross-section of the topography at maximum force Indentation depth required to reach F max Cross-section of the indentation depth Map of the real topography IX

10 X List of Figures 4.19 Cross-section of the real topography Dimensionless deformation factor ξ Cross-section of the map of ξ Sample compliance calculated by flat punch Histogram of the flat-punch map Sample compliance calculated by a JKR-like model D Histograms of effective modulus versus ξ Stress-strain curves of PMMA Stress-strain curves of rubber samples Sample compliance calculated by a cone-model Maximum indentation depth and post-flow distance Post-flow due to creep Cross-section of post-flow map Energy needed to indent Energy lost during one cycle Remanent depth Remanent depth of an indentation Map of the local adhesion of the dewetting polymers SBR and PMMA on silicon Detachment position Histogram of the detachment distance Energy needed for detachment Map of the energy of detachment A tip before use in DPFM The same tip after use The tip of the cantilever Sample reproducibility Reproducibility of the indentation depth Reproducibility of ξ Reproducibility of the flat punch compliance Reproducibility of the effective modulus Reproducibility of K versus ξ Reproducibility of the Young s modulus Reproducibility of the hysteretic losses Reproducibility of the dissipation ratio Reproducibility of the detachment force Reproducibility of the post-flow DPFM on Living Cells

11 List of Figures XI 6.1 Detection scheme for high speed friction measurements Position dependence of the lateral signal Modulation and response in case of sliding friction Absolute and relative tip-sample position Relative speed of tip and sample Friction contrasts on dewetting polymer mixtures Friction on dewetting polymer mixtures (zoom) Friction on dewetting polymer mixtures at 350 khz Procedure of µ-cp Octadecyltrichlorosilane Octadecanethiol Friction on SAMs of OTS D.1 Fischer Projection Patterns D.2 Fischer Projection Patterns E.1 Forces on the point of contact sliding upwards E.2 Forces on the point of contact sliding downwards E.3 Cantilever tip on a slope E.4 Exaggerated view for calculations

12 XII List of Figures

13 List of Tables 1.1 Typical geometrical dimensions of AFM cantilevers Characteristics of the sample materials Data tokens in the compressed DPFM file A.1 The Curve class A.2 The Image class A.3 The CALIBImage class A.4 The Map class A.5 The ScaleBlock class XIII

14 XIV List of Tables

15 List of Abbreviations AFM CCD CODYMode DDT DMA DMT DFFM DPFM E ECT FDC FFM FPGA FPP HDT HeLa cells HOPG JKR JM k K µ-cp MD ODT OTS PDMS PFM Atomic Force Microscope Charge-Coupled-Device Combined Dynamic X Mode Dodecanethiol Dynamical Mechanical Analysis Derjaguin Müller Toporov Dynamic Friction Force Microscopy Digital Pulsed Force Mode Young s Modulus Eicosanethiol Force Distance Curve Friction Force Microscopy Field-Programmable-Gate-Array Fischer-Projection-Pattern Hexadecanethiol cervix cancer cell-line from Henrietta Lacks Highly Ordered Pyrolithic Graphite Johnson Kendall Roberts Jumping Mode Spring Constant Effective Contact Stiffness Micro-Contact Printing Maugis Dugdale Octadecanethiol Octadecyltrichlorosilane Polydimethylsiloxane Pulsed Force Mode XV

16 XVI List of Abbreviations PMMA PS SAM SBR SBS SEM SFM SNOM or NSOM SPM STM TM ξ Polymethylmethacrylate Polystyrene Self-Assembled-Monolayer Styrene-Butadiene-Rubber Styrene-Butadiene-Styrene Scanning Electron Microscope Scanning Force Microscopy Scanning Nearfield Optical Microscopy Scanning Probe Microscopy Scanning Tunneling Microscopy Tapping Mode Deformation Ratio

17 Chapter 1 Introduction People always have longed to investigate the hidden features of nature. To do so, optical methods have been utilized such as telescopes and microscopes. While the astronomers look into deep-space, scientists in the field of microscopy gain more and more information about the smallest structures by increasing the resolution of their optical tools. However, it was shown by Lord Rayleigh in England and Ernst Abbe in Germany in the 19th century, that features smaller than about half the wavelength of the illuminating light source cannot be resolved by optics due to the diffraction limits. Nowadays, by methods such as near-field optics, a resolution as good as about one twentieth of the illumination wavelength can be reached. Since the eighties, however, nanoscale structures, that means structures much smaller than the wavelength can be accessed by means of Scanning Probe Microscopy (SPM). This type of microscopy is not based on an optical approach, but on a probe that is moved across the surface and which directly interacts with the sample. It is working like a tiny record-player that follows the topographical features representing the audio information. While optical instruments can only depict surfaces based on their optical properties like differences in the optical density or index of refraction, or opacity for example, in SPM, interactions between the sensor and the sample material can be determined and the surface can be investigated. Thus, microscopy has become a real hands-on the nanoscale experimental method. The resolution of such microscopes reaches down to the very smallest feature size, that are atomic steps and surface structure of a graphite sample for example, depending on the method of SPM in use, the type of stylus, and of course the abilities of the SPM operator. The name Scanning Probe Microscopy already gives away the basic principle of operation. A probe of micro- or nanoscale dimensions is moved over a sample in a scanning manner. Thus, the images that can be acquired are based on the upmost region of the sample, meaning, one can only expect to look into surface properties of the sample and not its bulk behavior. As Wolfgang Pauli once said: Gott schuf das Volumen, doch der 1

18 2 Chapter 1. Introduction Teufel die Oberfläche!. This statement means that one cannot expect to measure literature values for the bulk, when only looking at the surface of materials. Therefore, the boundary of the material may show a physical behavior totally different from the behavior of the bulk. The term SPM comprises many different methods such as Scanning Tunneling Microscopy (STM) Scanning Force Microscopy (SFM) Scanning Near-field Optical Microscopy (SNOM) Instead of Scanning Force Microscopy [5,30], the term Atomic Force Microscopy (AFM) can be used alternatively (Fig. 1.1). Figure 1.1: AFM cantilever, segmented photodetector, and a scratched polymer sample. The scratch reveals the silicon substrate below. This area was used as reference material. The development of Scanning Probe Techniques began in 1982 with the invention of the STM by Binnig and Rohrer from the IBM Zurich Research Labs and finally peaked in the nobel prize for Binnig and Rohrer for the invention of the STM [6] in 1986, sharing it with Ruska for his invention and description of electron optics and the electron microscope. The basic idea of Binnig and Rohrer, was to use the tunneling of an electron from a probe to a conducting surface or vice-versa as a means to determine the separation of the two. The tunneling-current, decreasing exponentially with the increase in probesample-distance, is converted by a logarithmic amplifier. This results in a linear response, correlating the distance and the control signal. A negative feedback loop is then used to maintain a pre-set tunneling-current. Thus, it is reproducing the surface structure by its control output. For a given distance from the surface, the electrical characteristics of the sample can be determined, that means imaging of the surface potential is possible. In

19 1.1. Scanning Force Microscopy 3 STM, the control parameter is the tunneling current. In Scanning Force Microscopy, however, it is the directly measured force by which the probe is pressed against the sample, but the general systematic approach is the same. Again, a negative feedback loop is keeping this control parameter at constant pre-set value. A schematic is shown in figure 1.2. Figure 1.2: A schematic of the detection principle. A feedback loop is controlling the positioning of the sample relative to the tip. This ensures permanent contact with the sample at a given pre-set force. Scanning Force Microscopy, again, has many different modes of operation. The most common ones together with their benefits and pitfalls shall be described in the following chapter. For a general overview, refer to the reviews of Colton [23], Sarid [71], and Van Landingham [84]. 1.1 Scanning Force Microscopy There are several competing possibilities to measure forces at the nanoscale. Firstly, and most obvious, one can use a rectangular beam clamped on one end and deflect it at the other end in contact with the sample surface. The force is then given by the slope at the end of the cantilever. The detection of this cantilever deflection is commonly done via the so called optical lever technique, which has been described by Meyer et al. [54 56], as well as Alexander et al. [1]. To first order, the deflection of the stylus upon a force applied at its end can be assumed to follow a linear force law (see [15,21,33,34] for more detail). z F = k(s)ds = kz (1.1) with 1 k = 1 k lever + 1 k sample (1.2)

20 4 Chapter 1. Introduction Secondly, the damping of a probe oscillated at its resonant frequency, causing a change in oscillation amplitude, frequency and phase in close proximity with the sample surface, can be used. It is commonly referred to as shear-force control (for the case of oscillation parallel to the sample surface) or Tapping, Near-Contact, or Non-Contact Mode (for the case of oscillation normal to the sample surface). The direction of the change in amplitude, phase and frequency can be increasing or decreasing, depending on the specific initial settings. An analysis of resonant measurement modes and the stability of measurement conditions can be found in a publication by Spatz et al. [75]. Even from the simple damped and driven harmonic oscillator, one can see the changes in amplitude, frequency and phase for a given damping and initial conditions, as shown in figure 1.3 and 1.4 respectively. The oscillation amplitude A at resonance is given by: A = F m 1 (ω 2 0 ω2 ext) 2 + (2δω ext ) 2 (1.3) where F is the external driving force, m the mass of the oscillator, δ the damping coefficient, ω 0 the eigenfrequency of the free oscillator, and ω ext the external driving frequency. Figure 1.3: Amplitude of a damped and driven harmonic oscillator. The driving frequency is plotted in units of the eigenfrequency of the free oscillator (ω 0 ). δ is increasing from top (δ = 0.2) to bottom (δ = 0.6). The phase lag φ between the external driving force and the oscillation of the oscillator is given by: ( ) 2δωext φ = arctan (ω0 2 ω2 ext) (1.4) The resonant frequency depends on the damping coefficient by: ω res = ω0 2 2δ 2 (1.5)

21 1.2. Modes of SPM-Operation 5 Figure 1.4: Phase of a damped and driven harmonic oscillator. The driving frequency is plotted in units of the eigenfrequency of the free oscillator (ω 0 ). The same δ range has been simulated as in figure 1.3. Thus, for the damping coefficients used for the calculation of figures 1.3 and 1.4, the resonant frequency of the oscillator is dropping like: damping resonant frequency relative to ω Modes of SPM-Operation In the following a beam of rectangular shape with a tip at the end will be assumed. Such a beam is called cantilever. Typical sizes are given in table 1.1. Mostly silicon or siliconnitride based cantilevers are used in Scanning Force Microscopes. A broad variety of modes for static and dynamic operation of an AFM can be found in literature [4, 17, 22, 24,42,48 50,59,69,76,89]. The most important modes will be described in the following.

22 6 Chapter 1. Introduction length width thickness tip-height frequency µm µm µm µm N khz 225 ± 5 48 ± ± ± ± 5 48 ± ± ± ± 5 50 ± ± ± Table 1.1: Typical geometrical dimensions of AFM cantilevers Static Modes Constant Height Mode To scan across the surface, the tip has to be kept in contact with the sample. The easiest implementation is to hold the base of the cantilever at a given height above the surface and to move the sample laterally. This is known as constant height mode. Crossing elevated features, the deflection of the stylus will increase. This corresponds to a higher force. One of the most widely used methods to detect the deflection of the cantilever, and thus to measure forces, is the optical-lever technique as described by Meyer et al. [54 56] or Alexander et al. [1]. A focused laser beam is directed towards the back of the stylus - or cantilever - and the beam is then reflected to the center of a position sensitive detector. One possible realization of this is using a segmented photodiode having for example four quadrants. The difference in current from the upper and lower or the left and right sides (as defined in figure 1.2) can be used to determine the position of the laser-spot on the detector. As the deflection is dependent on the bending of the probe, the differential signals give a direct measure for the forces acting on the contact. The difference between the top and bottom components is proportional to normal deflection, while a left-right signal corresponds to a torsional bending of the probe due to frictional forces arising when the sample is moved perpendicular to the cantilever. Constant Deflection Mode As the deflection in constant height mode is not controlled while scanning a sample surface at a given pre-set height, elevated features are subject to higher forces. These forces may be high enough to destroy or deform the sample permanently. Keeping the deflection constant resolves that problem. A control-electronics is used to minimize the force error signal, that means the deviation of the detected deflection from a given setpoint (desired deflection) by repositioning of the probe. Operating an SPM like this is known as constant deflection or constant force mode. This mode of operation offers one of the

23 1.2. Modes of SPM-Operation 7 most simple ways to characterize a sample. Firstly, the tip images the topographical features. Secondly, the lateral forces induced by the relative motion of tip and sample allow classification of the local material composition Dynamic Modes In dynamical operation, things are not quite that easy. Sustaining a constant average force or a constant peak force is necessary to scan across the surface. There are resonant and non-resonant modes which provide different degrees of insight into sample behavior, but, at the same time, have specific problems. Resonant modes of operation are for example non-contact, tapping, and self-excitation modes, while non-resonant modes are Pulsed Force Mode (PFM) or, using a similar strategy, Jumping Mode (JM). Non-contact mode allows scanning and imaging of very delicate samples in the attractive regime of the surface potential. It provides a good insight into short-range attractive forces. However, it is difficult to obtain mechanical information on the materials below the tip, due to the lack of immediate contact. Applying larger excitation amplitudes, one is operating in the so called intermittent-contact mode, which implies that the tip periodically comes into contact with the sample surface [47]. Coming to the repulsive regime of the interaction potential of the sample, the tapping mode is reached. Here, the tip impacts the surface once during each oscillation. The observables in this mode are the amplitude of the cantilever oscillation and the phase shift compared to the excitation of the cantilever [20, 92]. Both observables are typically analyzed by a lock-in amplifier. Since the cantilever is moved through the repulsive as well as the attractive regime during every oscillation period, amplitude and phase are affected by both regimes. Thus, the tipsample interaction affects those quantities in a non-trivial way, which is, except in special cases, not invertible. In tapping mode, the average amplitude of the cantilever oscillation is maintained at a constant level, thus there is no control of the actual force applied to the sample. Far below resonance, however, direct control of the peak force is possible, for example in PFM or JM. Resonant Modes Due to the movement within the surface potential of the sample (e.g. Lennard-Jones as the simplest potential), the oscillator is damped. The damping depends on properties of the materials close to the tip. This causes a shift in frequency as well as in phase. To compensate one of these effects, the probe is repositioned.

24 8 Chapter 1. Introduction The operating frequency is determined by the eigenfrequencies of the rectangular cantilever (see Budo [12] for details): ν n = (α nl) 2 2πL 2 Ed 3 b 12ρA. (1.6) E being Young s modulus, d the thickness, b the width, L the length, A the cross-section, and ρ the mass density of the cantilevered beam. The characteristic equation of such a beam is The coefficients for the resonance frequencies are cos(α n L) cosh(α n L) + 1 = 0. (1.7) α 0 L = 1.875, (1.8) α 1 L = 4.694, (1.9) α 2 L = 7.855,... (1.10) α n L (2n 1) π 2 n 4. (1.11) Thus, the frequency of the fundamental mode is given by ν 0 = (1.875)2 2πL 2 Ed 3 b 12ρA. (1.12) There are two different ways to drive the cantilever close to its resonance. In tapping mode, the cantilever is driven by an external oscillator at a frequency chosen reasonably close to its eigenfrequency. The eigenfrequency of the system is lowered and the amplitude is dropping, for example, when starting at maximum amplitude and directly on the free-eigenfrequency of the cantilever (see Spatz et al. [75] for a detailed discussion). However, the cantilevered beam is still driven at the pre-set external frequency, so a phase shift is observed depending on the material of the sample. The energy transferred to the cantilever in each period thus varies with position. This is the mechanism on which phase imaging of materials in tapping mode is based. The second way to achieve resonant excitation alleviates this problem. In the so-called self-excited oscillation mode, the measured deflection of the cantilever (due to thermal excitation or drive) is phase shifted by 90 degrees, amplified, and finally fed back to the drive-unit. Thereby, the positiove feedback is pumping energy into this oscillation of the cantilever. Thus, driving always happens at the actual eigenfrequency of the system. This method is self-adjusting, this means once started, the cantilever will oscillate at its eigenfrequency only. There is no phase shift in the detected oscillation signal. In this case, the surface potential can be determined by exploring the shift in frequency.

25 1.2. Modes of SPM-Operation 9 With the force being the first derivative of the potential, a modulated cantilever will always measure the second derivative of the interaction potential for small modulation amplitudes: 2 E pot z 2 = F z = F mod z mod (1.13) Force Modulation Mode As stated before, there is also a category of dynamic AFM modes which does not operate at the eigenfrequency of the cantilever. When scanning in contact with the surface, an additional oscillation can be applied to the position of the base of the cantilever. Thus, the force on the contact is modulated. This force modulation mode is very similar to a classical method in material science, namely dynamical mechanical analysis (DMA). As in DMA, F z is measured for a range of frequencies. One can obtain a quantity directly related to the elastic properties of the sample. This kind of nanoscale DMA with an AFM has been reported successful by Balooch [2]. Because the AFM is operated in permanent direct contact with the sample, however, it gives rise to wear of the surface. Nanoindentations Another method to investigate materials with an AFM is to perform indentations at discrete positions of the sample. In this case, the tip is impressedinto the sample in a controlled manner. Thus, repeat rates have to be far below the resonant frequency of the cantilever. Otherwise it would oscillate uncontrollably. When the peak forces exceed a sample and tip dependent level [8, 11, 26, 83, 84], the nanoindentation regime is reached. The resulting permanent deformation is a means to examine plasticity at the nanoscale. Experiments in this direction have been carried out by Hinz et al. [32], for example, who investigated the temperature dependence of the indentation formation process in thin polymer films (35 nm PMMA). They were able to measure the sample compliance below and above glass temperature. However, in most AFM experiments destruction of the surface is an unwanted side-effect of intermittent or full-contact AFM, which should be avoided.

26 10 Chapter 1. Introduction

27 Chapter 2 Theoretical Contact Mechanics A variety of forces is involved in the interaction of the tip and the surface. There are elasto-mechanical forces, adhesion, electrochemical forces in the presence of electrolytes, and hydrodynamic forces such as the liquid neck between the tip-apex and the sample in ambient conditions. It is necessary to distinguish between these effects to understand the interaction and to separate artifacts from real data. This chapter explores the interaction of tip and sample from a theoretical point of view. To understand the nature of the forces and signals in PFM, it is necessary to give an abridged description of contact mechanics as it can be found in several textbooks such as the ones by Maugis [52] or Johnson [35]. 2.1 Hertzian Modeling The contact mechanics of rigid bodies has first been described by Heinrich Hertz [31]. Assuming two perfectly elastic spheres made of an isotropic material (R m, E m, ν m, being radius, Young s modulus, and Poisson s ratio of each sphere of index m respectively), whose centers are being pushed towards each other (Fig. 2.1) by a load P and neglecting adhesion forces, he was able to calculate the analytical solution to the problem. Assuming that the spheres are only deformed inside the area of contact, one can calculate the geometric changes. Two spheres of radius R 1 and R 2 are given by R 2 1 = r 2 + (R 1 z 1 (r)) 2 (2.1) R 2 2 = r 2 + (R 2 z 2 (r)) 2 (2.2) z 1 and z 2 being the distances from the x-axis at position r from the z-axis, which is the axis of cylindrical symmetry of the system. For small distances r from the main axis z, z 1 (r) and z 2 (r) can be written as z 1 (r) = r R 1 (2.3)

28 12 Chapter 2. Theoretical Contact Mechanics z R 1 z (r)+z (r) 1 2 x R 2 a Figure 2.1: The situation of the impact of two fully elastic spheres of radius R 1 and R 2 as used for the derivation of the Hertzian contact model. At a given force, the centers of the spheres will be closer by a distance δ, resulting in a deformed contact area of radius a. The deformation of both spheres along the x-axis is δ 1 + δ 2 (z 1 (r) + z 2 (r)). z 2 (r) = r2 2 1 R 2. (2.4) The sum of equation 2.3 and 2.4 equals the distance of two opposing points (M, N) of the spheres. Before the impact, two points on the spheres, at the same distance r from the z-axis, have been separated by z 1 (r) + z 2 (r). The displacement δ is moving sphere 1 by δ 1 and sphere 2 by δ 2, while the distance between the points only changes by: Per definition, R is the effective radius: (r) = δ 1 + δ 2 (z 1 + z 2 ) (2.5) ( = δ r ) (2.6) 2 R 1 R 2 = δ r2 1 (2.7) 2 R 1 R = 1 R R 2. (2.8) The deformation of the spheres also has to be the result of an elastic deformation by pushing the spheres together. The pressure distribution for this problem is p(r) = p 0 1 r2 a 2, (2.9)

29 2.1. Hertzian Modeling 13 r being the actual position within the area A = π a 2 of radius a and p 0 being the maximum pressure in the center of this area (Fig. 2.1). Within the loaded circle, the local displacement of the sphere with index m along the z-axis is given by [35, 52] u zm (r) = (1 ν2 m) π p 0 E m 4 a (2a2 r 2 ) r a (2.10) Thus, the total deformation is - again - the sum of these displacements and has to equal (r) as calculated above (Eq. 2.7): (r) = ( (1 ν 2 1 ) + (1 ν2 2 ) ) π E 1 E 2 4 p 0 a (2a2 r 2 ). (2.11) The total load P on this area can be calculated as the integral of the pressure distribution over the entire area and yields P = p a2. (2.12) Comparison of the coefficients of the two equations for (r) gives ( (1 ν 2 δ = 1 ) + (1 ν2 2 ) ) π 2 a p 0 E 1 E 2 2 (2.13) and: r 2 2R = ( (1 ν 2 1 ) + (1 ν2 2 ) ) E 1 E 2 1 R = ( (1 ν 2 1 ) E 1 + (1 ν2 2 ) E 2 ) p 0 π 2 4a r2 (2.14) p 0 π 2 2a. (2.15) It is easy to see that by dividing of equations 2.13 and 2.15, the contact radius is a = Rδ. (2.16) Defining an effective modulus K as 1 K = 3 4 ( 1 v ) v2 2 E 1 E 2 (2.17) and using 2.12, this contact radius can also be written as a = ( ) 1 PR 3. (2.18) K Also, by solving for P, the dependency of the force on the deformation of the spheres is P = K Rδ 3. (2.19)

30 14 Chapter 2. Theoretical Contact Mechanics (A) P a (B) P P a P d a c Figure 2.2: Indentation of a sphere into an elastic half-space. (A) is valid for all contact models. The basic difference between DMT, JKR, and Maugis is the definition of the area that is involved in adhesive forces (B). In the DMT (upper image) it is only an annular zone around the Hertzian contact area which causes adhesion. JKR (middle) assumes that only the contact area contributes to the adhesive force, but the surface is compliant enough to adhere to the tip even beyond point zero. Both models are special cases of the more general Maugis model (bottom).

31 2.2. Sneddon s Extensions to the Hertzian Model 15 Given this load, the deformation can also be written as δ = ( P 2 K 2 R ) 1 3. (2.20) In the following effective radius R and the effective modulus K will be used for the description of contact mechanics: 1 R = (2.21) R 1 R 2 1 K = 3 ( 1 v ) v2 2 (2.22) 4 E 1 E 2 It is obvious that for a sphere of infinite radius R 2, the effective radius equals R 1. This describes a fully elastic sphere indenting into a fully elastic half-space. Furthermore, if one sphere is incompressible, this means means for example E 1 tends to infinity, only the other one will be deformed. 2.2 Sneddon s Extensions to the Hertzian Model In 1965, based on the calculations performed by Hertz in 1882, Sneddon [74] went on to extend the theoretical description to non-spherical, but also axi-symmetric, indenters. Later, many groups [13, 14, 18, 38, 44, 63, 64, 73] extended the theories to virtually every sufficiently smooth indenter shape with rotational symmetry. A brief summary of the approach will be given in the following. The basic assumption for the calculations is, that for a purely elastic material, the stiffness S (or the change in force upon a given change in indentation) will be directly proportional to the root of the contact area of indenter and surface at this position, this means: S = dp dδ A. (2.23) For some of the most important boundaries of indenters, the consequences will be shown in the following. One of the easiest shapes for an indenter is a cylinder that confers a flat-punch. In this case, the radius of the indenter is constant for all depths r(δ) R δ. (2.24) Clearly, the area of the punch is thus a constant, too: A π R 2. (2.25)

32 16 Chapter 2. Theoretical Contact Mechanics This implies S const., (2.26) which results in a linear dependency of the force P = Sdδ δ. (2.27) A spherical indenter with a boundary line of r(δ) = R 2 (R δ) 2 2Rδ (2.28) implies a stiffness S π2rδ, (2.29) which means that the force will be P = Sdδ δ 3 2, (2.30) which exactly reproduces the Hertzian behavior. Due to the approximation of the contact radius, this description is correct only for indentation depths small compared to the radius of the sphere (δ = R 3 already results in an error of 10%). Using a parabolic curve to describe the surface of the indenter, one finds r(δ) = δ, (2.31) S δ (2.32) and, hence, the force P δ 3 2, (2.33) which is exactly the same as the result of a sphere for small indentations (near the apex of a sphere it can be approximated as a paraboloid). Carrying out the same calculation for a cone which has a straight line as its boundary, results in a radius r(δ) = δ (2.34) and a stiffness This leads to the force law P = S δ. (2.35) Sdδ δ 2. (2.36)

33 2.2. Sneddon s Extensions to the Hertzian Model 17 For a curve of the arbitrary shape r(δ) = δ 1 α, (2.37) the stiffness will be proportional to δ 1 α as well, while the force on the contact will increase like P = Sdδ δ 1+ 1 α. (2.38) This last equation with all its simplicity has the power to explain measured indentation data. One often finds exponents greater than two which cannot be explained by standard indenter shapes. However, if the indenter would be formed as a root function, it would even allow for an exponent of three. The only problem of this formulation is, that one of the basic assumptions was to have a smooth surface, in order to avoid singularities in the pressure distribution at point zero. Thus, this theory would only be applicable at large indentations. The small indentation depths could be modeled with an indenter made of a small spherical part attached to the root shape. Thus, this would be a clever way to circumvent the problem of the singularity, while having an exponent greater than two. Anyway, this correction has been neglected throughout the published literature for the conical indenter, too. Only the shape of indentation curves at the initial contact are affected by this problem. From the analytical point of view, one simply has to be aware that basic assumptions are violated at this point. Recently, there have been several attempts for a more general description of an indenter by Borodich [9,10] or for more specialized, more realistic shapes, e.g. a Vicker s indenter which is a pyramidal tip, by Franco [29].

34 18 Chapter 2. Theoretical Contact Mechanics 2.3 Models Incorporating Adhesion The adhesive interaction between two bodies in contact is described by the theories of Maugis-Dugdale (MD), Johnson-Kendall-Roberts (JKR), and Derjaguin-Muller-Toporov (DMT). JKR and Maugis-Dugdale are the only theories describing negative indentation depths which explain the pull-off behavior of very compliant materials. In DMT the surface is not deforming upon adhesive forces. Thus only JKR and MD are able to give an explanation for the hysteresis in detachment processes. All three theories are described in the following. Maugis-Dugdale One of the shortcomings of the Hertzian model and, thus, the Sneddon extensions is that adhesion is not part of the model. A more complete approach was chosen by Maugis [51, 52] based on an earlier theory of Dugdale [27]. Maugis theory contains an adjustment parameter for the systematic inclusion of adhesion effects ( πϖk 2 ) 1 3 λ = 2σ 0. (2.39) R where ϖ is Dupré s work of adhesion to separate two solids reversibly and isothermally. It is assumed that the region in which adhesive forces are acting is larger than the contact area derived in the Hertzian model; m = c (2.40) a being the parameter determining the actual region of influence of adhesion (Fig. 2.2 B). The formulae describing the depth of indentation δ and the load P are: δ = a2 R 8aσ 0 m 2 1 (2.41) 3K P = a3 K ( ) R 2a2 σ 0 m m 2 arctan m 2 1 (2.42) σ 0 being the surface stress at the edge of the contact. For a given parameter λ and a contact radius a, the parameter m can be derived from the Griffith-relation: 1 = 1 ( 2 A2 λ m ( m 2 2 ) ) arctan m ( ) 3 A λ 2 1 m + m 2 1 arctan m 2 1. (2.43) There are two limiting cases where the Maugis-Dugdale model (MD) can be simplified to the Johnson-Kendall-Roberts model (JKR) [36, 37] and the Derjaguin-Muller-Toporov

35 2.3. Models Incorporating Adhesion 19 model (DMT) [25, 62]. JKR holds for the special case when adhesive forces are only acting on the region of immediate contact, while DMT describes the situation where these forces exclusively act along the circumference of the contact. In addition, JKR assumes that the sample is compliant enough to adjust to the detaching sphere while DMT claims that the surface will not deform on detachment. Johnson-Kendall-Roberts The main assumption of the JKR model is that the same maximum indentation force leads to the same radius of contact as the Hertzian model (a 3 = PR K ), while at the same time at zero load, the sphere still is in contact with the surface and adheres. For the situation of P = 0 the radius of contact is a 0 = ( 6πϖ R 2 ) 1 3 (2.44) K with a deformation of δ 0 = a2 0 3R. (2.45) For the general indentation process, the radius of contact is given by a 3 = PR ( ) ( ) ( ) 3πϖ R 3πϖ R 3πϖ R (2.46) K P P P It is obvious that for the non-adhesive case (ϖ = 0), the JKR model equals the Hertzian contact. Pulling indenter and surface further apart, one reaches an unstable position where the indenter finally detaches. The force at which this occurs is F adh = 3 πϖr (2.47) 2 with a contact radius of or a adh = ( 3 2 πϖr 2 ) 1 3 (2.48) K a adh = a0 = 0.63a 0. (2.49) The deformation of the material, at this position is δ adh = a2 R = ( π 2 ϖ 2 ) 1 R 3. (2.50) 12K 2

36 20 Chapter 2. Theoretical Contact Mechanics Derjaguin-Muller-Toporov In the DMT model, an adhesive contact of an elastic sphere with an elastic surface is described. It is assumed that adhesive forces originate from a ring-shaped zone around the contact radius and that the surface does not deform due to adhesion. The deformation profile is therefore identical with the one derived in the Hertzian model. The contact radius during an indentation is given by a 3 = PR K + 2πϖR2, (2.51) K where the non-adhesive case yields the Hertzian contact again. The depth of indentation is given by δ = a2 R = Under zero load the contact radius is given by ( P K R + 2πϖ ) 2 3 R. (2.52) K at an indentation of The pull-off force is a 3 0 = 2πϖR2 K (2.53) δ 0 = a2 0 R. (2.54) F adh = 2πϖR (2.55) at indentation depth δ adh = 0 and contact radius a adh = 0. The behavior of a DMT like indenter is simply the one of a Hertzian sphere, but shifted by the adhesive force. Introducing normalization parameters, one can compare the theoretical descriptions more easily. Summary in Reduced Variables One can normalize the variables to ease comparison: the reduced contact area; A = a ( πϖr 2 K ) 1 3, (2.56) P = P πϖr, (2.57)

37 2.3. Models Incorporating Adhesion 21 the reduced force on the contact; the reduced indentation depth; and = δ ( π 2 ϖ 2 R ) 1, 3 (2.58) K 2 λ = the degree of adhesion according to Maugis. 2σ 0 ) 1, (2.59) 3 ( πϖk 2 R Thus, the equations describing the contacts are simplified to: model Hertzian P = A 3 reduced load and indentation depth = A 2 DMT P = A 3 2 JKR = A 2 P = A 3 A 6A = A A ( m MD P = A 3 A 2 λ m 2 arctan ) m 2 1 = A 2 4 3( A λ m 2 1 m 1 = 1 2 A2 λ ( m 2 2 ) arctan ) m 2 ( A λ 2 1 m + m 2 1 arctan ) m 2 1 The reduced force curves versus indentation depth are shown in figure 2.3 for comparison.

38 22 Chapter 2. Theoretical Contact Mechanics Figure 2.3: With normalized parameters, all models describing contact mechanics can be plotted in one graph. It is obvious that DMT differs from the Hertzian curve only by a constant offset, while the JKR plot has a steeper increase. (taken from [52] with kind permission from Springer)

39 Chapter 3 Experimental Setup for Indentation Experiments 3.1 α-snom in AFM Operation The basis for the experimental work during this PhD-thesis was the α-snom setup by WITec GmbH ( A schematic drawing is shown in figure 3.1. By the name, one can tell that the original purpose for the microscope has been the development of a Scanning-Near-field-Optical-Microscope or SNOM - the technique is also known as NSOM in its US-English spelling. Meanwhile, it is a multi-purpose optical microscope, which also allows Scanning Probe Microscopical measurements. The range of possible measurements includes: Confocal optical microscopy, cantilever based SNOM measurements, measurements in transmission as well as reflection, fluorescence and spectral imaging, and - last not least - AFM (in all different measurement modes). The microscope carries a revolver-head for the microscope objectives, which allows switching of the type of microscopy above a certain sample position. The path of the light beams through the microscope is kept parallel for most distances, allowing for example the direction of several laser-beams to one focal point on a sample or, in this case on the AFM cantilever. In order to operate the α-snom in an AFM mode, there is a cantilever holder. Sample positioning and scanning is conducted by a scan-table from PI GmbH & Co. KG ( Karlsruhe, which allows lateral positioning over a range of 100 by 100 µm with 16-bit accuracy, which means that one digitization step equals 1.5 nm. Since the annex of the cantilever tip has a radius of some tens of nanometers, this resolution is definitely enough. In the z-direction, normal to the sample-plane, the scan-range is 20 µm, which equals 0.3 nm resolution per bit. Thus, depending on the stiffness of the cantilever, this uncertainty in height corresponds to the uncertainty in force applied to the sample. 23

40 24 Chapter 3. Experimental Setup for Indentation Experiments Figure 3.1: Beam path of the α-snom setup. The components shown are: 1 - laser, 2 - single mode fiber, 3 - cantilever-snom-sensor, 4 - multimode fiber with SMA connector, 5 - detector, 6 - b/w CCD camera, 7 - white light source for Köhler illumination, 8 - color CCD camera, 9 - SMA connector for signal pick-up in reflection, 10 - FC fiber connector for beam deflection laser, 11 - segmented photodiode. (figure used with kind permission from WITec GmbH)

41 3.2. Digital Pulsed Force Mode Implementation 25 Cantilever Holders In the original setup, the cantilever was held by a special zoom-objective, which allowed determined positioning of the laser spot on the back of the cantilever, since focusing is done while having visual feedback (CCD camera inside one of the objective-tubes). Underneath the magnetically held cantilever chip, a piezo element is installed to allow excitation of the cantilever and multiple dynamic measurement modes. During the first two years of this PhD work, this cantilever holder was in use for measurements in contact mode, tapping mode, and (Digital) Pulsed Force Mode, as well as friction measurements and it served very well for these purposes. Meanwhile, however, an improved setup became available. It allows remote positioning of the cantilever underneath any commercially available microscope objective. This is a major improvement, since the accuracy of a microscope objective outnumbers the focusing by a manual zoom-objective. So now the focal spot of the laser used in the beam deflection system is smaller and, thus, causes less uncertainty in the force measurements. For an investigation on laser focusing and positioning, see the PhD work by Jaime Colchero [22]. 3.2 Digital Pulsed Force Mode Implementation Various properties, measured simultaneously at the same point of the sample, facilitate the characterization of the behavior of the sample material. To keep the measurement process as non-destructive as possible, the tip must only be in contact for the shortest time possible. This can be achieved in Pulsed Force Mode (PFM) [41, 46, 68, 69, 76] or Jumping Mode (JM) [61]. Here, the tip follows a pre-defined approach and retract trajectory. Such a controlled positioning is not possible in any resonant mode of operation as discussed in the introduction. Although in PFM as well as JM most of the tips lateral scanning movement is out of contact, the shear forces acting between the tip and the sample cannot be fully neglected. An elaborate study on this was conducted for the Jumping Mode by Moreno et al. [61]. On the other hand, a controlled lateral excitation with simultaneous PFM operation can also be utilized to learn about further frictional properties of the sample surface. An example of CODYMode operation [41] is described later. At low speeds and triangular modulation the approach-retract cycle described above is known under the name of force distance curve (FDC) as seen in Fig In literature, FDCs have been well examined by many sources [7, 13, 14, 60, 64, 66]. The pull-off force, which is the peak attractive force on retract, provides an easy way to measure local adhesion. By applying a triangular modulation to the base of the cantilever, the tip is brought periodically into contact with the sample surface. Local adhesion, elastic, viscoelastic, and

42 26 Chapter 3. Experimental Setup for Indentation Experiments Figure 3.2: Force versus distance curve as it is acquired by AFM. The measured force is plotted versus the position of the base of the cantilever. Coming closer to the surface, the tip snaps into contact with the surface. On further approach, the cantilever is forced to bend backwards depending on the elastic, viscoelastic and plastic behavior of the sample material. A certain maximum force is reached, then the base is retracted causing relaxation of the cantilever. On the retract, a material dependent negative force has to be exceeded to pull-off the tip from the surface. All forces have to be measured relative to the free position of the cantilever at the leftmost point. plastic properties of the sample can be derived from the measured forces and positions. However, the range of accessible deformation rates is quite limited. In FDCs only repeatrates of a few Hertz can be used, since the bandwidth of the excitation electronics, in general, is not sufficiently wide for a triangular signal. Fourier decomposition of a triangular wave shows, that quite some higher harmonics have to be taken into account in order to synthesize a real triangular wave. Hence, the low-pass behavior of amplifiers and other components affects the transmitted curve-shape and, thus, cantilever positioning. For sinusoidal drive signals this dispersion does not occur. Therefore, a purely sinusoidal modulation will not suffer from this bandwidth limitation. In addition, sinusoidal drive eliminates the excitation of spurious resonances in the AFM. Nevertheless, phase shifts between the drive signal and the cantilever movement are present. A setup designed for minimal disturbance of the drive signals is the Pulsed Force Mode. The equivalence between this mode and FDCs can be shown by simple transformations. Examples of FDCs and force curves acquired using PFM are shown in Fig Both curves were measured with the same AFM setup at the same frequency of 100 Hz. It is obvious that at the turning point of the triangular drive, either the triangular wave shape is not transmitted correctly to the cantilever, or the cantilever is excited at its resonance, or both. Yet, it is clear that for repeat-rates of more than 100 Hz of a triangular modulation higher harmonics from

43 3.2. Digital Pulsed Force Mode Implementation 27 the wave shape couple to the cantilever. The interpretation of the results becomes very intricate. If the mechanical processes could be modeled analytically, it would be possible to simulate FDCs at those high repeat-rates. However, PFM offers a compromise, since it can be operated in a frequency range from 50 Hz to 20 khz, which has been verified experimentally. It has the advantage that the content of higher harmonics is negligible in the spectrum of the wave form, avoiding possible problems. Figure 3.3: A view of the development of the force within the time-domain. Trace A showing the response of the system upon modulation with a triangular trajectory. As in figure 3.2, snapin, maximum force, the repulsive regime, pull-off force, and the baseline can be assigned to the curve. Trace B, which corresponds to a sinusoidal movement of the cantilever, shows the same characteristics as trace A. However, in order to sustain the steep turns, the triangular wave is bandwidth-limited to a certain maximum velocity (or repetition rate). By coordinate transformation (assigning the trajectory of the base of the cantilever to the horizontal axis of the force plot), the modulation specific curve shape is removed from the curve shape. What is remaining is the classical FDC plot (Fig. 3.2). Given that each material has its own characteristic temporal response as shown in Fig. 3.3, one might conclude that the type of excitation function could affect the results. The coordinate transformation x(t) = A 0 sin(ω t + φ) (3.1) removes the trajectory from the signal. The resulting data turns out to be the equivalent to that obtained by triangular excitation. This can be seen in the sinusoidal trace in Fig. 3.2.

44 28 Chapter 3. Experimental Setup for Indentation Experiments Technical Implementation of Pulsed Force Mode The setup of a Pulsed Force Mode is rather straight forward. What is needed is the possibility to modulate the tip-sample distance in order to rupture the point of contact before a new approach curve is acquired. During operation, the contact time should be kept as short as possible to avoid sample destruction during the lateral scanning movement. This z-modulation is usually conducted by a separate piezo within the cantilever holder. Sampling and recording of the measurement data is another demanding topic. On the one hand, data-acquisition has to be sophisticated enough to allow the reduction of the amount of data to a reasonable scale, while, on the other hand, the data should only be compressed to a predetermined error-margin. In the analog implementation of the measurement system, this is accomplished by peakpickers and sample-and-hold circuitry, while with the digital version, it is possible to capture and store all data sets of an experiment. The digitization is therefore a very demanding topic and is implemented as a dedicated transfer and storage connection. The I/O system, for example, has to be able to constantly handle about 10 MB per second for a 1 khz experiment. For a typical experimental run with a scan-rate of about one line per second this adds up to a file-size of several GB. Anyway, the data also has to be processed. Here, the analog setup has an advantage over the digital version, since only the most important points and features of the interaction are investigated in this case. On the other hand, all the information contained in the remaining force curve is neglected. To also record this data means that a more powerful CPU has to process the streaming data. This is done by an Field Programmable Gate Array (FPGA), which is mimicking the analog behavior by operating simple algorithms such as peak-picking and sample and hold in real-time, while being able to analyze the data in an off-line data-evaluation more thoroughly. Another advantage of the digital implementation is that a synthesized curve can be used as modulation signal. Therefore, the curve shape is interpolated between supportpoints. Triangular and sinusoidal curves as well as for example trapezoidal excitations can be realized and material responses to changing excitations can be investigated. By simply switching to the one or the other modulation function inside the software, both experiments can be conducted at the identical position of the sample, allowing direct comparison of the results. With the analog setup this has not been possible without major modification.

45 3.2. Digital Pulsed Force Mode Implementation Analogies and Differences of PFM, JM & Force-Volume-Mode To conclude the overview of measurement modes, the so called Jumping Mode (JM) as proposed by de Pablo [24] has to be mentioned. In this mode, the tip is positioned at a certain discrete scanning position before running a single FDC instead of a permanent sinusoidal modulation of the tip-sample distance. Since this is done for every point of the scan, mapping of material parameters is possible, too. Both, PFM and JM are imaging techniques and, thus, three-dimensional in a sense that, by keeping the applied maximum force constant, the tip is following the topography under a predetermined stress. This 3D image can then be correlated with the material properties deduced from the indentation measurements and is therefore adding a fourth dimension - the material properties - to the topographical measurements. These measurement modes are usually referred to as Force-Volume-Mode Extended DPFM - Digital CODYMode While in general, the lateral motion of the cantilever should be conducted in off-contact to avoid the destruction of soft sample features, lateral forces acting between the scanning tip and the surface can also be exploited to study the frictional properties at the nanoscale. Friction Force Microscopy (FFM) is one of the possible scanning modes accessing these frictional properties - with the shortcoming of permanent contact with the surface. In FFM the tip is scanning across the surface and the lateral force signal is evaluated for friction information by subtracting the forward and backward line-scan. A calibration procedure has been described by Varenberg et al. [85]. Thus, the method is resulting in a map of friction parameters correlated with topographical features. However, piezo creep and drifts in the measurement result in an offset between the forward and backward traces, which causes problems in the data evaluation. A better method is to oscillate or shake the sample relative to the scanning tip and to use the torsional amplitude of the cantilever as a measure for the frictional behavior. In this mode, the tip is still in permanent contact with the surface with the problem of destruction at hand. Thus, it is reasonable to combine the Pulsed Force Mode with this Dynamic FFM method, resulting in the so called Combined Dynamic Mode (CODYMode) as described by Krotil et al. [41]. In this mode the sample is shaken permanently at a small amplitude and a high frequency, resulting in relative tipsample velocities in the order of centimeters or even meters per second. While the tip is in repulsive contact with the sample, the cantilever experiences the lateral forces causing torsion of the beam. Simultaneously - at a different frequency - it is also modulated normal to the sample surface to measure the elastic and viscoelastic material properties as well as adhesion. The correlation of mechanical features and frictional behavior at the

46 30 Chapter 3. Experimental Setup for Indentation Experiments nanoscale can be investigated by this technique, perhaps leading to a conclusive theory of friction processes at this length scale. Prior to understanding the processes which occur in a parallel indentation and dynamic friction, the processes have to be well understood separately. Thus, chapter 4 is completely devoted to data acquisition and evaluation in Digital Pulsed Force Mode. Chapter 6 will then be concerned with friction measurements in contact mode (FFM) and Digital- CODYMode.

47 Chapter 4 Measurements and Calibration Techniques for Digital Pulsed Force Mode An Atomic Force Microscope can be used to move an indenter towards and away from the sample surface with sub-nanometer precision. Furthermore it is possible to measure forces with a precision of several pico-newtons. Thus, the measured indentation curves, either in FDC or the PFM mode, can be understood in terms of the models of contact mechanics. However, effects of viscosity and plasticity are difficult to handle. The next sections are devoted to a thorough description of the data acquisition and analysis processes in PFM. 4.1 Force Curves For large tip sample separations, the cantilever is mostly influenced by long-range forces such as electrostatic or magnetic forces. There, the total force is varying only little. Thus, the average force measured far away from the surface serves as baseline of the force curve. From now on, this baseline force will be used as zero reference for all measured curves. Coming closer to the surface, Van-der-Waals and capillary forces become important. In case of a charged sample, the bending of the baseline during the approach, allows to measure the electrostatic interaction. The first contact will be established when the random oscillation of the cantilever is bringing the tip close enough to the surface to snap-in. In the repulsive regime, the models described in chapter 2 can be used for modeling. For purely elastic samples the theories of Hertz, JKR, DMT, and MD seem to be appropriate. During the indentation, the sample is deformed by the tip of the cantilever. The mechanical work is partially stored as elastic potential energy and partially dissipated in 31

48 32 Chapter 4. Measurements and Calibration Techniques for Digital Pulsed Force Mode viscoelastic processes, and in plastic deformation. Whereas during the approach, the material is pushed away, the sample pushes the tip back on release. The area enclosed by the force distance curve therefore corresponds to the energy dissipated in one cycle (hysteresis). The ratio of the dissipated energy to the total energy involved is a measure of the dissipative properties of the material. The detachment behavior can be observed at the snap-off. Depending on sample compliance and the mutual adhesive interaction forces between the sample and the tip, the tip can either detach as soon as the surface is back in the relaxed state, or it can even pull on the surface, resulting in a negative indentation. Except for the Hertzian model, which excludes adhesion at all, and the DMT model, that does not account for deformation on the pull-off, this process is described by all models. However, the physical origin of the adhesive behavior can be manifold. Wetting of the tip by the sample material is one of the effects, but Van-der-Waals forces, electrostatic forces, or even hydrodynamic forces such as the formation of a liquid neck between the apex of the tip and the sample in ambient conditions can also be involved [77,78,80,90]. Only the sum of all interactions can be measured by the AFM. To avoid the influence of the liquid necking, it is possible to conduct the measurements in vacuum or a liquid environment such as water or a buffer solution. The latter is especially suitable for the investigation of biological samples that has drawn considerable attention recently. Another possibility is to chemically functionalize the tip surface in order to achieve a hydrophobic behavior, for example by silanization. The sudden jolt on the cantilever upon snap-off excites the eigenfrequency of the cantilever. The damping of this oscillation depends on dissipative processes within the cantilever and external processes such as hydrodynamic damping in liquids. The frequency of this relaxation oscillation is usually lower than the eigenfrequency of the free cantilever. This can be due to the higher external damping close to the sample, or, hard to measure with large cantilevers, material transferred from the sample to the tip. In 1992, Oliver and Pharr [63, 64] described the evaluation of force curves on materials showing elastic and plastic deformation, in order to give a guideline how to evaluate force curves leaving permanent deformation marks on the surface (Fig. 4.1). Their strategy was to assume that the material, once deformed by the application of high forces, pushes back due to its elastic properties, only. This is to say that the retraction path of a force curve is only affected by the elastic recovery of the sample. The offset between the zero position of the force curves upon approach and retraction is assigned to a remanent depth, meaning the depth of the permanent deformation of the sample. Note, that no assumptions about the trajectory of the tip of the cantilever have been made so far. The description is valid for a linear as well as a sinusoidal movement. What is the difference between PFM operation and running FDCs? If one is interested in learn-

49 4.1. Force Curves 33 Figure 4.1: A very common method for evaluation of FDCs was developed by Oliver and Pharr [63, 64]. It is based on the assumption that the first part of the retract of the cantilever (unloading) will show purely elastic behavior, since the plastic deformation has to be caused on the indentation trace (loading). For purely elastic and plastic materials the analysis is in good agreement with the permanent deformations caused by nanoindentations [32]. Viscoelastic effects are not included in their description. (taken from [63] with kind permission from Materials Research Society)

50 34 Chapter 4. Measurements and Calibration Techniques for Digital Pulsed Force Mode ing about material parameters for quasi-static strains and stresses, FDCs are the tool of choice. Though, if the material parameters are to be mapped to the surface or if the rate dependence of the behavior under stresses and strains is to be investigated, much higher repetition rates are necessary. This is where the PFM, with its repetition rate from 50 Hz to 20 khz, comes in handy. At low rates it is acceptable to use triangular waves to drive the cantilever position. However, at higher rates the filter characteristics of amplifiers and piezo-ceramics cause different amplifications and phase shifts for the harmonics of the triangular drive, thus inducing distortion (remember: the higher harmonics in a triangular wave are necessary for the instantaneous change of slope at the turning points). Furthermore, mechanical resonances in the AFM might be excited by the triangular wave. 4.2 Calibration for Force Measurements In the following, a calibration technique for force curves of any kind is described. A detailed description of the implementation of the programs used for the calibration of the DPFM apparatus are given in chapter 5 and the corresponding appendices. Multiple experiments have been conducted with this thorough calibration and revealed that if the protocol is consequently followed, reproducibility of data is granted. One can imagine the cantilever as a beam fixed on one side at the AFM measurement head and supported by the sample on the other end. As described by the theories of contact mechanics (Ch. 2), the point of contact will deform. Therefore, when in contact with an infinitely hard substrate, every change in the cantilever-sample distance is causing exactly this deflection of the cantilever. The harder the substrate compared to the cantilever, the better the linearity. Thus, for high loads, one may assume a transfer function of 1 for the ratio of approach to deflection of the cantilever. This linear slope is called sensitivity of the AFM. Its units are Volts per meter. Dividing the cantilever stiffness by the sensitivity as described above gives the normal calibration factor of the system in units of Newtons per Volts. From an indentation measurement by force-distance curves or DPFM measurements, the indentation data on the hard substrate can be obtained. First, the time axis has to be transferred to the position of the cantilever relative to the sample surface. For a FDC it is the ramped approach that has to be used or in case of DPFM data, the sinusoidal trajectory: z(t) = t A PFM cos(2π ν t + φ). (4.1) However, after conversion, it is striking that for phase shift of φ = 0, the data is in general not at all linear in the range of the repulsive contact.

51 4.2. Calibration for Force Measurements 35 Figure 4.2: To calibrate the system, the temporal response of the cantilever on a hard and elastic substrate is used for calibration. Figure 4.3: Without phase correction, the approach and retract part of the repulsive contact are clearly not co-linear. Figure 4.4: Correcting the time lag between excitation and detection shrinks the area under the reference curve. Here: φ = 25. Figure 4.5: Correcting the time lag between excitation and detection shrinks the area under the reference curve. Here: φ = 18. Figure 4.6: Minimization of the area leads to a phase correction of 17. Figure 4.7: The area is clearly minimized and the result is a linear dependency on the position of the cantilever.

52 36 Chapter 4. Measurements and Calibration Techniques for Digital Pulsed Force Mode Shifting the phase in the transformation function until - for a hard and purely elastic reference surface - there is no more intersecting of the approach and retract paths, plus a minimized area underneath the approach and retract cycle, results in the systematic lag between modulation and detection. This time lag or phase shift is due to the run-times of the signals through the measurement system. Depending on the transfer functions of the components of the measurement setup, the phase shift changes with frequency and thus has to be corrected for every change. For a specific set of parameters for an experimental run, however, the phase shift can be considered constant. Furthermore, since all force traces are recorded relative to the first point of the modulation function, they can be phasecorrected by this shift without altering the physical content of the information stored. As the phase-correction sets the hysteresis of the reference material to zero, this correction eventually allows the analysis of viscoelastic effects on the actual sample materials. Afterwards, the deflection of the cantilever can always be converted into a force acting on the tip. The calibration signal accounts for both, the cantilever bending and the deformation of the tip-sample contact. Thus, the position of the tip on top of the sample surface, is the sum of the change in position due to cantilever bending and the actual indentation into the material. Therefore, the reference slope always has to be subtracted from the position imposed on the base of the cantilever by modulating the z-positioning piezo. After subtraction, the real position of the tip in the material of the surface is known. The force versus indentation plots can then be created and evaluated according to the theories for data evaluation described above. 4.3 Dewetting Polymer Mixtures A sample system that contains all different mechanical behaviors elastic, viscoelastic, adhesive, etc. at once is the most desirable for the experimentalist. Such a system allows exploration of the capabilities of the setup. For this purpose, a dewetting polymer mixture consisting of polymethylmethacrylate (PMMA) and styrene-butadiene-rubber (SBR) has been used. If thin films of these mixtures are produced, for example by spin-casting the solution onto a flat substrate such as a piece of a silicon wafer or a freshly cleaved piece of mica, the polymers can separate into different phases. For the mixtures of polystyrene and polymethylmethacrylate, there have been studies and publications [39, 45, 86, 88], where the mechanism of dewetting and phase-separation has been described into great detail. In the following a binary mixture of two parts by volume of PMMA and one part by volume SBR will be discussed. In general, polymers are not mixing, since the mixing entropy is too small, because of the huge amount of monomers - of low molecular weight - combined in long chains.

53 4.3. Dewetting Polymer Mixtures 37 However, together with a solvent, both components are dissolved by the short molecules of the solvent moving in between the long polymer chains. Thus, the solvent is mediating the mixing of the two previously immiscible polymers within a three component mixture. While in solution, the polymers are more or less homogeneously dispersed. Spin-casting the solution onto a solid substrate and evaporating the solvent then breaks the mixing equilibrium and the polymers are trying to separate in order to minimize the surface free energy. Together with the substrate, it is now a four component system: substrate, air and two polymers. The substrate surface is fixed, whereas the polymers can change shape. The affinity of the polymers towards the substrate decides, what amount of area is occupied by which polymer. In case both polymers show merely the same wetting behavior towards the substrate, both will extend down to the substrate, while, if one component has a minor tendency to wet the substrate, a stratified surface coverage will emerge. The polymer-polymer interface remains. Here, the polymers dewet due to the low mixing entropy, again. Thus, minimizing the surface energy causes separation of the polymer mixture into different phases with minimum contact surfaces. An example is shown in figure 4.8. For thin films as produced by spin-casting, these structures are accessible by AFM. Figure 4.8: Solutions of polymers phase-separate when spin-cast onto a favorable substrate. The morphology that is forming depends on the ratio of the polymers in the solution and their tendency to wet the substrate. The measurement shown here is the topographic image of one of the experiments evaluated in the following. Analyzing the mechanical properties can be nicely done in Pulsed Force Mode. The experiments are conducted on the surface and in extremely small regions of the materials. Generally, this may lead to results deviating from the literature values as measured for bulk samples, but since the length scale is hardly comparable to the macroscopic experiments by which these values are obtained, this has to be expected. Another interesting

54 38 Chapter 4. Measurements and Calibration Techniques for Digital Pulsed Force Mode behavior that can only be accessed by such local indentation experiments is the rupture of thin films. Depending on the load applied to the sample the AFM tip may penetrate all the way to the hard substrate. For AFM experiments, however, that are meant to probe the elastic properties of the sample, the maximum indentation depth should be one fifth of the its thickness. Otherwise the results can be strongly influenced by the substrate material. 4.4 DPFM on Polymers The indentation measurements shown in figure 4.9 were carried out on three different sample materials: silicon, polymethylmethacrylate (PMMA) and styrene-butadiene-rubber (SBR). The material properties can be found in table 4.1. SBR PMMA silicon wafer molecular weight 390 kg mol 100 kg mol - poly-dispersity radius of gyration 14 nm 7 nm - glass temperature -20 C 118 C - specialty not crosslinked - sonication cleaning 30% S + 70% BR 15 min toluene statistically arranged 15 min ethanol 15 min toluene Young s modulus MPa 3.3 GPa GPa (bulk, static) expected rubber-elastic elastic hard; marginal behavior clear relaxation glassy deformation polymer structure Table 4.1: Characteristics of the sample materials as used for the measurements shown in Fig. 4.9 and The experimental settings for the measurements were: modulation amplitude: 250 nm; modulation frequency: 1.0 khz; maximum normal load: 1.2 µn. The experiments were conducted in ambient conditions with cantilevers of m N force constant at a fundamental frequency of khz. The temporal response of the sample materials is

55 4.4. DPFM on Polymers 39 (A) force versus time (B) force versus position of the cantilever support (C) force versus indentation depth Figure 4.9: Three materials were probed in this measurement: SBR (circles), PMMA (diamond) and silicon (solid line). From the force versus time data as acquired by the DPFM (A), the known trajectory of the base of the cantilever can be removed resulting in plot of force versus position of the cantilever support (B) using equation 4.1. Silicon is taken as the reference material. The slope of the calibration curve is subtracted from all other data in order to re-scale the horizontal axis. Finally in (C), all three materials are plotted versus the indentation depth (the real physical quantity). Areas now correspond to work applied to the sample. The theories of continuum mechanics can be used to analyze the data, revealing Young s moduli or surface energies of the sample materials.

56 40 Chapter 4. Measurements and Calibration Techniques for Digital Pulsed Force Mode shown in figure 4.9 (A) as acquired by the DPFM. Following the calibration and evaluation strategy explained in chapter 4.2, the force versus time curves have been converted into classical FDCs as seen in figure 4.9 (B). The comparison of the silicon and PMMA curves is shown in figure 4.10 (A). Using the sample substrate (silicon) as an on-line reference (made visible by scratching the spin-coated surface in advance to the measurements), the data can be analyzed from the force versus indentation plots (Fig. 4.9 (C) and 4.10 (B)). For conversion, the position of the cantilever support deflected by the silicon sample is subtracted from the position of the cantilever support on the other materials. This yields the indentation depth as new plot axis Results from Individual DPFM Force Curves The creation of surface maps to pin physical properties to topographical features is based on single force curves measured by the DPFM setup. For an image, a vast amount of curves are collected and stored. What can be obtained from single curves is described in the following. (A) (B) Figure 4.10: Silicon is used as reference material. The linearity of the force curve of Si for high forces is used for calibration (A). The difference in position from the force trace on silicon compared to PMMA yields the indentation depth (B). At small forces the deformation of the contact can be observed. PMMA is compliant. Figure 4.10 shows a closeup view of figure 4.9 (B) and (C) for the silicon and PMMA force traces. In the force versus indentation graph (Fig (B)), silicon shows a small indentation which stops right after the point of contact, while PMMA allows some deformation. SBR does not seem to resist the indentation a lot as seen in figure 4.9 (C). The behavior of silicon is not surprising, since this hard material follows the Hertzian or JKR description quite perfectly. The scale of the indentation (1 nm) is small compared to the radius of the tip (approx. 50 nm, estimated from SEM imagery), so a Hertzian or JKR model is justified. It reveals an almost perfect elastic behavior for indentation and retraction. The area enclosed by the force trace, which corresponds to the hysteretic loss of one

57 4.4. DPFM on Polymers 41 cycle, performed on silicon is due to the uncertainty in cantilever deflection. This is the minimum uncertainty for all energies determined in the following. PMMA also behaves elastically, but with a much flatter slope than silicon. However, due to the application of quite high forces to the sample it is astonishing that no deformation remains. This is in contrast to FDC measurements [32] where residual indentations are observed. The main difference between those FDC measurements and our PFM data is the interaction time (or the frequency range) between the tip and the sample. It is known from classical DMA measurements [28] that polymers respond quite differently to stimuli at different frequencies. The theoretical side of DMA is covered by Christensen [19]. One can infer from those DMA measurements that PMMA is an almost perfect elastic material in the range of the PFM interaction times ( µs). For interaction times of 1 s as used in FDCs, viscoelastic processes are important. On SBR a large area under the force trace can be observed which means that large amounts of energy are dissipated by the material either by plasticity or viscoelasticity. However, since no surface destruction on the SBR areas is visible, one can conclude that the dissipative behavior has to be attributed to viscoelasticity alone. Otherwise the surface would have to show remanent plastic deformations. The timescale at which this relaxation happens has to be above 50 µs, since the sample does not push back quickly enough to observe a fully elastic behavior in the PFM curves. The timescale also has to be clearly below 1 ms, since the surface is restored within one single cycle. Another effect of time-dependent elasticity can be seen at the position where the maximum force is reached. While on silicon and PMMA the tip reaches its maximum indentation depth at this point, on SBR it indents further into the material even though the force is already lowered. This is due to the delayed deformation of viscous materials under strain. In figure 4.11 (A), the indentation depths at which F max is reached are compared for all three sample components. This is what can be measured directly from the curves in figure 4.9 (C). The post-flow, or creep, can be seen in figure 4.11 (B). As stated before, the area under a force curve corresponds directly to work involved in the motion. On the trace (approaching closer to the surface) elastic, plastic and viscoelastic forces are acting on the tip. In other words, work has to be done in order to push the tip into the sample. The elastic parts will be recovered on the release, while the plastic deformation will remain as a permanent deformation and the viscoelastic part heats the sample. The position at which the force equals zero again on this retrace gives an estimate of the depth of this remanent indentation as described by Oliver and Pharr [63, 64]. However, on SBR such a permanent deformation has not been observed in the scans. This happens, if the recovery rate of the material is smaller than the retraction rate of the tip. Thus, the surface can not be restored fast enough. In figure 4.11 (B), the energy deposited into the material on the trace part, the hysteresis of a single cycle, and the energy required for detachment from the surface are shown. For silicon, no hysteresis is observed, and only

58 42 Chapter 4. Measurements and Calibration Techniques for Digital Pulsed Force Mode (A) (B) (C) Figure 4.11: From the PFM curves in Fig. 4.9, one can derive the local behavior of the materials. (A) is showing the depth at which the pre-set maximum force was reached. From the curve describing SBR, one can clearly measure a post-flow (creep) distance (B), i.e. the difference between the position of maximum force to maximum indentation depth. (C) shows characteristic energies involved in the indentation process. Deformation energy is the energy that is deposited into the sample upon the loading path. During unloading, the surface recovers and energy is set free, except for the amount lost in the hysteresis of the material. In case of SBR, the ratio of energy lost in one cycle is above 90% while for the other materials it is close to zero. Detachment energy is also shown and reveals that SBR is the stickiest of the three materials.

59 4.4. DPFM on Polymers 43 a minuscule one for PMMA. For SBR the hysteresis is about 10 times larger. A similar result is found for detachment. Silicon and PMMA do not differ too much (PMMA being a bit less adhesive than silicon), while SBR firmly sticks to the tip until the contact ruptures Mapping Physical Quantities to the Topographical Features In order to analyze the maps created by the data-evaluation, boolean evaluation masks have been created including only the desired evaluation regions. This way, it was possible to exclude, for example, edge effects like the slipping away of the cantilever at a descend from one material to the other. Three different masks for each material have been created. Figure 4.12: Masks used for data evaluation. The areas allow selection of the materials when determining the histograms of the material parameters. On the left, the silicon region is marked (as used for calibration), in the middle, the PMMA islands, and on the right, the SBR regime. The masks are especially important when doing histogram analyses. In case that the relative areas of each material differ too much, one has to mask the area which has to be evaluated. Thus, the algorithms of the evaluation software are restricted to the predefined areas. Afterwards, the histograms have to be normalized to the number of pixels which they are based on, in order to be comparable. If any two materials, however, are too close in terms of their physical properties that a histogram of the entire image is not able to resolve the differences, the statistics of the separately evaluated regions will also tell that the regions overlap too much to distinguish the materials by their behavior. The number of pixels per each mask are huge, resulting in statistically backed data. For the masks shown in figure 4.12, the counts per mask are: Mask of Si pixels Mask of PMMA pixels Mask of SBR pixels

60 44 Chapter 4. Measurements and Calibration Techniques for Digital Pulsed Force Mode Maximum Force In order to maintain the maximum force which is exerted on the sample, the scanning unit is permanently repositioned by the feedback loop. There are approaches to enhance the performance of the feedback loops, for example by implementation of model-based feedbacks [72]. However, these approaches also suffer from the rule of cause and consequence, since without action there will be no reaction. Therefore, the change in topography or material compliance causes the cantilever to bend further back and thus the scanning unit to retract the sample. However, there is a certain time necessary for the setup to respond. Therefore, the force on the sample is higher when crossing an elevation and lower on the descend. The map of the maximum force during each force curve can be used to comment on the quality of the measurement settings. The oscillations visible in the image are due to 50 Hz mains noise. Deviations at edges only correspond to the delay in the feedback. The map of the maximum force relative to its average value is shown in figure It shows what deviations in the maximum force occurred during this measurement, but almost exclusively at edges. Since the edge regions of the materials will not yield any physical results, because the contact geometry is different from the one on a flat region, this map can be used to as an exclusion filter. A defect of the surface is also evident in the image. While scratching the surface to reveal the silicon wafer as a reference sample material, the scalpel also scratched parts of the sample surface that was intended to remain undamaged. The scratch runs parallel to the edge of the polymers. Probably, there was an edge in the scalpel, since this damage appeared on most of the samples that have been investigated in the course of this study. Figure 4.13: The relative error of the maximum force gives an estimate of the quality of the measurement settings.

61 4.4. DPFM on Polymers 45 Topography at Maximum Force During the scanning of a sample, a permanent maximum force is exerted on the sample materials. This force is given by the preset setpoint and is measured as described above, giving information on the quality of the settings. Measuring at a given force is also the reason why observation of the original and undisturbed sample surface will never be possible in any force controlled SPM mode. Hence, the topography that is recorded corresponds to the topography at the given force as shown in figure It seems that the presumably soft SBR surface (more details later) has a smoothly curved surface, while the PMMA islands are almost equally high. The silicon wafer, of course, is plainly flat and thus used as reference height zero. Figure 4.14: The topography of the surface as measured by the AFM setup. The surface gives the impression of an SBR region curved downwards towards the PMMA islands. The cross-section (Fig. 4.15) along the blue line in figure 4.14 reveals the deformed surface of the sample even clearer. The PMMA islands are rather flat and have a height of about (55±3) nm. SBR appears about 25 to 35 nm high with respect to the silicon substrate and bent towards the wafer. Indentation Depth and Real Topography The indentation depth where the tip reaches the maximum force within the sample material can be calculated as described in chapter 2. The map of the indentation depth at maximum force (Fig. 4.16) shows that the SBR regions become severely indented during the force cycles due to the rather high setpoint forces used during the experiment. On the

62 46 Chapter 4. Measurements and Calibration Techniques for Digital Pulsed Force Mode Figure 4.15: The cross-section of the topography as measured by the AFM setup. The islands of PMMA seem to be edgy and equally high, while the SBR regions in between appear much lower and bent towards the substrate. PMMA islands, this is not that obvious, but still distinguishable from the indentations into the silicon wafer. The cross-section of the indentation depth into the sample materials is shown in figure Using the calculated indentation depth as a correction to the deformation of the topography caused by the scan at maximum force, yields a rather surprising result if one is only used to images recorded at a given force. Both, the PMMA and the SBR have a very flat surface and are almost equally high. This, however, corresponds perfectly to what is known from the dewetting polymers. At the solid-air interface, polymers in almost every case have a very smooth and flat surface. The cross-section of the corrected and thus real surfaces is plotted in figure The corrected map of the surface height is shown in figure At edges (for example around the tiny islands standing out of the SBR background), one can still observe overshoots that are due to deviations in the feedback loop. Uniaxial Deformation With the corrected topography, it is possible to calculate the dimensionless deformation factor ξ which correlates the deformation - here: indentation depth - and the original thickness - here: corrected height. ξ is defined in analogy to the classical linear-strain parameter λ. In order to put the results into the context of classical polymer-science,

63 4.4. DPFM on Polymers 47 Figure 4.16: The topography measured by an AFM deviates from the real and undisturbed surface by the indentation depth required to reach the maximum preset force. Figure 4.17: The cross-section of the indentation depth.

64 48 Chapter 4. Measurements and Calibration Techniques for Digital Pulsed Force Mode Figure 4.18: Having corrected the topography by the indentation depth results in the real topography. The small overshoots at edges remaining in the topographical information are due to deviations in the feedback signal. Figure 4.19: The cross-section of the real topography after correction by the indentation depth.

65 4.4. DPFM on Polymers 49 this parameter is essential. Stress-strain curves are always plotted versus λ for example. Dividing the entire images pixel-by-pixel, one can obtain a map of the compression factor as shown in figure Figure 4.20: Dividing the indentation depth by the sum of indentation depth plus measured topography results in the deformation factor ξ. For the silicon region ξ is equal to zero, since the initial thickness is the thickness of the silicon wafer. Compared to this, all nanoscale deformations are negligibly small. For SBR and PMMA, however the results are clear: with an original height of (55±3) nm, the PMMA areas are indented by about 6 to 8% of their initial film thickness, while SBR, starting at an average height of about (47± 3) nm, is extremely deformed with ξ ranging from 20 to 65%. This behavior can also be found in the cross-section along the same line as before (see figure 4.21). Sample Compliance from a Flat-Punch Approach The easiest way to compare materials in terms of their compliance to external loading is to expose them to the same external load and observe the deformation caused. This means dividing the maximum force by the indentation depth needed to reach this loading. The model that resembles this situation is called flat-punch, since a flat circular stamp is pressed onto the material. The problem when modeling this is that infinite stresses would have to build up at the edge of the stamp. However, the model is good enough to compare materials in a relative manner. Hence, the results are shown in figure Also in the histogram (Fig. 4.23) of the entire image it is easy to identify the three peaks that correspond to the three shadings in the figure. The widening of the rightmost peak,

66 50 Chapter 4. Measurements and Calibration Techniques for Digital Pulsed Force Mode Figure 4.21: The deformation of the materials is obvious in the cross-section of the map of ξ. Figure 4.22: One of the easiest models - the flat punch - yields an quick differentiation of the materials. However, it is too simple to replace a deeper analysis.

67 4.4. DPFM on Polymers 51 which corresponds to the compliance of silicon, is due to the calculation of the compliance. Dividing F max by the indentation depth of silicon, which is on the order of one nanometer, very small deviations cause large scattering of the results. Thus, this scatter of the data is increasing with the increasing hardness of the material, which in turn widens the silicon peak. Figure 4.23: In the histogram plot, the three different materials clearly stand out. Left: SBR; middle: PMMA; right: silicon. This easy approach, how nice the images might be, is not a very physical one. Since only the secant is used to calculate the compliance of the material, it totally neglects the curvature of the approach curves as described by all contact mechanical models that have been proven more realistic. Yet, it at least suggests that the more realistic models have to yield similar results in terms of the differentiation between materials. Young s Modulus from JKR Calculations Using the theories of continuum mechanics (Ch. 2) to simulate or fit force curves, one has to choose the model very carefully. Since one can observe detachments that also cause the sample to be deformed, one definitely has to discard the Hertzian approach and turn to the models incorporating adhesive influences. Since the Maugis-Dugdale model has too many free parameters that might fit the theory to mostly every curve, it is not very practical in terms of automating an evaluation. The DMT approach, on the other hand, assumes that none of the surfaces is deforming during the detachment of the tip from the surface, so neither does that apply to our problem. The only one left is the JKR model, which

68 52 Chapter 4. Measurements and Calibration Techniques for Digital Pulsed Force Mode has been implemented in the data evaluation as described in chapter 5. By subtracting the adhesive contribution from the repulsive approach region in the force curve, what remains is a function of the indentation depth to the power of 3 2. Thus, this model can be easily applied to the data at hand. The calculations lead to the effective modulus, labeled K in the theoretical derivations in chapter 2. The resulting map of the effective modulus is shown in figure However, the Young s modulus can only be obtained by plugging reference values for the Young s modulus of the indenter and for the Poisson s ratios into the equation for K (Eq. 2.22) and then solving for the Young s modulus of the material that was indented. Figure 4.24: The JKR model, using a spherical indenter for the calculations, results in a map of effective moduli K. For each material the Poisson s ratio has to be known in order to calculate the real Young s modulus. Another behavior that was observed is that the effective modulus, and thus also the Young s modulus, decreases exponentially with increasing deformation ratio. The decrease in the effective modulus is shown in figure It can be explained by classical stress-strain experiments. Since there is no strictly linear range in either of the materials, the measured modulus is decreasing with increasing ξ. Even for small elongations of PMMA the linear Hookian region is left, since the stressstrain curve is bent from the very beginning. Stress-strain curves at different temperatures are shown in figure On SBR, the same behavior is found. However, the explanation is different, since rubber elasticity differs from the elasticity of glassy materials. Looking at a stress-strain experiment on a rubber sample yields a plateau area as shown in figure 4.27 for the unvulcanized sample. Stretching experiments with ξ from about 20% and up to 60% defi-

69 4.4. DPFM on Polymers 53 K versus ξ plotted linear and double-logarithmic Figure 4.25: The effective modulus decreases with the increase in deformation ratio. This can be explained looking at classical stress-strain-experiments. PMMA ranges for ξ from 0.05 to 0.1 and for K from 10 to 100 GPa. SBR, however is deformed between 0.2 to 0.6 and ranges from 0.1 to 10 GPa. Figure 4.26: Stress-strain curves of PMMA for different frequencies (from [16]). The curves are bent even for small elongations.

70 54 Chapter 4. Measurements and Calibration Techniques for Digital Pulsed Force Mode nitely work in this regime. This suggests that the SBR just like the PMMA above has to decrease in its Young s modulus for increasing deformations. Figure 4.27: Stress-strain curves for vulcanized and un-vulcanized rubber samples show a large plateau region. The absolute values for the Young s modulus assuming an elastic modulus of 160 GPa for silicon, and ν Si = 0.15, ν PMMA = 0.4, and ν SBR = 0.5 for the Poisson s ratios, are: Si - (110.1 ± 7.5) GPa PMMA - (42.6 ± 10.5) GPa SBR - (0.84 ± 0.64) GPa In comparison to literature values, these results appear by far too high to describe the materials present on the sample. Literature values for a bulk sample under static conditions are: 3.3 GPa for PMMA and 150 MPa for SBR. However, this is exactly the benefit of the AFM technique. The contact between any two bodies is always occurring on the surface. Thus, the mechanical properties of the surface are important. One is underestimating the elastic behavior if only using the lower literature values for bulk samples instead of the higher values obtained by a specialized surface-sensitive technique. PMMA will be discussed first. The reason, why the calculation of the Young s modulus yields a value that is too high is, that at a surface, the PMMA chain-molecules are not free to align as coils as in a bulk phase, but rather have to gain a flat arrangement due to the confined geometry. This increases the initial resistance towards indentation. As has been shown by Wilhelm and Frey [91] for actin cortex filaments of the cellular cytoskeleton that

71 4.4. DPFM on Polymers 55 the modulus decreases for increasing deformations if the geometrical freedom is reduced. A similar result has been reported by McHargue [53] for a diamond like carbon film on a softer glass substrate which also seems much harder at the beginning of a nanoindentation experiment, but then becomes more compliant upon increasing deformation, since the compliance of the substrate becomes increasingly important. Thus, indenting into a material like PMMA, which might have a coil structure in the bulk but a network or at least a stretched filament structure at its surface, must yield values that are higher than the ones from literature that have been obtained for a macroscopic experiment on a bulk material. That the values for the SBR are too big by about an order of magnitude is backed by the theory of heat capacity. As in thermodynamics, isothermal and adiabatic processes occur. Isothermal being characterized by energetic equilibration and adiabatic occurring in absence of balancing. Since the indentation experiment happens at non-zero velocities, one has to move from a isothermal view, that would be valid for a quasi static evolution of the forces applied, to an adiabatic model that is too fast for the exchange of energy. Energy diffusion can be described by the random walk. The average distance is given by: < x 2 >= 6Dt. (4.2) t being the interaction time, D the diffusion constant. For the DPFM experiments the interaction time is about 0.1 µs and D is about 10 7 m2 s. Thus, the average diffusion length-scale is about 250 nm, which is about the scale of the deformations in use. Therefore, the modulus of the SBR or PMMA cannot be calculated by an isothermal approach. This is why the Young s modulus has to be higher than for a purely static experiment. In case of the large deformations of SBR on the order of the initial film thickness, an influence of the substrate on the measurement of the Young s modulus also cannot be ruled out. Young s Modulus from a Conical Model Loading a sample with a spherical indenter as described above may not apply to all indentation experiments, especially in the case of indentation depths larger than the approximated radius of the tip of the cantilever, a conical indenter may be justified. SEM imagery suggests either a pyramidal or a conical shape for the adjacent form of the tip. Following the calculations by Sneddon for indenters with rotational symmetry as described in the theoretical part (Ch. 2), one can fit a force curve by a parabola. However, this model is exclusively applicable for indentations much larger than the tip radius, where the cone is really involved in the indentation process. As at the infinitely sharp tip in the case of a cone, infinite stresses occur, small indentations have to yield far too high values for the

72 56 Chapter 4. Measurements and Calibration Techniques for Digital Pulsed Force Mode effective modulus. In general, the effective modulus K can be transformed into the corresponding Young s modulus again. The values that can be determined in the measurements discussed here result in these Young s moduli: Si - ( 233 ± 200) GPa PMMA - (526 ± 10 5 ) GPa SBR - (1.95 ± 1.90) GPa assuming an opening angle of the cone of 40 degrees and again a Young s modulus of 160 GPa for silicon together with the Poisson s ratios as given above (ν Si = 0.15, ν PMMA = 0.4, and ν SBR = 0.5). Even though, the values for the effective modulus seem reasonable at first sight, the Young s moduli that are yielded by the calculation are rather disillusioning. However, as stated in chapter 2, the conical tip indents too little into silicon or PMMA. Thus, the region of the tip of the cantilever which has the slope of the cone and not the spherical shape will never be effective. This is why the values for E explode and from the calculations it is clear that E Si has to turn out negative, since the modulus for the silicon of the cantilever which was put into the formula is too small. The map of the effective modulus K that can be generated from this analysis is shown in figure Figure 4.28: A cone based model cannot resemble a correct Young s modulus if the indentation is too little. This model may be applied to large indentations only. The effective modulus K is plotted here.

73 4.4. DPFM on Polymers 57 Maximum Indentation Depth and Post-Flow The post-flow that has been observed in the single curve evaluation above serves as a basis for the following considerations on the creep of viscoelastic materials. Figure 4.29: For evaluation, several points of an indentation are especially interesting. The distance from δ (F max ) until maximum indentation is called post-flow distance. When a purely elastic material is considered, one would expect the force and indentation depth to rise until the turning point of the modulation and then decrease down along the curve of the approach. However, there are no purely elastic materials, but only materials having time-constants that are too small to be accessible by the experiment. In turn, so called viscoelastic materials or also rubbers may allow investigation of creep phenomena, since their spectrum of time-constants includes a region of several tens or hundreds of micro-seconds. Thus a DPFM experiment conducted at 1 khz might allow to estimate relaxation time-constants of these materials. In this case, SBR reveals some nanometers of creep, since after applying the maximum force externally to the material, the tip indents even more upon unloading as shown in figure Since, on the other hand, no remanent deformation has been observed in the next curve about 1 ms later, the relaxation time has to be between 50 and 500 µs. In terms of a map of the maximum indentation depth, there is nothing special compared to the map of maximum indentation. However, their difference, the post-flow of the material, looks very interesting, since it allows access of the deformation due to creep. The map of the post-flow is shown in figure 4.30 and the cross-section in figure 4.31.

74 58 Chapter 4. Measurements and Calibration Techniques for Digital Pulsed Force Mode Figure 4.30: Viscoelastic creep lets the tip of the cantilever indent further into the material, although the loading force is already decreased. The distance of creep is called post-flow. Figure 4.31: The post-flow is within the error margin for silicon and PMMA, but almost 1 nm for SBR covered regions.

75 4.4. DPFM on Polymers 59 Energy Transferred to the Sample Materials Integrating a force curve with respect to the indentation depth yields the energy needed for the displacement, which corresponds to the area marked in figure Figure 4.32: Area which corresponds to the energy needed for a complete indentation. Thus, the energy which is transferred to the sample materials is the integral of the force trace from the position of snap-in until the maximum indentation depth. Remember that for SBR, the maximum indentation depth strongly differs from the depth needed to achieve the maximum force. Therefore, the energy is also higher. Typical energies needed to indent the sample are: Si - (0.18 ± 0.05) J ˆ= (1.13 ± 0.32) kev PMMA - (0.98 ± 0.05) J ˆ= (6.13 ± 0.32) kev SBR - (5.6 ± 0.08) J ˆ= (35.00 ± 0.05) kev The errors have been determined as the average standard deviation for the same pixels as the average itself. The storage processes are rather complicated. First, there are elastic contributions which will be regained on the release. Second, there are dissipative processes such as generation of heat by internal friction within the sample material or plastic deformation. To compare these effects, one has to look at the energy lost during one cycle of the experiment, namely the hysteretic losses.

76 60 Chapter 4. Measurements and Calibration Techniques for Digital Pulsed Force Mode Energy Lost during one Cycle - Hysteresis Integration of the force curve from the point of snap-in until the point, where the force is equal to zero again on the retract path, yields the hysteretic loss during one force cycle. This is sketched in figure Figure 4.33: The hysteretic losses correspond to this area. The energies that are lost are: Si - ( 0.02 ± 0.024) J ˆ= ( 0.13 ± 0.15) kev PMMA - (0.30 ± 0.023) J ˆ= (1.88 ± 0.14) kev SBR - (5.2 ± 0.04) J ˆ= (32.50 ± 0.03) kev To obtain a negative value for silicon is due to the random noise on the force signal. From the standard deviation that is calculated for each pixel in the data-evaluation programs as described in section 5, one can calculate the accuracy to ± J. As above, the average standard deviation is used as error. Calculating the fraction of energy lost to hysteresis and energy transferred to the sample, one can see the tendency of the material to absorb energy: Si - ( 11.1 ± 13.7) % PMMA - (30.6 ± 2.8) % SBR - (92.9 ± 1.5) % The result for silicon is quite unsure, since the integration became corrupted by the noise of the signal yielding a negative hysteresis energy. The errors have been calculated using the Gaussian error propagation.

77 4.4. DPFM on Polymers 61 Remanent Depth Since hysteresis can be observed in the force curves, one also has to consider the possibility of leaving permanent damage on the sample. An estimate of the damage caused by an indentation is the remanent depth. This depth is given by the distance from the point of zero force right after the snap-in until the force is zero again on the retract part of the force trace (the same end-point as in the calculation of the energy of hysteresis). The points of F = 0N are labeled in figure Figure 4.34: The remanent depth corresponds to a possible plastic deformation. Calculation of the remanent depth yields a map as shown in figure 4.35 and average values of: Si - (0.0 ± 0.1) nm PMMA - (0.5 ± 0.1) nm SBR - (13.2 ± 0.15) nm On PMMA, the remanent depth of about half a nanometer suggests that the surface has been plowed a little bit, while for SBR there should really be huge deformations visible in the images. However this is has not been observed. Furthermore, the areas of the indentations always overlap, since the scanning speed was two seconds per each line of the image over a length of 15 µm sampled at 1 khz repeat rate, which means that the distance between two indentations is 7.5 nm. This is much smaller than the diameter of the tips in use, but a change in the curves has not been found. Thus, the remanent depth measured on the SBR regions is due to the viscoelastic delay in surface restoration during the retract of the tip.

78 62 Chapter 4. Measurements and Calibration Techniques for Digital Pulsed Force Mode Figure 4.35: The remanent depth of a force curve gives an estimate on the permanent damage that is caused during a force cycle. Local Adhesion A quantity that can be directly determined from the force traces is the force that is needed to detach the tip of the cantilever from the sample material. It is a direct measure for the adhesive properties as described for example by either model JKR, DMT, or MD. Determining the local adhesion for the polymer samples described above, results in a map (Fig. 4.36), where basically two regions can be clearly distinguished: First, the silicon (scratched area) and PMMA regions (round structures), that reveal a very similar detachment behavior, that can be tracked to the layer of water that covers most surfaces under ambient conditions. Second, the SBR areas, which obviously stick much more to the silicon tip. Actually the detachment force is about 3.3 times higher on the rubber surface than on the PMMA regions, while from PMMA to silicon it is only an additional 20 % of force needed to detach. In Newtons the adhesion forces are: Si - (119 ± 15) nn PMMA - (103 ± 11) nn SBR - (349 ± 13) nn In general, the adhesive properties of the materials comply with intuition. Silicon and PMMA demand about the same pull-off force to detach the tip from the surface, while SBR shows greater affinity to the silicon of the cantilever tip. Due to its large compliance, it is even possible to deform the surface outwards, resulting in a negative detachment position which will be described later. Thus, it is the stickiest of the three materials. At edge positions one has to be aware of the fact that topographical features have an influence

79 4.4. DPFM on Polymers 63 Figure 4.36: Map of the local adhesion of the dewetting polymers SBR and PMMA on silicon. on the contact area and thus on the adhesive behavior. As Stifter et al. showed in a series of papers [77, 78], the changes in adhesion in SPM measurements is mainly caused by the topographic shape of the samples. In turn, their model allows determination of the tip shape parameters when the shape of the sample is known, for example a step on an HOPG surface or a step on a calibration grating. Detachment Distance It is also interesting in terms of adhesion not only to look at the extremal force which allows detachment, but also to determine the separation of tip and sample, where the cantilever finally comes out of contact from the sample surface again. This distance corresponds to the ability of the surface to deform upon pulling the tip away. The distance is calculated as the difference in indentation depth from the point of force equal to zero after the snap-in until the point where the force equals zero after the force of adhesion had been reached on the retract as shown in figure The average results are: Si - (15 ± 2) nm PMMA - (15 ± 2) nm SBR - (30 ± 2), (60 ± 2), (90 ± 2) nm For silicon and PMMA this detachment position above the point of snap-in can be attributed to the water-layer which is present under ambient conditions. The liquid necking towards the rather big tips ( 50 nm as shown in figure 4.42) can explain that. For SBR,

80 64 Chapter 4. Measurements and Calibration Techniques for Digital Pulsed Force Mode Figure 4.37: The cantilever is completely out of contact at the detachment position. however, three accumulation points can be determined. This is due to the great affinity of SBR and silicon, which means that the tip is strongly glued to the surface on the SBR regions. Finding clear peaks in the histogram of the SBR region, however, suggests that there is even more to this effect. Material is dragged out of the surface and sticks to the tip. Thus, the distance depends on the amount of material pulled off. It even suggests a certain discreteness of the rubber, for example by coiled molecules, since the peaks are equidistant. DPFM may open up the doors to a new type of force spectroscopy, since it is able to add a statistical component to the search for singular detachment behavior like protein unfolding etc. One no longer has to put huge effort into finding the positions by tedious force distance curves, but is able to just scan the surface and then analyze the images for these events. In the case of SBR (Fig. 4.38), the histogram contains about pixels, which by themselves consist of about six PFM curves each. This vast amount of data has never been available before. Energy of Detachment Having seen the adhesive behavior in terms of detachment force and detachment distance, one can also calculate the energy needed to detach the tip from the surface. The integration is done on the region from the force equal to zero point on the retrace until the cantilever is free of force again. Thus, the entire region where the force curve is in an adhesive regime (negative force values) is used.

81 4.4. DPFM on Polymers 65 Figure 4.38: Histogram of the detachment distance for the SBR regions. The plot consists of the results of pixels. Figure 4.39: To rupture the tip sample contact the energy corresponding to the marked area is needed.

82 66 Chapter 4. Measurements and Calibration Techniques for Digital Pulsed Force Mode The energies also result in a surface map (Fig. 4.40). The average values that have been determined in this measurement are: Si - (0.82 ± 0.08) J ˆ= (5.13 ± 0.50) kev PMMA - (0.78 ± 0.07) J ˆ= (4.88 ± 0.48) kev SBR - (11.0 ± 0.2) J ˆ= (68.75 ± 1.25) kev Figure 4.40: Detachment from the surface requires a certain energy. On SBR this energy is much higher than on silicon or PMMA.

83 4.4. DPFM on Polymers Reproducibility Experiments In the previous sections the emphasis was clearly on the evaluation of a single measurement obtained by using the DPFM setup. However, it is always doubtful, if the same results can be reproduced in an independent experiment. Therefore, a set of ten different cantilevers of the same wafer have been used for DPFM experiments. The measurement parameters have always been chosen the same and the samples have been freshly spin-cast onto the silicon wafers before the experiment and scratched to reveal the substrate as a reference. The tips have been characterized before and after use in the SEM of the section for electron microscopy at Ulm university. For the recording of the data, it was always the first measurement after setting the gains of the feedback-loop, which has been used for evaluation. Thus, a total of fourteen measurements have been conducted. Each of which has been calibrated separately and the same protocol had been followed for each of the experiments. The DPFM measurements have also been conducted on different days over the course of two months to allow changes in the ambient conditions in order to make it a highly realistic study. SEM Images of the Tips of the Cantilevers The tips of the experiments described above has been characterized by means of SEM imagery. Thus, the tip radius can be estimated from a measurement and the claims of the manufacturer do not have to be taken for real. As an example, lever number three (overview in Fig is shown in figure 4.41 before and in figure 4.42 after use. It is obvious that the tip cracked due to the high maximum forces that were applied to the sample. However, it has never been observed that the lateral resolution remarkably changed during an experiment, which would suggest a breaking event during a scan. In fact, it seems that the tip broke on the first approach to the surface and adjusting the scanning parameters. What makes this quite special is, that the same has been observed for all the other tips, too. The resulting tip shape after the break, however, seems extremely stable and does not change anymore.

84 68 Chapter 4. Measurements and Calibration Techniques for Digital Pulsed Force Mode Figure 4.41: A fresh cantilever imaged by SEM before use. A radius of below 20 nm as claimed by the manufacturer seems justified. Figure 4.42: The tips obviously break when operated at elevated forces. However, the emerging large tip radius (approx. 50 nm) seems to be stable after the first break. Figure 4.43: The tip is positioned right at the end of the cantilever.

85 4.4. DPFM on Polymers 69 Check of Reproducibility The reliability of the method and the reproducibility of the DPFM experiments following the calibration and measurement technique described above will be demonstrated in the following. Especially the commonly most doubted results, namely the determination of the elastic modulus based on the JKR model, the previously unaccessible energy of hysteresis, and the dissipation fraction will be analyzed. The samples have been produced freshly for every experiment. Thus, one may wonder if the samples always generated the same patterns and structures. From the surface topography, it is clear that the islands that have formed from PMMA within the surrounding SBR dewetted resulting in very similar or even morphologically identical structures. Having scratched the sample, on the other hand, also allows the analysis of the thickness of the layer which were formed. The results of all 14 measurements is shown in figure Figure 4.44: The sample structures have to be highly reproducible in order to allow the comparison of the overall measurements. The average results on the materials are: Si - (1.7 ± 0.4) nm PMMA - (56.2 ± 1.0) nm SBR - (50.0 ± 2.3) nm The error margins of these measurements have been calculated as 3σ from the average value, thus comprising 99% of all values. Using the same maximum force for all experiments is necessary to maintain comparable conditions. Thus, on the same samples, the same indentations are expected. In figure 4.45, the indentation depth for all 14 measurements is plotted. It is clearly visible, that SBR

86 70 Chapter 4. Measurements and Calibration Techniques for Digital Pulsed Force Mode is indented between 15 and 30 nm until F max is reached. For PMMA, the indentation depth is constant at a level of 3 nm. On the scratched areas of the silicon wafer, the indentation depth is calculated as approximately 1.7 nm. However, this corresponds to the deformation of the tip and the sample until the linear regime of the force versus indentation plot is reached, which was used as calibration regime. The averages and three times the standard deviation are: Si - (1.7 ± 0.3) nm PMMA - (3.1 ± 1.1) nm SBR - (23.4 ± 2.7) nm Figure 4.45: Reproducibility of the indentation depth. As described above, the deformation ratio ξ is calculated as the fraction of indentation depth and corrected topography. For all the measurements conducted, the three materials clearly separate (Fig. 4.46). For Silicon, of course, the result is zero, since the initial thickness is infinitely large compared to variations on the order of nanometers. However, for PMMA and SBR, the results are clear. An average deformation of 0.05 is obtained for PMMA, while on SBR ξ ranges between 0.4 and 0.6, which are quite high values for straining experiments. Again for the 14 measurements, the averages are: Si - 0 PMMA - (0.056 ± 0.019) nm SBR - (0.460 ± 0.059) nm The simplest model to determine sample compliance is the flat punch approach. The results are shown in figure It is calculated by dividing the maximum force of the DPFM curve by the indentation depth at which this force is reached. For the hard materials PMMA and silicon, the values obtained are strongly fluctuating, which is because of the division by a small - and thus uncertain - quantity, namely the indentation depth. However,

87 4.4. DPFM on Polymers 71 Figure 4.46: Reproducibility of ξ. silicon ((603 ± 163) m N) turns out to be about twice as hard as PMMA ((390 ± 117) m N), which is reasonable. As on SBR, the indentation depth is bigger, the values obtained for the flat punch compliance do not scatter very much. (45.2 ± 6.6) m N is calculated as average hardness of the rubber. Figure 4.47: Reproducibility of the flat punch compliance. The model which is much better suited to describe the indentation measurements is the JKR-based calculation of a sphere indenting into the sample material. The effective modulus K which is determined is plotted in figure The average results of the measurements are: Si - (99.6 ± 18.4) GPa PMMA - (69.7 ± 25.5) GPa SBR - (1.77 ± 0.25) GPa

88 72 Chapter 4. Measurements and Calibration Techniques for Digital Pulsed Force Mode One can see that the materials can be distinguished, however, the real physical quantity is the Young s modulus of the material. Figure 4.48: Reproducibility of the effective modulus. Again as shown for the individual measurement above, one can plot a two-dimensional histogram of the effective modulus versus the deformation parameter ξ. Adding up all 14 layers of these histograms yields figure Here, it is easy to observe that the measurements were reproducible. Otherwise the accumulation would never be that strong. Figure 4.49: Reproducibility of K versus ξ.

89 4.4. DPFM on Polymers 73 Assuming Poisson s ratios and the elastic modulus for the silicon of the cantilever, one can derive the Young s modulus from the effective modulus. Like before, the parameters of 160 GPa for silicon together with the Poisson s ratios (ν Si = 0.15, ν PMMA = 0.4, and ν SBR = 0.5) have been used. The results thus are: Si - (141 ± 44) GPa PMMA - (73.8 ± 38.0) GPa SBR - (1.01 ± 0.14) GPa and also plotted in figure Figure 4.50: Reproducibility of the Young s modulus. The hysteretic loss during the indentation experiments corresponds to the energy dissipated by the material. For all of the experiments, the average losses are plotted in figure Silicon and PMMA do not show significant losses, which is due to their high elasticity. There are no viscoelastic effects visible. For SBR, however, most of the energy that is transferred to the sample during the deformation is lost: Si - ( 73.6 ± 35.8) J PMMA - (0.133 ± 0.397) J SBR - (8.24 ± 1.53) J This is especially obvious in figure 4.52, where the ratio of dissipated energy to energy stored in the material due to the deformation is shown.

90 74 Chapter 4. Measurements and Calibration Techniques for Digital Pulsed Force Mode Figure 4.51: Reproducibility of the hysteretic losses. Figure 4.52: Reproducibility of the dissipation ratio.

91 4.4. DPFM on Polymers 75 The dissipation ratio is highly fluctuating for silicon and PMMA. This is again due to the division by a very small number, the transferred energy in this case. Thus small changes in the denominator cause major changes in the result. However, the dissipation rate of 94.1% for the SBR regions complies very well with the dissipative behavior of rubber. The results for the materials are: Si - (0.62 ± 25.0) % PMMA - (3.3 ± 19.3) % SBR - (94.1 ± 1.6) % The detachment from a sample under ambient conditions is mostly governed by the water layer on top of the sample. Thus, the values that can be obtained for the detachment force change with the humidity of the surrounding: Si - (203 ± 70) nn PMMA - (206 ± 79) nn SBR - (419 ± 128) nn The reproducibility of the measurement of adhesion is not that good resulting in wide scattering. The comparison of the local adhesion of the materials (Fig. 4.53) shows that very well. However it is also obvious that SBR sticks much stronger to the tip than PMMA does. Figure 4.53: Reproducibility of the detachment force.

92 76 Chapter 4. Measurements and Calibration Techniques for Digital Pulsed Force Mode The creep to deeper indentations while reducing the normal force on the cantilever allows to estimate the viscoelastic behavior of the material (Fig. 4.54). For SBR, the post flow distance ranges between 1.0 and 1.6 nm. For PMMA and silicon the point where F max is reached is identical with the maximum indentation depth within the margin of error given by the uncertainty of the snap-in position: Si - (0.46 ± 0.06) nm PMMA - (0.42 ± 0.06) nm SBR - (1.22 ± 0.35) nm Figure 4.54: Reproducibility of the post-flow.

93 4.5. Recipe for Acquisition of Quantitative AFM-Data Recipe for Acquisition of Quantitative AFM-Data As shown above, reliable and reproducible acquisition of AFM data is possible using the Digital Pulsed Force Mode. Thorough calibration and evaluation methods are necessary to ensure this. Thus, a guideline to quantitative measurements is given in the following: 1. Determination of the radius of the tip that is used in the experiments before and after the measurements by SEM imagery. 2. Acquisition of the resonant frequency for the cantilever to calculate its stiffness. 3. Determination of the modulation amplitude of the DPFM system by interferometry, for example by using a Vibrometer. 4. Appropriate setting of the scan parameters (gains of the feedback system etc.). 5. Choice of a setpoint force that is at least as high as the detachment force in order to gain indentation measurements mostly independent of the adhesive contribution. 6. Phase correction of the detected signals so that for the (purely elastic!) reference surface the area under the force curve in repulsive contact is minimized and the approach and retract paths do not cross (within the margin of error). 7. Conversion of all force traces into force versus indentation depth plots using the sensitivity of the AFM. 8. Analysis based on contact mechanical models where appropriate (JKR has turned out to work fine, conical models are limited to huge indentation depths). 9. Statistics of the entire set of data results in very small error margins for the results of the evaluation.

94 78 Chapter 4. Measurements and Calibration Techniques for Digital Pulsed Force Mode 4.6 DPFM on Living Cells Having a setup that proved to be very well suited to measure soft materials, the measurement of living cells was taking the competition to the next level. A cell line which is well studied in biology are HeLa cells. The cell line was started in 1951 from malignant cervix tissue from the cervix of Henrietta Lacks, who died of this specific cancer. In culture, these cells grow aggressively what on the one hand makes them dangerous, but at the same time makes them easy to maintain, since they are more or less immortal and divide unlimitedly. Topography ( z = 1.5 µm) Effective modulus from calculations based on the JKR model Hysteretic loss Viscoelastic post-flow Figure 4.55: HeLa cells have been investigated using DPFM and the data evaluation algorithms developed for polymer samples. Being available in the Department, these cells have been investigated using the DPFM in buffer solution. Because of the strong damping by the liquid environment, the repeat rate of indentations was lowered to 175 Hz with a modulation amplitude of 175 nm.

95 4.6. DPFM on Living Cells 79 Furthermore, very soft cantilevers (k = 0.18 m N ) have been used, since the cells react allergic to indentation experiments at high loads. They lose contact with the substrate and then float away through the solution. The data recorded during the measurements has also been analyzed by the automated evaluation described in chapter 5. As for the silicon substrate of polymer coatings described above, it is also possible to use the surrounding substrate of the petri dish as reference surface. However, this surface is not perfectly hard, since it has to be coated to provide good adhesive properties for the growth of the cells. The algorithms can then be applied to the data recorded during the scan. The results show that the programs do not only work on handy polymers, but also on tricky samples like living cells. Imaging cells is a special challenge, since, first, they have been imaged in physiologic environment, second, they are several microns high, third, cells have an interesting mechanical structure underneath their outer membrane, namely their cytoskeleton. The investigation of these mechanical properties may lead to new kinds of medical diagnostics. One of the first steps towards this has been done by the experiments shown in figure The effective modulus obtained in the outer areas of the cell shows a certain substructure that is revealing the combined modulus of the cell membrane, the cytoskeleton, and the substrate underneath. Thus, in principle it is possible to investigate the mechanical properties of cells with the lateral resolution of some nanometers. The major advantage over the commonly used arrays of force distance curves on cells is that the DPFM records a vast amount of data during one scan. This information content is by far larger than what can be obtained in other modes. Most of all, however, positioning of the tip on the desired region for mechanical testing is not necessary, since the force traces are acquired on the entire scan field. Thus, even singular events are captured and do not have to be tediously hunted.

96 80 Chapter 4. Measurements and Calibration Techniques for Digital Pulsed Force Mode

97 Chapter 5 DPFM Data-Handling The handling of DPFM data is very time consuming if done by hand. First, the calibration of the measurement system has to be calculated from a DPFM experiment on a reference substrate. The main requirements for a reference sample are, that the material has to be harder than all other sample materials used in the later experiment. It should also respond purely elastic upon loading. Assuming a purely elastic material means that there is no hysteretic loss during a loading cycle and thus, the curves of the approach and retract fall together. However, this is not the case in general. At least it does not seem to be at first sight, since systematic delays in the measurement setup occur and cause a certain time lag between excitation and detection. As the excitation is sinusoidal, this time lag equals a certain phase shift. Since this shift, on the other hand, does not change for the same setup and measurement conditions, it can be used to compensate the systematic deviation for all curves of a measurement. Doing all this by hand is rather tedious. Furthermore, for statistical statements, it is necessary to analyze as many curves as possible, or in best practice every single recorded curve. Thus, a program has been developed, which allows an easy calibration of the DPFM apparatus and also an in depth physical analysis of the DPFM experiments in terms of local mechanical properties. Literally, this means an order of magnitude of about 10 6 curves to be loaded, evaluated and stored properly. To ease this evaluation procedure and to shorten the time needed to analyze the entire data, was the driving force behind the programming described in this chapter. The data-sets that had been exported manually into ASCII files from the WITec measurement software DPFM-Control have been analyzed using Origin 7.5 Pro. The algorithms developed for this semi-automated data evaluation have been written in the ANSI-C compiler included in Origin taking advantage of the easy plotting and fitting of data-sets using Origins graphical interface. However, the curve analysis was still very time-consuming. Thus, the algorithms have been transported into a command-line C++ programm, where the Borland C++ Builder 6 has been used as compiler. Where appropriate, optimized algorithms for fitting have been adapted from Numerical Recipes in C++ [65]. With this 81

98 82 Chapter 5. DPFM Data-Handling completely independent command-line tool, the entire measurement files can be analyzed. This is by far faster than the step by step procedure. To give an estimation: while using Origin with the same amount of evaluation algorithms, it took about one hour to analyze a set of 25 force traces, now using the new tool it is three hours for the processing of an entire file of curves and 12 GBytes of data. 45 minutes of these three hours are due to file access by the hardware, which is already targeted at high performance. In these terms, the expression fast data evaluation is relativized. For compression, calibration, and evaluation of the data separate dedicated tools have been produced to ease data analysis. Each of these tools is command line based and expects certain preliminary actions to be taken. Most important, an initialization file is needed that contains line-by-line the amplitude of the modulation of the DPFM in units of meters, then the stiffness in terms of Newtons per meter, followed by the radius of the tip in use and finally the opening angle of the corresponding conical approximation of the tip. For the calibration of the data, a boolean map-file has to be present, which contains the information on which pixels to use as reference curves. 5.1 Data-Compression In Digital Pulsed Force Mode, the force data during an experiment is entirely stored in one single file ( WITec Data Project.wdp ). The pixel and curve triggers are also written to the disk as internal and external clock files (files ending on IntClk.wmp and ExtClk.wmp, respectively). Since the DPFM hardware is already doing some basic data analysis, the results thereof are also slip-streamed into the data transfer protocol and thus also saved to disk. The data protocol consists of a fixed structure of tokens and values. Tokens are hard-coded to the lowest 64 values of the digitization, which does not interfere with the resolution of the stored data. 64 digitization steps are about 10 mv of the ±10 V input. This also does not limit the data quality. However, all protocol data may occur at all times and is then integrated into the data-stream. Whilst this random occurrences are appropriate for the realtime continuous data storage, for later evaluation, the tokens and values may be sorted to the beginning of each force trace. This is done in the Data-Compression tool. The Data-Compression tool is able to cut down the huge.wdp file into half of its size ( WITec Compressed Data.wcd ), since the curves on the retrace are not synchronized to Pixels and can thus be discarded. It turned out during data evaluation, that it makes sense to slightly modify the storage protocol. Hence, in the same run, which is compressing the data by a factor of two, the new protocol is used to store the data into a binary coded file, again. Also, a different clock file ( WITec Compressed Structure.wcs ) is

99 5.1. Data-Compression 83 written, which now contains the address of the first token of the first curve of the pixel, the number of curves stored for this specific pixel, and the number of force values that make up the force trace. Thus, walking along the clock file allows the access of every data block. This also cuts down the time needed for later evaluation. The tokens of the new protocol are listed in table 5.1. Token-name BASELINE_TOKEN LOSSANGLE_TOKEN FMAX_TOKEN STIFFNESS_TOKEN ADHESION_TOKEN BIAS_TOKEN EXT_ADC_1_TOKEN EXT_ADC_2_TOKEN EXT_ADC_3_TOKEN PIXEL_TOKEN MODULATION_TOKEN CONTENTS_TOKEN Hex-code 0x0 0x1 0x2 0x3 0x4 0x5 0x6 0x8 0x9 0xA 0xB 0xC Table 5.1: The modified data storage protocol as used in the further evaluation. The WITec original has been expanded and the original information is still maintained. In the new protocol, the PIXEL_TOKEN is used to store the absolute index of the pixel in the later image, the MODULATION_TOKEN is followed by the number of modulations that are stored between this pixel and its successor. The CONTENTS_TOKEN is containing the maximum index of data points per single force trace. Together with the information in the new clock file, this might seem redundant at first sight, but makes sense when filling the evaluation structure. As in the original data storage it are also the ultimate and usually unused 64 values that are used as triggering tokens, while the following force values are the evaluation results. While in the original data file, trigger values could possibly happen at all times, in the compressed file, the tokens are well sorted. They are all together occurring right at the beginning of each force data block and are in a fixed, predefined order.

100 84 Chapter 5. DPFM Data-Handling 5.2 Calibration of the DPFM Apparatus As said before, the AFM setup and the DPFM add-on system have to be calibrated thoroughly before trying to measure physical quantities. In the DPFM-Calibration program there is the choice to either use an entire image as a reference measurement or to use a predefined boolean map which determines pixels for calibration. In the first case, the entire image assumedly is the same hard purely elastic reference surface and the phase shift and sensitivity of the apparatus are determined as average values for the entire image. These values are then stored in the calibration file ending with Calib.ini. In the latter case, the boolean map is giving a true result for pixels that contain reference curves. The user of the program thus has to create this map before data-evaluation can be run by using the WITec Project software, drawing the true/false map and exporting it to an ASCII file. This file then has to be passed to the DPFM-Calibration tool as a command line option. This way, to calibrate DPFM data is especially well suited for measurements, where there is a large free area of reference material (for example a hard sample substrate) available. This can be achieved by scratching the surface as done in the experiments described in chapter 4, or it is implicated by the experiment as when measuring living cells in a liquid environment. 5.3 Data-Evaluation The evaluation of DPFM data can be split into various steps. Here, only the systematic ideas and the algorithmic background shall be described Data-Preparation At first, all force signal values have to be converted into forces in Newtons by dividing the detected voltages from the photo-detector by the sensitivity of the setup and multiplying with the stiffness of the cantilever. The sensitivity is obtained by calibration measurements on a hard substrate, such as silicon, as the slope of the voltage versus position plot. As shown by Cleveland et al. [21] the stiffness of the cantilever can be obtained from the design equation: k = 2π 3 l 3 w ρ 3 E ν3 0, (5.1) where l and w are the length and width of the cantilever respectively, ρ its mass-density, E its materials Young s modulus, and ν 0 its eigenfrequency. From the manufacturers data,

101 5.3. Data-Evaluation 85 one can calculate the constant pre-factor, which shortens the equation above to a cubic dependence of the spring constant on its eigenfrequency k = const ν0 3. The eigenfrequency can either be obtained by a direct frequency sweep using a dither piezo and detecting the driven oscillation amplitude, or by recording the frequency spectrum of the cantilever at ambient conditions, which means driven by thermal excitation. Furthermore, the baseline, which corresponds to the deflection of the free cantilever, has to be subtracted from the entire force curve in order to obtain real force values. This deflection at a position of the cantilever far away from the surface may be due to long range interactions such as Coulomb attraction by charges on the surface. However, it can also be that thermal drift or mechanical dislocation of the detector caused a displacement of the neutral position of the deflection signal Forces versus Indentation Depth After calibration, one can analyze certain mechanical aspects of the sample surface. Some of the mechanical properties can be easily accessed at this stage of the data evaluation, others need a more dedicated or sophisticated approach. For all of the following analysis, it came in very handy, that the window definitions that are used during the actual DPFM measurement are also stored in an INFormation file.inf. The window settings can be used in order to preselect certain regions of the force curves during data-evaluation, which again reduces the time for computation. When a cantilever is deflected during the indentation into a compliant sample material, the indentation depth can be used as the coordinate of the force plots. To do so, the slope of the calibration measurements is read in for each line of the image and is used in the analysis. Each force that is exerted on the cantilever in order to bend it backwards is causing both a deflection of the cantilever (which is measured) and an indentation of the tip into the material. Thus, one can calculate this indentation depth by subtracting the position of the cantilever deflected on a hard surface for a given normal force F from each position value for the cantilever on a compliant material. δ = z ref F sens k (5.2) where sens is the sensitivity of the AFM setup and k the cantilever stiffness. Of course, the position of the snap into contact with the surface has to be known before calculation of the indentation depth, since it is used as zero mark on the indentation depth. However, a simple search for a minimum force value before F max is reached does not work reliably enough, as the snap-in may be vanishing within the noise of the force signal. The

102 86 Chapter 5. DPFM Data-Handling approach used in the data evaluation program resolves that problem and worked without missing a snap-in so far. First, the force data is rigorously smoothed by a floating average calculation with a window of 55 indices. The width of 55 indices has been found to work very well and was thus hard-coded. Then the three point derivative is calculated for the force trace. Another smoothing, taking the derivative, and a final smoothing round prepare the force trace for finding the snap-in. Now, the second derivative is present and reveals a clear maximum for the narrow region of snap-in. The smoothing in between each derivative is necessary to avoid extreme results due to the random noise. Finally, the narrowed region is then searched for the minimum value of force, which corresponds to the snap-in. The numerical effort pays off, since the snap-ins are found very reliably. Adhesion Force Plotting the force trace versus this indentation depth is the actual plot that reveals the physics behind the experimental results. One of the most obvious points on a force curve is the point of detachment from the surface. The force needed for detachment corresponds to the local adhesion of the sample. The window settings for the adhesion detection by the hardware during the actual measurement is utilized to narrow this search region. The single curve result of the search is then buffered until all curves of the pixel have been processed. The average value of these results is then stored in the appropriate Result array in the Image class. Accordingly, the standard deviation from the average of the values is stored in the corresponding StdDev array of the same class. Maximum Force, Maximum Indentation Finding the maximum force is also quite easy when using the original search windows. Here, it is just the maximum force determined inside the window. Again, average and standard deviation for each pixel are calculated and stored for image creation. As a side product, the indentation depth at the maximum force is obtained in this analysis. This depth can be used to calculate the sample compliance, meaning the indentation needed to reach the maximum (setpoint) force. It is given in units of Newtons per meter, comparable to a local material stiffness. Furthermore, the indentation depth itself is an interesting quantity, since it tells us to what extend the surface has been deformed during the experiment. The averaged results and their standard deviation are collected for later images.

103 5.3. Data-Evaluation 87 Creep Effects and Post Flow For materials that are not responding purely elastic upon loading, one can observe further indentation into the sample even when the normal force is decreased again. This is due to creep or viscoelastic drift. In terms of distances, one can describe creep as the post flow distance, which is the distance from the position where the maximum normal force has been reached until the turning point of the indenting tip. Thus, the map of this post flow simply is the difference image of the indentation depth at maximum force and the absolute maximum of indentation. Elastic Energies and Hysteretic Losses Further interesting indices that have to be determined are the points where the force curve passes the zero force level. However, the concrete value F = 0N will only occur accidentally in a force trace. Thus, the index of an element which is next to the desired force value is used. Three points are calculated here: first - after the index of the snap-in, second and third - directly before and after the snap-off. All three are necessary for the following calculation of deformation energies. Since the force plot contains the force that occurred while moving the tip along a certain distance, the area under the curve is equal to an energy per definition. As described in section 4.4, one can find out about the hysteretic losses, elastic energies stored in the sample materials, and creep effects. The energies that are calculated are: The elastic energy brought into the sample materials during the approach, which is the integral from F = 0N after the snap-in until the maximum force is reached. The total energy transferred to the sample, which is the area until reaching the maximum indentation depth. The hysteretic loss during one loading cycle, which is the energy from F = 0N after the snap-in until F = 0N before the snap-off. Thus, when the sample is restoring its original shape elastically, the area is close to zero, while if energy is dissipated, it is clearly visible. The energy of adhesion, which equals the area from F = 0N before until F = 0N after the point of snap-off, which means until the cantilever is fully detached again. As can be seen for the SBR regions in chapter 4.4, this energy can be very large if the surface is able to deform and stick strongly to the tip of the cantilever.

104 88 Chapter 5. DPFM Data-Handling More Demanding Evaluation Fitting the models described in section 2 to the force data requires special data-handling. Since the JKR model is the most interesting for our experiments, this model calculation has been implemented in the evaluation programm. The JKR model (just like the Hertz, DMT, or Maugis-Dugdale model) is always calculated for a spherical indenter. However, if the tip is indenting too far into the sample, a conical indenter is better suited to describe the situation. The theoretical description of a cone can be obtained from the Sneddon extensions described in chapter 2.2 for a linear boundary, meaning an exponent of α = 1 and a cone opening angle of θ in the calculations. The dependency of the force on the indentation depth for the JKR model is given by: F JKR = K JKR R δ 2 2π Rϖ = KJKR R δ 2 Fadh, (5.3) where F adh is the force at zero indentation, which means the force value which is close to δ = 0m in the evaluation. Thus, this position is searched using the same search function described for the F = 0N above. For a given experiment, this is only an offset. For the conical indenter the force dependency is: F cone = K cone π 2 tan ( π 180 θ ) δ 2. (5.4) In order to use these dependencies for the calculation of the effective modulus K model, it is necessary to first convert the above equations to a linear dependency in δ. Doing so, eventually gives: f JKR = (F JKR + F adh ) 2 3 = K 2 3 JKR R 1 3 δ (5.5) ( ) 1 2 f cone = F 1 K 2 cone cone = π 2 tan ( π θ δ (5.6) 180 ) These functions can fitted linearly with respect to the indentation depth δ and the resulting coefficients of the fit correspond to the effective moduli K model times some constant systematic parameters like the radius of the tip which has to be obtained separately Zero-Assumption Data-Conversion One can use the calibration technique described above to evaluate DPFM measurements in a relative manner. For more than one material on the surface, one can give estimations

105 5.3. Data-Evaluation 89 of the sample compliance and adhesive behavior, without knowing anything about the surface. First, the hardest of the materials has to be identified. Then, the phase adjustment has to be conducted in order to achieve a linear relation between deflection signal (in volts) and cantilever position (in arbitrary units) for the assumedly elastic hard material. The relative sample compliance is then to be given in units of the hard material instead of Newtons per meter. Adhesion values, of course can be determined as before and are thus given in volts. again, a relative result can be given, even though absolute values will not be found. However, the relative comparison is often helpful enough, when determining material compositions. For the fitting algorithms, things are not quite that easy. Here, the conversion factors have to be removed from the theoretical equations, before an application makes sense.

106 90 Chapter 5. DPFM Data-Handling

107 Chapter 6 Lateral Forces Measured by AFM Measuring the frictional behavior of surfaces under realistic conditions is highly interesting both in a scientific as well in a technical sense. To understand wear at the nanoscale, however, the relative tip-sample velocity has to be increased. Thus, as a second topic of this work was the design and setup of a measurement system for sample behavior under high shear rates. Most AFMs are capable of measuring lateral forces by the torsional signal acquired as the trace and retrace signals given by the differential signal between the left and right segments of the segmented photodetector. Subtracting the trace and retrace signals results in a measure for the magnitude of lateral force occurring in the experiment. However, this scanning method is based on the assumption that, first, the surface is identical on the trace and retrace, and, second, that the sample will not be damaged by the scan. These assumptions, in general, are not satisfied for a real experiment. A possibility to measure lateral forces without these strong assumptions can be found in dynamic friction force microscopy. Here, the sample is oscillated at a given amplitude and frequency, thus, causing a relative movement between the tip and the sample surface. Oscillating at a small enough amplitude will allow to measure the local magnitude of the lateral forces directly at a given position. Yet, this method is very new and theoretical descriptions are still lacking the temporal resolution to simulate the results properly. A self-made model for the modeling of dynamic friction experiments based on the initial approach of Krotil [40] is introduced in chapter 7. In addition, a calibration method based on an ideas by Varenberg et al. [85] is calculated anew in appendix E. In this chapter the new setup for the measurement of lateral forces at high shear-rates will be described. 91

108 92 Chapter 6. Lateral Forces Measured by AFM 6.1 High-Speed-Friction Implementation To allow friction experiments that resemble realistic conditions, velocities in the range up to meters per second have to be reached. Common AFM setups are not capable of such measurements due to limitations in bandwidth in excitation as well as detection. To achieve such a highly integrated and powerful setup, the present α-snom has been extended by several parts. These parts are described in the following High-Bandwidth Photodetector For elevated relative velocities between tip and sample, the segmented photo-diode is not fast enough, because exciting the sample shaker at frequencies above khz exceeds the bandwidth of a segmented detector as used in the present setup. Thus, a setup was designed that allows for a detection of lateral forces with a bandwidth much higher than what is used for shaking the sample. The nominal bandwidth of the photodetector is 1.4 GHz. Having such a high bandwidth, one can be sure that the bottleneck of the experimental setup will not be the detector, but rather the excitation of the sample movement or other components like the lock-in amplifier. The idea is to use a fast pin photo-diode in combination with a sharp edge cutting off half of the intensity of the deflection laser. The schematic of the setup can be seen in figure 6.1. While a segmented diode relies on the differential measurement of the current from two lit areas, cutting off half of the Gaussian profile of the incident laser beam allows position sensitivity with a single segment device. The change in intensity collected by the diode corresponds to the deviation from the center position Vibrometer It is known from the basic Michelson interferometer setup, that variations in position of one of the mirrors and thus the change in optical path difference allow the measurement of tiny displacements. Anyhow, it is not easy to detect changes smaller than 5% of the wavelength of the laser in use. However, by modulating the position of the reference mirror on purpose and by a known frequency and amplitude, the detection can be improved by orders of magnitude. This is done in the Vibrometer setup which was acquired from SIOS GmbH, Ilmenau. Here, the detection of displacements works down to nominally 0.3 nm using a 632 nm laser-diode by measuring the phase shift of the interference pattern compared to the excitation. It is sampling at a maximum of 1 MHz, which means that according to the Nyquist sampling criteria, that a maximum frequency of 500 khz can be detected. The Vibrometer is used to calibrate the movement of the shaker piezo

109 6.1. High-Speed-Friction Implementation 93 cantilever function generator excitation signal laterally excited sample photo-diode microscope objective gaussian beam 1/2 I 0 I 0 microscope objective beam-splitter cubes laser reference signal L-R signal of lateral forces sharp-edge with xyz-positioning T-B signal of normal forces lock in amplifier SR 844 digitized data ADCs Figure 6.1: Detection scheme for high speed friction measurements. The benefit of this setup is that the common use of the α-snom did not have to be altered. underneath the sample in the dynamic friction measurements in situ. This is possible, since the measurement head is only about the size of a pack of cigarettes. Knowing the amplitude and frequency of the piezo movement then allows determination of the actual relative speed between tip and sample during the friction measurements Lock-In Technique Tiny signals in a noisy background require special measurement techniques. For AC signals the lock-in technique has been developed, which allows the tracking of a certain periodic signal within the noise. Lock-in amplifiers use an approach known as phase sensitive detection to single out a special frequency-component of the input signal. Noise signals at frequencies other than the reference frequency are rejected and do not affect the measurement [79]. In principle it is assumed, that the experiment is excited using a certain frequency. This frequency serves as reference frequency for the analysis. The lock-in detects the response of the system at this given reference frequency. It determines the amplitude and phase of the signal. It is also possible to determine the responses at higher harmonics, which is especially important for the investigation of nonlinear effects. Mathematically, the determination of the signal component at a given frequency is done by multiplication with a sinusoidal function at this special frequency. This way, the only component that survives

110 94 Chapter 6. Lateral Forces Measured by AFM averaging is the component following the periodicity of the reference input signal. The lock-in amplifiers used during this work were the SR-830 and SR-844 RF models from Stanford Research Systems Inc., Sunnyvale, CA ( The frequency range of these lock-in amplifiers is 1 mhz khz and 25 khz-200 MHz, respectively Acousto-Optical-Modulator In order to test the bandwidth of a photo-detector, one can switch on and off the illumination of the diode. However, mechanical transducers such as tiltable mirrors are not fast enough to test high-bandwidth detectors. An optical switch is suited much better, since there are no movable parts. In an acousto-optical-modulator, the acousto-optic effect is used to diffract and shift the passing laser beam using sound waves. By means of a piezo-element a certain stress is exerted on a piece of material like glass at a frequency in the radio frequency range. This vibration causes phonons inside the glass, which are coupling to the photons of the laser passing through. Scattering the photons at the periodically modulated refractive index in the glass and interference occurs, which basically makes the laser beam exit on two switching points of the glass. Blocking one of these spots results in a on-off modulation of the beam which can then be used to characterize optical detectors. The bandwidth of the modulators can reach up to 20 MHz as in our case. Technically, acousto-optical modulators are used in lasers for Q-switching, telecommunications for signal modulation, and in spectroscopy for frequency control The Actual Setup The parts described separately above were merged into the α-snom setup, under the requirement that the everyday operation may not be changed or hindered. Thus, a modular setup approach was chosen. The main parts are the Vibrometer, the stacked shear-piezos, the new photodetector with the sharp-edge setup, and the lock-in amplifier together with the Digital Pulsed Force Mode. First, the stack of shear piezos has to be fixed to the PIscanning stage and centered underneath the AFM cantilever holder. Then, the Vibrometer has to be adjusted in a way, that the detection beam is redirected to the measurement head, where the reflected beam is interfering with the the modulated reference beam. Upon correct alignment a precision measurement of oscillations up to 500 khz is possible with a sub-nanometer resolution of the excited modulation amplitude. Thus, the modulation of the excitation part can at least be measured, even though the piezo might have a certain frequency dependence which requires different drive amplitudes to achieve the same

111 6.1. High-Speed-Friction Implementation 95 modulation amplitude. However, as long as a quantity that cannot be controlled is at least measurable, the experiments can be conducted. Next is the alignment of the deflection detection laser onto the cantilever, which does not differ from any other procedure of setting up the AFM equipment at all. The α-snom is built with a mostly parallel beam path. Coupling the laser into the AFM setup is done via a beam splitting cube. The beam is directed towards the final microscope objective, which is used to focus the beam on the back of the cantilever. The reflected laser beam is then moving through all optical components again, before finally hitting the four quadrant photodiode. After passing the coupling beam-splitter cube, there is now an additional cube, which allows directing half of the beam towards the novel detector as shown in figure 6.1. Thus, a second beam carrying the same deflection information is created. This part of the beam is then allowed to exit the housing of the segmented photo-diode through a tiny orifice. Therefore, the additional optical noise for the α-snom is kept at a minimum when using the new detection system. The sharp-edge itself is held by a piezo element which allows a very accurate positioning of the edge in the center of the laser beam. The laser beam itself is focused by a microscope objective (Melles Girot 10x) onto the sharp edge, which increases the sensitivity of the system. After the edge, another focusing optics is used to collect all the remaining intensity onto the photo-detector. The detected intensity, which is now position-sensitive is read from the photo-diode by means of a lock-in amplifier which is picking the response signal out of the noisy background. The intensity depends on the position of the sharp edge like the error function as shown in figure 6.2. Thus it is obvious that at 50 % of the total intensity, the detector will have its maximum sensitivity. Figure 6.2: The dependence of the detected lateral signal has the shape of the error function. Thus the highest sensitivity is gained at 50 % of the total intensity.

112 96 Chapter 6. Lateral Forces Measured by AFM To detect the signals by a lock-in amplifier requires the supplement of the excitation frequency. This synchronization is achieved by connecting the TTL-output of the frequency generator to the reference input of the lock-in. The amplitude and phase signal as detected by the lock-in amplifier can then either be logged directly while scanning in contact mode across the surface, or the original real and imaginary parts can be recorded and converted into amplitude and phase later on. The latter method has the advantage, that the rear outputs of the lock-in amplifier can be used, which have a much higher update rate than the configurable, but hence slower, front plugs. Another step forward has been done by the implementation of an add-on module for the common Digital Pulsed Force Mode, which in a sense makes it a Digital CODYMode. Three additional channels can now be read in synchronized with the known Pulsed Force Signals. Thus, an exact positioning of the read-out timings is possible, while in the former analog CODYMode setup the stiffness or adhesion trigger had to be sacrificed in order to measure friction simultaneously with the indentation behavior.

113 Chapter 7 Approaches for Calibration and Evaluation of Lateral Force Experiments While scanning across a sample surface, energy is dissipated by friction. As it is known from the macroscopic length scale, a certain maximum force, known as sticking friction, is needed to move the probe for the first step forward. Afterwards, the movement is more steady and a certain lower force, known as sliding friction, has to be exerted to the probe in order to move it further. Anyway, there can also be a fluctuation between the two kinds of motion meaning a stop an go movement called stick-slip friction. However, this type of frictional behavior can mostly be observed by AFM experiments in vacuum or even UHV, where the adhesive layer of water in ambient conditions is gone and mostly atomic and molecular interactions occur. These interactions give rise to a certain surface interaction potential which is described in literature by the Tomlinson model. The model calculations are based on statistical jumps from one minimum in the potential surface to the next. In this case, the AFM is operated under ambient conditions, which means that the assumptions for the model may not be justified. Furthermore, the focus of this work is not on the atomically clean and flat surface, but on laterally structured oxide surfaces partially covered by oligomeric substance, namely silanes. Thus, it is more a view of a mechanical continuum then of a rather discrete physical environment. In the beginning of this PhD thesis, a method for calibration of the lateral signals had to be found in order to measure forces instead of just estimating physical parameters. A publication by Varenberg et al. [85] seemed very promising, since the authors presented a calibration method which is based on a simple scan of a silicon surface and analyzing the different sliding behavior of a cantilever tip on the sloped Si-(111) and flat Si-(100) crystallographic planes. Their analysis was based on the comparison of the friction coefficient µ which they determine on the flat and sloped surfaces. Since their theoretical 97

114 98 Chapter 7. Approaches for Calibration and Evaluation of Lateral Force Experiments views revealed a quadratic equation in µ they then use the set of coefficients which are the closest for the two areas. Thus, they find a conversion factor firstly from signals to friction forces and secondly from signals to absolute coefficients of friction. However, they neglect some contributing forces to the torque on a twisting cantilever, that actually may not be neglected without reasoning. Since the general approach of calculating force balances at the point of contact as well as moment balances for the torque on the twisted cantilever while sliding up and down a sloped surface, makes sense, the basic force and momentum analysis was recalculated. In appendix E, the considerations and calculations are demonstrated. Starting from a view of the forces on the point contact of tip and surfaces, a force balance is calculated. Then the moment arms are calculated from the geometry of the cantilever on a sloped surface. Finally, the moment balances are established, taking into account all the forces acting on the contact. From there, the coefficient of friction, µ can be calculated. For comparability, the variables used are the same as in [85]. Besides the calibration of an AFM setup based on theories of a static equilibrium of forces and moments, one also has to look at dynamic processes, especially when trying to explain high speed frictional behavior, thus far away from and static equilibrium. The theoretical modeling of such a situation is more complicated then it might seem at first sight. Taking into account Van-der-Waals forces and the viscoelastic behavior of polymer surfaces makes simulations more or less impossible. This is also, why most theoreticians prefer to not work in this field and rather focus on clean atomic interactions. However, one can describe the dynamic experiment by an empirical model. The basic approach had already been done by Krotil in his PhD thesis [40]. An extended modeling is described in the following. 7.1 Empirical Model for Dynamic Friction The situation which is described here is assuming an AFM tip in permanent contact with the surface. It is describing the sum of all lateral forces as developing during a friction loop at a given position on the sample. The scanning motion is assumed to be negligible in the relative tip sample velocity. A friction loop at a fixed point can be acquired by moving the sample sinusoidally at a very small modulation amplitude A m. In the experimental setup, this is done by a stack of shear piezos, which allow modulations of tens of nanometers at frequencies up to the MHz range. The modulation amplitude could be set higher as well, but the lateral resolution of the image may not be limited or dilated by the friction measurement. This means that the amplitude has to be kept smaller than the diameter of the tip in use or in numbers well below 40 nm.

115 7.1. Empirical Model for Dynamic Friction Modeling Sticking Friction As long as the modulation amplitude stays below a critical value which is given by the bending of the cantilever due to the maximum force of static friction F stat, the tip will be following the motion of the sample immediately. If the modulation of the sample is conducted sinusoidally with an amplitude A m like x(t) = A m cos(t), (7.1) the lateral force that is bending the cantilever will be given by F(t) = A m cos(t) k l, (7.2) where kl is the lateral force constant of the cantilever. The time domain in this model approach is scaled from 0 to 2π Modeling Kinetic Friction If the modulation, however, exceeds the critical distance, the tip will begin to slide across the sample surface. The point in the time domain where sliding is initiated is given by ( ) Fstat t 1 = π arccos. (7.3) k l A m After the sliding begins, the cantilever will relax into the bending caused by sliding friction. The relaxation time constant is modeled by τ = ln (π t 1 ) ( q(1 (1 ε)) 1 (1 ε) q arccos( Fstat = ln ) k l A m ( q(1 (1 ε)) 1 (1 ε) q ) (7.4) ) (7.5) ε being a systematic fit parameter which assures the asymptotic exponential decay to allow the value of F slide = qf stat to be reached and to continue into the next part of the cycle without jumps. For an entire friction loop, the force response of the system is given by a piecewise function: F stat (q+1)(cos(t) 1) ( ) qf F stat stat 0 t < t 1 k l Am 1 F stat (1 (1 ε)q) exp( t t 1 F(t) = τ ) + (1 ε)qf stat t 1 t < π F stat (q+1)(cos(t)+1) ( ) + qf F stat stat π t < π +t 1 k l Am 1 F stat (1 (1 ε)q) exp( t (π+t 1) τ ) (1 ε)qf stat π +t 1 t < 2π

116 100 Chapter 7. Approaches for Calibration and Evaluation of Lateral Force Experiments Figure 7.1: Modulation (blue curve) and response (red trace) of the simulated friction loop. Static friction is overcome at t = t 1 and again at t = π +t 1. The modulation (blue curve) and the response (red trace) are shown in figure 7.1. For the simulation the following values have been used: q = 0.8, A m = 20 nm, F stat = 15 nn, (1 ε) = and k l = 3 N m. As the cantilever is bent laterally, tip is sliding across the surface. The actual position of the tip can be calculated from the forces given above: A m cos(t) + A m q F stat k l 0 t < t 1 ( ) A m q F stat k P(t) = l exp ( t t ) 1 τ + (1 ε)q F stat k l t 1 t < π A m cos(t) + A m + q F stat k l π t < π +t 1 ( ) A m q F stat k l exp ( t t ) 1 τ (1 ε)q F stat k l π +t 1 t < 2π During a force cycle, the tip starts to slide at t = t 1. Thus its new starting position is shifted and the tip is pushed in fixed contact with the sample surface until the maximum static friction force is exceeded again. Thus, there is no relative motion of tip and sample until t 1. In figure 7.2, the absolute position of the tip on the sample is drawn in black, while the relative position is shown in green. For comparison, the modulation (blue) is also plotted. The corresponding relative tip-sample speed is thus given by the derivative of the relative tip-sample position. The result is plotted in figure 7.3. Multiplication by the modulation frequency yields the real velocity. As described for the measurement setup (chapter 6), the demodulation of the left-right signal of the segmented photodetector is achieved by means of a lock-in amplifier. Thus, the amplitude of the detected force signal and its phase shift with respect to the synchronizing TTL signal from the function generator can be obtained.

117 7.1. Empirical Model for Dynamic Friction 101 Figure 7.2: In case of sliding, the absolute (black) and relative (green) tip-sample positions can be calculated. Figure 7.3: The relative speed of tip and sample. Multiplication by the modulation frequency yields the real velocity.

118 102 Chapter 7. Approaches for Calibration and Evaluation of Lateral Force Experiments

119 Chapter 8 Experiments Using the Novel Setup As mentioned before, a novel setup for friction measurements under high shear rates has been developed. To prove that the measurement system is running properly, preliminary measurements have been conducted. To allow comparison with the measurements that have been obtained in Digital Pulsed Force Mode, the mixture of SBR and PMMA and a similar sample system consisting of styrene-butadiene-styrene (SBS) and PMMA have been used. The sample preparation was the same for all of these samples. Thus, the structures that can be expected are well known and it is easy to see if the apparatus works. However, the results obtained with the setup so far must be seen as a proof of principle of the operation rather than a detailed investigation. 8.1 Results on Polymer Blends The setup that has been created relies on the dynamic modulation of the sample. To reach high relative velocities between the tip and the sample one has to chose amplitude and frequency wisely. Tuning the velocity by increasing the amplitude of modulation is limited by the dimensions of the tip of the cantilever. In order to maintain the same lateral resolution of the measurements, the modulation may not exceed the radius of the tip of the cantilever in use. Thus, in this work, the maximum amplitude that has been used was 40 nanometers. On the other hand, tuning the velocity by setting a certain frequency can be difficult as well, since the excitation is done via a stack of shear piezos which by themselves show a frequency dependency. For the dewetting polymer mixture, various relative speeds between the tip and the sample have been given a try. As described in chapter 6, the Vibrometer does only allow calibration of the excitation for modulation frequencies up to 500 khz. Thus, measurements above this are currently uncalibrated and one has to rely on his own faith, to estimate the 103

120 104 Chapter 8. Experiments Using the Novel Setup amplitude at which the piezo is modulated. A method for calibration of these higher velocities could be Shearography. However, this method has not been available during the investigations for this PhD thesis. Topography ( z = 40nm) Stiffness Friction amplitude Friction phase Figure 8.1: Friction contrasts on dewetting polymer mixtures. In figure 8.1, the contrasts obtained on the dewetting polymers are shown. The measurement has been conducted in the extended Digital Pulsed Force Mode, meaning that the results for friction amplitude and phase have been stored within the stream of data that is recorded by the DPFM computer. The values are read out inside the window for the external inputs 1 and 2, and are stored as hardware evaluation results after the tokens EXT_ADC_1_TOKEN and EXT_ADC_2_TOKEN, as given in table 5.1 in chapter 5.

121 8.1. Results on Polymer Blends 105 Since the solution for the extension of the DPFM was developed as an add-on system, all other signals and tokens have not been changed or altered. The results are stored for each individual DPFM curve. However, the its entire development of the lateral force cannot be recorded so far except by using another data logger, as for example an additional DPFM, in a master-slave application. The measurement parameters of the measurements were A PFM = 150nm and ν PFM = 1kHz for the Pulsed Force Mode modulation and A lat = 20nm and ν lat = 82kHz for the lateral sample excitation. A zoom into a different region of the sample (Fig. 8.2) shows that the lateral resolution of the images has not been lost. Anyhow, it is obvious that parts of the sample have been collected and lost during the scans. This debris may have been due to the lateral excitation and thus the lateral forces acting on the sample causing wear on the polymers. However, it has been shown that contrasting of the different sample materials is possible and also higher excitation frequencies have been used. Another experiment has been conducted at ν lat = 350kHz, but the resolution on the polymer sample is rather poor, since the excitation amplitude was set to 60 nm to achieve a higher relative tip sample velocity. The contrasts, however, are shown in figure 8.3.

122 106 Chapter 8. Experiments Using the Novel Setup Topography ( z = 40nm) Friction amplitude Stiffness Friction phase Figure 8.2: Friction on dewetting polymer mixtures measured at 82 khz. Debris picked up and lost by the tip during scanning is visible.

123 8.1. Results on Polymer Blends 107 Topography ( z = 40nm) Friction amplitude Adhesion Stiffness Figure 8.3: Friction on dewetting polymer mixtures at 350 khz.

124 108 Chapter 8. Experiments Using the Novel Setup 8.2 Optimized Samples by Microcontact Printing As seen above for the friction measurements on polymers, topographical features cause crosstalk into the detected friction signals. Therefore, it is advantageous to have a perfectly flat sample that reveals a clear difference in its frictional behavior. Such lateral structuring without topography can be achieved by means of a lift-off technique based on micro contact printing (µ-cp ). In µ-cp, a compliant stamp is used to transfer patterns from a master to a substrate [58]. Large areas can be structured at once. The idea of µ-cp has first been described by Kumar and Whitesides [43]. As sketched in figure 8.4, a master structure, is replicated by pouring for example a liquid polydimethylsiloxane (PDMS) prepolymer onto a master structure. Then the PDMS is cured for several hours at 60 C [57]. After copolimerization is finished, the elastomer can be pulled-off from the master and used as a stamp. For PDMS as stamp material, the aspect ratio of width to height should be at least 1:1 and at most 3:1 [3]. The elastic properties of PDMS can be tuned by the ratio of prepolymer and hardener in the mixture before curing. A ratio of ten parts by weight of PDMS to one part of hardener is working very well and allows the stamp to adjust to the topology of the substrate which is also known as conformal contact [43]. After preparation of the stamp, it is inked for example by immersion of the stamp or by using an inker pad. As ink various alkanethiol solutions can be used; in this case it is a 1 mm solution of octadecanethiol (ODT, Fig. 8.6) in ethanol impregnating the inker pad for three hours. ODT: H 3 C (CH 2 ) 17 SH The inker pad then is dried by a stream of nitrogen. When brought into contact with the surface of a noble metal, the thiol group is strongly attracted. This is why silicon wafers covered with gold by means of thermal evaporation have been used as substrates. As an adhesive layer 1 nm of chromium has been evaporated to bridge from the native oxide layer to the metallic gold. This has been achieved by switching from the chromium source to gold without breaking the vacuum in the evaporation chamber. As soon as the concentration of thiols adsorbed on the gold surface reaches a critical density, self assembled monolayers (SAMs) are formed. Inside the SAMs, the alkanethiols are slightly tilted with respect to the surface-normal to allow for an optimal arrangement of the alkyl chains due to multiple Van-der-Waals interaction which makes the arrangement highly hydrophobic. The SAM itself then is closely packed and ordered. Because of this compactness, SAMs of alkanethiols are resistant towards certain etching solutions. In this case a wet etch of K 4 Fe(CN) 6, K 3 Fe(CN) 6, K 2 S 2 O 3 and KOH in

125 8.2. Optimized Samples by Microcontact Printing 109 Figure 8.4: Microcontact printing, modus operandi: a master structure, b casting the master in a mould, c stamp, d stamp on inker pad, e inked stamp, f SAM on gold, g after wet etch (figure used with kind permission from C. Gnahm)

126 110 Chapter 8. Experiments Using the Novel Setup de-ionized water was used [87, 95]. The dissolution works like: Au + 2CN + Fe(CN) 3 6 Fe(CN) Au(CN) 2 but the SAM of ODT is blocking the access of the cyanide ion CN to the gold surface, which prevents the etching of the alkanethiol covered areas. The best resistance to etching solutions is achieved using SAMs of nonpolar, methyl-group terminated alkanethiols [43]. For gold films of 20 nm, the optimum duration of etching is about 8 minutes [95]. Figure 8.5: Octadecyltrichlorosilane Figure 8.6: Octadecanethiol ( C - dark grey, H - silver grey, Si - red, Cl - (C - dark grey, H - silver grey, Sulfur - yellow) green) (figures used with kind permission from C. Gnahm) Having structured the gold surface by µ-cp of alkanethiols and having etched away the bare gold down to the oxidic surface of the substrate, allows the introduction of the actually desired monolayer of silanes to the oxide. This process is also known as lift-offprocedure and has been described by Walheim et al. [87]. This lift-off technique allows silanization of the surface from vapor or liquid phase instead of having to print the silanes directly onto the surface. The latter is rather impossible under ambient conditions, since the silanization reaction occurs as soon as water is present. In ambient, the silanes immediately crosslink amongst each other and there is no chance of bringing them into contact with the surface anymore. Using the pre-structured surfaces, however, allows application of the silanes from a dry solution. The most commonly used is octadecyltrichlorosilane (OTS, Fig. 8.5) dissolved in dry hexane [87] or heptane [67]. The concentration of OTS

127 8.3. Results on Oligomeric SAMs 111 may range between 25 µm and 2.5 mm [70]. As for the stamped monolayer of ODT, the OTS molecules also form SAMs, but the mechanism is different. A silicon wafer under ambient conditions is always covered by a layer of native oxide. Like any other oxide, this surface is hydrated [82]. As a result, the functional groups at the surface are silanol groups. The trichlorosilane group of the OTS is hydrolyzed as soon as it reaches the water film. The OTS is then coupled to the silanol groups as well as to the neighboring OTS by hydrogen bonds until H 2 O is separated and covalent bonds are formed. Thus, the SAM of OTS is crosslinked laterally and covalently bound to the surface, which makes is highly stable. The final step of sample preparation is the removal of the gold that was protected by the alkanethiols. Therefore, another wet etch is used, consisting of bromine (1 part by volume) and methanol (80 parts by volume). What remains is a patterned monolayer of silane on silicon oxide. This pattern on the one hand, shows negligible differences in the topography, but the differences in the friction behavior on the other hand, are very well gaugeable. 8.3 Results on Oligomeric SAMs The samples prepared had a lateral pitch of about 3 µm and a negligible height, which made finding them a tricky task. However, they do reveal clear contrast in all measurement channels, except for the stiffness measurements, of the extended Digital Pulsed Force Mode. Adhesion is of course lower on the areas layered with silanes than on the bare silicon oxide regions, since the SAM of silanes is highly hydrophobic. The stiffness or sample compliance, however does not result in a clear contrast, since the layer is made up of single molecules arranged in a compact layer on top of a hard substrate (silicon). Thus, the tip of the cantilever is directly sensing the underlying substrate and not the SAM on top. For the friction channels of friction amplitude and phase, on the other hand, things are clear again. The silane monolayers act as a lubricating layer and the cantilever is twisted by a much smaller angle than on the silicon oxide regions. Thus, friction is lower on the silanes. The phase contrast as a measure for the dissipative character of the frictional behavior is also obvious. On the lubricating SAM of silanes, much less energy is dissipated due to the creation of heat than on the oxidic areas. Measurements on the self assembled monolayers of silanes on oxidic surfaces are shown in figure 8.7. The measurements have been obtained in the extended DPFM with a lateral modulation amplitude of 20 nm and a frequency of 82 khz together with a normal modulation of 1 khz at 150 nm modulation amplitude. In all, contrasting the almost perfectly flat samples is possible and the high-speed fric-

128 112 Chapter 8. Experiments Using the Novel Setup Topography ( z = 2nm) Stiffness Friction amplitude Friction phase Adhesion Figure 8.7: Friction on self-assembled OTS monolayers. The samples were produced by the procedure based on µ-cp.

129 8.3. Results on Oligomeric SAMs 113 tion setup is ready to work on various sample systems. Much more measurement tasks lie ahead. What has been done during the diploma thesis of Claudia Gnahm were measurements with relative tip-sample-speeds in the range of millimeters per second. The results suggest that the difference in frictional behavior between the oxide and the silane SAM as well as between different silanes is decreasing. This behavior can be explained by the very recent dynamic friction model by Tambe and Bhushan [81]. They claim that for higher velocities (already in the high µm/s regime) friction is mainly governed by the elasticity of the samples and not by its adhesive behavior anymore. They claim that a soft material causes less friction since it is able to contain more energy within elastic deformation than hard materials which are subjected to wearing. Hence, silanes still seem very promising as protective layers. At low velocities, they do not cause much adhesion based friction and at elevated speeds, they protect the hard substrate form being worn by the hitting indenter.

130 114 Chapter 8. Experiments Using the Novel Setup

131 Chapter 9 Conclusions In the course of this thesis, two central topics have been treated: First, the quantitative and reliable acquisition and evaluation of AFM data, especially using the Digital Pulsed Force Mode technique; and second, the design and operation of a setup for high-speed friction measurements. Considering the first topic, a better understanding of the contact between the tip of an AFM cantilever and the sample surface it is indenting, has been achieved. The classical contact mechanical models have been applied successfully and reproducibly to a system of dewetting polymers. The sample system was composed of two polymers, polymethylmethacrylate (PMMA) and un-vulcanized styrene-butadiene-rubber (SBR). The molecular weights of these were 100 kg kg mol for the PMMA and 390 mol for the SBR. From macroscopic experiments like Dynamical Mechanical Analysis (DMA) as reported in literature, PMMA is known to be an elastic material with a Young s modulus of 3.3 GPa and SBR to be rubber-elastic with a Young s modulus ranging between 50 and 150 MPa. Thus, the materials allow observation of all aspects of dynamic indentation, like pure elasticity and viscoelastic behavior. The polymer solution was spin-cast onto a silicon wafer where the materials dewetted into a two-phase surface coating. By scratching the surface prior to a scan, silicon was revealed, and could therefore be used as a reference material. The samples have been investigated with respect to their dynamical behavior under cyclic loading. Examination of 14 samples under changing conditions revealed that all parameters that rely on the indentation of the tip into the material can be reproduced following a certain protocol. This protocol has been worked out during the thesis. However, parameters that are deduced from the detachment behavior are strongly influenced by the ambient conditions under which the experiments were conducted. A recipe for the acquisition of reliable AFM data has been described and used both on polymers and living cells. The data obtained during these experiments allow insight into the dynamical properties of the samples. The evaluation of the vast amount of data collected during each 115

132 116 Chapter 9. Conclusions of the experiments has been automated based on algorithms inferred from the models for contact mechanics. Since it turned out that the force signals of an AFM always include phase shifts due to the signal run-times through the AFM-hardware, a phase correction has been developed. Performing this phase compensation prior to the evaluation of the force traces allows computation of real physical results. As formulated by the JKR description of contact mechanics, an effective modulus was derived from the dependency of the indentation forces on the actual indentation depth. Furthermore, the Young s moduli of the investigated materials have been calculated. It was found that the absolute value of the Young s modulus is about one order of magnitude higher than the values reported for bulk samples in literature, but then decreases exponentially for increasing indentation into the material. This was determined by the evaluation of two dimensional histograms. The histograms were generated by plotting the effective modulus K on the y-axis versus the deformation ratio ξ on the x-axis for every pixel of the images. The density distribution revealed two accumulation regions for PMMA and SBR. The explanation of the softening under increasing load is given by the nonlinear stress-strain behavior of the materials. PMMA reveals a changing slope of its stress-strain curve even for very small elongations, while in the case of the SBR it is the plateau region of the un-vulcanized rubber that is responsible for the decrease. In comparison of the models for contact mechanics, it has been found that a conical indenter may not be applied to AFM indentation measurements if the indentation depth into the materials is too small. Especially in case of a hard surface like the silicon reference regions, the mathematical singularity of the conical model leads to non-physical results (negative elastic modulus). Other models than the JKR model have also been considered, but discarded for various reasons. First of all and most obvious, the Hertzian description does not hold for the evaluation of adhesive materials as presented here. Second, the DMT model does not describe any deformation of the sample upon the detachment of the tip from the sample, but this is exactly what was observed for the force traces on SBR. Finally the MD model has also been discarded, since it has too many free parameters that would have to be assumed. Reviewing the investigation of reproducibility of AFM data acquired in DPFM, one can conclude that all evaluations of quantities that arise from the repulsive contact can be beautifully reproduced. The uncertainties of these measurements are small due to the large statistics behind them. However, quantities that are obtained from the evaluation of data based on the detachment of the tip from the surface, are strongly influenced by the varying humidity under uncontrolled ambient conditions. The only possibilities for compensation of this variance are environmental control, measuring in a liquid environment, or under vacuum conditions. The second topic that was treated during this study was the setup and operation of a

133 117 measurement system that allows investigation of high relative speeds between the tip and the samples. Up to now, theoretical calculations and molecular dynamics simulations have been unable to describe the interactions for velocities between some micrometers and some meters per second. Experimental data could bridge this gap. In the current setup a maximum relative velocity of 0.15 m s has been achieved. The resonances of the piezo-excitation did not permit higher velocities. However, it is expected that using quartz modulators at their eigenfrequencies, one could also investigate speeds well in the meters per second regime. What remains to be done however is the calibration of the lateral excitation. In the present setup, a Vibrometer has been used to determine the frequency and amplitude of the piezo-drive, but this is only possible up to 500 khz excitation frequency. Increasing the amplitude of the oscillation might seem to be a solution to this speed-limit. However, since the lateral resolution of the measurements should still be limited by - at most - the diameter of the tip in use, this is only possible in a very limited range. Using the new setup, both the polymer samples used in the first part, as well as perfectly flat samples based on the well-established micro-contact printing (µ-cp ) technique, have been scanned. The latter became necessary to rule out topographical crosstalk into the friction data as an error-source. The flat samples have been prepared by a lift-off technique based on µ-cp and consisted of self-assembled monolayers (SAMs) of octadecyltrichlorosilane (OTS) on an oxidic substrate. In the process of sample preparation, silicon covered with chromium and gold by thermal evaporation has been used as starting point. Then a monolayer of octadecanethiol (ODT) molecules has been micro-contact printed to the gold surface. The SAM that is formed by the alkanethiols is protective against certain etchants. Thus, only the bare gold surfaces were dissolved in the subsequent ferro/ferricyanide wet etch step. This leaves a mask of gold on top of an oxide surface. The final monolayer of OTS molecules was then formed by liquid-phase silanization. The SAM of OTS is both crosslinked laterally between the OTS molecules and covalently bound to the oxidic substrate, which makes it highly stable. In a concluding wet etch by a bromine-methanol solution (concentration 1:80 by volume), the remaining gold, which was previously protected by the ODT monolayer, is lifted off. Afterwards, the final sample is laterally structured by its mechanical behavior but does not show topographical differences. The results of this study emphasize the importance of thorough calibration of the measurement system in order to obtain quantitative AFM data. DPFM has been proven to be one of the keys to the mechanical properties of surfaces at the nanoscale. Further insight into the dynamical properties of samples can be gained by the possibility of simultaneous measurements of samples under high shear rates.

134 118 Chapter 9. Conclusions

135 Chapter 10 Zusammenfassung Im Laufe der vorliegenden Arbeit wurden zwei zentrale Themen bearbeitet. Zum Einen, die quantitative und verläßliche Aufnahme und Auswertung von AFM-Daten, die mittels der Digital Pulsed Force Mode Technologie gewonnen wurde; und zum Anderen, der Entwurf und die Umsetzung eines Aufbaus zur Bestimmung des Reibungsverhaltens bei hohen Geschwindigkeiten. In Bezug auf das erste Teilgebiet wurde ein tieferes Verständnis über den Kontakt zwischen der AFM-Spitze und der mittels Indentation untersuchten Probenoberfläche gewonnen. Die Modelle aus der klassischen Kontaktmechanik konnten erfolgreich und reproduzierbar auf die Daten aus Experimenten mit entnetzenden Polymermischungen angewendet werden. Im Speziellen wurde ein Probensystem zweier Polymere - Polymethylmethacrylat (PMMA, oder Plexiglas) und nicht vulkanisiertem Styrol-Butadien-Kautschuk (SBR) - verwendet. Die Molekulargewichte der Substanzen waren 100 kg mol im Falle des PMMA und 390 kg mol für SBR. Aus Publikationen über makroskopische Experimente wie die Dynamisch-Mechanischen-Analyse, kurz DMA, ist bekannt, dass für PMMA ein Elastizitätsmodul von 3,3 GPa zu erwarten ist, während beim gummielastischen SBR der Modul zwischen 50 und 150 MPa liegen sollte. Somit erlauben die Materialien das ganze Spektrum der möglichen Auswertungen und Experimente auszuloten. Aspekte wie das dynamische Eindringverhalten, rein elastische Vorgänge und Viskoelatizität wurden untersucht. Die Polymermixtur wurde per Lackschleuder auf Siliziumstücke aufgebracht, wo die Polymere dann durch Entnetzung eine zweiphasige Oberflächenbeschichtung ergaben. Mittels eines Kratzers auf jeder Probe wurde jeweils das Siliziumsubstrat freigelegt und konnte somit als Referenz für die Kalibrierung der Indentationsexperimente verwendet werden. Die Proben wurden in Hinblick auf ihr dynamisches Verhalten unter zyklischer Beanspruchung erforscht. Die Untersuchungen an 14 Proben unter wechselnden Umgebungsbedingungen ergaben, dass alle Messgrössen, die auf dem Eindringen in das Probenma- 119

136 120 Chapter 10. Zusammenfassung terial beruhen hochgradig reproduzierbar sind, solange eine besondere Messvorschrift befolgt wird. Diese wurde im Laufe der Arbeit zunehmend verfeinert. Im Gegensatz zu den verläßlichen Indentationsexperimenten, hängen die Ergebnisse aus dem Ablöseverhalten der Spitze von der Probenoberfläche stark von den jeweiligen Umgebungsbedingungen, wie der Luftfeuchtigkeit, ab. Die Messvorschrift wurde sowohl auf Polymerproben als auch auf lebende Zellen angewendet. Die dabei gewonnenen Daten geben Aufschluss über das dynamische Verhalten der Proben. Die Auswertung der riesigen Datenmengen, die sich während eines jeden Experiments sammeln, wurde mittels einer speziell dafür entwickelten Software automatisiert. Die Software basiert auf den theoretischen Modellen der Kontaktmechanik. Wie sich herausstellte, besteht immer ine Laufzeitverzögerung der Signale im Messaufbau. Ein Algorithmus zur Korrektur dieser Phasenverschiebungen wurde ebenfalls entwickelt. Nach der entsprechenden Phasenkorrektur, erlaubt die Auswertung der Kraftsignale die Bestimmung realer physikalischer Größen. Wie durch das JKR-Model beschrieben, wurde ein effektiver Modul aus dem Verlauf der Kraft gegenüber der Eindringtiefe bestimmt. In einem weiteren Schritt, konnten daraus wiederum die Elastizitätsmoduln der einzelnen Probenmaterialien berechnet werden. Im ersten Moment des Eindringens der Spitze in die Probe berechnet sich ein Elastizitätsmodul, der im Vergleich zu Literaturwerten um etwa eine Größenordnung zu hoch liegt, danach jedoch exponentiell mit zunehmender Indentationstiefe abfällt. Dieses Verhalten wurde mittels zweidimensionaler Histogramme, also den Dichtediagrammen der Wertepaare zweier Messgrössen gegeneinander, studiert. In diesem Fall wurden für jeden Bildpunkt der effektive Modul K auf der y-achse und der Parameter der Deformation ξ auf der x-achse aufgetragen. Die Dichteverteilung ergab zwei Häufungspunkte für PMMA und SBR. Die Erklärung des Abfalls des Elastizitätsmodul über dem Deformationsverhältnis beruht auf dem nicht-linearen Verlauf der Spannungs-Dehnungs-Kurven der verwendeten Materialien. Schon für kleine Dehnungen fällt die Steigung der Kurve für PMMA, also der Elastizitätsmodul, ab. Ähnlich beobachtet man bei Gummielastizität ein Plateau in der Spannungs-Dehnungs-Kurve, wodurch der gemessene Modul ebenfalls kleiner ausfallen muß, als dies bei streng linearer Elastizität der Fall wäre. Im Vergleich der Modelle stellte sich heraus, dass ein kegelförmiger Indentationskörper nur für grosse Eindringtiefen in das Probenmaterial vernünftige Resultate ergeben kann und für die Beschreibung kleiner Deformationen ungeeignet ist. Speziell im Falle der harten Referenzoberfläche, also des Siliziums, wurde die mathematische Singularität des Kegelmodells offensichtlich; es ergab sich ein negativer Elastizitätsmodul. Neben dem JKR-Model wurden auch noch andere Beschreibungen der Kontaktmechanik in Betracht gezogen, aber nach eingehender Überlegung aus unterschiedlichen Gründen für ungeeignet befunden. Zu allererst wurde die Verwendung des Hertz schen Modells erwogen, das jedoch kein Adhäsionsverhalten beschreiben kann und damit ausschied. Das

137 121 DMT-Modell beschreibt wiederum nicht die Deformation der Probe beim Abreissen der Spitze von der Probenoberfläche, was jedoch experimentell beobachtet wurde. Schließlich wurde noch das Maugis-Dugdale Modell in Betracht gezogen, das jedoch zu viele Annahmen über die Wechselwirkungen voraussetzt. Aus den Versuchen zur Reproduzierbarkeit von mittels DPFM gewonnenen AFM-Daten, kann man schließen, dass eine Auswertung von Größen, die aus dem repulsiven Kontakt mit der Probe herrühren, mit grosser Genauigkeit reproduziert werden können. Die Messungenauigkeit der Ergebnisse ist sehr gering aufgrund der großen Anzahl von Messwerten. Hingegen können Messungen des Ablöseverhaltens nur mit großssen Fehlerbereichen angegeben werden, da die unkontrollierten Umgebungsbedingungen, wie zum Beispiel schwankende Luftfeuchtigkeit, die Ergebnisse stark beeinflussen. Die einzigen Lösungen für diese Problematik sind: die Kontrolle der Umgebungsbedingungen, die Messung in Flüssigkeit oder im Vakuum. Das zweite Themengebiet, das während dieser Arbeit behandelt wurde, war der Aufbau und der Betrieb eines neuartigen Messsystems zur Untersuchung der Spitze-Probe- Wechselwirkung bei hohen Relativgeschwindigkeiten. Bislang waren theoretische Rechnungen und Molekulardynamik-Simulationen noch nicht in der Lage das Verhalten dieses System für Geschwindigkeiten im Bereich von Mikrometern bis Nanometern pro Sekunde vorherzusagen. Mittels experimenteller Ergebnisse könnte diese Lücke überbrückt werden. Im bestehenden Aufbau wurden Geschwindigkeiten von bis zu 0,15 Metern pro Sekunde realisiert. Resonanzen in der Anregung mittels eines Piezostacks verhinderten die Messung bei höheren Geschwindigkeiten. Es wird jedoch erwartet, dass durch den Einsatz von Quarzen, getrieben auf ihrer Eigenfrequenz, noch weit höhere Geschwindigkeiten möglich sind. Eine offene Frage bleibt weiterhin die Kalibrierung der lateralen Modulation. Im bestehenden Aufbau wird ein Vibrometer zur Bestimmung von Amplitude und Frequenz dieser Modulation eingesetzt, die maximal detektierbare Frequenz ist allerdings 500 khz. Eine Steigerung der Modulationsamplitude mag dabei als möglicher Ausweg erscheinen, jedoch sollte die laterale Auflösung der Messungen nicht gemindert werden, wodurch die Modulationsamplitude auf maximal den Spitzendurchmesser begrenzt ist. Mit dem neuartigen Aufbau wurden sowohl die Polymerproben aus den Indentationsexperimenten als auch nahezu perfekt ebene Proben untersucht. Die Herstellung der letzteren basiert auf der wohlbekannten Technik des Mikrokontaktdrucks (µcp). Der Einsatz dieser Art von Proben war nötig, um das Übersprechen von Topographiesignalen in die Reibungsmessungen zu verhindern und somit als Fehlerquelle auszuschließen. Die flachen Proben wurden mittels einer Lift-Off Technik, also dem Abheben einer Hilfsmaske, die auf dem Prinzip des µ-cp beruht, hergestellt. Sie bestehen schlussendlich aus selbst-ordnenden Monoschichten (SAMs) aus Octadecyltrichlorosilan (OTS) auf einem

138 122 Chapter 10. Zusammenfassung oxidischen Substrat. Im Verlauf der Probenpräparation wird ein mit Chrom und Gold beschichteter Silizium Wafer als Ausgangsmaterial verwendet. Durch Stempeln einer von Octadecanethiol (ODT) formen sich SAMs auf der Goldunterlage. Diese Monolagen schützen das Gold vor der anschließenden selektiven Ferro/Ferrizyanid Ätzlösung. Die thiolfreien Bereiche werden jedoch zersetzt. Es bleibt eine Maske von Gold auf dem Substrat zurück. Die abschließende Silanisierung der Probe aus der Flüssigphase ergibt einen weiteren Monolayer, nun allerdings auf dem freigelegten Oxid. Der SAM aus OTS Molekülen ist sowohl mit seinen Nachbarmolekülen als auch mit dem Substrat kovalent gebunden. Dies macht die Monoschicht äußerst robust. Durch eine abschließende Ätzung mit einer Brom-Methanol-Lösung im Mischungsverhältnis 1:80 (Volumenanteile), wird das noch verbliebene Gold, das zuvor noch von den ODT SAMs geschützt war abgehoben. Danach verbleibt eine in ihrem mechanischen Verhalten lateral strukturierte Probe ohne jeglichen Höhenunterschied. Die Ergebnisse dieser Arbeit betonen die Notwendigkeit der genauest möglichen Kalibrierung des Messaufbaus zur quantitativen Datenaufnahme mittels AFM. Es konnte gezeigt werden, dass DPFM eine der Schlüsseltechnologien zur Untersuchung von Materialeigenschaften auf der Nanoskala ist. Weitere Einsicht in die Probendynamik kann durch die Option der gleichzeitigen Messung von Proben bei hohen Scherraten gewonnen werden.

139 References [1] S. Alexander, L. Hellemans, O. Marti, J. Schneir, V. Elings, P. K. Hansma, M. Longmire, and J. Gurley. An atomic-resolution atomic-force microscope implemented using an optical lever. J. Appl. Phys., 65:164 7, [2] G. Balooch, G. W. Marshall, S. J. Marshall, O. L. Warren, S. A. S. Asif, and M. Balooch. Evaluation of a new modulus mapping technique to investigate microstructural features of human teeth. Biomechanics, Vol. 37(8): , [3] H. A. Biebuyck, N. B. Larsen, E. Delamarche, and B. Michel. Lithography beyond Light: Microcontact Printing with Monolayer Resists. IBM J. Res. & Dev., 41(1/2), [4] A. Bietsch, M. A. Schneider, M. E. Welland, and B. Michel. Electrical testing of Gold Nanostructures by Conducting Atomic Force Microscopy. J. Vac. Sci. Technol. B, 18: , [5] G. Binnig, C. F. Quate, and Ch. Gerber. Atomic Force Microscope. Phys. Rev. Lett., 56:930 3, [6] G. Binnig and H. Rohrer. Scanning Tunneling Microscopy. Helvetica Physica Acta, 55:726 35, [7] U.-C. Boehnke, H. M. Brodowsky, H. Groothues, and F. Kremer. Investigations of the force-distance behaviour in polar liquids. J. Phys. Chem. B., 103(32): , [8] T. A. Bogetti, T. Wang, M. R. VanLandingham, R. F. Eduljee, and J. W. Jr. Gillespie. Characterization of nanoscale property variations in polymer composite systems part 2 Finite Element Modeling. Composites Part A, 30:85 94, [9] F. M. Borodich and L. M. Keer. Evaluation of elastic modulus of materials by adhesive (no-slip) nano-indentation. Proc. Mathematical, Physical and Engineering Sciences, 460(2042):507 14,

140 124 References [10] F. M. Borodich, L. M. Keer, and C.S. Korach. Analytical study of fundamental nanoindentation test relations for indenters of non-ideal shapes. Nanotechnology, 14:803 8, [11] E. Boschung, M. Heuberger, and G. Dietler. Energy dissipation during nanoscale indentation of polymers with an atomic force microscope. Appl. Phys. Lett., 64(26):3566 8, [12] A. Budó. Theoretische Mechanik. VEB Deutscher verlag der Wissenschaften, Berlin, 5 th edition, [13] S. I. Bulychev, V. P. Alekhin, M. Kh. Shorshorov, and A. P. Ternovskii. Mechanical properties of materials studied from kinetic diagrams of load versus depth of impression during microimpression. Strength Mater., 8:1084 9, [14] S.I. Bulychev, V.P. Alekhin, M.Kh. Shorshorov, A.P. Ternovskii, and G.D. Shnyrev. Determining Young s modulus from the indentor penetration diagram. Industrial Lab., 41: , [15] N. A. Burnham, S. P. Baker, and H. M. Pollock. Model for mechanical properties nanoprobes. J. Mat. Res., 15(9): , [16] W. D. Jr. Callister. Materials Science and Engineering an Introduction. Wiley, 6th edition edition, [17] B. Cappella and G. Dietler. Force-distance curves by atomic force microscopy. Surface Science Reports, 34:1 104, [18] R. W. Carpick, D. F. Ogletree, and M. Salmeron. A general equation for fitting contact area and friction vs load measurements. Colloid Interface Sci., 211: , [19] R. M. Christensen. Theory of viscoelasticity. Academic Press Inc., London, [20] J. P. Cleveland, B. Anczykowski, A. E. Schmid, and V. B. Elings. Energy dissipation in tapping-mode atomic force microscopy. Appl. Phys. Lett., 72:2613 5, [21] J. P. Cleveland, S. Manne, D. Bocek, and P. K. Hansma. A nondestructive method for determining the spring constant of cantilevers for scanning force microscopy. Rev. Sci. Instrum., 64:403 5, [22] J. Colchero. Reibungskraftmikroskopie. PhD thesis, Universität Konstanz, [23] R.J. Colton, A. Engel, J.E. Frommer, H.E. Gaub, A. Gewirth, R. Guckenberger, J. Rabe, W.M. Heckl, B. Parkinson, and Rabe. Procedures in Scanning Probe Microscopies. Wiley, 1998.

141 References 125 [24] P. J. de Pablo, J. Colchero, J. Gómez-Herrero, and A. M. Baró. Jumping Mode scanning force microscopy. Appl. Phys. Lett., 73:3300 2, [25] B. V. Derjaguin, V. M. Muller, and Yu. P. Toporov. Effect of contact deformations on the adhesion of particles. J. Colloid Interface Sci., 53:314 25, [26] D. Drechsler, A. Karbach, and H. Fuchs. Nanoindentation on polycarbonate/polymethyl methacrylate blends. Applied Physics A, 66:S825 9, [27] D. S. Dugdale. Yielding of steel sheets containing slits. J. Mech. Phys. Solids, 8:100 4, [28] J. D. Ferry. Viscoelastic properties of polymers. Wiley, New York, [29] A. R. Franco Jr., G. Pintaúde, A. Sinatora, and C. E. Pinedo. The use of a Vickers indenter in depth sensing indentation for measuring elastic modulus and Vickers hardness. Mater. Res., 7(3):483 91, [30] P. K. Hansma and J. Tersoff. Scanning Tunneling Microscopy. J. Appl. Phys., 61:R1 23, [31] H. Hertz. Über die Berührung fester elastischer Körper. J. Reine Angew. Mathematik, 92:156 71, [32] M. Hinz, A. Kleiner, S. Hild, O. Marti, U. Dürig, B. Gotsmann, U. Drechsler, T. R. Albrecht, and P. Vettiger. Temperature dependent nano indentation of thin polymer films with the scanning force microscope. European Polymer Journal, 40(5):957 64, [33] J. L. Hutter and J. Bechhoefer. Calibration of atomic-force microscope tips. Rev. Sci. Instrum., 64(7): , [34] J. L. Hutter and J. Bechhoefer. Erratum calibration of atomic-force microscope tips. Rev. Sci. Instrum., 64(11):3342, [35] K. L. Johnson. Contact Mechanics. Cambridge University Press, Cambridge, [36] K. L. Johnson and J. A. Greenwood. An adhesion map for the contact of elastic spheres. J. Colloid Interface Sci., 192:326 33, [37] K. L. Johnson, K. Kendall, and A. D. Roberts. Surface energy and the contact of elastic solids. Proc. R. Soc. Lond. A., 324:301 13, [38] R. B. King. Elastic analysis of some punch problems for a layered medium. Int. J. Solids Structures, 23(12): , [39] G. Krausch. Dewetting at the interface between two immiscible polymers. J. Phys.: COnd. Matter, 9: , 1997.

142 126 References [40] H.-U. Krotil. CODYMode Scanning Force Microscopy The Concurrent Measurement of Adhesion, Friction and Viscoelastics. PhD thesis, Universität Ulm, [41] H.-U. Krotil, T. Stifter, and O. Marti. Concurrent measurement of adhesive and elastic surface properties with a new modulation technique for scanning force microscopy. Rev. Sci. Instrum., 71: , [42] H. U. Krotil, T. Stifter, and O. Marti. Verfahren und Vorrichtung zur gleichzeitigen Bestimmung der Adhäsion, der Reibung und weiterer Materialeigenschaften einer Probenoberfläche. Patent No. PCT/DE00/00003, [43] A. Kumar, H. A. Biebuyck, and G. M. Whitesides. Patterning self-assembled monolayers: Applications on materials science. Langmuir, 10: , [44] H. M. Lankarani and Nikravesh P. E. Continuous contact force models for impact in multibody systems. Nonlinear Dynamics, 5: , [45] S. Ludwigs, G. Krausch, R. Magerle, Zvelindovsky A.V., and G.J.A. Sevink. Phase behavior of ABC triblock terpolymers in thin films: Mesoscale simulations. Macromolecules, 38: , [46] R. Luginbühl, M. D. Garrison, J. Christos, R. M. Overney, L. Weiss, H. Schieferdecker, and S. Hild. Chemical and contact mechanical characterization of thin plasma deposited hexafluoropropylene films. Fluorinated Surfaces, Coatings and Films, 787: , [47] M. Luna, J. Colchero, and A. M. Baró. Intermittent contact scanning force microscopy - the role of the liquid necks. Appl. Phys. Lett., 72(26):3461 3, [48] O. Marti, J. Colchero, and J. Mlynek. Combined scanning force and friction microscopy of mica. Nanotechnology, 1:141 4, [49] O. Marti, S. Gould, and P. K. Hansma. Atomic resolution atomic force microscopy of graphite and the "native oxide" on silicon. J. Vac. Sci. Technol. A, 6:287 90, [50] Y. Martin, D. W. Abraham, and H. K. Wickrmasinghe. High-resolution capacitance measurement and potentiometry by force microscopy. Appl. Phys. Lett., 52:1103 5, [51] D. Maugis. Adhesion of spheres the JKR-DMT transition using a Dugdale model. J. Colloid Interface Sci., 150:243 69, [52] D. Maugis. Contact, adhesion, and rupture of elastic solids. Springer, Berlin, [53] C. J. McHargue. Surface Mechanical Properties Using Nanoindentations in Micro/Nanotribology and its Applications, chapter Ch. 6, pages Kluwer Academic Press, 1997.

143 References 127 [54] G. Meyer and N. M. Amer. Erratum novel optical approach to atomic force microscopy. Appl. Phys. Lett., 53:2400 2, [55] G. Meyer and N. M. Amer. Novel optical approach to atomic force microscopy. Appl. Phys. Lett., 53:1045 7, [56] G. Meyer and N. M. Amer. Simultaneous measurement of lateral and normal forces with an optical-beam-deflection atomic force microscope. Appl. Phys. Lett., 57: , [57] B. Michel. Printing meets lithography. The Industrial Physicist, Aug/Sept:16 19, [58] B. Michel, A. Bernard, A. Bietsch, E. Delamarche, M. Geissler, D. Juncker, H. Kind, J.-P. Renault, H. Rothuizen, H. Schmid, P. Schmidt-Wolf, R. Stutz, and H. Wolf. Printing meets lithography: Soft approaches to high-resolution patterning. IBM J. Res. & Dev., 45(5): , [59] T. Miyatani, M. Horii, A. Rosa, M. Fujihira, and O. Marti. Mapping of electrical double-layer force between tip and sample surfaces in water by pulsed-force-mode atomic force microscopy. Appl. Phys. Lett., 71(18):2632 4, [60] H. A. Mizes, K.-G. Loh, R. J. D. Miller, S. K. Ahuja, and E. F. Grabowski. Submicron probe of polymer adhesion with atomic force microscopy - dependence on topography and material inhomogenities. Appl. Phys. Lett., 59:2901 3, [61] F. Moreno-Herrero, P.J. de Pablo, J. Colchero, J. Gómez Herrero, and A. Baró. The role of shear forces in scanning force microscopy a comparison between the jumping mode and tapping mode. Surface Science, 453:152 8, [62] V. M. Muller, V. S. Yushenko, and B. V. Derjaguin. On the influence of molecular forces on the deformation of an elastic sphere and its sticking to a rigid plane. J. Colloid Interface Sci., 77:91 101, [63] W.C. Oliver and G.M. Pharr. A new improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. Mat. Res., 7(6): , [64] G.M. Pharr, W.C. Oliver, and F.R. Brotzen. On the generality of the relationship among contact stiffness, contact area, and elastic modulus during indentation. Mat. Res., 7(3):613 7, [65] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipes in C++ - The Art of Scientific Computing. Cambridge University Press, 2 nd edition, 2002.

144 128 References [66] P. Prokhorenko, V. Rudnitsky, and A. Kren. Impact indentation testing of rubbers and plastics. Review of Quantitative Nondestructive Evaluation, 23:1135 9, [67] A. G. Richter, C.-J. Yu, A. Datta, J. Kmetko, and P. Dutta. In situ an interrupted-growth studies of the self-assembly of octadecyltrichlorosilane monolayers. Phys. Rev. E, 61(1):607 15, [68] A. Rosa-Zeiser. Der Pulsed Force Mode als kraftmikroskopische Untersuchungsmethode für unterschiedlich gefüllten Naturkautschuk bei linearer Deformation. PhD thesis, Universität Ulm, [69] A. Rosa-Zeiser, E. Weilandt, S. Hild, and O. Marti. The simultaneous measurement of elastic, electrostatic and adhesive properties by scanning force microscopy pulsed-force mode operation. Meas. Sci. Technol., 8:1333 8, [70] N. Rozlosnik, M. C. Gerstenberg, and N. B. Larsen. Effect of solvents and concentration on the formation of a self-assembled monolayer of octadecyltrichlorosilane on si(001). Langmuir, 19:1182 8, [71] D. Sarid. Scanning Force Microscopy. Oxford University Press, New York, [72] G. Schitter, F. Allgöwer, and A. Stemmer. A new control strategy for high-speed atomic force microscopy. Nanotechnology, 15:108 14, [73] M. Kh. Shorshorov, S.I. Bulychev, and V.P. Aleklin. Work of plastic and elastic deformation during indenter indentation. Sov. Phys.-Doklady, 26:769 70, [74] I. N. Sneddon. The relation between load and penetration in the axisymmetric boussinesq problem for a punch of arbitrary profile. In. J. Engng Sci., 3:47 57, [75] J. P. Spatz, S. Sheiko, M. Möller, R. G. Winkler, P. Reineker, and O. Marti. Tapping scanning force microscopy in air - theory and experiment. Langmuir, 13: , [76] P. M. Spizig. Dynamische Rasterkraftmikroskopie. PhD thesis, Universität Ulm, [77] T. Stifter, O. Marti, and B. Bhushan. Theoretical investigation of the distance dependence of capillary and van der waals forces in SFM. Phys. Rev. B, 62(20): , [78] T. Stifter, E. Weilandt, O. Marti, and S. Hild. Influence of the topography on adhesion measured by SFM. Appl. Phys. A, 66: , [79] Stanford Research Systems. SR844 RF Lock-In Amplifier Operating Manual and Programming Reference.

145 References 129 [80] D. Tabor. Surface forces and surface interactions. J. Colloid Interface Sci., 58:2 13, [81] N. S. Tambe and B. Bhushan. Friction model for the velocity dependence of nanoscale friction. Nanotechnology, 16: , [82] P. A. Thiel and T. E. Madey. The interaction of water with solid surfaces: Fundamental aspects. Surf. Sci. Rep., 7: , [83] M. R. VanLandingham, R. R. Dagastine, R. F. Eduljee, R. L McCullough, and J. W. Jr. Gillespie. Characterization of nanoscale property variations in polymer composite systems part 1 experimental results. Composites Part A, 30:75 83, [84] M. R. VanLandingham, J. S. Villarrubia, W. F. Guthrie, and G. F. Meyers. Nanoindentation of polymers an overview, recent advances in Scanning Probe Microscopy. Proceedings of the 220th American Chemical Scociety National Meeting, 44, August [85] M. Varenberg, I. Etsion, and G. Halperin. An improved wedge calibration method for lateral force in Atomic Force Microscopy. Rev. Sci. Instr., 74(7):3362 7, [86] S. Walheim, M. Böltau, J. Mlynek, G. Krausch, and U. Steiner. Structure formation via polymer demixing in spin-cast films. Macromolecules, 30: , [87] S. Walheim, R. Müller, M. Sprenger, E. Loser, J. Mlynek, and U. Steiner. Patterning self-assembled monolayers on oxide surfaces using a lift-off technique. Adv. Mater., 11(7):1431 3, [88] C. Wang, G. Krausch, and M. Geoghegan. Dewetting at a polymer-polymer interface: Film thickness dependence. Langmuir, 17: , [89] E. Weilandt. Kräfte an Graphitoberflächen unter elektrochemischer Potentialkontrolle. PhD thesis, Universität Ulm, [90] K. O. van der Werf, C. A. J. Putman, B. G. de Grooth, and J. Greve. Adhesion force imaging in air and liquid by adhesion mode atomic force microscopy. Appl. Phys. Lett., 65:1195 7, [91] J. Wilhelm and E. Frey. Elasticity of stiff polymer networks. Phys. Rev. Lett., 91(10): , [92] R. G. Winkler, J. P. Spatz, S. Sheiko, M. Möller, P. Reineker, and O. Marti. Imaging material properties by resonant tapping-force microscopy - a model investigation. Phys. Rev. B, 54: , 1996.

146 130 References [93] LFMR datasheet by Nanosensors. [94] TGF11 datasheet by MikroMasch. [95] Y. Xia, X.-M. Zhao, E. Kim, and G. M. Whitesides. A selective etching solution for use with patterned self-assembled monolayers of alkanethiols on gold. Chem. Mater., 7:2332 7, 1995.

147 Appendix 131

148

149 Appendix A Automated DPFM Data-Evaluation Data-Structure The data structure which is used for data evaluation has been well defined. The most fundamental class that has been designed is the Curve class. It is used to store all tokens as well as an array of double precision floating point force values. Evaluation of the data such as fits is exclusively done at this data level. After evaluation, the storage of all data is handled by the Image class. An image is a Pixel based structure, which is able to store all layers of an image in terms of evaluation results and standard deviation. The latter is possible, since multiple curves are stored for each Pixel and this data depth per pixel can be explored in a statistical sense. The Image class also contains the address of the first curve of each pixel, together with the number of curves for this pixel and the data length of each force trace, in order to be able to access each Curve directly. The entire list of entities in the evaluation class Curve is given in list A.1 and for the Image class in table A.2. Curve TokenData [MaxToken] = bit16 CurveData [MaxDataIndex] MaxToken MaxDataIndex = double = int = int Table A.1: Curves can be accessed in this type of class. MaxToken defines the maximum number of possible hex-tokens. MaxDataIndex contains the maximum length of a array of force values, i.e. the resolution of a force curve. As for calibration purposes, the Image class is too mighty, a derivative of this class, namely the CALIBImage class has been created. Here, the actual storage is done in variables called Angle and Slope, which belong to each CALIBPixel. All the entities used for navigation in the Image class are also present. Furthermore, the calibration program 133

150 134 Appendix A. Automated DPFM Data-Evaluation Image Pixel[PX*PY] Result [MAXLAYERS] = double PixelPerLine LinesPerImage StdDev [MAXLAYERS] FileOffset NoOfCurves MaxDataIndex = double = int64 = int = int = int = int Table A.2: Images are created by filling this data structure. PX and PY is the resolution number of pixels in each line and column, respectively. The values of PX and PY are stored in PixelPerLine and LinesPerImage. MAXLAYERS is a pre-set maximum number of possible evaluations. Thus, the maximum number of channels is twice this number due to the additional layers for the standard deviation images. utilizes a class called Map, which is able to store boolean true/false information for all Pixels. The purpose of this class is to allow calibration on selected areas of an image. This corresponds to the idea of having for example a scratch on the surface which reveals the hard substrate underneath the sample material that can then be used as a line-by-line calibration. Thus it is possible to either calibrate the DPFM system on a separate reference sample and use the calibration data thereof, or to choose areas on the sample as reference. This is especially important for measurements on bio-material, where substances like silicon are undesirable to have in the liquid keeping the sample alive. CALIBImage CALIBPixel[PX*PY] Angle = double PixelPerLine LinesPerImage Slope FileOffset NoOfCurves MaxDataIndex = double = int64 = int = int = int = int Table A.3: For calibration, an extra class has been devised. The phase correction angle and the reference slope of the calibration curves is stored. All the other variables are the same as in the Image class. Another data-structure that is filled consecutively during data evaluation, is the Scale- Block class together with original SXM-Image Header and Footer structures. This data structure is very well suited for storing images in various layers. First there is a data block of 16 bytes containing the information valid for all the layers like resolution, date, file-

151 135 Map BOOLPixel[PX*PY] setting = int PixelPerLine LinesPerImage = int = int Table A.4: To allow a line-by-line or a global calibration, it is necessary to have a boolean map of the measurement data, where to use pixels for calibration and where to neglect calibration. name, etc. Then followed by the image data in a block structure according to the layers. Afterwards, a scale block defines the color scaling of all the images in the same order of appearance as in the image blocks for each layer. The final information is given as a footer structure, containing the x-and y-scalings and offsets as well as the axis-labeling. The ScaleBlock class (table A.5) has been defined in order to ease the use inside the evaluation loops and to keep the hard-coded limitations for the evaluation software as small as possible. ScaleBlock SXMScale ZScale = float MLAYERS ZOffset ZMeasure ZName = float = 8 * char = 48 * char = int Table A.5: To export the evaluated results into images which can be further analyzed and post-processed by the WITec image analysis again, the scaling of the z-data has to be done in an appropriate way. For each layer, there is a ScaleBlock that is filled to match this output requirement. Upon completion of the evaluation procedures as described in the following, all maps of physical parameters that have been created during the data-evaluation are written to a WITec output file.wit. This file can then be imported into the original WITec Project software for further histogram analysis, calculations and image export. Thus, the programs developed for the data-compression, DPFM-calibration, and data-evaluation merge well into the sequence data-acquisition, data-storage, and data-evaluation as given by WITec.

152 136 Appendix A. Automated DPFM Data-Evaluation

153 Appendix B Composition of Inks and Etchants 1 mm Inks Alkanethiols of different chain lengths can be used for µ-cp. 1 mm solutions require these quantities: Alkanethiol Chemical formula Molecular weight DDT CH 3 (CH 2 ) 11 SH g/mol HDT CH 3 (CH 2 ) 15 SH g/mol ODT CH 3 (CH 2 ) 17 SH g/mol ECT CH 3 (CH 2 ) 19 SH g/mol Alkanethiol alkanethiol per alkanethiol per 50 ml ethanol 50 ml ethanol DDT ml mg HDT ml mg ODT ml mg ECT ml mg 137

154 138 Appendix B. Composition of Inks and Etchants Ferro/ferricyanide Wet Etch 40 ml of the etching solution the following has to be dissolved in 40 ml of Milli-Q water: Chemical formula Molecular ingredient of the ingredient weight per 40 ml Milli-Q KOH g/mol mg K 4 Fe(CN) 6 3H 2 O g/mol 16.9 mg K 3 Fe(CN) g/mol mg K 2 S 2 O 3 12 H 2O g/mol mg The sequence of the ingredients is crucial in the preparation of the solution. FIRST, KOH has to be completely dissolved in the water, then the remaining ingredients can be added. Otherwise cyanide gas is produced, which is extremely lethal. Silanization solution A 0.5 mm solution of OTS in dry hexane is composed by mixing: Silane Chemical formula Molecular silane per weight 50 ml hexane OTS CH 3 (CH 2 ) 17 Si Cl g/mol ml Final Wet Etch For a 1:80 solution of bromine and methanol the following is needed: Bromine Molecular bromine per weight 50 ml methanol Br g/mol ml

155 Appendix C Recipe for µ-cp The following recipe has been used for all micro-contact printed samples prepared during this work. stamp material: PDMS Sylgard 184 mixed 10:1 parts by weight of prepolymer and hardener stirring of the prepolymer mixture for one minute; afterwards removal of enclosed air bubbles by vacuum treatment for approx. 30 minutes curing the polymer on the master structure at 60 C for at least four hours inking solution: 1 mm solution of ODT in ethanol (solutions older than one week have been discarded) soaking of the PDMS inker pad for at least three hours inking and stamping time: 30 s each etching solution: 1 mm K 4 Fe(CN) 6, 10 mm K 3 Fe(CN) 6, 0.1 M K 2 S 2 O 3, and 1 M KOH in deionized water etching time: 8 min ± 10 s snow-jet cleaning of the etched surface silanization: 0.1 mm solution of OTS in dry hexane (solutions older than 12 hours have been discarded) silanization duration: three hours in a hermetically closed glass at room temperature final etching solution: 1:80 solution of bromine in methanol by volume 139

156 140 Appendix C. Recipe for µ-cp

157 Appendix D µ-cp of Fischer Projection Patterns Using a colloidal crystal of polystyrene spheres instead of a structured silicon wafer as a master structure, self-assembled structures can be replicated by the µ-cp based technique. Classic Production of Fischer Projection Patterns The classic way to produce Fischer Projection Patterns (FPPs) is to use a self-assembled colloidal crystal as a shadow-mask during thermal evaporation. Thus, the metal that is evaporated can only cover the surface through the holes left open. Meanwhile selfassembled areas of some square centimeters can be produced. An example of the quality of the triangular structures that can be obtained by this method is shown in figure D.1. Samples like the one shown here are used in the study of near-field optics and surfaceenhanced Raman scattering. FPPs based on µ-cp As described for the µ-cp of lines in chapter 8, the same technique can also be applied using the colloidal crystal as a master pattern. Curing the prepolymer on the PS spheres results in a stamp with elevated features of triangular shape. The printed triangular SAMs of ODT molecules are again protective against the wet etch and thus triangular gold particles in a hexagonal arrangement are produced on an oxide substrate. Since only the stamp was exchanged in the µ-cp process while all other parameters have been kept constant, the triangles had a thickness of 20 nanometers as shown in figure D.2. Compared to the classical technique, the stamping of the patterns does not require a new colloidal crystal for every single sample to be produced. Thus, the same structures can be replicated more easily. 141

158 142 Appendix D. µ-cp of Fischer Projection Patterns Figure D.1: Topography image of a Fischer Projection Pattern produced by thermal evaporation through a colloidal mask. Figure D.2: Topography image of a Fischer Projection Pattern produced µ-cp.