Atomic force microscopy

The atomic force microscopy (AFM) is the principal technique used in this study to investigate the conformation of DPs on solid substrates, their interaction with surfaces and mechanical response of individual polymer chains at the single-molecule level.

Since the invention of AFM in the mid 1980s by Binnig et al. ^[87], the technique has developed into a powerful and versatile tool intensely used for the study of the surface topography at atomic and nanometer scale, the chemical structure of a molecule ^[88], the high resolution imaging of DNA, proteins and polymers ^[89-91], the inter- and intramolecular interactions in surface-immobilized systems [27, 54, 92-94], the mechanical properties of polymers [28, 95-97], and more. In this thesis, AFM was employed to measure forces and surface topography; both approaches are discussed in the following.

Figure 1.6 (a) Schematic drawing of a force microscope and (b) illustration of a deflection-displacement (piezo position) experiment.

Typically AFM is carried out under ambient conditions, but a great advantage of AFM is the possibility to image the sample in almost any environment, ranging from vacuum, trough gas, to liquid.

AFM probes the surface of a sample with a sharp tip which is located at the free end of a cantilever spring. The cantilever acts as a sensor for the interaction between the tip and a sample (Figure 1.6a). The sample is mounted on a piezoelectric device, which allows the sample to be moved in the vertical direction (z-direction) and scanning the surface in the x-y direction. While the tip scans over the sample,

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the forces between the tip and the sample surface cause the cantilever to bend or deflect. The deflection of the cantilever is detected optically by the deflection of a laser beam focused at the back of the cantilever and reflected to a position-sensitive photodiode.

AFM force measurements. In a force measurement, the x-y position is fixed, while the sample (or tip) is moved up and down by applying a voltage to the piezoelectric device. The cantilever deflection and displacement of the piezo are recorded (Figure 1.6b). During an experiment, the tip is initially far away from the surface, forces are absent and the cantilever is not deflected (Figure 1.6b, Position 1).

Subsequently, attractive surface forces become apparent as the tip approaches the surface and eventually jumps into a close repulsive contact with the surface (Figure 1.6b, Position 2). While the tip and the sample are in contact, the pressure that the tip exerts on the surface increases with the sample displacement. In this region (Figure 1.6b, Position 2-3), known as the constant compliance region, the signal of the photodiode (voltage) and sample displacement are proportional, and the voltage can be directly converted into the cantilever deflection. Upon retraction of the tip and the substrate, the repulsive forces decrease continuously followed by the complete separation of the tip from the surface (Figure 1.6b, Position 4).

These deflection-piezopath data have to be transformed into force-distance curves. The measured piezo position (z0) can be converted into the real distance (z) between the AFM tip and the surface according to Eq. (1.11):

1

zz0s^ D (1.11)

Here, D is the measured cantilever deflection and s is the slope [voltage/length] of the linear part of the curve reflecting the bending of the cantilever upon contacting and indenting the substrate surface. The force F is then obtained by applying Hooke’s law [Eq. (1.12)] ^[98]:





F    kc z z (1.12)

where kc is the spring constant of the cantilever. The minus sign in the equation transforms the negative deflection into a positive force signal. Finally, the force acting on the cantilever is plotted against cantilever-surface distance (z), giving the true force-distance curve.

The cantilever spring constant kc is normally determined from the thermal oscillation spectrum of the cantilever ^{[99, 100]}, but many other methods ^[101], such as vibrational based on the geometry of the cantilever

[102], method measuring the resonance frequency of the cantilever before and after adding end masses ^[103], can be used.

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In the work reported in this thesis, the spring constant was obtained by thermal method in air ^[99]. The cantilever is positioned far away from the surface, is not affected by long range forces and only vibrates around its equilibrium position due to the thermal fluctuations. If a system is in thermal equilibrium, the ground oscillation of the cantilever has a mean energy equal to 1

2^{k T} (kis the Boltzmann constant, Tis the absolute temperature). Due to thermal motion, the cantilever oscillates with an amplitudex. If the cantilever is modeled as a harmonic oscillator, its resonance frequency is:

0 kc

  m (1.13)

where ₀is the resonance frequency, k_cthe spring constant, and mthe effective mass of the cantilever.

Considering just one degree of freedom for the cantilever (it can move only up and down) and the equipartition theorem:

2 2 2

0

1 1 1

2^m ^x 2^{kc x} 2^{k T} (1.14)

where x² is the mean square of the thermal cantilever fluctuations. The spring constant can be subsequently obtained as Eq. (1.15):

2 c

k k T

 x (1.15)

To exactly determine x² , it requires integration over the ground oscillation in the power spectrum of the cantilever measured over all frequencies ^[104].

AFM topography. AFM modes are generally classified as static or dynamic modes, based on the oscillation of the tip during the imaging ^[105].

In the static mode, the tip does not oscillate and the topography of the surface is generated from the cantilever deflection. There are two basic ways of operation, constant height and constant force. In the constant height mode, the cantilever deflection is detected without a feedback control as the height of the scanner is fixed as it scans and it is used directly to obtain the topographic data. This mode is applicable to very smooth, atomically flat surfaces where the variations in the cantilever deflections (i.e., in applied force) are small. In the constant force mode, the cantilever deflection is kept constant by moving the

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scanner up and down in z direction, generating the image from the scanner motion. Constant force mode is generally preferred as the total applied force to the sample is constant and well controlled.

In the dynamic modes of AFM, the system vibrates the cantilever at or near its free resonance frequency. According to the parameter used to establish the feedback mechanism, two major dynamic modes of AFM are amplitude modulation and frequency modulation AFM ^{[105, 106]}. In amplitude modulation AFM, the oscillation amplitude changes as the tip approaches the sample surface and is used as a feedback parameter to obtain the surface topography. On the other hand, in frequency modulation, the cantilever is kept oscillating with a fixed amplitude and the feedback parameter is a frequency shift between the resonance frequency far from the surface and the resonance frequency closer to the surface.

The resonance frequency depends on the forces acting between tip and sample surface. The dynamic (oscillating) AFM modes became widely popular, taking advantage of the signal-to-noise benefits associated with modulated signals. The imaging can be carried out with a small probe-sample force, thus preserving both the sample and the AFM tip.

Figure 1.7 Schematic illustration of the amplitude change as a function of the tip-surface separation.

Throughout this thesis, the amplitude modulation (AC) mode of AFM was used to image the topography of the sample. The cantilever is oscillated (with free oscillation amplitudeA₀) typically at or near its resonance frequency with an additional piezoelectric element. When the oscillating cantilever approaches the surface, the forces between the tip and the surfaces cause changes in the oscillation, a damping in the cantilever oscillation (Figure 1.7). The damping leads to a decrease in the resonance frequency and in turn in the amplitude of the oscillation. The feedback loop maintains the amplitude constant. This process involves comparison between the instantaneous value of the amplitude A_iwith respect to a reference value, the set point amplitudeA^. An error signal is generated. The goal is to keep

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this error signal as small as possible. Based on the error signal, an integral differential system moves the piezo scanner in zdirection in order to minimize the difference between A^ and A_i.

Depending on the oscillation amplitude used, AFM can be operated in different regimes, i.e. non-contact, and intermittent-contact regime (Figure 1.8). The interatomic force between the tip and the surface is either repulsive (contact mode) or attractive (non-contact mode). Using a small oscillation amplitude, the cantilever can be held in the attractive regime only. On the other hand, if a large oscillation amplitude is applied, the tip can move from ‘zero-force’ regime, through the attractive regime where is no tip-sample interaction, to the repulsive regime in each oscillation cycle. This technique is known as intermittent-contact AFM (IC-AFM). The cantilever tip, which vibrates at or near its resonance frequency, is brought closer to the sample surface and at the bottom of its travel it just barely hits the sample. IC-AFM has become an important technique as with its introduction it became possible to image soft structures such as polymers, since it is less likely to damage the sample than contact AFM by eliminating lateral forces and is more effective than non-contact AFM by overcoming its fundamental instability in air.

Figure 1.8 Different operating regimes for oscillating AFM modes. Force versus tip-to-sample separation curve illustrating the attractive and repulsive regimes.