Response curve, forced oscillations

If the process is second- or higher-order, we will not be able to make a discontinuous change in the slope of the response curve. Consequently we would expect a second-order process to overshoot the setpoint if we forced it to reach the setpoint in one sampling period. The output would oscillate between sampling periods and the manipulated variable would change at each sampling period. This is called rippling and is illustrated in Fig. 20.2c. 

In the contact mode, there are static modes (de-modes), and dynamic modes (ac-modes). In the former, a cantilever-type spring bends in response to the force which acts on the probing tip until a static equilibrium is established [1]. In the dynamic mode, the lever oscillates close to its resonance frequency. A distance-dependence force shifts the resonance curve. Another technique is to modulate the position of the sample at a frequency below the cantilever resonance but above the feedback-response frequency and send the response signal to a lock-in amplifier to measure the signal s amplitude and phase [4]. The lock-in output is connected to the auxiliary data acquisition channels to form an image - this approach is popularly known as force modulation (FM-mode). FM-mode imaging or force cmve is an AFM technique that identifies and maps differences in surface stiffness or elasticity. 

Another characteristics feature of the glass transition is the associated change in the modulus. The stress, elongation, is related to the strain, the force applied to a material by the modulus. Conventionally there are two approaches to the measurement of the modulus static and dynamic. The static method involves measurement of the stress strain profile and from the slope of the curve the elastic modulus can be determined. The dynamic method subjects the sample to a periodic oscillation and explores the variation of the amplitude and phase of the response of the sample as a function of temperature. A small sample of the test material is subjected to displacement as shown in Figure 7.3. 

In the local microscopy s field, a large effort is dedicated to study the mechanical properties which can be accessed with a nanotip. Within this context, soft materials are well adapted to probe mechanical response at the nanometer scale. After a discussion of some experimental and technical key points, we present three different types of experiments done on one model polymer polystyrene films with different molecular weights. In the experiments, the tip may scan the sample surface (friction loops), or move upward and downward in the vicinity of the sample -in contact mode (force curve) or in an oscillating mode (approach-retract curves)-. The comparison of the results shows the sensitiveness of the tip to local mechanical properties. New routes to explore mechanical properties without touching the sample are proposed. 

Figure 14. Description of the noncontact ac resonance loop control method. The frequency responses, both amplitude and phase, of a damped and a driven oscillator are shown. Note that both curves are shifted (a to b) by changes in the force gradient.

A second approach is to control the force on the tip directly, generally by means of a small magnet mounted on the back. This removes the need to model a complicated tip oscillation, but imposes stringent demands on the response and stability of the electronics controlling the force. Direct measurements of tip-sample potential curves have now been reported using this technique, but comparison with theory is still in its infancy. 

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