Force distance curve

Pulsed-force mode AFM (PFM-AFM) is a method introduced for fast mapping of local stiffness and adliesion with lower required data storage than recording force-distance curves at each point on the x-y plane [115]. A sinusoidal or triangular modulation is applied between the tip and sample (either via lever or sample piezo) at a lower frequency than that of either the piezo or cantilever resonance frequency. Tip and sample then come... 

Fig. 4.11. The energy curve (top), when differentiated (eqn. (4.6)) gives the force-distance curve (centre).

As we showed in Chapter 6 (on the modulus), the slope of the interatomic force-distance curve at the equilibrium separation is proportional to Young s modulus E. Interatomic forces typically drop off to negligible values at a distance of separaHon of the atom centres of 2rg. The maximum in the force-distance curve is typically reached at 1.25ro separation, and if the stress applied to the material is sufficient to exceed this maximum force per bond, fracture is bound to occur. We will denote the stress at which this bond rupture takes place by d, the ideal strength a material cannot be stronger than this. From Fig. 9.1... 

Quasi-static measurements force-distance curves and adhesion... 

Fig. 1. Schematic diagram illustrating the mechanical instability for (a) a weak spring (spring constant k) a distance D from the surface, experiencing an arbitrary surface force (after [19]) and (b) the experimentally observed force-distance curve relative to the AFM sample position (piezo displacement) for the same interaction.

Hertzian mechanics alone cannot be used to evaluate the force-distance curves, since adhesive contributions to the contact are not considered. Several theories, namely the JKR [4] model and the Derjaguin, Muller and Torporov (DMT) model [20], can be used to describe adhesion between a sphere and a flat. Briefly, the JKR model balances the elastic Hertzian pressure with attractive forces acting only within the contact area in the DMT theory attractive interactions are assumed to act outside the contact area. In both theories, the adhesive force is predicted to be a linear function of probe radius, R, and the work of adhesion, VFa, and is given by Eqs. 1 and 2 below. 

A series of force-distance curves for various materials pairs examined (gold/ nickel, diamond/graphite, diamond/diamond) are shown in Fig. 4 [39]. For an indentation, the unloading slope (dF/dr) of the force-displacement curve is a measure of the contact stiffness and can be used to determine the modulus if the contact area (A) is known using a variant of Eq. 3 below. 

At present, there are a variety of theoretical models to describe the mechanical contact between the two bodies under external load and many of such theories have been used to analyze force-distance curves.Among them. Hertz theory has been the most widely used because of its... 

FIGURE 21.10 (a) The schematic drawing of the sample deformation for an elastic sample, (b) The comparison between force-distance curves for stiff and elastic samples. 

Force curve gives the relationship between the z-piezo displacement and the cantilever deflection as shown in Figure 21.10b. When a cantilever approaches to a stiff sample surface, cantilever deflection. A, is equal to the z-piezo displacement, z — Zo- The value of zo is defined as the position where the tip-sample contact is realized. On the other hand, z-piezo displacement becomes larger to achieve the preset trigger value (set point) of the cantilever deflection in the case of an elastic sample due to the deformation of the sample itself. In other words, we can obtain information about a sample deformation, 8, from the force-distance curve of the elastic surface by the following relationship ... 

NR with standard recipe with 10 phr CB (NR 10) was prepared as the sample. The compound recipe is shown in Table 21.2. The sectioned surface by cryo-microtome was observed by AFM. The cantilever used in this smdy was made of Si3N4. The adhesion between probe tip and sample makes the situation complicated and it becomes impossible to apply mathematical analysis with the assumption of Hertzian contact in order to estimate Young s modulus from force-distance curve. Thus, aU the experiments were performed in distilled water. The selection of cantilever is another important factor to discuss the quantitative value of Young s modulus. The spring constant of 0.12 N m (nominal) was used, which was appropriate to deform at rubbery regions. The FV technique was employed as explained in Section 21.3.3. The maximum load was defined as the load corresponding to the set-point deflection. 

A force-distance curve between layers of the ammonium amphiphiles in water is shown in Figure 8. The interaction is repulsive and is attributed to the electric double-layer... 

The force-distance curves show the breaking strengths of gels. The evaluation of the curves surpassing the breaking strength can be difficult because of poor reproducibility. 

The breakdown of the semi-dilute brush theory was also revealed in the force-distance curve of the high-density PMMA brush. Using the scaling approach [138], de Gennes derived the equation concerning the interaction force between two parallel plates with a semi-dilute polymer brush layer, predicting that the force-distance profiles should be scaled by plotting (F/i )... 

Mapping of the elastic modulus of the glassy and rubbery blocks and clay regions was probed by employing Hertzian and Johnson-Kendall-Roberts (JKR) models from both approaching and retracting parts of the force-distance curves. In order to determine the elastic properties of SEBS nanocomposites in its different constituting zones, the corrected force-distance curve was fitted to the Hertz model ... 

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