Force-distance

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... 

It may be that in years to come, interatomic potentials can be estimated experimentally by the use of the atomic force microscope (Section 6.2.3). A first step in this direction has been taken by Jarvis et al. (1996), who used a force feedback loop in an AFM to prevent sudden springback when the probing silicon tip approaches the silicon specimen. The authors claim that their method means that force-distance spectroscopy of specific sites is possible - mechanical characterisation of the potentials of specific chemical bonds . 

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. 

In some activities metabolic energy may be converted to useful work (force distance). At steady state the rate of doing work P = force distance/ time and the thermal losses must balance with metabolism ... 

FIGURE 9.13 Schematic diagram of a force-distance experiment, illustrating the principle components and their movement. (From Huson, M.G. and Maxwell, J.M., Polym. Test., 25, 2, 2006.)... 

It has been shown [56] that if we measure the areas under the approach and retract curves of the force-distance plot we can get quantitative values of the resilience. Resilience is closely related to the ability of the polymer chain to rotate freely, and thus will be affected by rate and extent of deformation, as well as temperature. Different materials will respond differently to changes in these variables [46] hence, changing the conditions of testing will result in a change in absolute values of resilience and may even result in a change in ranking of the materials. Compared to more traditional methods of resilience measurement such as the rebound resiliometer or a tensUe/compression tester. 

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. 

The ratio (p/G) has the units of time and is known as the elastic time constant, te, of the material. Little information exists in the published literature on the rheomechanical parameters, p, and G for biomaterials. An exception is red blood cells for which the shear modulus of elasticity and viscosity have been measured by using micro-pipette techniques 166,68,70,72]. The shear modulus of elasticity data is usually given in units of N m and is sometimes compared with the interfacial tension of liquids. However, these properties are not the same. Interfacial tension originates from an imbalance of surface forces whereas the shear modulus of elasticity is an interaction force closely related to the slope of the force-distance plot (Fig. 3). Typical reported values of the shear modulus of elasticity and viscosity of red blood cells are 6 x 10 N m and 10 Pa s respectively 1701. Red blood cells typically have a mean length scale of the order of 7 pm, thus G is of the order of 10 N m and the elastic time constant (p/G) is of the order of 10 s. 

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... 

FIG. 8 Force-distance dependence for surfaces covered with fluorocarbon amphiphile 1 in pure water (1) and in aqueous solutions containing 0.7 mg/L poly (styrenesulfonate) (2) and 7.0 g/L poly (styrenesulfonate) (3). The molecular weight of the polymer is 5 X ICP. Lines are drawn as a visual guide. 

Figure 11a shows a force-distance profile measnred for poly(L-glutamic acid) brushes (2C18PLGA(44)) in water (pH = 3.0, 10 M HNO3) deposited at 40 mN/m from the water subphase at pH = 3.0. The majority of peptides are in the forms of an a-helix (38% determined from the amide I band) and a random coil. Two major regions are clearly seen in... 

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. 

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