Force-displacement curves

If we load a material in compression, the force-displacement curve is simply the reverse of that for tension at small strains, but it becomes different at larger strains. As the specimen squashes down, becoming shorter and fatter to conserve volume, the load needed to keep it flowing rises (Fig. 8.6). No instability such as necking appears, and the specimen can be squashed almost indefinitely, this process only being limited eventually by severe cracking in the specimen or the plastic flow of the compression plates. 

Numerous AFM imaging techniques have been developed and commercialized to monitor topography, friction, mechanical response, capacitance, magnetic properties, etc. However, adhesion measurements require the tip to come into, and out of, contact to measure attractive and adhesion forces. Therefore, other than to select an analysis region, most imaging techniques are not useful for adhesion studies. Instead, measurements are necessarily based on force-displacement curve approaches. 

In a force-displacement curve, the tip and sample surfaces are brought close to one another, and interact via an attractive potential. This potential is governed by intermolecular and surface forces [18] and contains both attractive and repulsive terms. How well the shape of the measured force-displacement curve reproduces the true potential depends largely on the cantilever spring constant and tip radius. If the spring constant is very low (typical), the tip will experience a mechanical instability when the interaction force gradient (dF/dD) exceeds the... 

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. 

We have recently been exploring this technique to evaluate the adhesive and mechanical properties of compliant polymers in the form of a nanoscale JKR test. The force and stiffness data from a force-displacement curve can be plotted simultaneously (Fig. 13). For these contacts, the stiffness response appears to follow the true contact stiffness, and the curve was fit (see [70]) to a JKR model. Both the surface energy and modulus can be determined from the curve. Using JKR analyses, the maximum pull off force, surface energy and tip radius are... 

Figure 11 Force displacement curves of a-PP, b-nonreac-tive PP-NBR blend, c-reactive blend containing 13 wt% GMA functionalized PP, and d-reactive blend containing 25 wt% GMA functionalized PP. Source Ref. 73.

In the current work a Digital Instmments Dimension 3000 SPM was operated in force-volume mode using a probe with stiffness selected to match the stiffness of the sample. Standard silicon nitride probes with a nominal spring constant of 0.12 or 0.58 N/m were used for recombinant and native resilin samples. These samples were characterized in a PBS bath at a strain rate of 1 Hz. For synthetic rubbers, silicon probes with a nominal spring constant of 50 N/m were used and the material was characterized in air. Typically, at least three force-volume plots (16 X 16 arrays of force-displacement curves taken over a 10 X 10 p.m area) were recorded for each of the samples. 

Fig. 18 A typical force-displacement curve. WF= work done overcoming die wall friction WD, work of elastic recovery Wp/, net work involved in tablet compact formation.

AM Mehta, LL Augsburger. Quantitative evaluation of force displacement curves in an automatic capsule filling machine. Presented to the IPT Section, A.Ph.A. Academy of Pharmaceutical Sciences, 128th Annual A.Ph.A. Meeting, St. Louis, March-April 1981. 

Force-Displacement Curves and Force-Displacement Curves Indentation (FDI)... 

A short introduction on force-displacement curves can be found in Sect. 2. A more detailed review about force-displacement curves and their application is given in [6]. In this section we will focus our attention on the use of force-displacement curves as a lithography tool in comparison to DPL. 

The use of force-displacement curves as a lithography tool presents two important drawbacks ... 

Since force distance curves cannot be acquired with any frequency (the normal frequency is 1 Hz above 3-4 Hz oscillations may engender several important artefacts), the acquisition of force-displacement curves is very time consuming, when compared with other methods. 

Since each force-displacement curve comprises normally 200 points, data files are 200-fold larger than in contact mode or in DPL. 

Since the first experiment, where Bhushan and Koinkar [265] have measured the hardness of silicon with a modified SFM, researchers have been using force-displacement curves more for the determination of certain sample properties, e.g. hardness and stiffness, than for the structuring of surfaces. 

Only in the last three years have some researchers begun to pay their attention to the structured surface and to parameters influencing the lithography with force-displacement curves (FDI) [266-268]. The big advantage of FDI is the possibility of gaining knowledge about the whole indentation process during FDI, the force and the indentation are known at every point, and not only stiffness and hardness, but also other important properties such as density, elasto-plastic behaviour, adhesion, time behaviour, etc. can be measured and calculated. 

Our method for the acquisition of force-displacement curves is described in details in [32]. Force-displacement curves are acquired following the surface profile and the sample is indented until a maximum force Fmax is reached. 

Fig. 26 a A hole written in PMMA with a single force-displacement curve (scan width 600 nm, z scale 142 nm, 60x60 pixels), b Depth of the carved holes as a function of the maximum force Fmax. The depth is proportional to the maximum force... 

Since the force-displacement curve contains information about the whole indentation process, the elastic deformation of the sample can be measured and used to calculate the stiffness S=dFldh at h=hmax, where F is the force and h is the indentation. As already explained in Sect. 3.1.1., in order to relate the stiffness to the Young s modulus, it is necessary to make assumptions about the contact area. The depth of the permanent indentation (plastic deformation), i.e. the depth DFdi shown in Fig. 26b, and the maximum indentation (sum of the plastic and of the elastic deformation) can be used to calculate a parameter that describes the relative weight of the elastic and of the plastic response. 

Hence, when a hole of the same depth is obtained with the two methods, the force applied in DPL could be much greater than the force applied in FDI. The Young s modulus at higher frequency cannot be measured with force-displacement curves in an approximated way because no force-displacement curves can be acquired at this frequency. 

In order to characterize the material in the border walls and to study its physico-chemical properties, force-displacement curves have been acquired on a surface where rectangles with large border walls, like the second square in Fig. 25, had previously been written [272]. 

Figure 29 shows the result of such a measurement. Figure 29a is the topography of the lithographed surface, acquired with force-displacement curves. In the bottom part of the figure, three typical force-displacement curves can be seen, acquired at the locations indicated in Fig. 29a. Approach (withdrawal) curves are plotted with full (open) circles.

When force-displacement curves are acquired with Fmaxdifferent hardness and to the different density of the polymer. 

Antikainen, O. K., and Yliruusi, J. K. (1997), New parameters derived from tablet compression curves. Part 2. Force-displacement curve, Drug Dev. Ind. Pharm., 23, 81-93. 

Fig. 17 Force-displacement curves corresponding to elementary material s behaviors a brittle, b semi-brittle, c semi-ductile and d ductile...

Relevant and complementary information about the damage process of polymers can be obtained among others by the analysis of the force-displacement curves, by the observation of the fracture surfaces (cf. Sects. 3.2.5 and 5.4) and, as will be shown in Sect. 6.2.2, by the determination of the amount of voids in a sample during and/or after deformation. However, a complete elucidation of the deformation mechanisms is only possible by their direct observation at the sub-micron level. Transmission electron microscopy is often used for this purpose. For convenience, the tests (which require experience and touch) are generally carried out at room temperature and at a low strain rate. 

Kgure 10.2. Schematic view of the three regions of the force-displacement curve of a typical cellular solid I, small deformation of the intact structure II, buckling and fracture of cell walls and III, compaction of what is increasingly collapsed cell wall material. 

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