Force-distance plot

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. 

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. 

Fig. 1 shows the force-distance plots for the carbon, (a), and the glass, (b) laminates with a notch depth of 5mm (for all samples except (p = 15° glass, which had 4mm notches). It can be seen that the (p = 15° and 30° laminates were stressed to failure without any significant plasticity effects being evident. The 45° coupons, on the other hand, had much quasi-stable deformation before failure, while the 60° and 75° laminates broke very easily. 

In force—displacement measurements, the sample is moved up and down (in and out of contact with the tip) at a fixed position (x, y) (compare also Chap. 2). The resulting force—distance plot (force—displacement curve) displays the bending of the cantilever as a result of tip—sample interaction forces. The interpretation of forces curves has been discussed in detail, e.g., by Cappella and Dietler [3] and by Butt and co-workers [4]. 

FIGURE 1.29 Force-distance plot for (a) polypyrrole (NO,-) and (b) polyaniline (HC1) on carbon foil. 

The AFM force-distance plots were obtained by oscillating the sample up and down and monitoring the response of the AFM cantilever. The sample had to come into contact with the AFM tip and retract from the tip with each oscillation. Under conditions of small cantilever bending, the force exerted on the cantilever is directly related to the extent of bending by F = kAx, where ax is the distance the tip of the cantilever has moved and k is the spring constant of the cantilever. 

The AFM force-distance plots are similar, but not identical to the force-distance profiles obtained by the surface force apparatus. The bending of the cantilever can follow the force exerted on it by the sample as long as the spring constant of the cantilever exceeds the force gradient of the exerted force. Otherwise, the cantilever jumps into contact with the surface in an attractive regime and in the repulsive regime, is merely pushed a distance equal to the distance the sample is mov, thus... 

Figure 2a depicts a typical force-distance plot in the absence of PEO derivatization. The abcissa is the response of the photodiode and is directly related to cantilever bending. The ordinate is the distance the sample has moved when oscillating up and down (larger separations are to the right). At large separation... 

Figure 2a. A schematic of a typical AFM force-distance plot using unmodified tips. The arrows show how the force plot is generated as the sample is advanced and retracted. At (a) the tip and sample are far apart. When the tip gets close enough to experience the attractive van der Waals force, the cantilever starts to bend (b). When the force gradient exceeds the cantilever spring constant, the tip jumps into contact with the surface (c). Once in contact, the tip and sample move the same amount as shown by the linear portion (d). Upon the retraction (e), the cantilever relaxes a distance equal to the amount the sample has been retracted. If there is an adhesive force, then there is hysteresis in the loop (f). The inset depict the state of cantilever bending. Positions 1 and 2 show how to calibrate the force scale. The distance the cantilever has moved, ax, multiplied by the spring constant, k, yields the force difference, aF, between the two positions.

The abcissa of the plots is already calibrated by the instrument, only the zero separation distance needs to be determined. The ordinate can be easily cdibrated to show force units because in the linear region of the force-distance plot, the slope equals k, the spring constant of the cantilever aF = kAx (see Figure 2a). Zero force is defined by the flat part of the plot. 

Figure 2b. A schematic of a typical AFM force-distance plot using a PEO modified tip. At (a), there is no interaction between tip and surface. As the chains begin to compress (b) a repulsive steric exclusion force is observed. At (c), the chains are compressed even more producing an even larger repulsive force that dominates the attractive van der Waals force. At (d), the chains arc so much compressed that the cantilever spring constant is much weaker than the spring constant of the PEO chains and the cantilever continues to bend upward the same amount as the sample has been moved due to the large repulsive force gradient. Upon retraction, no adhesion is observed (provided there is no bridging) and the curve coincides with the approach curve.

A typical AFM force-distance plot using a plasma cleaned silicon nitride tip and a freshly cleaved mica surface in the presence of 0.1 M KNO3 is shown in Figure 3 (plot A). The advancing curve and the receding curve should be coincident along the linear and flat portions, but the instrumentation does not always depict this superposition. In this plot, as is the case in all other plots, the advancing curve is... 

Plot B in Figure 3 shows the AFM force-distance curves obtained when a PEG 2 kDa covalently bound to the tip compresses against a freshly cleaved mica sample in the presence of 0.1 M KNO3. A small attractive force just prior to the linear portion of the force-distance plot is observed. This implies that the steric exclusion force at this position is weaker than the van der Waals attractive force. This is not unexpected, since such a small molecular weight PEG is bound to the surface. With a 2 kDa PEG polymer attached, the is 1.7 nm and the steric exclusion force should commence at about 4 nm (2.5 Rg). cS)mputer modelling by GOLIAD indicated that a force of 5.2 nN (5.7 minus 0.5 nN) can be expected at a 0.6 nm separation distance when one equivalent monolayer of PEG segments was present on the surface of the probe. Such a force is detectable by the AFM. One possibility why a repulsive force does not appear in this experiment is that a lesser amount of PEG 2 IdDa than one equivalent monolayer is bound to the surface. 

Figure 3. Plot A depicts force-distance curves recorded using a plasma cleaned tip and freshly cleaved mica in the presence of 0.1 N KNO3. These curves display a jump into contact upon approach and an adhesive force upon retraction of the tip. Plot B shows force-distance curves recorded using a PEG 2 IdDa treated tip and freshly cleaved mica in the presence of 0.1 N KNO3. This plot does not show a monotonically increasing repulsive force upon approach, but does show an adhesive force upon retraction and is similar to the force-distance plot of a tip plasma cleaned only. Plot C shows force-distance curves recorded using a PEG 9(X) kDa treated tip and freshly cleaved mica in the presence of 0.1 N KNO3. This plot shows no jump into contact upon approach and an adhesive force upon retraction. The cantilever spring constant was 0.064 nN/nm and the frequency of oscillation was 25 Hz in all cases.

Figure 6 shows two force-distance plots for a PEO 200 kDa adsorbed silicon nitride surface. The initial curve recorded at the pristine site (Figure 6, plot A) shows... 

Figure 4. This series of force-distance plots shows the measured forces when a plasma cleaned tip and mica are used to obtain force-distance plots in the presence of a 0.1 % w/v PEO 900 kDa solution containing O.IN KNO3 as a function of incubation time. Immediately after injection (plot A), following an 8 hour incubation (plot B), and after 2 minutes of oscillations following an 8 hour incubation (plot C). The cantilever spring constant was 0.064 nN/nm and the frequency of oscillation was 1 Hz.

Figure 5. Two force-distance plots obtained when the PEO 900 kDa has been physically adsorbed to the silicon nitride surface in the presence of 0.1 N KNO3. A pristine site compression cycle is shown by the curves A. Occasionally, a compression cycle would yield a force-plot similar to the curves B. Here, a series of a esive "snap-offs" upon retraction is most likely due to the ripping of PEO chains off from one of two surfaces after they have bridged the two surfaces. The cantilever spring constant was 0.064 nN/nm and the frequency of oscillation was IHz.

Fig. 9 Measurement of force distance curves (left) Schematic force-distance plot (right) The sample is approaching the tip (1, top) at some distance the gradient of the force overcomes the cantilever spring constant and the tip jumps into contact (2) further movement up causes a deflection of the cantilever (3) during retraction the tip sticks usually much longer to the surface (4) and snaps off when the spring constant overcomes the force gradient (5). The adhesion between tip and sample is characterized by the so-called pull-off or pull-out [156] force (snap off). (Adapted with permission from [143])...

Figure 1.26 (a) Force-distance plot for pol3 yrrole (NO3 ) on carbon foil, (b) Force-distance plot for jxjlyaniline (HCl) on carbon foil. 

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