Cantilever stiffness

One point, which is often disregarded when nsing AFM, is that accurate cantilever stiffness calibration is essential, in order to calculate accurate pull-off forces from measured displacements. Althongh many researchers take values quoted by cantilever manufacturers, which are usually calculated from approximate dimensions, more accurate methods include direct measurement with known springs [31], thermal resonant frequency curve fitting [32], temporary addition of known masses [33], and finite element analysis [34]. 

Fig. 13.5. Quantitative measurements of contact stiffness by UFM (a) theoretical calculation of the UFM response on Si (1) and Ge (2) surface, using the engineering parameters ESi = 164 GPa, EGe = 121 GPa, surface energy in an ambient environment y = 1 N m-1, and the manufacturer s data for cantilever stiffness kc = 2.8 nN nm-1 and tip radius Rt = 10 nm (b) experimentally measured UFM response of a silicon surface (c) schematic illustration of the differential UFM approach to the measurement of contact stiffness (Kolosov and Yamanaka 1993), from the threshold amplitude values (ai and a2) for two different normal force values (Fi and P2), the contact stiffness Seff is given by Seff = (F2 — -Fi)/(a2 — ) (d) experimental stiffness measurements... 

Figure 13.5(d) presents experimental stiffness measurements using differential UFM for three high modulus surfaces sapphire, Si(100) and LiF(lOO) (Dinelli et al. 2000b). The samples were probed with the same silicon tip on a V-shaped cantilever (nominally cantilever stiffness was kc - 2.8 nN nm 1,and radius of curvature R = 10 nm). The surface RMS roughness of the surfaces was less than 0.2 nm over a few square micrometres for all three samples. The relative difference between the three sets of data reveals that the elastic properties of these three materials can be distinguished by differential UFM the relative independence of the applied force may indicate the fact that the tip had been flattened by extended contact with such hard samples. 

To minimise the friction effect, it has been proposed to use smaller amplitudes and higher frequencies [122,137]. The so-called scanning local-acceleration microscopy (SLAM) is another modification of contact-mode SFM which was implemented by vibrating the sample at a frequency above the highest tip-sample resonance (region III in Fig. 13b). In this frequency range (around 1 MHz), the cantilever response to the sample excitations becomes independent of the cantilever stiffness and depends linearly on the contact stiffness and reciprocally on the cantilever mass m (Fig. 13b) ... 

Essentially, the sensitivity of FMM for stiffness variations is governed by the ratio of the cantilever stiffness to the stiffness of the tip-sample contact. Moreover, for strong disproportions between these two stiffnesses, the dynamic cantilever response was shown to be affected by frictional forces [57]. For typical cantilevers with bending stiffness, kc, of -0.1-50 N/m this condition is easily met when dealing with compliant polymers. In the case of isotactic polypropylene (PP) with the Young s modulus value Ewl.5 GPa, a tip radius of curvature 10 nm and an imaging force of %50 nN, the effective... 

When approaching the surface with the tip (A), the tip experiences an attractive force gradient " larger than the cantilever stiffness, it snaps to the surface by bending the cantilever (B). The AFM tip moves closer to the surface, the attractive forces between tip and sample become smaller. As the tip is approached further, the forces can even become repulsive (C). When the cantilever is then retrieved (D),... 

That means that for a produced Ao, the cantilever bending will be higher for longer and thinner cantilevers, i.e., with low stiffness. The parameter that characterizes the cantilever stiffness is the spring constant, which depends on the cantilever dimensions and Young s modulus (see Subheading 3.2). The biomolecular interactions could be difficult to detect with commercial micro-cantilevers due to the low surface stress induced on such micro-cantilevers. To improve the cantilever deflection, this parameter, and its dimensions dependency, needs to be simulated before fabrication. The simulation was performed using a finite element analysis (FEA) technique (with ANSYS software). 

Equations (5.25)-(5.28) demonstrate the importance of the cantilever stiffness with respect to the sample hardness. But to access quantitatively the work of adhesion and the elastic properties of the sample we have to know the spring constant accurately, since Pgff equals kZ with the cantilever deflection, or spring extension needed to unstick the tip. Unfortunately, because of the poor accuracy of the spring constant of the cantilever, rather than quantitative experiments, only comparative studies can be carried out. 

However, the nano-adhesion of PDMS 17 decreased after extraction. This result is the opposite of that for macro-adhesion, where an increase in adherence was observed after extraction. Important adsorption phenomena of the numerous and long free chains of PDMS 17 on the AFM tip could explain the higher nano-adhesion observed before extraction. The mobility of these free chains (greater than of pendant chains) allows a better adsorption on the tip, avoiding the separation (with the same experimental device, i.e., cantilever stiffness). 

The apparatus used for making experimental measurements was a cantilever bending tester (ASTM D1388-07A). Cantilever stiffness is given by Equation [16.5] ... 

Softer cantilevers are used for contact mode imaging of softer samples and tapping mode in liquid due to the lower resonance frequencies that reduce damping effects. In terms of force spectroscopy, the cantilever stiffness has to be... 

Figure 5 Variation of the friction deflection AZf with the tip-sample distance variation AZy. The friction coefficient p is given by = p P with F, the friction force, P the applied effective load, in this case F = k AZf and P = ky AZu + ky AZy with dtZyx a constant throughout the experiment and ki, ky are the lateral and vertical cantilever stiffness respectively, giving with h the tip height and L the cantilever length.

The limits of detection of surface deformation are determined by the ratio of materials elasticity and spring constant of cantilevers. If cantilever stiffness is too high then the vertical deflection of the SFM cantilever, Zdefi is very small as compared to Zpos the vertical displacement of the piezoelement. In this case, the ratio, T, of indentation depth, h Zpos Zdefl to total displacement reaches upper measurable limit ... 

On the other hand, if material is much harder the cantilever stiffness, then Zpos which results in lower limit of observation ... 

Only for the PDMS elastomer is the stiffness of the surface on the order of the cantilever stiffness used to probe the mechanical properties. All other surfaces show negligible penetration. The sensitivity to measure mechanical properties of polymer surfaces using AFM under static deflection depends on the relative stiffness of the cantilever and surface (14). The soft (k contact mode cantilevers were not able to distinguish between materials with bulk modulus in excess of IGPa (2). 

Force-Curves Measurements In force curve measurements, a vertical displacement of the sample, z, is imposed and the subsequent tip displacement, d, is measured. The tip-sample interaction force, F, is deduced by means of the Hooke s relation, F = -kcd, where is the cantilever stiffness. Force curves arc generally divided into different regions (7). If the part where the electrostatic repulsion forces are dominant is only considered, with silicon tips much stiffer than polymers, tips penetrate the sample surface and an indentation depth, 5, equal to Z d, can be measured. The lower the sample elastic modulus, the greater will be the indentation depth. By using the Hertz mechanical model adapted to the geometry of the tip-sample system (8,9) surface elastic modulus could be deduced from the following equations corresponding respectively to a spherical, a paraboloid and a conical tip ... 

In this relation, k. is the cantilever stiffness and ki is the tip-surface effective stiffness given by dFIBS. From the relations (1.1 to 1.3), it can be seen that k depends on the static contact force, the tip geometry and the surface elastic modulus. Knowing the cantilever stiffness, the static contact force and the tip geometry and dimensions, it is thus theoretically possible to determine the surface elastic modulus from the dynamic response. 

Bridge Drilled Non-linear Soil Equivalent Coupled Foundation Shaft System Spring Model Cantilever Stiffness Matrix... 

Figure 7.1. (a) Shirley Stiffness Tester (b) Principle of Cantilever Stiffness Test. 

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