Cantilever calibration

Fig. 2.30 Schematic of cantilever calibration on reference lever. Reprinted with permission from [16]. Copyright 2005. Elsevier...

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]. 

Quantitative evaluation of a force-distance curve in the non-contact range represents a serious experimental problem, since most of the SFM systems give deflection of the cantilever versus the displacement of the sample, while the experimentalists wants to obtain the surface stress (force per unit contact area) versus tip-sample separation. A few prerequisites have to be met in order to convert deflection into stress and displacement into tip-sample separation. First, the point of primary tip-sample contact has to be determined to derive the separation from the measured deflection of the cantilever tip and the displacement of the cantilever base [382]. Second, the deflection can be converted into the force under assumption that the cantilever is a harmonic oscillator with a certain spring constant. Several methods have been developed for calibration of the spring constant [383,384]. Third, the shape of the probe apex as well as its chemical structure has to be characterised. Spherical colloidal particles of known radius (ca. 10 pm) and composition can be used as force probes because they provide more reliable and reproducible data compared to poorly defined SFM tips [385]. 

When the experimentalist set an ambitious objective to evaluate micromechanical properties quantitatively, he will predictably encounter a few fundamental problems. At first, the continuum description which is usually used in contact mechanics might be not applicable for contact areas as small as 1 -10 nm [116,117]. Secondly, since most of the polymers demonstrate a combination of elastic and viscous behaviour, an appropriate model is required to derive the contact area and the stress field upon indentation a viscoelastic and adhesive sample [116,120]. In this case, the duration of the contact and the scanning rate are not unimportant parameters. Moreover, bending of the cantilever results in a complicated motion of the tip including compression, shear and friction effects [131,132]. Third, plastic or inelastic deformation has to be taken into account in data interpretation. Concerning experimental conditions, the most important is to perform a set of calibrations procedures which includes the (x,y,z) calibration of the piezoelectric transducers, the determination of the spring constants of the cantilever, and the evaluation of the tip shape. The experimentalist has to eliminate surface contamination s and be certain about the chemical composition of the tip and the sample. 

A homemade SMFS with a silicon nitride cantilever (Park, Sunnyvale, CA) was used. Each tip was calibrated by using a standard sample. The spring constants of these cantilevers were in the range 0.010-0.012 N/m. By moving the piezo tube, one could bring the sample into contact with the AFM tip so that some polymer chains were physically adsorbed onto the tip, resulting in a number of bridges . As the distance between the tip and the substrate increased, the chains were stretched and the elastic force deflected... 

Producing highly calibrated, trace quantities of explosive vapor is a challenging task. As the vapor pressures of most explosives are extremely small at room temperature, their vapors are often produced by maintaining the sources at higher temperatures. This leads to the condensation of vapor at cold spots, which should be avoided to deliver highly calibrated quantities of explosive vapors to the cantilever sensor. 

Calculation of the critical lateral force is more complicated because traditional LFM does not provide us with easy method to translate current units into force ones. There is no way to define the factor of proportionality until calibration algorithm was developed recently by ourselves [8], The required coefficient depends on design of a microscope, adjustment of the optical system, torsion force constant kL of cantilever and tip height lnp. 

In analogy to indentation experiments, measurements of the lateral contact stiffness were used for determining the contact radius [114]. For achieving this, the finite stiffness of tip and cantilever have to be taken into account, which imposes considerable calibration issues. The lateral stiffness of the tip was determined by means of a finite element simulation [143]. As noted by Dedkov [95], the agreement of the experimental friction-load curves of Carpick et al. [115] with the JKR model is rather unexpected when considering the low value of the transition parameter A(0.2Further work seems to be necessary in order to clarify the limits of validity of the particular contact mechanics models, especially with regard to nanoscale contacts. 

BENDING OF A BEAM. The complex dynamic Young s modulus can be determined from the forced, non-resonant oscillations of a single or double cantilever beam. The apparatus considered in this paper is the Dynamic Mechanical Thermal Analyzer (DMTA) (6), manufactured by Polymer Laboratories, Inc. Figure 3 shows the experimental setup for the single cantilever measurement. A thin sample is clamped at both ends. One end is attached to a calibrated shaker through a drive shaft. 

The Fitzpatrick Co. roller compactor (Fig. 20) features two feed screws in series one horizontal and the second vertical. This system is designed to transfer powder bulk into a set of cantilevered rolls positioned in the horizontal mode one is fixed, the other floats. Operator interface allows for on-line monitoring and controlling feed screw speeds, roll pressure, and roll gap. Features such as on-line help and diagnostic functions, maintenance, and calibration screens are displayed. All functions are interfaced and adjustable by PLC for process control and report generating. The machines have vacuum deaeration and roll cooling capabilities. The sizing unit is separate from the compactor and is either a rotor bar or a hammer mill. 

Cantilevers in AFM function as force transducers converting unknown force to measurable deflection. The value of the unknown force can then be expressed by Hookean mechanics following spring constant calibrations. In addition to static point loads, cantilevers can also be vibrated, e.g., by an oscillation piezo to which the fixed end of the beam is attached (or by other approaches). Excitation frequency, oscillation amplitude, and phase relationships are variables that govern dynamic tapping (intermittent contact) imaging. This problem will be discussed in the next section. 

Similarly challenging, but nevertheless important in the context of constant experimental conditions (e.g., imaging at the same force) or force mapping, is the calibration of spring constants in CM AFM. The nominal spring constants stated by the suppliers are mere indications and cannot be trusted. This is primarily due to the impact of the cantilever thickness on N. 

The sensitivity of the optical deflection detection system is easily calibrated by recording an f-d curve on a stiff substrate, e.g., a glass slide or a piece of silicon. The slope of the hard wall contact region in this photodetector - piezo displacement plot must be unity, as for the movement of 1 nm in z direction the tip and cantilever move upwards 1 nm as well. This function is typically implemented in the AFM software (Fig. 2.29). 

An alternative method relies on the acquisition of f-d curves on (1) a stiff substrate and (2) a reference lever with known spring constant. For this method, the reference lever should have a spring constant close to the one that will be calibrated. If we denote the deflection of the cantilever as Zc and the height of the piezoelectric translator as Zp (zero is defined for the situation, when the tip just touches the reference cantilever and no deflection has been detected), the spring constant is given by... 

With calibrated detection system and cantilevers at hand, the measurement of adhesive forces can be carried out as outlined in Chap. 4. 

A third group of calibration procedures, the (3) direct (single step) [23-25] techniques avoids (many of) these problems. In particular, the so-called improved wedge-calibration method, in which a tip/cantilever is scanned across a calibration sample with two well-defined slopes (see Fig. 2.31), allows one to calculate the calibration factors with an error of ca. 5% [18]. 

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