Gradient force

Use Equation VIII-1 to determine the effective mass of the cantilever if the cantilever has a spring constant C = 20 N/m, the minimum detectable force gradient is hF/dz = 4 X 10 N/m, and the frequency shift is 200 kHz. How does the frequency shift depend on distance from the surface if the force has a 1/z distance dependence ... 

Most NC-AFMs use a frequency modulation (FM) teclmique where the cantilever is mounted on a piezo and serves as the resonant element in an oscillator circuit [101. 102]. The frequency of the oscillator output is instantaneously modulated by variations in the force gradient acting between the cantilever tip and the sample. This teclmique typically employs oscillation amplitudes in excess of 20 mn peak to peak. Associated with this teclmique, two different imaging methods are currently in use namely, fixed excitation and fixed amplitude. 

Note Wh en you use Cl, HyperChem does not calculate forces (gradient). 

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... 

Ideally one would wish to remove the need for statistics by directly and reproduce-ably measuring a single bond only. One problem with the measurement of specific individual bond energies is that it is extremely difficult, even with a tip of small radius, to isolate a single bond species between the tip and the sample. To form a single bond in a controlled way requires the cantilever to be stiffer than the maximum force gradient experienced during the approach, but stiffer levers exhibit less sensitivity. If multiple bonds are formed, then it can be difficult to make an independent calculation of the contact area and hence the number of bonds involved. 

An alternative technique is noncontact AFM [18]. Figure 19 illustrates the concept. The tip oscillates above the surface, and the modulation in amplitude, phase, or frequency of the oscillating cantilever in response to force gradients from the sample can be measured to indicate the surface topography. Even without contact, the amplitude, phase, or frequency can be affected by the van der Waals forces of the sample within a nanometer range, which is the theoretical resolution however, this effect can be easily blocked by the fluid contaminant layer, which is substantially thicker than... 

Fig, 19, Differences between noncontact and tapping mode AFM, The signal for the former is dependent on the change in oscillation due to the force gradient, while the latter is dependent on the oscillation change due to the contact, (Courtesy of Digital Instruments, Veeco... 

Fig. 2.1. Correlation between tunneling conductance and force. The tunneling conductance and the force gradient are measured simultaneously. The origin of the abscissa is the equilibrium point, where the net force is zero. It corresponds to a tunneling conductivity of 10 O. The experimental setup is schematically shown in the inset. (Reproduced from Diirig et al., 1988, with permission.)...

Fig. 15.5. Force-gradient detection via mechanical resonance. Experimental and calculated resonance frequencies of a cantilever for various positions of the tunneling tip. Upper family of curves, for the copper, and lower, for the steel beam. (Reproduced from Diirig et al., 1986, with permission.)...

The dimensionless parameter a is introduced for convenience, which is almost always less than 1. The resonance frequeney too is related to the foree constant K through Eq. (15.3). The force gradient (in this operation mode usually negative) -dFJdz changes the resonance frequency too. 

The amplitude A in Eq. (15.5) changes. After a short amount of algebra, the relation between the force gradient and the amplitude reduction ratio a is... 

Figure 15.6 is a schematic diagram of an AFM with an optical interferometer (Erlandsson et al., 1988). The lever is driven by a lever oscillator through a piezoelectric transducer. The detected force gradient F is compared with a reference value, to drive the z piezo through a controller. In addition to the vibrating lever method, the direct detection of repulsive atomic force through the deflection of the lever is also demonstrated. 

In Fig. 1 (top right) we show a sloped conical intersection in the terminology of Ruedenberg et al (29). Here the cone is tilted due to the fact that the force (gradient) vectors on both the upper and lower surfaces point in the same direction. The first-order topology (sloped vs. peaked) controls the nature of the photochemical reaction dynamics, and whether reactants are regenerated or photoproducts are formed (23,24). 

Under centrifugal force, gradient redistributes and sample particles band at their... 

Uniform 2 Under centrifugal mixture of force, gradient... 

A force microscope actually measures the forces between two macroscopic bodies. The finite size and the macroscopic surface of the tip and the surface spot lead to a number of fundamental consequences in their interaction (Fig. 2). First, the net force is stronger than the intermolecular forces and it acts at much larger distances. Even in the 10-100 nm range, the interaction energy, which is proportional to the size of the tip, can exceed kBT. Secondly, the force between a spherical tip and a flat surface decays with the separation as F D 2 (Fig. 2b) compared to f r 7 for the attraction between two atoms (Fig. 2a). In combination with the finite tip size, the low force gradient increases the effective interaction area and limits the resolution (see Sect. 2.3.3). Third, the surrounding me-... 

FIGURE 1.8. (a) Schematic representation of the device used to study capillary surface instabilities. A polymer-air bilayer of thicknesses /ip and /ia, respectively, is formed by two planar silicon wafer held at a separation d by spacers. A capillary instability with wavelength k = 27t/q is observed upon applying a voltage U or a temperature difference AT. (b) Dispersion relation (prediction of Eq. (1.6)). While all modes are damped (r < 0) in the absence of an interfacial pressure pei, the application of an interfacial force gradient leads to the amplification of a range of k-values, with /.m the maximally amplified mode. 

FIGURE 1.14. Schematic representation of the pattern replication process. The topography of the top plate induced a lateral force gradient that focuses the instability towards the downward pointing protrusions of the master plate. 

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