PeakForce QNM Channels

The reduced Young’s Modulus, E*, is obtained by fitting the retract curve (green line in Figure 2.5a) using the Derjaguin, Muller, Toropov (DMT)Derjaguin B.V., Muller V.M., Toropov Yu.P., J. Colloid. Interface Sci. 53, 314 (1975).1 model given by

Where Ftip is the force on the tip, Fadh is the adhesion force, R is the tip end radius and d is the tip-sample separation.

Figure 1: Force vs. Separation plot

Figure 2 shows a DMT Modulus map of PS+LDPE blend.

Figure 2: DMT Modulus map of a PS+LDPE blend

The logarithm of the elastic modulus of the sample based on the DMT model.

The Sneddon model assumes a conical indenter as shown in Figure 3.

Figure 3: Contact between a cone and an elastic half-space

The force is:

Equation 4: Sneddon model

Equation 4 may be linearized by taking both sides to the 1/2 power, shown in Equation 5.

Equation 5: Linearized Sneddon equation

In this linearized model, Young's modulus is computed from the slope of Equation 5

You will need to enter the Tip Half Angle into the Cantilever Parameters panel. The Spring Constant and Tip Radius are taken from the Probe Database but may be changed in either the database or the Cantilever Parameters panel. You may also modify the Sample Poisson's Ratio.

Figure 6: The Cantilever Parameters panel in the Scan Parameters window

The logarithm of the Sneddon modulus, above, of the sample.

The peak force below the baseline, shown in Figure 7. Figure 8 shows an adhesion map of a PS+LPDE blend.

Figure 7: Adhesion on a PS+LDPE blend

Figure 8: Adhesion map of a PS+LDPE blend

This channel produces a map of the peak force (see Figure 11) measured during the scan. Because the PeakForce QNM mode uses peak force as the feedback signal, this channel is essentially the Peak Force Setpoint plus the error.Figure 9 and Figure 10 illustrate the peak force location.

Figure 9: The “heartbeat,” Force vs. Time

Figure 10: Force curve, Force vs. distance

Figure 11: Peak Force Error map of a PS+LDPE blend

Energy Dissipation (W) is given by the force times the velocity integrated over one period of the vibration:

F is the interaction force vector and dZ is the displacement vector. Because the Z motion and the velocity reverse direction in a half cycle, the integral is zero if the load and unload curves coincide. The dissipation is therefore the hysteresis between the load and unload curves. Pure elastic deformation has no hysteresis which corresponds to very low dissipation. Energy dissipated is displayed in electron volts as the mechanical energy lost per tapping cycle.

The Dissipation channel plots the dissipated energy in each cycle by integrating the area between the Trace (load or extend) and Retrace (unload or retract) curves as shown in the blue area in Figure 12.

Figure 12: Dissipation (shaded area) in a polystyrene (PS) and Low-density polyethylene (LPDE) blend

Figure 13 shows the dissipation image of a PS+LDPE blend.

Figure 13: Dissipation image of a PS+LDPE blend

The maximum deformation of the sample (defined as the distance from the Deformation Force Level position to the peak interaction force position) caused by the probe. See Figure 14. Figure 15 shows a deformation map of a PS+LPDE blend.

The total deformation will be slightly larger than the displayed deformation because the default Deformation Fit Region is 85% of the full deformation. See Deformation Force Level.

Figure 14: Deformation

Figure 15: Deformation map of a PS+LDPE blend