Model-based extraction of material properties in multifrequency atomic force microscopy

We present a method to reconstruct the nonlinear tip-surface force and extract material properties from a multifrequency atomic force microscopy (AFM) measurement with a high-quality-factor cantilever resonance. In a measurement time of $\sim$2 ms, we are able to accurately reconstruct the tip-surface force-displacement curve, allowing simultaneous high-resolution imaging of both topography and material properties at typical AFM scan rates. We verify the method using numerical simulations, apply it to experimental data, and use it to image mechanical properties of a polymer blend. We further discuss the limitations of the method and identify suitable operating conditions for AFM experiments.

Figure 1

Spectrum of the tip-surface force in intermodulation AFM. The force can be reconstructed from the partial spectrum of intermodulation products contained in the band $\Omega$ near resonance. (a) Experimental data acquired on poly(methyl methacrylate) with two drives, $f_1=270.4$ and $f_2=270.9$ kHz, producing a maximum free oscillation amplitude of 60 nm peak to peak. The amplitude at $f_1$ in the measurement was 85$\%$ of its free amplitude. The cantilever had $f_0=270.8$ kHz and $Q=416$, and noise calibration gave $k_c=21$ N/m. Many peaks of the intermodulation spectrum near $f_0$ are resolved above the noise. Intermodulation peaks near $2f_0$ are buried in the detector noise. (b) Simulated force spectrum using the DMT-Voigt model and parameters extracted from the experimental data. The partial spectrum in the band $\Omega$ contains enough information to reconstruct the full force spectrum.

Figure 2

Two-dimensional cuts of the multidimensional error space. The magnitude of the error function $E$ for the DMT-Voigt model is plotted in color against a few different pairs of parameters in the vicinity of the minimum. The white cross indicates the solution found by the numerical solver. The white area in (b) indicates that the error is larger than the range covered in the color bar.

Figure 3

Dependence of the extracted parameter values on approach distance. Extraction of the DMT-Voigt parameters is performed for a slow approach to a PMMA surface with maximum free oscillation amplitude of 60 nm peak to peak. Further away than 30 nm the tip is not interacting with the surface and no parameters can be extracted. In a shaded region between roughly 25 and 10 nm away from the surface, the DMT repulsion $R_{\mathrm{DMT}}$ and adhesion $F_{\mathrm{vdW}}$ are constant, indicating a suitable working distance for extraction based on the DMT model. The observed change of the surface damping parameter $D_V$ suggests that the Voigt damping has limited validity.

Figure 4

Dependence of the extracted parameter values on the drive amplitude. The drive amplitude is reflected in the measured maximum free oscillation amplitude. Parameters were extracted upon approach as in Fig. 3, and the mean value is plotted vs the free amplitude, where the error bars correspond to the standard deviation of the parameter values obtained in an interval where the amplitude at $f_1$ was between 85$\%$ and 50$\%$ of its free value.

Figure 5

Surface parameter maps and representative force curves obtained from one ImAFM scan. Soft droplets of LDPE are revealed in a much stiffer PS matrix. A 3D rendering of the surface topography is color coded to give the DMT repulsion factor $R_{\mathrm{DMT}}=E^{*}\sqrt{R}$, which is a tip-radius-dependent measure of the surface stiffness. 2D maps of the other parameters are shown together with a map of the fit error. Representative DMT force curves and Voigt damping curves are extracted and plotted for one point on the LPDE (blue curves) and one point on the PS (red curves).