Sensitivity of viscoelastic characterization in multi-harmonic atomic force microscopy

Abstract

Quantifying the nanomechanical properties of soft-matter using multi-frequency atomic force microscopy (AFM) is crucial for studying the performance of polymers, ultra-thin coatings, and biological systems. Such characterization processes often make use of cantilever's spectral components to discern nanomechanical properties within a multi-parameter optimization problem. This could inadvertently lead to an over-determined parameter estimation with no clear relation between the identified parameters and their influence on the experimental data. In this work, we explore the sensitivity of viscoelastic characterization in polymeric samples to the experimental observables of multi-frequency intermodulation AFM. By performing simulations and experiments we show that surface viscoelasticity has negligible effect on the experimental data and can lead to inconsistent and often non-physical identified parameters. Our analysis reveals that this lack of influence of the surface parameters relates to a vanishing gradient and non-convexity while minimizing the objective function. By removing the surface dependency from the model, we show that the characterization of bulk properties can be achieved with ease and without any ambiguity. Our work sheds light on the sensitivity issues that can be faced when optimizing for a large number of parameters and observables in AFM operation, and calls for the development of new viscoelastic models at the nanoscale and improved computational methodologies for nanoscale mapping of viscoelasticity using AFM.

Fig. 1. Schematic of the working principle of the IM-AFM. The cantilever is driven with a signal comprising two close frequencies ω₁ and ω₂, centered around its first resonance frequency. The intermodulation distortion caused by the nonlinear tip-sample interaction creates frequency comb at commensurate frequencies ω_IM = m₁ω₁ + m₂ω₂, with m₁,m₂ ∈ Z. The linear transfer function of the cantilever χ(ω) is measured via thermal calibration, and the amplitudes and phases of these intermodulation products are captured using a multi-lock-in amplifier. Here, d_c and d_s denote the tip cantilever and surface vertical displacements and h corresponds to the working distance between tip and sample. Finally, s = h + d_c − d_s represents the tip-sample distance.

Fig. 2. Experimental measurements performed on the PS-LDPE polymer blend. (a) Amplitude image at the second drive frequency (ω₂), which is part of the 32 different image pairs captured during the scanning operation. (b) Phase image at the second drive frequency. The image shows an island of LDPE within the PS matrix (red dashed box in Fig. 2(a)). The points of measurements are indicated with black crosses. (c–f) Experimental force quadratures obtained at the pixels marked by black crosses in the phase image. The quadratures in subfigures (c–f) are obtained on PS material, whereas the quadratures in sub figures (d–e) are obtained on LDPE material.

Fig. 3. Simulations of the cantilever (green) and sample (pink) surface dynamics based on the results provided in Table 1. (a and b) Simulated results for PS material with parameter values taken from IPs 1 and 55, respectively. (c and d) Simulated results for LDPE material with parameter values taken from IPs 1 and 87. (e–h) A close up visualization of the surface dynamics is reported in (a–d).

Fig. 4. Variation of the objective function in a 2-dimensional parameter space comprising ((k_s,η_s) or (k_v,η_v)), with the other parameters fixed in accordance with the best results found from the local minimization routine. (a–d) Visualizing the landscape of the minimization objective as a function of k_v and η_v for PS and LDPE material obtained at pixel (i) and (iii) of Fig. 2(b). The purple and orange lines indicate a 2D cross-sectional view of the objective function. (a–d) Visualizing the landscape of the minimization objective as a function of k_s and η_s for PS and LDPE material obtained at pixel (i) and (iii) of Fig. 2(b). Purple and orange lines indicate 2D cross-sectional views of the objective function.

Fig. 5. Estimated properties of the PS-LDPE sample obtained using the model without surface motion and the initial points selection procedure described in this work. The maps are of dimensions 2.5 μm × 2.5 μm. Left: Parameter maps of (a) adhesion force, (b) contact force stiffness, (c) contact force viscosity, (d) probe height. Right: Histogram distribution of the respective parameters (e–h).