Modeling and Measuring Viscoelasticity with Dynamic Atomic Force Microscopy

Abstract

The interaction between a rapidly oscillating atomic force microscope tip and a soft material surface is described using both elastic and viscous forces with a moving surface model. We derive the simplest form of this model, motivating it as a way to capture the impact dynamics of the tip and sample with an interaction consisting of two components: interfacial or surface force, and bulk or volumetric force. Analytic solutions to the piece-wise linear model identify characteristic time constants, providing a physical explanation of the hysteresis observed in the measured dynamic force quadrature curves. Numerical simulation is used to fit the model to experimental data and excellent agreement is found with a variety of different samples. The model parameters form a dimensionless impact-rheology factor, giving a quantitative physical number to characterize a viscoelastic surface that does not depend on the tip shape or cantilever frequency.

Nonlinear Phase Imaging of Gold Nanoparticles Embedded in Organic Thin Films

“…(51,52) These effects are intrinsically captured in the model described above. (40,38) The results shown in Figure 8 correspond to the simulated phase of the cantilever with respect to the driving phase, as also experimentally measured. The curves in Figure 8 are for various sample stiffness ks values, which were found by several numerical tests to be able to approach the experimental results from Figure 6 (see blue curve for instance).…”

“…Such unwanted phase instabilities are often caused by finite imaging bandwidths of the cantilever sensor or intrinsically by cantilevers with low spring constants. , Indeed, looking beyond the force gradient approximation introduced by Mertz et al, it was shown that the amplitude and phase signals may present discontinuities due to nonlinear tip–surface interactions. ,, Since then, the phase shift was recognized as an ample and stable imaging tool only for weak enough tip–sample interactions, which are distance dependent, that ensure a rather constant cantilever spring constant (linear response approximation). This is critical not only for amplitude-modulation (AM) imaging mode, − where the cantilever is externally driven at a constant excitation and frequency, but also for frequency-modulation (FM) mode when the excitation frequency is permanently adjusted by an external electronic device or by changing the probe–sample distance. , The situation is even more challenging when adhesion , and viscosity are considered, their effects being increasingly important when soft molecular surfaces are investigated. − …”

Multiparametric analytical quantification of materials at nanoscale in tapping force microscopy

“…AFM is used to calculate the Hamaker constant of materials as a quantitative measure of attractive van der Waals (vdW) interaction between the cantilever-tip ensemble and samples [8][9][10][11][12] . For the calculation of the elastic and viscoelastic properties of materials, such as stiffness, viscosity and loss tangent, different methods based on the force-curves, dissipation and virial concepts and multifrequency approaches have been proposed [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] . By applying bimodal AFM in air environment the stiffness and Hamaker constant of materials could be simultaneously quantified 15 .…”

“…Bengalia et al 20 quantified the viscoelastic properties of polymers including stiffness, viscosity and loss tangent by applying bimodal AFM. Thoren et al 18 proposed a dynamic AFM based on intermodulation technique to measure the elastic and viscous forces. The viscoelasticity of polymers and cells are measured using the force curves of tapping mode AFM 19,26 .…”

Probing nano-scale viscoelastic response in air and in liquid with dynamic atomic force microscopy

“…ImAFM has been thoroughly explained in our previous publications. [28][29][30][31] Fig. 1 and its caption describe the basic idea behind the technique.…”