A New Method for Obtaining Model-Free Viscoelastic Material Properties from Atomic Force Microscopy Experiments Using Discrete Integral Transform Techniques

Abstract

Viscoelastic characterization of materials at the micro- and nanoscales is commonly performed with the aid of force-distance relationships acquired using atomic force microscopy (AFM). The general strategy for existing methods is to fit the observed material behavior to specific viscoelastic models, such as generalized viscoelastic models or power-law rheology models, among others. Here we propose a new method to invert and obtain the viscoelastic properties of a material through the use of the Z-transform, without using a model. We present the rheological viscoelastic relations in their classical derivation and their Z-domain correspondence. We illustrate the proposed technique on a model experiment involving a traditional ramp-shaped force-distance AFM curve, demonstrating good agreement between the viscoelastic characteristics extracted from the simulated experiment and the theoretical expectations. We also provide a path for calculating standard viscoelastic responses from the extracted material characteristics. The new technique based on the Z-transform is complementary to previous model-based viscoelastic analyses and can be advantageous with respect to Fourier techniques due to its generality. Additionally, it can handle the unbounded inputs traditionally used to acquire force-distance relationships in AFM, such as “ramp” functions, in which the cantilever position is displaced linearly with time for a finite period of time.

Linear Viscoelasticity: Review of Theory and Applications in Atomic Force Microscopy

“…It has been determined that when this discrete result is compared to that of a Kelvin-Voigt and Maxwell-Wiechert models, the modified storage and loss behaviors have similar shapes to the analytical models in terms of the locations of the characteristic times; however, the modified storage and loss behaviors tend to be more spread across the frequency axis by a factor depending on the experimental sampling and 0 . This is discussed in detail elsewhere (Uluutku et al, 2021).…”

“…To derive the behavior of the Maxwell-Wiechert model, it is convenient to begin with the behavior of a single Maxwell arm (López-Guerra and Solares, 2014;Uluutku et al, 2021). The stress-strain relationship of the elastic spring element and viscous dashpot element can be written as Equation 9 and 10, respectively: = ( 9 When these two elements are connected in series, such as they are in a Maxwell arm, they will experience the same amount of stress as expressed in Equation 11.…”

“…Recent developments have allowed the viscoelastic contact mechanics equations to be used without a specific material model when analyzing experimental data (Uluutku et al, 2021). This approach requires the use of a frequency domain technique like the Laplace or Fourier transforms used in analyzing the previously discussed viscoelasticity models.…”

“…Of course, the storage and loss behaviors of a material may be obtained through Fourier transformations of data from harmonic, steady-state experiments. However, when used for a non-harmonic, and non-bounded functions such as those present in force-distance experiments, Fourier techniques give faulty results and do not correspond the analytical theory (Aspden, 1991;Uluutku et al, 2021). This mismatch can be said to be caused by the un-bounded nature of the signals and finite bandwidth of the experiments.…”

“…This mismatch can be said to be caused by the un-bounded nature of the signals and finite bandwidth of the experiments. To avoid the limitations of discrete Fourier techniques, it has recently been proposed to use the 'discrete Laplace transform,' known as the Z-Transform, to analyze experimental data (Oppenheim and Schafer, 1975;Uluutku et al, 2021). By using the Z-and modified Fourier transforms, one can directly access the storage and loss behaviors of a material.…”