Nanoindentation in finite thickness viscoelastic materials

Abstract

In this work, we present an analytical model to describe atomic force microscopy force curves of standard linear solid and power-law viscoelastic materials by taking indenter geometry and finite thickness effects into account. We show that conical/pyramidal cantilevers are less susceptible to finite thickness effects than other shapes, such as spherical and flat punch indenters. This is confirmed by finite element modeling of the stress field distribution within the sample. We also perform a systematic error analysis for the cases where finite thickness samples are analyzed with an infinite thickness force model. In particular, we show that for spherical indenters and indentation depth up to 20% of sample thickness, the mechanical response of viscoelastic materials will wrongly mimic a purely elastic behavior even within timescales where viscoelastic relaxation should appear.

Dynamically Mapping the Topography and Stiffness of the Leading Edge of Migrating Cells Using AFM in Fast-QI Mode

“…This point is essential to demonstrate the robustness of our analysis, especially regarding the finite size effect, i.e., the dependence of the cell stiffness over the higher stiffness of the underlying surface. 29,35,37 The finite size effect induces a large increase of the measured cell stiffness at the vicinity of the hard surface, typically when the ratio between indentation depth and cell height is over 10−20%, although a recent study indicated conical probes such as the ones we use in our experiments are less sensitive to that effect. 37 As we observed lower stiffnesses for areas with lower heights, and even if our measurements are affected by the finite size effect (see additional discussion in the Experimental (Materials and Methods) section), we unambiguously show that the dynamic lamellipodium at the leading edge is softer than the more stable lamellum that supports lamellipodial growth.…”

Measuring the viscoelastic relaxation function of cells with a time-dependent interpretation of the Hertz-Sneddon indentation model

“…This is in general justified in terms of contact area, which is smallest for the conical shape, and largest for the flat cylinder [24] . Besides, it has been shown analytically and numerically that conical indenters are less susceptible to finite thickness effects than other shapes [38] . A simple analysis of the HS model (Eq.…”

“…On the other hand, indenters with large radii (spherical/flat) are much less sensitive to small scale stiffer areas and only feel an averaged mechanical response of the cytoskeleton components and cytoplasm [24] . Other factor that may affect the measurements of with different geometries is that indenters with large contact radius are more susceptible to finite thickness effects, such that indentation depths larger that 10% of the sample thickness should be avoided, or properly treated with a viscoelastic force model that takes into account the finite thickness effects [38] , [46] .…”

Viscoelastic relaxation of fibroblasts over stiff polyacrylamide gels by atomic force microscopy

“…We used regular AFM cantilevers (pyramidal tip) with nominal spring constants of 0.08 N/m. Determining the viscoelastic properties of soft materials with pyramidal indenters is much less susceptible to finite thickness effects than with spherical and flat indenters [32,33]. A maximum force F 0 = 3 nN and a piezo extension frequency of f z = 1 Hz (cantilever speed of 6 μm/s) were employed in all measurements.…”

“…All measured cells exhibited at least h = 5 μm of height in their thickest part, and the deepest indentation achieved in our measurements ranged between 1 μm and 2 μm, depending on the type of substrate. According to the nanoindentation theory of thin viscoelastic materials of reference [33], this represents a maximum of 20% of indentation depths compared to the sample thicknesses, which exhibits nearly negligible errors in the determination of viscoelastic parameters when using DFC's.…”