### Dynamic force quadratures and the moving surface model

The standard approach to quantitative AFM reconstructs tip-surface force as a function of tip position, the force-distance curve $$F_{ts}(z)$$. Through analysis of these force curves one tries to understand something about material properties. The reconstruction assumes that tip and sample forces are in quasi-static equilibrium, thus neglecting viscous forces proportional to tip velocity. Quasi-static force curves give a conservative tip-surface force $$F_\mathrm{con}(z)$$, which can only tell us about the elastic nature of the material.

In contrast, dynamic force quadratures capture both elastic and viscous forces. But the force quadrature curves do not display these forces as a function of tip position, rather they show the integrated force over one oscillation cycle of the cantilever. Dynamical Mechanical Analysis (DMA) uses this same type of cyclic force analysis to characterize viscoelastic materials such as polymers. However, in contrast to DMA, which usually works with fixed oscillation amplitude, ImAFM force quadrature curves show the elastic (conservative) force $$F_I(A)$$ and the viscous (dissipative) force $$F_Q(A)$$ as functions of the amplitude $$A$$.

The first video (below) simulates the force quadratures for an AFM tip oscillating in constant contact with the sample, as is the case with 'contact resonance' AFM. To make it simple, we assume a linear force-distance curve in the contact region (black curve, upper left panel). The lower left panel shows the tip position $$z(t)$$ and tip-surface force $$F_{ts}(t)$$ as functions of time, plotted over one complete oscillation cycle. The relaxed position of the surface $$z_s =0$$ is indicated (red line, lower left panel). For each oscillation cycle the conservative (elastic) force quadrature $$F_I$$ and the dissipative (viscous) force quadrature $$F_Q$$ are plotted. Play the video to show these evolve as the oscillation amplitude is ramped up and down.

Note that $$F_Q=0$$ at all amplitudes, because we simulate a purely elastic tip-surface interaction. Also notice that the oscillating tip-surface force and tip position are simultaneously in exact opposition, i.e. phase angle $$\pi$$ with respect to one-another. The next video shows how this simulation changes when we add a linear velocity-dependent, or viscous tip-surface force.

The force-distance curve (magenta line, upper left panel) is no longer a single-valued function of tip position. There is no unique force-distance curve that explains viscoelastic response, because the force also depends on tip velocity. The tip oscillation lags behind the force oscillation with a phase angle larger than $$\pi$$. The tangent of this phase lag, the 'loss tangent' of DMA, is a quantity commonly used to characterize viscoelastic materials.

A viscous (dissipative) tip-surface interaction gives non-zero $$F_Q(A)$$. For linear viscoelastic response, both force quadrature curves are straight lines and from their slopes we could extract the elastic stiffness and viscous damping coefficient of the tip-surface contact. However, in AFM we often want to measure with intermittent contact, where the tip is tapping on the surface. The next video simulates intermittent contact with the DMT model, commonly used to analyze AFM force curves.

Short range van der Waals attraction cause rapid pulses of force when the tip when makes and breaks contact with the surface. The interaction model is a purely conservative force, so there is no dissipation and $$F_Q(A)=0$$ at all amplitudes. Note that $$F_{ts}(t)$$ is symmetric in time with respect to the lower turning point of the oscillation, as expected for any conservative force model.

The next video shows how this picture changes when we add a viscous damping force to intermittent contact. We modify the DMT model with a viscous force that turns on when the tip is in contact with the surface, $$z < z_s$$, where the surface position $$z_s = 0$$ remains fixed in time.

Again, the viscous nature of the interaction gives $$F_Q(A) \neq 0$$. The force curve $$F_{ts}(z)$$ is not single-valued and $$F_{ts}(t)$$ is no longer time-symmetric, i.e. the approaching tip feels a different force than the retracting tip. However, for this model the force quadrature curves are single-valued: $$F_I(A)$$ and $$F_Q(A)$$, are the same for increasing and decreasing amplitude. From their shape it is possible, under certain assumptions, to reconstruct a conservative force curve $$F_\mathrm{con}(z)$$ and viscous damping function $$\eta(z)$$. But these functions of tip position do not capture the true nature of viscoelastic material response in dynamic AFM. Rather, the shape and relative magnitude of the force quadrature cures themselves provide the quantitative information we need to understand the mechanical properties of the material and its surface.

To clarify this point we simulate force quadratures with a moving surface model that takes in to account the dynamics of the tip, $$z(t)$$, and the dynamics of a moving surface, $$z_s(t)$$. The force between the tip and surface is given as a function of the tip-surface separation, $$s(t) = z(t) - z_s(t)$$, and the rate of tip penetration $$\dot{s}$$. We assume simple linear functions of $$s$$ and $$\dot{s}$$ which, together with a fixed adhesion force, turn on instantly upon contact with the surface, $$s<0$$.

The short-range adhesion force rapidly lifts the soft surface and the viscoelastic interaction gives a net force that starts out repulsive and changes to attractive during each tap. The surface relaxation time is longer than the period of oscillation, so the lifted surface does not relax to it's equilibrium position before being lifted further with the next tap. The force quadrature curves are hysteretic: The amplitude at which the tip first touches the surface, is larger than the amplitude at which it stops touching the surface.

Hysteretic $$F_I(A)$$ and $$F_Q(A)$$ are common on soft materials, having many different shapes depending on material parameters. Often the curves are well-explained with a simple piece-wise-linear viscoelastic moving surface model. As an example we show force quadrature curves measured on amorphous polycaprolactone (grey curves, right panels). Fitting the parameters of the moving surface model, we achieve excellent agreement with the simulated curves (blue curves, right panels). The video below shows several oscillation cycles at various points on the force quadrature curves. The measured cantilever motion and simulated surface motion are shown in both time and space (left panels) where the surface profile is assumed to look like a capillary meniscus. Such 'solid capillarity' is expected when the sharp tip interface with very large curvature meets the soft material interface with very low curvature.

Force quadrature curves are reconstructed from measured ImAFM data with no assumed model, making them ideal primitive 'force curves' for quantitative analysis of dynamic AFM. Understanding their shape and magnitude leads to a deeper understanding of nanometer-scale viscoelastic impact, allowing us to develop models and extract parameters that describe physically meaningful material properties. You can read more about the moving surface model in our publication.

This video tutorial was prepared by David Haviland and Per Anders Thorén at Nanostructure Physics, KTH. The sample was provided by Phillipe Leclère at the University of Mons. You are free to copy and show these videos in your teaching and research presentations.