Modeling and Measuring Viscoelasticity with Dynamic Atomic Force Microscopy

The interaction between a rapidly oscillating atomic-force-microscope tip and a soft-material surface is described with use of both elastic and viscous forces in a moving-surface model. We present the simplest form of this model, motivating our derivation with the models ability 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 piecewise linear model identify characteristic time constants, providing a physical explanation for 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.

Figure 1

Coordinates and piecewise linear (PWL) interaction. The cantilever deflection $d=z-h$ is measured by the atomic force microscope detector, where $z(t)$ is the instantaneous tip position and the constant $h$ is the equilibrium (zero force) tip position. $z_0$ is the equilibrium position of the sample surface. (a) The traditional view of contact forces in AFM has the tip and surface moving together when they are in contact, $z=z_s$, and interaction force is considered to be a function of the surface indentation $(z_0-z)$. (b) The moving-surface model treats the surface position $z_s(t)$ as an independent dynamic variable. The model introduces elastic and viscous forces that depend on surface deflection $d_s=z_s-z_0$ and velocity $\dot{d_s}$, and an interaction force that depends on the separation $s=z-z_s$ and $\dot{s}$. Forces are balanced in the inertial reference frame where the cantilever has a fixed working distance to the sample, $w=h-z_0$. (c) The PWL interaction force plotted together with a calculated Derjaguin-Muller-Toporov (DMT) model frequently used in AFM [31]. The parameters for the DMT model are as follows: reduced modulus $E^{\ast }=100$ MPa, Hamaker constant $H=8\times {10}^{-20}\mathrm{J}$, tip radius $R=10\mathrm{nm}$, intermolecular spacing $a_0=0.16\mathrm{nm}$. The parameters for the PWL model are as follows: $F_{\mathrm{ad}}=5\mathrm{nN}$, $k_v=0.9\mathrm{N}/\mathrm{m}$. The range of separation shown is that typically covered by the cantilever oscillation for dynamic AFM on soft materials.

Figure 2

Contact formation and free relaxation. (a) At $t=0$ a tip, rigidly fixed in the laboratory frame, meets the sample surface. The adhesion force turns on, and the surface lifts by an amount $\delta$, forming the contact in a characteristic time ${\tau }_c$. (b) At $t=0$ the tip separates from the lifted surface. The surface relaxes to its equilibrium position in the characteristic time ${\tau }_s$.

Figure 3

Phase images at the first drive frequency of intermodulation AFM. (a) Scan size $0.8\mu \mathrm{m}$. Domains of polycaprolactone (PCL; blue) in polystyrene (PS; red). Pixels marked with o and $\times$ are analyzed in Fig. 4. (b) Scan size $1\mu \mathrm{m}$. Polydimethylsiloxane (PDMS; blue) with clusters of hydrophobic silica nanoparticles (NP; red). Pixels marked with o and $\times$ are analyzed in Fig. 5. (c) Scan size $1.5\mu \mathrm{m}$. A set-point study on a domain of low-density polyethylene (LDPE; blue) in PS. Pixels marked with $\times$ are analyzed in Figs. 6 and 8 at the set-point values shown. (d) Scan size $3\mu \mathrm{m}$. A suspended membrane of Au nanoparticles bound together in a monolayer by organic ligands. A pixel at the center of the membrane marked with $\times$ is analyzed in Fig. 7.

Figure 4

Polystyrene (PS) and polycaprolactone (PCL) blend. (a) The conservative force quadrature $F_I(A)$ and (b) the dissipative quadrature $F_Q(A)$. The curves are offset vertically for clarity, with dashed lines corresponding to zero force. The simulated motion of the tip $z(t)$ (blue) and surface $z_s(t)$ (orange) is shown for both PCL (c),(d) and PS (e),(f). The enlargements in (c),(e) show details of the surface motion around the time marked by the vertical dotted line in (d),(f), respectively. The surface motion of PS in (f) is magnified by a factor of 100. Note that the average surface position for PCL is approximately 12 nm above its rest position, while PS deviates by only approximately 0.05 nm from its equilibrium position. The best-fit parameters used in the simulation are given in the table.

Figure 5

Polydimethylsiloxane (PDMS) with hydrophobic silica nanoparticles. The details of this sample are described in a previous publication [36]. The curves labeled “Matrix” are obtained with the softer PDMS, where the simulation shows large-amplitude surface motion and deep tip penetration. The curves labeled “Particles” are obtained with a subsurface aggregation of nanoparticles, where the simulation reveals very little surface motion and lifting, yet equally deep tip penetration. (a) The conservative force quadrature $F_I(A)$ and (b) the dissipative quadrature $F_Q(A)$. The curves are offset vertically for clarity, with dashed lines corresponding to zero force. The simulated motion of the tip $z(t)$ (blue) and surface $z_s(t)$ (orange) is shown for both the PDMS matrix (c),(d) and a cluster of particles (e),(f). The enlargements in (c),(e) show details of the surface motion around the time marked by the vertical dotted line in (d),(f), respectively. The best-fit parameters used in the simulation are given in the table.

Figure 6

Low-density polyethylene measured with a Tap525 cantilever. Experimental (left) and simulated (right) force-quadrature curves, each offset vertically for clarity, with dashed lines corresponding to zero force. The experimental curves result from analysis of data at pixels marked with a cross of the corresponding color in Fig. 3. The working distance is changed in the experiment by adjustment of the amplitude-feedback set point to the value given in the left panels (percentage of free amplitude). For simulation the working distance $w$ shown in the right panels is found by numerical optimization with all other parameters of the model fixed at the values given in Table 1.

Figure 7

Nanoparticle membrane measured with Tap300 and AC55 cantilevers. Measurements are made in the middle of a membrane with diameter $2.5\mu \mathrm{m}$. We show results from two different membranes from the same batch, measured with two different cantilevers. (a) The conservative force quadrature $F_I(A)$ and (b) the dissipative quadrature $F_Q(A)$. The curves are offset vertically for clarity, with dashed lines corresponding to zero force. (c),(d) The simulated motion of the tip $z(t)$ (blue) and surface $z_s(t)$ (orange) for the AC55 cantilever (c),(d) and for the Tap300 cantilever (e),(f). The enlargements in (c),(e) show details of the surface motion around the time marked by the vertical dotted line in (d),(f), respectively. The best-fit parameters used in the simulation are given in the table.

Figure 8

Plots equivalent to those in Fig. 6 for the NSC15 cantilever (top) and the AC55 cantilever (bottom).