^{a)}Contribution of NIST, an agency of the US government; not subject to copyright in the United States.

^{1,b)}, Jason P. Killgore

^{1}and Donna C. Hurley

^{1}

### Abstract

We present a method to improve accuracy in measurements of nanoscale viscoelastic material properties with contact resonance atomic force microscope methods. Through the use of the two-dimensional hydrodynamic function, we obtain a more precise estimate of the fluid damping experienced by the cantilever-sample system in contact resonance experiments, leading to more accurate values for the tip-sample damping and related material properties. Specifically, we consider the damping and added mass effects generated by both the proximity of the cantilever to the sample surface and the frequency dependence on the hydrodynamic loading of the system. The theoretical correction method is implemented on experimental contact resonance measurements. The measurements are taken on a thin polystyrene film and are used to determine the viscoelastic loss tangent, tan δ, of the material. The magnitude of the corrections become significant on materials with low tan δ (<0.1) and are especially important for measurements made with the first flexural mode of vibration.

This research was performed while Ryan C. Tung held a National Research Council Research Associateship Award at the National Institute of Standards and Technology. We thank L. M. Cox and Y. Ding (University of Colorado-Boulder) for DMA measurements and time-temperature superposition analysis.

I. INTRODUCTION

II. THEORY

III. EXPERIMENTAL METHODS

IV. RESULTS AND DISCUSSION

V. CONCLUSIONS

### Key Topics

- Hydrodynamics
- 34.0
- Viscoelasticity
- 13.0
- Atomic force microscopy
- 12.0
- Vibration testing
- 10.0
- Materials properties
- 8.0

##### F15D

##### G01H

##### G01L

## Figures

Fluid damping effects that must be considered in CR spectroscopy. f far and Q far are the natural frequency and quality factor of the cantilever measured far from the sample surface, f near and Q near are the natural frequency and quality factor of the cantilever measured near to the sample surface, and f cont is the natural frequency of the cantilever when in contact with the sample surface. (a) Surface proximity effects. The quality factor decreases as the cantilever is brought closer to the surface. (b) Frequency dependence of fluid damping. Because f cont > f near, estimations of the fluid damping made at f near do not accurately portray the real fluid damping at f cont.

Fluid damping effects that must be considered in CR spectroscopy. f far and Q far are the natural frequency and quality factor of the cantilever measured far from the sample surface, f near and Q near are the natural frequency and quality factor of the cantilever measured near to the sample surface, and f cont is the natural frequency of the cantilever when in contact with the sample surface. (a) Surface proximity effects. The quality factor decreases as the cantilever is brought closer to the surface. (b) Frequency dependence of fluid damping. Because f cont > f near, estimations of the fluid damping made at f near do not accurately portray the real fluid damping at f cont.

Added mass and damping effects. (a) Damping term and added mass term M f vs. gap height g at the first free natural frequency of the cantilever f 1 ≈ 75 kHz and 5f 1. (b) Damping term and added mass term M f vs. frequency f at g = ∞. should not be confused with the actual quality factor Q. is inversely proportional to the damping experienced by the system at a particular frequency and gap height and is equal to Q in special cases.

Added mass and damping effects. (a) Damping term and added mass term M f vs. gap height g at the first free natural frequency of the cantilever f 1 ≈ 75 kHz and 5f 1. (b) Damping term and added mass term M f vs. frequency f at g = ∞. should not be confused with the actual quality factor Q. is inversely proportional to the damping experienced by the system at a particular frequency and gap height and is equal to Q in special cases.

Flowchart of the hydrodynamic correction procedure.

Flowchart of the hydrodynamic correction procedure.

Contact resonance results for tan δ of polystyrene sample obtained for the first three flexural modes with use of different inputs for the fluid damping estimation. “Far” represents tan δ values calculated with Q far and f far, “Near” represents tan δ values calculated with Q near and f near, and “Hydrodynamic” represents tan δ values calculated with Q hydro and f hydro.

Contact resonance results for tan δ of polystyrene sample obtained for the first three flexural modes with use of different inputs for the fluid damping estimation. “Far” represents tan δ values calculated with Q far and f far, “Near” represents tan δ values calculated with Q near and f near, and “Hydrodynamic” represents tan δ values calculated with Q hydro and f hydro.

Here, we have assigned values of (tan δ)hydro (x-axis) and sample stiffness normalized by cantilever stiffness α. We use the measured values of Q near, γ, and the calculated values of Q hydro for each mode. We assume that Q hydro is the actual fluid damping of the system, and compute the percent difference, ε, between (tan δ)near calculated with the damping measured close to the surface, Q near, and the prescribed theoretical values of (tan δ)hydro. (a) Results for α = 60. (b) Results for α = 240.

Here, we have assigned values of (tan δ)hydro (x-axis) and sample stiffness normalized by cantilever stiffness α. We use the measured values of Q near, γ, and the calculated values of Q hydro for each mode. We assume that Q hydro is the actual fluid damping of the system, and compute the percent difference, ε, between (tan δ)near calculated with the damping measured close to the surface, Q near, and the prescribed theoretical values of (tan δ)hydro. (a) Results for α = 60. (b) Results for α = 240.

## Tables

Experimental results for the cantilever's natural frequency f and quality factor Q. f far and Q far are measurements taken with the cantilever ∼1 mm from the sample surface. f near and Q near are measurements taken with the cantilever tip ∼250 nm from the sample surface. f cont and Q cont are measurements taken with the tip in contact with the polystyrene test sample.

Experimental results for the cantilever's natural frequency f and quality factor Q. f far and Q far are measurements taken with the cantilever ∼1 mm from the sample surface. f near and Q near are measurements taken with the cantilever tip ∼250 nm from the sample surface. f cont and Q cont are measurements taken with the tip in contact with the polystyrene test sample.

Values of tan δ calculated from the data in Table I with Eq. (5) . (tan δ)far is the viscoelastic loss tangent calculated with the values of Q far and f far as inputs. (tan δ)near is calculated with the values of Q near and f near, while (tan δ)hydro is calculated with the values of Q hydro and f hydro. Δ1 is the percent difference between (tan δ)far and (tan δ)near as defined in the text, and Δ2 is the percent difference between (tan δ)far and (tan δ)hydro.

Values of tan δ calculated from the data in Table I with Eq. (5) . (tan δ)far is the viscoelastic loss tangent calculated with the values of Q far and f far as inputs. (tan δ)near is calculated with the values of Q near and f near, while (tan δ)hydro is calculated with the values of Q hydro and f hydro. Δ1 is the percent difference between (tan δ)far and (tan δ)near as defined in the text, and Δ2 is the percent difference between (tan δ)far and (tan δ)hydro.

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