Why Measuring Viscoelasticity Using AFM Can Be A Challenge

After topography, nanomechanical measurements have become among the most important measurements conducted with atomic force microscopy (AFM). The fundamental mechanical interaction between the probe and sample permits an atomic force microscope to measure a sample’s various mechanical properties, while the size of the contact enables the measurement to be localized at the nanometer scale.

Overview of Viscoelastic Materials

Typical nanomechanical applications include measuring elastic modulus, friction and adhesion. Viscoelastic materials, which include polymers and biological materials, have elastic properties that exhibit time-dependent behavior such as creep and stress relaxation.

The viscoelastic properties are storage modulus, loss modulus, and loss tangent (tan δ). Nanoscale viscoelastic measurements are of particular interest due to their influence on the function and behavior of macroscopic material.

A considerable and important part of the chemicals industry is committed to developing heterogeneous materials such as polymer composites, blends, and multilayers. Establishing a structure property relationship for these materials routinely relies on bulk viscoelastic measurements, especially in the development of a material’s mechanical properties.

Research and Development of Materials

Research and development regarding materials often produces blends or composites that contain nano-sized domains of components that do not exist in the bulk or have properties that are influenced by the proximity of other components (interphase). Therefore, to optimize the attributes of such a heterogeneous material, it is important that the viscoelastic properties of the different components are well-characterized.

To measure such properties at the nanoscale, a specialized tool is required for effective development of structure-property relationships in next-generation materials. Using either shear or uniaxial loads enables the stress-deformation relationship of materials to be effectively measured, resulting in shear modulus (G) or tensile modulus (E), respectively.

Viscoelastic Moduli

For viscoelastic materials, the moduli are often complex, possessing both real and imaginary parts. The real part (e.g., tensile storage modulus, E’) is the elastic or in-phase element of the response and is given by (stress/strain)*cosδ where δ is the shifting phase between the stress and strain. The imaginary part (e.g., tensile loss modulus, E”) illustrates the viscous or out-of-phase part of the response and is given by (stress/strain)*sinδ. The ratio between loss modulus and storage modulus is the loss tangent (tan δ).

Significant variances can occur in the viscoelastic moduli with changes in frequency and/or temperature. For instance, in rubber-like polymers, the storage modulus usually has lower values at reduced frequencies. So, as the frequency increases, the storage modulus dramatically increases, plateauing at the modulus value of the glassy state at high frequencies.

For the same materials, both at low and high frequencies, the loss modulus is low (primarily elastic) and peaks in the frequency mid-range, reflecting the existence of a glass transition between the rubbery and glassy states.

Since the loss tangent is just the ratio of the loss modulus over the storage modulus (tan δ = E’’/E’), it follows a similar trajectory to the loss modulus. The empiric quality of the loss tangent is both practical and useful as a viscoelastic parameter, as when measured as a function of temperature or frequency it can precisely reveal thermal and structural transitions in a material.

Temperature and Time in Viscoelastic Materials

The equivalence between the behavior of temperature and time (frequency) in viscoelastic materials arises from the relaxation behavior of the molecules within the sample.1 This correlation is established in terms of the time-temperature superposition principle or TTS.

When modulus is measured over higher temperatures, comparable curves are obtained when the measurement is done at a range of frequencies as are those at a lower range of frequencies obtained and lower temperatures.

Due to these curves typically having the same shape, they can be superimposed onto one another to generate a “master curve” over an extended range of frequencies. Comparably, increasing the temperature is equivalent to shifting to a lower frequency, while lowering the temperature is analogous to shifting to a higher frequency.

When generating the master curve, the shift factors can be analyzed via different models (WLF or Arrhenius) to parameterize this time-temperature relationship. In this way, the data observed at one set of temperatures and frequencies is suitable to determine behavior at a different set of temperatures and frequencies, making TTS and generation of the master curve very useful.

Dynamic Mechanical Spectroscopy

In the bulk, dynamic mechanical analysis (DMA), also also known as dynamic mechanical spectroscopy (DMS), is used to measure viscoelastic properties. With the response of the entire sample measured as a function of oscillation frequency, this measurement applies oscillating stress to a macroscopic sample.

Depending on the sample properties and geometry, DMA can be carried out in a wide range of sample-mounting configurations, including three-point bending, compression, tension and shear.

Dynamic mechanical tensile analysis (DMTA) offers the most direct comparison to the motion of an AFM tip and sample as it requires the holding of the sample in tension.

AFM Imaging

Previously, AFM measurements of viscoelastic properties have been less than ideal for several reasons. The first two challenges involve frequency space. Rheologists running with DMA typically work at frequencies of less than 200Hz, while AFM imaging modes have been evolving to produce images more quickly, and are thus usually at much higher frequencies (kilohertz and higher).

In resonance-based techniques, the frequency of the AFM measurement, as dictated by the cantilever dimensions, is at especially high frequencies. While discrete, these frequencies are not tunable and are usually widely spaced (e.g., the second free eigenmode of a simple beam is 6.3 times the first).2-4

Perhaps even more significant than the frequency mismatch has been the adoption of methods where the tip dives into the sample and rapidly tears away from it, making contact in the minimum amount of time possible. This also helps speed up traditionally slow AFM imaging as well as permits more sample- and tip-friendly scanning. However, as the tip is making and breaking contact with the sample on each cycle, the tip-sample interaction is very non-linear, and the frequency of the measurement includes many harmonics of the nominal operating frequency that are ill-defined.5

The final challenge in designing an AFM technique to successfully measure viscoelastic properties concerns the models used to extract the storage and loss modulus. The most problematic part of this involves the measurement of and compensation for adhesion.

In AFM of polymers, the adhesion between tip and sample is comparatively greater than that of the other tip-sample forces, and it varies considerabley across heterogeneous polymer samples. Thus, adhesion cannot be measured or observed in many resonance-based AFM modes, which makes analyzing this important parameter, and ultimately compensating for it, nearly impossible.

Conclusion

The models of primary contact mechanics available for modeling adhesive tip-sample interactions for parabolic AFM tips are DerjaguinMuller-Toporov (DMT) and Johnson-Kendall-Roberts (JKR). These models are representative of two limits in the range of viable adhesive behaviors, with the JKR model accounting for stronger adhesion inside the tip-sample contact area, while the DMT model accounts for weak long-range adhesion (including outside the tip-sample contact area).6

Unfortunately, both models are purely elastic with no viscous component. Models that can extract both storage and loss modulus tend to ignore adhesion and cannot be deemed appropriate to AFM application of polymers in air.7,8

Difficulties such as these make it clear that a different approach is required to overcome such hurdles – one that produces well-defined rheological frequencies as a small perturbation to a carefully controlled preload.

References

  1. M. L. Williams, R. F. Landel, and J. D. Ferry, “The Temperature Dependence of Relaxation Mechanisms in Amorphous Polymers and Other Glass-forming Liquids,” J. Am. Chem. Soc., vol. 77, no. 14, pp. 3701–3707, Jul. 1955.
  2. S. Hu and A. Raman, “Analytical formulas and scaling laws for peak interaction forces in dynamic atomic force microscopy,” Appl. Phys. Lett., vol. 91, no. 12, 2007.
  3. U. Rabe, K. Janser, and W. Arnold, “Vibrations of free and surface-coupled atomic force microscope cantilevers: Theory and experiment,” Rev. Sci. Instrum., vol. 67, no. 9, p. 3281, 1996.
  4. B. Pittenger and D. G. Yablon, “Quantitative Measurements of Elastic and Viscoelastic Properties with FASTForce Volume CR,” Bruker Application Note AN148, doi: 10.13140/RG.2.2.25339.00806, 2017.
  5. O. Sahin, C. Quate, O. Solgaard, and A. Atalar, “Resonant harmonic response in tapping-mode atomic force microscopy,” Phys. Rev. B, vol. 69, no. 16, pp. 1–9, Apr. 2004.
  6. K. L. Johnson and J. A. Greenwood, “An adhesion map for the contact of elastic spheres,” J. Colloid Interface Sci., vol. 192, no. 2, pp. 326–333, 1997.
  7. M. Chyasnavichyus, S. L. Young, and V. V Tsukruk, “Probing of polymer surfaces in the viscoelastic regime.,” Langmuir, vol. 30, no. 35, pp. 10566–82, Sep. 2014. 8. Y. M. Efremov, W.-H. Wang, S. D. Hardy, R. L. Geahlen, and A. Raman, “Measuring nanoscale viscoelastic parameters of cells directly from AFM forcedisplacement curves,” Sci. Rep., vol. 7, no. 1, p. 1541, Dec. 2017.
  8. Y. M. Efremov, W.-H. Wang, S. D. Hardy, R. L. Geahlen, and A. Raman, “Measuring nanoscale viscoelastic parameters of cells directly from AFM force-displacement curves,” Sci. Rep., vol. 7, no. 1, p. 1541, Dec. 2017.

This information has been sourced, reviewed and adapted from materials provided by Bruker Nano Surfaces. For more information on this source, please visit the Bruker Nano Surfaces website.

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