From elasticity to capillarity in soft materials indentation

Jonathan T. Pham, Frank Schellenberger, Michael Kappl, and Hans-Jürgen Butt

Phys. Rev. Materials 1, 015602 – Published 19 June 2017

Abstract

For soft materials with Young's moduli below 100 kPa, quantifying mechanical and interfacial properties by small scale indentation is challenging because in addition to adhesion and elasticity, surface tension plays a critical role. Until now, microscale contact of very soft materials has only been studied by static experiments under zero external loading. Here we introduce a combination of the colloidal probe technique and confocal microscopy to characterize the force-indentation and force-contact radius relationships during microindentation of soft silicones. We confirm that the widespread Johnson-Kendall-Roberts theory must be extended to predict the mechanical contact for soft materials. Typically a liquid component is found within very soft materials. With a simple analytical model, we illustrate that accounting for this liquid surface tension can capture the contact behavior. Our results highlight the importance of considering liquid that is often associated with soft materials during small scale contact.

Received 20 February 2017
Revised 30 April 2017

DOI:https://doi.org/10.1103/PhysRevMaterials.1.015602

Physics Subject Headings (PhySH)

Adhesion Capillary interactions Elastic forces Surface & interfacial phenomena Wetting

Polymers & Soft Matter

Authors & Affiliations

Jonathan T. Pham, Frank Schellenberger, Michael Kappl^*, and Hans-Jürgen Butt

Max Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany

^*kappl@mpip-mainz.mpg.de

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Vol. 1, Iss. 1 — June 2017

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Images

Figure 1
Measuring force, indentation, and contact geometry of soft solid surfaces. (a) Schematic of a colloidal probe in contact with a soft solid surface. The relevant parameters of force $F$ , contact radius $a$ , indentation $δ$ , and the particle radius $R$ are labeled. (b) A representative force-indentation curve measured by AFM indentation on PDMS with $E_{bulk} = 5$ kPa and a maximum indentation depth $δ \approx 2.8 μ m$ . (c) Corresponding cross-sectional images obtained by confocal microscopy of the labeled points on the force-indentation curve in part (b). The green dashed line represents the interface between solid and air, which is determined by measuring reflection at the surface (Fig. S2 [35]). The gray circle represents the spherical probe as a guide to the eye. (d) Confocal images of indentation on PDMS with $E_{bulk} = 400$ kPa, illustrating the clear difference in contact deformation. Note that image III in (d) is not the maximum indentation but rather the same force as in (c) during retraction for comparison. The particle radius here is $R = 6.5 μ m$ . Scale bars: $5 μ m$ .
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Figure 2
Contact radius as a function of indentation and force for different moduli. (a) Contact radius as a function of indentation depth (both normalized by the particle radius $R = 6.5 μ m$ ), illustrating the rate at which $a / R$ approaches unity for different moduli (5, 15, 40, and 400 kPa). (b) Normalized contact radius vs normalized load, illustrating the rate at which $a / R$ increases with force for different moduli. See Fig. S4 [35] for retraction only and a zoomed in force range.
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Figure 3
Quasistatic force-indentation measurements. Contact radius $a$ , force $F$ , and indentation $δ$ as a function of time for the (a) 400 kPa and (b) 5 kPa samples. Each holding step has a duration of 15 minutes. Indentations steps are 500 nm and retraction steps are $1.5 μ m$ . Plots i, ii, and iii are zoomed in $F$ vs time plots to illustrate the different regimes of the labeled indentations. The $x$ axis is 16 minutes and the $y$ axis is $1 μ N$ for (a) and 100 nN for (b). Error bars on the contact radius are standard deviations of measured contact radii over the last five minutes of the holding step. (c) Snapshots of 1.1, 1.6, and $2.1 μ m$ indentation depths for a 15 kPa sample. The bottom dotted line represents the flat surface. The top dotted line is a reference to illustrate the motion of the cantilever (in yellow). Vertical red lines are the expected distances of the cantilever and indentation into the surface, respectively. The circle represents the spherical probe as a guide to the eye. Note the yellow reflection channel brightness is strongly increased for visualization of the cantilever distance at the expense of resolution. The particle radius is $R = 4 μ m$ . Scale bar: $5 μ m$ .
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Figure 4
Comparison of experiment with theory. (a) Measured force as a function of contact radius and indentation depth for quasistatic experiments after 15 minute holds. Dashed lines are linear fits, which provide modulus values $(E_{JKR})$ predicted by JKR from Eq. (1). (b) A zoomed-in plot of the red dashed boxed in part (a) of the lower modulus curves to illustrate linear behavior. As the material gets softer, the discrepancy between $E_{JKR}$ and $E_{bulk}$ systematically increases. Parts (a) and (b) show the results of three independent experiments for each modulus. (c) Plot of force vs indentation of experimental data (red points) for the 400 kPa PDMS and the JKR fit (dashed line) for approach and retract directions. (d) Similar plot to part (c) for the 5 kPa PDMS but also including the Maugis extension in Eq. (2) (blue line). The particle radius is $R = 4 μ m$ .
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Figure 5
Including a capillary correction to the force-indentation relationship. (a) Schematic illustrating the relevant variables for Eq. (3). The red line denotes the surface associated with $Υ$ . (b) $F$ vs $δ$ for the three softest materials with best fit $γ$ and $Υ$ and with the modulus fixed to $E_{bulk}$ . The kinks in the fit curves arise because $F_{total}$ is calculated with measured $a$ and $δ$ using three experiments for each modulus. The particle radius is $R = 4 μ m$ . (c) $F$ vs $a$ for the 400, 40, 15, and 5 kPa PDMS. (d) Zoom-in of the red box of part c for the lower moduli PDMS. The lines are calculated values using Eq. (3). For Figs. 5 and 5, single experiments are plotted for clarity.
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