Size effect in microcompression of epoxy micropillars

Abstract

Understanding the size effect on the mechanical properties of polymers is of great importance for a robust design of today’s polymer-based micro-devices. In this article, we propose the microcompression approach based on the focused ion beam milling technique to probe the possible size effect on the mechanical behavior of epoxy micropillars. By systematically reducing their size from the micrometer to submicron scale, these micropillars display a constant elastic modulus in their inner cores while exhibit an increasing yield and fracture strengths with decreasing diameters. Such a size effect is attributed to the intrinsic material heterogeneity at the submicron scale and the presence of a nano-scale stiff surface layer wrapping around the micropillars. This study provides a theoretical framework for the microcompression analysis of polymer-based micropillars, paving the way for future study of a variety of polymer-based advanced material systems by microcompression.

Appendices

Appendix A

The standard nanoindentation method based on the Oliver and Pharr’s approach was used to measure the Young’s modulus, E, of the bulk epoxy sample [13]. To exclude the creep effect, a holding section was included into the load function as seen from the inset of Fig. A1a. For consistency of experimental data, only those load–displacement curves following a mater curve in the loading portion were selected for the analysis. Furthermore, to attain a time-independent property, the duration of the holding section was varied such that the time dependence of the modulus could be monitored. The results showed that, once the holding time reached more than ~2 s, there was no systematic change in the distribution of the measured modulus, as shown in Fig. A1a. Based on our experimental results, the extracted Young’s modulus displays a normal distribution with a mean value of 3.9 Pa and a standard deviation of 0.1 GPa for the bulk epoxy sample (note that the Poisson’s ratio of the epoxy was assumed as ~0.35 in the data analysis).

Fig. A1

a The normal distribution of the Young’s modulus of the bulk epoxy sample measured from nanoindentation (the inset shows the experimental load–displacement curves in nanoindentation); and the variation of the measured b Young’s modulus, c hardness and d squared stiffness to load parameter (S ² /P) with the indentation contact depth

As complement to the histogram of the measured modulus, Fig. A1a and b shows its variation with the indentation contact depth, which seemingly reveals a surface effect, that is, the measured modulus appears slightly higher near the surface than in the bulk. Similar findings were reported by Briscoe and co-workers from nanoindentation of the surfaces of a variety of polymers [32]. Likewise, the measured indentation hardness, H, also exhibits a size effect, as shown in Fig A1c, which may be attributed to a confounding effect of surface stiffening and strain gradient. To assess the possible influence of material pileup, which may cause an overestimation of ~10 % in the above measurements [15, 21, 25], the parameter of S ²/P is plotted in Fig A1d, where S and P denote the stiffness and maximum indentation load, respectively, which can be extracted from an indentation load–displacement curve [13]. As S ² /P scales with E ² /H and is independent of the contact area, the slightly ascending tread of S ² /P in the near-surface region indicates a physically stiffened surface even though H could be regarded as a constant.

Appendix B

According to Ref. [27], the critical strain associated with the buckling of a uniform stiff elastic skin attached to a compliant elastic matrix, ε _c, is approximately:

$$ \varepsilon_{\text{c}} \approx 0.52\left( {\frac{{E_{\text{m}} }}{{E_{\text{f}} }}} \right)^{\frac{2}{3}} , $$

(A1)

where E _m and E _f are the Young’s modulus of the matrix and stiff skin, respectively. Furthermore, as shown in Fig. A2, the wavelength of the wrinkles, λ, of the stiff skin as normalized by its thickness t _f can be approximated as [27]:

Fig. A2

The schematics of the a undeformed and b wrinkled surface skin attached to the bulk epoxy resin

$$ \frac{\lambda }{{t_{\text{f}} }} \approx 4\left( {\frac{{E_{\text{f}} }}{{E_{\text{m}} }}} \right)^{\frac{1}{3}} . $$

(A2)

In microcompression, the epoxy core and the stiff skin are simultaneously compressed to the same amount of deformation. If the epoxy core has already yielded while the stiff skin is still within the elasticity regime, residual elastic deformation will remain even after full unloading. In such a case, surface wrinkling will take place as to minimize the overall elastic energy storage if the associated plastic strain in the epoxy core reaches the critical value. Note that almost all surface wrinkles were observed around the surface cracks in the plastically deformed micropillars (Fig. 3). This is because, as the constraint in the hoop direction is released on the two inner faces of the surface cracks, the stress state there in the stiff skin and its adjacent epoxy resin is closer to that for elastic wrinkling as predicted by Eqs. (A1) and (A2).

To estimate t _f and E _f, the plastically deformed epoxy micropillars were carefully examined using high-resolution scanning electron microscopy. The wavelengths λ were recorded with the corresponding plastic deformation in the micropillar and, then used together with the Young’s modulus of the bulk epoxy, as measured from nanoindentation (E _m ~ 3.9 GPa), to solve for t _f and E _f from Eqs. (A1) and (A2). Based on our experimental data, λ ~ 250 nm and ε _c ~ 0.12, which then leads to t _f ~ 30 nm and E _f ~ 30 GPa. As compared with amorphous-metal micropillars, such as Zr-based metallic glass whose FIB damage layer was measured to be a few nanometers in thickness [28], the epoxy micropillars possess a much thicker stiffened FIB damage layer, which may result from chain scission and formation of new cross-links in the polymeric materials because of ion irradiation [33].