Using capillary forces to determine the elastic properties of mesoporous materials

The capillary forces in mesoporous materials, when imbibed with liquid, are large enough to induce mechanical deformations. Using anisotropic porous silicon, we show that systematic measurements of strain as a function of the pore pressure can yield most of the elastic constants characterizing the porous matrix. The results of this poroelastic approach are in agreement with independent standard stress-strain measurements. The porosity dependence of Young's moduli as well as the values of Poisson's ratios are qualitatively consistent with porous silicon having a honeycomb structure. For a quantitative comparison, we performed finite element modeling of realistic pore geometries. The calculated elastic moduli are found to be much smaller than the measured ones. This is presumably due to both (i) finite-size effects, the Young's modulus of the 5-nm thick walls of the honeycomb could be notably smaller than the Young's modulus of bulk Si, and (ii) defects of the honeycomb structure along the pore axis.

Figure 1

Binarized TEM images of two porous silicon samples: (left) 50% porosity, (right) 85% porosity, from Ref. [22]. The pores of the honeycomb structure are straight and parallel to the [001] axis. The actual size of the images is $150\times 150{\mathrm{nm}}^2$.

Figure 2

Schematic view of the setup for transverse (a) and parallel (b) strain measurement. Transverse (c) and parallel (d) strains as a function of the pore pressure $P_L$ for a 50% sample (fluid: heptane).

Figure 3

Transverse biaxial modulus $B$ as a function of the normalized density $1-p$. Open and closed circles: experimental data (left scale); diamonds: results of finite elements modeling (right scale). The dashed line is only a guide for the eye.

Figure 4

Parallel Young's modulus $E_{\parallel }$ as a function of the normalized density $1-p$.