Wetting of Elastic Solids on Nanopillars

Solids and liquids are both known to exhibit Cassie-Baxter states, where a drop or a solid nanoparticle is maintained on top of pillars due to wetting forces. We point out that due to elastic strain, solid nanocrystals exhibit a behavior different from that of liquids. First, the equilibrium Cassie-Baxter state on a single pillar exhibits a spontaneous symmetry breaking due to elastic effects. The second consequence of elasticity is the existence of stable partially impaled states, resulting from a compromise between wetting forces which favor impalement and elastic strain which resists impalement. Based on kinetic Monte Carlo simulations which include elastic strain, we discuss these effects and we propose a global phase diagram for the stability of nanocrystals on nanopillars.

Figure 1

Analysis of the energy of CB states. (a) Schematics of the CB configuration. The lower part of the schematics shows the relative positions of the bottom facet and the pillar top for $n_s=0,1,2$, and the vector $r_p=(x_p,y_p)$. In (b-f), $K=50$ and the nanoparticle volume is $V={14}^3$, leading to a characteristic elastic length ${\ell }_{el}=J_1(1+4\zeta )/(5{\epsilon }^{2}Ka)\approx 0.82$ for $\epsilon =7\%$. (b) Variation of the total energy $F_{\text{tot}}$ in the CB state ($h^{*}=0$) as a function of the coordinate $x_p$. Solid lines correspond to displacements $r_p=x_p(1,1)$, and dashed-dotted lines to $r_p=x_p(1,0)$. We have used ${\ell }_p=6$, and $\chi =0.7$ with $\epsilon =0\%$ (lower curves, red), and $\epsilon =7\%$ (upper curves, black). (c) Limit of stability ${\chi }_c(n_s)$ of the CB state for $n_s=0,1,2$ without elasticity (lines without symbols) and ${\chi }_{el}$ with elasticity: Open circle for ${\ell }_p=4$, and square for ${\ell }_p=6$. (d) Data collapse of the graph (c) obtained from the derivative of the elastic energy $W$, see text. (e) Surface energy $F$ at $\chi =0.9$ and elastic energy $W$ for $\epsilon =7\%$ as a function of the impalement depth $h^{*}$. (f) Variation of the total energy $F_{\text{tot}}$ with $\epsilon =7\%$. We have used $\chi =0.95,0.85$ for $n_s=0$, $\chi =0.95,0.85,0.75$ for $n_s=1$, and $\chi =0.95,0.85,0.75,0.65$ for $n_s=2$.

Figure 2

Kinetic Monte Carlo simulations. (a) Phase diagram of the solid CB state with ${\chi }_c$, ${\chi }_{el}$, and ${\chi }^{*}$, for $V={14}^3$, ${\ell }_p=6$ and $\epsilon =7\%$ (lines with symbols), $\epsilon =0\%$ (lines without symbols). KMC simulation results: Open circle CB state, square partial impalement and triangle total impalement. (b) Snapshots of KMC simulations: (1) asymmetric CB at $\chi =0.5$, total simulation time $3.4\times {10}^7{\nu }_0^{-1}$; (2) partial impalement at $\chi =0.85$, total simulation time $6.1\times {10}^7{\nu }_0^{-1}$; (3) total impalement at $\chi =0.95$, total simulation time $3.0\times {10}^7{\nu }_0^{-1}$.