Comment on “Clusters of bundled nanorods in nanocarpet effect” [Appl. Phys. Lett. 88, 103123 (2006)] Free

A simple model was proposed in 2006 by Zhao and Fan¹ to explain the bundling of nano inverted frustums array in the nanocarpet effect. The model assumes that the inverted frustums can be described by spring constant of cylindrical rods. Explicit prediction for an exponent of 1.5 was obtained for the power law cluster size dependence on the nanorod heights h. An analysis of relevant experimental measurements for silicon nano inverted frustum arrays was also reported to be 1.2 ± 0.1 by Zhao and Fan.¹ Although the apparent difference (0.3) is outside the error estimates, the model did appeared to have captured the essential physics. The authors then assert that the model prediction is consistent with the experimental results. The noticeable discrepancy, however, remains an unsolved problem and raises doubts in the accuracy of the model.

Euler-Bernoulli theory (EBT) calculations² for the spring constant of inverted frustums show that the height dependence exponent can be very different for inverted frustums and cylindrical rods. The EBT theory was recently validated by simulations of model inverted frustums³ and applied to experiments.⁴ In this Comment, we show that the disagreement between the Zhao-Fan model predictions and experiments can be traced to the use of the cylindrical rod approximation for the inverted frustums. For the inverted frustums used in the experiments, EBT² predicts the spring constant k_eff to scale as $keff∼h−1.8$⁠. This is very different from that of a straight rod with $keff∼h−3$⁠, used by Zhao and Fan.¹ When the Zhao-Fan model is extended with the EBT scaling for the inverted frustums, an exponent of 0.9 is obtained for the power law cluster size dependence on the nanorod heights h.

Since the inverted frustums in the experiments are anchored to the substrate with nonzero cross-sections, one cannot use a simple log-log plot to analyze the cluster size dependence. The log-log plot used in Ref. 1 assumes the cluster diameter ξ scales with h as $ξ(h)∼hν$⁠. ν is some positive exponent. This will not be valid as $h→0$ and is not physical for very short frustums. EBT² and simulations³ indicate that a more general scaling of $ξ(h)∼(hν+b)$ should be used. b is a constant controlled by the frustum base structure near the substrate and scaling is expected only for sufficiently long frustums. We have taken the data from Figure 4 in Zhao-Fan's paper¹ and reanalyze them in Figures 1 and 2.

In Figure 1, a linear plot of the average cluster diameter ξ vs h^0.9 indicates a very good fit to h^0.9 dependences predicted by the model of Zhao-Fan¹ extended with EBT for the inverted frustums.² In Figure 2, the same data are plotted vs h^1.5. The solid line indicates a fit to h^1.5 scalings predicted by the original model of Zhao-Fan¹ using cylindrical approximation for the inverted frustum elastic constant. The dashed line is a fit to the data (to guide the eye) and clearly exhibits disagreements between data and the original Zhao-Fan model.

In summary, when the EBT results for the inverted frustums are used, the model of Zhao-Fan can accurately describe the experimental data. This resolves a long standing disparity between theory and experiments for the cluster size dependences in the important nanocarpet effect. It also shows that EBT can provide an accurate description of nano inverted frustum's elastic constant.

This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which was supported by National Science Foundation Grant No. OCI-1053575.