*where, setting out to provide a rationale for the Cassie relation, we had to conclude that it does not hold...*

*Previously, we have shown that line defects are key players for the propagation of the triple line. From this perspective, we naively expected that we would be able to adjust the mobility threshold of the triple line (i.e. the receding contact angle) by tuning the surface patterns. It turned out that this was not the case : in fact we found that the threshold is independent of the lattice type. A very puzzling result indeed, but which happens to answer a very old question : whether the Cassie relation (1948) should apply to patterned surfaces or not !*

## What we expected

Receding contact angle measured for various lattice types and plotted as a function of surface fraction φ. We observe a proportionality relation, with a significant offset at φ=0.In the case of metals, the mobility of dislocations is intimately connected to the type of atomic arrangement (lattice type). On our periodic hydrophobic surfaces, changing the lattice type, we found experimentaly that the receding contact angle ("mobility threshold") does not change *at all*. In fact, these disappointing results only confirmed prior knowledge : for years, people have rationalized receding contact angles against the **surface fraction Φ**. This is a very simple parameter which does not contain any information on the lattice geometry.

There is a very simple explanation for this relation : surface averaging. If the interaction energy scales with the real contact area between the solid and the liquid, and the threshold for mobility is proportional to the real interaction energy, then the receding contact angle must depend upon real contact area divided by apparent contact area, *i.e.* surface fraction. This is the Cassie relation (Cassie 1948).

Unfortunately, unlike many simple explanations, this one seems liable to many criticisms. In fact it has also long been suspected that the Cassie relation should not hold for these types of surfaces. This controversy over the Cassie relation is still very much open : the latest status report on the question lists 150 (mostly recent) references (Erbil 2014).

Receding contact angles simulated on various surface patterns plotted as a function of surface fraction φ. At very low surface fraction, a singular behaviour is manifest and a precipitous drop to zero is found. This singularity is the signature of pinning by triple line defects, and is also the source of the apparent offset in the data.

## What we found

Our simulations confirmed the dependence upon surface fraction, but also evidenced a very precipitous drop of the threshold at low surface fractions. This is quite unexpected from the Cassie point of view, but very consistent with most data. In fact, what we have shown is that the dependence upon surface fraction is not due to simple averaging, but to a very different process : pinning of the triple line by these specific line defects we have evidenced earlier. Indeed pinning results in a *strongly non-linear* dependence, and since kinks straddle two rows, they feel both lattice directions and surface fraction results.

### Publication

- Marco Rivetti, Jérémie Teisseire & Etienne Barthel, , Phys. Rev. Lett. 115, 016101 (2015) - Journal Page, [Hal]

Receding contact angle measured on various surface patterns, plotted as a funtion of line fraction - the ratio of the post size to the post spacing along the triple line direction. Line scaling does not comply with the experiments, as noted from the relevant plot. However, the partial success of line scaling theories can be explained by the fact that for rather isotropic lattices, the line fraction is the square root of the surface fraction, and this scaling introduces a singularity which bears some resemblance to the present one.