Identities for droplets with circular footprint on tilted surfaces.

Exact mathematical identities are presented between the relevant parameters of droplets displaying circular contact boundary based on flat tilted surfaces. Two of the identities are derived from the force balance, and one from the torque balance. The tilt surfaces cover the full range of inclinations for sessile or pendant drops, including the intermediate case of droplets on a wall (vertical surface). The identities are put under test both by the available solutions of a linear response approximation at small Bond numbers as well as the ones obtained from numerical solutions, making use of the Surface Evolver software. The subtleties to obtain certain angle-averages appearing in identities by the numerical solutions are discussed in detail. It is argued how the identities are useful in two respects. First is to replace some unknown values in the Young-Laplace equation by their expressions obtained from the identities. Second is to use the identities to estimate the error for approximate analytical or numerical solutions without any reference to an exact solution.

Contact angles of a drop pinned on an incline.

For a drop on an incline with small tilt angle α, when the contact line is a circle of radius r, we derive the relation mgsinα=γrπ/2(cosθ^{min}-cosθ^{max}) at first order in α, where θ^{min} and θ^{max} are the contact angles at the back and at the front, m is the mass of the drop and γ the surface tension of the liquid. We revisit in this way the Furmidge model for a large range of contact angles. We also derive the same relation at first order in the Bond number B=ρgR^{2}/γ, where R is the radius of the spherical cap at zero gravity. The drop profile is computed exactly in the same approximation. Results are compared with surface evolver simulations, showing a surprisingly large range of contact angles for applicability of first-order approximations.

Pinning of a drop by a junction on an incline.

The shape of a drop pinned on an inclined substrate is a long-standing problem where the complexity of real surfaces, with heterogeneities and hysteresis, makes it complicated to understand the mechanisms behind the phenomena. Here we consider the simple case of a drop pinned on an incline at the junction between a hydrophilic half plane (the top half) and a hydrophobic one (the bottom half). Relying on the equilibrium equations deriving from the balance of forces, we exhibit three scenarios depending on the way the contact line of the drop on the substrate either simply leans against the junction or overfills (partly or fully) into the hydrophobic side. We draw some conclusions on the geometry of the overlap and the stability of these tentative equilibrium states. In the corresponding retention force factor, we find that a major role is played by the wetted length of the junction line, in the spirit of Furmidge's observations. The predictions of the theory are compared with extensive molecular dynamics simulations.

Pareto genealogies arising from a Poisson branching evolution model with selection.

We study a class of coalescents derived from a sampling procedure out of N i.i.d. Pareto(α) random variables, normalized by their sum, including ß-size-biasing on total length effects (ß < α). Depending on the range of α we derive the large N limit coalescents structure, leading either to a discrete-time Poisson-Dirichlet (α, -ß) Ξ-coalescent (α ε[0, 1)), or to a family of continuous-time Beta (2 - α, α - ß)Λ-coalescents (α ε[1, 2)), or to the Kingman coalescent (α ≥ 2). We indicate that this class of coalescent processes (and their scaling limits) may be viewed as the genealogical processes of some forward in time evolving branching population models including selection effects. In such constant-size population models, the reproduction step, which is based on a fitness-dependent Poisson Point Process with scaling power-law(α) intensity, is coupled to a selection step consisting of sorting out the N fittest individuals issued from the reproduction step.

Is superhydrophobicity robust with respect to disorder?

We consider theoretically the Cassie-Baxter and Wenzel states describing the wetting contact angles for rough substrates. More precisely, we consider different types of periodic geometries such as square protrusions and disks in 2D, grooves and nanoparticles in 3D and derive explicitly the contact angle formulas. We also show how to introduce the concept of surface disorder within the problem and, inspired by biomimetism, study its effect on superhydrophobicity. Our results, quite generally, prove that introducing disorder, at fixed given roughness, will lower the contact angle: a disordered substrate will have a lower contact angle than a corresponding periodic substrate. We also show that there are some choices of disorder for which the loss of superhydrophobicity can be made small, making superhydrophobicity robust.

On the extended Moran model and its relation to coalescents with multiple collisions.

We study the asymptotics of the extended Moran model as the total population size N tends to infinity. Two convergence results are provided, the first result leading to discrete-time limiting coalescent processes and the second result leading to continuous-time limiting coalescent processes. The limiting coalescent processes allow for multiple mergers of ancestral lineages (Λ-coalescent). It is furthermore verified that any continuous time Λ-coalescent (with Λ any probability distribution) can arise in the limit. Typical examples of extended Moran models are discussed, with an emphasis on models being in the domain of attraction of beta coalescents or Λ-coalescents with Λ being log infinitely divisible.