arXiv subjects

François Dunlop

Publications and source records attributed to François Dunlop.

8 recordsLinked to original sources

arXiv2022

A Gibbsian random tree with nearest neighbour interaction

We revisit the random tree model with nearest-neighbour interaction as described in previous work, enhancing growth. When the underlying free Bienaymé-Galton-Watson (BGW) model is sub-critical, we show that the (non-Markov) model with interaction exhibits a phase transition between sub- and super-critical regimes. In the critical regime, using tools from dynamical systems, we show that the partition function of the model approaches a limit at rate $n^{-1}$ in the generation number $n$. In the critical regime with almost sure extinction, we also prove that the mean number of external nodes in the tree at generation $n$ decays like $n^{-2}$. Finally, we give a spin representation of the random tree, opening the way to tools from the theory of Gibbs states, including FKG inequalities. We extend the construction in previous work when the law of the branching mechanism of the free BGW process has unbounded support.

math.PR↗

arXiv2017

Pinning of a drop by a junction on an incline

The shape of a drop pinned in a local equilibrium on an incline is a long-standing problem. The substrate can be homogeneous or heterogeneous and we herewith consider a drop pinned on an incline at the junction between a hydrophilic half-plane (the top half) and a hydropho-bic one (the bottom half). Relying on the equilibrium equations deriving from the balance of forces, we exhibit three scenarii depending on the way the contact line of the drop on the substrate either simply leans against the junction or overfills (partly or fully) 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. The predictions of the theory are illustrated and supported by extensive molecular dynamics simulations.

cond-mat.soft↗

arXiv2013

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.

cond-mat.soft↗

arXiv2011

Metastable wetting

Consider a droplet of liquid on top of a grooved substrate. The wetting or not of a groove implies the crossing of a potential barrier as the interface has to distort, to hit the bottom of the groove. We start with computing the free energies of the dry and wet states in the context of a simple thermodynamical model before switching to a random microscopic version pertaining to the Solid-on-Solid (SOS) model. For some range in parameter space (Young angle, pressure difference, aspect ratio), the dry and wet states both share the same free energy, which means coexistence. We compute these coexistence lines together with the metastable regions. In the SOS case, we describe the dynamic transition between coexisting states in wetting. We show that the expected time to switch from one state to the other grows exponentially with the free energy barrier between the stable states and the saddle state, proportional to the groove's width. This random time appears to have an exponential-like distribution.

cond-mat.stat-mech↗

arXiv2010

Layering and wetting transitions for an SOS interface

We study the solid-on-solid interface model above a horizontal wall in three dimensional space, with an attractive interaction when the interface is in contact with the wall, at low temperatures. There is no bulk external field. The system presents a sequence of layering transitions, whose levels increase with the temperature, before reaching the wetting transition.

math-ph↗

arXiv2008

Random walk weakly attracted to a wall

We consider a random walk X_n in Z_+, starting at X_0=x>= 0, with transition probabilities P(X_{n+1}=X_n+1|X_n=y>=1)=1/2-δ/(4y+2δ) P(X_{n+1}=X_n+1|X_n=y>=1)=1/2+δ/(4y+2δ) and X_{n+1}=1 whenever X_n=0. We prove that the expectation value of X_n behaves like n^{1-(δ/2)} as n goes to infinity when δis in the range (1,2). The proof is based upon the Karlin-McGregor spectral representation, which is made explicit for this random walk.

math.PR↗

arXiv2006

Correlations of a bound interface over a random substrate

The correlation function of a one-dimensional interface over a random substrate, bound to the substrate by a pressure term, is studied by Monte-Carlo simulation. It is found that the height correlation < h_i ; h_{i+j} >, averaged over the substrate disorder, fits a form exp(-(j/b)^c) to a surprising precision in the full range of j where the correlation is non-negligible. The exponent c increases from 1.0 to 1.5 when the interface tension is taken larger and larger.

math-ph↗

arXiv2005

A necklace of Wulff shapes

In a probabilistic model of a film over a disordered substrate, Monte-Carlo simulations show that the film hangs from peaks of the substrate. The film profile is well approximated by a necklace of Wulff shapes. Such a necklace can be obtained as the infimum of a collection of Wulff shapes resting on the substrate. When the random substrate is given by iid heights with exponential distribution, we prove estimates on the probability density of the resulting peaks, at small density.

math.PR↗