119 research outputs found
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
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.Comment: 61 pages, 6 figures. Miscellaneous corrections and changes, primarily
in Section 4. Figure 5 added
Hard rods: statistics of parking configurations
We compute the correlation function in the equilibrium version of R\'enyi's
{\sl parking problem}. The correlation length is found to diverge as
when (maximum density) and as
when (minimum density).Comment: 9 pages, 1 figur
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
Bienaym\'e-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 in the generation number . In the critical regime with almost
sure extinction, we also prove that the mean number of external nodes in the
tree at generation decays like . 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
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
Layering in the Ising model
We consider the three-dimensional Ising model in a half-space with a boundary
field (no bulk field). We compute the low-temperature expansion of layering
transition lines
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-\delta/(4y+2\delta)
P(X_{n+1}=X_n+1|X_n=y>=1)=1/2+\delta/(4y+2\delta) and X_{n+1}=1 whenever X_n=0.
We prove that the expectation value of X_n behaves like n^{1-(\delta/2)} as n
goes to infinity when \delta is in the range (1,2). The proof is based upon the
Karlin-McGregor spectral representation, which is made explicit for this random
walk.Comment: Replacement with minor changes and additions in bibliography. Same
abstract, in plain text rather than Te
Renormalization Theory for Interacting Crumpled Manifolds
We consider a continuous model of D-dimensional elastic (polymerized)
manifold fluctuating in d-dimensional Euclidean space, interacting with a
single impurity via an attractive or repulsive delta-potential (but without
self-avoidance interactions). Except for D=1 (the polymer case), this model
cannot be mapped onto a local field theory. We show that the use of intrinsic
distance geometry allows for a rigorous construction of the high-temperature
perturbative expansion and for analytic continuation in the manifold dimension
D. We study the renormalization properties of the model for 0<D<2, and show
that for d<d* where d*=2D/(2-D) is the upper critical dimension, the
perturbative expansion is UV finite, while UV divergences occur as poles at
d=d*. The standard proof of perturbative renormalizability for local field
theories (the BPH theorem) does not apply to this model. We prove perturbative
renormalizability to all orders by constructing a subtraction operator based on
a generalization of the Zimmermann forests formalism, and which makes the
theory finite at d=d*. This subtraction operation corresponds to a
renormalization of the coupling constant of the model (strength of the
interaction with the impurity). The existence of a Wilson function, of an
epsilon-expansion around the critical dimension, of scaling laws for d<d* in
the repulsive case, and of non-trivial critical exponents of the delocalization
transition for d>d* in the attractive case is thus established. To our
knowledge, this provides the first proof of renormalizability for a model of
extended objects, and should be applicable to the study of self-avoidance
interactions for random manifolds.Comment: 126 pages (+ 24 figures not included available upon request),
harvmac, SPhT/92/12
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