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\begin{document}
\title{Wetting of Heterogeneous Surfaces at the Mesoscopic Scale}
\author{
Jo\"el De Coninck$^{1}$, Christophe Dobrovolny$^{2}$, \\
Salvador Miracle--Sol\'{e}$^{3}$, and Jean Ruiz$^{4}$}
\date{}
\maketitle
%\baselineskip 18pt
\begin{quote}
{\footnotesize \textsc{Abstract:} We consider the problem of
wetting on a heterogeneous wall with mesoscopic defects: i.e.\
defects of order $L^{\varepsilon}$, $0<\varepsilon<1$, where $L$ is
some typical length--scale of the system. In this framework, we
extend several former rigorous results which were shown for walls
with microscopic defects \cite{DMR,DMR2}. Namely, using
statistical techniques applied to a suitably defined semi-infinite
Ising-model, we derive a generalization of Young's law for rough
and heterogeneous surfaces, which is known as the generalized
Cassie-Wenzel's equation. In the homogeneous case, we also show
that for a particular geometry of the wall, the model can exhibit
a surface phase transition between two regimes which are either
governed by Wenzel's or by Cassie's law.
\\[3pt]
\textsc{Key words:} Wetting, Wenzel's law, Cassie's law,
roughness, interfaces.}
\end{quote}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\renewcommand{\thefootnote}{}
\footnote{Preprint CPT--2003/P.4532 revised}
\renewcommand{\thefootnote}{\arabic{footnote}}
\setcounter{footnote}{0}
\footnotetext[1]{%
Centre de Recherche en Mod\'elisation Mol\'eculaire, Universit\'e
de Mons--Hainaut, 20 place du Parc, B-7000 Mons, Belgium.
\hfill\break
E-mail
address: \texttt{Joel.De.Coninck@galileo.umh.ac.be\/}}
\footnotetext[2]{%
Centre de Recherche en Mod\'elisation Mol\'eculaire, Universit\'e
de
Mons--Hainaut, 20 place du Parc, B-7000 Mons, Belgium. %\hfill\break
Permanent address: Centre de Physique Th\'eorique, CNRS, Luminy
case 907,
F-13288 Marseille Cedex 9, France.
\hfill\break
E-mail address: \texttt{%
dobrovol@cpt.univ-mrs.fr\/}}
\footnotetext[3]{%
Centre de Physique
Th\'eorique, CNRS, Luminy case 907,
F-13288 Marseille Cedex 9, France.
% \hfill\break
E-mail address: \texttt{%
Salvador.Miracle-Sole@cpt.univ-mrs.fr\/}} \footnotetext[4]{%
Centre de Physique Th\'eorique, CNRS, Luminy case 907,
F-13288 Marseille Cedex 9, France. %\hfill\break
E-mail address: \texttt{%
Jean.Ruiz@cpt.univ-mrs.fr\/}}
\setcounter{footnote}{4}
\thispagestyle{empty}
\arraycolsep 2pt
% \baselineskip 24 pt
\section{Introduction}
Surface phenomena play an important role in many fundamental processes and,
among them, the wetting of surfaces is a subject of primary importance.
Consider a drop of liquid \( B \) in coexistence with a gas phase
\( A \) on top of the surface \( W \). The shape of this drop with
a fixed volume of liquid is obtained by minimizing the free energies
associated to the three interfaces under consideration. The solution
of the corresponding variational problem is given by the Winterbottom's
construction.
As a consequence, the contact angle of the droplet with the wall satisfies
in the isotropic case the well known Young's equation :
\begin{equation}
\label{eq:1.1}
\tau _{AB}\cos \theta =\tau _{AW}-\tau _{BW}\equiv \Delta {\tau }
\end{equation}
where \( \tau _{ij} \), \( \{i,j\}\in \{A,B,W\} \) is the surface
tension between the media \( i \) and \( j \). In the case of an
orientation dependent surface tension for the \( AB \)--interface,
the L.H.S.\ of the above equations have to be modified: e.g.\ in dimension
\( d=2 \), one should replace it, by
\( \cos \theta \, \tau _{AB}-\sin \theta \, \frac{d}{d\theta }\tau _{AB} \)
(see \cite{DD}).
\vspace{1cm}
\begin{center}
\unitlength 1mm
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\put(60,-18){\makebox{$\theta$}}
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\put(45,-25){\makebox{W}}
\put(70,-20){\line(-1,1){15}}
\end{picture}
\end{center}
\begin{center}
\footnotesize{Figure 1 : Young's contact angle}
\end{center}
%its contact-area with the surface. This naturally leads to a
%sessile drop, as shown in Figure~1.
The validity of Winterbottom's construction and Young's equations
in the frame of Statistical Mechanics has been established in several
works: see \cite{DD,DDR} for SOS-models and \cite{AK,PV} for Ising-like
models.
The substrate \( W \) is usually considered as perfectly flat and
homogeneous surface.
When the surface is homogeneous but rough, one usually introduce the
roughness as the ratio of the area \( A \) of the surface and the area \(
\bar{A} \)
of its projection on the horizontal plane: \( r=A/\bar{A} \). In
this case the differential wall tension \( \Delta \tau \) has to
be computed according to the Wenzel's law \cite{W}: \[
\Delta \tau =r(\Delta \tau )^{\text{flat}}\]
where \( (\Delta \tau )^{\text{flat}}=\tau^{\text{flat}}
_{AW}-\tau^{\text{flat}} _{BW} \)
is the differential wall tension
of the corresponding flat wall.
When the substrate is flat but made of two species \( W_{1} \) and
\( W_{2} \) with concentrations \( c_{1} \) and \( c_{2}=1-c_{1} \),
respectively, we will have: \[
\Delta \tau =c_{1}(\Delta \tau )^{\text{flat}}_{1}+c_{2}(\Delta \tau
)^{\text{flat}}_{2}\]
where $(\Delta \tau )^{\text{flat}}_{i}=\tau^{\text{flat}}
_{AW_i}-\tau^{\text{flat}} _{BW_i}$.
This relation is known as the Cassie's law \cite{C}.
When the substrate is both rough and heterogeneous the generalized
Cassie-Wenzel's law states: \begin{equation}
\label{eq:1.3}
\Delta \tau =r_{1}c_{1}
(\Delta\tau)^{\text{flat}}_{1}+
r_{2}c_{2})(\Delta\tau)^{\text{flat}}_{2}
\end{equation}
where \( r_{i}c_{i} \) is the ratio of the non planar surface covered
with material \( i \) to the total planar area.
This generalized Cassie-Wenzel's equation has been presented for macroscopic
defects using thermodynamical arguments in Ref.\ \cite{SL}. In Refs.\
\cite{DMR,DMR2},
the rigorous proof of this equation has been derived, within a SOS-like
model, for microscopic defects covering the surface with a certain
periodicity. In the later case the law is satisfied up to a small
temperature dependent correction (tending exponentially to zero with
the temperature). Namely,\[
\Delta \tau =r_{1}c_{1}(\Delta \tau )_{1}^{\text{flat}}+r_{2}c_{2}(\Delta
\tau )_{2}^{\text{flat}}+O(e^{-\beta C})\]
Let us now consider a surface \( z(x,y) \) over a certain area \(
\textrm{L}\times \textrm{L} \)
in atomic units. Combining the previous results, we know that we can
use the Cassie-Wenzel's equation for defects of order \( O(\textrm{L}) \)
or of order \( O(1) \). On the other hand, it is also obvious that
a real surface can present heterogeneities at all intermediate length-scales
\( \textrm{L}^{\varepsilon } \) with \( 0<\varepsilon <1 \). It
is thus interesting to extend the proof of the Cassie-Wenzel's relation
for such mesoscopic defects \( O(\textrm{L}^{\varepsilon }) \), \(
0<\varepsilon <1 \).
This is actually the aim of this paper. We consider an Ising-like
lattice gas model with mesoscopic defects.
We prove in Theorem ~1 below, the validity of the generalized
Cassie-Wenzel's
equation at low temperatures, within a
certain range of the coupling constants.
This equation reduces to
the Cassie's law when the wall is heterogeneous and flat
and to the
Wenzel's law when the wall is homogeneous and rough.
Let us stress that contrary to the case of microscopic defects, no
corrective term has to be added.
However, this result is only true when the strength of the interaction between
the particles and the wall is small.
We give then an important improvement of this law, showing
that when this strength is varied, the system exhibits surface phase
transitions between two regimes.
Namely, we show in Theorem ~2 that,
in the
homogeneous case, a transition takes place between a Wenzel's and Cassie's
behaviours for the
drop.
The paper is organized as follows. In Section~2, we introduce the
modified semi-infinite Ising model which describes the modeling
of the rough and heterogeneous surface, and we give the
microscopic definitions of the various surface-tensions. Our
results are stated in Section~3.. Finally,
Sections 4 and 5 are devoted to proofs.
\section{The model}
\setcounter{equation}{0}
To model the influence of roughness and heterogeneities on wetting
we use a suitable $3D$ half--infinite Ising model to describe the
drop and its vapor and an SOS surface to represent the boundary of
the wall. Namely, we will describe the wall by the boundary
$\partial W $ of a half infinite lattice $W\subset \mathbb{Z}^{3}
$ which represents the substrate, as shown in Figure~2.
This boundary will be rough (see below for the precise definition of $W $)
and we shall consider $W $ to be the union of two disjoint subsets $W_{1} $
and $W_{2} $. In this way we get an inhomogeneous wall $\partial W=\partial
W_{1}\cup \partial W_{2} $ composed of several pieces of the two different
substrates.
For the vessel containing the drop and the gas we take the complement $V=%
\mathbb{Z}^{3}\setminus W $.
To each site $x$ of the vessel $V$, we associate a variable
$\sigma _{x} $ which may take two values; $+1 $ associated to a
particle at $x $, and $-1 $ associated to an empty site. We assume
that the substrate is completely filled, i.e. $\sigma _{x}\equiv
+1 $ for all $x\in W $.
Inside the vessel, the variables are coupled with a nearest neighbour
coupling $J/2>0 $, representing a nearest neighbour attraction of particles
while at the boundary between the vessel and the substrate the spins of the
vessel are coupled with a nearest neighbour coupling constant, $K_{x}/2 $
with the particles of $W $: $K_{x}=K_{1} $ or $K_{2} $ according $x\in W_{1}
$ or $x\in W_{2} $.
Formally, for any finite set $\Omega \subset V $ these interactions are
described by the Hamiltonian
\begin{eqnarray}
H_{\Omega }^{\overline{\sigma }}(\sigma ) & = & -\frac{J}{2}\sum _{\atop{%
\langle xy\rangle }{x,y\in \Omega }}(\sigma _{x}\sigma _{y}-1) -\frac{J}{2}
\sum _{\atop{\langle xy\rangle }{x\in \Omega ,y\in \Omega ^{c}\setminus W}%
}(\sigma _{x}\overline{\sigma }_{y}-1) \notag \\
& & -\frac{K_{x}}{2}\sum _{\atop{\langle xy\rangle }{x\in \Omega ,y\in W}%
}(\sigma _{x}-1) \label{eq:2.1}
\end{eqnarray}
Here $\langle xy\rangle $ denotes nearest neighbour pairs, $\Omega ^{c}=%
\mathbb{Z}^{3}\setminus \Omega $ is the complement of $\Omega , $ and $%
\overline{\sigma } $ are the chosen boundary conditions defined as $%
\overline{\sigma }=+ $ or $- $, i.e.\ either $\overline{\sigma }_{y}=+1 $
for all $y\in \Omega ^{c}\setminus W $ or $\overline{\sigma }_{y}=-1 $ for
all $y\in \Omega ^{c}\setminus W $.
Let us now introduce the differential wall tension for the model (\ref%
{eq:2.1}). Considering a finite lattice $\Lambda (L)=\left\{
(x_{1},x_{2},x_{3})\in \mathbb{Z}^{3}:|x_{i}|\leq L,i=1,2,3\right\} $, we
let $Z_{W}^{+}(\Omega ) $ and $Z_{W}^{-}(\Omega ) $ be the partition
functions of the model (\ref{eq:2.1}) at inverse temperature $\beta $, in
the volume $\Omega =\Lambda (L)\cap V $, with respectively, $+ $ and $- $
boundary conditions on that part of the boundary of $\Lambda (L)\cap V $
which is not part of the wall (on the wall, the boundary conditions are
always $+1 $). We then define the wall free energy $\tau _{+W} $ \ (and
similarly $\tau _{-W} $) in term of $\log Z_{W}^{+}({\Omega }) $ by
subtracting the bulk term as well as the boundary terms associated with the
boundary $\partial \Omega \setminus \partial W $, and taking appropriate
limits. The differential wall tension
\begin{equation}
\Delta \tau =\tau _{+W}-\tau _{-W}
\end{equation}
is thus defined as \cite{FP,PP,BDKZ,BDK,DMR}:
\begin{equation}
\beta \Delta \tau =-\lim _{L\rightarrow \infty }\frac{1}{(2L+1)^{2}}\log
\frac{Z_{W}^{-}(\Omega )}{Z_{W}^{+}(\Omega )}
\end{equation}
For the usual surface tension $\tau _{^{+-}} $ between the $+ $
and $- $ phases we use the standard
definition \cite{GMM}. Namely,
let $Z^{+}(\Lambda (L)) $ be the partition function of the
standard Ising model with formal Hamiltonian
$$-\frac{J}{2}
\sum_{{\langle xy\rangle }}(\sigma _{x}\sigma _{y}-1) $$
in the volume
$\Lambda (L) $ with $+ $ boundary conditions on the boundary of
$\Lambda $ and $Z^{+-}(\Lambda (L))$ be the partition function
with $+ $ boundary conditions below the plane $x_{3}=1/2 $ and $-
$ boundary conditions above this plane. Then, the surface tension
$\tau _{^{+-}} $ is defined by the limit
\begin{equation} \label{eq:2.4}
\beta \tau _{+-}=-\lim _{L\rightarrow \infty }\frac{1}{(2L+1)^{2}}\log \frac{%
Z^{+-}(\Lambda (L))}{Z^{+}(\Lambda (L))}
\end{equation}
In the perfectly flat case, the set modeling the substrate will be just the
half space $W^{\text {flat}}=\{(x_{1},x_{2},x_{3})\in \mathbb{Z}^{3}\mid
x_{3}\leq 0\} $ and we let $(\Delta \tau )_{1}^{\text {flat}} $ (resp. $%
(\Delta \tau )_{2}^{\text {flat}} $) correspond to the case of the
homogeneous flat wall with $W_{1}=W^{\text {flat}} $, $W_{2}=\emptyset $
(resp. $W_{2}=W^{\text {flat}} $, $W_{1}=\emptyset $).
More generally, we consider a substrate surface $\partial W $ (defined as
the set of unit plaquettes, whose center intersects the bonds $xy $, $x\in W
$,
$y\in \mathbb{Z}^{3}\setminus W $, in their middle point) given by a
periodic Solid-On-Solid type interface, i.e.\ $\partial W $ corresponds to
the graph of a periodic function $x_{3}=x_{3}(x_{1},x_{2}) $.
For the sake of simplicity, we shall consider a boundary surface $\partial W
$ given by the graph of the function $x_{3}(x_{1},x_{2}) $ defined on the
cylinder $\big \{\frac{1}{2}\leq x_{1}\leq a+\frac{1}{2},\frac{1}{2}\leq
x_{2}\leq a+\frac{1}{2}\big \} $ by
\begin{equation*}
x_{3}(x_{1},x_{2})=\left\{
\begin{array}{ll}
-b+\frac{1}{2} & \text {for}\quad \frac{1}{2}\leq x_{1}\leq c+\frac{1}{2},%
\frac{1}{2}\leq x_{2}\leq c+\frac{1}{2} \\
\phantom {-b,}\frac{1}{2} & \text {otherwise}%
\end{array}%
\right.
\end{equation*}
and determined on the complement of this cylinder by the
periodicity (see Figure~2).
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\vspace{-4cm}
\begin{center}
\footnotesize{Figure 2: The substrate surface $\partial W$.}
\end{center}
We take a mesoscopic length-scale for the size of the pores.
Namely, we choose $a=a_{0}f(L) $, $b=b_{0}f(L) $, $c=c_{0}f(L) $, $%
d=d_{0}f(L) $, where %$f(L)=o(L) $, more precisely
$\lim_{L\rightarrow \infty }f(L)\\ =\infty $ and $\lim
_{L\rightarrow \infty }f(L)/L=0 $. The roughness of the wall is
$r=\lim_{L \rightarrow +\infty}(1+4bc/a^{2})
=1+4b_{0}c_{0}/a_{0}^{2} $.
Finally to describe heterogeneities, we take $W_{1} $ as the part
of the wall $W $ below the plane $x_{3}=-d+1/2 $ and $W_{2} $ as
the part of $W $ above this plane ($0\ \leq d\leq b$).
We use $A_{1} $ and $A_{2} $ to denote the area of the substrate surfaces $%
\partial W_{1} $ and $\partial W_{2} $ and $\bar{A}_{1} $ and $\bar{A}_{2} $
their projection onto the horizontal plane. The respective roughness $r_{1} $%
, $r_{2} $ and concentrations $c_{1} $, $c_{2} $, can then be defined by
\begin{equation}
r_{k}=\frac{A_{k}}{\bar{A}_{k}}\, ,\quad c_{k}=\frac{\bar{A}_{k}}{\bar{A}%
_{1}+\bar{A}_{2}}\, ,\quad k=1,2
\end{equation}
in terms of which the roughness reads $r =r_1 c_1 + r_2 c_2$.
\section{Results}
\setcounter{equation}{0}
Our first result establishes the validity of the generalized
Cassie--Wenzel's equation
for the model defined in the previous section.
\begin{theorem}
\label{T1} Assume that the parameters introduced above satisfy the
conditions
\begin{eqnarray}
C & \equiv & J[1-\max (\frac{1}{2}, \frac{r+c_1-1}{r+2c_1-1})]
\notag \\
& & -|K_{1}|\frac{r_1}{r_1+1} -|K_{2}|\max (\frac{1}{2}, \frac{r_2
c_2- c_2}{r_2 c_2- c_2+c_2-1})
>0
\label{eq:3.2}
\end{eqnarray}
and that the temperature is sufficiently low, namely $\beta C
>5.71 $, then
\begin{equation} \label{eq:3.1}
\Delta \tau =r_{1}c_{1}(\Delta \tau )_{1}^{\text{flat}}+r_{2}c_{2}(\Delta
\tau )_{2}^{\text{flat}}
\end{equation}
\end{theorem}
The condition (\ref{eq:3.2}) (which can be viewed as a condition of smallness
of $|K_{1}|/J$ and
$|K_{2}|/J$) ensures that the configurations \( + \) and \( -
\) are the respective ground states of \( H^{+} \) and \( \, H^{-}
\): $\min _{\sigma }H_{\Omega }^{+}(\sigma )\geq H_{\Omega
}^{+}(-) $ and $\min _{\sigma }H_{\Omega }^{-}(\sigma )\geq
H_{\Omega }^{-}(-) $. Let $h^{\pm }(\sigma )=\lim _{L\rightarrow
\infty }\frac{H_{\Omega }^{\pm }(\sigma )}{(2L+1)^{2}} $ be the
specific energies per unit surface. One has $h^{+}(+)=0 $ and $\,
\, h^{-}(-)=r_{1}c_{1}K_{1}+r_{2}c_{2}K_{2} $.
This implies that the law (%
\ref{eq:3.1}) holds true at the level of ground states.
Indeed, letting $%
\Delta e=\lim _{L\rightarrow \infty }\frac{1}{(2L+1)^{2}}\left[
\min
_{\sigma }H_{\Omega }^{-}(\sigma )-\min _{\sigma }H_{\Omega }^{+}(\sigma )%
\right] $, one has
\begin{equation}
\Delta e=r_{1}c_{1}K_{1}+r_{2}c_{2}K_{2}
\end{equation}
The proof of this result at the level of free energies is given in Section~%
\ref{A}.
%This condition is intimately related to the physics of the
%problem, and one may ask what happens whence increasing $K_1$ and
%$K_2$.
Let us mention the study on Cassie's law proposed in \cite{DT}
whose results do not rely on the knowledge of ground states.
Our second result concerns the homogeneous case. We will assume that $%
K_{1}=K_{2} $. We let $\rho =1+4b_{0}/c_{0} $ be the relative
roughness of the pores and let $c^{\prime }=(c_{0}/a_{0})^{2} $ be the
density of the pores.
\begin{theorem}
\label{T2}
\begin{description}
\item[i)] If $-J/\rho \ln 12^{2}+t+\frac{e^{-6t}}{1-e^{-2t}}\geq 5.71
\end{equation}
It implies
\begin{equation}
\sum _{X:X(\gamma )\geq 1}\left| \Phi ^{\pm }(X)\right| \leq \mu (\gamma )
\end{equation}
As a result of (\ref{eq:A8}) we can write%
\begin{equation}
\ln Z_{W}^{+}(\Omega )-\ln Z_{W}^{-}(\Omega )-\beta K_{1}A_{1}-\beta
K_{2}A_{2}=\sum _{X\in \chi (\Omega )}\left[ \Phi ^{+}(X)-\Phi ^{-}(X)\right]
\end{equation}
By definitions (\ref{eq:A1}) and (\ref{eq:A3}) the contributions of the
contours in the bulk are exactly the same for the $+ $ or $- $ b.c. Thus all
terms with $X $ supported by contours not touching the wall are canceled in
the above difference of the logarithms and only the sum over $X $ containing
contours touching the wall remains. We use $\chi _{W}(\Omega ) $ to denote
the set of all such multi--indexes $X $. Then,
\begin{equation}
\ln Z_{W}^{+}(\Omega )-\ln Z_{W}^{-}(\Omega )-\beta K_{1}A_{1}-\beta
K_{2}A_{2}=\sum _{X\in \chi _{W}(\Omega )}\left[ \Phi ^{+}(X)-\Phi ^{-}(X)%
\right]
\end{equation}
Using the fact that $z^{\pm }(\gamma ) $ are invariant under horizontal
translation by multiples of the periodicity constant $a $ and satisfy the
bound (\ref{eq:A5}), one get,
\begin{eqnarray}
\Delta \tau -r_{1}c_{1}K_{1}-r_{2}c_{2}K_{2} & = & \lim _{L\rightarrow
\infty }\frac{1}{\beta (2L+1)^{2}}\! \sum _{X\in \chi _{W}(\Omega )}\left[
\Phi ^{+}(X)-\Phi ^{-}(X)\right] \notag \\
& = & \lim _{a\rightarrow \infty }\frac{1}{\beta a^{2}}\sum _{X\in \chi
_{W}(\Omega _{a})}\left[ \Phi ^{+}(X)-\Phi ^{-}(X)\right] \label{eq:A13}
\end{eqnarray}
where $\Omega _{a}=V\cap \Lambda _{a} $, with
\begin{equation*}
\Lambda _{a}=\left\{ x\in \mathbb{Z}^{3}:0\leq x_{1}\leq a,0\leq x_{2}\leq
a,|x_{3}|\leq a\right\}
\end{equation*}
Let us now turn to the flat walls. Let $\Omega ^{\prime } $ be a box in the
semi-infinite lattice
\begin{equation*}
\mathbb{L}=\left\{ (x_{1},x_{2},x_{3})\in \mathbb{Z}^{3}:x_{3}>0\right\}
\end{equation*}
and let $\Pi =\partial W^{\text {flat}} $ be the plane $x_{3}=1/2 $. We let $%
Z_{W_{1}^{\text {flat}}}^{\pm }(\Omega ^{\prime }) $ and $Z_{W_{2}^{\text {%
flat}}}^{\pm }(\Omega ^{\prime }) $ be the partition functions corresponding
to the case of the flat walls. We define the contours as before and
introduce the weights
\begin{equation} \label{eq:A14}
z_{j}^{\pm }(\gamma )=e^{-\beta J(|\gamma _{\text {bk}}|\pm K_{j}|\gamma _{%
\text {f}}|)}
\end{equation}
Here $\gamma _{\text {f}}=\gamma \cap \Pi $ and $\gamma _{\text {bk}}=\gamma
\setminus \gamma _{\text {f}} $; we say that $\gamma $\textit{\ touches} $%
\Pi $ if it contains plaquettes of this plane. Then,
\begin{eqnarray}
Z_{W_{i}^{\text {flat}}}^{+}(\Omega ^{\prime }) & = & \sum _{\{\gamma
_{1},...,\gamma _{n}\}_{\text {comp}}}\prod _{i=1}^{n}z_{j}^{+}(\gamma _{i})
\\
Z_{W_{i}^{\text {flat}}}^{-}(\Omega ^{\prime }) & = & e^{-\beta
K_{i}A(\Omega ^{\prime })}\sum _{\{\gamma _{1},...,\gamma _{n}\}_{\text {comp%
}}}\prod _{i=1}^{n}z_{j}^{-}(\gamma _{i})
\end{eqnarray}
Here $\{\gamma _{1},...,\gamma _{n}\}_{\text {comp}} $ are collections of
compatible contours in $\Omega ^{\prime } $ and $A(\Omega ^{\prime }) $ is
the number of bonds $xy $, $x\in \Omega ^{\prime } $, $y\in \mathbb{Z}%
^{3}\setminus \Omega ^{\prime } $, that cross the plane $\Pi $. We let $%
\Phi _{i}^{\pm } $ be the truncated functional associated to the weights (%
\ref{eq:A14}). Then%
\begin{equation}
\ln Z_{W_{i}^{\text {flat}}}^{+}(\Omega ^{\prime })-\ln Z_{W_{i}^{\text {flat%
}}}^{-}(\Omega ^{\prime })-\beta K_{i}A(\Omega ^{^{\prime }})=\sum _{X\in
\chi _{\Pi }(\Omega ^{^{\prime }})}\left[ \Phi _{i}^{+}(X)-\Phi _{i}^{-}(X)%
\right]
\end{equation}
where $\chi _{\Pi }(\Omega ^{^{\prime }}) $ is the set of multi-indexes of $%
\chi (\Omega ^{^{\prime }}) $ whose support intersect the plane $\Pi $.
Using that the weights are now completely invariant with respect to
horizontal translations, we have
\begin{eqnarray}
(\Delta \tau )_{1}^{\text {flat}}-K_{1} & = & \lim _{L\rightarrow \infty }%
\frac{1}{\beta a^{2}}\! \sum _{X\in \chi _{\Pi }(\Omega _{a}^{\prime })}%
\left[ \Phi _{1}^{+}(X)-\Phi _{1}^{-}(X)\right] \notag \\
& = & \sum _{\atop{{X\in \chi _{\Pi }(\mathbb{L})}}{{p\in
X}}}\frac{\left[ \Phi _{1}^{+}(X)-\Phi _{1}^{-}(X)\right] }{\left|
X\cap \Pi \right| }\equiv \mathcal{F}_{1} \label{eq:A18}
\end{eqnarray}
and
\begin{eqnarray}
(\Delta \tau )_{2}^{\text {flat}}-K_{2} & = & \lim _{L\rightarrow \infty }%
\frac{1}{\beta a^{2}}\! \sum _{X\in \chi _{\Pi }(\Omega _{a}^{\prime })}%
\left[ \Phi _{2}^{+}(X)-\Phi _{2}^{-}(X)\right] \notag \\
& = & \sum _{\atop{{X\in \chi _{\Pi }(\mathbb{L})}}{{p\in
X}}}\frac{\left[ \Phi _{2}^{+}(X)-\Phi _{2}^{-}(X)\right] }{\left|
X\cap \Pi \right| }\equiv \mathcal{F}_{2} \label{eq:A19}
\end{eqnarray}
Here $\Omega _{a}^{\prime }=\Lambda _{a}\cap \mathbb{L} $ and the two last
sums in (\ref{eq:A18}) (\ref{eq:A19}) are over multi-indexes whose support
contains a given plaquette of the plane $\Pi $.
Our last step is to compare the R.H.S. of (\ref{eq:A13}) with (\ref{eq:A18})
and (\ref{eq:A19}). To this end, we split the sum over multi-indexes $X\in
\chi _{\Pi _{W}}(\Omega _{a}) $ in three terms $S_{1}(a) $, $S_{2}(a) $, and
$R(a) $. The first term $S_{1}(a) $ is the sum over $X $ that intersect only
one face of the part $(\partial W_{1})_{a} $ of the boundary of the wall
that separates $W_{1} $ from $\Omega _{a} $. Notice that for the
multi-indexes $X $ involved in this sum, one has $\Phi ^{+}(X)=\Phi
_{1}^{+}(X) $. Furthermore, since $(\partial W_{1})_{a} $ has five faces, $%
S_{1}(a) $ is the sum of five terms and each of them divided by the area of
corresponding face will actually equal $\mathcal{F}_{1} $ in the limit $%
a\rightarrow \infty $. Thus
\begin{equation*}
\lim _{a\rightarrow \infty }S_{1}(a)/\beta a^{2}=r_{1}c_{1}\mathcal{F}_{1}
\end{equation*}
The second term $S_{2}(a) $ is the sum over multi--indexes that intersect
only one face of the part $(\partial W_{2})_{a} $ of the boundary of the
wall that separates $W_{2}\, \, $from $\Omega _{a} $. In that case $\Phi
^{\pm }(X)=\Phi _{2}^{\pm }(X) $ and we get analogously to the previous
situation
\begin{equation*}
\lim _{a\rightarrow \infty }S_{2}(a)/\beta a^{2}=r_{2}c_{2}\mathcal{F}_{2}
\end{equation*}
Finally, the reminder $R(a) $ contains the terms where the supports of
multi-indexes intersect at least two faces of $(\partial W)_{a}=(\partial
W_{1})_{a}\cup (\partial W_{2})_{a} $. It thus can be bounded by a constant
times the length of the boundary of faces (for the adjacent ones) plus a
term proportional to the area of the vertical faces times a negative
exponential small correction with a power proportional to the length between
the opposed faces. Thus the ratio $R(a)/\beta a^{2} $ goes to $0 $ as $a $
goes to infinity and we get
\begin{equation*}
\Delta \tau -r_{1}c_{1}K_{1}-r_{2}c_{2}K_{2}=r_{1}c_{1}\mathcal{F}%
_{1}+r_{2}c_{2}\mathcal{F}_{2}
\end{equation*}
giving the desired result.
\section{Proof of Theorem~\ref{T2}}
\label{B} \setcounter{equation}{0}
We first consider the proof of Wenzel's regime stated in
(\ref{eq:3.3}) when \( \left| K\right| 5.71
\end{equation}
We now turn to the proof of Cassie's regime stated in (\ref{eq:3.4})
assuming that \( J/\rho From the definition of \( B^{0}(\sigma ) \) (defined as the
boundary of regions where the configuration differs from the
ground state \( \sigma _{0}^{-} \)), it follows that the
configuration \( \sigma _{\gamma } \) associated to the unique
contour \( \gamma \) satisfy: \begin{equation} H_{\Omega
}^{-}(\sigma _{\gamma })-H_{\Omega }^{-}(\sigma _{0}^{-})=J|\gamma
_{\text {bk}}|+K|\gamma _{\text {pr}}|-K|\gamma _{\text
{f}}|-J|\gamma _{0}|
\end{equation}
where \( \gamma _{\text {pr}}=\gamma \cap \) \( (\partial W)_{\text {pr}}
\),
\( \gamma _{\text {f}}=\gamma \cap (\partial W)_{\text {f}} \) is
the part of the contour that intersect \( (\partial W)_{\text
{pr}} \), respectively \( (\partial W)_{\text {f}} \), \( \gamma
_{0}=\gamma \cap \Pi _{0} \) is the part of the contour that
intersect \( \Pi _{0} \), and \( \gamma _{\text {bk}}=\gamma
\setminus (\gamma _{\text {pr}}\cup \gamma _{\text {fl}}\cup
\gamma _{0}) \) is the complement of these three sets. Introducing
now the weight \( z^{-}(\gamma ) \) as \begin{equation}
z^{-}(\gamma )=e^{-\beta (J|\gamma _{\text {bk}}|+K|\gamma _{\text
{pr}}|-K|\gamma _{\text {f}}|-J|\gamma _{0}|)}
\end{equation}
we get \begin{equation}
Z_{W}^{-}(\Omega )=e^{-\beta \lbrack KA_{\text {f}}(\Omega
)+JA_{0}(\Omega )]}\sum _{\{\gamma _{1},...,\gamma _{n}\}_{\text
{comp}}}\prod _{i=1}^{n}z^{-}(\gamma _{i})
\end{equation}
where \( A_{\text {f}}(\Omega ) \), is the number of bonds \( xy \),
\( x\in \Omega \), \( y\in W \) that crosses \( (\partial
W)_{\text {f}} \) and \( A_{0}(\Omega ) \) is the number of bonds
\( xy \), \( x\in \Omega \), \( y\in \Omega \) that crosses \(
\Pi _{0} \).
For \( Z_{W}^{+}(\Omega ) \) we keep the standard definitions of
contours, so that introducing the weight factors \begin{equation}
z^{+}(\gamma )=e^{-\beta (J|\gamma _{\text {bk}}|+K|\gamma _{\text
{pr}}|+K|\gamma _{\text {f}}|+J|\gamma _{0}|)}
\end{equation}
we get \begin{eqnarray}
Z_{W}^{+}(\Omega ) & = & \sum _{\{\gamma _{1},...,\gamma _{n}\}_{\text
{comp}}}
\prod _{i=1}^{n}z^{+}(\gamma _{i})
=\exp \left( \sum _{X\in \chi (\Omega )}\widetilde{\Phi }^{+}(X)\right)
\nonumber \\
& = & \exp \left( \sum _{S\in \chi (\Omega )}\Phi ^{+}(S)\right)
\label{zpmur}
\end{eqnarray}
where \( \widetilde{\Phi }^{+} \) is the truncated functional associated
to \( z_{\text {pr}}^{+} \), and as above we summed over all
multi-indexes with same support: \[ \Phi ^{+}(S)=\sum _{X:\text
{supp}X=S}\widetilde{\Phi }^{+}(X)\]
Note that the weights \( z^{+}(\gamma ) \) are bounded as \begin{equation}
|z^{+}(\gamma )|\leq e^{-\beta K|\gamma |}
\end{equation}
Since by definitions the weights of the contours not touching the
plane \( \Pi \) are exactly the same for \( + \) or \( - \) b.c.,
we have \begin{equation} Z_{W}^{-}(\Omega )=e^{-\beta [KA_{\text
{f}}(\Omega )+JA_{0}(\Omega )]}\sum _{{\{\gamma _{1},...,\gamma
_{n}\}_{\text {comp}}}\atop {\gamma _{i}\cap \Pi \neq \emptyset
}}\prod _{i=1}^{n}z^{-}(\gamma _{i})\sum _{{\{\gamma _{1}^{\prime
},...,\gamma _{m}^{\prime }\}_{\text {comp}}}\atop {\gamma
_{j}^{\prime }\cap \Pi =\emptyset ,\gamma _{j}^{\prime }\sim
\gamma _{i}}}\prod _{i=1}^{m}z^{+}(\gamma _{j}^{\prime })
\end{equation}
which gives by taking into account (\ref{zpmur}) \begin{eqnarray}
\frac{Z_{W}^{-}(\Omega )}{Z_{W}^{+}(\Omega )} &
= & e^{-\beta [KA_{\text {f}}(\Omega )+JA_{0}(\Omega )]}
\sum _{{\{\gamma _{1},...,\gamma _{n}\}_{\text {comp}}}
\atop {\gamma _{i}\cap \Pi \neq \emptyset }}
\prod _{i=1}^{n}z^{-}(\gamma _{i})
\exp \left( -\sum _{{S:S\in \chi _{\Pi }(\Omega )}
\atop {\text {or}S\nsim \gamma _{i}}}\Phi ^{+}(S)\right)
\nonumber \\
& = & e^{-\beta [KA_{\text {f}}(\Omega )+JA_{0}(\Omega )]}
\sum _{{\{\gamma _{1},...,\gamma _{n}\}_{\text {comp}}}
\atop {\gamma _{i}\cap \Pi \neq \emptyset }}
\prod _{i=1}^{n}z^{-}(\gamma _{i})\prod _{{S:S\in \chi _{\Pi }(\Omega )}
\atop {\text {or}S\nsim \gamma _{i}}}e^{-\Phi ^{+}(S)}
\end{eqnarray}
To expand the last product we introduce again aggregates \( A \)
as families of clusters \( S \) whose support is connected and
define the weights \( \widetilde{\rho }(A)={\prod _{S\in
S}}e^{-\Phi ^{+}(A)}-1 \) to get \begin{equation}
\frac{Z_{W}^{-}({\Omega })}{Z_{W}^{+}({\Omega })}=e^{-\beta
[KA_{\text {f}}(\Omega )+JA_{0}(\Omega )]}\sum _{{\{\gamma
_{1},...,\gamma _{n}\}_{\text {comp}}}\atop {\gamma _{i}\cap \Pi
\neq \emptyset }}\prod _{i=1}^{n}z^{-}(\gamma _{i})\sum _{{\left\{
A_{1},...,A_{m}\right\} _{\text {comp}}}\atop {A_{j}\cap \Pi \neq
\emptyset \; \text {or}\cap A_{j}\nsim \gamma _{i}}}\prod
_{j=1}^{m}\widetilde{\rho }(A_{j})
\end{equation}
Again, we sum over all aggregates with the same support by
defining the weights \[ \rho (S)=\sum _{A=\left\{
S_{1},...,S_{n}\right\} :\cup S_{i}=A}\widetilde{\rho }(A)\]
to get \begin{equation}
\label{zpmmur} \frac{Z_{W}^{-}({\Omega })}{Z_{W}^{+}({\Omega
})}=e^{-\beta [KA_{\text {f}}(\Omega )+JA_{0}(\Omega )]}\sum
_{{\{\gamma _{1},...,\gamma _{n}\}_{\text {comp}}}\atop {\gamma
_{i}\cap \Pi \neq \emptyset }}\prod _{i=1}^{n}z^{-}(\gamma
_{i})\sum _{{\left\{ S_{1},...,S_{m}\right\} _{\text {comp}}}\atop
{S_{j}\cap \Pi \neq \emptyset \; \text {or}\cap S_{j}\nsim \gamma
_{i}}}\prod _{j=1}^{m}\rho (S_{j})
\end{equation}
As in the previous section the weights \( z^{-}(\gamma ) \) have
good decaying properties only for contours touching the wall. To
control the ratio, we proceed as for the study of the surface
tension \( \tau _{+-} \). Namely, we first split the set \(
\{\gamma _{1},...,\gamma _{n}\}_{\text {comp}}\cup \left\{
S_{1},...,S_{m}\right\} _{\text {comp}} \) in connected components.
The components whose support touches the wall \( W \) are
called wall excitations and denoted \( \Gamma ^{\text {wall}} \).
We use \( B_{W} \) to denote the subset of \( \{\gamma
_{1},...,\gamma _{n}\}_{\text {comp}}\cup \left\{
S_{1},...,S_{m}\right\} _{\text {comp}} \) composed of wall
excitations and \( B_{\text {bk}} \) to denote its complement. For
the wall excitations, we define the weights \begin{equation}
\omega (\Gamma ^{\text {wall}})=\prod _{\gamma \in \Gamma ^{\text
{wall}}}z^{-}(\gamma )\prod _{S\in \Gamma ^{\text {wall}}}\rho (S)
\end{equation}
Since \( |z_{\text {f}}^{+}(\gamma )|\leq e^{-\beta K|\gamma |} \),
\( \rho _{\text {pr}}(S) \) satisfy the bound (\ref{borne}). On
the other hand, for these wall excitations one has \[ |z_{\text
{f}}^{-}(\gamma )|\leq \min (e^{-\beta (\frac{J-K}{2})|\gamma
|},e^{-\beta (\frac{\rho |K|-J}{1+\rho })|\gamma |})\]
for any \( \gamma \in \Gamma ^{\text {wall}} \). This implies that
the weights (\ref{wpr}) have good decaying properties for large \(
\beta \).
For the remaining part \( B_{\text {bk}} \) this is not the case
and we have to introduce the excitations differently. We shall
define them as in the study of the auxiliary tension \( (\Delta
\tau )_{\text {aux}} \). Namely, for any \( B\in B_{\text {bk}} \)
and any contour \( \gamma \in B \) touching the plane \( \Pi \),
we will divide the set of plaquettes of \( \gamma \) in two sets.
A plaquette \( p\in \gamma \) is called correct if it lies on the
plane \( \Pi \) or if the vertical lines that crosses it in its
middle crosses only two horizontal plaquettes of \( B_{\text {bk}}
\). All the other plaquettes of \( \gamma \) are called
incorrect: in particular, all the vertical plaquettes are
incorrect ones. We use \( I(B_{\text {bk}}) \) to denote the set
of incorrect plaquettes of \( B_{\text {bk}} \). Then the union of
\( I(B_{\text {bk}}) \) with the set clusters \( S\in B_{\text
{bk}} \) split into connected components \( \left\{
p_{1},..,p_{n};S_{1},...,S_{m}\right\} \) called elementary
excitations. A set \( B_{\text {bk}}=\{\gamma _{1},...,\gamma
_{n}\} \) such that \( \gamma _{i}\cap \Pi \neq \emptyset \) is
in one-to-one correspondence with a set of elementary excitations.
An elementary excitation \( \Gamma ^{\text {el}}=\left\{
p_{1},..,p_{n};S_{1},...,S_{m}\right\} \) is said in the standard
position if there exists a contour \( \gamma \) such that \(
\left\{ p_{1},..,p_{n}\right\} \) is the only elementary
excitation corresponding to \( \gamma \).
Let \( T_{h} \) denotes the vertical shift by a height \( h \): \(
T_{h}(x)=(x_{1},x_{2},x_{3}+h) \), \(
T_{h}(A)=\{x:T^{-1}_{h}(x)\in A\} \) . Then for any elementary
excitation, there is only one shifted excitation \( \Gamma ^{\text
{sh}}=T_{h}(\Gamma ^{\text {el}}) \) which is in the standard
position. We define the weights of any shifted or elementary
excitation by
\begin{equation} \omega (\Gamma ^{\text {sh}})
=\omega(\left\{ p_{1},..,p_{n};S_{1},...,S_{m}\right\} )=e^{-\beta
Jn}\prod _{j=1}^{m}\rho (S_{j})
\end{equation}
With these definitions, we get from (\ref{zpmmur}): \begin{equation}
\frac{Z_{W}^{-}(\Omega )}{Z_{W}^{+}(\Omega )}=e^{-\beta [KA_{\text
{f}}(\Omega )+JA_{0}(\Omega )]}\sum _{{\left\{ \Gamma _{1}^{\text
{wall}},...,\Gamma _{n}^{\text {wall}},\Gamma _{1}^{\text
{sh}},...,\Gamma _{m}^{\text {sh}}\right\} _{\text {comp}}}\atop
{\Gamma _{k}\cap \Pi =\emptyset }}\prod _{i=1}^{n}\omega (\Gamma
_{i}^{\text {wall}})\prod _{j=1}^{m}\omega (\Gamma _{i}^{\text
{sh}})
\end{equation}
where \( \left\{
\Gamma _{1}^{\text {wall}},...,\Gamma _{n}^{\text {wall}},
\Gamma _{1}^{\text {sh}},...,\Gamma _{m}^{\text {sh}}\right\} _{\text
{comp}} \)
are families of (compatible) wall or shifted excitations whose
support touches the plane \( \Pi \). We introduce as before the
multi-indexes \( C \) as non compatible families of excitations
and let \( \Psi \) be the corresponding truncated functional
associated to \( \omega \) to get
\begin{equation} \label{dmur}
\ln \frac{Z_{W}^{-}(\Omega)}{Z_{W}^{+}(\Omega )}+\beta [KA_{\text
{f}}(\Omega )+JA_{0}(\Omega )]=\sum _{C\in \chi _{\Pi
}(\Omega)}\Psi (C)
\end{equation}
Using the fact that \( \Psi (C) \) are invariant under horizontal
translation by multiples of the periodicity constant \( a \), one
gets, \begin{eqnarray}
\Delta \tau -(1-c^{\prime })K-c^{\prime }J
& = & \lim _{L\rightarrow \infty }
\frac{1}{\beta (2L+1)^{2}}\! \sum _{C\in \chi _{\Pi }(\Omega )}\Psi
(C)\nonumber \\
& = & \lim _{a\rightarrow \infty }
\frac{1}{\beta a^{2}}\sum _{C\in \chi _{\Pi }(\Omega _{a})}\Psi (C)
\end{eqnarray}
where \( \Omega _{a}=V\cap \Lambda _{a} \), with \[
\Lambda _{a}=\left\{ x\in \mathbb {Z}^{3}:0\leq x_{1}\leq a,0\leq
x_{2}\leq a,|x_{3}|\leq a\right\} \]
To fulfill the convergence conditions we need to take
\begin{equation}
2\nu \kappa \max (e^{-\beta (\frac{J-K}{2})},e^{-\beta (\frac{\rho
K-J}{1+\rho })},8e(e-1)\kappa \nu ^{2}e^{-\beta K})\leq 1
\end{equation}
Our last step is to compare the R.H.S.\ of (\ref{dmur}) with (\ref{dplat})
and (\ref{ddaux}). We shall take the box \( \Omega ^{\prime } \)
to be the complement of \( \Omega _{c,b} \) in \( \Omega _{a} \)
\[
\Omega ^{\prime }=\Omega _{a}\setminus \Omega _{c,b}\]
Then we split the sum over multi-indexes \( C\in \chi _{\Pi }(\Omega _{a}) \)
in three terms \( S_{\text {f}}(a) \), \( S_{\text {pr}}(a) \),
and \( R(a) \). The first sum \( S_{\text {f}}(a) \) is over
multi-indexes whose support lies inside \( \Omega ^{\prime } \).
Notice that for the multi-indexes \( C \) involved in this sum,
one has \( \Psi (C)=\Psi _{\text {f}}(C) \) and thus \[ \lim
_{a\rightarrow \infty }\frac{S_{\text {f}}(a)}{\beta
a^{2}}=\frac{a^{2}-c^{2}}{a^{2}}\mathcal{F}_{\text
{f}}=(1-c^{\prime })\mathcal{F}_{\text {f}}\]
The second sum \( S_{\text {pr}}(a) \) is over multi-indexes whose
support lies inside \( \Omega _{c,b} \). For the multi-indexes \(
C \) involved in this sum, one has \( \Psi (C)=\Psi _{\text
{pr}}(C) \) and thus \[ \lim _{a\rightarrow \infty }\frac{S_{\text
{f}}(a)}{\beta a^{2}}=\frac{c^{2}}{a^{2}}\mathcal{F}_{\text
{pr}}=c^{\prime }\mathcal{F}_{\text {pr}}\]
Finally the reminder \( R(a) \) contains the multi-indexes whose
support intersects both \( \Omega ^{\prime } \) and \( \Omega
_{c,b} \).\ This term is thus bounded by a constant times the
length of the separation line between and
\( \Pi _{0} \).
Therefore the limit \( R(a)/a^{2} \)goes to zero as \(
a\rightarrow \infty \) and we get \[ \Delta \tau -(1-c^{\prime
})K-c^{\prime }J=(1-c^{\prime })\mathcal{F}_{\text {f}}+c^{\prime
}\mathcal{F}_{\text {pr}}\]
giving the desired result.
The proof of (\ref{eq:3.5}) when \( -J