# Casimir contribution to the interfacial Hamiltonian for 3D wetting

Alessio Squarcini  Max Planck Institute for Intelligent Systems,
Heisenbergstr. 3, D-70569 Stuttgart, Germany  IV. Institut für Theoretische
Physik, Universität Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany
Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 21A,
A-6020, Innsbruck, Austria  José M. Romero-Enrique  Departamento de Física
Atómica, Molecular y Nuclear, Área de Física Teórica, Universidad de Sevilla,
Avenida de Reina Mercedes s/n, 41012 Seville, Spain  Andrew O. Parry
Department of Mathematics, Imperial College London, London SW7 2AZ, United
Kingdom

###### Abstract

Previous treatments of three-dimensional (3D) short-ranged wetting transitions
have missed an entropic or low temperature Casimir contribution to the binding
potential describing the interaction between the unbinding interface and wall.
This we determine by exactly deriving the interfacial model for 3D wetting
from a more microscopic Landau-Ginzburg-Wilson Hamiltonian. The Casimir term
changes the interpretation of fluctuation effects occurring at wetting
transitions so that, for example, mean-field predictions are no longer
obtained when interfacial fluctuations are ignored. While the Casimir
contribution does not alter the surface phase diagram, it significantly
increases the adsorption near a first-order wetting transition and changes
completely the predicted critical singularities of tricritical wetting,
including the non-universality occurring in 3D arising from interfacial
fluctuations. Using the numerical renormalization group we show that, for
critical wetting, the asymptotic regime is extremely narrow with the growth of
the parallel correlation length characterised by an effective exponent in
quantitative agreement with Ising model simulations, resolving a longstanding
controversy.

Interfaces between solids and fluids exhibit a wealth of physics which can
often be studied using effective models motivated by mesoscopic principles
e.g. the capillary-wave model of interfacial wandering [1] and models of
surface growth [2]. However, for critical wetting in 3D systems with short-
ranged forces the details of the interfacial model, and understanding how it
emerges from a microscopic framework, is of crucial importance. Critical
wetting refers to the continuous growth of a liquid phase (for example) at a
solid-gas interface (wall) as the temperature is increased towards a wetting
temperature and is associated with the divergence of a parallel correlation
length, $\xi_{\parallel}$, characterised by an exponent $\nu_{\parallel}$. For
comprehensive reviews see [3, 4, 5, 6]. In general, this can be described
accurately using a simple interfacial model incorporating the surface tension
(or stiffness) and a bind-potential determined by integrating the
intermolecular forces over the volume of liquid. Greater care is required in
3D with short-ranged forces where the binding potential itself arises from
density fluctuations and decays on the scale of the bulk correlation length.
3D is the upper critical dimension and the original renormalization group (RG)
studies predicted strong non-universal critical singularities [7, 8, 9]
implying that $\nu_{\parallel}\approx 3.7$ for Ising-like systems, very
different to the mean-field prediction $\nu_{\parallel}=1$. However, these
have never been seen in experiments [10, 11] nor very careful Ising model
simulations [12, 13, 14, 15, 16, 17] which observe instead an effective
exponent $\nu_{\parallel}^{{\rm eff}}=1.8\pm 0.1$. Allowing for a position
dependence to the stiffness exacerbated the problem since the transition is
driven first-order [18, 19, 20] – a prediction not seen in the simulations and
which also contradicts the expected Nakanishi-Fisher global surface phase
diagrams which connects consistently wetting to surface criticality [21]. A
likely factor in the resolution of this controversy is that the binding
potential is, in general, non-local arising from correlations within the
wetting layer [22, 23, 24, 25]. This can be expressed using a compact
diagrammatic formulation which can be used for walls of arbitrary shape. In
this way, the possibility that the wetting transition is driven first-order is
removed completely so that the global phase diagram is restored.

While this progress is encouraging there is a problem. Since wetting occurs
below the critical temperature, $T_{c}$, it has been assumed that bulk-like
fluctuations are unimportant, and it is only the (thermal) wandering of the
interface that leads to non-classical exponents. This is explicit in
derivations of interfacial Hamiltonians from more microscopic Landau-Ginzburg-
Wilson (LGW) models in which they are identified via a constrained
minimization, equivalent to a mean-field (MF) approximation of the trace over
microscopic degrees of freedom. This means that binding potentials have been
missing an entropic contribution arising from the multiplicity of microscopic
configurations that correspond to a given interfacial one – a feature which is
known to be important in molecular descriptions of free interfaces [26, 27,
28, 29]. The entropic contribution to the binding potential is akin to a
thermal Casimir effect – the force between two walls due to the restriction of
bulk fluctuations in a confined fluid [30, 31, 32, 33]. At $T_{c}$ this force
is long-ranged but it is always present, even away from the critical point
where it decays on the scale of the bulk correlation length [34, 35]. For
short-ranged wetting there is, therefore, an additional entropic or low-
temperature Casimir term in the binding potential, which is a similar range to
the MF contribution. In this paper we determine this using the non-local,
diagrammatic, formalism by performing properly the constrained trace for the
LGW model, which exactly determines the interfacial Hamiltonian for 3D
wetting. We show that the Casimir term plays an important role at wetting
transitions of all orders forcing a reappraisal of the accuracy of MF and
subsequent RG theory and altering even the values of critical exponents.

Our starting point is the LGW Hamiltonian based on a magnetization-like order-
parameter (see [21])

$H[m]=\int
d\mathbf{r}\left(\frac{1}{2}(\boldsymbol{\nabla}m)^{2}+\phi(m)\right)+\int_{\mathcal{S}_{\psi}}d\mathbf{s}\phi_{1}(m(\mathbf{s})),$
(1)

where $\phi(m)$ is a double well potential which we assume has an Ising
symmetry and denote $m_{0}$ the spontaneous magnetization and $\kappa$ the
inverse bulk correlation length. Here $\phi_{1}=-g(m-m_{s})^{2}/2$ is the
surface potential with $g$ the enhancement parameter and $m_{s}$ the favoured
order-parameter at the wall $\mathcal{S}_{\psi}$ with Monge parameterization
$(\mathbf{x},\psi)$. Equivalently $h_{1}=-gm_{s}$ is the surface field.
Minimizing $H[m]$ determines the MF phase diagram which for a planar wall
shows critical wetting (when $-g>\kappa$) and first-order wetting transitions
(for $-g<\kappa$). We can also use it to derive an interfacial model
$H_{I}[\ell]$ with the interfacial co-ordinate determined by a crossing
criterion so that $m(\mathbf{x},\ell(x))=0$ on the interface
$\mathcal{S}_{\ell}$. Formally, this is identified via $\exp(-\beta
H_{I}[\ell])=\int\mathcal{D}^{\prime}m\exp(-\beta H[m])$ where
$\beta=1/k_{B}T$ and the prime denotes a constrained trace over microscopic
degrees of freedom respecting the crossing criterion [36]. This yields

$H_{I}[\ell]=\gamma A_{l}+W[\ell,\psi]\,,$ (2)

where the first term is the surface tension times the interfacial area
describing the free interface (ignoring curvature terms) and $W[\ell,\psi]$ is
the binding potential functional describing the interaction with the wall.

To evaluate the constrained trace it is now customary to ignore bulk
fluctuations and make a MF approximation which identifies
$H_{I}[\ell]=H[m_{\Xi}]$ where $m_{\Xi}$ is the _unique_ profile that
minimizes the LGW Hamiltonian subject to the crossing criterion. Within the
reliable double parabola (DP) approximation
$\phi(m)=\kappa^{2}(|m|-m_{0})^{2}/2$ this can be done analytically. For
example, if the interface is a uniform thickness $\ell$ from a planar wall
($\psi=0$), of lateral area $L_{\parallel}^{2}$, then the binding potential
functional reduces to the binding potential function
$w_{MF}=W_{MF}/L_{\parallel}^{2}$ which has the well-known exponential
expansion [36]

$\frac{w_{MF}(\ell)}{\gamma}\approx-\frac{2tg}{g-\kappa}\textrm{e}^{-\kappa\ell}+\frac{g+\kappa}{g-\kappa}\textrm{e}^{-2\kappa\ell},$
(3)

where $\gamma=\kappa m_{0}^{2}$ is the surface tension and
$t=(m_{0}-m_{s})/m_{0}$ is the temperature-like scaling field for critical
wetting. The minimum of $w_{MF}$ determines the MF wetting layer thickness
while its curvature at this point determines $\xi_{\parallel}$. Both these
lengthscales diverge continuously as $t\to 0$ when $-g>\kappa$. Note that for
tricritical wetting ($g=-\kappa$) and first-order wetting ($-g<\kappa$) it is
necessary to include the next-order decaying exponential term. For non-planar
interfaces (and walls) the non-local MF functional $W_{MF}[\ell,\psi]$ can
also be determined exactly using boundary integral methods based on the Green
function in the wetting layer [24, 25]. However, within the DP approximation,
the constrained trace can be performed determining the exact binding potential
functional

$W[\ell,\psi]=W_{MF}[\ell,\psi]+W_{C}[\ell,\psi]\,,$ (4)

which contains a Casimir correction. The proof of this is rather technical but
is outlined in the Supplementary Material [37]. Full details will be published
elsewhere [38]. Here we limit ourselves to the final result for
$W_{C}[\ell,\psi]$ and the implications for wetting transitions of all orders.
The Casimir contribution can be represented diagrammatically similar to the
terms in the mean-field contribution but has a distinct topology. To this end
we introduce two kernels which connect positions, with respective transverse
co-ordinates s, $\textbf{s}^{\prime}$ (denoted by the open circles) on the
interface (upper wavy line) and wall (lower wavy line). We have

$\parbox{42.67912pt}{\psfig{width=42.67912pt}}=\mathbf{n}(\mathbf{s})\mathbf{\cdot}\frac{\mathbf{s}^{\prime}-\mathbf{s}}{|\mathbf{s}-\mathbf{s}^{\prime}|^{2}}\left(1+\frac{1}{\kappa|\mathbf{s}-\mathbf{s}^{\prime}|}\right)\textrm{e}^{-\kappa|\mathbf{s}-\mathbf{s}^{\prime}|}\,,$
(5)

which was introduced in [25] in the derivation of $W_{MF}[\ell,\psi]$. Here
$\textbf{n}(\textbf{s})$ is the normal at the wall. We also define

$\displaystyle\parbox{42.67912pt}{\psfig{width=42.67912pt}}=\frac{1}{2\pi}\int_{0}^{\infty}dq\
q\frac{g+\kappa_{q}}{g-\kappa_{q}}J_{0}(q\rho)\exp(-\kappa_{q}\ell)\,,$ (6)

where $\rho$ and $\ell$ are, respectively, the transverse and normal
coordinates of $\mathbf{s}-\mathbf{s}^{\prime}$, $J_{0}(z)$ is the Bessel
function of the first kind and zero order and
$\kappa_{q}\equiv\sqrt{\kappa^{2}+q^{2}}$. In terms of these the Casimir term
for a wetting film at a wall of arbitrary shape can be written

$\beta
W_{C}[\ell,\psi]=\frac{1}{2}\parbox{42.67912pt}{\psfig{width=42.67912pt}}-\frac{1}{4}\parbox{42.67912pt}{\psfig{width=42.67912pt}}+\ldots\,,$
(7)

which is our central result. Here the black dots simply imply integration over
the points on the wall and interface with the appropriate measure for the
local area. For thick wetting films, only the first term, which we refer to as
$(\Omega_{C})_{1}^{1}$ (see Supplementary Material [37]), is required.

Figure 1:  The Casimir contribution to the binding potential for a wetting
layer of uniform thickness, for increasing surface enhancements, showing the
qualitative change from attraction to repulsion near the MF tricritical point.
The dotted lines show the comparisons between the full result, Eq. (8), and
the leading term arising from $(\Omega_{C})_{1}^{1}$, which is near exact.

We now focus on wetting at planar walls. Before considering the role of
interfacial fluctuations we must first check if the MF predictions are altered
by determining the Casimir binding potential function
$w_{C}(\ell)=W_{C}/L_{\parallel}^{2}$ for a uniform wetting layer, which adds
to (3). This can be determined exactly as [38]

$\beta w_{C}(\ell)=\frac{1}{4\pi}\int dq\
q\ln\left(1-\frac{g+\kappa_{q}}{g-\kappa_{q}}\textrm{e}^{-2\kappa_{q}\ell}\right).$
(8)

This is similar in form to $w_{MF}(\ell)$ but controlled by $g$ rather than
$t$. For $\kappa\ell\ll 1$, $w_{C}(\ell)\propto 1/\ell^{2}$, which is the
familiar long-ranged Casimir limit [30]. More generally, for $-g>\kappa$ the
potential is repulsive at short-distances, attractive at large distances and
possesses a minimum which diverges continuously as $-g$ approaches $\kappa$.
For $-g<\kappa$, the potential is purely repulsive (see Fig. 1). In the
vicinity of the MF tricritical point, $-g\approx\kappa$, where this
qualitative change occurs, the Casimir potential behaves as

$\beta
w_{C}(\ell)\approx\frac{\textrm{e}^{-2\kappa\ell}}{32\pi\ell^{2}}(1+2(\kappa+g)\ell)\,,$
(9)

which is of a similar range to $w_{MF}(\ell)$. While some aspects of wetting
are unchanged, others are altered completely, even before we consider the role
of interfacial fluctuations. The Nakanishi-Fisher surface phase diagram is
unaffected qualitatively so that, for example, critical wetting still occurs
for $-g>\kappa$ as $t\to 0$ with $\xi_{\parallel}\sim t^{-1}$, as before.
However, there are two significant implications. Firstly, the tricritical
wetting transition is very different since it is the Casimir term that
determines the repulsion. In dimension $d>1$ this decays as
$\textrm{e}^{-2\kappa\ell}/\ell^{\frac{d+1}{2}}$ which, at finite $T$, always
dominates over the higher-order MF contribution. Thus, in dimension $d>3$,
where interfacial fluctuations are irrelevant, it follows that the parallel
correlation length diverges as $\xi_{\parallel}\sim 1/(t|\ln
t|^{\frac{d-1}{2}})$ in contrast to the strict MF prediction
$\xi_{\parallel}\sim 1/t^{\frac{3}{4}}$ which misses the Casimir term. MF is
only recovered on setting $T=0$ or $d=\infty$. There are also consequences for
first-order wetting and, in particular, the value of the film thickness
$\ell_{eq}$ at the transition which, recall, smoothly increases as we follow
the line of wetting transitions toward the tricritical point. MF theory
predicts $\kappa\ell_{eq}\approx-\ln(1+g/\kappa)$ while, in 3D, the Casimir
contribution alters this to

$\ell_{eq}\approx\frac{1}{\sqrt{16\pi\beta\gamma\left(1+\frac{g}{\kappa}\right)}}.$
(10)

The Casimir repulsion therefore dramatically increases the adsorption for
weakly first-order transitions and similarly enhances the parallel correlation
length.

Finally, we consider the non-universality occurring in 3D arising from
interfacial fluctuations, controlled by the wetting parameter
$\omega=k_{B}T\kappa^{2}/4\pi\Sigma$, where $\Sigma$ is the stiffness.
Critical exponents for critical wetting are unchanged since $W_{C}$ is higher-
order than $W_{MF}$. Thus we anticipate that, in the asymptotic critical
regime, $\xi_{\parallel}\sim t^{-\nu_{\parallel}}$, where
$\nu_{\parallel}=1/(1-\omega)$ for $0<\omega<1/2$,
$\nu_{\parallel}=1/(\sqrt{2}-\sqrt{\omega})^{2}$ for $1/2<\omega<2$ and
$\nu_{\parallel}=\infty$ for $\omega>2$ [8, 9]. For tricritical wetting
however the non-universality is different to previous predictions [13, 39] and
is very similar to critical wetting containing logarithmic corrections. For
$0<\omega<1/2$, the equilibrium film thickness and parallel correlation length
diverge as

$\kappa\langle\ell\rangle\approx(1+2\omega)\ln\xi_{\parallel}-\ln\ln\xi_{\parallel}$
(11)

and

$\xi_{\parallel}\sim(t|\ln t|)^{-\nu_{\parallel}},$ (12)

respectively, with $\nu_{\parallel}=1/(1-\omega)$. Note that on setting
$\omega=0$, corresponding to infinite stiffness, these do not recover MF
theory, as has been always assumed previously, but rather, the corrected
results based on minimising the total binding potential $w(\ell)$ allowing for
the Casimir contribution. For $\omega>1/2$, where the tricritical phase
boundary is shifted away from $-g=\kappa$, the exponents are identical to
those for critical wetting.

These features, together with the onset of asymptotic criticality, can be
illustrated most simply by setting $-g=\kappa$ and using the full diagrammatic
structure of the functionals $W_{MF}$ and $W_{C}$ to construct $H_{I}[\ell]$.
When the interface is non-planar the leading order MF attraction and Casimir
repulsion remain local, while the non-local MF repulsion vanishes.
Consequently, the interfacial Hamiltonian is local, $H_{I}[\ell]=\int
d\mathbf{x}\left[\frac{\Sigma(\ell)}{2}(\nabla\ell)^{2}+w(\ell)\right]$,
containing a Casimir modified binding potential

$w(\ell)\approx-t\gamma\textrm{e}^{-\kappa\ell}+\frac{\omega\Sigma}{8}\frac{\textrm{e}^{-2\kappa\ell}}{(\kappa\ell)^{2}}.$
(13)

Here $\Sigma(\ell)=\Sigma+w(\ell)$ is a position dependent stiffness which may
be fully accounted for in the RG analysis, although it plays no significant
role. Note, that for $0<\omega<1/2$ this potential models tricriticality
while, strictly speaking, it models critical wetting for $\omega>1/2$ since
the phase boundary is shifted (although the critical behaviour is identical to
that for tricritical wetting). In Fig. 2 we show the growth of the parallel
correlation length, obtained using the highly accurate numerical, non-linear,
RG [5, 40] for two values of $\omega$, the larger value corresponding to that
pertinent to the 3D Ising model [41, 42]. For $\omega=1/4$ there is excellent
agreement with the predictions Eqs. (11) and (12) over all lengthscales.
However, for $\omega=0.8$ the asymptotic regime, where $\nu_{\parallel}\approx
3.7$, is only reached when $\xi_{\parallel}$ is mesoscopically large. For
thinner wetting layers, for which $10^{2}\lesssim\kappa\xi_{\parallel}\lesssim
10^{3}$, we find an effective exponent $\nu^{\rm{eff}}_{\parallel}\approx 2$,
close to the value $\nu^{\rm{eff}}_{\parallel}=1.8\pm 0.1$ measured in the
Ising model simulations, which corresponded precisely to this range of
lengthscales [16].

Figure 2:  The divergence of the parallel correlation length and wetting film
thickness (see inset) obtained using the numerical RG for tricritical wetting,
with $\omega=1/4$ (red squares), and critical wetting, with $\omega=0.8$
(green circles), allowing for the Casimir term in the binding potential. The
continuous lines correspond to the predicted asymptotic behaviour
$\xi_{\parallel}\sim(t|\ln t|)^{-4/3}$ (see Eq. (12)) and
$\xi_{\parallel}\sim(t|\ln t|^{0.3})^{-3.7}$ (see Ref. [9]), respectively. The
dashed straight line has tangent $\nu_{\parallel}^{\rm{eff}}=2$ illustrating
the effective value of the critical exponent for thinner wetting films.

In summary, in this paper we have pointed out that previous theories of 3D
short-ranged wetting have missed a thermal Casimir, or entropic, contribution
to the binding potential, which we have determined exactly for the LGW model
within the DP approximation. This decays exponentially, similar to the MF
contribution and is qualitatively different for first-order and critical
wetting. Its presence changes the interpretation of thermal fluctuation
effects at wetting transitions which arise both from it and from capillary-
wave-like interfacial wandering. Both are missing in MF descriptions. The
Casimir term strongly affects first-order, critical, and tricritical wetting,
where it alters the exponents in all dimensions, including the non-
universality in 3D when we allow for interfacial fluctuations. Our central
conclusions remain valid beyond the present DP approximation, at least for
thick wetting layers. An entropic contribution will be present for other
systems and we anticipate it will be similar when there are short-ranged
fluid-fluid but long-ranged wall-fluid forces. This is also missing in MF
treatments of wetting and may influence surface phase behaviour in the
vicinity of the critical point [43].

###### Acknowledgements.

J.M.R.-E. acknowledges financial support from Junta de Andalucía through
grants US-1380729 and P20_00816, cofunded by EU FEDER. A.S. acknowledges
LPTMC, Sorbonne Université for a research stay.

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##  Appendix A Supplementary Material

Here we outline the derivation of the Casimir contribution using the boundary
integral method. This is based on the solution of the Green function
$K_{\psi}(\mathbf{r},\mathbf{r}^{\prime})$ in the wetting layer satisfying

$(-\nabla^{2}+\kappa^{2})K_{\psi}=2\kappa\delta(\mathbf{r}-\mathbf{r}^{\prime})$
(14)

together with the boundary condition on $\mathcal{S}_{\psi}$ [24, 25],

$-\frac{1}{g}\hat{\textbf{n}}(\textbf{s}_{\psi})\cdot\nabla
K_{\psi}(\textbf{s},\textbf{s}_{\psi})=K_{\psi}(\textbf{s},\textbf{s}_{\psi})\,.$
(15)

First we write

$H[m]=H[m_{\Xi}]+\Delta H[\delta m]\,,$ (16)

where $\delta m(\mathbf{r})=m-m_{\Xi}$. Here, within the wetting layer

$\Delta H[\delta m]=\int
d\mathbf{r}\left(\frac{1}{2}(\boldsymbol{\nabla}\delta
m)^{2}+\frac{\kappa^{2}}{2}(\delta
m)^{2}\right)-\frac{g}{2}\int_{\mathcal{S}_{\psi}}d\mathbf{s}\,(\delta
m)^{2}\,,$ (17)

is a Gaussian Hamiltonian with a surface term but only containing the
enhancement $g$. The trace over $\delta m$ identifies

$\beta W_{C}=\frac{1}{2}\textrm{tr}\ln[K_{\psi}(K_{\psi}^{\infty})^{-1}]\,,$
(18)

where the superscript corresponds to the unbound interface. The Green function
within the wetting layer can be determined using a rigorous perturbative
procedure by expanding about the free-space solution
$K(\mathbf{r},\mathbf{r}^{\prime})$, which is a simple Yukawa function - see
[24, 25, 38],

$K(\mathbf{r},\mathbf{r}^{\prime})=\frac{\kappa}{2\pi}\frac{\textrm{e}^{-\kappa|\mathbf{r}-\mathbf{r}^{\prime}|}}{|\mathbf{r}-\mathbf{r}^{\prime}|}\,.$
(19)

Combining the perturbative expansion of $K_{\psi}$ with the exact one-loop
result (18) gives

$\beta W_{C}[\ell,\psi]=\frac{1}{2}\sum_{n=1}^{\infty}(\Omega_{C})^{n}_{n}\,,$
(20)

where

$(\Omega_{C})_{n}^{n}\equiv\frac{1}{n}\textrm{tr}(\mathcal{N}^{n})\,,$ (21)

and the operator $\mathcal{N}$ is

$\displaystyle\mathcal{N}(\mathbf{s},\mathbf{s}^{\prime})$
$\displaystyle=\int_{\mathcal{S}_{\psi}}d\mathbf{s}_{1}\int_{\mathcal{S}_{\psi}}d\mathbf{s}_{2}\int_{\mathcal{S}_{\ell}}d\mathbf{s}_{3}\Big{(}K(\mathbf{s},\mathbf{s}_{1})+\frac{1}{g}\mathbf{n}(\mathbf{s}_{1})\mathbf{\cdot}\boldsymbol{\nabla}_{\mathbf{s}_{1}}K(\mathbf{s},\mathbf{s}_{1})\Big{)}X(\mathbf{s}_{1},\mathbf{s}_{2})K(\mathbf{s}_{2},\mathbf{s}_{3})K^{-1}(\mathbf{s}_{3},\mathbf{s}^{\prime})\,.$

In this expression, the inverse $K^{-1}(\mathbf{s},\mathbf{s}^{\prime})$ is
only required on $\mathcal{S}_{\ell}$ while
$X(\mathbf{s},\mathbf{s}^{\prime})$, which is the inverse of

$K(\mathbf{s},\mathbf{s}^{\prime})+(1/g)\mathbf{n}(\mathbf{s})\mathbf{\cdot}\boldsymbol{\nabla}_{\mathbf{s}}K(\mathbf{s},\mathbf{s}^{\prime})-(\kappa/g)\delta(\mathbf{s}-\mathbf{s}^{\prime})\,,$
(22)

(with ${\bf{n}}$ the normal), is only required on $\mathcal{S}_{\psi}$. Each
can be expressed as a series of convolutions involving

$U(\mathbf{s},\mathbf{s}^{\prime})\equiv
K(\mathbf{s},\mathbf{s}^{\prime})-\delta(\mathbf{s}-\mathbf{s}^{\prime})\,,$
(23)

and
$\mathbf{n}(\mathbf{s})\cdot\boldsymbol{\nabla}_{\mathbf{s}}K(\mathbf{s},\mathbf{s}^{\prime})$.
Here we focus on the leading terms in Eq. (20) for large distance between
$\mathcal{S}_{\ell}$ and $\mathcal{S}_{\psi}$ and large radii of curvature as
relevant to discussions of wetting.

We now recast the expression (A) diagrammatically using the same notation as
in [25]

$K(\mathbf{s},\mathbf{s}^{\prime})\equiv\parbox{42.67912pt}{\psfig{width=42.67912pt}}\qquad,\qquad
U(\mathbf{s},\mathbf{s}^{\prime})\equiv\parbox{42.67912pt}{\psfig{width=42.67912pt}}\qquad,\qquad\frac{1}{\kappa}\mathbf{n}(\mathbf{s})\cdot\boldsymbol{\nabla}_{\mathbf{s}}K(\mathbf{s},\mathbf{s}^{\prime})\equiv\parbox{42.67912pt}{\psfig{width=42.67912pt}}$
(24)

The integrals in $\mathcal{N}$ can also be represented diagrammatically. Each
connect the interface (top wavy line) with the wall (bottom wavy line). We
define

$\displaystyle\parbox{42.67912pt}{\psfig{width=42.67912pt}}=\int_{\mathcal{S}_{\psi}}d\mathbf{s}_{1}\left(K+\frac{1}{g}\mathbf{n}\mathbf{\cdot}\boldsymbol{\nabla}K\right)X$
(25)

where $K=K(\mathbf{s},\mathbf{s}_{1})$ and
$X=X(\mathbf{s}_{1},\mathbf{s}^{\prime})$. This new diagram can be written
$\alpha\mathcal{I}+(1-\alpha)\mathcal{J}$ where $\alpha=g/(g-\kappa)$,

$\mathcal{I}=\parbox{42.67912pt}{\psfig{width=42.67912pt}}-\alpha\parbox{42.67912pt}{\psfig{width=42.67912pt}}+\alpha^{2}\parbox{42.67912pt}{\psfig{width=42.67912pt}}+\ldots$
(26)

and

$\mathcal{J}=\parbox{42.67912pt}{\psfig{width=42.67912pt}}-\alpha\parbox{42.67912pt}{\psfig{width=42.67912pt}}+\alpha^{2}\parbox{42.67912pt}{\psfig{width=42.67912pt}}+\ldots\
,$ (27)

where the black dot means integration over the surface. These resum to the
explicit algebraic expression

$\displaystyle\parbox{42.67912pt}{\psfig{width=42.67912pt}}\approx\frac{1}{2\pi}\int_{0}^{\infty}dq\
q\frac{g+\kappa_{q}}{g-\kappa_{q}}J_{0}(q\rho)\exp(-\kappa_{q}\ell),$ (28)

where $\rho$ and $\ell$ are, respectively, the transverse and normal
coordinates of $\mathbf{s}-\mathbf{s}^{\prime}$, $J_{0}(z)$ is the Bessel
function of the first kind and zero order and
$\kappa_{q}\equiv\sqrt{\kappa^{2}+q^{2}}$. The last integral in $\mathcal{N}$
can be expressed

$\int_{\mathcal{S}_{\ell}}d\mathbf{s}_{1}K(\mathbf{s}^{\prime},\mathbf{s}_{1})K^{-1}(\mathbf{s}_{1},\mathbf{s})=-\parbox{42.67912pt}{\psfig{width=42.67912pt}},$
(29)

where the arrow diagram is the same as that appearing in the dictionary (24),
given explicitly by

$-\parbox{42.67912pt}{\psfig{width=42.67912pt}}=\mathbf{n}(\mathbf{s})\mathbf{\cdot}\frac{\mathbf{s}-\mathbf{s}^{\prime}}{|\mathbf{s}-\mathbf{s}^{\prime}|^{2}}\left(1+\frac{1}{\kappa|\mathbf{s}-\mathbf{s}^{\prime}|}\right)\textrm{e}^{-\kappa|\mathbf{s}-\mathbf{s}^{\prime}|}.$
(30)

This leads to the diagrammatic expansion

$\beta
W_{C}[\ell,\psi]=\frac{1}{2}\left(\parbox{42.67912pt}{\psfig{width=42.67912pt}}-\frac{1}{2}\parbox{42.67912pt}{\psfig{width=42.67912pt}}+\ldots\right),$
(31)

which is our central result.