# Meniscus osculation and adsorption on geometrically structured walls

Martin Pospíšil  Department of Physical Chemistry, University of Chemical
Technology Prague, Praha 6, 166 28, Czech Republic;
The Czech Academy of Sciences, Institute of Chemical Process Fundamentals,
Department of Molecular Modelling, 165 02 Prague, Czech Republic  Andrew O.
Parry  Department of Mathematics, Imperial College London, London SW7 2BZ, UK
Alexandr Malijevský  Department of Physical Chemistry, University of Chemical
Technology Prague, Praha 6, 166 28, Czech Republic; The Czech Academy of
Sciences, Institute of Chemical Process Fundamentals, Department of Molecular
Modelling, 165 02 Prague, Czech Republic

###### Abstract

We study the adsorption of simple fluids at smoothly structured, completely
wet, walls and show that a meniscus osculation transition occurs when the
Laplace and geometrical radii of curvature of locally parabolic regions
coincide. Macroscopically, the osculation transition is of fractional, $7/2$,
order and separates regimes in which the adsorption is microscopic, containing
only a thin wetting layer, and mesoscopic, in which a meniscus exists. We
develop a scaling theory for the rounding of the transition due to thin
wetting layers and derive critical exponent relations that determine how the
interfacial height scales with the geometrical radius of curvature. Connection
with the general geometric construction proposed by Rascón and Parry is made.
Our predictions are supported by a microscopic model density functional theory
(DFT) for drying at a sinusoidally shaped hard-wall where we confirm the order
of the transition and also an exact sum-rule for the generalized contact
theorem due to Upton. We show that as bulk coexistence is approached the
adsorption isotherm separates into three regimes: a pre-osculation regime
where it is microscopic, containing only a thin wetting layer, a mesoscopic
regime, in which a meniscus sits within the troughs, and finally another
microscopic regime where the liquid-gas interface unbinds from the crests of
the substrate.

##  I Introduction

Capillary condensation nakanishi81 ; nakanishi83 ; evans84 ; evans85 ; evans86
; hend ; gelb , wetting sullivan ; dietrich ; row ; schick ; forgacs ; bonn
and wedge-filling hauge ; rejmer ; wood99 ; abraham02 ; delfino ; binder03 ;
bernardino ; our_prl ; our_wedge  are but a few examples of surface phase
transitions in which the surface tension plays a crucial role determining the
location of the phase boundary and even the nature of the transition itself.
Recent studies of capillary condensation in open slits, possessing corners or
edges, have also highlighted another phase transition involving the depinning
of the meniscus, located at the end(s) of the capillary as the pressure is
increased md1 ; md2 . At a macroscopic level, meniscus depinning is continuous
with the order of the transition depending on the wetting properties of the
confining walls. For example, it is third-order if the walls are completely
wet and second-order if they are partially wet. In each case, the transition
is rounded or smoothed by a mesoscopic length scale, specifically the parallel
correlation length, associated with wetting layers at the capillary walls. The
cross-over scaling describing the rounding of meniscus depinning shows a
remarkable consistency with the underlying theory of wetting transitions,
including conjectured critical exponent relations, giving physically intuitive
results for mesoscopic corrections to macroscopic predictions for the
adsorption at the phase boundary.

In this paper we point out that there is another type of phase transition
involving a meniscus in a different confining geometry, which can be
understood macroscopically, which is also rounded by mesoscopic wetting length
scales. We refer to this as meniscus osculation and is associated with the
vanishing of the meniscus, that is the change from macroscopic to microscopic
adsorption, when its curvature coincides with the geometrical curvature of a
confining sculpted wall that is completely wet. As we shall show,
macroscopically, meniscus osculation is continuous but of fractional order,
different to depinning, and that the rounding, while also due to wetting
behaviour, is described by a different cross-over scaling theory. This cross-
over scaling will allow us to make connection with very general expectations
of how substrate geometry influences wetting behaviour.

Our paper is arranged as follows. In the next section we recall briefly the
macroscopic and cross-over scaling theory for meniscus depinning, emphasizing
the underlying consistency with the theory of (complete) wetting. We then turn
to meniscus osculation beginning first with the macroscopic description before
discussing the rounding due to complete wetting layers. This allows us to make
a connection with a more general proposal for how geometry influences
adsorption on structured surfaces nature . In the second part of our paper, we
present the results of a microscopic model Density Functional Theory (DFT)
study of adsorption at a corrugated, sinusoidal, wall which shows not only
meniscus osculation but also a modified complete wetting transition. Indeed,
as the chemical potential approaches bulk coexistence, we show that the
adsorption changes from being determined by microscopic length scales (pre-
osculation) to macroscopic (geometry dominated) length scales and back to
microscopic length scales when it eventually unbinds from the crests of the
sinusoid.

##  II Meniscus Depinning and Meniscus Osculation

Before describing the osculation transition, we recall the macroscopic and
mesoscopic cross-over scaling theory for meniscus depinning in an open
capillary, following closely the presentation in Refs. md1 ; md2 . This will
act as a guide for developing and interpreting the scaling theory for the
osculation transition occurring in parabolic and sinusoidally corrugated
geometries.

###  II.1 Meniscus Depinning

####  II.1.1 Macroscopics

Figure 1: Schematic illustration of two possible condensed capillary liquid
phases in an open slit. In the top panel (a) the two circular menisci of the
Laplace radius of curvature are pinned at the upper edges which they meet at
an edge contact angle $\theta_{e}$. The bottom of the menisci meets the
horizontal, lower, wall at the equilibrium contact angle $\theta$. In the
lower panel (b) the two circular menisci are unpinned, spilling out into the
right-angle corners and meet both the vertical and lower walls at the contact
angle $\theta$.

Consider an open capillary, made from two opposing planar walls separated by a
distance $L$. The bottom wall is of infinite extent while the top wall is of
finite length $H$, which for our present purposes we suppose is much larger
than $L$. Each end of the capillary is a right-angle edge and is in contact
with a bulk reservoir of gas at temperature $T$, far below the bulk critical
temperature $T_{c}$, and pressure $p$ (or chemical potential $\mu$) below bulk
saturation $p_{\rm sat}$ (see Fig. 1). We denote $\delta p=p_{\rm sat}-p$. All
the walls are considered completely wet by the liquid phase corresponding to
an equilibrium contact angle $\theta=0$. We suppose that the pressure is
sufficiently close to bulk saturation that the fluid within the capillary has
already condensed to a liquid-like phase. For very long capillaries, this
capillary condensation will occur near $\delta p_{\rm cc}\approx 2\gamma/L$,
as dictated by the macroscopic Kelvin equation, where $\gamma$ is the surface
tension of the liquid-gas interface. The full $H$ dependence of the pressure
shift $\delta p_{\rm cc}$ can be determined but is not needed for the present
purposes md1 ; md2 . The capillary liquid phase is characterized by two
menisci each of which are arcs of circles of Laplace radius $R=\gamma/\delta
p$ located near the ends of the capillary. Since the equilibrium contact angle
$\theta=0$ these meet the bottom wall tangentially. However, there are two
possibilities for how they attach to the top wall. For $\delta p>\delta p_{\rm
md}$, with $\delta p_{\rm md}=\gamma/L$, the menisci are pinned at the corners
and are characterized by an edge contact angle $\theta_{e}$. This is pressure
dependent, given by

$\theta_{e}=\cos^{-1}\left(\frac{L-R}{R}\right)\,.$ (1)

As the pressure is increased, the edge contact angle increases until at
$\delta p=\delta p_{\rm md}$, equivalent to $R=L$, it reaches its maximum
value $\theta_{e}=\pi/2$, at which point it is tangential to the vertical
sides of the upper wall and hence the menisci are no longer pinned. Clearly,
at the point of this depinning the menisci are each quarter circles which just
fit inside the open ends of the capillary. For $\delta p<\delta p_{\rm md}$
the menisci are unpinned and meet the side walls tangentially at a distance
$R-L$ above the corners. For complete wetting, meniscus depinning is a
continuous third-order phase transition. This is most easily seen by
determining the excess adsorption $\Delta\Gamma$ beyond the contribution
$\Delta\rho HL$ from the volume of liquid within the capillary, with
$\Delta\rho$ the difference in the bulk liquid and gas densities. For the
pinned phase, with $R<L$ this is given by

$\Gamma=2\Delta\rho R^{2}\left[\sin\theta_{e}+\frac{\sin
2\theta_{e}}{4}+\frac{1}{2}(\theta_{e}-\pi)\right]\,,$ (2)

while in the unpinned phase, for which $R>L$, it is given, trivially, by

$\Gamma=2\Delta\rho\left(1-\frac{\pi}{4}\right)R^{2}\,.$ (3)

It is then easy to check that the second derivative of the adsorption is
discontinuous at the depinning transition satisfying

$\frac{\partial^{2}\Gamma}{\partial
R^{2}}=\left\\{\begin{array}[]{cc}\Delta\rho(4-\pi)\,;&\delta p=\delta
p_{md}^{-}\,,\\\ \Delta\rho(2-\pi)\,;&\delta p=\delta
p_{md}^{+}\,.\end{array}\right.$ (4)

This is associated with a discontinuity in the third-derivative of the grand
potential (per unit length of the capillary) showing that the transition is
indeed third-order, at the present macroscopic level.

####  II.1.2 Mesoscopics

The third-order meniscus depinning transition is rounded by length scales
associated with complete wetting layers which coat the bottom and side walls.
The mechanism for this rounding is different to the standard finite-size
scaling theory of the rounding of phase transitions when the available volume
is finite (or the dimensionality is below that of the lower critical
dimension) which is due to fluctuations barber ; privman . The rounding of the
meniscus depinning transition is not due to fluctuation effects but rather due
to mesoscopic length scales which smooth the transition from pinned to
unpinned states. Meniscus depinning is not associated with coexisting phases,
symmetry breaking or a diverging order-parameter and therefore there cannot be
any true critical behaviour. The rounding of this transition is present as
soon as we go beyond macroscopics, for example, in mean-field DFT or even
simpler interfacial models which accommodate the presence of wetting layers.
The length scales of relevance here are the wetting layer thickness
$\ell_{\pi}\sim\delta p^{-\beta_{s}^{\rm co}}$ and parallel correlation length
$\xi_{\parallel}\sim\delta p^{-\nu_{\parallel}^{\rm co}}$, arising from
interfacial fluctuations, which characterize wetting layers at planar walls,
close to bulk saturation. We note that the critical exponents, depend on the
range of the intermolecular forces, but always satisfy the exact exponent
relation sullivan ; dietrich ; row ; schick ; forgacs ; bonn

$1+\beta_{s}^{\rm co}=2\nu_{\parallel}^{\rm co}\,,$ (5)

which is of relevance to the cross-over scaling theory for the rounding of the
meniscus depinning transition. This exponent relation is the consequence of an
exact sum rule which determines that for wetting transitions
$\partial\Gamma/\partial\mu\propto\xi_{\parallel}^{2}$ evans_tar84 . As
described above, macroscopically, the depinning transition occurs when $R=L$ .
Certainly, we can anticipate that the presence of the wetting layer at the
bottom wall decreases the effective width of the slit by $\ell_{\pi}$.
However, of greater relevance is the rounding arising from the parallel
correlation $\xi_{\parallel}$, associated with the wetting layer at the
vertical side walls, which smooths the point of contact of the upper part of
the meniscus with the corner (see Fig. 2). Since $\xi_{\parallel}$ is larger
than $\ell_{\pi}$ we can therefore expect that the depinning begins to occur
when

$R-L\approx\xi_{\parallel}\,.$ (6)

This simple, finite-size scaling, observation is the basis for the cross-over
theory. The purely macroscopic result (4) implies that the grand potential
contains a singular contribution $\Omega_{\rm sing}=\gamma(R-L)^{3}/6L^{2}$.
To allow for the rounding due to complete wetting layers we modify this by a
multiplicative scaling function $W_{\rm md}(x)$ whose argument is the
dimensionless scaling variable $x=(R-L)/\xi_{\parallel}$. Thus, the
appropriate scaling ansatz is

$\Omega_{\rm sing}=\frac{(R-L)^{3}}{L^{2}}W_{\rm
md}\left(\frac{R-L}{L^{\nu_{\parallel}^{\rm co}}}\right)\,,$ (7)

where we have ignored all constants and metric factors and substituted that,
near the depinning transition, the correlation length scales with the slit
width as $\xi_{\parallel}\approx L^{\nu_{\parallel}^{\rm co}}$. We require
that $W_{\rm md}(x)\to 0$ as $x\to\infty$ and $W(x)\to 1$ as $x\to-\infty$
which represent the macroscopic unpinned and pinned states, respectively. The
form of $W_{\rm md}(x)$ describes the smooth crossover between these two
states when the mesoscopic wetting length-scale $\xi_{\parallel}$ is allowed
for. In order that $\Omega_{\rm sing}$ is non-zero at the macroscopic meniscus
depinning transition, $R=L$, we require that $W_{\rm md}(x)\propto 1/x^{3}$,
which leads to a singular or mesoscopic contribution. The derivative of
$\Omega_{\rm sing}$ w.r.t. $\delta p$ determines the singular contribution to
the adsorption, over and above the macroscopic contribution $\Gamma\propto
L^{2}$. Since $\partial\Omega/\partial\delta p\propto
R^{2}\partial\Omega/\partial R$ it follows that the adsorption contains a
singular contribution

$\Gamma_{\rm sing}=(R-L)^{2}\Lambda_{\rm
md}\left(\frac{R-L}{L^{\nu_{\parallel}^{\rm co}}}\right)\,,$ (8)

where $\Lambda_{\rm md}(x)$ is a suitable new scaling function. We require
that $\Lambda_{\rm md}(x)\to 0$ as $x\to\infty$, $\Lambda_{\rm md}(x)\to 1$ as
$x\to-\infty$ and $\Lambda_{\rm md}(x)\propto 1/x^{2}$ as $x\to 0$. It follows
that exactly at the (macroscopic) depinning phase boundary, $R=L$, the excess
adsorption contains a mesoscopic contribution $\Gamma_{\rm sing}\propto
L^{2\nu_{\parallel}^{\rm co}}$ in addition to the leading-order macroscopic
term $\Gamma=2\Delta\rho\left(1-\frac{\pi}{4}\right)L^{2}$ determined earlier.
Using the exact exponent relation (5) it follows that the mesoscopic term can
be written

$\Gamma_{\rm sing}\propto\ell_{\pi}L\,,\hskip 28.45274ptR=L\,,$ (9)

which is, of course, simply the additional contribution to the adsorption from
the meniscus when we shift its position by the thickness of a wetting layer,
$\ell_{\pi}\sim L^{\beta_{s}^{\rm co}}$ coating the side walls (see Fig. 2).
Explicitly, for systems with dispersion forces this yields $\Gamma_{\rm
sing}\propto L^{4/3}$.

Figure 2: Schematic illustration of the length scales determining the
mesoscopic rounding of the continuous meniscus depinning transition, due to
wetting layers adsorbed along the bottom and side walls. The macroscopic phase
boundary $R=L$ for meniscus depinning is rounded by the wetting layer
thickness $\ell_{\pi}$ and parallel correlation length $\xi_{\parallel}$ along
the bottom and top walls, respectively. The gray area depicts an additional
contribution to the excess adsorption $\Gamma\propto\ell_{\pi}L$ due to the
presence of the wetting layers.

Before moving on to consider the meniscus osculation transition we point out
that the above cross-over scaling theory has a remarkable internal consistency
which appears to connect, very naturally, the macroscopics of the meniscus
with the microscopic physics associated with wetting behaviour. Suppose, for
example we did not know what length scale was responsible for rounding the
depinning transition. Instead, we simply marry the macroscopic result that
$\Gamma\approx(R-L)^{2}$ with the direct geometrical requirement that at $R=L$
there must be a higher order contribution $L\ell_{\pi}$, due to the wetting
layers at the side walls which just shift the position of the meniscus. To do
this we, we write $\Gamma_{\rm sing}=(R-L)^{2}\Lambda_{\rm
md}({(R-L)}/\lambda_{\rm md})$ which contains an, as yet, undetermined length
scale $\lambda_{\rm md}$ responsible for the rounding. At $R=L$, it follows
that the singular contribution must scale as $\lambda_{\rm md}^{2}$ which we
identify with $L\ell_{\pi}$. Substituting that $\ell_{\pi}\sim\delta
p^{-\beta_{s}^{\rm co}}$ and that $R=\gamma/\delta p$ it follows that the
length scale responsible for the rounding, must behave as $\lambda_{\rm
md}\sim\delta p^{-(1+\beta_{s}^{\rm co})/2}$. Using the exact exponent
relation (5) it follows that length scale responsible for the rounding must be

$\lambda_{\rm md}=\xi_{\parallel}\,,$ (10)

consistent with Eq. (6). This is a remarkable self consistency – adding a
geometrical shift to the adsorption, due to the wetting layer thickness
$\ell_{\pi}$, we have generated a length scale, associated with wetting, which
is the parallel correlation length. In other words, crossover scaling demands
that the exponent relation $1+\beta_{s}^{\rm co}=2\nu_{\parallel}^{\rm co}$ is
true.

###  II.2 Osculation transition

####  II.2.1 Macroscopics

Figure 3: Geometrical construction determining the macroscopic meniscus of in
a parabolic, completely wet wall. The dashed circle is of $R=\gamma/\delta p$
and connects with the wall tangentially at points $(-x_{0},z_{0})$ and
$(x_{0},z_{0})$. The height of the meniscus above the wall is $\ell_{0}$. The
construction is only possible if $R$ is larger than the geometrical radius of
curvature at the bottom., $R_{w}$.

We now turn attention to the phenomena of meniscus osculation and suppose that
the confining wall has a cross-section of a parabola

$\psi(x)=\frac{x^{2}}{2R_{w}}\,,$ (11)

with translational invariance assumed in the other direction. We restrict
ourselves entirely to walls that are completely wet corresponding to
equilibrium contact angle $\theta=0$. The wall is again supposed to be in
contact with a bulk gas at subcritical temperature $T$ and pressure $p<p_{\rm
sat}$. As the pressure is increased to bulk coexistence the adsorption
diverges due to the growth of a meniscus near the bottom of the parabola. At a
macroscopic level this once again takes the shape of a circular arc of Laplace
radius $R=\gamma/\delta p$ which meets the walls tangentially at $\pm x_{0}$
at height $z_{0}$. We denote by $\ell_{0}$ the height of the meniscus above
the bottom (see Fig. 3). These length scales are trivially determined yielding

$\displaystyle x_{0}$ $\displaystyle=$
$\displaystyle\sqrt{R^{2}-R_{w}^{2}}\,,$ (12) $\displaystyle z_{0}$
$\displaystyle=$ $\displaystyle\frac{R^{2}-R_{w}^{2}}{2R_{w}}\,,$ (13)
$\displaystyle\ell_{0}$ $\displaystyle=$
$\displaystyle\frac{(R-R_{w})^{2}}{2R_{w}}\,.$ (14)

As pointed out in Ref. nature , as $\delta p\to 0$ these length scales diverge
with universal exponents which are geometry-dominated, independent of the
range of the intermolecular forces. More generally, for walls which have a
power-law cross-section $\psi(x)\propto|x|^{\phi}$, the divergence of the
adsorption as $\delta p\to 0$ is geometry dominated provided that
$\phi>\beta_{s}^{\rm co}/\nu_{\parallel}^{\rm co}$. For $\phi<\beta_{s}^{\rm
co}/\nu_{\parallel}^{\rm co}$ the adsorption diverges similar to complete
wetting at a planar wall which depends strongly on the range of intermolecular
forces. Returning to the parabola, we focus here on the _disappearance_ of the
meniscus as the pressure is reduced to point at which

$R=R_{w}\,.$ (15)

We refer to this as meniscus osculation. Macroscopically, for $R<R_{w}$ there
is no adsorption of liquid at the walls since no meniscus can be fitted into
the geometry. More generally, beyond macroscopics, meniscus osculation
separates regimes in which the adsorption is geometry-dominated and
microscopic, respectively. Associated with the osculation is a singular
contribution to the grand potential $\Omega$ per unit length of the wall. At a
purely macroscopic level this is determined by

$\Omega=\delta pA+\gamma(\ell_{m}-\ell_{w})\,,$ (16)

where $A$ is the area of liquid, $\ell_{m}$ is the meniscus length and
$\ell_{w}$ is contact length of the liquid with the wall. Again, these are
very simply determined as

$\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{x_{0}}{3R_{w}}(2R^{2}+R_{w}^{2})-R^{2}\sin^{-1}\left(\frac{x_{0}}{R}\right)\,,$
(17) $\ell_{m}=2\sin^{-1}\left(\frac{\sqrt{R^{2}-R_{w}^{2}}}{R}\right)$ (18)

and

$\ell_{w}=\frac{R}{R_{w}}\sqrt{R^{2}-R_{w}^{2}}+R_{w}\sinh^{-1}\left(\frac{\sqrt{R^{2}-R_{w}^{2}}}{R_{w}}\right)\,.$
(19)

This implies that the grand potential is given explicitly by

$\displaystyle\frac{\Omega}{\gamma}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{R^{2}-R_{w}^{2}}}{3R_{w}}\left(\frac{R_{w}^{2}}{R}-R\right)+R\sin^{-1}\left(\frac{\sqrt{R^{2}-R_{w}^{2}}}{R}\right)$
(20) $\displaystyle-
R_{w}\sinh^{-1}\left(\frac{\sqrt{R^{2}-R_{w}^{2}}}{R_{w}}\right)\,.$

As the pressure is decreased, and $R$ reduces to $R_{w}$, this exhibits the
(macroscopic) critical singularity

$\frac{\Omega}{\gamma}\approx-\frac{16\sqrt{2}}{105}\frac{(R-R_{w})^{\frac{7}{2}}}{R_{w}^{\frac{5}{2}}}\,,$
(21)

which is of fractional, higher-order, than the third-order meniscus depinning.
The derivative of the grand potential determines the associated singularity in
the adsorption which disappears as

$\Gamma_{\rm
sing}\approx\frac{8\sqrt{2}}{15}\frac{\Delta\rho(R-R_{w})^{\frac{5}{2}}}{\sqrt{R_{w}}}\,,$
(22)

which is simply $\Delta\rho$ times the meniscus area $A$. For $R<R_{w}$ there
is no macroscopic adsorption.

Meniscus osculation is closely related to the phenomena of capillary emptying
in horizontal capillaries, under the influence of gravity, of elliptical
cross-section empty . Specifically, the rich phase behaviour associated with
emptying is related to whether menisci, which now represent the cross-
sectional shape of horizontal liquid tongues, can be locally inscribed within
the ellipse – see e.g. the cross-sections shown in Figs. 1 and 2 in Ref. empty
.

####  II.2.2 Mesoscopics

Similar to depinning, the continuous osculation transition must be rounded by
length scales due to mesoscopic complete wetting layers. There will always be
some residual microscopic adsorption even when $R<R_{w}$ implying that the
cross-over from the geometry-dominated regime, $R>R_{w}$, must be smooth since
there is neither symmetry-breaking nor any diverging order parameter or length
scale. For the meniscus depinning the rounding of the macroscopic phase
boundary $R=L$ can be understood directly, by considering how the macroscopic
meniscus attaches to the edge of the opening when wetting layers are present,
leading to $R-L\approx\xi_{\parallel}$. This line of reasoning is not so
obvious for the rounding at osculation since the meniscus itself is
disappearing. The simplest length scale and observable to focus on is the
value of the interfacial height $\ell_{0}$ when $R=R_{w}$ for which there is
no macroscopic contribution. We wish to determine how, exactly at osculation,
$R=R_{w}$, the interfacial height scales with radius of curvature and
introduce a new osculation critical exponent, $\beta_{\rm osc}$, to
characterise this;

$\ell_{0}\approx R_{w}^{\beta_{\rm osc}}\,,\hskip 28.45274ptR=R_{w}\,,$ (23)

where we have omitted any unimportant dimensional pre-factors. To determine
the osculation exponent, we suppose that in the vicinity of the osculation
transition the interfacial height shows crossover scaling

$\ell_{0}=\frac{(R-R_{w})^{2}}{2R_{w}}\mathcal{L}_{\rm
osc}\left(\frac{R-R_{w}}{\lambda_{\rm osc}}\right)\,,$ (24)

with $\lambda_{\rm osc}$ the undetermined rounding length scale and
$\mathcal{L}_{\rm osc}(x)$ the appropriate scaling function. The macroscopic
limit, corresponding to Eq. (14), is recovered by imposing that
$\mathcal{L}_{\rm osc}(x)\to 1$, as $x\to\infty$. We also require that
$\mathcal{L}_{\rm osc}(x)\sim x^{-2}$, as $x\to 0$, in order that at
osculation the value of $\ell_{0}$ remains finite implying
$\ell_{0}\propto\lambda_{\rm osc}^{2}/R_{w}$. Since the wall is completely wet
we also require that this diverges as $R_{w}\to\infty$, i.e. as we follow the
line of osculation transitions to bulk coexistence. This means that we can
eliminate the possibility that the rounding length scale is the planar wetting
layer thickness $\ell_{\pi}$, since in that case $\ell_{0}$ does not diverge.
In addition, the identification $\lambda_{\rm osc}\approx\xi_{\parallel}$,
while appropriate for meniscus depinning, is unsatisfactory since, on using
the exact exponent relation (5), this only yields $\ell_{0}\approx\ell_{\pi}$.
This is most likely only a lower bound since it is physically reasonable that
the residual interfacial height at osculation is greater than the planar
wetting layer thickness at the same pressure.

To identify the rounding length scale we instead focus on the value of
$\ell_{0}$ deep in the pre-osculation regime, $R_{w}\gg R$, and ensure that
our scaling ansatz is compatible with the necessary curvature induced
enhancement of the adsorption. For wetting on spheres and cylinders of radius
$R_{c}$ (say), which have a negative curvature, it is well known that the
increase in the surface tension contribution thins the wetting layer
equivalent to a shift in the effective value of the partial pressure from
$\delta p$ to $\delta p+\gamma/R_{c}$ holyst ; bieker ; stewart ; morgan ;
nold . For the parabola it is therefore reasonable to expect that when
$R_{w}\gg R$ the local interface height is enhanced by the positive curvature
so that $\ell_{0}$ is the same as the planar wetting layer thickness but
evaluated at an effective reduced partial pressure $\delta p-\gamma/R_{w}$,
i.e. $\ell_{0}\to(1/R-1/R_{w})^{-\beta_{s}^{co}}$. This requirement is highly
restrictive and is only compatible with the crossover scaling ansatz (24)
provided that the scaling function $\mathcal{L}_{\rm
osc}(x)\propto|x|^{-2-\beta_{s}^{co}}$, as $x\to-\infty$, and identifies that
the rounding length-scale scales with $R_{w}$ and $R$ according to

$\lambda_{\rm osc}^{2+\beta_{s}^{\rm co}}=R_{w}^{1+\beta_{s}^{\rm
co}}R^{\beta_{s}^{\rm co}}$ (25)

where again we have omitted unimportant microscopic length scales. Using this
value for $\lambda_{\rm osc}$ it follows that the osculation and wetting
exponents are related via

$\beta_{\rm osc}=\frac{3\beta_{s}^{\rm co}}{2+\beta_{s}^{\rm co}}$ (26)

which is the central prediction of our mesoscopic scaling theory. In 3D and
with dispersion forces, and also in 2D with short-ranged forces, for which
$\beta_{s}^{\rm co}=1/3$ this predicts that $\beta_{\rm osc}=3/7$ pointing to
a rather non trivial interplay between geometry and wetting at osculation. In
3D and with short-ranged forces it is likely that the prediction $\beta_{\rm
osc}=0$ corresponds to a logarithmic dependence of $\ell_{0}$ on $R_{w}$.

Figure 4: A schematic illustration of three possible adsorption regimes of a
completely wet ($\theta=0$) sinusoidal wall: At low-adsorption, pre-osculation
regime (a), far away from bulk coexistence, a microscopic amount of liquid
coats the wall and the liquid-gas interface follows the wall shape. Within the
second regime (b), the voids of the wall become gradually filled with liquid.
In the third regime (c), the liquid-gas interface unbinds from the wall and
its height diverges as the coexistence is approached, similarly, but not
identically, to planar walls. While the first and third regimes are governed
by microscopic forces, the second regime is controlled by the wall geometry
and can be described macroscopically.

Figure 5: Schematic illustration of (a) geometrical construction for
macroscopic meniscus in a sinusoidal wall; (b) modified, RP construction where
the wall is first coated by a uniform layer of thickness $\ell_{\pi}$.

Having identified the rounding length scale $\lambda_{\rm osc}$ the crossover
scaling theory can be applied to other quantities. For example, we can expect
that the singular contribution to the grand potential per length scales as

$\Omega=\frac{(R-R_{w})^{7/2}}{R_{w}^{5/2}}W_{\rm
osc}\left(\frac{R-R_{w}}{\lambda_{\rm osc}}\right)\,,$ (27)

where we have ignored the pre-factors. The scaling function $W_{\rm osc}(x)$
must satisfy $W_{\rm osc}(\infty)=1$ in order to recover the macroscopic
singularity in the geometry-dominated regime. On approaching the osculation
transition at $R=R_{w}$, the scaling function must also behave as $W_{\rm
osc}\propto x^{-7/2}$ in order to leave a finite, singular contribution, which
we define as

$\Omega\approx R_{w}^{2-\alpha_{\rm osc}}\,,$ (28)

This identifies the exponent as

$2-\alpha_{\rm
osc}=\frac{3}{2}\left(\frac{3\beta_{s}^{co}-1}{2+\beta_{s}^{co}}\right)\,,$
(29)

again, relating it to the singularities at complete wetting. This exponent
relation is noteworthy because it predicts that $2-\alpha_{\rm osc}=0$ when
$\beta_{s}^{co}=1/3$, as is pertinent to 3D systems with dispersion forces and
2D systems with short-ranged forces, and is presumably indicative of a
marginal logarithmic singularity. We mention this because precisely the same
marginal singularity is predicted for meniscus depinning for systems with
dispersion forces md1 ; md2 . Exactly at the depinning transition $R=L$ the
singular contribution to the grand potential is predicted to be $\Omega\approx
L^{2-\alpha_{md}}$ where $2-\alpha_{md}=(3\beta_{s}^{co}-1)/2$ which obviously
also vanishes when $\beta_{s}^{co}=1/3$. The vanishing of both these exponents
is consistent with the known marginal logarithmic contribution to the free-
energy of a finite-size droplet in these systems stripe17 . This consistency
between osculation, depinning and complete wetting for systems with dispersion
forces is not achieved if we had chosen $\lambda_{\rm
osc}\approx\xi_{\parallel}$ which, as mentioned above, we believe is a lower-
bound for the rounding length scale at osculation.

The prediction of the mesoscopic crossover scaling theory, that at osculation
$\ell_{\pi}$ is a lower bound for the value of $\ell_{0}$, is consistent with
a simple idea proposed by Rascón and Parry (RP) who suggested that the total
adsorption on a completely wet sculpted surface can be determined by first
coating the wall with a wetting layer of thickness, $\ell_{\pi}$, defined
normal to each surface point nature . This coating modifies the original wall
shape, $\psi(x)\to\tilde{\psi}(x)$, on to which we then fit a meniscus of
Laplace radius $R$. This simple modified macroscopic construction reproduces
with remarkable accuracy different scaling regimes, arising due to the
competition between microscopic forces and geometry, obtained from numerical
studies of interfacial Hamiltonian models and experiments exp1 ; exp2 ; exp3 .
We shall return to this in the next section.

##  III Adsorption on a corrugated wall

In preparation for studying meniscus osculation in a microscopic DFT we turn
our attention to the adsorption occurring near a sinusoidally corrugated wall
described by the height function

$\psi(x)=A\left[1-\cos\left(\frac{2\pi x}{L}\right)\right]\,,$ (30)

where $A$ is amplitude and $L$ is the period. The axes here are chosen such
that $\psi=0$ at the bottom at wall. The adsorption and wetting properties on
such corrugated surfaces have been studied previously, but usually
concentrating on the partial wetting regime where, for simple fluids, pre-
wetting and unbending transitions (equivalent to a local condensation of
liquid within the troughs) compete rascon ; rejmer2000 ; kubalski2001 ;
rejmer2002 ; rejmer2007 ; rodriguez . The surface behaviour is very rich for
complex fluids (liquid crystals) where dislocations near the crests must be
taken into account patricio1 ; patricio2 . Here we restrict ourselves to
simple fluids and to complete wetting ($\theta=0$) where none of these
features are present. In spite of this, the adsorption isotherm is still quite
rich and, we anticipate, falls into three regimes (see Fig. 4):

###  III.1 Microscopic, pre-osculation regime

When the pressure is low, such that the Laplace radius of curvature $R\ll
R_{w}$ where $R_{w}=L^{2}/4\pi^{2}A$, no meniscus is present. In this case the
adsorption is entirely microscopic comprising a thin wetting layer of
approximate thickness $\ell_{\pi}$ which coats the wall. As the pressure is
increased, a rounded meniscus osculation transition occurs near
$R=R_{w}-\lambda_{\rm osc}$ where the adsorption changes from microscopic to
macroscopic due to the growth of the meniscus. In three dimensional systems
and for short-range forces $\lambda_{\rm osc}\approx\sqrt{R_{w}}$ and
therefore is of a similar size to $\xi_{\parallel}$. A signature of this
rounded transition would be a dramatic change in the behaviour of the second
and third derivatives, $\partial^{2}\Gamma/\partial\mu^{2}$ and
$\partial^{3}\Gamma/\partial\mu^{3}$, which reflect the fractional order of
the macroscopic transition.

###  III.2 Macroscopic, geometry-dominated regime

When $R<R_{w}$ a meniscus sits near the troughs of the corrugated wall
connecting tangentially with its sides. As the pressure is increased the
meniscus grows until it is near the crests of the sinusoid. Macroscopically
this, geometry-dominated, regime extends from osculation ($R=R_{w}$),
equivalent to

$\delta p_{\rm osc}=\frac{4\pi^{2}A\gamma}{L^{2}}\,,$ (31)

to saturation, $\delta p=0$. It can again be shown easily that for the
sinusoidal wall (30) the height of the meniscus above the bottom is

$\ell_{0}=z_{0}+\sqrt{R^{2}-x_{0}^{2}}-R\,.$ (32)

where $\pm x_{0}$ and $z_{0}$ are the coordinates denoting the contact of the
meniscus with the wall (see Fig. 3) which are given implicitly by solving
simultaneously:

$A\sin\left(\frac{2\pi
x_{0}}{L}\right)=\frac{Lx_{0}}{2\pi\sqrt{R^{2}-x_{0}^{2}}}$ (33)

and

$z_{0}=A\left[1-\cos\left(\frac{2\pi x_{0}}{L}\right)\right]\,.$ (34)

The presence of wetting layers qualitatively and quantitatively affects this
and is described approximately by the RP construction, in which we coat the
wall with the wetting layer prior to fitting inside the circular meniscus.

Similarly, the approach to saturation is more properly a cross-over from a
geometry-dominated regime, where the meniscus lies within the wells, to a
second microscopic regime where the liquid-gas interface unbinds from the
crests. This second cross-over occurs when the period
$L\approx\xi_{\parallel}$, i.e. when the characteristic scale of the
interfacial fluctuations is similar to the geometric scale.

###  III.3 Microscopic, interfacial unbinding regime

On approaching saturation, the liquid-gas interface must unbind from the wall
and will display properties which are very similar to complete wetting at a
planar surface. The divergence of the film thickness in this limit cannot be
described macroscopically and we must allow for the intermolecular forces and
the effective binding that results between the interface and the wall. The
shape of the wall does have some influence on the equilibrium shape of the
liquid-gas interface, but this is negligible, at leading-order, when the
parallel correlation length is larger than the period $L$, corresponding to
the regime $\delta p\ll L^{-1/\nu_{\parallel}^{\rm co}}$ in which case the
interface is effectively planar. Critical exponents which characterise the
divergence of the film thickness and parallel correlation length are
unchanged. The question whether there is any signature of the wall shape on
the complete wetting layer, as it unbinds from the crests, is rather subtle
and is related to the nature of the binding potential. This is particularly
subtle for systems with short-ranged forces and we intend to focus on this
problem in a separate work future .

##  IV DFT analysis of adsorption on sinusoidal hard wall

Here we present the results for adsorption on sinusoidal hard wall obtained
using a microscopic DFT. We present first the model whose accuracy we test
using an exact sum rule due to Upton upton . Numerical results illustrating
the meniscus osculation transition and different adsorption regimes are then
presented.

###  IV.1 Density functional theory

Consider a simple fluid which is a subject of an external field
$V({\mathbf{r}})$ due to a presence of a confining wall and which is at
contact with a bulk reservoir at temperature $T$ and chemical potential $\mu$.
Within classical DFT evans79  the equilibrium density profile
$\rho({\mathbf{r}})$ of the fluid is obtained by minimisation of the grand
potential functional

$\Omega[\rho]=F[\rho]+\int{\rm d}{\bf
r}\rho({\mathbf{r}})\left[V({\mathbf{r}})-\mu\right]\,,$ (35)

where $F[\rho]$ is the intrinsic free energy functional containing all the
information about the fluid interactions. Its approximation, which is a
crucial part of any DFT model, depends largely on the choice of the fluid
model and particularly for simple fluids the intrinsic free energy can be
treated in the following perturbative manner:

$F[\rho]=F_{\rm id}[\rho]+F_{\rm hs}[\rho]+F_{\rm att}[\rho]\,,$ (36)

which splits the functional into the ideal gas, $F_{\rm id}$, repulsive hard-
sphere, $F_{\rm hs}$, and attractive, $F_{\rm att}$, contributions.

The ideal gas part is known exactly and is given by

$\beta F_{\rm id}[\rho]=\int{\rm d}{\bf
r}\rho({\mathbf{r}})\left[\ln(\rho({\bf r})\Lambda^{3})-1\right]$ (37)

where $\Lambda$ is the thermal de Broglie wavelength which can be set to unity
and $\beta=1/k_{B}T$ is the inverse temperature.

The repulsive interaction between fluid molecules is approximated by the hard-
sphere potential and its contribution to the free energy is described
accurately using Rosenfeld’s fundamental measure theory ros

$F_{\rm hs}[\rho]=\frac{1}{\beta}\int{\rm
d}{\mathbf{r}}\,\Phi(\\{n_{\alpha}\\})\,,$ (38)

where $\\{n_{\alpha}\\}$ denotes a set of six weighted densities

$n_{\alpha}({\mathbf{r}})=\int{\rm d}{\bf
r}^{\prime}\rho({\mathbf{r}}^{\prime})\omega_{\alpha}({\mathbf{r}}-{\mathbf{r}}^{\prime})\,,\;\;\alpha=\\{0,1,2,3,v1,v2\\}\,,$
(39)

given by convolutions between one-body fluid density $\rho({\bf r})$ and the
weight functions $\\{\omega_{\alpha}\\}$ which characterize so called
fundamental measures of the hard-sphere particles of diameter $\sigma$:

$\displaystyle\omega_{3}({\mathbf{r}})$ $\displaystyle=$
$\displaystyle\Theta({\mathcal{R}}-|{\mathbf{r}}|)\,,\;\;\;\;\omega_{2}({\mathbf{r}})=\delta({\mathcal{R}}-|{\mathbf{r}}|)\,,$
(40) $\displaystyle\omega_{1}({\mathbf{r}})$ $\displaystyle=$
$\displaystyle\omega_{2}({\mathbf{r}})/4\pi{\mathcal{R}}\,,\;\;\;\,\omega_{0}({\mathbf{r}})=\omega_{2}({\mathbf{r}})/4\pi{\mathcal{R}}^{2}\,,$
(41) $\displaystyle\omega_{v2}({\mathbf{r}})$ $\displaystyle=$
$\displaystyle\frac{{\mathbf{r}}}{{\mathcal{R}}}\delta({\mathcal{R}}-|{\mathbf{r}}|)\,,\omega_{v1}({\mathbf{r}})=\omega_{v2}({\mathbf{r}})/4\pi{\mathcal{R}}\,.$
(42)

Here, $\Theta$ is the Heaviside function, $\delta$ is Dirac’s delta function
and ${\mathcal{R}}=\sigma/2$. Among various possible prescriptions for
describing the free energy density $\Phi$ for inhomogeneous hard-sphere fluid,
we adopt the original Rosenfeld approximation ros  which accurately describes
short-range correlations between fluid particles and satisfies exact
statistical mechanical sum rules hend .

For separations $r>\sigma$, a pair of fluid particles is assumed to interact
via the attractive part of the Lennard-Jones potential, $u_{\rm att}(r)$,
which is truncated at a cut-off which we set to be $r_{c}=2.5\,\sigma$, i.e.:

$u_{\rm att}(r)=\left\\{\begin{array}[]{cc}0\,;&r<\sigma\,,\\\
-4\varepsilon\left(\frac{\sigma}{r}\right)^{6}\,;&\sigma<r<r_{c}\,,\\\
0\,;&r>r_{c}\,.\end{array}\right.$ (43)

This attractive contribution is modelled in simple mean-field fashion:

$F_{\rm att}[\rho]=\frac{1}{2}\int\int{\rm d}{\mathbf{r}}{\rm
d}{\mathbf{r}}^{\prime}\rho({\bf r})\rho({\mathbf{r}}^{\prime})u_{\rm
att}(|{\mathbf{r}}-{\mathbf{r}}^{\prime}|)\,.$ (44)

We assume that the external potential is a hard wall of sinusoidal shape:

$V(x,z)=\left\\{\begin{array}[]{ll}\infty,&z<\psi(x)\,,\\\
0,&z>\psi(x)\,,\end{array}\right.$ (45)

where $\psi(x)$ is given by Eq. (30). A purely hard is known to be completely
dry, i.e. completely wet by vapour, corresponding to contact angle
$\theta=\pi$. By studying drying we are also able to avoid the complications
arising from layering since liquid layers are absent near the surface of the
wall. We assume that the wall is of a macroscopic extent in the remaining
Cartesian direction and the system is thus translationally invariant along the
$y$ axis.

The minimization of (35) leads to the self-consistent equation for the
equilibrium density profile

$\rho({\mathbf{r}})=\Lambda^{-3}\exp\left[\beta\mu-\beta
V({\mathbf{r}})+c^{(1)}({\mathbf{r}})\right]\,,$ (46)

where
$c^{(1)}({\mathbf{r}})=c^{(1)}_{\mathrm{hs}}({\mathbf{r}})+c^{(1)}_{\mathrm{att}}({\mathbf{r}})$
is the one-body direct correlation function, which can be split into the hard-
sphere contribution

$c^{(1)}_{\mathrm{hs}}({\mathbf{r}})=-\sum_{\alpha}\int{\rm
d}{\mathbf{r}}^{\prime}\;\frac{\partial\Phi(\\{n_{\alpha}\\})}{\partial
n_{\alpha}}\,\omega_{\alpha}({\mathbf{r}}^{\prime}-{\mathbf{r}})$ (47)

and the attractive part

$c^{(1)}_{\mathrm{att}}({\mathbf{r}})=-\beta\int{\rm
d}{\mathbf{r}}^{\prime}\;u_{\rm
att}(|{\mathbf{r}}-{\mathbf{r}}^{\prime}|)\,\rho({\mathbf{r}}^{\prime})\,.$
(48)

We solve Eq. (46) numerically on a rectangular discrete grid of spacing of
$0.1\,\sigma$ using Picard iteration where the convolutions in (39), (47), and
(48) are determined using a fast Fourier transform frink .

From the equilibrium density profile $\rho(x,z)$ we can extract the local
height of the liquid-gas interface identified as

$\rho(x,\ell(x))=\frac{\rho_{g}+\rho_{l}}{2}$ (49)

and also the excess adsorption

$\Gamma=\int_{-L/2}^{L/2}{\rm d}x\int_{\psi(x)}^{\infty}{\rm
d}z(\rho_{b}-\rho(x,z))\,,$ (50)

where $\rho_{b}$ is the bulk density. In our numerical study, we are, of
course, not able to model a semi-infinite system, instead, we set the density
equal to the bulk density at a fixed distance of $50\,\sigma$ above the crests
of the sinusoid.

Figure 6: Three-dimensional plot of the contact density profile for a
corrugated hard wall, (30), over a single period with $L=20\,\sigma$ and
$A=10\,\sigma$, which is in contact with an supersaturated bulk liquid at
$\delta\mu=0.0023\,\varepsilon$. The black line with red dots represents the
value of the local contact density $\rho({\bf s})$ where ${\bf
s}=(x,\psi(x))$. The corresponding full DFT density profile, in the form of a
colormap in the $xz$-plane, is also shown. Figure 7: Numerical DFT results for
the dimensionless ratio $\beta p/\bar{\rho_{w}}$ as a function of the bulk
fluid supersaturation $\delta\mu$ for two different wall amplitudes and
periods showing good agreement with Upton’s generalized contact theorem (52).

Figure 8: Numerical DFT results for the equilibrium density profiles for the
sinusoidal hard wall with $L=30\,\sigma$ and $A=30\,\sigma$ illustrating a)
microscopic, pre-osculation regime, b) macroscopic, geometry-dominated regime
and c) microscopic, interface unbinding regime which correspond to the three
points highlighted in Fig. 9. The green line denote the equilibrium height,
$\ell(x)$, of the liquid-gas interface.  Figure 9: Growth of the height of the
liquid-gas interface, $\ell_{0}$, measured above the wall bottom, for the
sinusoidal hard wall with $L=30\,\sigma$ and $A=30\,\sigma$. The points a), b)
and c) correspond to the density profiles shown in Fig. 8 representative of
the three adsorption regimes.

###  IV.2 DFT results

####  IV.2.1 Upton sum rule

The contact sum rule for a planar hard wall in contact with a bulk fluid (i.e.
in a semi-infinite geometry) elegantly relates the fluid density at contact,
$\rho_{w}$, to the bulk pressure hend :

$\beta p=\rho_{w}\,.$ (51)

For non-planar hard walls, the contact density becomes position-dependent and
the theorem generalizes to upton :

$\beta p=\bar{\rho}_{w}\,.$ (52)

Here, $\bar{\rho}_{w}$ is the geometrically averaged contact density defined
as

$\bar{\rho}_{w}=\frac{1}{A_{w}}\int\rho({\bf s})ds\,,$ (53)

where the integration is over the wall surface of total area $A_{w}=\int ds$.
Here, $\rho({\bf s})$ is the value of the contact density at position $x$ with
${\bf s}=(x,\psi(x))$ and $ds=dx\sqrt{1+\psi^{\prime}(x)^{2}}$ is the local
area element.

Being exact, the contact theorem is a useful test of the accuracy of the
numerical methods used to solve Eq. (46). Errors arise in a number of ways
including the discretization used in the Piccard iteration and also from
finite-size effects, since we have to impose that the density takes its bulk
value at some finite distance (which we set to be $50\,\sigma$) above the
crests. In Fig. 7 we plot the numerically determined value of $\beta
p/\bar{\rho_{w}}$ as a function of the chemical potential for two different
wall amplitudes and periods which we use to study the meniscus osculation
transition and adsorption isotherms. For the shallower geometry, with
$L=100\,\sigma$ and $A=20\,\sigma$ there is a very good agreement with the
prediction of the Upton sum rule with the worst relative error about $2\%$.
For the more corrugated wall with $L=30\,\sigma$ and $A=30\,\sigma$ the
agreement is still good but a little worse with the maximum relative error
about $7\%$.

####  IV.2.2 Osculation transition and meniscus shapes

We now turn our attention to the adsorption occurring on sinusoidal walls for
a range of periodicities and amplitudes. All the DFT calculations were
performed at a temperature $T=0.925\,T_{c}$, where $T_{c}$ is the critical
temperature of the bulk fluid ($k_{B}T_{c}=1.41\,\varepsilon$). At this
temperature, the corresponding densities of the coexisting bulk gas and liquid
phases are $\rho_{g}=0.104\,\sigma^{-3}$ and $\rho_{l}=0.431\,\sigma^{-3}$,
respectively, and the chemical potential of saturation is $\mu_{\rm
sat}=-3.943\,\varepsilon$.

At least qualitatively, the equilibrium density profile $\rho(x,z)$, and
adsorption, fall into the three regimes as described earlier. This is
illustrated in Fig. 8 for the more corrugated wall where $L=30\,\sigma$ and
$A=30\,\sigma$. In the density profile we also highlight the local height of
the liquid-gas interface, $\ell(x)$, which is shown as the green line. Of
particular interest is the local height above the wall bottom, $\ell_{0}$,
whose dependence on the chemical potential is shown in Fig. 9. Figures 8 a)-c)
and the corresponding points on the adsorption of Fig. 9 correspond to the i)
microscopic, pre-osculation regime, ii) macroscopic, geometry-dominated regime
and iii) microscopic, interfacial unbinding regime, respectively.

Now we seek to be more quantitative. Numerical evidence for a (rounded)
osculation transition is shown in Fig. 10 for the shallower geometry with
$L=100\,\sigma$ and $A=20\,\sigma$ where we plot the second and third
derivatives of the excess adsorption as a function of $\mu$. The corresponding
growth of the mid-point height $\ell_{0}$ is shown in Fig. 11 together with
state points whose profiles we shall compare with the predictions of the
geometric construction. It can be seen from Fig. 10 that in contrast to the
second derivative, $\partial^{2}\Gamma/\partial\mu^{2}$, the third derivative,
$\partial^{3}\Gamma/\partial\mu^{3}$, undergoes a dramatic increase near
$\delta\mu/\varepsilon\approx 0.01$. This qualitative difference between the
second and third derivatives is consistent with the macroscopic, $5/2$
singularity predicted for the osculation transition – recall Eq. (22). The
sudden increase in the third derivative occurs close to the value of the
chemical potential where the meniscus lifts off the trough, see plots (3) and
(4) in Fig. 12.

Figure 10: Numerical DFT results comparing the second (blue line) and third
(orange line) derivatives of the excess adsorption $\Gamma$ w.r.t. $\mu$ for a
sinusoidal hard wall with $L=100\,\sigma$ and $A=20\,\sigma$. The dramatic
increase in the third derivative near $\delta\mu/\varepsilon\approx 0.01$
closely coincides with the growth of a meniscus near the bottom of the wall
corresponding to a rounded meniscus transition. Figure 11: Comparison of the
numerical DFT results (red symbols) for $\ell_{0}$ as a function of
$\delta\mu$ with the predictions of the purely macroscopic theory (green line)
and RP mesoscopic construction (blue line) for a sinusoidal hard wall with
$L=100\,\sigma$ and $A=20\,\sigma$. The continuation of the green and blue
lines beyond the osculation transition are denoted by the respective dotted
lines.

Figure 12: Equilibrium DFT density profiles over a single period for the
sinusoidal hard wall with the periodicity $L=100\,\sigma$ and amplitude
$A=20\,\sigma$ corresponding to the points in Fig. 11. For each of these
profiles three curves are highlighted. The solid green curve is the DFT result
for the location of the interface, $\ell(x)$, defined as the locus where the
local fluid density is $(\rho_{g}+\rho_{l})/2$. This is compared with two
other curves. The solid green circle has the Laplace radius $R$ and has been
placed so that its lowest point coincides with the true value of $\ell_{0}$.
The green dashed line is the RP prediction for $\ell(x)$.

Figure 13: Top row: Comparison of the meniscus height, $\ell_{0}$, as a
function of $\delta\mu$ obtained from DFT (red symbols), the macroscopic
theory (green line) and the mesoscopic RP construction (green line) for a
sinusoidal hard walls with the fixed periodicity $L=100\,\sigma$ and varying
amplitude: $A=5\,\sigma$ (left panel), $A=10\,\sigma$ (middle panel), and
$A=50\,\sigma$ (right panel). Bottom row: The same comparison but for hard
sinusoidal wall of fixed amplitude $A=20\,\sigma$ and varying periodicity:
$L=50\,\sigma$ (left panel), $L=30\,\sigma$ (middle panel), and $L=20\,\sigma$
(right panel). Note that the scale of the abscissa in the bottom row is an
order of magnitude larger than in the top row.

Next, we focus on the geometry dominated regime and test the macroscopic and
mesoscopic predictions of section III.2 for the shape and height of the
meniscus. In Fig. 11, we compare the growth of the meniscus height,
$\ell_{0}$, with the macroscopic prediction, Eq. (32), and its mesoscopic
correction following the RP construction. The purely macroscopic prediction is
only in semi-quantitative agreement with the DFT results. The predicted height
$\ell_{0}$ is systematically lower than the DFT results, which suggests that
the discrepancy is mainly due to the presence of thin drying layers which are
allowed for, approximately, in the RP construction. The accuracy of this
construction for the meniscus shape is illustrated in Fig. 12 for five
representative DFT density profiles corresponding to the thermodynamic points
highlighted in Fig. 11. In these figures we can compare the true shape of the
interfacial profile $\ell(x)$ (solid green line) with the prediction of the RP
construction (green dashed line). The RP construction gives significantly
better agreement with the DFT results, see the blue line in Fig. 11. The RP
construction is particularly accurate for the state points (1) and (2) where
the meniscus is higher up the trough, meeting the wall above the points of
inflection of the sinusoid. It is probably not coincidental that the
comparison between the DFT results and the prediction of the RP construction
is worst near the osculation transition. In these plots we have also
highlighted a green circle of Laplace radius $R$ which we have purposely
placed to touch the _true_ value of $\ell_{0}$. It is apparent that the
interfacial shape $\ell(x)$ is always well described by the arc of the
circular meniscus in the central region of the trough. However, it is clear
from these plots that the simple RP scheme of first coating the wall with the
uniform layer of $\ell_{\pi}$ slightly overestimates the adsorption at the
crests and slightly underestimates the true interfacial height above the
troughs. It might be possible to modify the RP construction by allowing for
some local geometric dependence to the wetting that we first coat the wall
with before inserting the arc of the circular Laplace meniscus.

Further comparisons between the purely macroscopic and the RP predictions with
the DFT results are presented in Fig. 13 for various wall parameters. In the
upper row, we vary the wall amplitude for the fixed periodicity
($L=100\,\sigma$), while in the bottom row the periodicity is varied for the
fixed amplitude ($A=20\,\sigma$). In general, the agreement between the
mesoscopic theory and the DFT is close but somewhat deteriorates as the ratio
$A/L$ increases. This is most likely because for highly curved walls any small
discrepancy in $\ell_{\pi}$ or $R$ has a significant impact on the resulting
value of $\ell_{0}$. It appears that for sinusoidal walls the condition $A=L$
represents the limit beyond which even the mesoscopic RP construction ceases
to be quantitatively reliable.

##  V Summary

In this paper we have studied the adsorption of fluids at smoothly corrugated
completely wet walls using macroscopic theory, mesoscopic scaling theory and
microscopic DFT. In particular, we have focused on a rounded meniscus
osculation transition occurring near the trough of the sinusoid which is
associated with the appearance of a meniscus as the chemical potential is
increased towards bulk saturation. Macroscopically, this transition is of
$7/2$ order but is rounded due to the influence of a thin wetting layer
arising from microscopic interactions. The scaling theory that we developed
for this predicts a non-trivial relationship between the interfacial height at
the osculation and the wetting layer thickness. The simple RP construction for
mesoscopic corrections within the geometry-dominated regime is tested in our
DFT where we show that it accurately describes the adsorption on the
corrugated surface, particularly where the meniscus is above the points of
infection of the wall. For a sinusoidal wall the adsorption falls into three
regimes – the pre-osculation regime, where the adsorption is determined by the
microscopic interactions, a post-osculation regime where the meniscus sits
within the troughs of the wall and finally a complete wetting regime where the
interface unbinds from the crests. This final regime can again be only
understood by taking into account the microscopic interactions. Our work can
be extended in many ways. In particular, it will be necessary to test the
prediction for the value of the osculation exponent $\beta_{\rm osc}=3/7$ in
systems with dispersion forces and also in 2D; these studies are in progress.
Also, recall that for meniscus depinning the order of the phase transition
changes when the walls are partially wet. This may also occur for meniscus
osculation where the phase transition also competes with wetting and unbending
transitions rascon . The surface phase diagram may be very rich in these
systems similar to prediction for wetting on chemically patterned surfaces.
Returning to the case of complete wetting, our DFT results are suggesting that
one may modify and improve the RP construction by allowing curvature
corrections to the thin wetting layer that first coats the wall. Finally, it
would be interesting to examine in more detail how the shape of the wall
influences the divergences of the adsorption due to the unbinding of the
liquid-gas interface as bulk coexistence is approached. To correctly
understand this, it is necessary to model the effect of interaction between
the interface and the wall using the fully non-local binding potential
nonlocal .

###### Acknowledgements.

This work was financially supported by the Czech Science Foundation, Project
No. 20-14547S.

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