# Edge Wetting:
Steady State of Rivulets
in Wedges

Nikolai Kubochkin111corresponding author<EMAIL_ADDRESS>
Tatiana Gambaryan-Roisman222corresponding author<EMAIL_ADDRESS>
Institute for Technical Thermodynamics, Technische Universität Darmstadt
Alarich-Weiss-Straße 10, 64287, Darmstadt, Germany

###### Abstract

The geometry of rough, textured, fractured and porous media is topologically
complicated. Those media are commonly represented by bundles of capillary
tubes when modeled. However, angle-containing geometries can serve as a more
realistic portrayal of their inner structure. A basic element abidingly
inherent to all of them is an open wedge-like channel. The classical theory of
capillarity ignoring intermolecular interactions implies that liquid entering
the wedge must propagate indefinitely along its spine when the liquid-gas
interface is concave. The latter is well-known as a Concus-Finn condition. In
the present paper, we show that steady-state rivulets violating Concus-Finn
condition can be formed in such channels when the surface forces are taken
into account. We present a simple model based on the disjoining pressure
approach and analyze the shape of the rivulets in the wedges. Besides, we
consider a case when the walls of the wedge are soft and can be deformed by
the liquid.

Keywords: edge wetting, nanochannels, wedge, disjoining pressure, Concus-Finn
condition

##  1 Introduction

Most of surfaces, both natural and man-made, are rough or/and contain pores or
cracks [1]. Usually, they are exposed to a vapor or brought in a contact with
a liquid [2]. Thus, investigation of the capillary transport in rough, porous
or cracked media is beneficial for various technologies, including, by not
limited to, ink-jet printing and ink production [3], biophysics [4], and
micro- and nanofluidics [5, 6]. Combating the climate change stimulates
development and improving the techniques of the oil and gas recovery [7] and
sequestration of carbon dioxide [8], which, in turn, requires the further
comprehension of physics of fluid transport in porous and fractured media.

The geometry of the rough, textured and porous media is generally
topologically complicated. In mathematical description, the porous media are
commonly and easily represented as bundles of capillary tubes [9]. However,
angle-containing geometries can serve as a more realistic portrayal of inner
structure of those media [10]. A basic element abidingly inherent to all of
them is an open wedge-like channel [11, 12, 13, 14, 15, 16]. Statics and
dynamics of wetting of the wedges (corners) pioneered by works of Taylor and
Hauksbee [17] have been studied for decades. One of the first important
insights into the liquid behavior inside the wedge has been reported by Concus
and Finn [18]. They showed that the steady shape of the liquid interface is
only possible when the liquid-gas interface is convex. Thus, a simple
geometrical condition $\alpha+\theta_{w}>\cfrac{\pi}{2}$, where $\alpha$ is a
half of the opening angle of the wedge and $\theta_{w}$ is the contact angle,
with which the liquid meets the wall of the wedge (Figure 1), should be
fulfilled. Otherwise, when the liquid-gas interface is concave, according to
the authors [18], the solution is unbounded and the capillary pressure
gradient drives the rivulet along the corner both in the absence of gravity
and, against it, when the it is present. Their result has been augmented by
consideration of the wedges with the walls having different wetting properties
[19]. Besides, the maps for stability in the case of more complicated
geometries have been plotted in works [20, 21, 22, 23]. In order to derive the
Concus-Finn condition, a simplified approach ignoring the actual non-plane
liquid interface shape was developed by Berthier et al. [24, 25].

Figure 1: Wetting of the symmetrical wedge channel: (a) the wedge wetted by a
liquid within the classical capillarity; no adsorbed film is present. $\alpha$
is the half-opening angle of the wedge, $\beta=\pi/2-\alpha$ is the angle of
inclination of the wedge wall, $\theta_{w}$ is the contact angle, $X_{w}$ and
$Y_{w}$ are the half-width and length of the wedge, respectively; (b) the
wedge wetted by a liquid whose bulk relaxes to the wetting/adsorbed film.
Variables $h$ and $w$ measure the distance from the $xy$-plane to the liquid
interface and to the wedge wall, respectively.

The result of Concus and Finn clearly showed the principal difference between
physics of the liquid ascending in round capillary tubes and wedges. Indeed,
it is well known that if a vertically oriented capillary tube is brought in
contact with a liquid pool ($\theta_{w}<\cfrac{\pi}{2}$), the meniscus starts
to climb up over the walls of the tube. The maximal height the meniscus can
reach is dictated by the gravity and can be evaluated with the Jurin law [14].
Counter-intuitively, if a liquid enters a vertically oriented sharp wedge with
the contact angle $\theta_{w}<\cfrac{\pi}{2}-\alpha$, gravity cannot stop the
ascending rivulet flow, although changes the dynamics of the rivulet
propagation. Thus, the rivulet has to propagate infinitely if the wedge is in
contact with an infinitely large liquid reservoir. The interface shape in that
case is expected to be a hyperbola as was derived in a number of works [26,
27, 28]. The predicted infinite rise of the rivulet is obviously caused by
infinite curvature of the corner, which is not the case for real systems [28].
Only recently, Gerlach et al. [29] showed that in the real systems, the finite
curvature, determined by the manufacturing process, allows the spreading
rivulet to stop.

The rivulet flows are omnipresent in the porous and rough media, for which
macroscopic considerations of the matter are often insufficient. In order to
gain further comprehension of the wetting processes in those media, the
nanoscale effects must be incorporated into the models. For the nanoscaled
channels, the impact of the surface forces arising from electrostatic,
electrodynamic, hydrophobic, steric, and structural interactions is tremendous
and cannot be ignored [9]. Despite the fact that there are a number of works
on the wetting of wedges employing the surface force approach, all of them aim
to discuss either the critical wetting (filling behavior) [30, 31, 32, 33] or
the stability of rivulets [15]. That leads to consideration of a simplified
geometry of the problem (two dimensional instead of three dimensional) and no
question on the possible rivulet flow arises since the liquid profile is
translationally invariant along the channel. The aforementioned works are of
undoubted interest; nevertheless, the problems studied do not allow for the
answer whether the surface forces stop the rivulet flow in the case when the
Concus-Finn condition is violated, and what the shape of such a steady rivulet
is. Meanwhile, understanding the limits of the applicability of the Concus-
Finn condition can significantly help to understand physics of imbibition,
since it is directly related to the extent of the liquid front propagation in
porous, textured, or fractured materials.

To the best of our knowledge, the only work discussing the stable shapes of
the rivulets in the case when the Concus-Finn condition is violated has been
performed by Wong et al. [34] for the capillaries with the regular polygon
cross-sections. It has been shown by them that accounting for the surface
forces renders the existence of the steady-states which are not possible
within the classical theory of capillarity. However, the disjoining pressure
in their work was used primarily as a mathematical tool in order to resolve
the contact line problem.

In the present paper, we study the steady shape of the rivulets in
corners/wedges. We demonstrate that introduction of the surface forces
embodied in the disjoining pressure leads to the appearance of the steady
state of the rivulet in the wedge, although the commonly used Concus-Finn
condition is violated. We discuss the influence of the parameters of the
disjoining pressure alongside the corner geometry on the steady rivulet
profile. Besides, we pay attention to the case when the wedge material is soft
and, hence, the wedge walls can be deflected by the traction exerted by the
liquid onto the walls of the wedge. The paper is organized as follows. In
section 2, we present the mathematical model and discuss the surface force
model. The simulation results for the rivulets in rigid and soft wedges are
presented in section 3.

##  2 Mathematical model

###  2.1 Governing equations and system parameters

We consider a concave wedge-shaped channel, which is symmetrical with respect
to the $yz$-plane and has a width of $2X_{w}$ and length of $Y_{w}$. The walls
of the wedge are chemically and physically homogeneous. They are inclined by
angle $\beta$ with respect to the $xy$-plane. The opening angle is
$2\alpha=\pi-2\beta$. The variable $h=h(x,y)$ measures the distance between
the $xy$-plane and the position of the liquid interface. The variable $H=h-w$,
with $w(x)=|x|\tan\beta$ being the profile of the wall, measures the thickness
of the liquid layer inside the wedge. The geometry and the orientation of the
wedge are shown in Figure 1. The channel is brought in contact with an
infinite reservoir (in practice, it can be a droplet which radius is much
larger than the wedge width) located at plane $xz$ the way that a liquid
profile $H(x,0)=I(x)$ at the entrance of the channel is maintained. The wedge
is, hence, filled with a liquid of the surface tension $\gamma$. The
surrounding gas phase contains the vapor of the liquid so it is favorable for
the equilibrium adsorbed/wetting film of thickness $H_{ads}$ to form on the
walls of the wedge [35, 36]. Wetting films can be introduced as a tool for
elimination of the sharp transition between the liquid-gas interface and
solid-gas interface. However, in many cases they are present in real systems
and affect their behavior drastically [37]. The importance of accounting for
the wetting films when considering wetting of nanopores has been also reported
recently by Zhang et al. [38].

For the small angles $\beta$, the interfacial Hamiltonian of the system given
can be written within the gradient-squared approximation as

$\mathscr{H}[h(x,y)]=\iint\left(\frac{\gamma}{2}|\nabla
h|^{2}+\int_{H}^{\infty}\Pi(\tilde{H})d\tilde{H}+\Pi\left(H_{ads}\right)H\right)dxdy,$
(1)

where the first term is so-called Dirichlet energy, $\Pi(H)$ is the disjoining
pressure related to the intermolecular interactions inevitably affecting the
behavior of the nanoscale systems and the second integral term is the
intefacial potential
$\mathfrak{S}(H)=\displaystyle\int_{H}^{\infty}\Pi(\tilde{H})d\tilde{H}$. The
last term $\Pi\left(H_{ads}\right)$ is the Lagrange multiplier. We note again
that similar to other works on wedges, we assume small $\beta$ allows for
assumption that thickness of the adsorbed film $H_{ads}$ in the wedge
(measured parallel to $z$-axis as is shown in Figure 1) is equal to that over
the planar wall [39, 32]. That leads to the error in the evaluation of
$H_{ads}$ which is smaller than $10\%$, where the error of $10\%$ is reached
only in the case of the largest $\beta$ considered in the present paper.

The disjoining pressure is considered to encompass a non-retarded repulsive
van der Waals (London, Debye, Keesom) term and an attractive electrostatic
term. It reads $\Pi(H)=\cfrac{A}{H^{3}}-Ke^{-H/\chi}$ with $A,K$ and $\chi$
being the effective Hamaker constant, strength of electrostatic, and Debay-
Hückel, respectively. The Hamaker constant reads as $A_{h}=6\pi A$. For the
sake of simplicity, we assume that the electrostatic component of the
disjoining pressure can be taken in a weak overlap approximation [8]. Using
the exponential term also allows for going beyond DVLO theory as it is a fair
approximation of the structural (hydration) pressure, which agrees well with
the experimental data [40]. In that case, $K$ is the magnitude of the
structural disjoining pressure component and $\chi$ is the hydration layer
thickness [41]. The chosen expression for the disjoining pressure allows to
obtain an S-shaped isotherm having both alpha and beta-branches [42] (not to
be confused with the inclination angles of the wedge).

If the liquid films are ultrathin ($H<1$\mathrm{nm}$$), $\gamma$ starts to
depend on their thickness [43] and can be written in a simplified form as
$\gamma(H)=\displaystyle\cfrac{1}{2}\int_{d_{0}}^{H}\widehat{\Pi}(\tilde{H})d\tilde{H}$
with $d_{0}$ being the effective cut-off distance and the disjoining pressure
$\widehat{\Pi}(H)$ resulting from the interactions liquid-vacuum-liquid. The
general assumption is that the hydrodynamic framework can still capture the
system behavior when the confined region is of $H>1$ $\mathrm{nm}$ [38, 8].
Therefore, in the following, we assume that the hydrodynamics approach is
still valid and that the surface tension $\gamma$ is a constant.

The minimization of (1) leads to the Derjaguin equation defining the steady-
state shape of the rivulet in the wedge:

$\nabla^{2}H=\frac{\Pi\left(H_{ads}\right)-\Pi(H)}{\gamma}.$ (2)

The following boundary conditions have been set: $\cfrac{\partial H}{\partial
x}=-\tan\beta$ at $x=0$, $H=H_{ads}$ at $x=X_{w}$, $H=I(x)$ at $y=0$, and
$H=V(x)$ at $y=Y_{w}$ where the functions $I(x)$ and $V(x)$ for the inlet and
limiting profiles, accordingly, are to be defined later. Equation (2) can also
be derived from the thin-film equation. We note that setting the wetting film
thickness $H_{ads}$ can also be interpreted as exposure of the wedge to the
atmosphere of vapor of pressure $p_{ads}$. That will be discussed later as
well.

Figure 2: Disjoining pressure isotherms used in the present work (Table 1): (a) disjoining-conjoining pressure transition defining $H_{1}$; (b) conjoining-disjoining pressure transition defining $H_{2}$.  System | $A\times 10^{21},$\mathrm{J}$$ | $K\times 10^{4},$\mathrm{Pa}$$ | $\chi,$\mathrm{nm}$$ | $H_{1},$\mathrm{nm}$$ | $H_{2},$\mathrm{nm}$$ | $\bar{A}\times 10^{6}$ | $\bar{K}\times 10^{2}$ | $\bar{\chi}\times 10^{2}$
---|---|---|---|---|---|---|---|---
Case A | $2.48$ | $5.25$ | $10.00$ | $4.15$ | $99.43$ | $3.84$ | $7.25$ | $10.06$
Case B | $2.48$ | $15$ | $5.00$ | $3.79$ | $42.10$ | $19.47$ | $8.76$ | $11.89$
Table 1: Parameters of the disjoining pressure isotherms.

In the following, we consider a system consisting of an aqueous phase, a solid
hydrophilic phase and a gas phase. We assume that changes of the system’s
parameters are caused mainly by the changes of electrostatic/structural
component of the surface force. In practice, that can be reached, for example,
by varying the ionic strength of the aqueous phase. As has been reported, the
Hamaker constant varies weakly for aqueous solutions and we keep it,
therefore, constant. For the liquid and solid interacting across air, $A$ can
be evaluated using the Hamaker constants of corresponding liquid (l) and solid
(s) media interacting with themselves across the vacuum (v) as
$A_{h}=\sqrt{A_{h(lvl)}A_{h(svs)}}$ [44]. We set $A_{h(lvl)}=3.73\times
10^{-20}$J, which corresponds to water [8, 44]. For the solid substrate, we
choose the Hamaker constant to be $A_{h(svs)}\approx 5.86\times 10^{-20}$J,
which finally results in $A_{h}\approx 4.68\times 10^{-20}$J or $A\approx
2.48\times 10^{-21}$J. The disjoining pressure parameters for two cases – case
A and case B – are presented in Table 1. The corresponding disjoining pressure
isotherms are shown in Figure 2(a,b). In contrast to the sessile droplets, the
van der Waals forces alone still allow for the equilibrium wetting film
existence. The difference between cases $A\neq 0$, $K=0$ and $A\neq 0$, $K\neq
0$ is expected only for relatively thick alpha-films.

Wetting behavior of a system is known to be interrelated with the interfacial
potential/disjoining pressure isotherms. Employing the Frumkin-Derjaguin
theory [45], the equilibrium contact angle of the liquid meeting the surface
covered with the adsorbed film can be evaluated as

$\cos{\theta_{w}}=1+\frac{1}{\gamma}\int_{H_{1}}^{\infty}\Pi(H)dH=1+\frac{\mathfrak{S}(H_{1})}{\gamma}$
(3)

where $H_{1}=-3\chi\mathcal{W}_{0}\left(-\cfrac{A^{1/3}}{3K^{1/3}\chi}\right)$
is the first root of equation $\Pi(H)=0$ written with the use of the Lambert
$\mathcal{W}$-function (see Supplementary Material), and $\mathfrak{S}(H_{1})$
defines the minimum of the interfacial potential. For both isotherms
presented, the contact angle is $\theta_{w}\approx 5^{\circ}$. This contact
angle can be observed for a macroscopic meniscus. As can be seen from equation
(3), keeping the contact angle $\theta_{w}$ constant means preserving the
minimal value of the interfacial potential while allowing for the variations
of its shape. The Hamaker constant $A_{h(svs)}$ presented above is a value
adjusted to meet the chosen $\theta_{w}$. Nevertheless, it is still very close
to $A_{h(svs)}$ corresponding to mineral surfaces [8] or a polymer surface
such as rubber [46].

Before the solution, equation (2) is written in a dimensionless form using the
following dimensionless variables: $\bar{x}=x/H_{2}$, $\bar{y}=y/H_{2}$,
$\bar{H}=H/H_{2}$, $\bar{\Pi}(\bar{H})=\Pi(H)H_{2}/\gamma$ with
$H_{2}=-3\chi\mathcal{W}_{-1}\left(-\cfrac{A^{1/3}}{3K^{1/3}\chi}\right)$
being the second root of equation $\Pi(H)=0$ (see Figure 2, b). Since
generally the amplitude of the attractive (conjoining the interfaces) forces
is much stronger than those repulsive (Figure 2, a-b), choosing $H_{2}$ for
the role of the scaling parameter defining the range of the surface force
action is natural. The dimensional width of the wedge was $X_{w}=700$
$\mathrm{nm}$ and was scaled with respect to the $H_{2}$ of the isotherm
chosen. The dimensional length was $Y_{w}>500$ $\mathrm{nm}$ and,
analogically, was scaled with respect to the $H_{2}$ of the isotherm chosen.

The equation for the equilibrium shape of the rivulet in the dimensionless
form is:

$\nabla^{2}\bar{H}=\bar{\Pi}\left(\bar{H}_{ads}\right)-\bar{\Pi}(\bar{H}).$
(4)

The equation (4) is solved numerically using finite element analysis. It has
been checked that the solution converges as maximal element size decreases. It
has been checked additionally that the variation of $\bar{Y}_{w}$ as well as
the number of points, over which the solution has been interpolated, led to
only small (less than $2\%$) differences in a decay length $\bar{D}_{cr}$ (for
the definition of $\bar{D}_{cr}$, see subsection
$\ref{SB:Steady_states_of_rivulets})$.

###  2.2 Inlet profile and the limiting profile of the rivulet

As we stated before, we assume that at the inlet $\bar{y}=0$, the wedge is
connected to the reservoir of infinite volume and the profile of the liquid-
gas interface $\bar{h}(\bar{x},0)=\bar{I}(\bar{x})+|\bar{x}|\tan\beta$ does
not change in time. In real-life systems that can be a nanopore with a
macroscopic droplet at its entrance. Assuming that at the entrance, away from
the wall of the channel, the shape of the interface $\bar{I}(\bar{x})$ can be
obtained from the macroscopic considerations, one can employ the Young-Laplace
equation. One integrates it twice and obtain
$\bar{h}(\bar{x})=\cfrac{\tan(\beta-\theta_{w})(\bar{x}^{2}-\bar{X}_{w}^{2})}{2\bar{X}_{w}}+\bar{H}_{ads}$.
Since the inlet profile must satisfy the boundary conditions and also relax to
the wetting film, we introduce a coordinate $\bar{x}_{t}$ of the point, where
the macroscopic meniscus meets the wetting film with the contact angle
$\theta_{w}$. The inlet profile is, hence, a piece-wise function defined as

$\bar{h}(\bar{x})=\begin{cases}\cfrac{\tan(\beta-\theta_{w})}{\bar{x}_{t}}\cfrac{\left(\bar{x}^{2}-\bar{x}_{t}^{2}\right)}{2}+\bar{x}_{t}\tan\beta+\bar{H}_{ads}+c_{1}\bar{H}_{ads},&\text{if
$\bar{x}<\bar{x}_{t}$}.\\\ \bar{H}_{ads},&\text{if
$\bar{x}>\bar{x}_{t}+c_{2}\left(\bar{X}_{w}-\bar{x}_{t}\right)$}.\end{cases}$
(5)

The value of $\bar{x}_{t}$ has been kept equal to
$\bar{x}_{t}=0.95\bar{X}_{w}$, unless other is stated (see also Supplementary
Materials). Real constants $c_{1}$ and $c_{2}$ are chosen in a way to allow
for a smooth transition from the profile to the wetting film. The constant
$c_{2}$ allowed to change the width of the transition region without changes
in $\bar{I}(0)$. Two regions – the macroscopic profile and the film – have
been matched (see Supplementary material). Spline interpolation showed
overshoots and undulations in the transition region. Therefore, piece-wise
cubic Hermite polynomials have been used. Giving less smooth transition zones,
it nevertheless allowed to minimize the undesirable waviness there.

The function $H(\bar{x},\bar{Y}_{w})=\bar{V}(\bar{x})$ describes the interface
shape far away from the inlet. This function is unknown a priori. Far away
from the inlet, the interface is translationally invariant along the $Y$-axis.
In that case, the reduced one-dimensional Hamiltonian $\mathscr{H}[h(x)]$ can
be written and the liquid profile, hence, can be evaluated using one-
dimensional Derjaguin equation

$\frac{d^{2}\bar{H}}{d\bar{x}^{2}}=\bar{\Pi}\left(\bar{H}_{ads}\right)-\bar{\Pi}(\bar{H}),$
(6)

which solution reads

$\int_{\bar{H}_{ads}}^{\bar{H}}\frac{d\tilde{H}}{\sqrt{\displaystyle
2\int_{\bar{H}_{ads}}^{\tilde{H}}\left(\bar{\Pi}\left(\bar{H}_{ads}\right)-\bar{\Pi}(\hat{H})\right)d\hat{H}}}=\bar{x}_{ads}-\bar{x}.$
(7)

We introduce the limiting thickness of the interface at $\bar{x}=0$:
$\bar{H}_{w}=\bar{V}(0)$. The value of $\bar{H}_{w}$ depends on the disjoining
pressure and on the slope of the wall as follows from the equation

$\frac{\tan^{2}\beta}{2}=\bar{\Pi}\left(\bar{H}_{ads}\right)\left(\bar{H}_{w}-\bar{H}_{ads}\right)-\int_{\bar{H}_{ads}}^{\bar{H}_{w}}\left(\bar{\Pi}(\bar{H})\right)d\bar{H}.$
(8)

As long as the disjoining pressure isotherm contains the exponential tail, the
equation (8) is transcendental and the integral in (7) cannot be taken
analytically. Therefore, equations (7) and (8) were integrated numerically,
subject to boundary conditions $\bar{H}=\bar{H}_{ads}$ at
$\bar{x}=\bar{x}_{w}$ and $\cfrac{d\bar{H}}{d\bar{x}}=\tan\beta$ at
$\bar{x}=0$. In order to avoid a singularity when the adsorbed film thickness
is reached, the integration was started from $\bar{H}_{ads}+\epsilon$ where
for $\epsilon$ the condition $\epsilon/\bar{H}_{ads}\ll 1$ is fulfilled. The
function $V(x)$ was obtained as $V(x)=\bar{H}(\bar{x})+|x|\tan\beta$. The
solution of (6) has been checked by the direct differentiation as well as by
solving it as a boundary value problem fully numerically.

When the wetting film is very thin: $\bar{H}_{ads}=\phi\bar{H}_{1}$ with
$\phi\ll 1$, one can expect the limiting height of the rivulet $\bar{H}_{w}$
to be smaller than the height
$\bar{H}_{c1}=-4\bar{\chi}\mathcal{W}_{0}\left(-\cfrac{(3\bar{A})^{1/4}}{4\bar{K}^{1/4}\bar{\chi}^{3/4}}\right)$,
where the disjoining pressure isotherm finds its minimum. In this case, the
disjoining pressure isotherm can be linearized and equation (8) can be easily
solved analytically. The limiting height $\bar{H}_{w}$ can be obtained as

$\bar{H}_{w}=\bar{H}_{ads}+\tan\beta\sqrt{\frac{\bar{H}_{1}-\bar{H}_{ads}}{\bar{\Pi}\left(\bar{H}_{ads}\right)}}$
(9)

The derivation and discussion of (5) and (9) are presented in Supplementary
Materials.

##  3 Results and discussion

###  3.1 Rigid Wedges

The presence of the surface forces stabilizing the adsorbed/wetting film
allows for the steady state meniscus in the wedge even when the Concus-Finn
condition is not fulfilled. The solution of equation (4) for the adsorbed film
thicknesses $\bar{H}_{ads}=0.0125$, ($\phi=0.3$) and $\bar{H}_{ads}=0.0334$,
($\phi=0.8$) are illustrated in Figure 3. The thickness of the liquid layer
$\bar{H}$ and and the liquid-gas interface $\bar{h}$ are shown in Figure 3(a,
b) and (c, d), correspondingly. As is set by the boundary conditions, the
thickness of the liquid layer is maximal at the entrance of the wedge. The
liquid layer thickness at the plane $\bar{x}=0$ decays until the limiting
thickness $\bar{H}_{w}$ dictated by the solution of equation (8) is reached.
One can see that when the adsorbed film is thinner (Figure 3, a, c), the
thickness of the liquid filling the wedge diminishes down to the limiting
profile $\bar{V}(\bar{x})$ very close to the entrance (schematically shown by
the arrows). In contrast to that, when the adsorbed film is thicker (Figure 3,
b, d), the liquid thickness decreases gradually and reaches the limiting
profile $\bar{V}(\bar{x})$ far away from the entrance.

Figure 3: Wetting of the symmetrical wedge-shaped channel. The thickness of
the liquid $\bar{H}$ in the channel when the equilibrium adsorbed film
thickness (a) $H_{ads}=0.0125$ ($\phi=0.3$) and (b) $H_{ads}=0.0334$
($\phi=0.8$). The profile of the liquid-gas interface $\bar{h}$ in the channel
when the equilibrium adsorbed film thickness (c) $H_{ads}=0.0125$ ($\phi=0.3$)
and (d) $H_{ads}=0.0334$ ($\phi=0.8$). The inclination angle is
$\beta=2\theta_{w}$. The isotherm A has been used for the calculations. The
color bars are to show the thickness/height distribution. Less number of
points than the solutions contain was used for plotting in order to decrease
rendering time and avoid low-level graphic errors. Figure 4: Wetting of the
wedge channel; (a) thickness of the rivulet at the symmetry axis ($\bar{x}=0$)
calculated with the different adsorbed film thicknesses $\bar{H}_{ads}$; (b)
dependence of the decay length $\bar{D}_{cr}$ on the adsorbed film thickness;
the decay lengths are illustrated in (a); (c) limiting profiles
$\bar{V}(\bar{x})$ of the rivulets calculated with the different adsorbed film
thicknesses $\bar{H}_{ads}$; (d) dependence of the limiting curvature
$\bar{\kappa}_{w}$ and limiting height $\bar{H}_{w}$ of the rivulet at the
symmetry axis ($\bar{x}=0$) on the adsorbed film thickness (or parameter
$\phi$). The inclination angle is $\beta=2\theta_{w}$. The system behavior is
defined by the isotherm A. The lines in (b) and (d) are to guide an eye.

The smoothness of the transition from the inlet to the limiting profile can be
illustrated more clear by plotting the thickness $\bar{H}$ of the liquid layer
at $\bar{x}=0$ (Figure 4, a). We quantify this transition by introducing the
decay length $\bar{D}_{cr}$. We define $\bar{D}_{cr}$ as the length, at which
the relative difference between $\bar{H}(0,\bar{y})$ and $\bar{H}_{w}$ is less
than $1\%$. The decay length is shown in Figure 4(a) by the black arrows and
squared markers. The calculated values of $\bar{D}_{cr}$ are presented in
Figure 4(b). One can see that $\bar{D}_{cr}$ increases with the increasing
ratio $\phi$ (increasing $\bar{H}_{ads}$). The observed trend is obviously
related to changes in the limiting profile $\bar{V}(\bar{x})$ of the rivulet
resulting from equation (6). The limiting profile is in turn dictated by
interplay of curvature-induced and disjoining pressures. In Figure 4(c-d), the
limiting rivulet profiles as well as the dependency of the dimensionless
curvature
$\bar{\kappa}_{w}=\cfrac{\partial^{2}\bar{H}}{\partial\bar{x}^{2}}(0,\bar{Y}_{w})$
and dimensionless limiting height of the rivulet $\bar{H}_{w}=\bar{V}(0)$ on
the ratio $\phi$ are shown. The limiting height increases as the adsorbed film
$\bar{H}_{ads}$ increases, while the opposite trend is observed for the
dimensionless curvature: it decreases with the increasing $\bar{H}_{ads}$. In
Figure 5, decay length $\bar{D}_{cr}$ is plotted against the limiting height
$\bar{H}_{w}$. As the adsorbed/wetting film thickness increases causing a rise
of $\bar{H}_{w}$, the decaying length $\bar{D}_{cr}$ increases as well.

Figure 5: Dependence of the decay length $\bar{D}_{cr}$ on the limiting
thickness $\bar{H}_{w}$ of the profile $\bar{V}(\bar{x})$. The inclination
angle is $\beta=2\theta_{w}$. The system behavior is defined by the isotherm
A. Figure 6: Influence of the relative humidity on the thickness of the
adsorbed/wetting film at $T=293$ K ($\nu_{m}=18\times 10^{-3}$ kg/m) for
isotherms A and B.

As has been briefly mentioned before, the adsorbed/wetting film of a constant
equilibrium thickness $\bar{H}_{ads}$ is formed on the walls of the wedge when
it is exposed to the vapor atmosphere of pressure $\bar{p}_{ads}$. The value
of the corresponding vapor pressure or relative humidity
$RH=\cfrac{p_{ads}}{p_{s}}$ (where $p_{s}$ is the pressure of the saturated
vapor) can be calculated using Derjaguin-Churaev-Müller theory of the surface
forces [35]:

$RH=\exp{\left(-\cfrac{\nu_{m}\Pi(H_{ads})}{RT}\right)}$ (10)

where $\nu_{m}$ is a molar volume of liquid, $R$ is a gas constant, $T$ is
temperature. We plot the dependence of $\phi$ on $RH$ in Figure 6. One can see
that very thin wetting films can exist even when humidity is close to zero.
The isothermal increase of the relative humidity $RH$ leads to the increasing
limiting height $\bar{H}_{w}$ and, therefore, to the increasing decay length
$\bar{D}_{cr}$ of the rivulet.

Figure 7: Wetting of the wedges with different inclination angle $\beta$: (a)
thickness of the liquid layer at the symmetry axis ($\bar{x}=0$) calculated
with the different inclination angles $\beta$ ($\phi=0.3$); (b) influence of
the inclination angle $\beta$ on the decay length $\bar{D}_{cr}$; (c)
influence of the inclination angle $\beta$ on the on the limiting thickness
$\bar{H}_{w}$ of the profile $\bar{V}(\bar{x})$ for different $\phi$; (d)
influence of the ratio $\phi$ (adsorbed film thickness) on the decay length
$\bar{D}_{cr}$ for wedges with the walls inclined by different angles $\beta$.
The system behavior is defined by the isotherm A.

The rivulet shape is drastically affected by the geometry of the wedge as
well. In order to show that, we solve equation (4) for five different
inclination angles $\beta$ chosen in the way that the steady liquid profile in
the sense of the Concus-Finn consideration does not exist. The ratio
$\beta/\theta_{w}$ varies from $1$ to $5$.

In Figure 7(a), the liquid profiles corresponding to different inclination
angles $\beta$ (ratios $\beta/\theta_{w}$) are presented. Here, we first
consider the case when increasing $\beta$ leads to increasing of the maximal
film thickness at the inlet. That naturally follows from inlet condition (5)
with $\bar{x}_{t}$ kept constant. Interestingly, when closing the wedge in a
described way, the decay length of the rivulet $\bar{D}_{cr}$ increases
linearly with $\beta$ (Figure 7, b). In contrast to that, the dependence of
the limiting height of the rivulet $\bar{H}_{w}$ on the ratio of the wedge
inclination angle $\beta$ and contact angle $\theta_{w}$ demonstrates the non-
linear growth (Figure 7, c). The dependencies $\bar{H}_{w}(\beta)$ are plotted
in Figure 7(c) for different values of $\phi$. It can be seen that when the
wetting film is thin, the effect of the the wedge inclination is not strong,
while it becomes more pronounced when the adsorbed film is thicker ($RH$ is
higher). It is shown in Figure 5 that the decay length $\bar{D}_{cr}$
increases with the increasing $\bar{H}_{w}$. Therefore, one should expect
$\bar{D}_{cr}$ is more significantly affected by the wedge angle when the
adsorbed film is thicker ($RH$ is higher). In Figure 7(d), the decay length
$\bar{D}_{cr}$ is plotted against $\phi$ for different ratios
$\beta/\theta_{w}$ and the expected trend can be observed.

As has been stated afore and shown in Figure 7(a), the thickness of the liquid
layer at the center of the inlet $\bar{I}(0)$ has not been preserved when
closing the wedge (increasing wedge angle $\beta$). The variation of
$\bar{I}(0)$ can affect the dependence $\bar{D}_{cr}(\beta)$. In order to
eliminate the effect of $\bar{I}(0)$ on $D_{cr}(\beta)$, we perform
calculations, in which $\beta$ has been varied, while $\bar{I}(0)$ is kept
constant. The latter was possible by varying the value of $\bar{x}_{t}$ from
$\approx 0.29\bar{X}_{w}$ ($\beta=5\theta$) to $\approx 0.95\bar{X}_{w}$
($\beta=\theta$). The values of $\bar{x}_{t}$ as well as of the parameters
$c_{1}$ and $c_{2}$ of the inlet profiles are given in Supplementary
materials. The inlet profiles $\bar{I}(\bar{x})$ employed are shown in Figure
8(a). The resulting thicknesses of the rivulets along the symmetry axis of the
wedge are presented in Figure 8(b). Despite the fact that $\bar{I}(0)$ is now
the same, the rivulet in the wedge, where the wetting/adsorbed film is
thicker, still extends over the longer distance. The dependencies of
$\bar{D}_{cr}$ on $\beta/\theta_{w}$ for the cases of both same and different
$\bar{I}(0)$ are shown in Figure 8(c). Keeping thickness of the rivulet at the
inlet $\bar{I}(0)$ constant does not break the linear dependence of
$\bar{D}_{cr}$ on $\beta/\theta_{w}$, although makes it less pronounced.

Figure 8: Wetting of the wedges with the different inclination angles $\beta$:
(a) inlet profiles $\bar{I}(\bar{x})$ for different inclination angles $\beta$
($\phi=0.3$). The thickness of the liquid layer at the inlet, at the symmetry
axis ($\bar{x}=0$) is fixed; (b) thickness of the liquid layer at the symmetry
axis ($\bar{x}=0$) calculated with the different inclination angles $\beta$
($\phi=0.3$); (c) influence of the inclination angle $\beta$ on the decay
length $\bar{D}_{cr}$. The system behavior is defined by the isotherm A.
Figure 9: Wetting of the wedges of different size: Thickness of the rivulets
at the symmetry axis ($\bar{x}=0$). The inclination angle is
$\beta=2\theta_{w}$. The system behavior is defined by the isotherm A.

According to the Washburn law, the extent of the liquid front in the porous
media is related to the size of the pores [9]. Moreover, the kinetics of the
liquid front in corner geometry depends on the meniscus height at the inlet
[16]. Therefore, it is also of interest to study how the size of the wedge
affects the decay length $\bar{D}_{cr}$ of the static rivulet. The width of
the wedge was varied from $\bar{X}_{w}\approx 7$ to $\bar{X}_{w}\approx 28$,
the length of the wedge $\bar{Y}_{w}$, the adsorbed film thickness
$\bar{H}_{ads}$, and the wedge inclination angle $\beta$ were kept constant.
The profiles of the liquid $\bar{h}(\bar{y})$ filling corresponding wedges are
presented in Figure 9. In the larger wedges, the liquid thickness at the inlet
is larger and the rivulet extends along a larger distance. Therefore, the
decay length $\bar{D}_{cr}$ increases with increasing size of the wedge. All
the rivulets have been initially computed with the inlet located at
$\bar{y}=0$. However, the shape of the profiles can give us an idea that
$\bar{h}(\bar{y})$ being plotted and shifted along $\bar{y}$ with respect to
the profile of the liquid in the widest wedge might match. Indeed,
$\bar{h}(\bar{y})$ for the wedges with the widths of up to fourfold difference
coincide almost perfectly when plotted with a subsequent shift (Figure 9).
That observation can be of significant help when a profile of a liquid rivulet
is known for a large angular pore and the length of a smaller rivulet is to be
evaluated.

Figure 10: Influence of the disjoining pressure on the wetting of the wedge;
(a) limiting profiles $\bar{V}(\bar{x})$ of the rivulets calculated with the
adsorbed film thicknesses $\bar{H}_{ads}=0.0334$; (b) dependence of the
limiting height $\bar{H}_{w}$ on the adsorbed film thickness; (c) thickness of
the rivulet at the symmetry axis ($\bar{x}=0$) calculated with the adsorbed
film thicknesses $\bar{H}_{ads}=0.0334$; (d) dependence of the decay length
$\bar{D}_{cr}$ on the adsorbed film thickness. The inclination angle is
$\beta=2\theta_{w}$. The lines in (b) and (d) are to guide an eye.

As can be expected, the shape of the rivulet will be different for different
liquid systems. It is, however, of substantial interest how the surface forces
affect the steady state of nanorivulets in the systems having the same
macroscopic wetting behavior (same $\theta_{w}$ and
$\mathfrak{\bar{S}}(\bar{H}_{1})$). In the following, we show how the
disjoining pressure isotherm influences the shape of the meniscus in the
wedge.

Two isotherms shown in Figure 2 have been employed. We note that dimensional
values of $H_{ads}$ for isotherm B were smaller than $1$\mathrm{nm}$$.
However, evaluation of the surface tension according to [47] accounting for
the hydrogen bonds of water [48] shows that its alteration does not exceed
$1.5\%$. Therefore, the assumption of the constant surface tension is still
valid.

In Figure 10(a), the limiting profiles $\bar{V}(\bar{x})$ are shown for both
isotherms. Despite the fact that the contact angle $\theta_{w}$ is preserved,
at the nanoscale, the limiting profiles demonstrate significant differences in
the limiting heights $\bar{H}_{w}$. That is caused by the interplay of long-
range and short-range surface forces. Since wedges of the same $\bar{X}_{w}$,
$\bar{Y}_{w}$, $\bar{H}_{ads}$ and $\beta$ are under consideration, limiting
height $\bar{H}_{w}$ is the main parameter impacting the decay length (Figure
10, b). The influence of the disjoining pressure isotherm on $\bar{H}_{w}$ can
be understood from the simplified consideration based on (9). Since
$\bar{H}_{ads}$ are the chosen to be the same for both wedges, the differences
in the limiting height $\bar{H}_{w}$ are mainly rendered by the value of
$\bar{\Pi}(\bar{H}_{ads})$ which is larger for the isotherm B. The latter
causes smaller $\bar{H}_{w}$. The same can also be comprehended in a more
graphical way – namely, considering the slope of the alpha-branch of the
disjoining pressure isotherm (see Supplementary Material). Since the values of
$\bar{H}_{w}$ are much smaller for the case of isotherm B, the rivulet is
forced to reach the limiting profile very close to the inlet (Figure 10, c)
and has smaller values of $\bar{D}_{cr}$, as is shown in Figure 10(d). Thus,
tuning the parameters of the disjoining pressure isotherm can be a way for
control of the length, over which the rivulets can extend. Similarly to the
trend observed for influence of the inclination angles, the differences
between $\bar{H}_{w}$ and $\bar{D}_{cr}$ for isotherms A and B are larger when
the adsorbed film is thicker (humidity is higher).

###  3.2 Soft wedges

In many cases, surfaces, which are in contact with liquid, can respond to its
action by deformation, dissolution, or swelling [41]. Understanding of
possible bending/folding of solid structures is of utmost importance in
different branches of industry such as micro- and nanofabrication [49].
Shrinkage of cement-based materials when drying also lacks understanding,
which is paramount for its quality improvement and control [50].

Figure 11: Schematic illustration of (a) wedge with rigid walls and (b) wedge
with deformable walls. The position of the liquid-gas interface is $h$, the
position of the undeformed wall and deformed wall is $w$ and $w^{\dagger}$,
accordingly. Thickness of the liquid rivulet filling the wedge is $H$.

Let us consider the case when a soft wedge is left in the atmosphere of vapor
pressure $p_{ads}$, so that an equilibrium adsorbed/wetting film of thickness
$H_{ads}$ is formed on its walls. The wedge walls are of constant thickness
$b_{w}$. We assume that the wedge response is purely elastic and instant. It
can be defined by its Young modulus $E$ and Poisson ratio $\sigma$. The
bending energy of the wedge can be written in the form [51]

$\mathscr{E}[w^{\dagger}(x,y)]=\iint\left(D\left(\nabla^{2}w^{\dagger}\right)^{2}+Tw^{\dagger}\right)dxdy,$
(11)

where the first term is similar to Euler-Bernoulli elastica with $w^{\dagger}$
being a deflection of the wall of the wedge, and a constant $D$ being the
flexural rigidity, $D=\cfrac{Eb_{w}^{3}}{12(1-\sigma^{2})}$. The second term
is a Lagrange multiplier accounting for a capillary pressure-induced traction
$T=\gamma\nabla^{2}h$ applied to the wall of the wedge. Minimization of (11)
leads to the biharmonic Kirchhoff-Love equation defining the steady-state of
the wedge [51]:

$D\nabla^{2}\nabla^{2}w^{\dagger}-T=0.$ (12)

Since the wedge in the following consideration is not connected to any liquid
pool, the liquid profile along the wedge is translationally invariant:
$w^{\dagger}=w^{\dagger}(\bar{x})$. Therefore, equation (12) is reduced to

$D\frac{d^{4}w^{\dagger}}{dx^{4}}-T=0.$ (13)

Equation (13), albeit with a different traction, has been recently used by
Svetovoy et al. [52] to model an adhered cantilever bent by the van der
Waals/Casimir forces. In our case of the soft wedge, the traction can be only
obtained from the solution of the Derjaguin equation accounting for the
possible deformation of the walls. The position of the liquid-gas interface is
defined as $\bar{h}=\bar{H}+\bar{w}+\bar{w}^{\dagger}$ (Figure 11) and,
therefore, equation reads

$\frac{d^{2}\bar{H}}{d\bar{x}^{2}}=\bar{\Pi}\left(\bar{H}_{ads}\right)-\bar{\Pi}(\bar{H})-\frac{d^{2}\bar{w}^{\dagger}}{d\bar{x}^{2}},$
(14)

Equation (14) is solved as a boundary value problem. In the case of the rigid
wedge, the numerical solution can be controlled by comparing with analytical
solution (7).

To write equation (13) in a dimensionless form, we use the scaling related to
the range of the surface force action $H_{2}$ used in subsection 3.1. The
deformation of the wedge wall and the wedge’s wall thickness reads
$\bar{w}^{\dagger}=\cfrac{w^{\dagger}}{H_{2}}$ and
$\bar{b}_{w}=\cfrac{b_{w}}{H_{2}}$, accordingly. Defining the dimensionless
elastocapillary length
$\mathcal{\bar{L}_{C}}=\gamma\cfrac{1-\nu^{2}}{EH_{2}}$, we can write the
dimensionless traction exerted onto the wall by the liquid as
$\bar{T}={\mathcal{\bar{L}_{C}}}\cfrac{d^{2}\bar{h}}{d\bar{x}^{2}}$. Using
equation (14), the traction $\bar{T}$ can also be written as
$\bar{T}={\mathcal{\bar{L}_{C}}}\left[\bar{\Pi}\left(\bar{H}_{ads}\right)-\bar{\Pi}(\bar{H})\right]$.
Thus, equation (13) can be rewritten using dimensionless variables as

$\frac{d^{4}\bar{w}^{\dagger}}{d\bar{x}^{4}}-\mathcal{S}\cfrac{d^{2}\bar{h}}{d\bar{x}^{2}}=0,$
(15)

where
$\mathcal{S}=\cfrac{12\mathcal{\bar{L}_{C}}}{\bar{b}_{w}^{3}}=\cfrac{12\gamma(1-\sigma^{2})H_{2}^{2}}{Eb_{w}^{3}}$
is a softness parameter. The equation is solved as a boundary value problem.
The boundary conditions at the symmetry axis are
$\cfrac{d\bar{w}^{\dagger}}{d\bar{x}}\biggr{|}_{\bar{x}=0}=0$ and
$\cfrac{d^{3}\bar{w}^{\dagger}}{d\bar{x}^{3}}\biggr{|}_{\bar{x}=0}=0$. The
first boundary condition is due to the symmetry. The second boundary condition
stands for the absence of a shear force at the symmetry axis. The right end of
the wedge is considered to be simply supported and, therefore, we impose the
following boundary conditions: $\bar{w}^{\dagger}(\bar{X}_{w})=0$ and
$\cfrac{d^{2}\bar{w}^{\dagger}}{d\bar{x}^{2}}\biggr{|}_{\bar{x}=\bar{X}_{w}}=0$.
The latter condition corresponds to the absence of the bending moment.

Figure 12: Wetting of the wedges with soft walls: (a) Rivulet profiles for
soft ($\mathcal{S}=0.031$) and rigid ($\mathcal{S}=0$) walls; (b) Deflection
of the wedge’s wall for different values of softness parameter $\mathcal{S}$;
(c) Dependence of the maximal deformation $\bar{w}^{\dagger}_{max}$ on the
softness parameter $\mathcal{S}$; (d) Thickness of the rivulet in the deformed
wedges. The system behavior is defined by the isotherm A, $\beta=2\theta_{w}$,
$\phi=0.7$. Figure 13: Wetting of the wedges with the soft walls. Dependence
of the maximal deformation $\bar{w}^{\dagger}_{max}$ on the (a) inclination
angle ($\mathcal{S}=0.031$, $\phi=0.7$), (b) on the wetting film thickness
(parameter $\phi$) ($\mathcal{S}=0.031$, $\beta=2\theta_{w}$); (c) Traction
exerted onto the walls of the wedge inclined by different $\beta$
($\mathcal{S}=0.031$, $\phi=0.7$); (d) Traction exerted onto the walls of the
wedge covered by the wetting film of different thickness
($\mathcal{S}=0.031,\beta=2\theta_{w}$). The system behavior is defined by the
isotherm A.

We note that in order to predict the large deformations, equation (15) has to
be solved in iterations, so that the deformation corresponds to the traction
induced by the liquid layer covering the deflected wall. Generally, 100
iterations were enough to reach a good convergence of the traction. The
solutions of both equations (14) and (15) have been checked directly by
numerical differentiation. The results have been in good agreement with those
obtained by the numerical differentiation. The solver for obtaining the
deflection has also been checked by comparison with the corresponding
analytical solution of (15)

$\bar{w}^{\dagger}=\frac{\mathcal{S}\bar{T}}{24}\left(5\bar{X}_{w}^{4}-6\bar{X}_{w}^{2}\bar{x}^{2}+\bar{x}^{4}\right),$
(16)

valid in the case when the traction $\bar{T}$ is uniformly distributed over
the wall of the wedge.

In Figure 12(a), the effect of the softness parameter on the shape of the
rivulet is demonstrated. When the wedge walls are soft, the traction resulted
from the interplay of the curvature-induced pressure and disjoining pressure
deflects them and pull the wedge tip up. The deformation of the wedge’s wall
$\bar{w}^{\dagger}$ is shown in Figure 12(b). As can be expected, the
deformation becomes more pronounced for larger $\mathcal{S}$ and is maximal at
the symmetry axis, since the wedge ends are supported. The dependence of the
maximal deformation of the wedge
$\bar{w}^{\dagger}_{max}=\sup_{\bar{x}\in[0,\bar{X}_{w}]}\bar{w}^{\dagger}$ on
the softness parameter $\mathcal{S}$ is illustrated in Figure 12(c). It can be
seen that the dependence is nearly linear for small $\mathcal{S}$ (the linear
trend is shown by a dashed black line). The changes of
$\bar{w}^{\dagger}_{max}$ depart from the linear trend for the softer wedges
($\mathcal{S}>0.05$). Bending the wall of the wedge renders the region with
$\bar{H}<\bar{H}_{ads}$ (Figure 12, d), on which the negative traction is
exerted. The departure from the linear trend is, hence, related to the shape
of the function $\mathcal{S}\bar{T}$.

The influence of the wedge inclination angle $\beta$ and wetting film
thickness $\bar{H}_{ads}$ (corresponding parameter $\phi$) on the maximal
deformation and traction distribution $\bar{T}$ is illustrated in Figure 13
(a-d). The increasing inclination angle $\beta$ leads to the increasing
deflection at the symmetry axis of the wedge. As can be seen in Figure 13(a),
this dependence is linear. However, the effect of the inclination angle does
not manifest itself via only the limiting thickness $\bar{H}_{w}$, as could be
suggested. The limiting thickness $\bar{H}_{w}$ increases with the increasing
$\beta$. For $\beta/\theta_{w}>2$, the limiting height is
$\bar{H}_{w}\in\left(\bar{H}_{c1};1\right)$ (see schematic illustration in
Supplementary Materials). Therefore, such increase of $\bar{H}_{w}$ produces
smaller traction at the symmetry axis (Figure 13, c). The maximal traction in
those cases is $\bar{\Pi}(\bar{H}_{c1})$. The coordinate $\bar{x}$, where it
is exerted onto the wall, is farther from the symmetry axis for the larger
$\bar{H}_{w}$. That renders wider distributed (spread) traction and, hence,
larger deformation, even though the the traction at the symmetry axis is
smaller. For the case when $\beta/\theta_{w}=1$, the limiting height
$\bar{H}_{w}$ lies very close to $\bar{H}_{c1}$, where the disjoining pressure
isotherm finds its minimal value. Therefore, the maximal value of the traction
is reached already at the symmetry axis which, in turn, renders a less spread
traction.

The influence of the adsorbed/wetting film thickness (parameter $\phi$) on the
deformation $\bar{w}^{\dagger}_{max}$ is presented in Figure 13(b). Since the
formation of the film of thickness $\bar{H}_{ads}$ is induced by the
adsorption onto the walls of the wedge, we also plot a dependence of the
maximal deformation of the wall of the wedge on the relative humidity $RH$ of
the atmosphere (Supplementary Materials). Unexpectedly, increasing humidity
and, therefore, the adsorbed/wetting film thickness leads to the barely
observable changes in the maximal deflection. The same trend was obtained for
different values of $\mathcal{S}$. One can see from Figure 13(d) that for
smaller $\bar{H}_{ads}$, the traction has a bigger amplitude $\bar{T}_{max}$,
although it acts mainly in the vicinity of the symmetry axis. In turn, when
$\bar{H}_{ads}$ is larger, the traction has smaller $\bar{T}_{max}$ albeit
wider region of action. Thus, the negligibly small changes of
$\bar{w}^{\dagger}_{max}$ can result from the interplay of the traction
amplitude and distribution.

Figure 14: (a) The $\mathcal{S}\bar{T}_{max}$-$\bar{a}$-map of deformations
obtained from solution (17); (b) Magnified region of the map for the small
deformations. The analytical solutions for the cases with
$\phi=0.4,0.5,0.6,0.7$ ($\mathcal{S}=0.03$), $\phi=0.4,0.5,0.6$
($\mathcal{S}=0.008$) and $\phi=0.4,0.5,0.6$ ($\mathcal{S}=0.006$) are shown
by the orange squares, diamonds and triangles, accordingly. They can be
compared to the numerically obtained deformations presented in Figure 13(b).

The interplay of breadth of the region $\bar{a}$, over which the traction
$\bar{T}$ is distributed, and the traction amplitude $\bar{T}_{max}$ can be
better understood from the simpler case suggested by Svetovoy et al. [52]. The
traction can be modeled as a Heaviside function
$\bar{T}_{max}\Theta(\bar{a}-\bar{x})$. Solving the corresponding equation
$\cfrac{d^{4}\bar{w}^{\dagger}}{d\bar{x}^{4}}-\mathcal{S}\bar{T}_{max}\Theta(\bar{a}-\bar{x})=0$,
one can arrive at the deformation at the symmetry axis (see Supplementary
Material for the solution):

$\bar{w}^{\dagger}_{max}=\cfrac{\mathcal{S}\bar{T}_{max}}{24}\left[5\bar{X}_{w}^{4}+(\bar{a}-\bar{X}_{w})^{4}-6\bar{X}_{w}^{2}(\bar{a}-\bar{X}_{w})^{2}\right].$
(17)

The calculated deformations in the form of a
$\mathcal{S}\bar{T}_{max}$-$\bar{a}$-map are shown in Figure 14(a, b). One can
see that a concentrated traction with higher amplitude can still cause the
same deformation as a spread traction having much smaller amplitude. In Figure
14(b), the analytical solutions for the cases with $\phi=0.4,0.5,0.6,0.7$ are
shown by orange squares and triangles for $\mathcal{S}=0.03$ and
$\mathcal{S}=0.006$, accordingly. The value of breadth $\bar{a}$ cannot be
obtained analytically. It has been evaluated using the numerical results as
$\bar{a}=\cfrac{\int_{0}^{\bar{X}_{w}}\bar{T}d\bar{x}}{\sup_{\bar{x}\in[0,\bar{X}_{w}]}\bar{T}}$,
so the forces per unit length exerted onto the wall of the wedge by the real
traction (obtained numerically) and the Heaviside traction are equal. Although
the deformations obtained analytically slightly depart from the numerical
values (Figure 13, b), they generally support the findings, being located
along the lines of the equal deformations.

##  4 Conclusions

In the present work, we model edge wetting and discuss the shape of rivulets
in rigid and soft open wedge channels. We show that when intermolecular
interactions (surface forces) are taken into account, the validity of the
well-known Concus-Finn condition breaks down. The steady-state rivulets can
exist in the wedges even when the liquid-gas interface is concave. In order to
obtain and quantify the rivulet profiles, we use the disjoining pressure
concept and solve the Derjaguin equation. We demonstrate that the extent of
the rivulet drastically depends on the humidity of the surrounding atmosphere,
wedge geometry and surface force parameters.

We use Kirchhoff-Love theory in order to model the deformation of the soft
wedge induced by the presence of the rivulet. We show that the relative
humidity does not affect the deflection of the walls of the wedge, whereas the
geometry of the wedge does.

We present simple analytical models allowing to predict the limiting height of
the rivulet in the wedge and to estimate the deformation resulting from the
interplay of the amplitude of the traction exerted onto the walls of the wedge
and the breadth of the region, over which the traction is distributed.

Acknowledgements
The authors gratefully acknowledge the financial support from the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) – Priority Program
“Dynamic Wetting of Flexible, Adaptive and Switchable Surfaces” (SPP 2171),
Project Number 422792679.

## Supplementary Materials

### Disjoining pressure isotherm

The advantage of using the disjoining pressure isotherm containing a
dispersive power-law part $\cfrac{A}{H^{n}}$ ($n$ defines the dispersive
repulsion/attraction) and a weak-overlap electrostatic/structural exponential
tail $Ke^{-H/\chi}$ is the easier evaluation of $H_{1}$, $H_{2}$ as well as
its minima and maxima. The function $\Pi(H)$ is differentiable for
$H\in(0;\infty)$ with the derivatives

$\Pi^{(k)}(H)=(-1)^{k}\left(A\frac{\prod_{i=0}^{i=k-1}(n+i)}{H^{n+k}}-K\frac{e^{-H/\chi}}{\chi^{k}}\right),$
(18)

where $k$ is the order of the derivative. For the van der Waals forces, we
have $n=3$, whereas for the Casimir forces, we have $n=4$. The roots of
equation $\Pi^{(k)}(H)=0$, that is

$A\frac{\prod_{i=0}^{i=k-1}(n+i)}{H^{n+k}}-K\frac{e^{-H/\chi}}{\chi^{k}}=0,$
(19)

where $k\geq 1$, can be found using Lambert $\mathcal{W}$-function with
branches $l=0$ and $l=-1$, read as

$H_{c\\{k,l\\}}=-(n+k)\chi\mathcal{W}_{l}\left[-\frac{1}{(n+k)\chi}\left(\frac{A\chi^{k}\prod_{i=0}^{i=k-1}(n+i)}{K}\right)^{\frac{1}{(n+k)}}\right]$
(20)

For the case when $k=0$, we obtain zeros of the disjoining pressure isotherm

$H_{c\\{0,0\\}}=H_{1}=-n\chi\mathcal{W}_{0}\left[-\frac{1}{n\chi}\left(\frac{A}{K}\right)^{\frac{1}{n}}\right]$
(21)

$H_{c\\{0,-1\\}}=H_{2}=-n\chi\mathcal{W}_{-1}\left[-\frac{1}{n\chi}\left(\frac{A}{K}\right)^{\frac{1}{n}}\right]$
(22)

When $k=1$, zeros of the derivative of the disjoining pressure isotherm read

$H_{c\\{1,0\\}}=H_{c1}=-(n+1)\chi\mathcal{W}_{0}\left[-\frac{1}{(n+1)\chi}\left(\frac{nA\chi}{K}\right)^{\frac{1}{(n+1)}}\right]$
(23)

$H_{c\\{1,-1\\}}=H_{c2}=-(n+1)\chi\mathcal{W}_{-1}\left[-\frac{1}{(n+1)\chi}\left(\frac{nA\chi}{K}\right)^{\frac{1}{(n+1)}}\right]$
(24)

### Inlet profile construction

The inlet profile is assumed to be macroscopic. Thus, it results from the
solution of the Young-Laplace equation:

$\frac{d^{2}\bar{h}(\bar{x})}{d\bar{x}^{2}}=\bar{p}_{c},$ (25)

where $\bar{p}_{c}$ is the capillary pressure. The boundary conditions are
$\cfrac{d\bar{h}}{d\bar{x}}=0$ at $\bar{x}=0$,
$\bar{h}=\bar{H}_{ads}+\bar{X}_{w}\tan\beta$ at $\bar{x}=\bar{X}_{w}$.
Integrating (25), one arrives at

$\bar{h}(\bar{x})=\bar{p}_{c}\frac{\left(\bar{x}^{2}-\bar{X}_{w}^{2}\right)}{2}+\bar{X}_{w}\tan\beta+\bar{H}_{ads}.$
(26)

This solution, however, does not satisfy the boundary condition for the
Derjaguin equation for the rivulet, since it does not contain wetting film, to
which the interface relaxes (Figure 15). Therefore, we must introduce a point
$x_{t}$, where the transition to the wetting film occurs. Since the
macroscopic profile does not immeadiately reaches the wetting film, at the
transition point $\bar{x}_{t}$ it has the thickness
$\bar{H}_{ads}+c_{1}\bar{H}_{ads}$, where $c_{1}$ is a constant. Equation (26)
will be rewritten as

$\bar{h}(\bar{x})=\bar{p}_{c}\frac{\left(\bar{x}^{2}-\bar{x}_{t}^{2}\right)}{2}+\bar{x}_{t}\tan\beta+\bar{H}_{ads}+c_{1}\bar{H}_{ads}.$
(27)

Figure 15: Schematic representation of the inlet profile.

The value of the capillary pressure can be found from the condition on the
contact angle $\theta_{w}$:
$\cfrac{d\bar{h}}{d\bar{x}}=\tan(\beta-\theta_{w})$ at $\bar{x}=\bar{x}_{t}$.
The capillary pressure is obtained as
$\bar{p}_{c}=\cfrac{\tan(\beta-\theta_{w})}{\bar{x}_{t}}$, and the shape of
the interface for $\bar{x}\leq\bar{x}_{t}$ is written

$\bar{h}(\bar{x})=\frac{\tan(\beta-\theta_{w})}{\bar{x}_{t}}\frac{\left(\bar{x}^{2}-\bar{x}_{t}^{2}\right)}{2}+\bar{x}_{t}\tan\beta+\bar{H}_{ads}+c_{1}\bar{H}_{ads}.$
(28)

In order to make the transition smooth, we assume that the interface is the
plane wetting film when
$\bar{x}\geq\bar{x}_{t}+c_{2}\left(\bar{X}_{w}-\bar{x}_{t}\right)$ and
interpolate the region
$\bar{x}\in\left(\bar{x}_{t};\bar{x}_{t}+c_{2}\left(\bar{X}_{w}-\bar{x}_{t}\right)\right)$.

The parameters $\bar{x}_{t}$, $c_{1}$, and $c_{2}$ of the inlet profile 28
used for the simulations are presented in Table 2. Slight variation of the
parameters did not lead to significant changes in $\bar{H}(0,\bar{y})$.

$\bar{x}_{t}$ | $c_{1}$ | $c_{2}$ | Geometry
---|---|---|---
Data in Figures 3-5, 7, 10 |  |  |
$0.95\bar{X}_{w}$ | $0.6$ | $0.650$ | All configurations presented
Data in Figure 8 |  |  |
$0.950\bar{X}_{w}$ | $0.6$ | $0.650$ |  $\bar{X}_{w}=7.0$, $\beta=\theta$, $\phi=0.3$
$0.631\bar{X}_{w}$ | $0.3$ | $0.030$ |  $\bar{X}_{w}=7.0$, $\beta=2\theta$, $\phi=0.3$
$0.465\bar{X}_{w}$ | $0.3$ | $0.030$ |  $\bar{X}_{w}=7.0$, $\beta=3\theta$, $\phi=0.3$
$0.364\bar{X}_{w}$ | $0.3$ | $0.025$ |  $\bar{X}_{w}=7.0$, $\beta=4\theta$, $\phi=0.3$
$0.294\bar{X}_{w}$ | $0.3$ | $0.015$ |  $\bar{X}_{w}=7.0$, $\beta=5\theta$, $\phi=0.3$
Data in Figure 9 |  |  |
$0.950\bar{X}_{w}$ | $0.6$ | $0.650$ |  $\bar{X}_{w}=7.0$, $\beta=2\theta$, $\phi=0.5$
$0.950\bar{X}_{w}$ | $0.6$ | $0.600$ |  $\bar{X}_{w}=10.6$, $\beta=2\theta$, $\phi=0.5$
$0.950\bar{X}_{w}$ | $0.6$ | $0.550$ |  $\bar{X}_{w}=14.1$, $\beta=2\theta$, $\phi=0.5$
$0.950\bar{X}_{w}$ | $0.6$ | $0.500$ |  $\bar{X}_{w}=17.6$, $\beta=2\theta$, $\phi=0.5$
$0.950\bar{X}_{w}$ | $0.6$ | $0.450$ |  $\bar{X}_{w}=21.1$, $\beta=2\theta$, $\phi=0.5$
$0.950\bar{X}_{w}$ | $0.6$ | $0.380$ |  $\bar{X}_{w}=24.6$, $\beta=2\theta$, $\phi=0.5$
$0.950\bar{X}_{w}$ | $0.6$ | $0.350$ |  $\bar{X}_{w}=28.2$, $\beta=2\theta$, $\phi=0.5$
Table 2: Parameters of the inlet profile 28 used for the simulations.

### Limiting height of the rivulet for
the linearized disjoining pressure isotherm

When the wetting film is very thin: $\bar{H}_{ads}=\phi\bar{H}_{1}$ with
$\phi\ll 1$, one can expect the limiting height of the rivulet $\bar{H}_{w}$
to be smaller than the height $\bar{H}_{c1}$ (from 23 with $n=3$)

$H_{c1}=-4\chi\mathcal{W}_{0}\left[-\frac{1}{4\chi^{\frac{3}{4}}}\left(\frac{3A}{K}\right)^{\frac{1}{4}}\right],$
(29)

where the disjoining pressure isotherm finds its minimum.

Figure 16:  (a) Schematic representation of the disjoining pressure isotherm
(dashed dark blue line) and its linearized branch (thick solid red line); (b)
region used for the evaluation of integral in (30).

In this case, consideration of only the first stable branch of the disjoining
pressure isotherm is sufficient. Due to its shape, this branch can be fairly
represented by a straight line connecting points
$(\bar{H}_{ads};\Pi\left(\bar{H}_{ads}\right))$ and
$(\bar{H}_{w};\Pi\left(\bar{H}_{w}\right))$ where
$\Pi\left(\bar{H}_{w}\right)$ (Figure 16, a). We also assume that simplified
isotherm crosses $\bar{H}$-axis in the immediate vicinity to $\bar{H}_{1}$ so
it can be assumed that the point $\left(\bar{H}_{1},0\right)$ belongs to this
straight line.

Equation

$\frac{\tan^{2}\beta}{2}=\bar{\Pi}\left(\bar{H}_{ads}\right)\left(\bar{H}_{w}-\bar{H}_{ads}\right)-\int_{\bar{H}_{ads}}^{\bar{H}_{w}}\bar{\Pi}(\bar{H})d\bar{H}$
(30)

can be rewritten in the following form

$\tan^{2}\beta=2\bar{\Pi}\left(\bar{H}_{ads}\right)\left(\bar{H}_{w}-\bar{H}_{1}\right)+2\Pi\left(\bar{H}_{ads}\right)\left(\bar{H}_{1}-\bar{H}_{ads}\right)-2\int_{\bar{H}_{ads}}^{\bar{H}_{w}}\bar{\Pi}(\bar{H})d\bar{H},$
(31)

where the last term can be easily evaluated when the disjoining pressure
isotherm is linearized (Figure 16, b)

$2\int_{\bar{H}_{ads}}^{\bar{H}_{w}}\bar{\Pi}(\bar{H})d\bar{H}=\bar{\Pi}\left(\bar{H}_{ads}\right)\left[\left(\bar{H}_{1}-\bar{H}_{ads}\right)-\frac{\left(\bar{H}_{w}-\bar{H}_{1}\right)^{2}}{\bar{H}_{1}-\bar{H}_{ads}}\right].$
(32)

After some elementary arithmetical operations with (31) and (32), one obtains

$\left(\bar{H}_{w}-\bar{H}_{ads}\right)^{2}=\frac{\tan^{2}\beta\left(\bar{H}_{1}-\bar{H}_{ads}\right)}{\bar{\Pi}\left(\bar{H}_{ads}\right)},$
(33)

or, introducing the parameter $\phi$,

$\bar{H}_{w}=\phi\bar{H}_{1}+\tan\beta\sqrt{\cfrac{\bar{H}_{1}(1-\phi)}{\bar{\Pi}\left(\phi\bar{H}_{1}\right)}}=\bar{H}_{ads}+\tan\beta\sqrt{\zeta^{-2}},$
(34)

where $-\bar{\zeta}^{2}$ is the slope of the first branch of the simplified
disjoining pressure isotherm. Thus, among two rivulets having the same wetting
film thickness $\bar{H}_{ads}$, the limiting height $\bar{H}_{w}$ can be
expected to be smaller for the rivulet, whose disjoining pressure isotherm has
a larger slope of the first stable branch (alpha-branch).

Figure 17: Dependence of the limiting height $\bar{H}_{w}$ of the rivulet on
$\phi$. Blue solid line corresponds to the solution of equation (34), teal
circle markers correspond to $\bar{H}_{w}$ obtained from the analytical
solution of the Derjaguin equation (as described in the manuscript, Eq. (7)
and Eq. (8)), solid red line corresponds to $H_{w}$ obtained from the
numerical solution of the Derjaguin equation (solved as a boundary value
problem). The isotherm A has been employed for the calculations.

The limiting height $\bar{H}_{w}$ evaluated using equation (34) is presented
in Figure 17 alongside the numerical results. The dashed black line depicts
$\bar{H}_{c1}$ setting the limiting $\phi$, at which the assumptions used for
(34) are valid. It can be seen that (34) generally predicts the dependence
well for $\phi\lessapprox 0.5$, even despite the fact that the actual shape of
the disjoining pressure isotherm has been ignored. We highlight that for the
sake of simplicity, we have not accounted for the height dependence of the
surface tension, although films with $\phi<0.25$ are thinner than $1$ nm.

### Deformation of the soft wedge:
Solution of Kirchhoff-Love equation for a Heaviside traction

Let us consider the case when a step-like traction exerted onto the substrate
near its end, in the region $\bar{x}\in[0;\bar{a}]$. The traction can be
modeled as a Heaviside function $\bar{T}=\bar{T}_{max}\Theta(\bar{a}-\bar{x})$
of amplitude $\bar{T}_{max}$. The solid will respond to the traction by
deflecting. The deflection $\bar{w}^{\dagger}$ satisfies the Kirchhoff-Love
equation:

$\cfrac{d^{4}\bar{w}^{\dagger}}{d\bar{x}^{4}}-\mathcal{S}\bar{T}_{max}\Theta(\bar{a}-\bar{x})=0.$
(35)

We rewrite 35 using Laplace transforms. For the fourth derivative, we have

$\mathcal{L}\left[\cfrac{d^{4}\bar{w}^{\dagger}(\bar{x})}{d\bar{x}^{4}}\right]=s^{4}\mathcal{L}[\bar{w}(\bar{x})]-s^{3}\bar{w}^{\dagger}(0)-s^{2}\bar{w}^{\dagger^{\prime}}(0)-s\bar{w}^{\dagger^{\prime\prime}}(0)-\bar{w}^{\dagger(3)}(0),$
(36)

where primes denote the derivatives with respect to $s$. From the boundary
conditions, we have $\bar{w}^{\dagger^{\prime}}(0)=0$ and
$\bar{w}^{\dagger(3)}(0)=0$. The Heaviside function can be written in the form

$\Theta(\bar{a}-\bar{x})=1-\Theta(\bar{x}-\bar{a}),$ (37)

and its Laplace transform reads

$\mathcal{L}[1-\Theta(\bar{x}-\bar{a})]=\left(\cfrac{1}{s}-\cfrac{e^{-as}}{s}\right).$
(38)

Therefore, we arrive at the equation in $s$-domain

$\mathcal{L}[\bar{w}^{\dagger}(\bar{x})]=\frac{\mathcal{S}\bar{T}_{max}\left(1-e^{-as}\right)}{s^{5}}+\frac{w^{\prime\prime}(0)}{s^{3}}+\frac{w(0)}{s},$
(39)

which can be easily written in the original $\bar{x}$-domain by the reversed
Laplace transform

$\bar{w}^{\dagger}(\bar{x})=\frac{1}{24}\mathcal{S}\bar{T}_{max}\left[\bar{x}^{4}-(\bar{a}-\bar{x})^{4}\Theta(\bar{x}-\bar{a})\right]+\frac{\bar{x}^{2}}{2}\bar{w}^{\dagger^{\prime\prime}}(0)+\bar{w}^{\dagger}(0).$
(40)

Due to the presence of the Heaviside function, it is more convenient to
consider two intervals:

$\bar{w}^{\dagger}(\bar{x})=\begin{cases}\cfrac{1}{24}\mathcal{S}\bar{T}_{max}\left[\bar{x}^{4}-(\bar{a}-\bar{x})^{4}\right]+\cfrac{\bar{x}^{2}}{2}\bar{w}^{\dagger^{\prime\prime}}(0)+\bar{w}^{\dagger}(0),&\text{if
$\bar{x}\leq\bar{a}$}\\\
\cfrac{1}{24}\mathcal{S}\bar{T}_{max}\bar{x}^{4}+\cfrac{\bar{x}^{2}}{2}\bar{w}^{\dagger^{\prime\prime}}(0)+\bar{w}^{\dagger}(0).&\text{if
$\bar{x}>\bar{a}$}.\end{cases}$ (41)

From the two boundary conditions left $\bar{w}^{\dagger}(\bar{X}_{w})=0$ and
$\bar{w}^{\dagger^{\prime\prime}}(\bar{X}_{w})=0$, we obtain

$\bar{w}^{\dagger}(0)=-\frac{\mathcal{S}\bar{T}_{max}\bar{X}_{w}^{4}}{24}+\frac{\mathcal{S}\bar{T}_{max}\left(\bar{a}-\bar{X}_{w}\right)^{4}}{24}-\frac{\mathcal{S}\bar{T}_{max}\bar{w}^{\dagger^{\prime\prime}}(0)\bar{X}_{w}^{2}}{2},$
(42)

$\bar{w}^{\dagger^{\prime\prime}}(0)=\frac{\mathcal{S}\bar{T}_{max}\left(\bar{a}-\bar{X}_{w}\right)^{2}}{2}-\frac{\mathcal{S}\bar{T}_{max}\bar{X}_{w}^{2}}{2}.$
(43)

The deformation at the symmetry axis reads

$\bar{w}^{\dagger}(0)=\frac{\mathcal{S}\bar{T}_{max}}{24}\left[5\bar{X}_{w}^{4}+(\bar{a}-\bar{X}_{w})^{4}-6\bar{X}_{w}^{2}(\bar{a}-\bar{X}_{w})^{2}\right].$
(44)

Figure 18: (Schematic representation (a) of the wedge; (b) of the distribution
of the disjoining pressure over the wedge; (c) influence of the increasing
wedge inclination angle on the distribution of the disjoining pressure over
the wedge.  Figure 19: Influence of the relative humidity $RH$ on the
deformation of the wedge.

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