# Capillary Bridges on Liquid Infused Surfaces

Alvin C. M. Shek  Department of Physics, Durham University, Durham, DH1 3LE,
UK  Ciro Semprebon  Department of Mathematics, Physics and Electrical
Engineering, Northumbria University, Newcastle upon Tyne NE1 8ST, UK  Jack R.
Panter  Department of Physics, Durham University, Durham, DH1 3LE, UK  Halim
Kusumaatmaja  Email<EMAIL_ADDRESS>Department of Physics,
Durham University, Durham, DH1 3LE, UK

###### Abstract

We numerically study two-component capillary bridges formed when a liquid
droplet is placed in between two liquid infused surfaces (LIS). In contrast to
commonly studied one-component capillary bridges on non-infused solid
surfaces, two-component liquid bridges can exhibit a range of different
morphologies where the liquid droplet is directly in contact with two, one or
none of the LIS substrates. In addition, the capillary bridges may lose
stability when compressed due to the envelopment of the droplet by the
lubricant. We also characterise the capillary force, maximum separation and
effective spring force, and find they are influenced by the shape and size of
the lubricant ridge. Importantly, these can be tuned to increase the effective
capillary adhesion strength by manipulating the lubricant pressure, Neumann
angle, and wetting contact angles. As such, LIS are not only “slippery”
parallel to the surface, but they are also “sticky” perpendicular to the
surface.

##  I Introduction

The subject of capillary bridges has received much attention in the literature
due to their ubiquity in nature and engineering applications. For example,
capillary bridges between parallel flat plates Carter (1988); Klingner et al.
(2004); Broesch and Frechette (2012) have been considered for applications in
liquid transfer Kang et al. (2008); Dodds et al. (2011), wet adhesion device
Lee et al. (2007); Vogel and Steen (2010), and for the formation of curved
polymeric particles Wang and McCarthy (2015), while those between curved solid
bodies Kohonen et al. (2004); De Lazzer et al. (1999); Pakarinen et al. (2005)
occur in wet granular materials Mitarai and Nori (2006); Scheel et al. (2008)
and in atomic force microscopy experiments Xu et al. (1998); Xiao and Qian
(2000); Wei and Zhao (2007). Capillary bridges also play important roles in
the physiology of numerous insects and animals, such as in the adhesive pads
of Asian Weaver ants Beutel and Gorb (2001); Federle et al. (2001) and how
shorebirds trap and consume prey inside their beaks Prakash et al. (2008).
Furthermore, the stability of liquid bridge shapes and their force-separation
relations have been investigated for smooth and patterned surfaces De Souza et
al. (2008); Nosonovsky and Bhushan (2008); Zhuang and Ju (2015); Banerjee et
al. (2012), as well as in the presence of one and multiple droplets Banerjee
et al. (2012); Gui and Marchon (1995); De Souza et al. (2008).

In contrast to previous works, where the capillary bridges comprise of a
single liquid component, here we study the case where they have two liquid
components, especially in the context of the so-called liquid infused surfaces
(LIS) or slippery liquid infused porous surfaces (SLIPS). These are a novel
class of functional surfaces constructed by infusing textured or porous
materials with wetting lubricants Lafuma and Quéré (2011); Smith et al.
(2013); Wong et al. (2011). Many applications of LIS exploit the fact that, on
LIS, liquids move easily parallel to the substrate Lafuma and Quéré (2011);
Smith et al. (2013); Wong et al. (2011); Kim et al. (2012); Anand et al.
(2012); Wexler et al. (2015); Villegas et al. (2019). This “slippery” nature
results in numerous advantageous properties, such as self-cleaning, enhanced
heat transfer, anti-fouling, and anti-icing. Here, we focus instead on the
adhesive properties of droplets when displaced perpendicularly to the
substrate. Since the adhesive force now has contributions which originate not
just from the liquid droplet but also from the lubricant, we will argue that
LIS are “sticky” in the perpendicular direction.

Figure 1: Capillary bridge geometry and parameters. (a) Capillary bridge
between two LIS, where $H$ is the separation distance, and $\hat{r}_{perp}$,
$\hat{r}_{par}$ are the unit vectors perpendicular and parallel to the
substrate. Panels (b) and (c) show the capillary bridge cases where the
lubricant partially and completely wetting the solid corrugations,
respectively. Additionally, panel (b) illustrates the definition of the
Neumann angles, $\theta_{l}$, $\theta_{d}$, $\theta_{g}$, and the effective
wetting contact angles, $\theta^{\rm eff}_{ld}$ and $\theta^{\rm eff}_{lg}$.
Panel (c) also gives a geometrical interpretation of the apparent contact
angle, $\theta_{app}$, and the lubricant film thickness, $d$.

To study capillary adhesion on LIS, we will consider a liquid droplet
sandwiched between two parallel LIS, as illustrated in Fig. 1 (a). For
simplicity, we assume that gravity is negligible and consider symmetric
surface properties such that the Neumann ($\theta_{l}$, $\theta_{d}$ and
$\theta_{g}$) and wetting contact angles ($\theta^{\rm eff}_{ld}$ and
$\theta^{\rm eff}_{lg}$), illustrated in Fig. 1 (b) and (c), will be the same
for the top and bottom plates. In this work, we focus on equilibrium rather
than dynamic effects, obtained by quasi-statically varying the separation
between the substrates.

The fact that capillary bridges on LIS involve two liquid components, instead
of just one, leads to several interesting phenomena. Firstly, two-component
liquid bridges can exhibit a wider range of interfacial morphologies and
topologies. Morphological transitions can be triggered by compressing and
stretching the capillary bridges. Secondly, to characterise the “stickiness”
of LIS, we calculate the capillary force and effective spring constant of the
liquid bridges. We find both the size and shape of the lubricant ridge to play
an important role. In the limit of vanishing ridge size, we observe that the
adhesion force converges to that of a perfectly smooth surface. Since
perfectly smooth surfaces are challenging to realise, LIS can be considered as
an excellent alternative to mimic their properties in the limit of small
ridge. Additionally, LIS can also be tuned to have stronger adhesion than an
equivalent smooth surface.

This paper is organized as follows. In Section II we introduce our theoretical
framework to study the capillary bridges. Then, in Section III, we organize
our results in terms of the morphology diagrams, capillary forces and spring
constants of the capillary bridges as function of the relevant control
parameters. Finally, we summarize our results in the Conclusion Section.

##  II Model and Method

This section is divided into three parts. In the first part we introduce the
model for capillary bridges between two LIS. Then, we describe our numerical
scheme to compute the equilibrium bridge morphologies. Finally, we provide the
expressions for calculating the capillary force and spring constant.

###  II.1 Model

We begin by considering the case of a one-component liquid bridge. The total
free energy Brinkmann and Lipowsky (2002); Kusumaatmaja and Lipowsky (2010) is
given by

$\displaystyle E_{\rm{smooth}}$ $\displaystyle=$
$\displaystyle\gamma_{dg}A_{dg}+\gamma_{ds}A_{ds}+\gamma_{gs}A_{gs}-\Delta
P_{dg}V_{d},$ $\displaystyle=$
$\displaystyle\gamma_{dg}A_{dg}-\gamma_{dg}\cos\theta_{dg}A_{ds}-\Delta
P_{dg}V_{d}+{\rm{C}_{1}},$

where the subscripts $s$, $d$, and $g$ stand for solid, drop and gas
respectively; $\gamma_{ij}$ and $A_{ij}$ are the surface tension and
interfacial area between components $i$ and $j$, $V_{d}$ is the drop volume;
and $\Delta P_{dg}$ is the pressure difference between the drop and gas
components. We assume the liquid drop is non-volatile, so that its volume is a
conserved quantity. For a fixed drop volume $V_{d}$, the term $\Delta P_{dg}$
can be interpreted as a Lagrange multiplier. Rearranging the terms, and
assuming the capillary bridge is confined by ideal and smooth surfaces, the
liquid wettability is expressed by the material contact angle,
$\cos\theta_{dg}=(\gamma_{gs}-\gamma_{ds})/\gamma_{dg}$. The remaining terms
sum up to a constant, ${\rm{C}_{1}}=\gamma_{gs}(A_{gs}+A_{ds})$, which do not
alter the liquid bridge morphology, and can be neglected.

Now we turn our attention to a capillary bridge sandwiched between two LIS
substrates, as depicted in Fig. 1(a). Here the lubricant ridges are formed
around the droplet due to capillary action. As in our previous works Semprebon
et al. (2017); Sadullah et al. (2018, 2020), we divide the total free energy
contributions into two parts, the fluid-fluid and fluid-solid contributions,
such that $E_{\rm LIS}=E_{\rm FF}+E_{\rm FS}$. The fluid-fluid contributions
are

$E_{\rm FF}=\gamma_{dg}A_{dg}+\gamma_{ld}A_{ld}+\gamma_{lg}A_{lg}-\Delta
P_{dg}V_{d}-\Delta P_{lg}V_{l}.$ (2)

Similar to the one-component liquid bridge case, the drop volume is assumed to
be constant. For the lubricant (subscript $l$), we assume the rest of the
substrate infused by the lubricant provides a virtually infinite reservoir. In
experiments the lubricant exchange between the ridge and surrounding substrate
can occur on a rather slow timescale due to the strong viscous dissipation in
the thin lubricant layer Kreder et al. (2018). However, we will assume this
exchange is still much faster than the typical variation in the control
parameters used in this work, such as the separation between the two LIS.
Consequently, we employ the pressure ensemble for the lubricant, parameterised
by the pressure jump $\Delta P_{lg}$ at the lubricant-gas interface. The term
$-\Delta P_{lg}V_{l}$ in the free energy represents the energy cost for
drawing additional lubricant from the reservoir.

When the lubricant only partially wets (PW) the solid surface, see Fig. 1(b),
we can write the fluid-solid contributions as

$\displaystyle E_{\rm FS(PW)}$ $\displaystyle=$ $\displaystyle\gamma^{\rm
eff}_{ds}A^{\rm pr}_{ds}+\gamma^{\rm eff}_{ls}A^{\rm pr}_{ls}+\gamma^{\rm
eff}_{gs}A^{\rm pr}_{gs}$ $\displaystyle=$ $\displaystyle\gamma_{ld}A^{\rm
pr}_{ds}\cos\theta^{\rm eff}_{ld}-\gamma_{lg}(A^{\rm pr}_{ds}+A^{\rm
pr}_{ls})\cos\theta^{\rm eff}_{lg}+{\rm{C_{2}}},$

where $A^{\rm pr}_{is}$ is the projected interfacial area between fluid $i$
and the substrate. The index $i=d,g$ represents either the drop or gas phase.
It is worth emphasizing that the substrate is effectively a composite of the
underlying rough solid surface and the imbibed lubricant. For simplicity, we
will not resolve the details of the composite surface. Instead, we simply
assume this gives rise to an effective average surface tension $\gamma^{\rm
eff}_{is}=f\gamma_{is}+(1-f)\gamma_{il}$, with $f$ the fraction of the
projected solid area exposed to the drop or gas phase. We also define the
effective contact angle $\cos\theta^{\rm eff}_{ij}=(\gamma^{\rm
eff}_{js}-\gamma^{\rm eff}_{is})/\gamma_{ij}$ between phases $i$ and $j$ on
the composite solid-lubricant substrate. The constant term
${\rm{C}_{2}}=\gamma^{\rm eff}_{gs}(A^{\rm pr}_{ds}+A^{\rm pr}_{ls}+A^{\rm
pr}_{gs})$ can be neglected for the same reason that it will not affect the
resulting morphology.

The limit of $\theta_{ld}$, $\theta_{lg}\rightarrow 0^{\circ}$ describes the
complete wetting (CW) case, Fig. 1(c), where the lubricant will form a thin
layer above the surface. The thickness of the lubricant layer is determined by
the intermolecular interactions between lubricant and solid Daniel et al.
(2017). These interactions are also often called the disjoining pressure term.
To extend our calculations to this limit, we introduce

$E_{\rm FS(CW)}=\int\frac{B}{12\pi d(\mathbf{r})^{2}}\text{ d}A,$ (4)

where $B$ is a constant and $d$ is the thickness of the lubricant layer. A
different choice for this contribution would not alter the results of this
study, as long as the film thickness is small compared to the size of both the
drop and the lubricant ridge. In practice, we also do not observe major
differences in the results for full wetting and small but finite contact
angles.

On LIS, the drop-gas interface does not come in contact with the solid, due to
the ubiquitous presence of a lubricant ridge, see Fig. 1. At the top of this
lubricant ridge, there is a triple contact line where the drop-gas (dg)
interface meets the lubricant-gas (lg) and lubricant-drop (ld) interfaces. The
three Neumann angles, $\theta_{l}$, $\theta_{d}$ and $\theta_{g}$, are related
to the interfacial tensions via

$\displaystyle\frac{\gamma_{ld}}{\sin\theta_{g}}=\frac{\gamma_{dg}}{\sin\theta_{l}}=\frac{\gamma_{lg}}{\sin\theta_{d}}.$
(5)

Consequently, on LIS the definition of a material contact angle needs to be
adapted. Two alternatives are possible: either taking the slope of the drop-
gas interface immediately above the triple line, or by estimating the slope of
the (virtual) drop-gas interface if it were to continue within the lubricant
ridge. Following Semprebon et al. Semprebon et al. (2017), we employ the first
definition, as illustrated in Fig. 1(c). This definition has an advantage that
the triple line is directly visible, and therefore it is easy to be measured
in experiments. The second definition instead would better approximate the
overall drop shape in the case of large ridges Gunjan et al. (2020). In the
limit of vanishing lubricant ridges, the two definitions coincide and the
apparent contact angle can be expressed as an average of the effective
lubricant-drop and lubricant-gas contact angles, weighted by the ratios of
surface tensions:

$\displaystyle\cos\theta_{app}$ $\displaystyle=-\cos\theta^{\rm
eff}_{ld}\frac{\gamma_{ld}}{\gamma_{dg}}+\cos\theta^{\rm
eff}_{lg}\frac{\gamma_{lg}}{\gamma_{dg}}$ $\displaystyle=-\cos\theta^{\rm
eff}_{ld}\frac{\sin\theta_{g}}{\sin\theta_{l}}+\cos\theta^{\rm
eff}_{lg}\frac{\sin\theta_{d}}{\sin\theta_{l}}.$ (6)

Unless specified otherwise, this equation defines the apparent angle used
throughout the paper. It allows us to compare the wettability of droplets on
LIS with those on homogeneous surfaces, employing $\theta_{app}$ for the
former and $\theta_{dg}$ for the latter.

Throughout the paper, we non-dimensionalize the separation by the
characteristic length scale of the deposited droplet,
$s=\left(3V_{d}/4\pi\right)^{1/3}$; the lubricant pressure by a characteristic
Laplace pressure for a droplet with surface tension $\gamma_{dg}$ and volume
$V_{d}$, given by $\gamma_{dg}/s$; and the capillary force by a characteristic
force, $2\pi\gamma_{dg}s$.

###  II.2 Numerical Method

We compute the liquid bridge morphologies in mechanical equilibrium by
numerically minimising the free energy. To do this, we employ the public
domain software Surface Evolver Brakke (1992, 1996). Owing to the symmetry of
the problem, we employ an effective 2D model with rotational symmetry
Semprebon et al. (2017); De Souza et al. (2008, 2008, 2008) .

The three fluid interfaces between drop, lubricant and gas are modelled by
discrete segments joining in a point representing the drop-lubricant-gas
triple line. The discrete segments are weighted appropriately by the relevant
surface tensions.

In this effective 2D model, each of the lubricant-drop and lubricant-gas
interface also meets the substrate at a point corresponding to the substrate-
lubricant-drop and substrate-lubricant-gas contact lines in the case of finite
wettability. The relevant fluid-solid energy terms can then be calculated
given the positions of the substrate-lubricant-drop and substrate-lubricant-
gas contact lines.

In the case of full wetting, there are no substrate-lubricant-drop and
substrate-lubricant-gas contact lines. Instead, the fluid-solid energy is
accounted by numerically integrating the disjoining pressure term. In this
case we ensure that the lubricant layer has reached a constant thickness far
away from the droplet in the case of full wetting. For this reason, we also
apply a symmetric boundary condition with the external wall for the lubricant-
gas interface.

Importantly, for all the results presented in this work, the drop volume is
fixed by an integral constraint, while the lubricant exchange between the
ridge and the surrounding reservoir is modelled by the $-\Delta P_{lg}V_{l}$
term as described in Eq. (2).

###  II.3 Calculating the Forces and the Spring Constant

Figure 2:  Schematic of the capillary bridge profile. Here, $y_{1}(x)$ (solid
line) is the drop-gas interface, $y_{2}(x)$ (dashed line) is the lubricant-gas
interface, and $y_{3}(x)$ (dotted line) is the lubricant-drop interface.

A natural measure of the “stickiness” of capillary bridges between LIS
surfaces is given by the capillary force exerted by the bridges on the two
substrates. This force can be directly calculated from the profile of a liquid
bridge in mechanical equilibrium, as illustrated in Fig. 2. The detailed
derivation of the expression for the force is reported in the Appendix. There
are two natural locations to evaluate the capillary force. At the substrate,
the capillary force is given by

$\displaystyle F$
$\displaystyle=2\pi\left[\gamma_{lg}\frac{y_{2}}{(1+{y_{2}^{\prime}}^{2})^{\frac{1}{2}}}+\gamma_{ld}\frac{y_{3}}{(1+{y_{3}^{\prime}}^{2})^{\frac{1}{2}}}\right]_{x=L}$
$\displaystyle\quad-\pi\left[\Delta P_{dg}y_{3}^{2}+\Delta
P_{lg}(y_{2}^{2}-y_{3}^{2})\right]_{x=L}.$ (7)

The first term can be interpreted as the surface tension forces of both the
lubricant-gas and lubricant-droplet interfaces acting perpendicular to the
substrate, while the second term corresponds to force contributions due to the
droplet and lubricant pressures relative to the surrounding gas pressure.

Alternatively, the capillary force can be conveniently computed at the
symmetry plane located at $x=0$ in the schematic diagram in Fig 2, in which
$y_{1}^{\prime}=0$. In this case, the force expression simplifies to

$\displaystyle F$ $\displaystyle=$ $\displaystyle
2\pi\left[\gamma_{dg}\frac{y_{1}}{(1+{y_{1}^{\prime}}^{2})^{\frac{1}{2}}}\right]_{x=0}-\pi\left.\Delta
P_{dg}y_{1}^{2}\right|_{x=0},$ (8) $\displaystyle=$ $\displaystyle
2\pi\gamma_{dg}r_{d}-\pi\Delta P_{dg}r_{d}^{2}.$

Here $r_{d}$ is the radial distance of the droplet-gas interface. The correct
implementation of these calculations has been validated by comparing them to
numerical derivatives of the energy with respect to the plate separation.

A complementary measure of the “stickiness” is given by the spring constant of
the bridge, $K_{eq}$, when the capillary force vanishes. To calculate
$K_{eq}$, we can employ a simple linear relation between the calculated force
and the displacement from the capillary bridge equilibrium distance. Following
Hooke’s Law,

$\displaystyle F=-K_{eq}(H-H_{eq}),$

where $H_{eq}$ is the capillary bridge separation when $F=0$. By fitting how
$F$ varies with $H$ near the equilibrium distance, $K_{eq}$ can be obtained.

##  III Results: Study of capillary bridges between liquid infused surfaces

###  III.1 Morphology Classes

Figure 3: Three broad categories of the energy minimization results. The top
and bottom rows illustrate the typical morphologies for hydrophilic
($\theta_{app}<90^{\circ}$) and hydrophobic ($\theta_{app}>90^{\circ}$)
capillary bridges, respectively. First, at small separation, the capillary
bridge is unstable and the droplet is enveloped by the lubricant. Here, the
droplet can be (a) detached from or (b) attached to the substrates. Second,
stable capillary bridges are formed. For hydrophilic bridges, the droplet can
be (c) detached from both substrates or (e) attached to both substrates; while
for hydrophobic bridges, the droplet can be (d) attached to both substrates or
(f) attached to only one substrate. Third, at large separation, the capillary
bridges can become unstable by (g) breaking the droplet or (h) transferring to
one of the substrates.

In this work we performed systematic numerical energy minimization varying the
plate separation $H$, the apparent contact angle $\theta_{app}$, the lubricant
Neumann angle $\theta_{l}$, and the lubricant pressure $\Delta P_{lg}$. We can
broadly group the bridge morphologies on LIS into three categories. Referring
to panels in Fig. 3 these are: (a-b) unstable bridges due to envelopment
instability, (c-f) stable bridges, and (g-h) unstable bridges due to capillary
break up or detachment. Strictly speaking, the minimization routine fully
converges only for morphologies labelled as stable. The unstable morphologies
are depicted to illustrate the capillary bridges as the instabilities take
place. Due to the large number of control parameters, we have computed
morphology diagrams in the form of 2D slices of the parameter space, as
depicted in Fig. 4. For each slice, all the remaining parameters are fixed.

To start, let us vary the apparent angle $\theta_{app}$ and plate separation
$H/s$ in Fig. 4(a). Here we have fixed the wetting contact angles for the
lubricant, $\theta_{ld}^{\rm eff}=\theta_{lg}^{\rm eff}=5^{\circ}$, the
lubricant Neumann angle, $\theta_{l}=160^{\circ}$, and the normalised
lubricant pressure, $\Delta P_{lg}s/\gamma_{dg}=-0.62$. Such lubricant
pressure results in lubricant ridge that is small compared to the droplet size
for the majority of the results shown in Fig. 4(a), as commonly observed in
experiments Smith et al. (2013); Schellenberger et al. (2015); McHale et al.
(2019). However, the lubricant ridge can become comparable in size with the
droplet at small separations.

Figure 4:  Morphology phase diagrams for a capillary bridge between two LIS.
In panel (a), we vary the apparent contact angle $\theta_{app}$ and normalised
separation $H/s$, and fix the lubricant Neumann angle $\theta_{l}=160^{\circ}$
and normalised lubricant pressure $\Delta P_{lg}s/\gamma_{dg}=-0.62$. In panel
(b), we vary $H/s$ and $\theta_{l}$, keeping $\theta_{app}=100^{\circ}$ and
$\Delta P_{lg}s/\gamma_{dg}=-0.62$. In panel (c), $H/s$ and $\Delta
P_{lg}s/\gamma_{dg}$ are varied, with $\theta_{app}=100^{\circ}$ and
$\theta_{l}=10^{\circ}$. In all panels, the wetting contact angles for the
lubricant are set at $\theta_{ld}^{\rm eff}=\theta_{lg}^{\rm eff}=5^{\circ}$.
The red region denotes stable morphologies when the droplet remains attached
to both LIS substrates, blue when the droplet is detached from both
substrates, and purple when the droplet is detached from one of the
substrates. The green and yellow regions in panels (a) and (c) represent
regions where multiple morphologies are possible.

The largest, red colored region in Fig. 4(a) identifies the most common case
of a stable liquid bridge in contact with both LIS substrates, as illustrated
in Fig. 3(d-e). Moving to the bottom-left of the phase diagram by decreasing
both the apparent angle and the plate separation, there also exists a
morphology in which the droplet is detached from both substrates for
hydrophilic capillary bridges ($\theta_{app}<90^{\circ}$), and where lubricant
menisci connect the droplet and the substrates. Here, as we reduce the plate
separation, we find the Neumann triangle at the top of the lubricant ridge
rotates such that the drop-gas interface becomes more aligned to the direction
normal to the substrate. This in turn leads to a detachment of the lubricant-
drop interface from the substrate. Such morphology is shown in Fig. 3(c) and
corresponds to the blue region on the phase diagram in Fig. 4(a). The green
region highlights the parameter space in which both morphologies are possible.

Further decreasing the plate separation leads to an instability related to the
coalescence of the two lubricant menisci. The lubricant floods the space in
between the LIS substrates and envelopes the droplet. In this work, since we
use the pressure ensemble for the lubricant, this continues indefinitely.
However, in practice, especially in the limit of starved lubricant regime, it
will do so until the lubricant from the drop surrounding is depleted. This
envelopment instability is unique to LIS where the capillary bridge has two
liquid components. Inspecting the data reported in Fig. 4(a), we further
observe that the envelopment instability occurs at larger separation as we
decrease the apparent angle. Typically, the envelopment occurs with the
droplet detached from the surfaces for hydrophilic bridges
($\theta_{app}<90^{\circ}$), as illustrated in Fig. 3(a); and with the droplet
attached to the surfaces for hydrophobic bridges ($\theta_{app}>90^{\circ}$),
illustrated in Fig. 3(b).

The capillary bridges also become unstable at large plate separation. Similar
to conventional one-component capillary bridges on non-infused surfaces, this
can occur in two ways. The drop can break in two as in Fig. 3(g), typically
observed for hydrophilic capillary bridges. Alternatively, the droplet may
detach from one of the substrates, as illustrated in Fig. 3(h), commonly found
for hydrophobic bridges. Since we ignore gravity in this work, the droplet can
move to the top or bottom substrate with equal probability triggered by
numerical noise during minimization. The detailed dynamics of these
instabilities are beyond the scope of this work.

Next, we study the role of the lubricant Neumann angle $\theta_{l}$, in
particular to characterize the envelopment instability at small $H/s$. Fig.
4(b) shows a phase diagram as a function of $H/s$ and $\theta_{l}$. We now fix
$\theta_{app}=100^{\circ}$, while the other parameters are set as before in
Fig. 4 (a). It is worth noting that the other two Neumann angles are no longer
independent variables when an apparent angle and one of the Neumann angles are
defined. Here we employ the lubricant Neumann angle as control parameter, as
it provides a good measure of the shape of the lubricant ridge.

Figure 5:  (a) and (b) Force contour plots as a function of contact angle
($\theta_{dg}$ or $\theta_{app}$) and substrate separation for smooth solid
surfaces and LIS, respectively. For the latter, we have set
$\theta_{l}=160^{\circ}$, $\theta_{ld}^{\rm eff}=\theta_{lg}^{\rm
eff}=5^{\circ}$ and $\Delta P_{lg}s/\gamma_{dg}=-0.62$. The white lines trace
the separation for which the force is zero. (c) Capillary force as a function
of the substrate separation for contact angles $\theta_{app}=60^{\circ}$,
$100^{\circ}$ and $120^{\circ}$. The LIS cases are represented by triangles
and circles for when the droplet is attached and detached from the substrates,
while those for smooth surfaces are represented by solid lines.

We clearly observe that the system is more prone to envelopment instability
for smaller $\theta_{l}$. Considering the force balance at the droplet-
lubricant-gas contact line, this is because smaller $\theta_{l}$ implies the
droplet-gas surface tension becomes more dominant in magnitude over the
droplet-lubricant and lubricant-gas surface tensions. As such, the system can
lower its free energy by reducing any droplet-gas interface. In contrast, the
detachment instability at large $H/s$ is relatively insensitive to
$\theta_{l}$.

In Fig. 4(a) we observe that the stability region for bridge morphologies
where the drop is detached from both substrates, as depicted in Fig. 3(c), is
limited to $\theta_{app}<90^{\circ}$. We now investigate whether this
morphology can occur also for $\theta_{app}>90^{\circ}$ for a different
combination of parameters. We find the critical parameter is the lubricant
pressure $\Delta P_{lg}$. Thus, we present a morphology phase diagram as a
function of separation $H/s$ and $\Delta P_{lg}s/\gamma_{dg}$ in Fig. 4(c).
Here we fix $\theta_{ld}^{\rm eff}=\theta_{lg}^{\rm eff}=5^{\circ}$,
$\theta_{app}=100^{\circ}$, and $\theta_{l}=10^{\circ}$. Interestingly, the
morphology where the droplet is detached from both substrates is not found to
be stable for hydrophobic bridges. Rather, we find a morphology where the
droplet is only directly in contact with one of the substrates, as illustrated
in Fig. 3(f), when we increase $\Delta P_{lg}s/\gamma_{dg}$ (becoming less
negative). Physically, increasing $\Delta P_{lg}s/\gamma_{dg}$ corresponds to
reducing the energy costs of drawing lubricant from the reservoir, leading to
lubricant meniscus size which is comparable to the droplet size. In the phase
diagram of Fig. 4(c), this morphology is indicated by the purple region,
bounded from below by the envelopment instability and from above by the
droplet detachment instability. The two instability boundaries terminate in a
cusp located at $\Delta P_{lg}s/\gamma_{dg}=-0.053$ and $H/s=2.896$. It is
also worth commenting that the morphology in Fig. 3(f) is observed for a wide
range of $\theta_{l}$. Here we have fixed $\theta_{l}=10^{\circ}$ because
lower $\theta_{l}$ allows a broader range of plate separation in which the
morphology is stable. The yellow region corresponds to parameter regime where
the morphologies of Fig. 3(d) and (f) are both possible.

###  III.2 Capillary Forces

Having discussed the spectrum of different liquid bridge morphologies between
two LIS substrates, we now address the key question of how the forces exerted
by stable capillary bridges are affected by the presence of the lubricant. We
will start by considering cases in which the lubricant ridge is small and
flat, which we can achieve by setting large lubricant Neumann angle and low
lubricant pressure. Here we use $\theta_{l}=160^{\circ}$, and $\Delta
P_{lg}s/\gamma_{dg}=-0.62$. Fig. 5(a) and (b) show the (normalised) force
contour plots as a function of the (apparent) contact angle and plate
separation for a capillary bridge on smooth surfaces and LIS, respectively.
For the latter, we focus on cases where the droplet is directly in contact
with both LIS substrates. The parameter space explored here is the same as in
the morphology diagram in Fig. 4(a).

In this limit in which the lubricant ridge is small and flat, the capillary
forces obtained are quantitatively very similar for LIS and for smooth solid
surfaces. For hydrophilic capillary bridges ($\theta_{app}<90^{\circ}$), the
force is always attractive and monotonically decreases as one increases $H/s$;
while for hydrophobic capillary bridges ($\theta_{app}>90^{\circ}$), the force
is repulsive at short separation and attractive at large separation. Following
Eqs. II.3 and 8, this is because while the surface tension contributions are
always attractive, the pressure contributions can be either attractive or
repulsive. The contour of zero force, corresponding to the equilibrium
separation distance, $H_{eq}$, is marked by the white lines in Fig. 5 (a) and
(b). From these results, we can also conclude that the apparent contact angle
is a suitable control parameter to compare capillary bridges on conventional
smooth, non-infused surfaces and LIS in the limit of small lubricant ridges.

The main difference between one-component liquid bridges on non-infused
surfaces and two-component liquid bridges on LIS, as shown in Fig. 5, is found
at small $H/s$ for $\theta_{app}<90^{\circ}$, as we approach the parameter
space in which the detached droplet and enveloped morphologies in Figs. 3(a)
and (c) are preferred. To better illustrate this point, Fig. 5(c) plots the
capillary force $F$ as a function of separation $H$ for three contact angles,
$60^{\circ}$, $100^{\circ}$ and $120^{\circ}$. The LIS cases are shown with
triangular and circular markers for when the droplet is attached and detached
from the substrates, while those for one-component capillary bridge on smooth
solid surfaces are shown with solid lines. As illustrated in Fig. 5(c), we see
the force deviates strongly at small $H/s$ for $\theta_{app}=60^{\circ}$. This
is due to significant amount of lubricant being drawn from the reservoir,
increasing the attractive force of the capillary bridge.

This deviation illustrates the importance of the lubricant ridge in the
resulting capillary force. Since the shape and size of the ridge depends on
the Neumann and lubricant contact angles, as well as the lubricant pressure,
we further investigate how these parameters affect the strength of the
capillary adhesion of the droplet on LIS. To study these effects we have
chosen to employ an apparent angle of $\theta_{app}=100^{\circ}$ as an
exemplary case. Such contact angle value is quite typical in experiments Smith
et al. (2013); Schellenberger et al. (2015), though of course the apparent
contact angle can be varied depending on the materials involved.

Figure 6:  (a-c) Force contour plots as a function of the lubricant Neumann
angle $\theta_{l}$ and normalised separation $H/s$. In panels (a-c), we have
respectively set $\theta^{\rm eff}_{ld}$, $\theta^{\rm eff}_{lg}=0^{\circ}$,
$5^{\circ}$, and $30^{\circ}$. In all cases, we have
$\theta_{app}=100^{\circ}$ and $\Delta P_{lg}s/\gamma_{dg}=-0.62$. For the
complete wetting case, we have also chosen a Hamaker constant of
$B/\gamma_{dg}s^{2}=2.6\times 10^{-8}$. (d) Force contour plot as a function
of $H/s$ for a one-component capillary bridge with $\theta_{dg}=100^{\circ}$.

The effect of altering the wetting contact angles for the lubricant
$\theta_{lg}^{\rm eff}$, $\theta_{ld}^{\rm eff}$ and the Neumann angle
$\theta_{l}$ at fixed apparent angle is shown in Fig. 6. Fig. 6(a)-(c) show
contour plots of the (normalised) force as function of $\theta_{l}$ and $H/s$
for different lubricant contact angles. In all three plots we have chosen
$\Delta P_{lg}s/\gamma_{dg}=-0.62$. For comparison, we also show the force
contour plot for one-component capillary bridge with $\theta_{dg}=100^{\circ}$
in Fig. 6(d).

The case with completely wetting lubricant is shown in Fig. 6(a). Here we need
to include the physics of disjoining pressure, and we have chosen a Hamaker
constant of $B/\gamma_{dg}s^{2}=2.6\times 10^{-8}$ to ensure the thickness of
the lubricant layer is several orders of magnitude, $O(10^{3})$, smaller than
the droplet size $V_{d}^{1/3}$, as is typical in experiments Daniel et al.
(2017); Kreder et al. (2018). We have also studied cases with larger
disjoining pressure (data not shown), and the only difference is that the
lubricant layer thickness offsets the value of $H/s$ when comparing the
capillary forces. From the plot in Fig. 6(a), the maximum capillary force and
separation distance increases with decreasing $\theta_{l}$.

The same tendency in the variation of $\theta_{l}$ is observed in Fig. 6(b)
and (c), where we now employ partially wetting lubricants with
$\theta_{lg}^{\rm eff}=\theta_{ld}^{\rm eff}=5^{\circ}$ and $30^{\circ}$. We
note that, for panel (c), there is no available solution for
$\theta_{l}>120^{\circ}$ because the three angles forming the lubricant ridge
must sum to less than $180^{\circ}$. Comparing the results in Fig. 6(a)-(d),
it is also clear that the capillary adhesion is stronger on LIS than for one-
component capillary bridge on smooth surfaces, and that this “stickiness” is
amplified for more wetting lubricants. We find the capillary force on LIS can
be up to $40\%$ higher than that on smooth solid surfaces. Taking the results
in Fig. 6 together, the shape of the lubricant ridge is an important factor to
manipulate the strength of the capillary force.

Figure 7: (a) Normalised capillary force as a function of normalised
separation for several values of normalised lubricant pressure and an apparent
angle of $\theta_{app}=100^{\circ}$. For comparison, the force curves for one-
component capillary bridge is also shown for $\theta_{dg}=90^{\circ}$,
$100^{\circ}$. (b-c) Force contour plots as a function of the normalised
separation and normalised lubricant pressure when the droplet is connected to
(b) both LIS substrates and (c) only one of the LIS substrates. Here we have
used $\theta_{l}=10^{\circ}$, and $\theta^{\rm eff}_{ld},\theta^{\rm
eff}_{lg}=5^{\circ}$.

Next, we address the issue of the size of the lubricant ridge, which can be
tuned by varying the lubricant pressure. Fig. 7 (a) illustrates the case for a
droplet with $\theta_{app}=100^{\circ}$, $\theta_{l}=10^{\circ}$,
$\theta_{ld}^{\rm eff}=\theta_{lg}^{\rm eff}=5^{\circ}$, while $\Delta
P_{lg}s/\gamma_{dg}$ is varied. Here, the force curves only exist over a
certain range of $H/s$, bounded by the envelopment instability from below and
the detachment instability from above, as discussed in the previous section.

It is known that the apparent angle as defined in Eq. II.1 is valid for the
limit of vanishing ridge. As we make $\Delta P_{lg}s/\gamma_{dg}$ less
negative, the lubricant ridge increases in size in comparison to the droplet
size, and this can lead to a reduction in the measured geometric apparent
angle Semprebon et al. (2017); Sadullah et al. (2018). For hydrophobic cases,
this angle will always be bounded from below by $90^{\circ}$. Therefore, to
test whether the increase in the capillary force is simply due to changes in
the effective apparent angle, we also plot force curves for one-component
capillary bridges in Fig. 7(a) for contact angles $\theta_{dg}=90^{\circ}$,
$100^{\circ}$, to allow for easy comparison. We find that we can even exceed
the force from the $90^{\circ}$ case for certain separation distances. This
implies that the apparent angle alone is insufficient to fully describe the
increase in capillary force.

Fig. 7(b) and (c) summarise the force contour plots as a function of $\Delta
P_{lg}s/\gamma_{dg}$ and $H$ for the stable morphologies where the droplet is
directly attached to both LIS substrates and only one of the substrates
respectively. In both scenarios, the capillary force increases with increasing
$\Delta P_{lg}s/\gamma_{dg}$ at constant $H/s$, and decreases with $H/s$ at
constant $\Delta P_{lg}s/\gamma_{dg}$. However, comparing the two contour
plots, the capillary force is affected differently. This is most clearly
illustrated by the force curves in Fig. 7(a). The cases of $\Delta
P_{lg}s/\gamma_{dg}=-0.17$ and $-0.12$ have a different shape, with a greater
dip in force near their maximum separation where droplet detachment occurs.
This is because these cases correspond to morphology in Fig. 3(f), where the
droplet is only directly connected to one of the LIS substrates. Here, the
capillary force is dominated by the drawn lubricant. In contrast, the cases of
$\Delta P_{lg}s/\gamma_{dg}=-0.56$ and $-0.37$ correspond to the morphology in
Fig. 3(d), where the droplet dominates the capillary force response.

###  III.3 Equilibrium Separation and Spring Constant

To further corroborate our findings on the adhesion of a liquid droplet on
LIS, in this subsection we will consider how the equilibrium separation,
$H_{eq}$, and spring constant, $K_{eq}$, of the capillary bridge compare to
those for one-component capillary bridge on non-infused, smooth solid
surfaces.

Figure 8: In both plots, the circles and triangles correspond to $\theta^{\rm
eff}_{ld},\theta^{\rm eff}_{lg}=5^{\circ}$ and $30^{\circ}$, respectively;
while the straight and dashed lines represent $\Delta
P_{lg}s/\gamma_{dg}=-0.31$ and $-0.62$. The apparent contact angle is set at
$\theta_{app}=120^{\circ}$. (a) Equilibrium separation $H_{eq}$ as a function
of the lubricant Neumann angle. (b) Effective spring constant, $K_{eq}$ as a
function of the lubricant Neumann angle. The results for one-component
capillary bridge on smooth surfaces are shown by the dotted lines.

We first note that, for one-component capillary bridge on non-infused
surfaces, there is always an equilibrium separation in which the capillary
force $F=0$ for $\theta_{dg}>90^{\circ}$. In contrast, this is not the case on
LIS. In particular, the envelopment instability often kicks in before the
capillary bridge can reach its equilibrium separation distance for small but
negative value of $\Delta P_{lg}s/\gamma_{dg}$. For
$\theta_{app}=100^{\circ}$, this instability significantly limits the range of
parameters where the capillary bridge can reach $F=0$. As such, in this
subsection, we have chosen to focus on results with
$\theta_{app}=120^{\circ}$. The qualitative behaviour is the same for
different apparent angles.

Fig. 8 summarises our findings. In panel (a), we first study how the
equilibrium separation is affected by the Neumann angle, $\theta_{l}$, the
wetting contact angles for the lubricant, $\theta_{lg}^{\rm eff}$,
$\theta_{ld}^{\rm eff}$, and the lubricant pressure, $\Delta
P_{lg}s/\gamma_{dg}$. For comparison, the one-component liquid bridge case is
also shown as the black dotted line. Overall, we find the equilibrium
separation is slightly smaller for LIS than for a one-component bridge. From
the four curves for $H_{eq}$, we also observe that we approach the one-
component liquid bridge case as we increase $\theta_{l}$ and make $\Delta
P_{lg}s/\gamma_{dg}$ more negative. This is consistent with our observation in
the previous subsection that the lubricant ridge morphology impacts the
capillary force, so that as the ridge vanishes, LIS are equivalent to smooth
surfaces in this limit.

We also study the effective spring constant, $K_{eq}$, around the equilibrium
separation in Fig. 8(b). $K_{eq}$ is always greater than that for the one-
component liquid bridge implying that, not only capillary forces are larger on
LIS, but also the resulting capillary bridges are stiffer due to the lubricant
drawn from the LIS substrates.

##  IV Conclusion

In summary we have studied two-component capillary bridges formed when a
liquid droplet is sandwiched between two LIS. The lubricant ridge morphology
was shown to be important in determining the capillary force, the maximum
substrate separation, and the effective spring constant, which are all
enhanced by lowering the lubricant-gas pressure difference $\Delta P_{og}$,
the lubricant Neumann angle $\theta_{l}$, and the lubricant contact angles
$\theta^{\rm eff}_{ld},\theta^{\rm eff}_{lg}$ relative to the drop and gas
phases. The parameter $\Delta P_{lg}$ affected the overall size of the
lubricant ridge, $\theta_{l}$ changed the shape of the ridge, and $\theta^{\rm
eff}_{ld},\theta^{\rm eff}_{lg}$ tuned the adhesion between the lubricant and
the solid surface. Varying these parameters we found that, for a given drop
apparent contact angle, a capillary force could be achieved on LIS which was
up to 40% higher than that for the one-component case. Conversely, in the
limit of a vanishing lubricant ridge, the properties of the two-component
capillary bridge tended to those of a one-component bridge between smooth,
solid substrates.

In contrast to the one-component capillary bridge, a two-component capillary
bridge also had a richer set of possible morphologies. Due to the presence of
the lubricant, the capillary bridge could be stable with the liquid droplet
directly in contact with two, one or none of the LIS substrates. At small
separation, we have further identified a new envelopment instability, where
the lubricant fills the space in between the two substrates.

While the facts that LIS have low contact angle hysteresis and are thus
“slippery” parallel to the surface have been extensively discussed in the
literature, here we point out that, at the same time, they are potentially
“sticky” perpendicular to the surface. This suggests LIS are a unique class of
liquid repellent surfaces. For instance, on superhydrophobic surfaces, liquid
droplets are suspended on top of the surface textures in the Cassie-Baxter
state, and as a result, they move easily both parallel and perpendicular to
the surface Roach et al. (2008); Quéré (2008); Kusumaatmaja and Yeomans
(2007). In the Wenzel state, liquid drops penetrate the surface corrugations
leading to both large contact angle hysteresis and strong adhesion Roach et
al. (2008); Quéré (2008); Kusumaatmaja and Yeomans (2007), and so they are
“sticky” in both directions.

There are a number of directions for future work. In this work we have assumed
the lubricant reservoir to be virtually infinite. However, LIS substrates can
be prepared so that the lubricant layer has minimal thickness. In this case,
the amount of lubricant which can be gathered will be limited, and this can
affect the resulting liquid bridge morphologies. Another avenue is to extend
our study to include cases where the droplet is encapsulated by a thin film of
lubricant. This occurs when the wetting parameter,
$S_{dl}=\gamma_{dg}-\gamma_{ld}-\gamma_{lg}<0$. It is also interesting to
consider potential applications exploiting the “sticky”-“slippery” behaviour
of LIS, such as for microfluidic devices, fog harvesting and heat transfer.
For instance, Launay et al. recently showed that LIS can capture droplets
against gravity while allowing droplet transport via self-propulsion driven by
topographical gradients Launay et al. (2020). For these problems, it is
important that we not only consider the static liquid bridge morphologies, but
also the rich dynamics of the droplet deformation and lubricant flow.

Acknowledgments \- We thank Yonas Gizaw for useful discussions. ACMS is
supported by EPSRC’s Centre for Doctoral Training in Soft Matter and
Functional Interfaces (EP/L015536/1; ACMS). HK and JRP acknowledge funding
from Procter & Gamble. CS acknowledges support from Northumbria University
through the Vice-Chancellor’s Fellowship Programme and EPSRC (EP/S036857/1)
for funding.

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##  Appendix A Thermodynamic Derivation of the Force

The capillary force exerted by the liquid bridge can be directly calculated
from the equilibrium profile. To derive the force, consider variation of the
free energy of the system induced by varying the plate separation,
$H\rightarrow H+\delta H$, following a similar approach carried out for one-
component liquid bridges Kusumaatmaja and Lipowsky (2010); Carter (1988);
Brinkmann and Lipowsky (2002). Since we are considering symmetric surfaces, it
is mathematically more convenient to set the origin of our coordinate system
to be at the middle of the liquid bridge. If we define the $y_{i}(x)$ to
represent each interface’s radial distance from the axial centre of the
bridge, then the surfaces are located at $\pm H/2$, as defined in Fig 2. As
such, exploiting symmetry, we only need to consider the profile curves from
$x=0$ to $x=L=H/2$. Under this variation, the position of the drop-lubricant-
gas triple contact line varies from $l\rightarrow l+\delta l$, and
correspondingly the profile curves also vary as
$y_{i}(x)\rightarrow\bar{y}_{i}(x)=y_{i}(x)+g_{i}(x)$.

The free energy upon extending the plate separation by $\delta H=2\delta L$ is
given by

$\displaystyle\frac{E_{\rm LIS}(L+\delta L)}{2\pi}$
$\displaystyle=\int_{0}^{l+\delta
l}\gamma_{dg}\bar{y}_{1}(x)(1+({\bar{y}_{1}^{\prime}})^{2}(x))^{1/2}-\frac{\Delta
P_{dg}}{2}\bar{y}_{1}^{2}(x)\text{ d}x$ $\displaystyle+\int_{l+\delta
l}^{L+\delta
L}\gamma_{lg}\bar{y}_{2}(x)(1+({\bar{y}_{2}^{\prime}})^{2}(x))^{1/2}-\frac{\Delta
P_{lg}}{2}\bar{y}_{2}^{2}(x)\text{ d}x$ $\displaystyle+\int_{l+\delta
l}^{L+\delta
L}\gamma_{ld}\bar{y}_{3}(x)(1+({\bar{y}_{3}^{\prime}})^{2}(x))^{1/2}-\frac{\Delta
P_{dl}}{2}\bar{y}_{3}^{2}(x)\text{ d}x$
$\displaystyle+\gamma_{ld}\cos\theta^{\rm eff}_{ld}\bar{y}_{3}^{2}(L+\delta
L)-\gamma_{lg}\cos\theta^{\rm eff}_{lg}\bar{y}_{2}^{2}(L+\delta L).$ (9)

The first variation of the free energy leads to the well-known Euler-Lagrange
or shape equation for each of the fluid interface, $y_{i}(x)$, as derived in
Kusumaatmaja and Lipowsky (2010); Carter (1988); Brinkmann and Lipowsky
(2002). In addition, it leads to boundary conditions at $x=l$ and $x=L$. For
the former, considering the variation in $\delta l$ gives us

$\displaystyle\left[\frac{\gamma_{dg}}{(1+({y_{1}^{\prime}})^{2})^{1/2}}-\frac{\gamma_{lg}}{(1+({y_{2}^{\prime}})^{2})^{1/2}}-\frac{\gamma_{ld}}{(1+({y_{3}^{\prime}})^{2})^{1/2}}\right]y(l)$
$\displaystyle-\left[\frac{\Delta P_{dg}}{2}-\frac{\Delta
P_{lg}}{2}-\frac{\Delta P_{dl}}{2}\right]y(l)^{2}=0.$ (10)

where we have used $y(l)=y_{i}(l)$. The second term in the above equation is
identically zero, since the pressure terms cancel one another. Furthermore,
the square bracket in the first term corresponds to the force balance of the
surface tensions projected in the $x$-direction. Similarly, the variation in
$\delta y(l)$ results in

$\left[\frac{\gamma_{dg}y_{1}^{\prime}}{(1+({y_{1}^{\prime}})^{2})^{1/2}}-\frac{\gamma_{lg}y_{2}^{\prime}}{(1+({y_{2}^{\prime}})^{2})^{1/2}}-\frac{\gamma_{ld}y_{3}^{\prime}}{(1+({y_{3}^{\prime}})^{2})^{1/2}}\right]=0,$
(11)

which is the force balance of the surface tensions projected in the
$y$-direction. Note that for these derivations we have used the geometrical
relation $g_{i}(l+\delta l)=\delta y_{i}(l)-y_{i}^{\prime}(l)\delta l$.

We can do a similar analysis for the boundary condition at $x=L$. Considering
the variation in $\delta y_{2}(L)$ and $\delta y_{3}(L)$, we recover the
expected contact angle equations for the lubricant-gas and lubricant droplet
interfaces,

$\displaystyle\left[\gamma_{lg}\frac{y_{2}{y_{2}^{\prime}}}{(1+{y_{2}^{\prime}}^{2})^{\frac{1}{2}}}\right]_{x=L}-\gamma_{lg}\cos\theta_{lg}^{\rm
eff}y_{2}(L)=0,$ (12)
$\displaystyle\left[\gamma_{ld}\frac{y_{3}{y_{3}^{\prime}}}{(1+{y_{3}^{\prime}}^{2})^{\frac{1}{2}}}\right]_{x=L}+\gamma_{ld}\cos\theta_{ld}^{\rm
eff}y_{3}(L)=0.$ (13)

Finally, the variation in $\delta L$ leads to the equation for the force
exerted by the capillary bridge, given by

$\displaystyle F$
$\displaystyle=\left.\frac{\delta(2E)}{\delta(2L)}\right|_{x=L}$
$\displaystyle=2\pi\left[\gamma_{lg}\frac{y_{2}}{(1+{y_{2}^{\prime}}^{2})^{\frac{1}{2}}}+\gamma_{ld}\frac{y_{3}}{(1+{y_{3}^{\prime}}^{2})^{\frac{1}{2}}}\right]_{x=L}$
$\displaystyle\quad-\pi\left[\Delta P_{dg}y_{3}^{2}+\Delta
P_{lg}(y_{2}^{2}-y_{3}^{2})\right]_{x=L}$ (14)

At equilibrium, the force is constant everywhere along the capillary bridge.
Therefore, it is often convenient to consider the force at the mid-point of
the liquid bridge, where $x=0$ and $y_{1}^{\prime}=0$. In this case, the force
expression simplifies to

$\displaystyle F$ $\displaystyle=$ $\displaystyle
2\pi\left[\gamma_{dg}\frac{y_{1}}{(1+{y_{1}^{\prime}}^{2})^{\frac{1}{2}}}\right]_{x=0}-\pi\left[\Delta
P_{dg}y_{1}^{2}\right]_{x=0},$ (15) $\displaystyle=$ $\displaystyle
2\pi\gamma_{dg}r_{d}-\pi\Delta P_{dg}r_{d}^{2}.$

where $r_{d}$ is the radial distance of the drop-gas interface at the middle
of the bridge as defined as in Fig 1.