# Capillary condensation between non-parallel walls

Alexandr Malijevský  Research group of Molecular Modelling, The Czech Academy
of Sciences, Institute of Chemical Process Fundamentals, 165 02 Prague, Czech
Republic; Department of Physical Chemistry, University of Chemical Technology
Prague, 166 28 Prague, Czech Republic  Jiří Janek  Department of Physical
Chemistry, University of Chemical Technology Prague, 166 28 Prague, Czech
Republic

###### Abstract

We study the condensation of fluids confined by a pair of non-parallel plates
of finite height $H$. We show that such a system experiences two types of
condensation, termed single- and double-pinning, which can be characterized by
one (single-pinning) or two (double-pinning) edge contact angles describing
the shape of menisci pinned at the system edges. For both types of capillary
condensation we formulate the Kelvin-like equation and determine the
conditions under which the given type of condensation occurs. We construct the
global phase diagram revealing a reentrant phenomenon pertinent to the change
of the capillary condensation type upon varying the inclination of the walls.
Asymptotic properties of the system are discussed and a link with related
phase phenomena in different systems is made. Finally, we show that the change
from a single- to a double-pinned state is a continuous transition, the
character of which depends on the wetting properties of the walls.

##  I Introduction

Confining fluids exhibiting liquid-gas phase separation below a critical
temperature $T_{c}$ can profoundly change their phase behaviour RW ; croxton ;
charvolin ; henderson92 ; gelb . If the confinement is formed by a pair of
parallel walls whose dimensions are macroscopic (effectively infinite) that
are a distance $L$ apart, the shift in the liquid-gas phase boundary from the
vapour pressure $p_{\rm sat}$ to $p_{\rm CC}(L)=p_{\rm sat}-\delta p_{\rm
CC}(L)$, due to interplay between surface and finite-size effects, can be
described by the classical Kelvin equation, according to which

$\delta p_{\rm CC}(L)=\frac{2\gamma\cos\theta}{L}\,.$ (1)

Here, $\gamma$ is the liquid-gas surface tension and $\theta$ is Young’s
contact angle pertinent to a macroscopic liquid drop sitting on one of the
isolated walls. This phenomenon of capillary condensation, which persists up
to the capillary critical point at a temperature $T_{c}(L)<T_{c}$ which is
itself a subject of a finite-size change NF81 ; NF83 , has been a subject of
numerous theoretical studies, simulations and experiments, which revealed that
the Kelvin equation remains quantitatively reliable even for slits several
molecular diameters wide fisher ; binder08 ; van ; zhong ; kim ; yang ; geim .

(a) Single-pinned state.

(b) Double-pinned state.

Figure 1: Schematic 2D illustration of two possible condensed states in a slit
formed of two identical walls of finite length $H$, which both make an angle
$\alpha$ with the vertical plane. The widths of the slit openings are $L_{1}$
and $L_{2}>L_{1}$ and the walls (with Young’s contact angle $\theta$) are
macroscopically deep. Panel a) refers to a single-pinned state which can be
characterized by one edge contact angle $\theta_{e}$, whilst panel b) shows a
double-pinned state characterized by two contact angles $\theta_{e}$ and
$\theta_{e}^{\prime}$.

However, the recent advances in nanoscale physics and the challenges
associated with the fabrication of nano-devices called for further elucidation
of phase phenomena induced by objects of microscopic dimensions nano1 . In
particular, a description of condensation in such systems requires a
modification of the original Kelvin equation accounting properly for the
effects at walls boundaries and the system geometry. For instance, for plates
of finite height $H$, but still parallel, it turns out that condensation
appears always closer to saturation (i.e. at a higher pressure) compared to
infinite ($H\to\infty$) walls and that the condensed state can be
macroscopically characterized by an occurrence of two symmetric menisci that
meet the ends of the walls at the _edge contact angle_ $\theta_{e}$, which is
always larger than the corresponding Young contact angle $\theta$ edge_slit .
This can be described using the modified Kelvin-like equation edge_slit ;
depin21

$\delta p_{\rm CC}(L,H)=\frac{2\gamma\cos\theta_{e}}{L}\,,$ (2)

determining the condensation pressure inside the finite slit $p_{\rm
CC}(L,H)=p_{\rm sat}-\delta p_{\rm CC}(L,H)$, at which the edge contact angle
satisfies

$\cos\theta_{e}=\cos\theta-\frac{L}{2H}\left[\sin\theta_{e}+\sec\theta_{e}\left(\frac{\pi}{2}-\theta_{e}\right)\right]\,.$
(3)

In this paper we show that the phase behaviour of confined fluids is
significantly enriched if the confining walls are _non-parallel_ forming a
_truncated wedge_ geometry. In this case, the confinement (sketched in Fig. 1)
is formed of two identical plates of length $H$ that are both inclined
(relative to the vertical, say) by an angle $\alpha$. The model capillary is
thus of finite height with $L_{1}$ and $L_{2}>L_{1}$ denoting the “bottom” and
“top” opening widths. We assume that the depth of the capillary is macroscopic
and that the system is translation-invariant along the walls, implying that
only one principal radius of curvature of the menisci is relevant. Hence, the
model can be thought of either as an extension of that of the finite slit
mentioned above or as a part sliced off from a linear wedge with an opening
angle $2\alpha$ hauge ; rejmer ; wood ; bruschi ; our_wedge . Owing to the
presence of two openings of different widths, the system allows for
condensation in two distinct ways: i) to a _single-pinned_ state (see Fig.
1a), which is characterised by the presence of two menisci, such that one is
pinned at the narrow slit end, while the other is located inside the slit or
ii) to a _double-pinned_ state (see Fig. 1b), in which case both menisci are
located at the slit ends. It is then natural to ask, under which conditions
the given type of capillary condensation, if any, occurs. Associated with it,
there are further natural questions. What are the analogues of the Kelvin
equation for both types of condensation? When does the system cease to
condense? What is the role of the walls surface properties? What is the nature
of the change from a single-pinned to a double-pinned state?

The remainder of the paper is organized as follows. In section II, we present
a macroscopic theory of condensation, separately describing both possible
types. Some of these results are then employed in section III for an analysis
of condensation inside a semi-infinite system terminology  where we primarily
focus on the asymptotic behaviour of the system in the limit of $\alpha\to 0$.
The main findings of our work are described in Section IV where we discuss the
phenomenology of condensation in the general case, construct the global phase
diagram, determine the nature of the depinning transition and present some
analytic predictions that we compare with the numerical calculations. We
conclude in section V with a detailed summary of the results.

##  II Two types of condensation

In this section we formulate the Kelvin-like equations corresponding to both
single-pinning and double-pinning condensation between non-parallel walls. To
this end, we treat our capillary model as an open system which is in
equilibrium with a reservoir of the same subcritical temperature $T$ and the
chemical potential $\mu$. Within this grand-canonical treatment an equilibrium
state corresponds to such a configuration, which minimizes the grand potential
$\Omega$ which we will approximate using a simple macroscopic description.
Clearly, the condition for condensation requires the value of $\Omega$ of the
“empty” (i.e., a gas-like) configuration and the pertinent condensed state, to
be the same. Throughout the paper we will assume that the walls are
“hydrophilic” with Young’s contact angle $\theta<\pi/2$. In the case of
“hydrophobic” walls, $\theta>\pi/2$, capillary condensation would be replaced
by capillary evaporation occurring at a pressure $p>p_{\rm sat}$, with the
role of gas and liquid phases being swapped. Some more details regarding the
derivation of the Kelvin-like equations is given in Appendix A.

###  II.1 Single pinning

We start off with a description of condensation to a single-pinned state. The
condensed system is characterized by the presence of two menisci, with one
located at the narrow end of the walls and the other inside the capillary (see
Fig. 1a). Macroscopically, the shape of the menisci is determined by two
characteristics: the Laplace radius of curvature $R$ and the angle at which
the menisci meet the walls. The Laplace radius, $R=\gamma/\delta p$, is of
course the same for both menisci, where $\delta p=p_{g}-p_{l}$ is the pressure
difference between the ambient gas and the (metastable) condensed liquid
phase. However, while the meniscus inside the capillary meets the walls with
the equilibrium Young contact angle, $\theta$, the other meniscus which is
pinned connects the walls at the _edge_ contact angle, $\theta_{e}$. It is
also clear, that while the location of the pinned meniscus is fixed, the one
inside the capillary is free and its location $X$, varying with $\delta p$,
determines the portion of the capillary filled with liquid (cf. Fig. 1a). The
macroscopic excess grand potential, per unit length, of the condensed system
(relative to the state where only gas is present), can be written as

$\Omega^{\rm ex}_{\rm SP}=\delta pS+\gamma(\ell_{1}+\ell_{2}-2X\cos\theta)\,.$
(4)

Here, the first term is the free-energy cost due to the presence of the
metastable liquid, the next two terms correspond to the surface free energies
due to the presence of the menisci and the final term is the interfacial
liquid-wall free energy where Young’s law has been used. Furthermore,

$\ell_{1}=R(\pi+2\alpha-2\theta_{e})\,,$ (5)

and

$\ell_{2}=R(\pi-2\alpha-2\theta)$ (6)

denote the arc-lengths of the pinned and the free menisci, respectively, and

$S=S_{\rm tot}-S_{1}-S_{2}$ (7)

is the area corresponding to the region occupied by liquid. The latter was
separated into the part $S_{\rm tot}$ of the trapezoid of length $X$,

$S_{\rm tot}=\frac{4R^{2}\cos^{2}(\alpha+\theta)-L_{1}^{2}}{4}\cot\alpha\,,$
(8)

and the parts

$S_{1}=\frac{R^{2}(\pi+2\alpha-2\theta_{e})-RL_{1}\sin(\theta_{e}-\alpha)}{2}$
(9)

and

$S_{2}=\frac{R^{2}(\pi-2\alpha-2\theta)}{2}-R^{2}\sin(\alpha+\theta)\cos(\alpha+\theta)\,,$
(10)

corresponding to circular segments formed by the respective menisci. Finally,
the length of the walls at the contact with liquid is

$X=\frac{2R\cos(\alpha+\theta)-L_{1}}{2\sin\alpha}\,.$ (11)

The location of the coexistence between the low-density state (filled only
with gas) and the single-pinned state is determined from the condition
$\Omega^{\rm ex}_{\rm SP}=0$. From this, it follows that the single-pinned
condensation occurs at the pressure $p_{\rm SP}=p_{\rm sat}-\delta p_{\rm
SP}$, where

$\delta p_{\rm
SP}(L_{1},H,\alpha)=\frac{2\gamma\cos(\theta_{e}-\alpha)}{L_{1}}\,,$ (12)

is the Kelvin-like equation for the condensation partial pressure with the
edge contact angle $\theta_{e}$ given implicitly by the equation

$\cot\alpha\left[\cos^{2}(\alpha+\theta)-\cos^{2}(\alpha-\theta_{e})\right]+\pi-\theta-\theta_{e}+\frac{\sin(2\alpha+2\theta)-\sin(2\alpha-2\theta_{e})}{2}=\frac{2\cos\theta\left[\cos(\alpha+\theta)-\cos(\alpha-\theta_{e})\right]}{\sin\alpha}\,.$
(13)

Note that the location of the condensation can also be expressed in terms of
the chemical potential $\mu_{\rm SP}=\mu_{\rm sat}(T)-\delta\mu_{\rm SP}$,
where $\mu_{\rm sat}(T)$ is the value of the chemical potential at saturation
and $\delta p_{\rm SP}\approx\delta\mu_{\rm SP}(\rho_{l}-\rho_{g})$ where
$\rho_{l}$ and $\rho_{g}$ are the number densities of coexisting liquid and
gas at the given temperature marconi .

###  II.2 Double pinning

Similar considerations lead to the Kelvin-like equation for double-pinned
condensation, where, in contrast to the previous case, both menisci are pinned
at the capillary ends, so that the walls are at contact with liquid along
their whole length (see Fig. 1b). Now, the excess grand potential per unit
length can be written as

$\Omega^{\rm ex}_{\rm DP}=\delta pS+\gamma(\ell_{1}+\ell_{2}-2H\cos\theta)\,,$
(14)

where the arc-length of the shorter meniscus, $\ell_{1}$, is given by (5),
while the arc-length of the meniscus pinned with the edge contact angle
$\theta_{e}^{\prime}(<\theta_{e})$ at the wider end of the capillary is

$\ell_{2}=R(\pi-2\theta_{e}^{\prime}-2\alpha)\,.$ (15)

The area $S=S_{\rm tot}-\Delta S$ corresponding to the volume occupied by
liquid is given by the total area between the walls, $S_{\rm
tot}=H(L_{1}+L_{2})\cos\alpha/2$, reduced by the area of circular segments due
to the menisci

$\Delta
S=R^{2}(\pi-\theta_{e}-\theta_{e}^{\prime})-\frac{R\left[L_{1}\sin(\theta_{e}-\alpha)+L_{2}\sin(\theta_{e}^{\prime}+\alpha)\right]}{2}\,.$
(16)

The edge contact angles are related to the Laplace radius of the menisci
according to

$R=\frac{L_{1}}{2\cos(\theta_{e}-\alpha)}$ (17)

and

$R=\frac{L_{2}}{2\cos(\theta_{e}^{\prime}+\alpha)}\,,$ (18)

which simply follows from the system geometry.

The phase boundary between low-density and double-pinned states occurs at the
pressure $p_{\rm DP}=p_{\rm sat}-\delta p_{\rm DP}$, where

$\delta p_{\rm DP}(L_{1},H,\alpha)=\frac{\gamma}{R_{\rm DP}}\,.$ (19)

Here, the Laplace radius $R_{\rm DP}$ must, besides the geometric relations
(17) and (18), satisfy the thermodynamic condition, $\Omega^{\rm ex}_{\rm
DP}=0$, which yields

$\frac{L_{2}^{2}-L_{1}^{2}}{4}\cot\alpha+R^{2}_{\rm
DP}(\pi-\theta_{e}-\theta_{e}^{\prime})-\frac{(L_{2}-L_{1})R_{\rm
DP}\cos\theta}{\sin\alpha}+\frac{R_{\rm
DP}}{2}\left[L_{1}\sin(\theta_{e}-\alpha)+L_{2}\sin(\theta_{e}^{\prime}+\alpha)\right]=0\,.$
(20)

The phase boundary is thus determined by solving simultaneously Eqs. (17),
(18), and (20) for $\theta_{e}$, $\theta_{e}^{\prime}$ and $R_{\rm DP}$; it is
straightforward to verify that for $\alpha=0$ Eqs. (19) and (20) reduce to (2)
and (3), respectively.

##  III Condensation in semi-infinite slits

Prior describing phase transitions in systems shown in Fig. 1, we begin with
an analysis of semi-infinite systems that correspond to the limit of
$H\to\infty$. In this case, which serves as a pre-requisite to our main
analysis, the situation is rather simple owing to the fact that only single-
pinned condensation is possible. The condition under which the condensation
may occur is determined by an interplay between the thermodynamic angle
$\theta$ and the geometric angle $\alpha$ (and does not depend on $L_{1}$,
which is now the only macroscopic length scale). For $\alpha=0$, the
condensation is possible for any value of the contact angle $\theta\leq\pi/2$
as in the case of a standard infinite slit and the location of the meniscus,
which forms at condensation, is arbitrary semi_inf . However, as soon as
$\alpha>0$, the free-energy balance between the volume and interfacial
contributions stabilizes the distance $X$ between the (lower) meniscus pinned
at the narrow end and the other one inside the capillary. This separation
depends on $\alpha$ in a non-monotonic fashion, which is illustrated in Fig. 2
and which has further repercussions for finite capillaries as will be
discussed later. Now for the given value of $\theta$, the maximum opening
angle $\alpha_{\rm max}$ still allowing for condensation is the one, for which
the condensation occurs at the highest possible pressure below the bulk
coexistence, $\delta p_{\rm SP}=0$, corresponding to flat menisci, implying
that

$\alpha_{\rm max}=\frac{\pi}{2}-\theta\,.$ (21)

Alternatively, the maximum contact angle $\theta_{\rm max}$ allowing for
condensation in the semi-infinite capillary with the fixed opening angle
$\alpha$ is

$\theta_{\rm max}=\frac{\pi}{2}-\alpha\,.$ (22)

The latter condition interpolates between that for capillary condensation in
an infinite narrow slit ($\theta_{\rm max}=\pi/2$) and that for wetting
transition on a planar wall ($\theta_{\rm max}=0$). Also note that it is just
in the marginal case, $\alpha=\alpha_{\rm max}$, when $\theta_{e}=\theta$ and
when _both_ menisci can freely move along the walls without any free energy
change.

Figure 2: Illustration of a non-monotonic behaviour of the meniscus height $X$
at capillary condensation in a semi-infinite system as a function of the
capillary opening angle $\alpha$; for $\theta=0$ and $\theta=45\degree$.
Figure 3: The behaviour of $\theta_{e}$ for small values of the opening angle
$\alpha$ in semi-infinite slits for partially wet walls with $\theta=45$° and
completely wet walls ($\theta=0$). The dotted lines refer to the numerical
solution of Eq. (13), while the solid lines represent the asymptotic results
given by Eqs. (23) and (25), respectively.

Next, we want to make a link between capillary condensation in our system and
that in an infinite narrow slit. To this end, let us write $\Delta p_{\rm
SP}(L_{1},\infty,\alpha)=p_{\rm SP}(L_{1},\infty,\alpha)-p_{\rm CC}(L_{1})$
for the pressure difference between the respective condensation pressures. The
difference is positive for any $\alpha>0$ and we wish to know how it vanishes
in the limit of narrowing the capillary. Here, we have to distinguish between
partially and completely wet walls. For partially wet walls, $\theta>0$, we
find that to lowest order in $\alpha$, the edge contact angle approaches
Young’s contact angle according to

$\theta_{e}=\theta+c(\theta)\sqrt{\alpha}\,,\;\;\;{\rm as}\;\alpha\to 0$ (23)

with the amplitude $c(\theta)=\sqrt{(\pi-2\theta+\sin(2\theta)}/\sin\theta$
and which implies that

$\Delta p_{\rm SP}(L_{1},\infty,\alpha)=\frac{2\gamma
c(\theta)\sin\theta\sqrt{\alpha}}{L_{1}}+{\cal{O}}(\alpha)\,.$ (24)

This contrasts with the complete wetting regime, $\theta=0$, for which we
obtain

$\theta_{e}=\theta+(4\pi\alpha)^{\frac{1}{4}}\,,\;\;\;{\rm as}\;\alpha\to 0$
(25)

and

$\Delta p_{\rm
SP}(L_{1},\infty,\alpha)=\frac{2\gamma\sin\theta(4\pi\alpha)^{\frac{1}{4}}}{L_{1}}+{\cal{O}}(\sqrt{\alpha})\,.$
(26)

The test of the asymptotic relations (23) and (25) against the numerical
results obtained from Eq. (12) for both partial and complete wetting is shown
in Fig. 3.

##  IV Condensation in finite slits

###  IV.1 Phase diagrams

(a)

(b)

Figure 4: a) Global phase diagram specifying phase boundaries displayed in the
$\alpha$-$\theta$ projection for several values of $H$. The areas bounded by
the solid curves referring to a given value of $H$ correspond to a set of
parameters, for which the system condenses to single-pinned states. The area
outside the given solid curve and below the pertinent dashed line corresponds
to a set of parameters, for which the system condenses to a double-pinned
state. Above the dashed lines, representing the curves given by Eq. (27), the
opening angle is too large for the system to condense; these curves tend in
the limit of $H\to\infty$ to the solid black line $\pi/2-\theta$ (cf. Eq.
(27)). b) Same as a) except that in place of the lines showing the
condensation limit, the line connecting the “turning points” corresponding to
$X_{\rm min}(\theta)$ shown in Fig. 6 is displayed. Figure 5: The meniscus
height $X$ at capillary condensation as a function of the opening angle
$\alpha$ for $H/L_{1}=10$ and $\theta=45\degree$. For $\alpha<\alpha_{1}$ and
$\alpha>\alpha_{2}$, $X=H$, meaning that the system is in a double-pinned
state. For $\alpha_{1}<\alpha<\alpha_{2}$, the system condenses to a single-
pinned state and $X(\alpha)$ is non-monotonic with a minimum at
$\alpha=\alpha_{\rm min}\approx 9.7$°, at which $X=X_{\rm min}\approx
5.8\,L_{1}$ (cf. Fig. 6).  Figure 6: The dependence of the minimal meniscus
height $X$ of a condensed, single-pinned state on $\theta$.

We now turn to non-parallel slits formed of walls of finite length $H$. In
contrast to the semi-infinite case, there are now two types of condensation
and we wish to specify conditions determining their realization. These will be
expressed in the parameter space of $H$, $\alpha$ and $\theta$, with
$L_{2}=L_{1}+2H\sin\alpha$ and $L_{1}$ taken as a unit of length. Prior to it,
however, we start again with a formulation of requirements that the system
must obey to experience capillary condensation at all, expressed in terms of
the geometrical parameters $\alpha$ and $H$, for a given value of $\theta$.
The marginal conditions allowing for capillary condensation follow from the
condition $\Delta\Omega^{\rm ex}_{\rm DP}=0$ at the maximal pressure,
$p=p_{\rm sat}$, i.e. when the menisci are flat. This implies that the maximal
value $\alpha_{\rm max}$ of the opening angle is (cf. Appendix B)

$\sin\alpha_{\rm max}=\cos\theta-\frac{L_{1}}{H}\,,$ (27)

with $H\geq L_{1}\sec\theta$. For $H<L_{1}\sec\theta$ no condensation is
possible for any value of $\alpha$.

Note that Eq. (27) is a generalization of the condition (21) for the semi-
infinite case and is more stringent. The presence of the last term in Eq. (27)
implies that along the saturation path $p=p_{\rm sat}(T)$ the system will no
longer condense at the wedge-filling phase boundary but at a temperature
$T_{f}(\alpha)<T(\alpha,H)<T_{w}$.

(a) 0°

(b) 1°

(c) 1.5°

(d) 5°

(e) 10°

(f) 20°

(g) 29°

(h) 37°

Figure 7: A sequence illustrating capillary condensed states of a system with
$H=10\,L_{1}$ and $\theta=45$° for increasing $\alpha$.

Alternatively, Eq. (27) can be recast to the form determining the minimal
length of the walls, $H_{\rm min}$, required for the presence of capillary
condensation for the given value of the opening angle $\alpha<\alpha_{\rm
max}$:

$H_{\rm min}=\frac{L_{1}}{\cos\theta-\sin\alpha}\,.$ (28)

We recall that these conditions were obtained by assuming that condensation at
$p=p_{\rm sat}$ is of a double-pinned type. This must be the case, since
condensation to a single-pinned state at $p_{\rm sat}$ would require that
$\sin\alpha<\cos\theta$ (in order $\Omega^{\rm ex}_{\rm SP}<\Omega^{\rm
ex}_{\rm DP}$), in which case, however, $\Omega^{\rm ex}_{\rm SP}>0$
precluding the condensation.

The main results summarising the macroscopic predictions from section II are
shown in Fig. 4. Here, the global phase diagram in the $\alpha$-$\theta$ plane
is presented for several representative values of $H$. Fig. 4a specifies, for
each ratio of $H/L_{1}$, the values of $\alpha$ and $\theta$ corresponding
(same colour) to the given type of capillary condensation. The regions bounded
by the solid lines correspond to the parameters, for which the system
condenses to single-pinned states. The area outside these regions but below
the corresponding dashed line refers to double-pinning; the dashed lines show
the maximal opening angle $\alpha_{\rm max}(\theta)$ above which no
condensation is possible. Note that these lines straighten up and shift to
higher and higher values of $\theta$ as $H$ increases, approaching the line
$\alpha=\pi-\theta$ pertinent to the semi-infinite case, which is also
displayed.

These results reveal two noteworthy features which we want to point out:

_Firstly_ , the condensation exhibits a re-entrant phenomenon along the path
of constant $\theta$ on varying the opening angle $\alpha$. This follows from
the very fact that the boundaries of this path always correspond to double-
pinned states: the case $\alpha=\alpha_{\rm max}$ was discussed above, while
for $\alpha=0$ only double-pinned states are accessible edge_slit . Hence,
within the interval of $\theta(H)$ allowing for single-pinned states, the
increase of $\alpha$ will first switch from double- to single-pinning before
turning back to double-pinning condensation.

The change in condensation type occurs for the values of $\alpha$, solving the
equation

$c^{2}\cot\alpha(1-q^{2})+\pi-\theta-\cos^{-1}(qc)-\alpha+c\sqrt{1-c^{2}}+qc\sqrt{1-q^{2}c^{2}}=\frac{2c(1-q)\cos\theta}{\sin\alpha}\,,$
(29)

where we used the abbreviations $q=L_{1}/L_{2}$ and $c=\cos(\theta+\alpha)$
(see Appendix C for more details). Eq. (29) determines the parameters for
which both types of condensation coincide, i.e., when
$\theta_{e}^{\prime}=\theta$, or equivalently $X=H$. This can be illustrated
in Fig. 5 where a typical dependence of $X(\alpha)$ is shown. As for the semi-
infinite case, the dependence is non-monotonic but now, of course, is bounded
by the length of the walls. The highlighted values of the opening angle
$\alpha_{1}$ and $\alpha_{2}$ are those for which
$X(\alpha_{1})=X(\alpha_{2})=H$, both solving Eq. (29).

The third significant value of the opening angle is $\alpha_{\rm min}$, for
which $X(\alpha)$ reaches its minimum. This value can be determined
analytically by solving $\partial X/\partial\alpha=0$, using

$\left(\frac{\partial
R}{\partial\alpha}\right)_{\theta}=-\frac{\left(\frac{\partial\Omega^{\rm
ex}_{\rm SP}}{\partial\alpha}\right)_{\theta}}{\left(\frac{\partial\Omega^{\rm
ex}_{\rm SP}}{\partial\theta_{e}}\right)_{\theta}}\cdot\left(\frac{\partial
R}{\partial\theta_{e}}\right)_{\alpha}\,,$ (30)

where $(\partial
R/\partial\theta_{e})_{\alpha}=L_{1}\sec(\theta_{e}-\alpha)\tan(\theta_{e}-\alpha)/2$,
as follows from Eq. (12). The dependence of $X_{\rm min}$ on $\theta$ is shown
in Fig. 6 and can be interpreted as the minimal length of walls $H$ allowing,
for the given contact angle $\theta$, condensation to a single-pinned state.
The dependence corresponds to the line shown in Fig. 4b connecting the
“turning points” of the lines bounding the regions of single-pinning
condensation, which are the single solutions of Eq. (29). Note that the graph
$X_{\rm min}(\theta)$ itself exhibits a minimum for $\theta\approx
11.65\degree$ for which $X_{\rm min}\approx 4.5\,L_{1}$.

_Secondly_ , the lines specifying the region of single-pinned condensation
always possess a “turning point” at $\theta^{+}(H)$, which can be interpreted
as the contact angle above which only double-pinning is possible for arbitrary
$\alpha$ (for the given $H$). However, there exists a specific value of the
system size $\tilde{H}\approx 4.63\,L_{1}$, such that for $H<\tilde{H}$ the
lines enclosing single-pinned condensation exhibit two such points –
$\theta^{+}(H)$ and $\theta^{-}(H)$ – meaning that single-pinning is only
possible within the interval $\theta^{-}(H)<\theta<\theta^{+}(H)$. As $H$
further decreases, the single-pinning region shrinks and ultimately vanishes
when $H=H_{c}\approx 4.5\,L_{1}$, which corresponds to the minimum of $X_{\rm
min}(\theta)$ shown in Fig. 6.

We briefly summarise the present results. At first, our system formed of a
pair of walls of length $H$ with Young’s contact angle $\theta$ may only ever
experience capillary condensation provided the walls are long enough, such
that $H\geq L_{1}\sec\theta$. Under these conditions two types of capillary
condensation are in principle possible: i) to a single-pinned state
characterized by an edge contact angle $\theta_{e}$ at the pressure given by
the generalized Kelvin equation (12) or, more commonly, ii) to a double-pinned
state characterized by two edge contact angles $\theta_{e}$ and
$\theta_{e}^{\prime}$ at the pressure given by the generalized Kelvin equation
(19). The latter type always takes place for acute angles when the walls are
almost parallel; however, as $\alpha$ increases the upper edge contact angle
$\theta_{e}^{\prime}$ decreases and eventually depins from the top edges when
$\theta_{e}^{\prime}=\theta$, in which case the bottom edge contact angle
$\theta_{e}$ takes the value given by Eq. (29). Under further widening of the
system, the condensation will lead to a single-pinned state, so that a portion
of the system filled with liquid (proportional to $X$) decreases rapidly with
$\alpha$. This decrease continues up to the point given by the solution of Eq.
(30), at which the upper meniscus reaches its minimal position, $X_{\rm min}$,
and beyond which the meniscus location approaches the upper edges again. The
re-entrance to the double-pinned condensation is accomplished at such a value
of $\alpha$ that corresponds to the second solution of Eq. (29) (when $X=H$).
The condensation to a double-pinned state persists up to $\alpha_{\rm max}$ as
given by Eq. (27), which generalizes the condition for the filling transition
in a linear wedge. For $\alpha>\alpha_{\rm max}$, the free-energy cost for the
presence of the metastable liquid becomes too high for the system to condense.
This phenomenology is illustrated in Fig. 7, where a sequence of equilibrium
configurations of a system with $H=10\,L_{1}$ and $\theta=45$° is depicted for
several values of $\alpha$. A corresponding movie is shown in the
Supplementary Material SM .

###  IV.2 Asymptotic behaviour of capillary condensation

(a)

(b)

(c)

Figure 8: The numerical test of the asymptotic results given by a) Eq. (31),
b) Eq. (34), and c) Eq. (35). The analytic predictions (solid lines) are
compared with the numerical solutions of Eqs. (19)–(20) (symbols) for
partially wet walls ($\theta=45$°) and completely wet walls ($\theta=0$°) with
$H=20\,L_{1}$.

In this section, we analyze asymptotic features of capillary condensation,
starting with its behaviour in the limit of $\alpha\to 0$. This, for any
finite value of $H$, contrasts with the case of semi-infinite capillaries
examined in section III, since now the relevant type of condensation is
double-pinning. From Eqs. (17) and (18) it follows that to first order in
$\alpha$ the relation between the edge contact angles is

$\theta_{e}-\theta_{e}^{\prime}\approx
2\alpha\left(1+\frac{\cot\theta_{e}}{a}\right)\,,$ (31)

where we have temporarily introduced the abbreviation, $a=L_{1}/H$. To same
order, the corresponding Laplace radius is

$R\approx\frac{L_{1}}{2}\sec\theta_{e}\left(1-\alpha\tan\theta_{e}\right)\,.$
(32)

In what follows, we will further assume that the length-to-width ratio of the
slits is large and will drop the terms of the order of $a\alpha$. The free
energy balance, Eq. (20), can then be written as

$\displaystyle\alpha+a+\frac{a^{2}}{4}\sec^{2}\theta_{e}(\pi-2\theta_{2})-a\cos\theta\sec\theta_{e}$
$\displaystyle+\frac{a^{2}}{2}\tan\theta_{e}+{\cal{O}}(\alpha^{2},a\alpha)=0\,,$
(33)

which yields the relation between $\theta_{e}(\alpha)$ and the edge contact
angle for parallel slits, $\theta_{e}(0)$,

$\cos\theta_{e}(\alpha)=\cos\theta_{e}(0)\left(1-\frac{\alpha
H}{L_{1}}\right)\,,\;\;\;(\alpha\to 0)\,.$ (34)

This implies that $\theta_{e}(\alpha)$ decays to $\theta_{e}(0)$ linearly,
and, in contrast to the semi-infinite case, the asymptotic behaviour is the
same both for partially and completely wet walls. Furthermore, we can also
write for the shift in the corresponding partial pressures:

$\delta p_{\rm DP}=\delta p_{\rm
CC}(L_{1},H)\left[1-\left(\frac{H}{L_{1}}-\tan\theta_{e}(0)\right)\alpha\right]\,,\;\;\;(\alpha\to
0)\,.$ (35)

These asymptotic relations are confirmed by comparing with the numerical
results obtained from Eqs. (19)–(20) as shown in Fig. 8.

Figure 9: The numerical test of the power law (38) showing the divergence of
the Laplace radius of the menisci upon widening the pore up to the
condensation limit $\alpha_{\rm max}$. The analytic predictions (solid lines)
are compared with the numerical solutions of Eqs. (19)–(20) (dotted) for
systems with $H=10\,L_{1}$ and $\theta=0$ and 45°.

Next, we wish to investigate the growth of the equilibrium Laplace radius $R$
of the menisci along the double-pinned phase boundary, as the maximal opening
angle $\alpha_{\rm max}$ is approached. From Eqs. (17) and (18) it follows
that as $\alpha\to\alpha_{\rm max}$ (and $R\to\infty$), $\theta_{e}$ and
$\theta_{e}^{\prime}$ tend to adopt their respective asymptotic values,
$\pi/2+\alpha$ and $\pi/2-\alpha$, according to

$\theta_{e}\sim\frac{\pi}{2}+\alpha-\frac{L_{1}}{2R}\,,\;\;\;\theta_{e}^{\prime}\sim\frac{\pi}{2}-\alpha-\frac{L_{2}}{2R}\,.$
(36)

In this limit, Eq. (20) reduces, upon substituting from (36), to

$L_{1}+L_{2}+\frac{(L_{1}-L_{2})\cos\theta}{\sin\alpha}+\frac{(L_{2}^{2}-L_{1}^{2})\cot\alpha}{4R}=0\,,$
(37)

bearing in mind that $L_{2}$ itself is a function of $\alpha$. This implies
the power law for the growth of the menisci

$R\sim\frac{H\sin\alpha}{2}(\alpha_{\rm
max}-\alpha)^{-1}\,,\;\;\;(\alpha\to\alpha_{\rm max})\,,$ (38)

which is verified by a comparison with the exact numerical solution of Eqs.
(17)–(20) as shown in Fig. 9.

###  IV.3 Depinning transition

(a)

(b)

Figure 10: The illustrative behaviour of first (a) and second (b) derivatives
of the grand potential of the single-pinned phase (for $R<R_{\rm depin}$) and
the double-pinned phase (for $R>R_{\rm depin}$) where $R_{\rm
depin}\equiv\gamma/\delta p_{\rm depin}$ (denoted by the vertical line)
corresponds to the depinning transition of the system formed of partially wet
walls with $\theta=0.3$. The other parameters of the system are
$L_{2}=2\,L_{1}$ and $\alpha=0.03$.

(a)

(b)

(c)

Figure 11: The illustrative behaviour of first (a), second (b) and third (c)
derivatives of the grand potential of the single-pinned phase (for $R<R_{\rm
depin}$) and the double-pinned phase (for $R>R_{\rm depin}$) where $R_{\rm
depin}\equiv\gamma/\delta p_{\rm depin}$ (denoted by the vertical line)
corresponds to the depinning transition of the system formed of completely wet
walls ($\theta=0$). The other parameters of the system are $L_{2}=2\,L_{1}$
and $\alpha=0.08$.

The final remark regards the transition from a single- to a double-pinned
state in a fixed system geometry by varying the pressure. This _depinning
transition_ , referring to a meniscus (de)pinning at the wider capillary end,
occurs when $\theta_{e}^{\prime}=\theta$, i.e., at the partial pressure

$\delta p_{\rm depin}=\frac{2\gamma\cos(\theta+\alpha)}{L_{2}}\,,$ (39)

as follows from Eq. (18). The free energy change (per unit length) associated
with the transition is

$\Delta\Omega(R)\equiv\Omega_{\rm DP}^{\rm ex}-\Omega_{\rm SP}^{\rm
ex}=\frac{\gamma}{4R}\left[L_{2}^{2}\cot\alpha+4R^{2}\left(\frac{\cos\theta\cos(\alpha+\theta)}{\sin\alpha}+\alpha+\theta-\cos^{-1}\left(\frac{L_{2}}{2R}\right)\right)-4L_{2}R\frac{\cos\theta}{\sin\alpha}+L_{2}\sqrt{4R^{2}-L_{2}^{2}}\right]\,.$
(40)

It is straightforward to show that both $\Delta\Omega(R)=0$ and ${\rm
d}\Delta\Omega(R)/{\rm d}R=0$ – under the necessary condition that
$\theta+\alpha<\pi/2$ – for $R=R_{\rm depin}$, where $R_{\rm
depin}\equiv\gamma/\delta p_{\rm depin}$ is the Laplace radius corresponding
to the transition. For the second derivative of $\Delta\Omega(R)$ at the
transition, one obtains

$\left.\frac{{\rm d}^{2}\Delta\Omega}{{\rm d}R^{2}}\right|_{R=R_{\rm
depin}}=\frac{4\gamma\cos^{3}(\alpha+\theta)(\cot\alpha-\cot(\alpha+\theta))}{L_{2}}\,,$
(41)

which vanishes _only_ for $\theta=0$. Finally, the third derivative of
$\Delta\Omega(R)$ for completely wet walls at the depinning transition is

$\left.\frac{{\rm d}^{3}\Delta\Omega}{{\rm d}R^{3}}\right|_{R=R_{\rm
depin}}=\frac{8\gamma\cos^{2}\alpha\cot^{3}\alpha}{L_{2}^{2}}\,,\;\;\;(\theta=0)\,,$
(42)

which is already finite.

Hence, the change from a single- to a double-pinned state is accompanied by a
continuous (depinning) phase transition, which (according to the classical
Ehrenfest classification) is of second order for systems formed of partially
wet walls and of third order for completely wet walls. The behaviour of the
excess grand potential near the transition is illustrated in Fig. 10 (for
partially wet walls) and in Fig. 11 (for completely wet walls).

##  V Summary and Concluding Remarks

In this work, we studied condensation in capillaries formed of a pair of non-
parallel walls, each of length $H$ and making an angle $\alpha$ with the
vertical plane (say). We showed that the rotation of the walls by the angle
$\alpha$, breaking the reflection up-down symmetry, enriches the phase
behaviour of the confined fluid substantially. The main features of the system
phase behaviour are as follows:

  1. 1.

The system exhibits capillary condensation only if the walls are sufficiently
long, such that $H\geq L_{1}\sec\theta$ where $L_{1}$ is the capillary width
at the narrow end and $\theta$ Young’s contact angle of the walls.

  2. 2.

Furthermore, if the condition $H\geq L_{1}\sec\theta$ is obeyed, capillary
condensation may occur only for systems with the opening angle
$\alpha\leq\alpha_{\rm max}$, where the marginal angle $\alpha_{\rm max}$ is
given by the relation $\sin\alpha_{\rm max}=\cos\theta-L_{1}/H$. This
generalizes the condensation condition for the semi-infinite system,
$H\to\infty$, which is $\alpha_{\rm max}=\pi/2-\theta$, representing the
wedge-wetting (filling) phase boundary.

  3. 3.

Capillary condensation is one of two types, termed single-pinning and double-
pinning:

    1. (a)

In the former case, the capillary is only partially filled with liquid, such
that its lower meniscus is pinned at the narrow capillary end and makes edge
contact angle $\theta_{e}$ with the walls as given by Eq. (13); the upper
meniscus is inside the capillary and thus meets the walls with Young’s contact
angle $\theta$. The pressure, at which the system condenses to a single-pinned
state, is given by the modified Kelvin equation (12) in terms of $\theta_{e}$.

    2. (b)

If the condensation leads to a double-pinned state, the whole capillary is
filled with liquid and the upper meniscus is pinned at the wider capillary end
with a different edge contact angle $\theta_{e}^{\prime}$. In this case, the
condensation pressure is given by the modified Kelvin equation (19), which can
be expressed by either of the edge contact angles using Eqs. (17), (18), and
(20).

  4. 4.

The condensation scenario (when $H\geq L_{1}\sec\theta$) is as follows:

    1. (a)

For $H>\tilde{H}(\approx 4.6\,L_{1})$, there exists a marginal value of
Young’s contact angle $\theta^{+}$, so that the condensation always leads to a
double-pinned state for any $\alpha\leq\alpha_{\rm max}$. For
$\theta<\theta^{+}$, the condensation may be of either type depending on
$\alpha$, the variation of which leads to the re-entrant phenomenon:

       * •

For $\alpha<\alpha_{1}$, the condensation leads to a double-pinned state.

       * •

For $\alpha_{1}<\alpha<\alpha_{2}$, the condensation leads to a single-pinned
state.

       * •

For $\alpha_{2}<\alpha<\alpha_{\rm max}$, the condensation leads to a double-
pinned state again. As $\alpha$ tends to $\alpha_{\rm max}$, the menisci
flatten as $R\sim(\alpha_{\rm max}-\alpha)^{-1}$.

The opening angles $\alpha_{1}$ and $\alpha_{2}$, for which single- and
double-pinning coincide, are the solutions of Eq. (29).

    2. (b)

In a narrow interval of $H$, such that $H_{c}\leq H\leq\tilde{H}$ with
$H_{c}\approx 4.5\,L_{1}$, the interval of $\theta$ allowing for single-
pinning is restricted by two marginal contact angles:
$\theta^{-}<\theta<\theta^{+}$.

    3. (c)

For $H<H_{c}$, condensation always leads to a double-pinned state.

  5. 5.

Upon narrowing the slit, i.e., in the limit of $\alpha\to 0$, the asymptotic
properties of capillary condensation are as follows:

    1. (a)

For semi-infinite walls, the condensation pressure, $p_{\rm SP}$, tends to the
one corresponding to capillary condensation in an infinite parallel slit,
$p_{\rm CC}$, as $p_{\rm SP}-p_{\rm CC}\sim\alpha^{1/2}$ for partially wet
walls, but substantially slowly, $p_{\rm SP}-p_{\rm CC}\sim\alpha^{1/4}$, for
completely wet walls. Analogous behaviour applies to the way the edge contact
angle $\theta_{e}$ approaches Young’s contact angle $\theta$.

    2. (b)

For $H$ finite, the asymptotic behaviour is different because the system can
now lower its free energy by condensing to a double-pinned state, so that the
upper meniscus meets the walls with the edge contact angle
$\theta_{e}^{\prime}$ rather than $\theta$. Now, the condensation pressure
approaches the one pertinent to a finite parallel slit linearly in $\alpha$—as
does the difference in the edge contact angles,
$\theta_{e}-\theta_{e}^{\prime}$—both for partially and completely wet walls.

  6. 6.

For geometries allowing for single-pinned condensation, there is a continuous
transition between a single-pinned and a double-pinned state by varying the
pressure as given by Eq. (39). The transition is of second order for partially
wet walls and of third order for completely wet walls.

We conclude with a few remarks concerning possible extensions of this study.
Throughout this work we implicitly assumed that the capillary forces dominate
the system behaviour and that gravity effects can be neglected. This is the
case when the pertinent capillary length is much larger than the
characteristic lengths of the system (such as $L_{1}$). Allowing for gravity
would make possible to consider significantly larger ($\sim\mu m$) systems and
supposedly would substantially enrich the system phase behaviour in view of
the competing surface and gravity effects; this is experimentally accessible
using e.g. interferometric technique, as demonstrated by Moldover and Gammon
for measuring elevation and adsorption thickness of $SF_{6}$ in wedge-like
cavities moldover . On the other hand, employing more microscopic approaches
that would allow to link the system behaviour with its molecular origin would
also be desirable; this would, most notably, be important for the case of
$\theta=0$, when wetting layers adsorb at the walls effectively reducing the
system size, something not accounted for within our analysis. Here, the direct
link with real experiments is less straightforward but possible using e.g.
atomic force microscopy and we believe that a simple geometric modification of
the recent experiments by Geim et al. geim  of capillary condensation of water
on mica or graphite “walls” of an atomic scale (few ångstöms) is feasible.
More sophisticated treatments would be needed to capture further relevant
aspects, such as interfacial fluctuations, surface roughness or bulk
criticality. Possible extensions also include modification of the system
geometry by, for example, breaking the mirror symmetry, which can further
increase the number of condensation types. Some of these extensions will be
subjects of our future work.

Figure 12: Detailed sketch of a single-pinned state.

##  Appendix A Derivation of formulae from Section II

Here, we present a detailed description of the geometric expressions that lead
to the modified Kelvin equations for both single- and double-pinning
condensations.

###  A.1 Single-pinning condensation

All the geometric measures that appear in Eq. (4) for the excess grand
potential can be determined easily using Fig. 12 where all the important
angles and distances are denoted. Specifically, for the angles $\omega_{1}$
and $\omega_{2}$ we have that

$\omega_{1}=\pi+2\alpha-2\theta_{e}$ (43)

and

$\omega_{2}=\pi-2\alpha-2\theta\,,$ (44)

where $\theta$ is Young’s contact angle at which the upper meniscus meets the
walls and $\theta_{e}$ is the edge contact angle at which the lower meniscus
meets the walls. From here, the expressions (5) and (6) for $\ell_{1}$ and
$\ell_{2}$ follow immediately. Furthermore, the shaded areas, $S_{1}$ and
$S_{2}$, can now be determined as a difference between the areas of the
corresponding circular sectors, $R^{2}\omega_{1}/2$ and $R^{2}\omega_{2}/2$,
and those of the isosceles triangles with the apex angle $\omega_{1}$ and
$\omega_{2}$, respectively, leading to Eqs. (9) and (10). Finally, for the
length $X$ of the walls which at contact with liquid, it holds

$X=\frac{2R\cos(\alpha+\theta)-L1}{2\sin\alpha}\,.$ (45)

After substituting for $\ell_{1}$, $\ell_{2}$ $S=S_{\rm tot}-S_{1}-S_{2}$,
where the trapezoid area $S_{\rm tot}$ is given by Eq. (8), and $X$ into the
thermodynamic relation $\Omega_{\rm SP}^{\rm ex}=0$, the condition (13), at
which the single-pinning condensation occurs, is obtained. The second relation
between the Laplace radius of curvature, $R$, and $\theta_{e}$ reads

$R\cos(\theta_{e}-\alpha)=\frac{L_{1}}{2}\,,$ (46)

as required by the system geometry.

###  A.2  Double-pinning condensation

Figure 13: Detailed sketch of a double-pinned state.

Similar relations apply for the double-pinning condensation, which is
illustrated in Fig. 13. While the expression (43) for $\omega_{1}$ remains
unchanged, the one for $\omega_{2}$ is now given by the second edge contact
angle, $\theta_{e}^{\prime}$, as
$\omega_{2}=\pi-2\alpha-2\theta_{e}^{\prime}$. Hence, for the (shaded) areas
of the circular segments we now have that

$S_{1}=\frac{R^{2}}{2}\left(\pi-2\theta_{e}+2\alpha\right)-\frac{\sin(\theta_{e}-\alpha)RL_{1}}{2}$
(47)

and

$S_{2}=\frac{R^{2}}{2}\left(\pi-2\theta_{e}^{\prime}-2\alpha\right)-\frac{\sin(\theta_{e}^{\prime}+\alpha)RL_{2}}{2}\,,$
(48)

which determines the area occupied by liquid
$S=H\cos\alpha(L_{1}+L_{2})/2-S_{1}-S_{2}$.

The phase boundary for the double-pinning condensation is given by the
thermodynamic condition, $\Omega_{\rm DP}^{\rm ex}=0$, leading, after
substituting for all the geometric measures, to Eq. (20), which is
complemented by two geometric relations

$R\cos(\theta_{e}-\alpha)=\frac{L_{1}}{2}$ (49)

and

$R\cos(\theta_{e}^{\prime}+\alpha)=\frac{L_{2}}{2}\,,$ (50)

forming now the set of three equations for the unknowns $R$, $\theta_{e}$ and
$\theta_{e}^{\prime}$.

##  Appendix B Derivation of Eq. (27)

At the saturation, $\delta p=0$, both menisci in a double-pinned state must be
flat and the simple geometry dictates that in this case
$\theta_{e}-\alpha=\pi/2$ and $\theta_{e}^{\prime}+\alpha=\pi/2$. The manner
at which the edge contact angles, $\theta_{e}$ and $\theta_{e}^{\prime}$,
acquire their limiting values follows by expanding of (17) and (18), which
yields

$\theta_{e}\to\frac{\pi}{2}+\alpha-\frac{L_{1}}{2R}\,,\;\;\;({\rm
as}\;R\to\infty)$ (51)

and

$\theta_{e}^{\prime}\to\frac{\pi}{2}-\alpha-\frac{L_{2}}{2R}\,,\;\;\;({\rm
as}\;R\to\infty)\,,$ (52)

implying that $\pi-\theta_{e}-\theta_{e}^{\prime}\sim(L_{1}+L_{2})/(2R)$, as
$R\to\infty$. Dividing Eq. (20) by $R_{\rm DP}$ and considering the limit
$R_{\rm DP}\to\infty$ one obtains that in this limit the condition for a
double-pinning condensation reduces to

$L_{1}+L_{2}+\frac{(L_{1}+L_{2})\cos\theta}{\sin\alpha}=0\,,$ (53)

which in terms of the parameter $H=(L_{2}-L_{1})/(2\sin\alpha)$ can be
expressed in the form of Eq. (27).

##  Appendix C Derivation of Eq. (29)

The change in the condensation type from single- to double-pinning occurs,
when $X=H$ (cf. Fig. 1). In view of Eq. (45) and the relation
$H=(L_{2}-L_{1})/2\sin\alpha$, the condition can be written as

$\cos(\theta_{e}-\alpha)=cq$ (54)

with $q\equiv L_{1}/L_{2}$ and $c\equiv\cos(\theta+\alpha)$. Using Eq. (54)
and substituting for $\theta_{e}=\cos^{-1}(cq)+\alpha$ and
$\sin(\theta_{e}-\alpha)=\sqrt{1-c^{2}q^{2}}$ into Eq. (13) for $\theta_{e}$
at single-pinning condensation, Eq. (29) is obtained.

###### Acknowledgements.

This work was financially supported by the Czech Science Foundation, Project
No. 21-27338S. J. J. acknowledges a financial support from the grant of
Specific university research – grant No A1_FCHI_2023_001.

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#  Meta-Policy Learning over Plan Ensembles for
Robust Articulated Object Manipulation

Constantinos Chamzas  Rice University
Houston, USA
<EMAIL_ADDRESS>Caelan Garrett  NVIDIA Research
Seattle, USA
<EMAIL_ADDRESS>Balakumar Sundaralingam  NVIDIA Research
Seattle, USA
<EMAIL_ADDRESS>Lydia E. Kavraki  Rice University
Houston, USA
<EMAIL_ADDRESS>Dieter Fox  NVIDIA Research
Seattle, USA
<EMAIL_ADDRESS>
##  I Introduction

Model-based robotic planning techniques, such as inverse kinematics and motion
planning, can endow robots with the ability to perform complex manipulation
tasks, such as grasping, object manipulation, and precise placement. However,
these methods often assume perfect world knowledge and leverage approximate
world models. For example, tasks that involve dynamics such as pushing or
pouring are difficult to address with model-based techniques [1] as it is
difficult to obtain accurate characterizations of these object dynamics.
Additionally, uncertainty in perception prevents them populating an accurate
world state estimate.

Recent works have shown that integrating learning with model-based planning
can address some of these limitations [2, 3, 4]. Following a similar
direction, we integrate a geometric model-based motion planner with learning
in a mixture-of-experts fashion to solve manipulation problems where the world
model and dynamics are only approximately known.

Specifically, we propose using a model-based motion planner to build an
ensemble of plans under different environment hypotheses. Then, we train a
meta-policy to decide online which plan to track based on the current history
of observations. By leveraging history, this policy is able to switch ensemble
plans to circumvent getting “stuck” in order to complete the task. We tested
our method on a 7-DOF Franka-Emika robot pushing a cabinet door in simulation,
as shown in Fig. 1. We demonstrate that a successful meta-policy can be
trained to push a door in settings high environment uncertainty, all while
requiring little data ($\leq$ 1000 episodes).

The main contributions of this work are:

  1. 1.

An ensemble approach for manipulation policies, which outperforms single-
trajectory planners.

  2. 2.

Learning a meta-policy to select among the ensemble.

  3. 3.

Simulated experiments that show our method results in $40\%$ higher success
rate than the non-learning baseline.

Figure 1: Example of the panda robot pushing the left door of the cabinet. In
this setting, the robot knows its own state and geometry along with the state
and geometry of the cabinet. However, the position of the cabinet is only
approximately estimated, and the dynamic properties of the objects are
unknown. In this work, we attempt to solve this task by combining a model-
based planner and learning from experience.

##  II Related Work

At the core of model-based planning techniques for robot manipulation lies
motion planning [5]. Such techniques include constrained motion planning [6],
Multi-Modal Motion Planning [7] (MMMP), and Task and Motion Planning (TAMP)
[8]. The main advantages of these methods are that they are general, i.e. work
with any geometries, and have theoretical guarantees with respect to their
models, e.g. probabilistic completeness [9]. However, they depend critically
on the fidelity of their models, limiting their effectiveness when there are
state estimation errors and unmodeled constraints, e.g. dynamic constraints
such as limited pushing force or friction between objects and the environment.
Finally, these methods often have long (several seconds) planning times,
making it difficult for them to react online in real-time to unexpected
events.

Recent advances in learning have prompted researchers to combine learning with
model-based planning to address these limitations. Some techniques try to
learn world models [10, 11, 12, 13], warm start policy-learning with model-
based techniques [14, 15], or improve planning efficiency [16, 3]. We also
leverage learning to improve the downstream performance of a model-based
planner. However, we focus on approximate world models with uncertain object
poses and geometries and problems that involve dynamics such as pushing.

##  III Methodology

In this work, we aim to endow a robotic agent with advanced manipulation
skills regarding grasping, pushing, and even throwing objects under realistic
assumptions. The realistic assumptions that we consider are that a world model
can be engineered but is not entirely accurate. These inaccuracies could be
due to pose uncertainty, lack of dynamic modeling, or approximate geometric
representations. Our method uses a model-based planner to produce candidate
plans using the approximate world model and then uses learning from experience
to learn a meta-policy that improves task performance.

We will use the toy problem shown in Fig. 2 a) to demonstrate the key concepts
in this work. In this toy problem, the robot aims to push the red object to
the shaded red goal location. The modeled constraints include collision
avoidance and joint limits. Unknown aspects include dynamics, namely how the
object moves upon contact with the end-effector. Task failure can occur if the
object falls over due to the pushing angle. Uncertainty exists in the object’s
x-axis position estimation and its exact geometry.

The world state is the prorpioceptive state of the robot along with the
potentially uncertain state of the world objects. For example, in the problem
shown in Fig. 2 a), the state includes the positions of the robot’s five
joints and an approximate estimate of the position and geometry of the red
object along the x-axis. We consider such state estimation realistic as robots
usually have accurate joint encoder measurements but perceive the objects in
the world around them with some uncertainty.

The proposed approach is composed of two main components. First, a geometric
model-based planner creates an ensemble of plans as described in subsection
III-A and shown in Fig. 2 b). Second, given the currently observed state, we
learn a meta-policy that chooses plans from the ensemble as described in
subsection III-B and shown in Fig. 2 b) (the large black arrows). The proposed
hybrid approach leverages the approximate world model to create plan-ensembles
that satisfy known constraints, such as geometric-constraints. Then we use
learning to account for unmodeled constraints, e.g. dynamics, and be robust to
estimation errors.

Figure 2:  a) An illustrative example of a task that combines geometric,
dynamic reasoning and uncertainty. The robot is tasked with pushing the red
object to the goal location, without toppling it over. The robot can control
its 5 joints and estimate within some error the position and geometric height
of the red object. b) The proposed method uses model-based planning to create
several path policies. The path shown with green is produced with a model-
based geometric planner and satisfies the geometric (known) constraints of the
problem. The black arrows represent the plan, which is the combination of the
path and a position controller. Note that the plan is shown in the end-
effector 2D space but its state space is the 5D space of the joints of the
robot. Algorithm 1 Plan Ensemble Pseudocode

1:procedure Plan-Ensemble($G$, ${\Pi_{p}}$, $K$)

2: ${\Pi_{p}}$ $\leftarrow\emptyset$ $\triangleright$ Plan-Ensembles

3: while $i\leq N$ do $\triangleright$ Create Plan-Ensembles

4: $\hat{s}\leftarrow\textsc{sample-world}()$

5: $p\leftarrow\textsc{plan}(\hat{s},G)$ $\triangleright$ Plan Geometric Path

6: $\pi_{p}\leftarrow\textsc{make-policy}(p)$ $\triangleright$ Create Plan

7: ${\Pi_{p}}$ $\leftarrow$${\Pi_{p}}$ $\cup~{}\pi_{p}$

8: while $t\leq T$ do $\triangleright$ Iteratively Reselect Plan

9: $o\leftarrow\textsc{observe}()$

10:
$\pi_{p}^{chosen}\leftarrow\underset{\pi_{p}\in\Pi_{p}}{\mathrm{argmax}}~{}\textbf{{\color[rgb]{0,0.88,0}best}}(o,\pi_{p})$

11: while $\textbf{{\color[rgb]{0,0.88,0}progress}}(o,p)$ do

12: $o\leftarrow\textsc{observe}()$ $\triangleright$ Observe State

13: if $o\in G$ then

14: return True$\triangleright$ Success!

15: $r\leftarrow\textsc{execute}(\pi^{chosen}_{p})$

16: return False$\triangleright$ Failure

###  III-A Create Plan-Ensembles

Creating plan-ensembles is described in Line 3 to Line 7 of Alg. 1. In Line 4,
sample-world() samples a possible world state according to a given uncertainty
distribution. In Line 5, given the currently estimated state of the world
$\hat{s}$ and a goal $G$, a model-based planner computes a candidate path $p$.
A path $p$ is as sequence of waypoints where each waypoint specifies the
position of the robot and other known movable objects in the world. The first
waypoint is the current state, while the last waypoint must satisfy the goal
specification. Examples of paths are the green lines shown in Fig. 2 b).

In Line 6, the path $p$ is converted into a plan $\pi_{p}$. The path is time-
parameterized and a waypoint-tracking trajectory controller is used to follow
it. In this context, a plan is converted into an atomic policy that provides
an action for any state. Example plan actions are shown as the black arrows
that attract towards the green path $p$ in Fig. 2 b).

Note that each plan is computed for an estimation of the world state
$\hat{s}$. This is illustrated in Fig. 2 b), where the objects for each
computed path policy are either in different locations (inaccurate pose
estimation) or have different geometry (inaccurate geometry estimation).
Finally, in Line 7, the plan is added in the plan-ensemble ${\Pi_{p}}$.

###  III-B Selecting Plans

Figure 3: Success Rate, Number of Steps, and Distance to Goal different
configurations of method. All configurations were tested on a 100 different
environments, with a maximum of 200 timesteps. single refers to using just a
single path policy while multi means that 10 path-policies were used for the
path-policy set ${\Pi_{p}}$ respectively. fixed/progress denotes whether a
fixed interval or the progress rule was used to choose a new path-policy.

Selecting and executing a plan is described in Line 8 to Line 15 of Alg. 1.
Given a plan-ensemble ${\Pi_{p}}$, we need to choose the most appropriate plan
to follow for the current observed state $o$. In this context, the observed
state includes the same information as each waypoint of the path. We choose a
simple supervised approach to learn how to select a plan.

In Line 9, the known state of the robot and the world is observed, which can
be used to determine if we have reached a terminal state. An episode
terminates when the goal is reached or when maximum number of steps has
passed. An important implementation choice is the decision-frequency at which
a new plan is chosen. One end of the spectrum would be choosing a single plan
and following it until the end of the episode. The opposing end would be
choosing a new plan at every timestep. In this work, we propose using a simple
rule that leverages the computed path $p$ to decide when to switch. Taking
advantage of this knowledge, in Line 11, the progress function monitors if
progress towards the planned path $p$ is made and returns False if no progress
has been made in the last $k$ timesteps. When this function returns False, a
new plan is chosen, as shown in Line 10.

To generate training data, we first create the plan-ensemble ${\Pi_{p}}$ and
then execute plans $\pi_{p}$ randomly. After execution, we collect the state,
action, and progress triplet and then store it to later train the network
best. We define the distance to goal that was covered by the execution of the
plan as progress. We use the collected data to train the network in supervised
manner. The network learns to predict the progress that will be achieved, if
we execute the plan $\pi^{chosen}_{p}$ from the current observed state $o$.
During testing, we use the trained network best Line 10 to choose the most
promising plan and execute it. Execution continues until the goal is reached
or the progress returns False.

##  IV Experiments

We applied our framework to a pushing a cabinet door problem, as shown in Fig.
1. The robot task is to push the left cabinet door from the open state to the
closed state.

###  IV-A Experiment Setup

The initial state uncertainty we consider lies in the position of the cabinet
relative to the robot. Thus, the sample-world() function samples a random
position of the cabinet relative to the robot according to the given
uncertainty distribution. Here, we only consider translational uncertainty.

Given the random sample from this distribution $\hat{s}$, the planner produces
a geometric path $p$ that maintains contact perpendicular to the door and
intends to push it until it closes. The pushing contact location on the
cabinet door is chosen randomly; different contact points are sampled for each
computed path. We use TRAC-IK [17] combined with a pid controller to convert
the geometric path $p$ to a policy which we refer to as a plan $\pi_{p}$. To
represent the geometric world and generate the plan-ensembles ${\Pi_{p}}$, we
use PyBullet [18]. We use IsaacGym [19] for physics and control simulation.

We conducted 2 experiments to examine the performance of our proposed method.
The first experiment(subsection IV-B), studies the choices of the switching
frequency rule and the use of ensembles. Specifically, we investigate the
effect of the progress rule, and choosing to use a plan-ensemble instead of a
single plan . The second experiment (subsection IV-C), benchmarks the proposed
method with respect to a non-learning baseline and ablates different input
features to the best neural network. We evaluated with these metrics:

  * •

Distance to goal (Progress): This simply measures how much the door closed. A
distance of 0 means that the door closed (i.e. the goal was reached), and a
distance of $\pi/2$ means that the door did not move at all.

  * •

Success rate: The percentage of tasks solved successfully within the given
horizon. The task is considered successful if the door is closed up to some
error threshold.

  * •

Number of steps: The number of timesteps to complete the episode. The episode
terminates if the door is closed or if the maximum of 200 timesteps is
reached.

Figure 4: The 1) Success Rate, 2) Number of Steps, and 3) Distance to Goal for
both the two random baselines, with and without the progress
(RandNoProgress/RandProgress) rule, as well as the two different feature sets
(LearnedF1/LearnedF2) used for training the best network. The x-axis denotes
the translational estimation error on the cabinet position. The error bars
indicate 0.25 of the standard deviation.

###  IV-B Using Plan Ensembles and Progress Rule

Here, the uncertainty was emulated as a uniform distribution $\pm$10 $cm$ on
the position of the cabinet and applied across x,y,z dimensions independently.
We tested the following:

  1. 1.

single/multi paths: single means that only one plan was used, while multi
means that we had a plan-ensemble of 10 plans. The plan to follow at each
time-step was chosen randomly. This comparison motivates why using multiple
plans(an ensemble) is better than a single one.

  2. 2.

fixed/progress rule: fixed means that the decision to switch plan was every 10
fixed timesteps, and progress means that the rule described in subsection IV-C
was used to determine the switching frequency. In both cases, the new plan
$\pi_{p}$ is chosen randomly. For single, fixed has no effect, while progress
added noise $\pm$ 10cm in the end-effector space to help unstuck the robot.

The results demonstrate that the progress rule and using an ensemble
significantly improves performance. Intuitively as long as progress is made
along the path, the current plan is good and it should not be switched. The
advantage of using an ensemble of plans lies in the fact that part of one plan
could apply for part of the episode and part of another plan can apply to
another part of the episode. Thus an ensemble can complete tasks that a single
plan could never complete. The multi/progress combination achieves both the
best distance to goal and also requires the least number of steps to complete
the task. The main takeaways from these results are first that the progress
rule helps significantly in all uncertainty settings.

###  IV-C Learning a Meta Policy

In this experiment, we investigated the improvements in task performance of
learning the subroutine best. First, we use a random strategy to choose and
execute path-policies for the pushing task in Fig. 1 using the same setting as
the experiment setup of subsection IV-B with multi paths and the progress rule
enabled. We collected data from 1000 episodes.

We trained a regressor neural network best, which takes as input the plan
$\pi_{p}$ as well as the current observation $o$ and possibly the history of
observations and predicts the progress toward the goal after executing
$\pi_{p}$. The neural network architecture has three fully connected layers
with batch normalization[20]. Each hidden layer has 64 units, and the network
was trained to minimize Mean Squared Error (MSE).

An important decision is how to represent the path and history of observations
as the input to the neural network. We use the following values are input
features:

  * •

Current End-Effector(EE) State: the position, expressed in Cartesian
coordinates (3 dim) and the orientation expressed as a quaternion (4 dim).

  * •

Current Robot State: joint values of the robot (7 dim).

  * •

Current Door State: the joint value of the door (1 dim).

  * •

Final End-Effector State (7 dim).

  * •

Next Immediate End-Effector State (7 dim).

  * •

Midpoint End-effector State: the midpoint between Current and Final End-
Effector States (7 dim), using spherical linear orientation interpolation.

The observed state of the world includes the current end effector state, robot
joint state, and door state (15 dim). The first set of features $F1$ includes
the observation, the next end-effector goal, and the end effector goal (29
dim). The second set of features $F2$ extends $F1$ by adding the midpoint end-
effector state, the difference between the current end-effector state and the
next end-effector state, and the difference between the current end-effector
state and the final end-effector state. Additionally, the last 5 observations
are included (175 dim).

We trained the neural network with the same data from $\pm 10$ cm uncertainty
and tested it in 100 environments under $\pm 0,\pm 2,\pm 5,\pm 10$ cm of
translational uncertainty. We also tested a random strategy similar to the
previous section when using and not using the progress rule. In Fig. 4, the
success rate, number of steps included, and distance to goal are shown for the
four uncertainty settings.

The learned strategies are better than the random strategy, and the F2 set of
features seems to help best in the lower uncertainty regime. We hypothesize
that the $\pm 10$ cm regime is too uncertain to leverage the observation
history or potentially over-fits, while in the lower regime the observation
history can help disambiguate the error in position estimation. Overall,
incorporating the observation history helps, and it is possible to learn the
meta policy over the plan-ensemble. A video demonstration of the experiments
is available
111https://www.dropbox.com/s/8gosmt9e838icqg/RSS2023-LTAMP.mp4?dl=0.

##  V Conclusion

In this work, we proposed a method for manipulation that learns a meta policy
over plan-ensembles using approximate world models. In our preliminary
results, we have demonstrated the efficacy of the proposed method over simple
baselines in a simulated environment where the robot is tasked with pushing
the door of cabinet. The two main assumptions of the proposed method are that
an approximate model of the world is given and a quantification of the
uncertainty of the model is also available. In the future, we would like to
apply the proposed method to a real-world setting and investigate the
feasibility of these assumptions. We would also like to investigate extending
to more manipulation skills, multi-modal planning, and image-based
predictions.

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