# A Markovian Formalism for Active Querying

Sid Ijju
University of California Berkeley
<EMAIL_ADDRESS>

###### Abstract

Active learning algorithms have been an integral part of recent advances in
artificial intelligence. However, the research in the field is widely varying
and lacks an overall organizing leans. We outline a Markovian formalism for
the field of active learning and survey the literature to demonstrate the
organizing capability of our proposed formalism. Our formalism takes a
partially observable Markovian system approach to the active learning process
as a whole. We specifically outline how querying, dataset augmentation, reward
updates, and other aspects of active learning can be viewed as a transition
between meta-states in a Markovian system, and give direction into how other
aspects of active learning can fit into our formalism.

_K_ eywords Active Learning $\cdot$ Markov $\cdot$ Inverse Reinforcement
Learning $\cdot$ Query

##  1 Introduction

Current reinforcement learning policies are heavily dependent upon the usage
of a reward function. However, for advanced problems such as learning behavior
for a self-driving car or a robot with many degrees of freedom, the reward
function is very difficult or impossible to design to effectively encompass
all human considerations. Inverse reinforcement learning algorithms are a
class of algorithms that attempt to solve this issue by learning a reward
function from expert demonstrations, and then subsequently using any manner of
standard reinforcement learning algorithms to learn an optimal policy based
off that learned reward function [1].

Inverse reinforcement learning algorithms require a dataset of expert
demonstrations to derive a reward function from. However, this reward function
can often be inaccurate and highly variable. Furthermore, the learned reward
function can often be misaligned with the true intentions of the human user,
even if it does accurately account for the demonstrated data it learned from
[2]. One recent advance in the training of inverse reinforcement learning
algorithms is to use additional expert data about preferences between
demonstrations [3]. The general idea is to first train an inverse
reinforcement learning algorithm on a dataset of demonstrations, and then
subsequently actively query a human expert to select a preference between
generated demonstrations based off the learned prior of trajectories from the
dataset of demonstrations. The policy can then be updated using the new expert
data.

We identify two general categories of research direction within the field of
active learning. First, one could learn how to more effectively utilize the
information provided by the human expert with each query. Second, one could
learn how to increase the amount of information provided by each individual
query. We can call the two directions information derivation (for learning how
to maximize the information derived from a single query), and information
optimization (for learning how to ask more optimal future queries)
respectively. Information optimization approaches can be further divided into
approaches that take a policy/learning algorithm-based approach as opposed to
a data-based approach.

The key observation of our work is that both research directions can be viewed
as an informational choice made by the robot at each step. More formally, we
can view the active learning querying process itself as a Markov Decision
Process (MDP) with non-Markovian reward. The action space and the transition
function encompass both the choices of information type the robot can choose
to query as well as the information derived by the robot from the query. In
other words, both directions of research focus on determining the best setup
for the querying MDP.

We organize the field of query-based active learning using this formalism.
Furthermore, we outline future research directions using the proposed
formalism and discuss applications of the outlined methods.

##  2 Formalism

#### Model

We model our actual learning environment as a deterministic, fully-observable
Markov decision process. We define a set $\mathcal{S}$ to represent the state
of the system, and a set $\mathcal{A}$ to denote the action input to the
system. At a time $t$, the system will be at a state $s_{t}\in\mathcal{S}$ and
the robot will be tasked with taking an action $a_{t}\in\mathcal{A}$. We
further define a trajectory $\xi\in\Xi$ as a finite sequence of states and
actions (we define the horizon $\mathcal{H}$ of a trajectory $\xi$ as the
maximum number of states). Therefore, we can write a trajectory
$\xi=((s_{t},a_{t}))^{T}_{t=0}$, where $\max_{\xi\in\Xi}{T}=\mathcal{H}$. We
are provided an initial dataset $\mathcal{D}\subset\Xi$ of trajectories $\xi$.
Finally, we assume a reward function
$r\in\mathcal{R}:\Xi\rightarrow\mathbb{R}$ that encapsulates how a human wants
the system to behave. The goal of inverse reinforcement learning is to learn
the reward function $r$ that best represents the human intentions shown in the
dataset $\mathcal{D}$.

We also make a common simplifying assumption about the reward function, $r$
[4]. We let $r$ be a linear combination of selected features
$\Phi:\Xi\rightarrow\mathbb{R}^{d}$. Now, we can write
$r(\xi)=\omega^{T}\Phi(\xi)$. With this simplification, we no longer need to
learn a potentially high dimensional $\mathcal{R}$ and instead can learn a
finite dimensional ($d$) vector $\omega$ that describes the human’s reward
function.

#### Markovian Formulation

The central aspect of the active learning process is the idea of querying the
human expert. To define and formalize the notion of a query, we first define
an "information Markov Decision Process". We can then represent the querying
process as an aspect of this new system.

We define the state space $\mathcal{S}^{\prime}\in\Xi\times\mathcal{R}$. The
state of our system at time will consist of a dataset $\mathcal{D}$ and a
reward function $r\in\mathcal{R}$. We further define the action space
$\mathcal{A^{\prime}}$ as the set of potential queries (we use the term action
and query interchangeably moving forward) we can ask using the current state
$s\in\mathcal{S}^{\prime}$ 111For our discussion on the scope of the domain
for $\mathcal{A}^{\prime}$, refer to 3.1, an observation space $\Omega$ that
represents the set of human responses to a "query" $a\in\mathcal{A^{\prime}}$,
and a set of conditional observation probabilities $\mathcal{O}:\Omega\times
S^{\prime}\times A^{\prime}\rightarrow[0,1]$ that represent the probability of
a response $o\in\Omega$ given the action $a\in\mathcal{A}^{\prime}$ and the
new state $s^{\prime}\in\mathcal{S}^{\prime}$. We also have
$\mathcal{R}^{\prime}:\mathcal{S}^{\prime}\times\mathcal{A}^{\prime}\rightarrow\mathbb{R}$
and
$\mathcal{T}^{\prime}:\mathcal{S}^{\prime}\times\mathcal{A}^{\prime}\rightarrow\mathcal{S}^{\prime}$,
and $\gamma$ (the discount factor). In this new system, the reward function
$\mathcal{R}^{\prime}$ represents the benefit of asking our query for the
state of the system and its capability to learn the original reward function
$r\in\mathcal{R}$. Furthermore, the transition function encapsulates the
changes in system information that receiving a response to a query (action)
entails.

Define our "information Markov Decision Process" as a partially observable
Markov Decision Process
$\mathcal{M}^{\prime}=(\mathcal{S}^{\prime},\mathcal{A}^{\prime},\mathcal{T}^{\prime},R^{\prime},\Omega,\mathcal{O},\gamma)$.
Our claim is that you can represent any active querying algorithm as a policy
that optimizes on an "information Markov Decision Process" .

##  3 Survey

We now survey the active learning literature using the formalism above with
different selections of
$\mathcal{A^{\prime}},\mathcal{T^{\prime}},\mathcal{R^{\prime}},\mathcal{O}$.
Many methods fall into a single one of the above categories. However, there
are some other approaches that are more subtle and which require multiple
changes across the different categories defined to fit under the formalism,
speaking to the nontrivial nature of identifying a common formalism for this
field.

###  3.1 Action Space: $\mathcal{A^{\prime}}$

To preface this section, we identify a simplifying assumption used by our
formalism. We define the set $\Omega$ as the set of possible human responses
to a "query" $a^{\prime}\in\mathcal{A}^{\prime}$ which uniquely ties the
definition of the set $\Omega$ to the definition of $\mathcal{A}^{\prime}$,
and so in all further references we will discuss $\mathcal{A^{\prime}}$ and
$\Omega$ in a joint context. Furthermore, any modification to the action space
$\mathcal{A}^{\prime}$ will require a modification to the transition
probability space $T^{\prime}$ since a transition function $T(s^{\prime}|s,a)$
inherently depends on the action $a\in\mathcal{A^{\prime}}$, so we also
jointly discuss $\mathcal{T^{\prime}}$. For simplicity, we will consider the
transition function $T$ as a complete description of the set
$\mathcal{T}^{\prime}$ in this section. Since the transition function $T$
operates as a tuple, we can decompose the transition function using a dataset
transformation function $F\in\mathcal{F}:\Xi\rightarrow\Xi$ and a learning
algorithm $L\in\mathcal{L}:\Xi\times\mathcal{R}\rightarrow\mathcal{R}$ that
updates a learned reward function $r\in\mathcal{R}$ using a dataset
$\mathcal{D}\in\Xi$.

Most research in this field considers changes to the algorithm used for
selection of the action $a\in\mathcal{A}^{\prime}$. Barring trivial selection
strategies such as random selection, uncertainty sampling remains the most
popular strategy for this application [5], which involves simply selecting the
sample the training algorithm is least certain about for labeling. Other
strategies include query-by-committee [6] and expected model change [7], among
others [8, 9]. Some research also considers the ease of answering by the human
for the query within the querying strategy [10]. Below, we will discuss the
changes made to the system itself, but note that most if not all of the
considered research above focuses heavily on the algorithm to solve the
"information MDP" and not the changes required to the actual Markov system
itself. For the full list of considered changes to the Markov model to account
for changes in query type that will be discussed in this section, refer to 1.

Table 1: The choice of $\mathcal{A^{\prime}}$, and $T$ for different types of queries as described in 3.1 Query Type | $\bm{\mathcal{A^{\prime}}}$ | $\bm{T(s^{\prime}|s,a)}$
---|---|---
Labels | $\xi\notin\mathcal{D}$ | $I(s^{\prime}=(\mathcal{D}^{\prime},L(\mathcal{D}^{\prime},r)),\mathcal{D}^{\prime}=\mathcal{D}\cup\\{\xi\\}$
Comparisons | $(\xi_{a},\xi_{b})\in\mathcal{D}$ | $I(s^{\prime}=(\mathcal{D},L(D,r))$
Demonstrations | $s\in S$ | $I(s^{\prime}=(\mathcal{D}^{\prime},L(\mathcal{D}^{\prime},r)),\mathcal{D}^{\prime}=\mathcal{D}\cup\\{o\\}$
Feature Labels | $f(\xi)_{i}$ | $I(s^{\prime}=(\mathcal{D},L(D,r))$
Language | $\xi\in\mathcal{D}$ | $I(s^{\prime}=(\mathcal{D}^{\prime},L(\mathcal{D}^{\prime},r)),\mathcal{D}^{\prime}=(\mathcal{D}\setminus\\{\xi\\})\cup\\{o\\}$

#### Labels

The vast majority of traditional AL methods [11, 12, 13] utilize a simple
querying strategy. The algorithm initially learns from a dataset
$\mathcal{D}\subset\Xi$. Then, the algorithm selects an unlabeled instance
$\xi\notin\mathcal{D}$ (or several [14, 15, 16]) - this is our definition of
the action space $\mathcal{A}^{\prime}$. The algorithm also requests a label
for the selected sample - this label can be a binary indication of the
selected samples quality or in some cases [17] a more fine-grained rating such
as classifications or numerical scales. The choice of rating defines the set
$\Omega$ \- since there are many variations of labeling methods spanning
discrete, continuous, categorical, and numerical sets, we refrain from exactly
specifying the possible values for $\Omega$. We do note that the selection of
$\Omega$ highly varies from application to application but in general simply
encompasses a measure of quality for the samples $\xi\in\mathcal{D}$. After
the algorithm receives the response, it will add the selected sample to its
dataset $\mathcal{D}$ and relearn the reward function $r\in\mathcal{R}$, which
can be stated as a deterministic transition between two states in our new
system.

This avenue has received much attention in the active learning literature due
to the simplicity of the action space - the complexity of strategies utilizing
this type of action space usually falls within the query selection algorithm
[18]. However, recently more methods have begun to propose changes to the
action space itself [19], paving the way for more intelligent query selection
algorithms by moving complexity into the action space.

#### Comparisons

We define a comparison as a type of query where the human responder is given
two trajectories $\xi_{a},\xi_{b}$ and is asked to select the better
trajectory of the two. Comparison queries have received wide study in
literature due the removal of the need to maintain an online learning
procedure [20, 21]. Formally, the dataset $\mathcal{D}$ remains constant
throughout, and the function $L(\mathcal{D},r)$ can be modified to only update
the reward function $r$ as a result of the response to the given query. For
example, one could update $r$ to minimize the probability of $\xi_{b}$
receiving a high reward while maximizing the probability of $\xi_{a}$
receiving a high reward (assuming $\xi_{a}$ was preferred to $\xi_{b}$). Under
the linearity assumption, this becomes even simpler and can be succinctly
written as a gradient update on the learned human reward vector $\omega$.

However, comparison query methods tend to suffer from lack of information
value - the value of a binary comparison between two queries depends largely
on the queries selected. Furthermore, due to the fixed nature of the dataset
$\mathcal{D}$ used to select queries, the improvement of the active learning
algorithm depends heavily on the original dataset $\mathcal{D}$. Other
research attempts to address this by utilizing $K-$wise comparisons and other
more informational comparisons such as rankings [22]. Another interesting
approach in this vein is one where the comparison is between two candidate
labels instead of two candidate trajectories [23]. We refrain from including
all manner of specifications of the action space in Table 1 due to the
similarity they maintain to the default comparison specification but all can
be fit in under our formalism.

#### Demonstrations

There is also a large body of work on the task of actively querying for
demonstrations themselves [24, 25, 26]. Instead of relying on a set of
trajectories to ask for labels from, this type of querying asks the human to
generate a trajectory from a starting state provided by the robot. In contrast
to comparison queries, which use more features in the action space, this
approach minimizes action space features and instead moves complexity to the
response space $\Omega$, which in this case would be equivalent to the space
of all trajectories $\Xi$.

An interesting extension to this work is the idea of providing multiple states
for the human to use of waypoints in the requested trajectory generation [27].
This method simply uses a larger action space (several states instead of one)
but still follows the same reasoning for fitting under the formalism as above.

#### Feature Labels

Although feature labels as a query type are a relatively new addition to the
research body in the field of active learning, they have still been the
subject of several studies [19, 28, 29]. The general approach involves
querying the human expert about the usefulness of a particular feature in
relation to a data point or overall objective. This methodology takes
advantage of the fact that in practice, many active learning algorithms opt to
use a feature transformation $f:\Xi\rightarrow\mathbb{R}^{d}$ where
$d<<\mathcal{H}*dim(\mathcal{S})*dim(\mathcal{A})$. We query about the
usefulness of one of the dimensions $d$ and one of its corresponding values
towards the overall objective and use the human response to reweight our
model’s learned importance of the feature $d$. This can essentially be viewed
as a transformation on the learning algorithm $L$ to consider a weighted
version of the dataset $\mathcal{D}$.

#### Language

Many recent advances in active learning have been in the domain of natural
language processing [30, 31]. Specifically, Reinforcement Learning from Human
Feedback algorithms for language models has been tremendously successful. In
this body of work, the requested query for the human labeler is either a
selection between different language outputs or a correction on a generated
output [32]. If we view sentences as a type of trajectory, we can view these
options for actions as either a comparison (which we have already covered) or
a correction. We can formalize a correction for a trajectory query $\xi$ as
essentially the creation of a new trajectory $\xi^{\prime}$. The dataset
$\mathcal{D}$ will then be modified to replace $\xi$ with $\xi^{\prime}$.
Modifying an existing element of the dataset as part of the transition
function lends this category of action a unique place under our framework.

###  3.2 Reward Space: $\mathcal{R^{\prime}}$

Implicitly, many methods do contain specifications of the reward space
$\mathcal{R}^{\prime}$. Methods like expected model change and variance
reduction, while primarily created for the purpose of action selection, could
also feasibly be used as a measure of the reward a given action has on the
state of the system. In contrast, there are relatively few methodologies that
actively specify the reward space $\mathcal{R}^{\prime}$. One method is
information gain, used in [24, 25] which essentially computes the expected
gain in information of a query given a distribution over the predicted
response and the actual response.

###  3.3 Observation Probability Space: $\mathcal{O}$

Most of the literature on active learning utilizes a simple model for human
rationality that has received widespread acceptance in the field - the
Boltzmann rationality model [33]. This model assumes that the human’s choice
of response will be proportional to the exponentiated return for that
response. Under our formalism, any modification to the modeling used for
modeling the probability of a human response would fall under the domain of
$\mathcal{O}$. As such, this portion of the Markov system encompasses a large
variety of research across fields such as psychology and computer science.

##  4 Discussion

For all aspects of active learning, instead of viewing each aspect of the
overall system as separate, our formalism ties them all together under the
partially observable Markovian framework. the robot can extract information
from humans by modeling them as making approximate reward-rational choices. We
believe this unifying lens can enable us to better understand how to contend
with all these system components. In the future, we hope this formalism will
enable research on active querying with more integrated and complex learning
algorithm as well as making it easier to do active learning research in new
directions.

In our survey, we discuss the body of work in the active learning field. We
observe that the vast majority of work in the field occurs in the context of
the action space $\mathcal{A}^{\prime}$. In most cases, the transition
function $\mathcal{T}^{\prime}$ and $\Omega$ follow naturally as a result of
the choice of $\mathcal{A}^{\prime}$. We believe that many promising
directions of research in the future could occur by accounting for the other
aspects of the Markovian formalism, i.e. $\mathcal{R}^{\prime}$ and $O$.
Below, we outline some potential considerations and future directions of work.

#### Non-deterministic $\mathcal{T^{\prime}}$

In our discussion of query types in 3.1, we can observe that the transition
function is deterministic in all of the examples covered. However, it does not
necessarily have to be. Under our formalism, we can propose several new
avenues of research in terms of creating non-deterministic transitions between
states in $\mathcal{S}^{\prime}$. There are two main avenues to approach this.

First, one could develop methods that add data to $\mathcal{D}$ in a non-
deterministic manner e.g. use a parameter $\alpha$ as a probability threshold
for adding a new query and response pair to the dataset
$\mathcal{D}^{\prime}$. If we treat this parameter as part of the query
selection algorithm, we could effectively treat $\alpha$ as a selection metric
for how valuable a query could be to ask a human expert. Normal methods also
require many human samples to function properly - this approach allows for
consideration for the human expert’s time by probabilistically determining
whether or not a given query would be valuable to the learning process as a
whole.

Second, we could also conceive of methods that non-deterministically change
the learned reward function $r$ as a result of the changes to the state. If
our learning algorithm $L$ has an element of randomness such that the output
of the update $L(\mathcal{D},r)$ is not deterministic, then we could instead
represent $r$ as a distribution over possible reward functions that $L$ could
output as a result of a query and data update instead of just one possibility.

#### Dataset Tuning

Consider the modifications to the dataset $\mathcal{D}$ that occur in the
transition function for our examples above. In all of the aforementioned
research, the changes either consisted of an addition or no change at all. A
new research direction could involve modifying old examples using new data
gained from querying - for example, reducing the labeled reward for a query
deemed bad by the human expert.

#### Meta-RL

Many methods ignore the specification of the reward space
$\mathcal{R}^{\prime}$ in this Markovian setup. Implicitly, most methods do
however use some sort of quantitative assessment during the query selection
phase. One potential future research direction could involve the design of a
meta-RL algorithm that operates and optimizes on the "information MDP" by
utilizing an approximation of query value as the reward function and then
optimizing based on that.

##  5 Conclusion

As the capability of artificial intelligence advances, it has become
increasingly more important to develop a nuanced understanding of how
intelligence systems learn and work. For all types of learning systems, from
self-driving cars to assistive robots, we believe that robots cannot rely on
simply specified reward functions and systems to guarantee long term user
satisfaction. By considering and formalizing the full complexities of the
active learning process, we believe we take a small step towards better
understanding how learning processes can function as a whole, and we hope our
work contributes to a future where in which robots fully take advantage of all
the complexities inherent within their learning processes. We are optimistic
that future research will account for the nuances in the formalism and seek
out algorithms and modifications that fully take advantage of the complicated
relationships within the Markovian system. Overall, we hope that our work
offers a unifying perspective on the field of active learning as a whole and
provides insight towards creating a truly intelligent robot.

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# Contact line depinning from sharp edges.

J. Graña Otero<EMAIL_ADDRESS>279 RGAN. Dept. of Mechanical Engineering.
University of Kentucky, KY, USA  I. E. Parra Fabián  Center for Computational
Simulation (CCS). Universidad Politécnica de Madrid. España

###### Abstract

With aim of finding mathematical criteria for contact line depinning from
sharp corners, we have studied the equilibrium and stability of a semi-
infinite planar liquid layer pinned at the vertex of a wedge. Equilibrium is
compatible with a fan of apparent contact angles $\theta_{0}$ bracketed by the
equilibrium contact angles of both flanks of the wedge, so the contact line
could remain pinned if $\theta_{0}$ is within this fan. However, the analysis
of the stability of these solutions, studied exploiting the variational
structure of the problem through turning-point arguments, shows that the
equilibrium becomes unstable at critical depinning advancing $\theta_{0}^{a}$
and receding $\theta_{0}^{r}$ contact angles, which are found as subcritical
saddle-node bifurcations. Equilibrium is thus possible (stable) within the
interval $\theta_{0}^{a}<\theta_{0}<\theta_{0}^{a}$ but the contact line
depins from the vertex beyond these critical angles if they occur within the
equilibrium fan.

Contact line, Pinning, Superhydrophobicity, Bond number, Eigenvalue problems

###### pacs:

47.55.np, 47.61.-k

##  I Introduction

Static liquid free interfaces meet smooth solid surfaces along the contact
line, also called triple line, with a well-defined (uniquely determined)
angle, the so-called contact angle, which is a thermodynamic property at
equilibriumde Gennes (1985); Adamson and Gast (1997); de Gennes  _et al._
(2003). If instead the triple line glides over the surface, the contact angle
is no more a thermodynamic property, although it is still characterized by
well-defined advancing and receding values, typically functions of the
instantaneous apparent relative velocityConinck and Blake (2008); Bonn  _et
al._ (2009); Snoeijer and Andreotti (2013); Eral  _et al._ (2013).

However, the contact angle is not uniquely determined by a local equilibrium
analysis on surface singularities, i.e. points or lines where the vector
normal to the surface is not well-defined, such as sharp corners. Instead, in
these cases a fan of angles is compatible with the pinned triple line, so it
can pivot around the singularity while still pinned, even when the liquid and
its free interface around the triple point are in motionSchellenberger  _et
al._ (2016); Koishi  _et al._ (2009); Liu and Kim (2014a); Hao  _et al._
(2015). For instance, in the case of the planar configuration of Fig. 1, with
an interface pinned at the vertex of a wedge so the triple line is just a
point, it was first shown by GibbsGibbs (1961), and later by others both
experimentally and theoretically Dyson (1988); White (1980); Oliver  _et al._
(1977); Bayramli and Mason (1978), that, at equilibrium, the contact angle can
span the fan defined by the equilibrium contact angles of the two flanks
forming the wedge.

Triple line configurations with more complex geometry than that considered by
Gibbs are considerably more difficult to analyze. This is the case for
instance of the interaction of contact lines with superhydrophobic
substratesQuéré (2008); Watson  _et al._ (2010). This type of surfaces is
typically characterized by complex micro-roughness composed of discrete
structures such as posts with a variety of cross section shapes, and with
characteristic scales much smaller than the capillary lengthShirtcliffe  _et
al._ (2004); Bhushan and Jung (2011); Extrand and Moon (2014); Yamamoto  _et
al._ (2015); Fernández  _et al._ (2017); Li  _et al._ (2013); Wang  _et al._
(2017); Barthlott  _et al._ (2017); Kreder  _et al._ (2016); Paxson and
Varanasi (2013); Hensel  _et al._ (2016); Jun and Bhushan (2008). In these
cases, the contact line has typically two different length scales. Locally
close to the solid substrate, the interface is deformed and develops wrinkles
with the same characteristic length as the micro topology. However, at the
macroscopic scale, for instance the diameter of the drop or the thickness of
the liquid layers resting on the substrate, the interface can be characterized
by well-defined apparent contact anglesDong  _et al._ (2018); Erbil (2014).

Recently, studies solving the details of the geometry of the interface around
the contact line in complex configurations have been publishedChen  _et al._
(2005); Mognetti and Yeomans (2010); Blow and Yeomans (2011); Semprebon  _et
al._ (2012); Iliev and Pesheva (2016); Lu  _et al._ (2017). However, the more
common approach is to use homogenized mesoscopic analysesKoishi  _et al._
(2009); Erbil (2014); Sarkar and Kietzig (2015); Prakash  _et al._ (2016);
Dong  _et al._ (2018), often based on energy arguments, which somehow
circumvent the difficulties associated with the detailed description of the
interface. Important contributions from this point of view are the studies due
to Pomeau and VannimeniusPomeau and Vannimenius (1985) and to Joanny and de
GennesJoanny and de Gennes (1984). Both analyzed the limit of weak
heterogeneities on the substrate, the former in static conditions, the latter
in the case of a moving triple line pinning on an isolated defect.

Two other seminal contributions using homogenized viewpoints are the
phenomenological and now classical models due to Cassie-BaxterCassie and
Baxter (1944) and to WenzelWenzel (1936), originally proposed to address the
specific case of superhydrophobic substrates. Wenzel’s model is appropriate
for the so-called wetting conditions, when the liquid wets the entire exposed
solid surface; in this case the apparent contact angle at the triple line is
given in terms of the ratio of the wetted to the projected areas of the solid
surface. Cassie-Baxter’s model applies instead to the so-called non-wetting
conditions, when the liquid entraps underneath its free surface gas pockets
which prevent its direct contact with the solid; in this case, the apparent
contact angle is given in terms of ratio of the wetted to the exposed areas of
the solid surface.

These mesoscopic homogenized models provide guidance understanding
experimental work. However, precise quantitative criteria predicting depinning
still remains vague despite their critical role in phenomena involving liquid
free interfaces in contact with solids. For instance, the transition between
wetting and non-wetting states on superhydrophobic surfaces is well-known to
be mediated by depinning transitions, both in staticPapadopoulos  _et al._
(2013), and in the dynamics conditionsSchellenberger  _et al._ (2016). Or in
boiling liquids, vapor bubbles grow on microscopic wall cavities up to final
bubble sizes at detachment which depend critically on whether the triple line
of the bubble remains pinned at the rim or it glides and spreads over the wall
around the cavityBetz  _et al._ (2013); Jo  _et al._ (2011); Shoghl  _et al._
(2014a, b); Hazi and Markus (2009); Liu and Kim (2014b).

This paper is devoted to the mathematical analysis of depinning from sharp
edges. We show how depinning criteria can be precisely defined, illustrating
the methodology in a simple configuration so the mathematical treatment
remains affordable. The essential idea is to formulate the problem for the
interface shape as an eigenvalue problem, with the Bond number $Bo$ as the
eigenvalue in the specific case studied. Depinning can be found then as a
subcritical saddle-node bifurcation for a critical Bond number. This basic
idea is still generally applicable in more complex configurations despite the
considerable mathematical difficulties.

##  II Advancing triple lines.

Figure 1 shows a sketch of the 2D configuration we study here, namely a semi-
infinite horizontal liquid layer in static equilibrium under the action of
gravity and surface tension forces and pinned at the vertex of a wedge. The
material of the front and back flanks of the wedge can be different, and the
corresponding equilibrium contact angles are $\theta_{ef}$ and $\theta_{eb}$
respectively. In order to ensure equilibrium, the solid substrate must become
horizontal (perpendicular to gravity) far upstream from the wedge, so the
liquid layer also levels off to a constant thickness $h_{\infty}$, which
ultimately sets the pressure distribution inside the liquid layer. The actual
details of the substrate between the wedge and this horizontal zone are
actually irrelevant in this static problem.

Initially the interface meets the smooth substrate upstream from the wedge’s
vertex with the corresponding equilibrium contact angle
$\theta_{0}=\theta_{ef}$. In these conditions, as shown below, the balance
between gravity and surface tension uniquely determinesde Gennes  _et al._
(2003) the liquid layer thickness $h_{\infty}$ in terms of $\theta_{ef}$ as:

$\displaystyle h^{2}_{\infty}=2(\sigma/\rho g)(1-\cos\theta_{ef}).$ (1)

Angles are measured through the liquid, relative to the local horizontal,
positive clockwise as shown in Figure 1.

Figure 1: Geometry of the liquid layer interface (pink line) pinned at the
vertex of a wedge of inner angle $\phi$. The plotted interface is the
equilibrium shape corresponding to an angle $\theta_{0}=240^{\circ}$, or
equivalently to a Bond number $Bo=\rho gh_{\infty}^{2}/\sigma=3$, i.e. a
thickness upstream $h_{\infty}=\sqrt{3}\ell_{c}$ with
$\ell_{c}^{2}=\sigma/(\rho g)$ the capillary length. As shown below, this is
an equilibrium but unstable configuration.

The liquid layer is then forced quasi-statically to slowly glide – for
instance by continuously supplying liquid far upstream from the wedge – over
the front flank of the wedge until the triple line touches the vertex. Pushing
the liquid layer further against the vertex forces the contact angle
$\theta_{0}$ to increase beyond $\theta_{ef}$. With the triple line pinned,
the angle $\theta_{0}$ is now a free parameter, which is no longer determined
by thermodynamic equilibrium. Instead, the liquid layer thickness $h_{\infty}$
and the angle $\theta_{0}$ are now arbitrary, only linked by the balance
between gravity and surface tension forces which leads to the same
relationship as in (1) but with $\theta_{0}$ replacing $\theta_{ef}$. In terms
of the capillary length $\ell^{2}_{c}=\sigma/(\rho g)$ and of the Bond number
$Bo=(h_{\infty}/\ell_{c})^{2}$ this expression can be alternatively written
as:

$\displaystyle
Bo=\left(\frac{h_{\infty}}{\ell_{c}}\right)^{2}=2(1-\cos\theta_{0})$ (2)

which is represented in Figure 2.

Thus, forcing the liquid layer against the wedge’s vertex, as trying to
overcome it, requires at equilibrium to increase the thickness $h_{\infty}$ of
the liquid layer, and accordingly the angle $\theta_{0}$, as Eq. (2) shows.
The triple line can be expected to still remain pinned at the vertex until, in
principle, the equilibrium contact angle with the back flank is reached. Once
this condition is reached, the triple line would continue gliding over this
flank. The apparent contact angle $\theta_{0}$ spans therefore, with the
triple line pinned, a fan of angles bracketed by the lower
$\theta_{0}^{min}=\theta_{ef}-\alpha$ and the upper
$\theta_{0}^{max}=\pi-\alpha-\phi+\theta_{eb}$ limits, as Gibbs first and
others later showedGibbs (1961); Dyson (1988); White (1980).

However, as we show below, the interface becomes unstable at a critical angle
$\theta_{0}^{a}$ (the superscript $a$ stands for advancing front). In this
case, the triple line will depin if $\theta_{0}^{a}$ is contained within the
Gibbs’ fan. More specifically, for the configuration considered here the
interface destabilizes at $\theta_{0}^{a}=\pi$ so if
$\theta_{0}^{min}<\pi<\theta_{0}^{max}$, then the interface depins before
starting to glide along the back surface. This depinning condition is
suggested by Figure 2, which shows that the height $h_{\infty}$, or
equivalently the Bond number $Bo$, as functions of the angle $\theta_{0}$,
exhibit a maximum defined by:

$\displaystyle Bo^{a}=4\qquad\text{for}\qquad\theta^{a}_{0}=\pi,$ (3)

or equivalently $h_{\infty}^{a}=2\ell_{c}$.

Thus, forcing more liquid into the liquid layer, trying to increase its
thickness and the contact angle in order to overcome the wedge’s edge,
ultimately fails at the critical angle $\theta^{a}_{0}=\pi$ because
equilibrium is not any more possible for thicknesses greater than
$h^{a}_{\infty}$. The triple line depins therefore. This argument can be made
rigorous by studying the stability of the equilibrium solutions to show that
$\theta^{a}_{0}=\pi$ actually represents a bifurcation point where the
equilibrium solutions for $\theta^{a}_{0}>\pi$ become unstable as shown in
Section IV.

Figure 2: (Color online) The continuous blue and dashed red lines represent
the equilibrium (thickness) Bond number $Bo=\rho gh_{\infty}^{2}/\sigma$ as a
function of the apparent angle $\theta_{0}$ at the edge, as given by the
expression (2). The blue line represents the stable equilibrium solutions
branch, whereas the red dashed line are unreachable equilibrium solutions
because depinning occurs before. The inserts show the shape of the interfaces
for some representative values of $\theta_{0}$ on the stable branch.

###  II.1 Formulation.

The analysis has been straightforward so far due to the simple configuration
considered. However, a more general and systematic approach will be used
below, in order to show how to analyze more complex geometries. At
equilibrium, the pressure distribution in the liquid is given by the
hydrostatic balance $p(z)=p_{\infty}+\rho g(h_{\infty}-z)$, so the shape of
the interface $\mathcal{S}$ is determined by the condition that $p(z)$,
evaluated at $\mathcal{S}$, is $p(\mathcal{S})=p_{\infty}+\sigma\mathcal{K}$,
where $\sigma$ is the surface tension of the liquid in contact with the
ambient gas, and $\mathcal{K}$ is twice the mean curvature of the interface.

The geometry of the liquid interface will be represented parametrically in the
form $(x_{\mathcal{S}}(t),z_{\mathcal{S}}(t))$, where $x_{\mathcal{S}}$,
$z_{\mathcal{S}}$ are, respectively, the horizontal and vertical coordinates
of the interface in the system of reference of Figure 1. The parameter $t$ is
the arc length, so $x_{\mathcal{S}}^{\prime 2}(t)+z_{\mathcal{S}}^{\prime
2}(t)=1$, with the primes representing the $t$-derivative. Lengths are made
dimensionless with the far-upstream liquid thickness $h_{\infty}$, leading to
the dimensionless variables $\xi=x_{\mathcal{S}}/h_{\infty}$,
$\zeta=z_{\mathcal{S}}/h_{\infty}$ and $s=t/h_{\infty}$; the s-derivative is
represented below with a dot, $\dot{\phi}=d\phi/ds$. With this
parametrization, the dimensionless interface curvature can be written asdo
Carmo (1976); Eisenhart (2004)
$\kappa_{\mathcal{S}}=h_{\infty}\mathcal{K}=\dot{\xi}_{\mathcal{S}}\ddot{\zeta}_{\mathcal{S}}-\dot{\zeta}_{\mathcal{S}}\ddot{\xi}_{\mathcal{S}}$,
so the shape of the interface is determined by the fourth order differential
system (dropping for simplicity the subscript $\mathcal{S}$):

$\displaystyle\dot{\xi}\ddot{\zeta}-\dot{\zeta}\ddot{\xi}=Bo(1-\zeta)$ (4a)
$\displaystyle\dot{\xi}^{2}+\dot{\zeta}^{2}=1$ (4b)

The appropriate boundary conditions are those enforcing pinning at the edge of
the substrate:

$\displaystyle\xi(s=0)=\zeta(s=0)=0,$ (4c)

and leveling off to $h_{\infty}$ far upstream from the edge:

$\displaystyle\xi\rightarrow-\infty,\quad\zeta\rightarrow
1\qquad\text{as}\qquad s\rightarrow\infty.$ (4d)

This problem has a continuum of solutions parametrized by the Bond number,
which enters due to the presence of two characteristic lengths, namely the
liquid layer thickness $h_{\infty}$, and the capillary length $\ell_{c}$. In
particular, the apparent contact angle $\theta_{0}$ of the interface at the
edge can then be determined from either $\cos\theta_{0}=-\dot{\xi}(0)$ or
$\sin\theta_{0}=\dot{\zeta}(0)$, which give $\theta_{0}$ as a function of $Bo$
as shown in Figure 2. This Figure reveals that $\theta_{0}$ is a multi-valued
function of the Bond number $Bo$. On the contrary, $Bo$ is univocally
determined as a function of $\theta_{0}$. Thus, the problem of finding the
shape of the interface is more conveniently formulated as an eigenvalue
problem, i.e. that of finding the Bond number $Bo$ that corresponds to a given
angle $\theta_{0}$ at the edge. In this case, the boundary conditions must be
augmented with $\dot{\xi}(0)=-\cos(\theta_{0})$,
$\dot{\xi}(0)=\sin(\theta_{0})$.

Formulated as an eigenvalue problem, (4) can thus be written as:

$\displaystyle\dot{\xi}\ddot{\zeta}-\dot{\zeta}\ddot{\xi}=Bo(1-\zeta)$ (5a)
$\displaystyle\dot{\xi}^{2}+\dot{\zeta}^{2}=1$ (5b) with boundary conditions
$\displaystyle\xi(s=0)=\zeta(s=0)=0$ (5c)
$\displaystyle\dot{\xi}(0)=-\cos(\theta_{0}),\quad\dot{\zeta}(0)=\sin(\theta_{0})$
(5d) $\displaystyle\xi\rightarrow-\infty,\quad\zeta\rightarrow
1\qquad\text{as}\qquad s\rightarrow\infty$ (5e)

where $Bo(\theta_{0})$ is the eigenvalue which must be obtained as part of the
solution for each value of the contact angle $\theta_{0}$ of the pinned
interface at the wedge’s vertex.

###  II.2 Solution.

The problem just formulated can be solved analytically in closed form as shown
in Appendix A. Here we will just use a first integral that can be obtained as
follows. Combining equations (5a) and (5b) to eliminate $\xi$ gives:

$\displaystyle\ddot{\zeta}=-Bo\,(1-\zeta)\sqrt{1-\dot{\zeta}^{2}},$ (6)

which is an ordinary differential equation for the height $\zeta$ of the
interface in terms of the arc-length $s$. Integrating it once and matching the
solution with the constant thickness $\zeta\rightarrow 1$,
$\dot{\zeta}\rightarrow 0$ far upstream from the edge gives the first
integral:

$\displaystyle\pm\sqrt{1-\dot{\zeta}^{2}}=1-\frac{Bo}{2}(1-\zeta)^{2}.$ (7)

Evaluating it at the triple point, namely $\zeta(s=0)=0$,
$\pm\sqrt{1-\dot{\zeta}^{2}(s=0)}=\cos\theta_{0}$, gives the Bond number in
terms of $\theta_{0}$, i.e. equation (2):

$\displaystyle
Bo=\left(\frac{h_{\infty}}{\ell_{c}}\right)^{2}=2(1-\cos\theta_{0})$ (8)

Figure 3: (Color online) Interface shapes for different values of the
effective thickness $\Delta h_{\infty}=h_{\infty}-h_{0}$ of the liquid layer.
The thick vertical blue arrow and the thin curved green one indicate the
direction of increasing values of $\Delta h_{\infty}$ and $\theta_{0}$
respectively. $\theta_{0}$ range from negative (the two lowest darker blue
interfaces) to positive (the highest light blue one) values, with
$\theta_{0}=0$ corresponding to a straight horizontal interface at the wedge’s
vertex height. The arrows also indicate the evolution as the liquid layer is
forced to advance against the vertex (green curve starting at $b$ in the
$\theta_{0}$-Bo plane of Figure 4), whereas for a layer receding (purple curve
starting at $a$ in Figure 4) back from the vertex the evolution is reversed.

In order to keep the reasoning simple we defer the detail analysis of the
stability of these equilibrium solutions to Section IV. However, it can be
anticipated that, as common experience shows, the equilibrium configurations
are stable for small $\theta_{0}$ contact angles, although increasing it
ultimately results in the destabilization and depinning at the critical
$\theta_{0}=\pi$, so the solutions corresponding to $\theta_{0}>\pi$ are
unstable.

We note here that the key point is formulating the problem of the equilibrium
of the liquid layer as an the eigenvalue problem (5), which naturally leads to
the interpretation of Figure 2 as the bifurcation diagram of the equilibrium
solutions with the eigenvalue $Bo$, or equivalently the liquid layer thickness
$h_{\infty}$, as bifurcation parameter. For each value of the Bond number
$Bo<4$ there are two solutions, the stable branch with $\theta_{0}<\pi$, and
the unstable one $\theta_{0}>\pi$. These two branches merge at $Bo=4$ and
disappear for $Bo>4$. Therefore, the triple line depinning occurs as a
subcritical saddle-node or fold bifurcation.

##  III Receding triple lines.

The solution of the previous section can be generalized to cases in which the
edge height $h_{0}$ relative to the substrate far upstream is non-zero as
shown in Figure 3. Now, in addition to the liquid layer thickness
$h_{\infty}$, the problem has a new length scale $h_{0}$. However, as shown in
Appendix C, it is easy to see that the problem only depends on the difference.
Thus, scaling lengths with the effective thickness $\Delta
h_{\infty}=h_{\infty}-h_{0}>0$ (the negative case is discussed below), leads,
to a problem with the exact same form as (5), except that now the eigenvalue
is the effective Bond number:

$\displaystyle Bo_{eff}=\left(\frac{\Delta
h_{\infty}}{\ell_{c}}\right)^{2}=2(1-\cos\theta_{0})$ (9)

Figure 4: (Color online) The complete bifurcation diagram including positive
as well as negative apparent contact angles. The purple arrow, starting at
$a$, represents the evolution path for a receding interface, eventually
depining at $\theta_{0}=-180^{\circ}$. The green one, starting at $b$,
corresponds to the evolution of an advancing liquid front with an initial
negative effective thickness $\Delta h_{\infty}$, eventually depining at
$\theta_{0}=180^{\circ}$. The solution branches in dash-dot lines are
inaccessible because they are unstable and deppining occurs before, at
$\theta_{0}=\pm 180^{\circ}$.

As suggested in Figure 3, this configuration admits both positive and negative
values of the effective height $\Delta h_{\infty}$, when the liquid layer is,
respectively, above or below the wedge’s vertex. Due to the symmetries of the
problem the solutions in this latter case of negative values of $\Delta
h_{\infty}$ (i.e. $h_{\infty}<h_{0}$) can be obtained from symmetry
considerations by simply flipping the interface about the $\xi$ axis, which
changes the sign of $\theta_{0}$, but leaves the effective Bond number
$Bo_{eff}$ unchanged. Therefore, the bifurcation diagram in this generalized
case can be obtained from that in Figure 2, extending it to negative values of
$\theta_{0}$ by a reflection about the axis $\theta_{0}=0$ as shown in Figure
4.

###  III.1 Depinning conditions.

In order to derive the deppining criterion as the liquid layer recedes, one
can consider as initial condition a point in the bifurcation diagram with
positive values of $\Delta h_{\infty}$ and $\theta_{0}$, as the point $a$ in
Figure 4. The thickness of the liquid layer, and accordingly the apparent
contact angle, can be forced to decrease by removing liquid from far away
upstream from the vertex. $\Delta h_{\infty}$ traverses the equilibrium line
$Bo_{eff}(\theta_{0})$ in the direction represented by the blue path, reaching
eventually $\Delta h_{\infty}=0$ for $\theta_{0}=0$, and then becoming
negative, forcing accordingly negative apparent contact angles $\theta_{0}$.
The evolution in the physical space is that schematically represented in
Figure 3, from the light top to the darker blue interfaces. This thinning of
the liquid layer can be continued until $\theta_{0}=-180^{\circ}$. Beyond this
point it is not possible keep on decreasing $h_{\infty}$, because
$Bo_{eff}=((h_{0}-h_{\infty})/\ell_{c})^{2}$ has a local maximum, and
therefore $h_{\infty}$ a local minimum. Consequently, the triple line depins
if forced to recede further. The critical deppining condition is thus
$Bo_{eff}=4$, or equivalently $h_{\infty}=h_{0}-2\ell_{c}$, and
$\theta_{0}^{r}=-180^{\circ}$.

The green path on the other hand, starting at $b$ represents the opposite
evolution, namely starting with a liquid layer below the vertex, and therefore
pinned with an angle $\theta_{0}$ below the horizontal. Adding liquid so to
increase its thickness will force the growth of $\theta_{0}$, all the way up
through $\theta_{0}=0$, until the limit $\theta_{0}^{a}=180^{\circ}$, beyond
which the triple line depins. The reason is again that
$\theta_{0}^{a}=180^{\circ}$ is a local maximum of
$Bo_{eff}=((h_{\infty}-h_{0})/\ell_{c})^{2}$, and therefore of $h_{\infty}$,
so it is not possible to increase it any further, as the advancing condition
being forced would require.

##  IV Stability

The heuristic arguments given above showing the loss of stability beyond the
critical angles $\theta_{0}=\pm 180^{\circ}{}$ can be made rigorous through an
analysis of the stability of the equilibrium solutions. We exploit for that
purpose the variational structure of the problem, derived for reference in
Appendix B. From this point of view, stable solutions correspond to local
minima of the energy functional Gelfand and Fomin (1963), i.e. to solutions
with the second variation of the energy functional is positive definite. We
will proceed in two steps: we first show that the solutions with
$|\theta_{0}|<180^{\circ}{}$ are stable; then, we use Turning Point
methodsMaddocks (1987); Bostwick and Steen (2015) to prove the loss of
stability at $|\theta_{0}|=180^{\circ}{}$, and therefore the instability of
the solutions branch corresponding to $\theta_{0}>180^{\circ}{}$.

###  IV.1 Stability for $|\theta_{0}|<180^{\circ}{}$.

We consider here the stability of the equilibrium solutions subject to planar
perturbations leaving the interface pinned at the vertex. In addition, we
restrict this analysis to apparent contact angles $\theta_{0}$ smaller than
$\theta_{0}<180^{\circ}{}$. In this case, using the same system of coordinates
as in Figure 1, the interface $\mathcal{S}$ can be described explicitly in the
form $\xi=\xi_{\mathcal{S}}(\zeta)$, with $\xi_{\mathcal{S}}(\zeta=0)=0$ to
enforce pinning at the vertex. This representation is admissible for contact
angles in the range $[-180^{\circ},180^{\circ}]$. Beyond these limits the
function $\xi_{\mathcal{S}}(\zeta)$ would be doubled-valued and some more
general representation should be used.

The energy functional becomes thus:

$\displaystyle\mathcal{E}=\int_{0}^{1}{\left\\{\sqrt{1+\xi_{\mathcal{S}}^{{}^{\prime}2}(\zeta)}-Bo(1-\zeta)(\xi_{\mathcal{S}}(\zeta)-H)\right\\}d\zeta}$

where $H$ is a constant. Strictly, $H$ should be let go to infinity, which
would make the integral infinite. However, it is easy to check that the value
of this constant is immaterial and can actually be set to zero.

We write the perturbed interface as $\xi_{\mathcal{S}}(\zeta)+h(\zeta)$, with
the perturbation $h$ bounded and compatible with the pinned interface, that
is, $h(0)=0$, and with a horizontal interface far upstream from the edge, i.e.
$h^{\prime}(\zeta\rightarrow 1)=0$. Otherwise $h(\zeta)$ is unrestricted, so
the perturbed solutions can accommodate variations in the contact angle so
$h^{\prime}(0)$ is arbitrary, although the interface
$\xi_{\mathcal{S}}(\zeta)+h(\zeta)$ must still be a well-defined, single
valued function of $z$.

With this parametrization the second variation of $\mathcal{E}$ is:

$\displaystyle 2\delta^{2}\mathcal{E}=\int_{0}^{1}{\frac{h^{\prime
2}}{(1+\xi_{\mathcal{S}}^{{}^{\prime}2})^{3/2}}\,d\zeta}$

which is clearly non-negative for any admissible perturbation $h(x)$. Thus,
the equilibrium solutions found for contact angles $\theta_{0}$ within
$]-180^{\circ},180^{\circ}[$ are stable under planar pinned perturbations.

###  IV.2 Stability for $|\theta_{0}|\geq 180^{\circ}{}$.

The previous analysis shows the stability of the solutions with a contact
angle smaller than 180∘. The question of the stability for larger contact
angles can be addressed using the elegant results due to MaddocksMaddocks
(1987) for problems with a variational structure as in the present case. In
order to in this analysis apparent contact angles larger than $180^{\circ}{}$
we will use the parametric representation of the interface introduced in the
text in Section II.1. The energy functional $\mathcal{E}(u)$ with
$u=(\xi,\zeta,\dot{\xi},\dot{\zeta})$ can be written in that case as:

$\displaystyle\mathcal{E}=\mathcal{A}+Bo\mathcal{P}$ (10a) with
$\displaystyle\mathcal{A}=\int_{0}^{\infty}{\left(\sqrt{\dot{\xi}^{2}+\dot{\zeta}^{2}}-1\right)ds}$
(10b)
$\displaystyle\mathcal{P}=\int_{0}^{\infty}{\left((\zeta-\zeta^{2}/2)\dot{\xi}+1/2\right)ds}$
(10c) with boundary conditions $\displaystyle\xi(s=0)=\zeta(s=0)=0$ (10d)
$\displaystyle\dot{\xi}(0)=-\cos(\theta_{0}),\quad\dot{\zeta}(0)=\sin(\theta_{0})$
(10e) $\displaystyle\xi\rightarrow-\infty,\quad\zeta\rightarrow
1\qquad\text{as}\qquad s\rightarrow\infty$ (10f)

The Bond number $Bo(\theta_{0})$ is an eigenvalue of the problem and the
bifurcation parameter. Following MaddocksMaddocks (1987) and translating his
results into our nomenclature, the changes of stability can be studied in the
plane $Bo-\mathcal{E}_{Bo}$, where $\mathcal{E}_{Bo}=\mathcal{P}$ is the
derivative of the energy functional with respect to the bifurcation parameter
$Bo$. In this plane, the equilibrium solutions are represented by curves
parametrized in our case by the contact angle $\theta_{0}$, i.e.
$(Bo(\theta_{0}),\mathcal{P}(\theta_{0}))$ as shown in Figure 5. The changes
of stability are associated with folds of the solution branchesMaddocks
(1987), defined as points where the derivative $dBo/d\theta_{0}$ of the
bifurcation parameter $Bo$ with respect to $\theta_{0}$ vanishes, such as the
point $\theta_{0}=180^{\circ}{}$ shown in Figures 4 and 5.

Figure 5: The solid blue and dash-dots red lines represent respectively the
stable and unstable equilibrium solutions in the plane $Bo-\mathcal{P}$, with
the contact angle as parameter along them. This is the preferred planeMaddocks
(1987) to analyze the stability of the solutions of the problem defined by
(10). The fold ($dBo/d\theta_{0}=0$) at $|\theta_{0}|=180^{\circ}{}$ ($Bo=4$)
represents the loss of stability of the blue stable branch which correspond to
$|\theta_{0}|<180^{\circ}{}$.

This can be seen as follows (see MaddocksMaddocks (1987) for rigorous proofs).
The stable equilibrium solutions correspond to minima of (10a), i.e. to zeros
of the first Fréchet $u$-derivative $\delta\mathcal{E}$ where the second
Fréchet $u$-derivative $\delta^{2}\mathcal{E}$ is non-negative. Since
$\delta^{2}\mathcal{E}$ is a real symmetric quadratic form in the perturbation
$(\delta\xi,\delta\zeta,\delta\dot{\xi},\delta\dot{\zeta})$ of the equilibrium
solution, its spectrum is real and form an increasing discrete sequence of
eigenvalues. Thus, the second variation being non-negative is equivalent to
its critical (smallest) eigenvalue $\mu$ being non-negative as well, and the
question of the stability of the equilibrium solutions is reduced to the
analysis of the evolution of the critical eigenvalue as the contact angle
$\theta_{0}$ is varied. As shown by MaddocksMaddocks (1987), changes of
stability can only occur at folds, where the critical eigenvalue crosses zero.
This can be easily seen by taking the derivative of the equilibrium condition
$\delta\mathcal{E}=0$ along the solution branch (i.e. taking the derivative
with respect to the contact angle $\theta_{0}$, denoted with a prime hereafter
$d\phi/d\theta_{0}=\phi^{\prime}$). Performing the derivative leads to
$\delta^{2}\mathcal{E}\cdot u^{\prime}+\delta{\mathcal{E}}_{Bo}\cdot
Bo^{{}^{\prime}}=0$, with $\delta{\mathcal{E}}_{Bo}$ the derivative of
$\delta{\mathcal{E}}$ with respect to $Bo$. Thus $\delta^{2}\mathcal{E}$ has a
zero eigenvalue when $Bo^{{}^{\prime}}=0$. This shows that in our problem,
changes of stability can only occur at the folds $|\theta_{0}|=180^{\circ}{}$,
where $Bo^{{}^{\prime}}=0$ as Figures 4 and 5 reveal. Furthermore, it can be
shownMaddocks (1987) that the derivative $\mu^{\prime}$ of the critical
eigenvalue with respect to the parameter, $\theta_{0}$ in this problem, is
given, up to a positive factor, by:

$\displaystyle\mu^{\prime}\approx Bo^{\prime\prime}\cdot\mathcal{P}^{\prime}$
(11)

where the primes represent the derivatives with respect to the contact angle
$\theta_{0}$. Figure 5 shows that $\mathcal{P}$ is an increasing function of
$\theta_{0}$ around the fold $Bo=4$, whereas $Bo(\theta_{0})$ features a
maximum (see also Figure 2), and therefore $Bo^{\prime\prime}$ is negative.
Thus $\mu^{\prime}$ is negative at the fold. Consequently, since as seen the
branch $|\theta_{0}|<180^{\circ}{}$ is stable and therefore the critical
eigenvalue $\mu$ is positive, $\mu$ crosses zero at $Bo=4$
($\theta=180^{\circ}{}$) and becomes negative along the branches
$|\theta_{0}|>180^{\circ}{}$. This shows therefore that the equilibrium
solutions are unstable for apparent contact angles
$|\theta_{0}|>180^{\circ}{}$.

##  V Conclusions.

We have introduced a general methodology capable of providing precise
mathematical criteria for depining transitions. In the specific problem
analyzed here, and in other cases with similar symmetries, the transition
occurs as a subcritical bifurcation, with no equilibrium solutions beyond a
critical value of the bifurcation parameter, typically an effective Bond
number. This methodology leads to straightforward criteria, which in the
analyzed configuration is easily obtained due to the relatively uncomplicated
system of ordinary differential equations governing the analyzed 2D
configuration. Axisymmetric configurations are still governed by a system of
differential equations and precise mathematical depinning conditions, which
again arise as subcritical saddle-node bifurcations, can be derived. However,
general cases of 3D non-symmetrical configurations lead to free boundary
problems involving intricate non-linear partial differential equations, with
triple lines that are not known in advanced, but must instead be found as part
of the solution. One immediate open question for instance is about the
robustness of the saddle-node bifurcation in these more complex problems, i.e.
if the saddle-node bifurcation is the generic route to triple line depinning.
However, despite these technical difficulties, this new approach seems to
uncover an exciting new line of research, which could be helpful towards
understanding open problems related to wetting-dewetting transitions, and more
broadly the dynamics of free interfaces gliding over solid surfaces.

##  VI Acknowledgments

We gratefully acknowledge the funding support from NASA award number
NNX13AB12A via NASA Kentucky EPSCoR.

##  Appendix A Equilibrium solutions.

As shown in the text, the problem (5) has the first integral:

$\displaystyle\pm\sqrt{1-\dot{\zeta}^{2}}=1-\frac{Bo}{2}(1-\zeta)^{2}.$ (12)

from which, the problem for $\zeta$ turns to be:

$\displaystyle\dot{\zeta}=\pm\sqrt{\frac{Bo}{2}}(1-\zeta)\left(2-\frac{Bo}{2}(1-\zeta)^{2}\right)^{1/2}$
(13a) $\displaystyle\zeta(s=0)=0$ (13b) where the positive or negative signs
apply for contact angles $0\leq\theta_{0}\leq\pi$ or $\pi\leq\theta_{0}\leq
2\pi$ respectively

This equation can be simplified using the auxiliary independent variable
$u^{2}=1-(Bo/4)(1-\zeta)^{2}$, which, in the case $0\leq\theta_{0}\leq\pi$,
transforms the problem for $\zeta$ in:

$\displaystyle\dot{u}=\sqrt{Bo}\,(1-u^{2})\qquad(0\leq\theta_{0}\leq\pi)$
(14a) to be integrated with $\displaystyle
u(s=0)=\left(1-\frac{Bo}{4}\right)^{1/2}$ (14b)

The integration is straightforward in terms of the hyperbolic trigonometric
functions as for instance:

$\displaystyle\zeta=1-\frac{2}{\sqrt{Bo}\,\cosh\left(\sqrt{Bo}\,s+\tanh^{-1}\sqrt{1-Bo/4}\right)}$
(15) $\displaystyle\text{for}\qquad(0\leq\theta_{0}\leq\pi)$

In the case of contact angles in the interval $\pi\leq\theta_{0}\leq 2\pi$,
the negative sign in equation (13) must be chosen initially. However,
$\dot{\zeta}$ eventually vanishes at a turning point
$s=s_{tp}=Bo^{-1/2}\tanh^{-1}\sqrt{1-Bo/4}$, corresponding to the minimum
negative height of the drop. The integration then proceeds for $s>s_{tp}$ with
positive values of $\dot{\zeta}$, so the sign in (13) must be switched back to
plus. The solution can be written thus as:

$\displaystyle\zeta=1-\frac{2}{\sqrt{Bo}\,\cosh\left(-\sqrt{Bo}\,s+\tanh^{-1}\sqrt{1-Bo/4}\right)}$
(16a) $\displaystyle\text{for}\qquad(\pi\leq\theta_{0}\leq 2\pi)$

On the other hand, the equation for $\xi$ follows from (12) as:

$\displaystyle\dot{\xi}=-1+\frac{Bo}{2}(1-\zeta)^{2}.$ (16b)

to be integrated with the condition $\xi(s=0)=0$. Thus, using the previous
solution for $\zeta$ gives:

$\displaystyle\xi=-s+\frac{2}{\sqrt{Bo}}\left[\tanh\left(\sqrt{Bo}(s+s_{tp}\right)-\sqrt{\left(1-\frac{Bo}{4}\right)}\right]$
(16c) $\displaystyle\text{for}\qquad(0\leq\theta_{0}\leq\pi)$
$\displaystyle\xi=-s-\frac{2}{\sqrt{Bo}}\left[\tanh\left(\sqrt{Bo}(-s+s_{tp}\right)-\sqrt{\left(1-\frac{Bo}{4}\right)}\right]$
(16d) $\displaystyle\text{for}\qquad(\pi\leq\theta_{0}\leq 2\pi)$

##  Appendix B Variational formulation.

The system (5) used in the text can be as well obtained as the Euler-Lagrange
equationsGelfand and Fomin (1963) for the shape that minimizes the total
energy Landau and Lifshitz (1987); Bostwick and Steen (2015) $E=\int{\\{\sigma
dA+PdV\\}}$, including the surface and the pressure-gravitational potential
energies. In the specific case of the 2D configuration being studied, it is
more convenient to consider the energy per unit length (normal to the paper in
Figure 1), so the volume integral of the potential energy becomes a surface
integral extended to the area enclosed between the interface and the
substrate, whereas the surface integral reduces to a line integral along the
interface. Assuming a uniform surface tension along the interface, and writing
$\mathcal{E}$ for the energy per unit length scaled with $\sigma h_{\infty}$,
the appropriate energy functional can be written as the line integral:

$\displaystyle\mathcal{E}=\int_{0}^{\infty}{\left(\sqrt{\dot{\xi}^{2}+\dot{\zeta}^{2}}+Bo(\zeta-\zeta^{2}/2)\dot{\xi}-\mathcal{E}_{0}\right)ds}$
(17)

where $\mathcal{E}_{0}=1-Bo/2$ is a constant added to render the integral
convergent as $s\rightarrow\infty$, where $\dot{\xi}\rightarrow-1$,
$\zeta\rightarrow 1$. However this constant plays no role in the solution and
can be dropped.

Standard methods give the system:

$\displaystyle\ddot{\xi}=-Bo\,\dot{\zeta}\,(1-\zeta)$ (18a)
$\displaystyle\ddot{\zeta}=Bo\,\dot{\xi}\,(1-\zeta)$ (18b)

which can also easily be derived from Eq. 4.

It can be noticed that integral of the term $\zeta\dot{\xi}$ in (17)
represents the volume (per unit length) of the liquid layer. $Bo$ is therefore
the Lagrange multiplier of the constrained problem of finding the shape of
minimum surface area (per unit length) subject to a constant volume (per unit
area). In general cases, the Lagrange multiplier is not known in advance and
must be obtained as part of the solution, so in these cases it is more
advantageous to formulate the problem as that of finding the interface shape
of minimum surface area constrained to a constant volume. However, due to the
special configuration of our problem, namely with a constant liquid thickness
far upstream, the multiplier can be determined beforehand and the problem can
thus be formulated as the unconstrained minimization of the energy functional
(17).

##  Appendix C Triple line pinned at a non-zero height.

We derive here the formulation of the problem to find the equilibrium shape of
the interface in cases where the height of the edge where the interface is
pinned is not the same as that of the horizontal substrate far upstream. We
use the formulation introduced in the previous Appendix, based on a
variational approach, which is equivalent to (5) used in the text. The origin
of coordinates is again chosen at the wedge’s vertex where the interface is
pinned. The effective height of the liquid layer far upstream from the triple
line is therefore $\Delta h_{\infty}=h_{\infty}-h_{0}>0$ (we consider below
the case of negative values of $\Delta h_{\infty}$), where the absolute
heights, $h_{\infty}$ and $h_{0}$, are measured relative to some horizontal
reference, for instance the substrate far upstream. The pressure distribution
is thus given by:

$\displaystyle p-p_{\infty}=\rho g(\Delta h_{\infty}-z),$ (19)

Proceeding as before, lengths are made dimensionless with the effective liquid
layer thickness $\Delta h_{\infty}$, whereas the Bond number is defined now in
terms of $\Delta h_{\infty}$, namely $Bo_{eff}=\rho g\Delta
h_{\infty}^{2}/\sigma$. Using the more symmetrical form of the differential
equations derived from the variational formulation (18), the shape of the
interface is thus given by the solution of an eigenvalue problem with the
form, exactly equivalent to that in the main text (5):

$\displaystyle\ddot{\xi}=-Bo_{eff}\,\dot{\zeta}\,(1-\zeta)$ (20a)
$\displaystyle\ddot{\zeta}=Bo_{eff}\,\dot{\xi}\,(1-\zeta)$ (20b) with boundary
conditions $\displaystyle\xi(s=0)=\zeta(s=0)=0$ (20c)
$\displaystyle\dot{\xi}(0)=-\cos(\theta_{0}),\quad\dot{\zeta}(0)=\sin(\theta_{0})$
(20d) $\displaystyle\xi\rightarrow-\infty,\quad\zeta\rightarrow
1\qquad\text{as}\qquad s\rightarrow\infty$ (20e)

The solution of this problem is therefore the same as that for zero height the
wedge’s edge, but using now as eigenvalue the effective Bond number, given by:

$\displaystyle\left(\frac{h_{\infty}-h_{o}}{\ell_{c}}\right)^{2}=Bo_{eff}=2(1-\cos\theta_{0})$
(21)

In particular, the bifurcation diagram remains unchanged.

###  C.1 Negative heights

This configuration admits as well negative values of the relative height
$\Delta h_{\infty}$ as shown in Figure 3. The solution in these cases can be
obtained proceeding similarly. Using as length scale the absolute value
$|\Delta h_{\infty}|$ of the relative height leads to

$\displaystyle\ddot{\xi}=-Bo_{eff}\,\dot{\zeta}\,(-1-\zeta)$ (22a)
$\displaystyle\ddot{\zeta}=Bo_{eff}\,\dot{\xi}\,(-1-\zeta)$ (22b) subject to
$\displaystyle\xi(s=0)=\zeta(s=0)=0$ (22c)
$\displaystyle\dot{\xi}(0)=-\cos(\theta_{0}),\quad\dot{\zeta}(0)=\sin(\theta_{0})$
(22d)
$\displaystyle\xi\rightarrow-\infty,\quad\zeta\rightarrow-1\qquad\text{as}\qquad
s\rightarrow\infty$ (22e)

This problem can be brought back to the same form as (20) with the change of
variables $\zeta\rightarrow-\zeta$, $\theta_{0}\rightarrow-\theta_{0}$, which
represent the reflection about the $\xi$ axis of the solutions
$[\xi(s),\zeta(s)]$ corresponding to positive values of $\Delta h_{\infty}$,
i.e. a reflection about the horizontal line through the vertex of the wedge as
anticipated. $Bo_{eff}$ has been left unchanged, and therefore it must be an
even function of $\theta_{0}$, which means the bifurcation diagram is even in
$\theta_{0}$ as represented in Figure 4.

This derivation could have been based on the appropriate energy functional in
the case of negative heights, namely

$\displaystyle\mathcal{E}=\int_{0}^{\infty}{\left(\sqrt{\dot{\xi}^{2}+\dot{\zeta}^{2}}\right.\left.-Bo(\zeta-\zeta^{2}/2)\dot{\xi}-1+(3/2)Bo\right)ds}$
(23)

which accounts for the opposite sign (convexity) of the interface curvature in
these cases of negative height.

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