Young and Young–Laplace equations for a static ridge of nematic liquid
crystal, and transitions between equilibrium states

J. R. L<EMAIL_ADDRESS>/
<EMAIL_ADDRESS>B. R<EMAIL_ADDRESS>S. K.
<EMAIL_ADDRESS>N. J.
<EMAIL_ADDRESS>
1Department of Mathematics and Statistics, University of Strathclyde,
Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, United Kingdom
2School of Mathematics and Statistics, University of Glasgow, University
Place, Glasgow G12 8QQ, United Kingdom
(Dated: 15th November 2021)

###### Abstract

Motivated by the need for greater understanding of systems that involve
interfaces between a nematic liquid crystal, a solid substrate, and a passive
gas that includes nematic–substrate–gas three-phase contact lines, we analyse
a two-dimensional static ridge of nematic resting on a solid substrate in an
atmosphere of passive gas. Specifically, we obtain the first complete
theoretical description for this system, including nematic Young and
Young–Laplace equations, and then, under the assumption that anchoring
breaking occurs in regions adjacent to the contact lines, we use the nematic
Young equations to determine the continuous and discontinuous transitions that
occur between the equilibrium states of complete wetting, partial wetting, and
complete dewetting. In particular, in addition to continuous transitions
analogous to those that occur in the classical case of an isotropic liquid, we
find a variety of discontinuous transitions, as well as contact-angle
hysteresis, and regions of parameter space in which there exist multiple
partial wetting states that do not occur in the classical case.

##  1 Introduction

For the past 50 years or so, technological interest in liquid crystals has
largely been focused on the visual display market, where Liquid Crystal
Displays (LCDs) are still the dominant technology [1]. In recent years,
however, the push to exploit the optical, dielectric, and viscoelastic
anisotropies of liquid crystals has led to the development of devices used in
medicine, flow processing, microelectronic production, and adaptive-lens
technologies [2, 3, 4, 5, 6]. These devices often involve liquid crystal
droplets or films, which are complicated multiphase systems that involve
interfaces between the liquid crystal, a solid substrate, and a passive gas,
and often include liquid crystal–substrate–gas three-phase contact lines.
Theoretical studies of liquid crystal droplets or films often use theories of
wetting and dewetting for isotropic droplets and films which do not account
for the full anisotropic nature of liquid crystals [7, 8, 9, 10, 11, 12, 13,
14].

###  1.1 Wetting and dewetting phenomena

Simply stated, _wetting_ and _dewetting_ are the phenomena in which a liquid
advances and retreats, respectively, over a substrate [15]. When a finite
volume of liquid advances or retreats over a flat horizontal substrate, it
will eventually reach an equilibrium state. This equilibrium state is known
as: the _complete wetting_ state (sometimes also called the perfectly wetting
state), which we denote by $\displaystyle\mathbb{W}$, when the liquid
completely coats the substrate; the _complete dewetting_ state, which we
denote by $\displaystyle\mathbb{D}$, when the substrate completely repels the
liquid; and the _partial wetting_ state, which we denote by
$\displaystyle\mathbb{P}$, when the liquid partially coats the substrate.
Transitions between these equilibrium states can occur as a result of changes
in the liquid or substrate material properties (due to, for example, changes
in temperature) that cause the liquid to advance or retreat over the substrate
and/or change its contact angle. The classification of the equilibrium states
and the transitions between them is well known for an isotropic liquid [15].

Wetting and dewetting phenomena have been of scientific interest for
centuries, and are now of increasing technological importance [16]. For
systems in which creating a uniform liquid film (i.e. complete wetting) is
required, wetting is essential and dewetting is undesirable [15]. However, in
other situations, dewetting can be desirable, and can be initiated in a
variety of ways, such as amplification of thermal fluctuations on the liquid
free surface, nucleation at impurities, chemical treatment of the substrate,
and non-uniform evaporation [17]. In recent years there has been considerable
research in the area of tailored dewetting of liquid films to produce
patterned films [2, 18, 6]. The thermal, mechanical, and chemical stability of
liquid films is therefore an area of considerable research effort, and
understanding and controlling the onset of dewetting is crucial for creating
and maintaining both uniform and patterned films.

###  1.2 Wetting and dewetting phenomena for liquid crystals

For liquid crystals, which are anisotropic liquids that typically consist of
either rod-like or disc-like molecules that tend to align locally to minimise
molecular interaction energies, wetting and dewetting phenomena can be more
complicated than they are for isotropic liquids. The local orientational order
of liquid crystal molecules allows for a mathematical description of the
average molecular orientation of the liquid crystal in terms of a unit vector
called the director $\displaystyle\bm{n}$ [19]. As well as an orientational
order, many liquid crystal phases also possess positional order; for example,
smectic liquid crystals (smectics) self-organise into two-dimensional layers,
and this positional ordering may affect the wetting and dewetting behaviour
[20]. However, in the present work we consider only nematic liquid crystals
(nematics), which possess orientational but not positional ordering.

A variety of effects, including spinodal dewetting and nucleation at
impurities [9, 12, 21], can cause the dewetting of nematic films. In
particular, such dewetting can involve competition between many effects,
including internal elastic forces, alignment forces on the interfaces,
gravity, van der Waals forces, and in cases in which an external
electromagnetic field is applied, electromagnetic forces [22]. Many
experimental studies have considered delicate balances between a number of
these effects in different situations, for instance close to the
isotropic–nematic phase transition [23, 13, 24], near to a contact line [25,
26, 27], or in the presence of an external electromagnetic field [28, 29, 30].
Since in the present work we consider length-scales greater than a nanometre-
scale, it is appropriate to neglect van der Waals forces [15] and we consider
only the competition between elastic forces, alignment forces on the
interfaces, and gravity.

###  1.3 Liquid crystal anchoring

As mentioned above, the alignment forces on the interfaces between the gas and
the nematic (the gas–nematic interface) and the nematic and the substrate (the
nematic–substrate interface) can play an important role in wetting and
dewetting behaviour [31]. The dependence of these interactions on the
orientational anisotropy typically results in energetic preferences for the
director to align either normally or tangentially to the interfaces, which
leads to interfacial energies that are anisotropic; this is known as _weak
anchoring_. An energetic preference for the director to align normally to an
interface is known as _weak homeotropic anchoring_ , and an energetic
preference for the director to align tangentially to an interface is known as
_weak planar anchoring_. The strength of the energetic preference for a
homeotropic or planar alignment of the director on an interface is measured by
a parameter called the anchoring strength. Infinite anchoring strength
represents a situation where the director on an interface is fixed at the
preferred alignment. This situation is known as _infinite anchoring_
(sometimes also called strong anchoring). Zero anchoring strength corresponds
to a situation where the director on an interface has no preferred alignment.
This situation is known as _zero anchoring_.

Perhaps the most important effect of weak anchoring in a nematic film occurs
when there is weak homeotropic anchoring on the gas–nematic interface and weak
planar anchoring on the nematic–substrate interface, or vice versa. In this
situation, which is known as _antagonistic anchoring_ , competition between
the different preferred alignments on the interfaces can introduce elastic
distortion in the bulk of the nematic leading to a spatially–varying director
field [32, 26, 33], with an associated non-zero elastic energy, which can have
a destabilising effect on the film [10, 14]. For situations with antagonistic
anchoring, it has long been known that there exists a critical film thickness,
which we term the Jenkins–Barratt–Barbero–Barberi critical thickness [34, 35]
(often just called the Barbero–Barberi critical thickness), below which the
energetically favourable state has a uniform director field in which the
director aligns parallel to the preferred director alignment of the interface
with the stronger anchoring. For film thicknesses above this critical
thickness, the energetically favourable state has a director field that varies
continuously across the film; this state is known as a hybrid aligned nematic
(HAN) state [1].

The theoretical study of nematic systems that include contact lines often
avoids the consideration of antagonistic anchoring at the contact lines, by,
for example, imposing infinite anchoring on the nematic–substrate interface
which overrides the weak anchoring on the gas–nematic interface at the contact
line (see, for example, Lam et al. [9, 8]), or assuming the existence of a
thin precursor film on the substrate to remove the contact line entirely (see,
for example, Lin et al. [11]). While there have been relatively few studies of
nematic contact lines, Rey [36, 37] considered two rather specific two-
dimensional scenarios, namely either infinite planar anchoring or equal weak
planar anchoring, on both interfaces. Although neither infinite anchoring nor
equal weak anchoring is likely to occur in practice, these studies highlight
the possibility that anchoring breaking, i.e. the process by which the
preferred orientation of the nematic molecules on one of the interfaces is
overridden by that on the other, occurs in a region adjacent to the contact
line. Rey [36, 37] also discusses the possibility of the formation of a
defect, or a disclination line in his two-dimensional scenarios, located at
the contact line. At such disclination lines, a description of the nematic
only in terms of the director is no longer valid and there is a high degree of
elastic distortion associated with increased elastic energy [38]. In the
present work we will assume that the energy associated with anchoring breaking
in a region adjacent to the contact line is lower than the energy associated
with the formation of a disclination line [39] and, therefore, that such
disclination lines do not occur.

###  1.4 A static ridge of liquid crystal

Motivated by a need for increased understanding of situations involving the
wetting and dewetting of nematics, in the present work we consider a two-
dimensional static ridge of nematic resting on an ideal (i.e. flat, rigid,
perfectly smooth, and chemically homogeneous) solid substrate surrounded by a
passive fluid. In order to make comparisons with the most commonly studied
experimental situation, we consider the case in which the passive fluid
surrounding the nematic is an atmosphere of passive gas, although the
subsequent theory and results may be readily generalised to a ridge of nematic
surrounded by a static isotropic liquid. There are many applications of liquid
crystals that may benefit from an increased understanding of this situation.
For instance, the patterning of discotic liquid crystals (discotics) into
precise and controllable ridges has been demonstrated [18, 6], and this
technology, together with the excellent charge-transport properties of
discotics, has led to them being used as printable nanometre-scale wires for
applications in electronics [40]. The controlled formation of static ridges of
liquid crystal also has applications in optics, particularly for creating
self-organised diffraction gratings [41, 42].

The nematic ridge is bounded by a gas–nematic interface and a
nematic–substrate interface. The theoretical description of a nematic bounded
by such interfaces has previously been considered by Jenkins and Barratt [34],
who obtained general forms of the interfacial conditions and the force per
unit length on a contact line, and Rey [43, 44], who obtained a general form
of the nematic Young and Young–Laplace equations. In the present work, we
combine aspects of these two approaches to derive the first complete
theoretical description for a static ridge of nematic, which includes the bulk
elastic equation, the nematic Young equations, the nematic Young–Laplace
equation, the weak anchoring conditions, and the other relevant boundary
conditions. We provide full details of a readily accessible derivation of the
governing equations in Sections 2, 3 and 4, which may, in principle, be
generalised to include electromagnetic forces, additional contact-line
effects, non-ideal substrates, or more detailed models for the nematic
molecular order.

We proceed by constructing the free energy of the system as a function of both
the shape of the gas–nematic interface (i.e. the nematic free surface) and the
director field, and then minimise the free energy using the calculus of
variations. In order to determine the free energy of the system, we use a
well-established continuum theory to consider contributions from elastic
deformations of the director $\displaystyle\bm{n}$, the gravitational
potential energy, and interface energies associated with the three interfaces
(for a full account of this continuum theory of nematics, see [19]). We use
the standard Oseen–Frank bulk elastic energy density $\displaystyle W$ (energy
per unit volume) which depends on $\displaystyle\bm{n}$ and its spatial
gradients [19]. The interface energies associated with the gas–nematic and
nematic–substrate interfaces will be described using the standard
Rapini–Papoular interface energy density (energy per unit area)
$\displaystyle\omega$ which depends on $\displaystyle\bm{n}$ and the interface
normal $\displaystyle\bm{\nu}$ [45].

Although we proceed in Sections 2, 3 and 4 by deriving the governing equations
of the most commonly occurring experimental situation of the partial wetting
state, $\displaystyle\mathbb{P}$, the same governing equations also describe
the complete wetting state, $\displaystyle\mathbb{W}$, and the complete
dewetting state, $\displaystyle\mathbb{D}$. In the $\displaystyle\mathbb{W}$
state, in which the nematic forms a film that completely coats the substrate,
there is no gas–substrate interface and hence no contact lines. In the
$\displaystyle\mathbb{D}$ state, in which the gas–nematic interface forms a
cylinder (which, because of anisotropic effects, is not necessarily circular),
there is no nematic–substrate interface, and the gas–nematic interface meets
the gas–substrate interface at a single contact line. For an isotropic ridge,
described briefly in Section 5, the classification of the equilibrium states
and the transitions that occur between them are well known and can be obtained
by solving the classical isotropic Young–Laplace equation and comparing the
free energies of the possible equilibrium states [46, 15]. For a nematic
ridge, the free energy of the equilibrium states cannot be determined
analytically; however, by comparison with the classical results for the
isotropic ridge, the classification of the equilibrium states and the
transitions between them can still be obtained. In particular, in Sections 6
and 7, we use the nematic Young equations obtained in Section 4 to determine
the continuous and discontinuous transitions between the equilibrium states of
complete wetting, partial wetting, and complete dewetting. Previously, Rey
[44] found that a general form of the general nematic Young equations allow
for discontinuous transitions between partial wetting and complete wetting and
between partial wetting and complete dewetting. However, without the
assumption made in the present work that anchoring breaking occurs in regions
adjacent to the contact lines, an explicit description of these transitions
was not possible. Under this assumption, in Sections 6 and 7 we find not only
continuous transitions analogous to those that occur in the classical case of
an isotropic liquid, but also a variety of discontinuous transitions, as well
as contact-angle hysteresis, and regions of parameter space in which there
exist multiple partial wetting states that do not occur in the classical case.

##  2 Model formulation

Figure 1: A schematic of a static ridge of nematic (N) resting on an ideal
solid substrate (S) at $\displaystyle z=0$, $\displaystyle L^{-}\leq x\leq
L^{+}$, in an atmosphere of passive gas (G), with gas–nematic interface at
$\displaystyle z=h$ and contact lines at $\displaystyle x=d^{\pm}$. The
Cartesian coordinates $\displaystyle x$, $\displaystyle y$ and $\displaystyle
z$ (where the $\displaystyle y$-direction is into the page), the region of
nematic in the $\displaystyle(x,z)$-plane $\displaystyle\Omega$ bounded by the
interface $\displaystyle\Gamma$, the outward unit normals
$\displaystyle\bm{\nu}$ and the contact angles $\displaystyle\beta^{\pm}$ are
also indicated.

As described in the previous section, we consider a static ridge of nematic
(N) resting on an ideal solid substrate (S) in an atmosphere of passive gas
(G), as shown in Figure 1, which also indicates the Cartesian coordinates
$\displaystyle x$, $\displaystyle y$ and $\displaystyle z$ that we use. The
region of nematic in the $\displaystyle(x,z)$-plane $\displaystyle\Omega$ is
bounded by the interface $\displaystyle\Gamma$, which consists of the
gas–nematic interface at $\displaystyle z=h(x)$, denoted by
$\displaystyle\Gamma_{\rm GN}$, and the nematic–substrate interface at
$\displaystyle z=0$, denoted by $\displaystyle\Gamma_{\rm NS}$, and has two
nematic–substrate–gas three-phase contact lines at $\displaystyle x=d^{-}$ and
$\displaystyle x=d^{+}$. We assume that the ridge height $\displaystyle h$ and
the position of the contact lines do not vary in the $\displaystyle
y$-direction, so that the contact lines form two infinitely-long parallel
lines in the $\displaystyle y$-direction and the ridge height $\displaystyle
h$ is subject to the boundary conditions $\displaystyle h(d^{\pm})=0$. We also
assume that the director $\displaystyle\bm{n}$ is confined to the
$\displaystyle(x,z)$-plane, and hence takes the form

$\displaystyle\displaystyle\bm{n}=\cos\theta\,\bm{\hat{x}}+\sin\theta\,\bm{\hat{z}},$
(1)

where $\displaystyle\bm{\hat{x}}$ and $\displaystyle\bm{\hat{z}}$ are the
Cartesian coordinate unit vectors in the $\displaystyle x$\- and
$\displaystyle z$-directions, respectively, and
$\displaystyle\theta=\theta(x,z)$ is the director angle, which also does not
vary in the $\displaystyle y$-direction.

The outward unit normals of the interfaces $\displaystyle\Gamma_{\rm GN}$ and
$\displaystyle\Gamma_{\rm NS}$, which we denote by $\displaystyle\bm{\nu}_{\rm
GN}$ and $\displaystyle\bm{\nu}_{\rm NS}$, are given by

$\displaystyle\displaystyle\bm{\nu}_{\rm GN}$
$\displaystyle\displaystyle=-\dfrac{{h}_{x}}{\sqrt{1+{h}_{x}^{2}}}\bm{\hat{x}}+\dfrac{1}{\sqrt{1+{h}_{x}^{2}}}\bm{\hat{z}},$
(2) $\displaystyle\displaystyle\bm{\nu}_{\rm NS}$
$\displaystyle\displaystyle=-\bm{\hat{z}},$ (3)

respectively, where the subscript $\displaystyle x$ denotes differentiation
with respect to $\displaystyle x$. These two interfaces meet the gas–substrate
interface, denoted by $\displaystyle\Gamma_{\rm GS}$, at the two contact lines
$\displaystyle x=d^{\pm}$. The left-hand and right-hand edges of the substrate
are at $\displaystyle x=L^{-}(<d^{-})$ and $\displaystyle x=L^{+}(>d^{+})$,
respectively, as shown in Figure 1. The contact angles formed between
$\displaystyle\Gamma_{\rm GN}$ and $\displaystyle\Gamma_{\rm NS}$ at
$\displaystyle x=d^{-}$ and $\displaystyle x=d^{+}$ are denoted by
$\displaystyle\beta^{-}$ and $\displaystyle\beta^{+}$, respectively, and
satisfy

$\displaystyle\displaystyle\tan\beta^{\pm}$ $\displaystyle\displaystyle=\mp
h_{x}\quad\text{at}\quad x=d^{\pm}.$ (4)

We note that there is, in general, no requirement for $\displaystyle h$ to be
symmetric about its midpoint and, in particular, no requirement for the
contact angles to be the same.

In general, we do not fix either the contact line positions or the contact
angles, and allow $\displaystyle d^{\pm}$ and $\displaystyle\beta^{\pm}$ to be
unknowns. However, if the substrate has been treated in such a way as either
to pin the contact lines or to fix the contact angles, then either
$\displaystyle d^{\pm}$ or $\displaystyle\beta^{\pm}$, respectively, are
prescribed and the nematic Young equations, which will be derived shortly, are
not relevant. The ridge has a prescribed constant cross-sectional area
$\displaystyle A$ in the $\displaystyle(x,z)$-plane, so that

$\displaystyle\displaystyle\iint\,\differential\Omega$
$\displaystyle\displaystyle=A.$ (5)

As mentioned in Section 1, we include the effects of gravity. Specifically, we
assume the gravity acts in the $\displaystyle(x,z)$-plane but, in order to
keep the setup as general as possible, do not specify its direction.

In Section 3 we obtain the complete theoretical description for this system
using the calculus of variations assuming that the ridge height $\displaystyle
h$ is a single-valued function of $\displaystyle x$. A necessary, but not
sufficient, condition for this to be valid is that the contact angles are
acute (i.e. that $\displaystyle 0\leq\beta^{\pm}\leq\pi/2$). We have performed
the corresponding derivation when the ridge height $\displaystyle h$ is a
double-valued function of $\displaystyle x$. However, since this derivation
involves either splitting the gas–nematic interface into three parts in each
of which $\displaystyle h$ is a single-valued function of $\displaystyle x$ or
using a different coordinate system, for simplicity of presentation, and
because many of the situations described in Section 1 involve small contact
angles, in the present work we describe the details of the derivation only
when $\displaystyle h$ is a single-valued function of $\displaystyle x$. The
details of the corresponding derivation when $\displaystyle h$ is a double-
valued function of $\displaystyle x$ are given by Cousins [47].

##  3 Constrained minimisation of the free energy

Using the calculus of variations we minimise the free energy of the system
$\displaystyle E$ (per unit length in the $\displaystyle y$-direction) subject
to the area constraint 5 and the boundary conditions $\displaystyle
h(d^{\pm})=0$ to obtain the governing equations for a ridge of nematic in
terms of the four unknowns $\displaystyle\theta(x,z)$, $\displaystyle h(x)$,
$\displaystyle d^{-}$ and $\displaystyle d^{+}$, and a Lagrange multiplier
associated with the area constraint 5, the latter of which we denote by
$\displaystyle p_{0}$. The unknown contact angles $\displaystyle\beta^{\pm}$
are obtained from the slope of the ridge height $\displaystyle h_{x}$ using 4.
The free energy of the system $\displaystyle E$ is the sum of the bulk elastic
energy of the nematic, denoted by $\displaystyle E_{\rm bulk}$, and the
interface energies, denoted by $\displaystyle E_{\rm GN}$, $\displaystyle
E_{\rm NS}$ and $\displaystyle E_{\rm GS}$, for the interfaces
$\displaystyle\Gamma_{\rm GN}$, $\displaystyle\Gamma_{\rm NS}$ and
$\displaystyle\Gamma_{\rm GS}$, respectively, where

$\displaystyle\displaystyle E_{\rm bulk}$
$\displaystyle\displaystyle=\int^{d^{+}}_{d^{-}}\int^{h}_{0}\Big{(}W(\theta,\theta_{x},\theta_{z})+\psi_{\rm
g}\Big{)}\,\,\differential z\,\differential x,$ (6)
$\displaystyle\displaystyle E_{\rm GN}$
$\displaystyle\displaystyle=\int^{d^{+}}_{d^{-}}\sqrt{1+h_{x}^{2}}\,\big{[}\omega_{\rm
GN}(\theta,h_{x})\big{]}^{z=h}\,\differential x,$ (7)
$\displaystyle\displaystyle E_{\rm NS}$
$\displaystyle\displaystyle=\int^{d^{+}}_{d^{-}}\big{[}\omega_{\rm
NS}(\theta)\big{]}^{z=0}\,\,\differential x,$ (8) $\displaystyle\displaystyle
E_{\rm GS}$ $\displaystyle\displaystyle=\int^{d^{-}}_{L^{-}}\big{[}\omega_{\rm
GS}\big{]}^{z=0}\,\,\differential x+\int^{L^{+}}_{d^{+}}\big{[}\omega_{\rm
GS}\big{]}^{z=0}\,\,\differential x.$ (9)

In 6 the bulk elastic energy density $\displaystyle
W(\theta,\theta_{x},\theta_{z})$ is assumed to depend on the director angle
$\displaystyle\theta$ and on elastic distortions of the director via the
spatial derivatives of $\displaystyle\theta$ [19]. Also in 6, the
gravitational potential energy density $\displaystyle\psi_{\rm g}(x,z)$ is
allowed to depend on one or both of the Cartesian coordinates $\displaystyle
x$ and $\displaystyle z$. In 7 and 8 the interface energy densities
$\displaystyle\omega_{\rm GN}(\theta,h_{x})$ and $\displaystyle\omega_{\rm
NS}(\theta)$ are assumed to be in the form of the Rapini–Papoular energy
density [45], which depends on the angle between the director 1 and the
outward unit normal of the interfaces, namely 2 and 3, respectively. In 9 the
interface energy density $\displaystyle\omega_{\rm GS}$ takes a constant
value.

We define the functional $\displaystyle
F=F(\theta,\theta_{x},\theta_{z},h,h_{x},d^{-},d^{+})=E+\mathcal{C}_{\rm
area}$ as the sum of the free energy of the system $\displaystyle E$ and a
term $\displaystyle\mathcal{C}_{\rm area}$, corresponding to the area
constraint 5, given by

$\displaystyle\displaystyle\mathcal{C}_{\rm area}$
$\displaystyle\displaystyle=p_{0}\times\left(A-\int^{d^{+}}_{d^{-}}\int^{h}_{0}\,\differential
z\,\differential x\right),$ (10)

so that the functional $\displaystyle F$ is given by

$\displaystyle\displaystyle F=E_{\rm bulk}+E_{\rm GN}+E_{\rm NS}+E_{\rm
GS}+\mathcal{C}_{\rm area}.$ (11)

We now consider the variation of $\displaystyle F$, given by 11 with 6, 7, 8,
9 and 10, with respect to small variations of the variables
$\displaystyle\theta$, $\displaystyle h$, $\displaystyle d^{-}$ and
$\displaystyle d^{+}$ of the form

$\displaystyle\displaystyle\theta\to\theta+\delta_{\theta},\quad h\to
h+\delta_{h},\quad d^{-}\to d^{-}+\delta_{d^{-}},\quad d^{+}\to
d^{+}+\delta_{d^{+}}.$ (12)

There are no constraints on the director angle $\displaystyle\theta$, and
therefore there are no constraints on its variation
$\displaystyle\delta_{\theta}$. There is, however, a constraint on the ridge
height $\displaystyle h$ because of the boundary conditions $\displaystyle
h(d^{\pm})=0$, so that the variation of the ridge height
$\displaystyle\delta_{h}$ at the contact lines satisfies

$\displaystyle\displaystyle\delta_{h}$
$\displaystyle\displaystyle=-{h}_{x}\delta_{d^{-}}=-\tan\beta^{-}\,\delta_{d^{-}}\quad\text{at}\quad
x=d^{-},$ (13) $\displaystyle\displaystyle\delta_{h}$
$\displaystyle\displaystyle=-{h}_{x}\delta_{d^{+}}=+\tan\beta^{+}\,\delta_{d^{+}}\quad\text{at}\quad
x=d^{+}.$ (14)

The variation of the functional $\displaystyle F$, denoted by
$\displaystyle\delta F$, is given by

$\displaystyle\displaystyle\delta F$
$\displaystyle\displaystyle=F\left(\theta+\delta_{\theta},(\theta+\delta_{\theta})_{x},(\theta+\delta_{\theta})_{z},h+\delta_{h},(h+\delta_{h})_{x},d^{-}+\delta_{d^{-}},d^{+}+\delta_{d^{+}})\right.$
$\displaystyle\displaystyle\qquad\qquad\left.-F(\theta,\theta_{x},\theta_{z},h,h_{x},d^{-},d^{+}\right).$
(15)

We now consider the variation of each term in 11 in turn, and neglect terms in
15 that are quadratic in the variations $\displaystyle\delta_{\theta}$,
$\displaystyle\delta_{h}$, $\displaystyle\delta_{d^{-}}$ and
$\displaystyle\delta_{d^{+}}$.

For the bulk elastic energy $\displaystyle E_{\rm bulk}$, given by 6,
$\displaystyle\delta E_{\rm bulk}$ is given by

$\displaystyle\displaystyle\delta E_{\rm bulk}$
$\displaystyle\displaystyle=\int^{d^{+}}_{d^{-}}\int^{h}_{0}\delta_{\theta}\partialderivative{W}{\theta}+{\delta_{\theta}}_{x}\partialderivative{W}{\theta_{x}}+{\delta_{\theta}}_{z}\partialderivative{W}{\theta_{z}}\,\,\differential
z\,\differential x+\int^{d^{+}}_{d^{-}}\delta_{h}\left[W+\psi_{\rm
g}\right]^{z=h}\,\,\differential x$
$\displaystyle\displaystyle\quad\quad-\delta_{d^{-}}\Bigg{[}\int^{h}_{0}(W+\psi_{\rm
g})\,\,\differential
z\Bigg{]}^{x=d^{-}}+\delta_{d^{+}}\Bigg{[}\int^{h}_{0}(W+\psi_{\rm
g})\,\,\differential z\Bigg{]}^{x=d^{+}}.$ (16)

Since $\displaystyle h(d^{\pm})=0$, the last two terms in 16 are identically
zero. The terms in 16 containing derivatives of
$\displaystyle\delta_{\theta}$, namely $\displaystyle{\delta_{\theta}}_{x}$
and $\displaystyle{\delta_{\theta}}_{z}$, are transformed into terms involving
$\displaystyle\delta_{\theta}$ by using the divergence theorem, namely

$\displaystyle\displaystyle\int\int{\delta_{\theta}}_{\alpha}\partialderivative{W}{\theta_{\alpha}}\,\differential\Omega$
$\displaystyle\displaystyle=\oint_{\Gamma}\delta_{\theta}\partialderivative{W}{\theta_{\alpha}}\bm{\hat{\alpha}}\cdot\bm{\nu}\,\differential\Gamma-\int\int\delta_{\theta}\partialderivative{\alpha}\left(\partialderivative{W}{\theta_{\alpha}}\right)\,\differential\Omega,$
(17)

where $\displaystyle\alpha=x$ or $\displaystyle\alpha=z$. The line integral
along $\displaystyle\Gamma$ in 17 is composed of a component along
$\displaystyle\Gamma_{\rm GN}$ from $\displaystyle x=d^{+}$ to $\displaystyle
x=d^{-}$ on $\displaystyle z=h$ with
$\displaystyle\differential\Gamma=-\sqrt{1+{h}_{x}^{2}}\,\differential x$ and
outward unit normal 3, and a component along $\displaystyle\Gamma_{\rm NS}$ at
$\displaystyle z=0$ from $\displaystyle x=d^{-}$ to $\displaystyle x=d^{+}$
with $\displaystyle\differential\Gamma=\differential x$ and outward unit
normal 2, and is given explicitly by

$\displaystyle\displaystyle\oint_{\Gamma}\delta_{\theta}\partialderivative{W}{\theta_{\alpha}}\bm{\hat{\alpha}}\cdot\bm{\nu}\,\differential\Gamma$
$\displaystyle\displaystyle=\int^{d^{-}}_{d^{+}}\Bigg{[}\delta_{\theta}\partialderivative{W}{\theta_{\alpha}}\Bigg{]}^{z=h}\bm{\hat{\alpha}}\cdot(h_{x}\bm{\hat{z}}-\bm{\hat{z}})\,\,\differential
x-\int^{d^{+}}_{d^{-}}\Bigg{[}\delta_{\theta}\partialderivative{W}{\theta_{\alpha}}\Bigg{]}^{z=0}\bm{\hat{\alpha}}\cdot\bm{\hat{z}}\,\,\differential
x.$ (18)

Equations 16–18 can be combined and rearranged to express the variation
$\displaystyle\delta E_{\rm bulk}$ as

$\displaystyle\displaystyle\delta E_{\rm bulk}$
$\displaystyle\displaystyle=\int^{d^{+}}_{d^{-}}\int^{h}_{0}\delta_{\theta}\left(\partialderivative{W}{\theta}-\partialderivative{x}\left(\partialderivative{W}{\theta_{x}}\right)-\partialderivative{z}\left(\partialderivative{W}{\theta_{z}}\right)\right)\,\differential
z\,\differential x$
$\displaystyle\displaystyle\quad\quad+\int^{d^{+}}_{d^{-}}\delta_{h}\left[W+\psi_{\rm
g}\right]^{z=h}\,\,\differential
x-\int_{d^{-}}^{d^{+}}\left[\delta_{\theta}h_{x}\partialderivative{W}{\theta_{x}}\right]^{z=h}\,\,\differential
x$ (19)
$\displaystyle\displaystyle\quad\quad+\int_{d^{-}}^{d^{+}}\left[\delta_{\theta}\partialderivative{W}{\theta_{z}}\right]^{z=h}\,\,\differential
x-\int_{d^{-}}^{d^{+}}\left[\delta_{\theta}\partialderivative{W}{\theta_{z}}\right]^{z=0}\differential
x.$

For the gas–nematic interface energy $\displaystyle E_{\rm GN}$, given by 7,
carrying out integration by parts on the terms involving
$\displaystyle{\delta_{h}}_{x}$ shows that $\displaystyle\delta E_{\rm GN}$ is
given by

$\displaystyle\displaystyle\delta E_{\rm GN}$
$\displaystyle\displaystyle=\int^{d^{+}}_{d^{-}}\left[\delta_{\theta}\sqrt{1+{h}_{x}^{2}}\partialderivative{\omega_{\rm
GN}}{\theta}+\delta_{h}\left(\sqrt{1+{h}_{x}^{2}}\partialderivative{\omega_{\rm
GN}}{\theta}\partialderivative{\theta}{z}-\partialderivative{x}\Big{[}\partialderivative{{h}_{x}}\big{(}\sqrt{1+{h}_{x}^{2}}\,\,\omega_{\rm
GN}\big{)}\Big{]}\right)\right]^{z=h}\,\differential x$
$\displaystyle\displaystyle\quad\quad-\delta_{d^{-}}\left[\sqrt{1+{h}_{x}^{2}}\,\omega_{\rm
GN}\right]^{x=d^{-}}+\delta_{d^{+}}\left[\sqrt{1+{h}_{x}^{2}}\,\omega_{\rm
GN}\right]^{x=d^{+}}$ (20)
$\displaystyle\displaystyle\quad\quad-\left[\delta_{h}\partialderivative{{h}_{x}}\left(\sqrt{1+{h}_{x}^{2}}\,\,\omega_{\rm
GN}\right)\right]^{x=d^{-}}+\left[\delta_{h}\partialderivative{{h}_{x}}\left(\sqrt{1+{h}_{x}^{2}}\,\,\omega_{\rm
GN}\right)\right]^{x=d^{+}}.$

Substituting for the variation of the ridge height $\displaystyle\delta_{h}$
at the contact lines, given by 13 and 14, then yields

$\displaystyle\displaystyle\delta E_{\rm GN}$
$\displaystyle\displaystyle=\int^{d^{+}}_{d^{-}}\Bigg{[}\delta_{\theta}\sqrt{1+{h}_{x}^{2}}\partialderivative{\omega_{\rm
GN}}{\theta}+\delta_{h}\left(\sqrt{1+{h}_{x}^{2}}\partialderivative{\omega_{\rm
GN}}{\theta}\partialderivative{\theta}{z}-\partialderivative{x}\Big{[}\partialderivative{{h}_{x}}\big{(}\sqrt{1+{h}_{x}^{2}}\,\,\omega_{\rm
GN}\big{)}\Big{]}\right)\Bigg{]}^{z=h}\,\differential x$
$\displaystyle\displaystyle\quad\quad-\delta_{d^{-}}\left[\sqrt{1+{h}_{x}^{2}}\,\omega_{\rm
GN}-{h}_{x}\partialderivative{{h}_{x}}\left(\sqrt{1+{h}_{x}^{2}}\,\omega_{\rm
GN}\right)\right]^{x=d^{-}}$ (21)
$\displaystyle\displaystyle\quad\quad+\delta_{d^{+}}\left[\sqrt{1+{h}_{x}^{2}}\,\omega_{\rm
GN}-{h}_{x}\partialderivative{{h}_{x}}\left(\sqrt{1+{h}_{x}^{2}}\,\omega_{\rm
GN}\right)\right]^{x=d^{+}}.$

For the nematic–substrate interface energy $\displaystyle E_{\rm NS}$, given
by 8, $\displaystyle\delta E_{\rm NS}$ is given by

$\displaystyle\displaystyle\delta E_{\rm NS}$
$\displaystyle\displaystyle=\int^{d^{+}}_{d^{-}}\left[\delta_{\theta}\partialderivative{\omega_{\rm
NS}}{\theta}\right]^{z=0}\,\,\differential x-\delta_{d^{-}}\left[\omega_{\rm
NS}\right]^{x=d^{-}}+\delta_{d^{+}}\left[\omega_{\rm NS}\right]^{x=d^{+}}.$
(22)

For the gas–substrate interface energy $\displaystyle E_{\rm GS}$, given by 9,
$\displaystyle\delta E_{\rm GS}$ is given by

$\displaystyle\displaystyle\delta E_{\rm GS}$
$\displaystyle\displaystyle=\delta_{d^{-}}\left[\omega_{\rm
GS}\right]^{x=d^{-}}-\delta_{d^{+}}\left[\omega_{\rm GS}\right]^{x=d^{+}}.$
(23)

Finally, for the area constraint term $\displaystyle\mathcal{C}_{\rm area}$,
given by 10, using the boundary conditions $\displaystyle h(d^{\pm})=0$ shows
that $\displaystyle\delta\mathcal{C}_{\rm area}$ is given by

$\displaystyle\displaystyle\delta\mathcal{C}_{\rm area}$
$\displaystyle\displaystyle=-\int^{d^{+}}_{d^{-}}\delta_{h}\,p_{0}\,\,\differential
x.$ (24)

The variation of $\displaystyle F$ is obtained by adding the terms from each
of the individual variations, given by 19 and 21–24, so that

$\displaystyle\displaystyle\delta F=\delta E_{\rm bulk}+\delta E_{\rm
GN}+\delta E_{\rm NS}+\delta E_{\rm GS}+\delta\mathcal{C}_{\rm area}.$ (25)

Since we seek extrema of the free energy $\displaystyle E$ for which
$\displaystyle\delta F=0$, and the variations $\displaystyle\delta_{\theta}$,
$\displaystyle\left[\delta_{\theta}\right]^{z=0}$,
$\displaystyle\left[\delta_{\theta}\right]^{z=h}$, $\displaystyle\delta_{h}$,
$\displaystyle\delta_{d^{-}}$ and $\displaystyle\delta_{d^{+}}$ are
independent and arbitrary, their coefficients in $\displaystyle\delta F$,
given by 25, must be zero. Together with the area constraint 5 and the
boundary conditions $\displaystyle h(d^{\pm})=0$, the coefficients of each
variation yield the governing equations for a nematic ridge, as described in
the next section.

##  4 Governing equations for a nematic ridge

Each of the six governing equations derived from setting the coefficients of
$\displaystyle\delta_{\theta}$,
$\displaystyle\left[\delta_{\theta}\right]^{z=0}$,
$\displaystyle\left[\delta_{\theta}\right]^{z=h}$, $\displaystyle\delta_{h}$,
$\displaystyle\delta_{d^{-}}$ and $\displaystyle\delta_{d^{+}}$ in 25 to zero
has a distinct physical interpretation, namely the balance of elastic torque
within the bulk of the nematic, the balance-of-couple conditions on the
gas–nematic and nematic–substrate interfaces, the balance-of-stress condition
on the gas–nematic interface, and the balance-of-stress conditions at the
contact lines, respectively. These equations are summarised below.

The balance of elastic torque within the bulk of the nematic, i.e. the
Euler–Lagrange equation, for the elastic free energy density $\displaystyle
W$, is

$\displaystyle\displaystyle\partialderivative{W}{\theta}-\partialderivative{x}\left(\partialderivative{W}{\theta_{x}}\right)-\partialderivative{z}\left(\partialderivative{W}{\theta_{z}}\right)$
$\displaystyle\displaystyle=0.$ (26)

The balance-of-couple conditions on the gas–nematic interface and the
nematic–substrate interface, namely the weak anchoring conditions [19, 48],
are given by

$\displaystyle\displaystyle\partialderivative{W}{\theta_{z}}-{h}_{x}\,\partialderivative{W}{\theta_{x}}+\sqrt{1+{h}_{x}^{2}}\,\partialderivative{\omega_{\rm
GN}}{\theta}$ $\displaystyle\displaystyle=0\quad\text{on}\quad z=h$ (27)

and

$\displaystyle\displaystyle-\partialderivative{W}{\theta_{z}}+\partialderivative{\omega_{\rm
NS}}{\theta}$ $\displaystyle\displaystyle=0\quad\text{on}\quad z=0,$ (28)

respectively.

The balance-of-stress condition on the gas–nematic interface is given by

$\displaystyle\displaystyle W+\psi_{\rm
g}-p_{0}+\sqrt{1+{h}_{x}^{2}}\,\partialderivative{\omega_{\rm
GN}}{\theta}\partialderivative{\theta}{z}-\partialderivative{x}\left(\partialderivative{{h}_{x}}\left(\sqrt{1+{h}_{x}^{2}}\,\,\omega_{\rm
GN}\right)\right)$ $\displaystyle\displaystyle=0\quad\text{on}\quad z=h.$ (29)

To distinguish equation 29 from the classical isotropic Young–Laplace equation
[15], henceforth it is referred to as the _nematic Young–Laplace equation_.

The balance-of-stress conditions at the contact lines are given by

$\displaystyle\displaystyle\omega_{\rm NS}-\omega_{\rm
GS}+\sqrt{1+{h}_{x}^{2}}\,\omega_{\rm
GN}-{h}_{x}\partialderivative{{h}_{x}}\left(\sqrt{1+{h}_{x}^{2}}\,\omega_{\rm
GN}\right)$ $\displaystyle\displaystyle=0\quad\text{at}\quad x=d^{\pm}.$ (30)

To distinguish equations 30 from the classical isotropic Young equations [15],
henceforth they are referred to as the _nematic Young equations_.

Once explicit forms of the energy densities $\displaystyle W$,
$\displaystyle\psi_{\rm g}$, $\displaystyle\omega_{\rm GN}$,
$\displaystyle\omega_{\rm NS}$ and $\displaystyle\omega_{\rm GS}$ have been
prescribed, the balance of elastic torque within the bulk of the nematic 26,
the three interface conditions 28, 27 and 29, the two nematic Young equations
30, the area constraint 5, and the two boundary conditions $\displaystyle
h(d^{\pm})=0$ provide the complete theoretical description for a static ridge
of nematic in terms of the five unknowns $\displaystyle\theta(x,z)$,
$\displaystyle h(x)$, $\displaystyle d^{-}$, $\displaystyle d^{+}$ and
$\displaystyle p_{0}$.

###  4.1 The bulk elastic energy density and the interface energy densities

As mentioned in Section 1, for the bulk elastic energy density $\displaystyle
W$ we use the standard Oseen–Frank bulk elastic energy density [19] for which

$\displaystyle\displaystyle W$
$\displaystyle\displaystyle=\dfrac{1}{2}K_{1}(\nabla\cdot\bm{n})^{2}+\dfrac{1}{2}K_{2}(\bm{n}\cdot\nabla\times\bm{n})^{2}+\dfrac{1}{2}K_{3}(\bm{n}\times\nabla\times\bm{n})^{2}$
$\displaystyle\displaystyle\quad\quad+\dfrac{1}{2}(K_{2}+K_{4})\nabla\cdot\big{[}(\bm{n}\cdot\nabla)\bm{n}-(\nabla\cdot\bm{n})\bm{n}\big{]},$
(31)

where $\displaystyle K_{1}$, $\displaystyle K_{2}$, $\displaystyle K_{3}$, and
the combination $\displaystyle K_{2}+K_{4}$ are called the splay, twist, bend,
and saddle-splay elastic constants, respectively, and
$\displaystyle\nabla=(\partial/\partial x,\partial/\partial
y,\partial/\partial z)$. Substituting 1 into 31 yields

$\displaystyle\displaystyle W(\theta,\theta_{x},\theta_{z})$
$\displaystyle\displaystyle=\dfrac{K_{1}}{2}\left(\theta_{z}\cos\theta-\theta_{x}\sin\theta\right)^{2}+\dfrac{K_{3}}{2}\left(\theta_{x}\cos\theta+\theta_{z}\sin\theta\right)^{2},$
(32)

which depends only on the splay and bend elastic deformations. Although we use
the full Oseen–Frank energy density 32, we note that the commonly used one-
constant approximation of the elastic constants [19] can be implemented in 32
by setting $\displaystyle K=K_{1}=K_{3}$, in which case $\displaystyle
W(\theta_{x},\theta_{z})=K(\theta_{x}^{2}+\theta_{z}^{2})/2$.

As also mentioned in Section 1, for $\displaystyle\omega_{\rm GN}$ and
$\displaystyle\omega_{\rm NS}$ we use the standard Rapini–Papoular form [45]
for which

$\displaystyle\displaystyle\omega_{\rm GN}$
$\displaystyle\displaystyle=\gamma_{\rm GN}+\dfrac{C_{\rm
GN}}{4}\left(1-2(\bm{\nu}_{\rm GN}\cdot\bm{n})^{2}\right),$ (33)
$\displaystyle\displaystyle\omega_{\rm NS}$
$\displaystyle\displaystyle=\gamma_{\rm NS}+\dfrac{C_{\rm
NS}}{4}\left(1-2(\bm{\nu}_{\rm NS}\cdot\bm{n})^{2}\right),$ (34)

where $\displaystyle C_{\rm GN}$ and $\displaystyle\gamma_{\rm GN}$ are the
anchoring strength and isotropic interfacial tension for the gas–nematic (GN)
interface, respectively, and $\displaystyle C_{\rm NS}$ and
$\displaystyle\gamma_{\rm NS}$ are the anchoring strength and isotropic
interfacial tension for the nematic–substrate (NS) interface, respectively.
The Rapini–Papoular form ensures that the interface energy densities
$\displaystyle\omega_{\rm GN}$ and $\displaystyle\omega_{\rm NS}$ are at a
minimum when $\displaystyle\bm{n}$ and $\displaystyle\bm{\nu}$ are parallel
for $\displaystyle C_{\rm GN}>0$ and $\displaystyle C_{\rm NS}>0$,
respectively, and at a minimum when $\displaystyle\bm{n}$ and
$\displaystyle\bm{\nu}$ are perpendicular for $\displaystyle C_{\rm GN}<0$ and
$\displaystyle C_{\rm NS}<0$, respectively. Therefore weak homeotropic
anchoring occurs on the gas–nematic interface when $\displaystyle C_{\rm
GN}>0$ and on the nematic–substrate interface when $\displaystyle C_{\rm
NS}>0$, and weak planar anchoring occurs on the gas–nematic interface when
$\displaystyle C_{\rm GN}<0$ and on the nematic–substrate interface when
$\displaystyle C_{\rm NS}<0$. Substituting 2 and 3 into 33 and 34 yields

$\displaystyle\displaystyle\omega_{\rm GN}(\theta,{h}_{x})$
$\displaystyle\displaystyle=\gamma_{\rm GN}+\dfrac{C_{\rm
GN}}{4}\left[\dfrac{1-{h}_{x}^{2}}{1+{h}_{x}^{2}}\cos
2\theta+\dfrac{2{h}_{x}}{1+{h}_{x}^{2}}\sin 2\theta\right],$ (35)
$\displaystyle\displaystyle\omega_{\rm NS}(\theta)$
$\displaystyle\displaystyle=\gamma_{\rm NS}+\dfrac{C_{\rm NS}}{4}\cos
2\theta.$ (36)

Experimental techniques for the measurement of $\displaystyle C_{\rm NS}$ are
well established [49, 50, 51], and values in the range $\displaystyle|C_{\rm
NS}|=10^{-6}$–$\displaystyle 10^{-3}\,$N m-1 have been reported for a variety
of nematic materials and substrates with planar or homeotropic anchoring [49,
50, 7]. Measurements of $\displaystyle C_{\rm GN}$ are less common [7];
however, the reported values of $\displaystyle C_{\rm GN}>10^{-5}\,{\rm
Nm}^{-1}$ between air and the nematic mixture ZLI 2860 [52] and of
$\displaystyle C_{\rm GN}>10^{-4}\,{\rm Nm}^{-1}$ between air and the nematic
p-methoxy-benzylidene-p-n-butyl aniline (MBBA) [53] suggest that
$\displaystyle C_{\rm GN}$ and $\displaystyle C_{\rm NS}$ can be of comparable
magnitude. In standard low-molecular-mass nematics, the isotropic interfacial
tensions (i.e. $\displaystyle\gamma_{\rm GN}$ and $\displaystyle\gamma_{\rm
NS}$) are typically much larger than the magnitudes of the anchoring strengths
(i.e. $\displaystyle|C_{\rm GN}|$ and $\displaystyle|C_{\rm NS}|$) [7]. For
example, the isotropic interfacial tension of an interface between air and the
nematic 4-Cyano-4’-pentylbiphenyl (5CB) has been measured as
$\displaystyle\gamma_{\rm GN}=4.0\times 10^{-2}\,{\rm N\,m}^{-1}$, and the
isotropic interfacial tension of an interface between the substrate
poly(methyl methacrylate) (PMMA) and 5CB has been measured as
$\displaystyle\gamma_{\rm NS}=4.051\times 10^{-2}\,{\rm N\,m}^{-1}$ [54].

The gas–substrate interface has constant energy density

$\displaystyle\displaystyle\omega_{\rm GS}=\gamma_{\rm GS},$ (37)

where $\displaystyle\gamma_{\rm GS}$ is the isotropic interfacial tension of
the gas–substrate interface.

###  4.2 Governing equations using the Oseen–Frank bulk elastic energy density
and the Rapini–Papoular interface energy densities

Using 32 in 26 yields the balance of elastic torque within the bulk of the
nematic,

$\displaystyle\displaystyle(K_{1}\sin^{2}\theta+K_{3}\cos^{2}\theta)\theta_{xx}+(K_{1}\cos^{2}\theta+K_{3}\sin^{2}\theta)\theta_{zz}$
(38)
$\displaystyle\displaystyle\quad\quad+(K_{3}-K_{1})\Big{[}\left(\theta_{z}\cos\theta-\theta_{x}\sin\theta\right)\left(\theta_{x}\cos\theta+\theta_{z}\sin\theta\right)+\theta_{xz}\sin
2\theta\Big{]}=0.$

Using 32 and 35 in 27 yields the balance-of-couple condition on the
gas–nematic interface,

$\displaystyle\displaystyle\left(K_{1}\cos^{2}\theta+K_{3}\sin^{2}\theta\right)\theta_{z}+\dfrac{1}{2}(K_{3}-K_{1})(\theta_{x}-h_{x}\theta_{z})\sin
2\theta-\left(K_{1}\sin^{2}\theta+K_{3}\cos^{2}\theta\right)h_{x}\theta_{x}$
$\displaystyle\displaystyle\quad\quad+\dfrac{C_{\rm
GN}}{2\sqrt{1+{h}_{x}^{2}}}\left[({h}_{x}^{2}-1)\sin 2\theta+2{h}_{x}\cos
2\theta\right]=0\quad\text{on}\quad z=h,$ (39)

while using 36 in 28 yields the balance-of-couple condition on the
nematic–substrate interface,

$\displaystyle\displaystyle-\left(K_{1}\cos^{2}\theta+K_{3}\sin^{2}\theta\right)\theta_{z}-\dfrac{1}{2}(K_{3}-K_{1})\theta_{x}\sin
2\theta-\dfrac{C_{\rm NS}}{2}\sin 2\theta=0\quad\text{on}\quad z=0.$ (40)

Using 32 and 35 in 29 yields the nematic Young–Laplace equation,

$\displaystyle\displaystyle p_{0}-W-\psi_{\rm g}+\gamma_{\rm
GN}\dfrac{{h}_{xx}}{\left(1+{h}_{x}^{2}\right)^{3/2}}+\dfrac{C_{\rm
GN}}{4\left(1+{h}_{x}^{2}\right)^{5/2}}\Bigg{[}3{h}_{xx}\Big{[}({h}_{x}^{2}-1)\cos
2\theta-2{h}_{x}\sin 2\theta\Big{]}$
$\displaystyle\displaystyle\quad\quad+(1+{h}_{x}^{2})\bigg{(}4\cos
2\theta\Big{[}\theta_{x}-h_{x}(1+{h}_{x}^{2})\theta_{z}\Big{]}+2\sin
2\theta\Big{[}(1-{h}_{x}^{4})\theta_{z}+{h}_{x}(3+{h}_{x}^{2})\theta_{x}\Big{]}\bigg{)}\Bigg{]}=0$
$\displaystyle\displaystyle\hskip 271.72389pt\text{on}\quad z={h}.$ (41)

In order to express the nematic Young equations 30 in terms of the contact
angles $\displaystyle\beta^{\pm}$, we use the relations 4. Then, using 35, 36
and 37 in 30 yields

$\displaystyle\displaystyle\gamma_{\rm GS}-\gamma_{\rm NS}-\gamma_{\rm
GN}\cos\beta^{-}$ $\displaystyle\displaystyle=\dfrac{C_{\rm NS}}{4}\cos
2\theta+\dfrac{C_{\rm GN}}{4}\left[\cos 2(\theta-\beta^{-})\cos\beta^{-}-2\sin
2(\theta-\beta^{-})\sin\beta^{-}\right]$
$\displaystyle\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{at}\quad
x=d^{-},$ (42) $\displaystyle\displaystyle\gamma_{\rm GS}-\gamma_{\rm
NS}-\gamma_{\rm GN}\cos\beta^{+}$ $\displaystyle\displaystyle=\dfrac{C_{\rm
NS}}{4}\cos 2\theta+\dfrac{C_{\rm GN}}{4}\left[\cos
2(\theta+\beta^{+})\cos\beta^{+}-2\sin
2(\theta+\beta^{+})\sin\beta^{+}\right]$
$\displaystyle\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{at}\quad
x=d^{+}.$ (43)

The terms on the left-hand sides of 42 and 43 appear in the classical
isotropic Young equations, while the terms on the right-hand sides are due to
the anisotropic nature of the nematic and arise from the weak anchoring on the
gas–nematic interface and on the nematic–substrate interface, respectively. In
particular, the classical isotropic Young equations are recovered from the
nematic Young equations 42 and 43 by setting $\displaystyle C_{\rm NS}=C_{\rm
GN}=0$.

We note that, as previously mentioned, while the nematic Young equations 42
and 43 were derived assuming the ridge height $\displaystyle h$ is a single-
valued function of $\displaystyle x$, they also hold when the ridge height
$\displaystyle h$ is a double-valued function of $\displaystyle x$ [47].

###  4.3 The equilibrium states of complete wetting and complete dewetting

The governing equations derived thus far in the present manuscript describe
the partial wetting state, $\displaystyle\mathbb{P}$. As mentioned in Section
1, these equations can also be used to describe the equilibrium states of
complete wetting, $\displaystyle\mathbb{W}$, and of complete dewetting,
$\displaystyle\mathbb{D}$. In the $\displaystyle\mathbb{W}$ state, in which
the nematic forms a film that completely coats the substrate, the nematic
Young equations 42 and 43 and the boundary conditions $\displaystyle
h(d^{\pm})=0$ are not relevant. The behaviour of the director and gas–nematic
interface for nematic films has been studied previously (see, for example,
Sonin [7] and Manyuhina [14]). Similarly, for the $\displaystyle\mathbb{D}$
state, in which the gas–nematic interface forms a cylinder, the nematic Young
equations 42 and 43, the boundary conditions $\displaystyle h(d^{\pm})=0$, and
the balance-of-couple condition on the nematic–solid interface 40 are not
relevant. In the special case in which the gas–nematic interface is a circular
cylinder, the possible director configurations are the same as those in the
case of a nematic confined within a circular capillary and have been
extensively studied (see, for example, Kleman [55]). The limiting cases
$\displaystyle\beta^{\pm}=0$ and $\displaystyle\beta^{\pm}=\pi$ correspond to
the $\displaystyle\mathbb{W}$ and the $\displaystyle\mathbb{D}$ states,
respectively.

As we will show in what follows, using just the nematic Young equations 42 and
43 under the assumption that anchoring breaking occurs in regions adjacent to
the contact lines, we can determine the continuous and discontinuous
transitions that occur between the equilibrium states of complete wetting,
partial wetting, and complete dewetting. We first briefly review the behaviour
of an isotropic ridge in Section 5, before analysing the corresponding
behaviour of a nematic ridge in Sections 6 and 7.

##  5 The equilibrium states and transitions of an isotropic ridge

For a static ridge of isotropic liquid resting on an ideal solid substrate in
an atmosphere of passive gas, a much simpler version of the derivation
presented in Sections 2, 3 and 4, or a simple horizontal force balance at the
contact lines, shows that the well-known classical isotropic Young equations
[15] are given, in the present notation, by

$\displaystyle\displaystyle\gamma_{\rm GS}-\gamma_{\rm IS}-\gamma_{\rm
GI}\cos\beta^{\pm}=0\quad\text{at}\quad x=d^{\pm},$ (44)

where $\displaystyle\gamma_{\rm GI}$ and $\displaystyle\gamma_{\rm IS}$ denote
the isotropic interfacial tensions of the gas–isotropic liquid and isotropic
liquid–substrate interfaces, respectively. In particular, 44 shows that in
this case the left-hand and right-hand contact angles are always the same,
i.e. $\displaystyle\beta^{-}=\beta^{+}=\beta$, say. The classical isotropic
Young equations 44 can be written in terms of a single non-dimensional
parameter, namely the classical isotropic spreading parameter $\displaystyle
S_{\rm I}$, which is defined by

$\displaystyle\displaystyle S_{\rm I}=\dfrac{\gamma_{\rm GS}-\gamma_{\rm
IS}}{\gamma_{\rm GI}}-1,$ (45)

as

$\displaystyle\displaystyle S_{\rm I}+1-\cos\beta=0.$ (46)

Specifically, 46 shows that the $\displaystyle\mathbb{P}$ state exists only
when $\displaystyle-2\leq S_{\rm I}\leq 0$ and that the contact angle is then
given by $\displaystyle\beta=\cos^{-1}\left(S_{\rm I}+1\right)$.

As mentioned in Sections 1 and 4.3, the equilibrium state can also be the
$\displaystyle\mathbb{W}$ state, which corresponds to $\displaystyle\beta=0$,
or the $\displaystyle\mathbb{D}$ state, which corresponds to
$\displaystyle\beta=\pi$, for both of which equation 46 is not relevant. The
classification of the $\displaystyle\mathbb{W}$, $\displaystyle\mathbb{P}$ and
$\displaystyle\mathbb{D}$ states can be obtained by solving the classical
isotropic Young–Laplace equation and expressing the minimum energy state in
terms of $\displaystyle S_{\rm I}$ [46, 15]. In particular, the minimum energy
state is the $\displaystyle\mathbb{W}$ state for $\displaystyle S_{\rm I}>0$,
the $\displaystyle\mathbb{P}$ state for $\displaystyle-2\leq S_{\rm I}\leq 0$,
and the $\displaystyle\mathbb{D}$ state for $\displaystyle S_{\rm I}<-2$. The
contact angle $\displaystyle\beta$ of the minimum energy state of an isotropic
ridge is plotted as a function of $\displaystyle S_{\rm I}$ in Figure 2.

Figure 2:  Summary of the solution for the contact angle $\displaystyle\beta$
as a function of the isotropic spreading parameter $\displaystyle S_{\rm I}$.
The transition points are denoted by stars
$\displaystyle(\mathop{\scalebox{1.5}{\raisebox{-0.86108pt}{$\displaystyle\ast$}}})$.
The solid line denote the local minimum energy state and the dashed lines
denote the local maximum energy states. Sketches of the minimum energy state
are also shown.

We define the values of $\displaystyle S_{\rm I}$ at which there is a change
in the number of possible equilibrium states as _transition points_. At these
points, a transition occurs as $\displaystyle S_{\rm I}$ increases or
decreases if the previous minimum energy state ceases to exist or a new
minimum energy state comes into existence. In particular, as Figure 2 shows,
for an isotropic ridge there is a change in the number of equilibrium states
at the transition points $\displaystyle S_{\rm I}=-2$ and $\displaystyle
S_{\rm I}=0$ which leads to continuous transitions to a new minimum energy
state as $\displaystyle S_{\rm I}$ increases or decreases. For consistency
with the notation used in Section 7, we denote a continuous transition between
two equilibrium states for both increasing and decreasing $\displaystyle
S_{\rm I}$ with a double arrow $\displaystyle(\Leftrightarrow)$. At
$\displaystyle S_{\rm I}=-2$ there is a continuous transition from complete
dewetting to partial wetting or vice versa, which is denoted by
$\displaystyle\mathbb{D}\Leftrightarrow\mathbb{P}$. Similarly, at
$\displaystyle S_{\rm I}=0$ there is a continuous transition from complete
wetting to partial wetting or vice versa, which is denoted by
$\displaystyle\mathbb{P}\Leftrightarrow\mathbb{W}$.

We also note that the behaviour of the contact angle for an isotropic ridge is
non-hysteretic. The well-known phenomenon of isotropic contact-angle
hysteresis occurs only in isotropic systems with _non-ideal_ substrates [15],
and therefore does not occur for the isotropic ridge on an ideal substrate
discussed in this section.

##  6 The nematic Young equations

As for the isotropic ridge discussed in the previous section, for the nematic
ridge considered in the present work we can use the nematic Young equations 42
and 43 to determine the continuous and discontinuous transitions that occur
between the equilibrium states of complete wetting, partial wetting, and
complete dewetting. At first sight, determining these transitions would appear
to involve solving the governing equations for $\displaystyle\theta$ in the
bulk of the nematic ridge, which would, in turn, involve solving for the ridge
height $\displaystyle h$, the contact line positions $\displaystyle
x=d^{\pm}$, and the Lagrange multiplier $\displaystyle p_{0}$. However, under
the assumption that anchoring breaking occurs in regions adjacent to the
contact lines, we can determine these continuous and discontinuous transitions
from just the nematic Young equations 42 and 43.

###  6.1 The director orientation at the contact lines

At the contact lines the preferred director orientations on the gas–nematic
and the nematic–substrate interfaces are, in general, different. Even when the
anchoring is non-antagonistic (i.e. when either planar or homeotropic
anchoring is preferred on both interfaces), since the preferred director
orientation of both interfaces is measured relative to that interface, and the
two interfaces meet at the non-zero contact angles $\displaystyle\beta^{\pm}$,
the orientations are, in general, not the same. Hence the director cannot, in
general, align with the preferred orientations of both interfaces. In such a
situation there are three possibilities for the director orientation at the
contact lines: (i) the contact angles are such that the preferred orientations
on the two interfaces coincide exactly; (ii) there may be defects
(disclination lines in this two-dimensional case) at one or both of the
contact lines; (iii) the weak anchoring on both interfaces allows anchoring
breaking to occur in regions adjacent to the contact lines and the director(s)
on one or both of the interfaces deviates from the preferred alignment(s) and
attains the same orientation on both interfaces.

Case (i) is a very special situation in which the contact angles are such that
the preferred director orientations on the two interfaces coincide exactly at
the contact lines. For instance, when the preferred orientations on the two
interfaces are antagonistic, the contact angles must be exactly
$\displaystyle\beta^{\pm}=\pi/2$ to allow the director to be tangent to one
interface and perpendicular to the other. Since this type of special case is
highly unlikely to occur in practice, we do not consider it any further in the
present work.

As discussed in Section 1, case (ii) has been considered in [36] in which
infinite planar anchoring was assumed on the gas–nematic and nematic–substrate
interfaces. In this case, since the infinitely-strong anchoring cannot be
broken, the director must adopt a splayed configuration (for a full account of
splayed director configurations, see [19]) in a region adjacent to the contact
line, with a disclination line located at the contact line [36]. For the
finite anchoring strengths considered in the present work, we assume that the
energy associated with anchoring breaking is less than the energy associated
with the formation of a disclination line, and therefore that such
disclination lines do not occur.

Having ruled out cases (i) and (ii), we are left with case (iii). In this case
the weak anchoring on the interfaces allows anchoring breaking to occur in
regions adjacent to the contact lines so that the director(s) on one or both
of the interfaces deviates from the preferred alignment(s) and attains the
same orientation on both interfaces.

As discussed in Section 1, for nematic films with antagonistic anchoring, when
the film thickness is less than the Jenkins–Barratt–Barbero–Barberi critical
thickness the energetically favourable state has a uniform director field in
which the director aligns parallel to the preferred director alignment of the
interface with the stronger anchoring. For a nematic ridge, close to the
contact lines, where the ridge height approaches zero and hence the separation
between the gas–nematic and nematic–substrate interfaces is always less than
the critical thickness, anchoring breaking occurs and the director aligns
parallel to the preferred alignment of the interface with the stronger
anchoring. Specifically, if the nematic–substrate interface has the stronger
anchoring (i.e. if $\displaystyle|C_{\rm NS}|>|C_{\rm GN}|$) then the director
at the contact lines aligns parallel to the nematic–substrate interface with
$\displaystyle\theta=0$ at $\displaystyle x=d^{\pm}$ in the case of planar
anchoring corresponding to $\displaystyle C_{\rm NS}<0$ or perpendicular to
the nematic–substrate interface with $\displaystyle\theta=\pi/2$ at
$\displaystyle x=d^{\pm}$ in the case of homeotropic anchoring corresponding
to $\displaystyle C_{\rm NS}>0$; we term both of these scenarios
“nematic–substrate (NS) dominant anchoring”. Correspondingly, if the
gas–nematic interface has the stronger anchoring (i.e. if
$\displaystyle|C_{\rm GN}|>|C_{\rm NS}|$) then the director at the contact
lines aligns parallel to the gas–nematic interface with
$\displaystyle\theta=\beta^{\pm}$ at $\displaystyle x=d^{\pm}$ in the case of
planar anchoring corresponding to $\displaystyle C_{\rm GN}<0$ or
perpendicular to the gas–nematic interface with
$\displaystyle\theta=\beta^{\pm}+\pi/2$ at $\displaystyle x=d^{\pm}$ in the
case of homeotropic anchoring corresponding to $\displaystyle C_{\rm GN}>0$;
we term both of these situations “gas–nematic (GN) dominant anchoring”.

There are two special situations in which anchoring breaking cannot occur as
described above because the interfaces have either equal anchoring strengths
($\displaystyle C_{\rm NS}=C_{\rm GN}$) or equal and opposite anchoring
strengths ($\displaystyle C_{\rm NS}=-C_{\rm GN}$). In both of these
situations, anchoring breaking occurs on both interfaces and the director
orientation adopts the average of the preferred orientations [47, 36]. In
particular, when the anchoring strengths of the interfaces are equal and
planar anchoring is preferred, the director angles are
$\displaystyle\theta=\beta^{\pm}/2$ at $\displaystyle x=d^{\pm}$, as discussed
by Rey [36], and when the anchoring strengths of the interfaces are equal and
homeotropic anchoring is preferred, the director angles are
$\displaystyle\theta=\beta^{\pm}/2+\pi/2$ at $\displaystyle x=d^{\pm}$. When
the anchoring strengths of the interfaces are equal and opposite, the director
angles are $\displaystyle\theta=\beta^{\pm}/2+\pi/4$ or
$\displaystyle\theta=\beta^{\pm}/2-\pi/4$ at $\displaystyle x=d^{\pm}$.

Since for an ideal substrate the material properties of the substrate are the
same at both contact lines, anchoring breaking must occur in the same way, and
hence the director angles at the two contact lines must be the same. However,
as we will show below, in some situations the nematic Young equations 42 and
43 allow for more than one possible contact angle for the same parameter
values, and so the contact angles $\displaystyle\beta^{\pm}$ do not, in
general, have to be the same and so the ridge can be asymmetric. Moreover, the
contact angles $\displaystyle\beta^{\pm}$ could be different if the substrate
is non-ideal and the material properties of the substrate are different at the
two contact lines (for example, if the substrate was manufactured so that the
values of $\displaystyle C_{\rm NS}$ at $\displaystyle x=d^{\pm}$ were
different, or if gradients in the temperature of the gas or adsorption of a
surfactant from the gas lead to different values of $\displaystyle C_{\rm GN}$
at $\displaystyle x=d^{\pm}$ [56]). Without loss of generality, for the
remainder of the present work, we consider only the left-hand contact line,
which is described by the nematic Young equation 42, and write
$\displaystyle\beta^{-}=\beta$ for simplicity. The corresponding results for
the right-hand contact line can be obtained in the same way.

###  6.2 Nematic spreading parameters

For NS-dominant anchoring (for which either $\displaystyle\theta=0$ or
$\displaystyle\theta=\pi/2$) the nematic Young equation 42 reduces to a cubic
equation for $\displaystyle\cos\beta$, namely either

$\displaystyle\displaystyle\gamma_{\rm GS}-\left(\gamma_{\rm
NS}+\tfrac{1}{4}C_{\rm NS}\right)-\left(\gamma_{\rm GN}+\tfrac{1}{4}C_{\rm
GN}\right)\cos\beta=-\tfrac{1}{2}C_{\rm
GN}\cos\beta\left(\cos^{2}\beta-1\right)$ (47)

when $\displaystyle\theta=0$ or

$\displaystyle\displaystyle\gamma_{\rm GS}-\left(\gamma_{\rm
NS}-\tfrac{1}{4}C_{\rm NS}\right)-\left(\gamma_{\rm GN}-\tfrac{1}{4}C_{\rm
GN}\right)\cos\beta=\tfrac{1}{2}C_{\rm
GN}\cos\beta\left(\cos^{2}\beta-1\right)$ (48)

when $\displaystyle\theta=\pi/2$. On the other hand, for GN-dominant anchoring
(for which either $\displaystyle\theta=\beta$ or
$\displaystyle\theta=\beta+\pi/2$) the nematic Young equation 42 reduces to a
quadratic equation for $\displaystyle\cos\beta$, namely either

$\displaystyle\displaystyle\gamma_{\rm GS}-\left(\gamma_{\rm
NS}+\tfrac{1}{4}C_{\rm NS}\right)-\left(\gamma_{\rm GN}+\tfrac{1}{4}C_{\rm
GN}\right)\cos\beta=\tfrac{1}{2}C_{\rm NS}\left(\cos^{2}\beta-1\right)$ (49)

when $\displaystyle\theta=\beta$ or

$\displaystyle\displaystyle\gamma_{\rm GS}-\left(\gamma_{\rm
NS}-\tfrac{1}{4}C_{\rm NS}\right)-\left(\gamma_{\rm GN}-\tfrac{1}{4}C_{\rm
GN}\right)\cos\beta=-\tfrac{1}{2}C_{\rm NS}\left(\cos^{2}\beta-1\right)$ (50)

when $\displaystyle\theta=\beta+\pi/2$. Each of the equations 47, 48, 49 and
50 may each be written in terms of just two parameters as follows: 47 and 48
may be written as

$\displaystyle\displaystyle S_{\rm N}+1-\cos\beta$
$\displaystyle\displaystyle=-\Delta_{\rm
GN}\cos\beta\left(\cos^{2}\beta-1\right),$ (51)

while 49 and 50 may be written as

$\displaystyle\displaystyle S_{\rm N}+1-\cos\beta$
$\displaystyle\displaystyle=\Delta_{\rm NS}\left(\cos^{2}\beta-1\right),$ (52)

where $\displaystyle S_{\rm N}$, $\displaystyle\Delta_{\rm NS}$ and
$\displaystyle\Delta_{\rm GN}$ are defined by

$\displaystyle\displaystyle S_{\rm N}$
$\displaystyle\displaystyle=\dfrac{4\gamma_{\rm GS}-\left(4\gamma_{\rm
NS}-|C_{\rm NS}|\right)}{4\gamma_{\rm GN}-|C_{\rm GN}|}-1,$ (53)
$\displaystyle\displaystyle\Delta_{\rm NS}$
$\displaystyle\displaystyle=\dfrac{2C_{\rm NS}}{4\gamma_{\rm GN}-|C_{\rm
GN}|},$ (54) $\displaystyle\displaystyle\Delta_{\rm GN}$
$\displaystyle\displaystyle=\dfrac{2C_{\rm GN}}{4\gamma_{\rm GN}-|C_{\rm
GN}|},$ (55)

respectively. Note that while the nematic spreading parameter $\displaystyle
S_{\rm N}$ is the appropriate generalisation of the isotropic spreading
parameter $\displaystyle S_{\rm I}$ defined in 45, the scaled anchoring
coefficients $\displaystyle\Delta_{\rm NS}$ and $\displaystyle\Delta_{\rm GN}$
have no isotropic counterparts. We also note that when
$\displaystyle\Delta_{\rm NS}=\Delta_{\rm GN}=0$ (i.e. when $\displaystyle
C_{\rm GN}=C_{\rm NS}=0$) then $\displaystyle S_{\rm N}=(\gamma_{\rm
GS}-\gamma_{\rm NS})/\gamma_{\rm GN}-1=S_{\rm I}$, and both of the nematic
Young equations 51 and 52 reduce to the classical isotropic Young equation 46.

Each of the right-hand sides of the nematic Young equations 51 and 52 involve
only one parameter, namely the scaled anchoring coefficients
$\displaystyle\Delta_{\rm NS}$ and $\displaystyle\Delta_{\rm GN}$,
respectively. At first sight, it may seem counter-intuitive that
$\displaystyle\Delta_{\rm NS}$ appears in the case of GN-dominant anchoring
and $\displaystyle\Delta_{\rm GN}$ appears in the case of NS-dominant
anchoring. However, for GN–dominant anchoring the director is aligned with the
preferred director orientation of the gas–nematic interface, and so the
corresponding anchoring energy, and therefore the couple on the director, is
zero. The non-zero contribution to the anchoring energy therefore derives from
the breaking of the nematic–substrate interface anchoring. The corresponding
explanation applies to the NS-dominant case. The right-hand sides of equations
51 and 52 may therefore be interpreted physically as the contribution to the
balance of stress at the contact line associated with the breaking of the
anchoring on the interface with the weaker anchoring.

##  7 The equilibrium states and transitions of a nematic ridge

With the director angle determined in regions adjacent to the contact lines,
we can now use the nematic Young equations 51 and 52 to determine the
continuous and discontinuous transitions between the
$\displaystyle\mathbb{W}$, $\displaystyle\mathbb{P}$ and
$\displaystyle\mathbb{D}$ states. As 51 and 52 are cubic and quadratic
equations for $\displaystyle\cos\beta$, respectively, they can have up to
three real solutions for $\displaystyle\beta$ and up to two real solutions for
$\displaystyle\beta$, respectively. Each of these solutions for
$\displaystyle\beta$ corresponds to a different $\displaystyle\mathbb{P}$
state, and therefore, unlike for the isotropic ridge described in Section 5, a
nematic ridge can have multiple $\displaystyle\mathbb{P}$ states.

Following the same approach as for the isotropic ridge in Section 5, the
values of $\displaystyle S_{\rm N}$ and $\displaystyle\Delta_{\rm NS}$ (for
GN-dominant anchoring) or $\displaystyle S_{\rm N}$ and
$\displaystyle\Delta_{\rm GN}$ (for NS-dominant anchoring) at which there is a
change in the number of possible equilibrium states are again called the
transition points. Specifically, a transition occurs as $\displaystyle S_{\rm
N}$, $\displaystyle\Delta_{\rm NS}$ or $\displaystyle\Delta_{\rm GN}$
increases or decreases if the previous minimum energy state ceases to exist or
a new minimum energy state comes into existence. In an analogous manner to
that in the isotropic case, at $\displaystyle S_{\rm N}=-2$ and $\displaystyle
S_{\rm N}=0$ the number of equilibrium states changes, which leads to
transitions to a new equilibrium state as $\displaystyle S_{\rm N}$ increases
or decreases through these values. However, unlike in the isotropic case, in
which only continuous transitions occur, in the nematic case discontinuous
transitions can also now occur, i.e. the contact angle can transition
discontinuously.

In both NS-dominant and GN-dominant anchoring, the nature of the different
transitions, the contact-angle transitions, and the transition points can be
obtained from just the nematic Young equations 51 and 52. In NS-dominant
anchoring the transition behaviour depends on whether
$\displaystyle\Delta_{\rm GN}<-4$, $\displaystyle-4\leq\Delta_{\rm GN}<-1$,
$\displaystyle-1\leq\Delta_{\rm GN}\leq 1/2$ or $\displaystyle\Delta_{\rm
GN}>1/2$, whereas in GN-dominant anchoring the transition behaviour depends on
whether $\displaystyle\Delta_{\rm NS}<-1/2$, $\displaystyle-1/2\leq\Delta_{\rm
NS}\leq 1/2$ or $\displaystyle\Delta_{\rm NS}>1/2$. Figures 3 and 4 show
summaries of the solutions for the contact angle $\displaystyle\beta$ as a
function of the nematic spreading parameter $\displaystyle S_{\rm N}$ for
these four ranges of $\displaystyle\Delta_{\rm GN}$ for NS-dominant anchoring
and for these three ranges of $\displaystyle\Delta_{\rm NS}$ for GN-dominant
anchoring, respectively.

|
---|---
(a) | (b)
|
(c) | (d)
Figure 3:  Summaries of the solutions for the contact angle $\displaystyle\beta$ as a function of the nematic spreading parameter $\displaystyle S_{\rm N}$ for NS-dominant anchoring according to 51 for the four ranges of $\displaystyle\Delta_{\rm GN}$: (a) $\displaystyle\Delta_{\rm GN}<-4$, (b) $\displaystyle-4\leq\Delta_{\rm GN}\leq-1$, (c) $\displaystyle-1\leq\Delta_{\rm GN}\leq 1/2$, and (d) $\displaystyle\Delta_{\rm GN}>1/2$. The transition points are labelled and shown by stars $\displaystyle(\mathop{\scalebox{1.5}{\raisebox{-0.86108pt}{$\displaystyle\ast$}}})$ for a continuous transition and by dots ($\displaystyle\bullet$) for a discontinuous transition, where $\displaystyle s_{1}=\sqrt{4(1+\Delta_{\rm GN})^{3}/(27\Delta_{\rm GN})}$, $\displaystyle b_{1}=\sqrt{(1+\Delta_{\rm GN})/(3\Delta_{\rm GN})}$ and $\displaystyle b_{2}=-1/2+\sqrt{\Delta_{\rm GN}(\Delta_{\rm GN}+4)}/(2\Delta_{\rm GN})$. The arrows shows the directions of the associated transitions in $\displaystyle\beta$. The solid lines denote the hypothesised local minimum energy states and the dashed lines denote the hypothesised local maximum energy states.  |
---|---
(a) | (b)
(c)
Figure 4:  Summaries of the solutions for the contact angle
$\displaystyle\beta$ as a function of the nematic spreading parameter
$\displaystyle S_{\rm N}$ for GN-dominant anchoring according to 52 for the
three ranges of $\displaystyle\Delta_{\rm NS}$: (a) $\displaystyle\Delta_{\rm
NS}<-1/2$, (b) $\displaystyle-1/2\leq\Delta_{\rm NS}\leq 1/2$, and (c)
$\displaystyle\Delta_{\rm NS}>1/2$. The transition points are labelled and
shown by stars
$\displaystyle(\mathop{\scalebox{1.5}{\raisebox{-0.86108pt}{$\displaystyle\ast$}}})$
for a continuous transition and by dots ($\displaystyle\bullet$) for a
discontinuous transition, where $\displaystyle s_{2}=-1-\Delta_{\rm
NS}-1/(4\Delta_{\rm NS})$ and $\displaystyle b_{3}=-1/(2\Delta_{\rm NS})$. The
arrows shows the directions of the associated transitions in
$\displaystyle\beta$. The solid lines denote the hypothesised local minimum
energy states and the dashed lines denote the hypothesised local maximum
energy states.

In Figures 3 and 4, and what follows, a rightward arrow
($\displaystyle\Rightarrow$) denotes a discontinuous transition for increasing
$\displaystyle S_{\rm N}$, and a leftward arrow ($\displaystyle\Leftarrow$)
denotes a discontinuous transition for decreasing $\displaystyle S_{\rm N}$.
Thus, for example, a discontinuous transition from complete wetting to partial
wetting for increasing $\displaystyle S_{\rm N}$ is denoted by
$\displaystyle\mathbb{W}\Rightarrow\mathbb{P}$, and a discontinuous transition
from partial wetting to complete wetting for decreasing $\displaystyle S_{\rm
N}$ is denoted by $\displaystyle\mathbb{W}\Leftarrow\mathbb{P}$. In addition,
we denote a discontinuous transition in the contact angle using the same
notation, so that, for example, the contact-angle transition for a
$\displaystyle\mathbb{W}\Rightarrow\mathbb{P}$ transition, for which the
contact angle transitions discontinuously from $\displaystyle\beta=0$ to
$\displaystyle\beta=\beta^{*}$, is denoted by $\displaystyle
0\Rightarrow\beta^{*}$. Summaries of all of the possible transitions shown in
Figures 3 and 4 are given in Tables 1 and 2 for NS-dominant and GN-dominant
anchoring, respectively.

Range of $\displaystyle\Delta_{\rm GN}$ |  $\displaystyle S_{\rm N}$ value at transition | Nature of the transition | Contact-angle transition
---|---|---|---
$\displaystyle\Delta_{\rm GN}<-4$ | $\displaystyle-1-s_{1}\>(<-2)$ | $\displaystyle\mathbb{D}\Leftarrow\mathbb{P}$ | $\displaystyle\pi\Leftarrow\cos^{-1}b_{1}$
$\displaystyle-2$ | $\displaystyle\mathbb{D}\Leftrightarrow\mathbb{P}$ | Continuous with $\displaystyle\beta=\pi$
$\displaystyle 0$ | $\displaystyle\mathbb{P}\Leftrightarrow\mathbb{W}$ | Continuous with $\displaystyle\beta=0$
$\displaystyle(0<)\,-1+s_{1}$ | $\displaystyle\mathbb{P}\Rightarrow\mathbb{W}$ | $\displaystyle\pi-\cos^{-1}b_{1}\Rightarrow 0$
$\displaystyle-4\leq\Delta_{\rm GN}<-1$ | $\displaystyle-2$ | $\displaystyle\mathbb{D}\Leftrightarrow\mathbb{P}$ | Continuous with $\displaystyle\beta=\pi$
$\displaystyle(-2\leq)\,-1-s_{1}\,(<-1)$ | $\displaystyle\mathbb{P}\Leftarrow\mathbb{P}$ | $\displaystyle\pi-\cos^{-1}2b_{1}\Leftarrow\cos^{-1}b_{1}$
$\displaystyle(-1<)\,-1+s_{1}\,(\leq 0)$ | $\displaystyle\mathbb{P}\Rightarrow\mathbb{P}$ | $\displaystyle\pi-\cos^{-1}b_{1}\Rightarrow\cos^{-1}2b_{1}$
$\displaystyle 0$ | $\displaystyle\mathbb{P}\Leftrightarrow\mathbb{W}$ | Continuous with $\displaystyle\beta=0$
$\displaystyle-1\leq\Delta_{\rm GN}\leq 1/2$ | $\displaystyle-2$ | $\displaystyle\mathbb{D}\Leftrightarrow\mathbb{P}$ | Continuous with $\displaystyle\beta=\pi$
$\displaystyle 0$ | $\displaystyle\mathbb{P}\Leftrightarrow\mathbb{W}$ | Continuous with $\displaystyle\beta=0$
$\displaystyle\Delta_{\rm GN}>1/2$ | $\displaystyle-1-s_{1}\,(<-2)$ | $\displaystyle\mathbb{D}\Leftarrow\mathbb{P}$ | $\displaystyle\pi\Leftarrow\pi-\cos^{-1}b_{1}$
$\displaystyle-2$ | $\displaystyle\mathbb{D}\Rightarrow\mathbb{P}$ | $\displaystyle\pi\Rightarrow\pi-\cos^{-1}b_{2}$
$\displaystyle 0$ | $\displaystyle\mathbb{P}\Leftarrow\mathbb{W}$ | $\displaystyle\cos^{-1}b_{2}\Leftarrow 0$
$\displaystyle(0<)\,-1+s_{1}$ | $\displaystyle\mathbb{P}\Rightarrow\mathbb{W}$ | $\displaystyle\cos^{-1}b_{1}\Rightarrow 0$
Table 1:  Summary of all of the possible transitions for NS-dominant anchoring obtained from 51. The four ranges of values of $\displaystyle\Delta_{\rm GN}$, the $\displaystyle S_{\rm N}$ value at which transitions occur, where $\displaystyle s_{1}=\sqrt{4(1+\Delta_{\rm GN})^{3}/(27\Delta_{\rm GN})}$, and the nature of the different transitions and the contact-angle transitions, where $\displaystyle b_{1}=\sqrt{(1+\Delta_{\rm GN})/(3\Delta_{\rm GN})}$ and $\displaystyle b_{2}=-1/2+\sqrt{\Delta_{\rm GN}(\Delta_{\rm GN}+4)}/(2\Delta_{\rm GN})$, are shown.  Range of $\displaystyle\Delta_{\rm NS}$ |  $\displaystyle S_{\rm N}$ value at transition | Nature of the transition | Contact-angle transition
---|---|---|---
$\displaystyle\Delta_{\rm NS}<-1/2$ | $\displaystyle-2$ | $\displaystyle\mathbb{D}\Leftrightarrow\mathbb{P}$ | Continuous with $\displaystyle\beta=\pi$
$\displaystyle 0$ | $\displaystyle\mathbb{P}\Leftarrow\mathbb{W}$ | $\displaystyle\cos^{-1}\left(-1+2b_{3}\right)\Leftarrow 0$
$\displaystyle(0<)\,s_{2}$ | $\displaystyle\mathbb{P}\Rightarrow\mathbb{W}$ | $\displaystyle\cos^{-1}b_{3}\Rightarrow 0$
$\displaystyle-1/2\leq\Delta_{\rm NS}\leq 1/2$ | $\displaystyle-2$ | $\displaystyle\mathbb{D}\Leftrightarrow\mathbb{P}$ | Continuous with $\displaystyle\beta=\pi$
$\displaystyle 0$ | $\displaystyle\mathbb{P}\Leftrightarrow\mathbb{W}$ | Continuous with $\displaystyle\beta=0$
$\displaystyle\Delta_{\rm NS}>1/2$ | $\displaystyle s_{2}\,(<-2)$ | $\displaystyle\mathbb{D}\Leftarrow\mathbb{P}$ | $\displaystyle\pi\Leftarrow\cos^{-1}b_{3}$
$\displaystyle-2$ | $\displaystyle\mathbb{D}\Rightarrow\mathbb{P}$ | $\displaystyle\pi\Rightarrow\cos^{-1}\left(1+2b_{3}\right)$
$\displaystyle 0$ | $\displaystyle\mathbb{P}\Leftrightarrow\mathbb{W}$ | Continuous with $\displaystyle\beta=0$
Table 2:  Summary of all of the possible transitions for GN-dominant anchoring
obtained from 52. The three ranges of values of $\displaystyle\Delta_{\rm
GN}$, the $\displaystyle S_{\rm N}$ value at which transitions occur, where
$\displaystyle s_{2}=-1-\Delta_{\rm NS}-1/(4\Delta_{\rm NS})$, and the nature
of the different transitions and the contact-angle transitions, where
$\displaystyle b_{3}=-1/(2\Delta_{\rm NS})$, are shown.

Although for a nematic ridge, unlike for an isotropic ridge, the free energy
of each equilibrium state cannot be determined analytically, we can
hypothesise the local minimum energy states for the nematic ridge by
comparison with those for the isotropic ridge described in Section 5. Hence we
hypothesise that the $\displaystyle\mathbb{D}$ state is a local minimum energy
state for $\displaystyle S_{\rm N}<-2$ and a local maximum energy state for
$\displaystyle S_{\rm N}\geq-2$. Similarly, we hypothesise that the
$\displaystyle\mathbb{W}$ state is a local minimum energy state for
$\displaystyle S_{\rm N}>0$ and a local maximum energy state for
$\displaystyle S_{\rm N}\leq 0$. Assuming that there will always be at least
one local minimum energy state, within the range $\displaystyle-2\leq S_{\rm
N}\leq 0$, where the $\displaystyle\mathbb{W}$ and the
$\displaystyle\mathbb{D}$ states are local maximum energy states, the local
minimum energy state must be a $\displaystyle\mathbb{P}$ state. The local
minimum and maximum energy states are shown in Figures 3 and 4 by solid lines
and dashed lines, respectively. In the absence of a full dynamical theory, we
also hypothesise that the continuous and discontinuous transitions shown in
Figures 3 and 4 each correspond to classical pitchfork or fold bifurcations
[57]. In particular, the transitions at $\displaystyle S_{\rm N}=-2$ and
$\displaystyle S_{\rm N}=0$ are pitchfork bifurcations, where a change in
$\displaystyle S_{\rm N}$, $\displaystyle\Delta_{\rm GN}$ or
$\displaystyle\Delta_{\rm NS}$ leads to a local minimum energy state becoming
a local maximum energy state, forcing the system to transition continuously
(through a super-critical pitchfork bifurcation) or discontinuously (through a
sub-critical pitchfork bifurcation) to a new local minimum energy state.
Furthermore, the discontinuous transitions at $\displaystyle S_{\rm N}=-1\pm
s_{1}$ and $\displaystyle S_{\rm N}=s_{2}$, where

$\displaystyle\displaystyle s_{1}$
$\displaystyle\displaystyle=\sqrt{4(1+\Delta_{\rm GN})^{3}/(27\Delta_{\rm
GN})},$ (56) $\displaystyle\displaystyle s_{2}$
$\displaystyle\displaystyle=-1-\Delta_{\rm NS}-1/(2\Delta_{\rm NS}),$ (57)

are associated with fold bifurcations, where a change in $\displaystyle S_{\rm
N}$, $\displaystyle\Delta_{\rm GN}$ or $\displaystyle\Delta_{\rm NS}$ leads to
a local minimum energy state combining with a local maximum energy state,
forcing the system to transition discontinuously to a different local minimum
energy state.

Figures 3(a) and (b) and Figures 4(a) and (c) also show that there are ranges
of $\displaystyle S_{\rm N}$ values for which there are two local minimum
energy states (shown by solid lines). Perhaps most interestingly, we see from
Figure 3(a) that when $\displaystyle-2\leq S_{\rm N}\leq 0$ and from Figure
3(b) that when $\displaystyle-1-s_{1}\leq S_{\rm N}\leq-2$ there are two local
minimum energy $\displaystyle\mathbb{P}$ states. This implies that the effects
of anchoring breaking can give rise to two local minimum energy
$\displaystyle\mathbb{P}$ states, a situation that does not occur in the
isotropic case.

From the results summarised in Tables 2 and 1 the asymptotic behaviour of the
contact-angle transitions in the limits of large anchoring coefficients
relative to the isotropic interfacial tension, namely the limits
$\displaystyle\Delta_{\rm GN}\to\pm\infty$ and $\displaystyle\Delta_{\rm
NS}\to\pm\infty$, may be determined. For example, for NS-dominant anchoring,
as $\displaystyle\Delta_{\rm GN}\to\infty$ the contact-angle transition for
the $\displaystyle\mathbb{P}\Leftarrow\mathbb{W}$ transition approaches a
discontinuous transition in contact angle from $\displaystyle\beta=0$ to
$\displaystyle\beta=\pi/2$, and the contact-angle transition for the
$\displaystyle\mathbb{D}\Rightarrow\mathbb{P}$ transition approaches a
discontinuous transition in the contact angle from $\displaystyle\beta=\pi$ to
$\displaystyle\beta=\pi/2$. This limiting behaviour shows that for GN-dominant
anchoring in the limit $\displaystyle\Delta_{\rm NS}\to\infty$ the contact-
angle transition for the $\displaystyle\mathbb{P}\Leftarrow\mathbb{W}$
transition approaches a discontinuous transition in the contact angle from
$\displaystyle\beta=0$ to $\displaystyle\beta=\pi$, i.e. it approaches a
discontinuous transition from the $\displaystyle\mathbb{W}$ state directly to
the $\displaystyle\mathbb{D}$ state, which bypasses the
$\displaystyle\mathbb{P}$ state. Similarly, in the limit
$\displaystyle\Delta_{\rm NS}\to-\infty$ the contact-angle transition for the
$\displaystyle\mathbb{D}\Rightarrow\mathbb{P}$ transition approaches a
discontinuous transition in the contact angle from $\displaystyle\beta=\pi$ to
$\displaystyle\beta=0$, i.e. it approaches a discontinuous transition from the
$\displaystyle\mathbb{D}$ state directly to the $\displaystyle\mathbb{W}$
state.

The discontinuous transitions shown in Figures 3(a), (b), and (d) and Figures
4(a) and (c) show that the behaviour of the contact angle is hysteretic. This
nematic contact-angle hysteresis, which occurs for an ideal substrate, is
fundamentally different from the well-known phenomenon of isotropic contact-
angle hysteresis which, as we have previously mentioned, occurs only for a
non-ideal substrate. However, we note that when
$\displaystyle-1\leq\Delta_{\rm GN}\leq 1/2$ for NS-dominant anchoring, as
shown in Figure 3(c), and when $\displaystyle-1/2\leq\Delta_{\rm NS}\leq 1/2$
for GN-dominant anchoring, as shown in Figure 4(b), the behaviour is similar
to the isotropic case and no contact-angle hysteresis occurs.

##  8 Conclusions

In the present work we analysed a two-dimensional static ridge of nematic
resting on an ideal solid substrate in an atmosphere of passive gas. In
Sections 2 and 3, we obtained the first complete theoretical description for
this system by minimising the free energy, which is given by the sum of the
bulk elastic energy, gravitational potential energy and the interface
energies, subject to a prescribed constant cross-sectional area. We then, in
Section 4, chose explicit forms of the bulk elastic energy density and the
interface energy densities, namely the standard Oseen–Frank bulk elastic
energy density and the standard Rapini–Papoular interface energy densities,
and obtained the governing equations 38, 39, 40, 4.2, 42 and 43. Specifically,
these equations determine the director angle $\displaystyle\theta(x,z)$, the
ridge height $\displaystyle h(x)$, the contact line positions $\displaystyle
x=d^{\pm}$, and the Lagrange multiplier $\displaystyle p_{0}$, in terms of the
physical parameters, namely the splay and bend elastic constants
$\displaystyle K_{1}$ and $\displaystyle K_{3}$, the corresponding isotropic
interfacial tensions $\displaystyle\gamma_{\rm GN}$, $\displaystyle\gamma_{\rm
NS}$ and $\displaystyle\gamma_{\rm GS}$, and the anchoring strengths
$\displaystyle C_{\rm GN}$ and $\displaystyle C_{\rm NS}$. These governing
equations can, in principle, be generalised to include electromagnetic forces,
additional contact line effects, non-ideal substrates, or more detailed models
for the nematic molecular order, such as Q-tensor theory [58].

After briefly reviewing the behaviour of an isotropic ridge in Section 5 and
discussing the nematic Young equations 42 and 43 in Section 6, in Section 7,
under the assumption that anchoring breaking occurs in regions adjacent to the
contact lines, we used the nematic Young equations 42 and 43 to determine the
continuous and discontinuous transitions that occur between the
$\displaystyle\mathbb{W}$, $\displaystyle\mathbb{P}$ and
$\displaystyle\mathbb{D}$ states. In particular, it was shown that the nematic
Young equations in the cases of NS-dominant and GN-dominant anchoring, which
are given by 51 and 52, respectively, can each be written in terms of two
parameters, namely the nematic spreading parameter $\displaystyle S_{\rm N}$
and one of the scaled anchoring coefficients $\displaystyle\Delta_{\rm GN}$
and $\displaystyle\Delta_{\rm NS}$. In both situations, we found continuous
transitions analogous to those that occur in the classical case of an
isotropic liquid, but also a variety of discontinuous transitions, as well as
contact-angle hysteresis, and regions of parameter space in which there exist
multiple partial wetting states that do not occur in the classical case of an
isotropic liquid. Summaries of all the transitions for NS-dominant and GN-
dominant anchoring are given in Figures 3 and 4, respectively, and in Tables 1
and 2, respectively.

For simplicity, in Section 7 we considered only the left-hand contact line,
which is described by the nematic Young equation 42. Corresponding results can
be obtained for the right-hand contact line, and, since we have shown that
there is more than one possible contact angle value for the same parameter
values, $\displaystyle\beta^{\pm}$ do not, in general, have to be the same and
so the ridge can be asymmetric. This is in agreement with observations by
Vanzo et al. [59], who found that anisotropic effects can lead to multiple
contact-angle values and asymmetry of elongated sessile nematic droplets.

Concerning potential future comparisons with the results of physical
experiments of the situation modelled in the present work, we have shown that
discontinuous transitions and contact-angle hysteresis will occur if the
parameters are such that $\displaystyle\Delta_{\rm GN}>1/2$ or
$\displaystyle\Delta_{\rm GN}<-1$, or $\displaystyle|\Delta_{\rm NS}|>1/2$.
Inspection of 55 and 54 shows that one of these inequalities may be satisfied
when the isotropic interfacial tension and the anchoring strength are of
similar magnitude at one of the interfaces, i.e. $\displaystyle\gamma_{\rm
GN}\simeq|C_{\rm GN}|$ or $\displaystyle\gamma_{\rm NS}\simeq|C_{\rm NS}|$.
For standard low-molecular-mass nematics, for which the isotropic interfacial
tension is typically larger than the anchoring strength [7], this may be
difficult to achieve. For example, for the typical parameter values given in
Section 4, $\displaystyle|\Delta_{\rm GN}|\ll 1$ and
$\displaystyle|\Delta_{\rm NS}|\ll 1$. Therefore, the present analysis
indicates that, as many previous authors have implicitly or explicitly
assumed, for standard low-molecular-mass nematics the classical isotropic
Young equations 44 are a good approximation for the nematic Young equations 42
and 43 and discontinuous transitions and contact-angle hysteresis will not be
observed. However, for high-molecular-mass nematics, e.g. nematic polymers, or
systems with particularly strong anchoring, the anchoring strengths would be
considerably higher, and the discontinuous transitions could potentially be
observed experimentally. For example, the use of polymeric compounds to
produce tailored anchoring [31] leads to a strong preference for polymers to
align at interfaces [60, 61] and may result in large anchoring strengths,
which could lead to $\displaystyle|\Delta_{\rm GN}|=O(1)$ and
$\displaystyle|\Delta_{\rm NS}|=O(1)$ and hence the transitions predicted in
the present work could potentially be observed. Alternatively, the situation
in which the surrounding fluid is the isotropic melt of the nematic could
lower the isotropic interfacial tension $\displaystyle\gamma_{\rm GN}$. In
this situation, the isotropic interfacial tension for the isotropic–nematic
interface $\displaystyle\gamma_{\rm IN}$ would be much smaller than the
gas–nematic interfacial tension $\displaystyle\gamma_{\rm GN}$ and may become
comparable with the anchoring strength $\displaystyle C_{\rm GN}$. For
instance, $\displaystyle\gamma_{\rm IN}$ was measured for the nematic MBBA as
$\displaystyle\gamma_{\rm IN}=10^{-5}\,{\rm Nm}^{-1}$ [62], which is three
orders of magnitude smaller than typical isotropic interfacial tension for a
gas–nematic interface $\displaystyle\gamma_{\rm GN}$ [54]. Such a situation
could be realised experimentally by using controlled heating and cooling of
regions of a substrate coated in a nematic film [31, 63].

The range of anisotropic wetting and dewetting phenomena occurring in this
nematic system may also be useful from a technological perspective, for
instance, for tailored dewetting of liquid films, as discussed in Section 1
[2, 18, 6, 41, 42]. The variety of possible transitions between two-
dimensional equilibrium states will have similar forms in three dimensions,
which may be relevant to applications such as the One Drop Filling method of
LCD manufacturing [64, 65, 66] and adaptive-lens technologies [4, 5]. In order
to explore such applications, further theoretical investigations, particularly
into the dynamics of transitions, and experimental investigations would be
needed.

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