# Shape Bistability in Squeezed Chromonic Droplets

Silvia Paparini<EMAIL_ADDRESS>Epifanio G. Virga<EMAIL_ADDRESS>Dipartimento di Matematica, Università di Pavia, Via Ferrata 5, 27100 Pavia,
Italy

(August 27, 2024)

###### Abstract

We study droplets of chromonic liquid crystals squeezed between parallel
plates inducing degenerate tangential anchoring on the nematic director. In
the coexistence regime, where droplets in the nematic phase are at equilibrium
with the surrounding melt, our two-dimensional theoretical model predicts a
regime of shape bistability for sufficiently large bipolar droplets, where
_tactoids_ (pointed shapes) and _discoids_ (smooth shapes) coexist in
equilibrium. This phenomenon has not yet been observed. Two-dimensional
droplets of disodium cromoglycate (DSCG) have been the object of a thorough
experimental study [Y.-K. Kim _et al._ , J. Phys.: Condens. Matter 25, 404202
(2013)]. We show that our theory is in good quantitative agreement with these
data and extract from them what promises to be a more accurate estimate for
the isotropic surface tension at the nematic/melt interface of DSCG.

##  I Introduction

Chromonic liquid crystals (CLC), also called _chromonics_ , form a very
peculiar class of liquid crystals (LC). These latter are systems constituted
by basic _mesogens_ (either single molecules or molecular assemblies), which
possess orientational ordering, and sometimes—under appropriate
circumstances—also positional ordering. Basically, there exist two classes of
LCs: _thermotropic_ LCs, in which the ordering process is temperature-driven,
and _lyotropic_ LCs, in which the mesogenic units are molecular assemblies in
a solvent, and concentration is the driving parameter for ordering. These
latter are found in detergents and living organisms alike [1].

In particular, CLCs are lyotropics formed by certain dyes, drugs, and short
nucleic-acid oligomers in aqueous solutions [2, 3, 4, 5, 6, 7] Since most
biological processes take function normally in these types of solutions, it is
no wonder that interest in CLCs has recently surged for possible applications
in medical sciences. But this is not the only reason that makes them special
(or rather unique). A number of informative, updated reviews are available on
this topic [8, 9, 10, 2, 11]; they are all highly recommended.

In ordinary lyotropic LCs, the aggregated molecules are bonded so strongly
that neither size nor shape of the assemblies change appreciably upon varying
moderately both temperature and concentration. Quite to the contrary, in CLCs,
aggregation starts at very low concentrations.

CLC molecules are typically plank-shaped with aromatic cores and polar groups
on their peripheries [12]. They tend to stack face-to-face so as to form
columns that order into a fluid nematic phase or (for higher concentrations or
lower temperatures) even in a solid-like medium phase, where columns are
organized parallel to one another with their centres arranged in an hexagonal
pattern [2]. The former phase is just the analogue of the nematic phase in
ordinary thermotropics, whilst the latter is the analogue of the columnar
phase in discotic thermotropics. It is the variability in size and shape of
the supra-molecular columns that makes CLCs so unique. There is also a wide
variety of situations in the molecular organization: some are stacks of single
molecules, others have more than a molecule in their cross-sections [1]. These
might seem to be minor details, but they may have momentous consequences at
the macroscopic scale [13].

The aggregation process is often called _isodesmic_ , because the energy gain
in adding a unit to a preexisting column (typically between $5$ and $10$ $kT$)
does not depend on the length of the column. The isodesmic nature of the
process results in a broad length column distribution, which is prone to the
action of temperature. When the temperature is increased, the concentration of
longer assemblies decreases. This is reflected by the elastic properties of
the phase, in a way that ordinary lyotropics do not exhibit [12]. Further
increasing the temperature results into a first order nematic-isotropic
transition with a wide coexistence region ($5$-$10\,^{\circ}\mathrm{C}$).
Conversely, when the temperature is decreased, short, disordered columns in
the isotropic phase tend to grow and aggregate, eventually separating from the
parent isotropic melt to form islands of ordered phase.

As customary, the nematic phase is described by the extra director field
$\bm{n}$, which represents at the macroscopic level the local orientation of
the supra-molecular rod-like aggregates. At the nematic/isotropic interface,
an orientation-dependent surface tension arises that favours the (degenerate)
tangential orientation of $\bm{n}$, in agreement with the purely entropic
argument of Onsager [14], according to which a tangential anchoring would
enhance the local translational freedom of columns at the interface.

We shall only be concerned with a two-dimensional problem, inspired by the
experimental setting explored by Kim et al. [15]. There, CLC droplets in the
nematic phase appeared surrounded by the isotropic phase, their shape being
the characteristic _tactoids_ (spindle-like) that were observed in aqueous
dispersions of tobacco mosaic viruses since the seminal paper by Bernal and
Fankuchen [16]. Most reported droplets were _bipolar_ , with point defects of
$\bm{n}$ at the pointed poles.

In this paper, we adapt to the two-dimensional setting our theory for the
representation of tactoids recently presented in [17]. We shall achieve two
main results.

First, we predict a _shape bistability_ , which seems characteristic of the
two-dimensional setting. We identify a range of droplet’s areas where two
distinct shapes could be observed, one tactoidal, as expected, and the other
_discoidal_ (smooth). A sort of _shape_ coexistence thus parallels the phase
coexistence observed in these materials. This regime manifests itself for
droplets larger than those reported in [15]; to our knowledge, it has not yet
been observed.

Second, we use the very detailed data of [15] to compare the observed shapes
with those predicted by our theory. We extract an estimate for the isotropic
component of the surface tension at the droplet’s interface, which turns out
to comparable in order of magnitude to the typical values measured for
standard thermotropic materials ($\sim 10^{-5}\,\mathrm{J/m}^{2}$, see [18, p.
495]).

The paper is organized as follows. In Sec. II, we recall from [17] our theory
and adapt it to the present two-dimensional setting. The optimal shapes of
bipolar droplets that minimize the total free-energy functional are derived
and discussed in Sec. III, where we illustrate in detail the bistability
scenario that we envision. We show, in particular, how the critical values of
the droplet’s area that delimit the corresponding shape hysteresis depend on
the elastic constants of the material. Section IV is devoted to the comparison
with the experimental study [15] that has motivated this work. We contrast our
predicted shapes to the observed ones and, encouraged by their agreement, we
estimate the isotropic component of the surface tension. Finally, in Sec. V,
we summarize our conclusions and comment on possible further extensions of our
study. The paper is closed by two mathematical appendices, where we collect
computational details and auxiliary results needed in the main text, but
inessential to its comprehension.

##  II Two-dimensional setting

Here, we set our theoretical scene; we shall first recall the energetics of a
CLC drop squeezed between two parallel plates and we shall then describe both
its outer profile and inner direction field.

###  II.1 Energetics

In classical liquid crystal theory, the nematic director field $\bm{n}$
describes the average of the molecules that constitute the material. The
elastic distortions of $\bm{n}$ are locally measured by its gradient
$\nabla\bm{n}$, which may become singular where the director exhibits
_defects_ , that is, discontinuities in the field $\bm{n}$. The bulk free
energy is the following functional,

$\mathscr{F}_{\mathrm{b}}[\mathscr{B},\bm{n}]:=\int_{\mathscr{B}}f_{\mathrm{OF}}(\bm{n},\nabla\bm{n})\operatorname{d}\\!V,$
(1)

where $\mathscr{B}$ is the region occupied by the material,
$\operatorname{d}\\!V$ is the volume element, and $f_{\mathrm{OF}}$ is the
Oseen-Frank free-energy density. Letting the latter be the most general frame-
indifferent function, even in $\bm{n}$ and quadratic in $\nabla\bm{n}$, one
arrives at (see, e.g., [19, Ch. 3])

$f_{\mathrm{OF}}:=\frac{1}{2}K_{11}(\operatorname{div}\bm{n})^{2}+\frac{1}{2}K_{22}(\bm{n}\cdot\operatorname{curl}\bm{n})^{2}+\frac{1}{2}K_{33}|\bm{n}\times\operatorname{curl}\bm{n}|^{2}+K_{24}\big{(}\operatorname{tr}(\nabla\bm{n})^{2}-(\operatorname{div}\bm{n})^{2}\big{)},$
(2)

where $K_{11}$, $K_{22}$, $K_{33}$, and $K_{24}$ are the _splay_ , _twist_ ,
_bend_ , and _saddle-splay_ elastic constants, respectively. They are material
moduli characteristic of each nematic liquid crystal, corresponding to four
independent orientation fields, each igniting a single distinctive term in
(2).111Recently, an equivalent modal decomposition has been put forward for
$f_{\mathrm{OF}}$ [20], which has also been given a graphical representation
in terms of an _octupolar_ tensor [21].

The energy $f_{\mathrm{OF}}$ is meant to penalize all distortions of $\bm{n}$
away from the unifom alignment (in whatever direction); to this end, the
elastic constants in (2) must satisfy Ericksen’s inequalities [22],

$K_{11}\geqq K_{24}\geqq 0,\quad K_{22}\geqq K_{24}\geqq 0,\quad K_{33}\geqq
0.$ (3)

CLCs in three-dimensional space exhibit a different behavior: in cylindrical
capillary tubes subject to degenerate tangential boundary conditions,222That
is, when $\bm{n}$ is bound to be tangent to the boundary, but oriented in any
direction. the director $\bm{n}$ has been seen to steer away from the uniform
alignment along the cylinder’s axis [23, 24, 25, 26, 27]. Two symmetric
twisted configurations (left- and right-handed) have been observed in
capillaries, each variant occurring with the same likelihood, as was to be
expected from the lack of chirality in the molecular aggregates that
constitute these materials.

Despite the clear indication that CLC’s ground state differs from the uniform
alignment presumed in $f_{\mathrm{OF}}$, the Oseen-Frank theory has been
applied to rationalize the experiments with capillary tubes [23, 24, 25, 26],
at the cost of taking $K_{22}<K_{24}$, which violates one of Ericksen’s
inequalities (3). Since such a violation would make $f_{\mathrm{OF}}$
unbounded below, the legitimacy of these theoretical treatments is threatened
by a number of mathematical issues that will be tackled elsewhere [28]. Here,
we can spare this issue, as our setting is two-dimensional. The region
$\mathscr{B}$ will be a thin blob representing a droplet of CLC surrounded by
its isotropic melt, squeezed between two parallel plates at the distance $h$
from one another.

Formally, $\mathscr{B}=\mathscr{R}\times[-\frac{h}{2},\frac{h}{2}]$, where
$\mathscr{R}$ is a region with piecewise smooth boundary $\partial\mathscr{R}$
in the $(x,y)$ plane of a Cartesian frame $(\bm{e}_{x},\bm{e}_{y},\bm{e}_{z})$
(see Fig. 1).

Figure 1: Two-dimensional domain $\mathscr{R}$ with area $A_{0}$ in the
$(x,y)$ plane. It represents the cross-section of a drop squeezed between two
parallel substrates $h$ apart and surrounded laterally by the isotropic phase.
In accord with the experimental observations in [15], both shape and director
field are uniform across the gap of thickness $h$ along $\bm{e}_{z}$.

The field $\bm{n}$ lies in the same plane as $\mathscr{R}$ and it is thought
of as being extended uniformly through the blob’s thickness. For a _planar_
field of this sort, the twist term in (2) vanishes identically, as
$\operatorname{curl}\bm{n}\parallel\bm{e}_{z}$, and so the twist constant
$K_{22}$ will play no role here (and those Ericksen’s inequalities that
involve it can altogether be ignored). Thus, leaving aside the possibly
controversial issue arising in general from the violation of one Ericksen’s
inequality, we remain assured that in the present setting the Oseen-Frank
energy-density $f_{\mathrm{OF}}$ is bounded below and the free energy
$\mathscr{F}_{\mathrm{b}}$ can be safely minimized.

We assume that a given mass of material constitute the blob $\mathscr{B}$, and
so its volume $V_{0}$ is prescribed by the incompressibility constraint (which
is satisfied to a large degree of approximation). Consequently, the area
$A_{0}$ of $\mathscr{R}$ is prescribed, as $V_{0}=A_{0}h$. We further assume
that the plates squeezing both $\mathscr{B}$ and the surrounding melt exert a
degenerate tangential anchoring on the nematic director $\bm{n}$, so that, in
light of the constraint on the area of $\mathscr{R}$, the additional anchoring
energy can be treated as an inessential additive constant.

This is not the case for the surface energy $\mathscr{F}_{\mathrm{s}}$ at the
interface between $\mathscr{B}$ and the surrounding melt. Here we assume that
this energy is represented by the Rapini-Papoular formula [29],

$\mathscr{F}_{\mathrm{s}}[\mathscr{R},\bm{n}]=h\int_{\partial\mathscr{R}}\gamma(1+\omega(\bm{n}\cdot\bm{\nu})^{2})\operatorname{d}\\!A,$
(4)

where $\gamma>0$ is the _isotropic_ surface tension of the liquid crystal in
contact with its melt, $\omega$ is a dimensionless _anchoring strength_ ,
which we take to satisfy $\omega\geqq 0$, $\bm{\nu}$ is the outer unit normal
to $\partial\mathscr{R}$, and $\operatorname{d}\\!A$ is the area element. Thus
$\mathscr{F}_{\mathrm{s}}$ is minimized when $\bm{n}$ lies tangent to
$\partial\mathscr{R}$. Henceforth, we shall enforce this minimum requirement
as a constraint on $\bm{n}$,

$\bm{n}\cdot\bm{\nu}\equiv 0,\quad\hbox{on}\quad\partial\mathscr{R},$ (5)

save checking ultimately this assumption with appropriate energy comparisons
(see Appendix B) which ensures that such a _tangential anchoring_ is not
broken. In short, the validity of (5) requires that the droplet is not too
_small_ in a sense that will be made precise below.

To sum up, the total free energy
$\mathscr{F}:=\mathscr{F}_{\mathrm{b}}+\mathscr{F}_{\mathrm{s}}$ of the system
is given by the functional

$\mathscr{F}[\mathscr{R},\bm{n}]:=h\left(\int_{\mathscr{R}}f_{\mathrm{OF}}\operatorname{d}\\!A+\gamma\ell(\partial\mathscr{R})\right),$
(6)

subject to (5) and to the isoperimetric constraint

$A(\mathscr{R})=A_{0},$ (7)

where $A$ and $\ell$ are the area and length measures, respectively.

###  II.2 Admissible Shapes

No preferred direction is present on the substrates that squeeze the drop, and
so, contemplating tactoids among the possible equilibrium shapes of
$\mathscr{R}$ (as well as other smooth shapes), we assume that these are
mirror symmetric about two orthogonal axes, one joining the possible sharp
tips of the boundary $\partial\mathscr{R}$. We denote by $y$ the latter axis
and by $x$ the orthogonal symmetry axis.333Clearly, due to the absence of a
privileged orientation on the bounding substrates, the $y$ axis could indeed
be oriented in any direction. If several drops are present, their axes would
be isotropically distributed. Thus, only half of the curve that bounds
$\mathscr{R}$ needs to be described, the other half being obtained by mirror
symmetry. We take this curve to be represented as $x=R(y)$, where $y$ ranges
in the interval $[-R_{0},R_{0}]$, with $R_{0}$ to be determined, and $R$ is a
smooth, even function such that

$R(R_{0})=0\quad\text{and}\quad R^{\prime}(0)=0,$ (8)

where a prime ′ denotes differentiation with respect to $y$ (see Fig. 2).

Figure 2: The function $R(y)$ and its retraction $R_{t}(y)=g(t)R(y)$ represent
(half) the boundary $\partial\mathscr{R}$ and retracted inner curves
$\partial\mathscr{R}_{t}$ for generic $t\in[0,1]$. The nematic director
$\bm{n}$ is the unit vector field everywhere tangent to the retracted curves;
$\bm{n}_{\perp}=\bm{e}_{z}\times\bm{n}$ is the orthogonal field, and
$\bm{\nu}$ is the outer unit normal to $\partial\mathscr{R}$, where
$\bm{\nu}=-\bm{n}_{\perp}$. The function $R$ represents half a drop; the other
half is obtained by mirror symmetry about the $y$ axis.

The points where $R$ vanishes correspond to the _poles_ of $\mathscr{R}$.
Whenever $R^{\prime}(R_{0})$ is finite, $R$ represents a _tactoid_ , as the
outer unit normal $\bm{\nu}$ to $\partial\mathscr{R}$ is discontinuous at the
poles. Smooth shapes correspond to $R^{\prime}(R_{0})=-\infty$.

We call $R_{\mathrm{e}}$ the radius of the _equivalent_ disc, which has area
$A_{0}$, and we denote by $\mu$ the dimensionless length of the semi-axis of
the drop,

$\mu:=\frac{R_{0}}{R_{\mathrm{e}}}.$ (9)

Hereafter, we shall rescale all lengths to $R_{\mathrm{e}}$ (while keeping
their names unchanged, to avoid typographical clutter). With this
normalization, the area constraint (7) reads simply as

$\int_{-\mu}^{\mu}R(y)\operatorname{d}\\!z=\frac{\pi}{2}.$ (10)

###  II.3 Director Retraction

The shape of $\mathscr{R}$ is unknown and needs to be determined. Since
$\bm{n}$ is tangent to $\partial\mathscr{R}$, following [17] we device a
method that also derives $\bm{n}$ inside $\mathscr{R}$ from the knowledge of
$\partial\mathscr{R}$, thus reducing the total free energy $\mathscr{F}$ to a
pure _shape_ functional. This is achieved by _retracting_
$\partial\mathscr{R}$ inside $\mathscr{R}$ with its tangent field $\bm{n}$.

Formally, we define a function $R_{t}(y)=g(t)R(y)$, where $t\in[0,1]$ and $g$
is an increasing monotonic function such that $g(0)=0$ and $g(1)=1$. The graph
of $R_{t}$, shown in Fig. 2, represents the _retraction_ of
$\partial\mathscr{R}$ that borders an inner domain
$\mathscr{R}_{t}\subseteq\mathscr{R}$. All domains $\mathscr{R}_{t}$ are
nested one inside the other as $t$ decreases towards $0$. For $t=0$,
$\mathscr{R}_{t}$ reduces to the $y$ axis. The advantage of this method is
that it also affords to describe the inner director $\bm{n}$ as the field
everywhere tangent to the family of curves $\partial\mathscr{R}_{t}$. All
director fields obtained by this geometric construction are _bipolar_ , in
that they have two point defects at the poles; in the language of Mermin [30],
they are _boojums_ with topological charge $m=+1$ (see also [18, p. 501]).

It is shown in Appendix A how to compute the area element
$\operatorname{d}\\!A$ in the $(t,y)$ coordinates and how to express
$\nabla\bm{n}$ in the orthonormal frame $(\bm{n},\bm{n}_{\perp})$, where
$\bm{n}_{\perp}:=\bm{e}_{z}\times\bm{n}$, in terms of the functions $R(y)$ and
$g(t)$.

In the rescaled variables $y$ and $R(y)$, an appropriate dimensionless form of
$\mathscr{F}$ in (6) is then given by

$\displaystyle
F[\mu;R]:=\frac{\mathscr{F}[\mathscr{B}]}{K_{11}h}=\int_{-\mu}^{\mu}$
$\displaystyle\left\\{\left[\frac{R^{\prime}}{R}-\dfrac{R^{\prime\prime}}{R^{\prime}}+\frac{1}{8}\dfrac{RR^{\prime\prime
2}}{R^{\prime 3}}\left(3+k_{3}\right)\right]\arctan
R^{\prime}+\frac{R^{\prime\prime}}{1+R^{\prime 2}}+\right.$
$\displaystyle\left.+\frac{1}{8}\frac{RR^{\prime\prime 2}}{(1+R^{\prime
2})^{2}}\left[(k_{3}-5)-\frac{1}{R^{\prime
2}}(k_{3}+3)\right]\right\\}+2\alpha\sqrt{1+R^{\prime
2}}\operatorname{d}\\!y,$ (11)

where

$k_{3}:=\frac{K_{33}}{K_{11}}$ (12)

is a _reduced bend_ constant and

$\alpha:=\frac{\gamma R_{\mathrm{e}}}{K_{11}}$ (13)

is a _reduced area_. Equivalently, $\alpha=R_{\mathrm{e}}/\xi_{\mathrm{e}}$,
where $\xi_{\mathrm{e}}$ is the de Gennes-Kleman _extrapolation length_ [18,
p. 159].444Thus, when we say that a drop is either small or large, we mean
precisely that either $\alpha\ll 1$ or $\alpha\gg 1$, respectively.

The reduced functional $F$ in (II.3) suffers from a typical pathology of two-
dimensional director theory: it diverges logarithmically to $+\infty$ near
defects. Here, the culprit is the integrand $\frac{R^{\prime}}{R}\arctan
R^{\prime}$, which is not integrable at $y=\pm\mu$. Following a well
established practice (see, e.g.,[31, p. 171]), we imagine that the energy
concentration near defects causes a localized transition to the isotropic
phase, which constitutes a _defect core_ (whose fine structure is better
explored within Ericksen’s theory [32]). The energy associated with such a
phase transition is proportional to the core’s area and will be taken as
approximately fixed. Moreover, for simplicity, instead of considering a
circular core, which in the most common choice, we take it in the shape of the
tapering drop’s tip. Letting $r_{\mathrm{c}}$ denote the core’s size, we set
$r_{\mathrm{c}}=\varepsilon R_{\mathrm{e}}$ and restrict $y$ to the interval
$[-\eta,+\eta]$, where $\eta$ is defined by

$R(\eta)=R(-\eta)=\varepsilon,$ (14)

and so depends indirectly on $\varepsilon$ (see Fig. 3).

Figure 3: Isotropic defect core (in red) at the tip of a drop (only a quarter
is shown). In our dimensionless units (rescaled to $R_{\mathrm{e}}$), $\eta$
is defined by (14). A sensible value for the parameter $\varepsilon$, which
here is out of scale, is $\varepsilon\approx 10^{-3}$. A similar construction
also applies to a drop with smooth poles (not shown here).

For $R_{\mathrm{e}}$ of the order of $10\mu\mathrm{m}$, it is reasonable to
take $\varepsilon\approx 10^{-3}$, as we shall do here, which corresponds to
$r_{\mathrm{c}}$ of the order of $10\mathrm{nm}$. The integral in (II.3) will
hereafter be limited to the interval $[-\eta,+\eta]$, so that it will always
converge. The extra energy stored in the defects, being approximately
constant, will play no role in our quest for the equilibrium shape of squeezed
drops.

###  II.4 Special Family of Shapes

Here, we follow closely [17], albeit in a two-dimensional setting, in an
attempt to restrict the admissible shapes of drops to a special family
amenable to a simple analytical treatment. The admissible drop profiles will
be described by the function555For $b=0$, $R$ in (15) reduces to he parabolic
profile considered in [33].

$R(y)=a(\mu^{2}-y^{2})+b\sqrt{\mu^{2}-y^{2}},$ (15)

where $a$ and $b$ are real parameters that must be chosen subject to the
requirements that $R(y)\geqq 0$ for all $-\mu\leqq y\leqq\mu$ and that (10) is
satisfied. It is a simple matter to show (see also [17]) that $a$ and $b$ can
be expressed in terms of the free parameters $(\phi,\mu)$ that span the
configuration space $\mathsf{S}:=\\{(\phi,\mu):0\leqq\phi\leqq\frac{3\pi}{4},\
\mu>0\\}$. Precisely,

$a=\frac{1}{\mu^{3}}\frac{\pi\cos\phi}{h(\phi)},\quad
b=\frac{1}{\mu^{2}}\frac{\pi\sin\phi}{h(\phi)}\quad\text{with}\quad
h(\phi):=\frac{8}{3}\cos\phi+\pi\sin\phi>0,\quad
0\leqq\phi\leqq\frac{3\pi}{4}.$ (16)

Shapes with different qualitative features correspond to different regions of
$\mathsf{S}$, as illustrated in Fig. 4. _Prolate_ shapes are characterized by

$\mu\geqq\varpi(\phi):=\sqrt{\frac{\pi(\cos\phi+\sin\phi)}{h(\phi)}},$ (17)

whereas _oblate_ shapes are characterized by $\mu<\varpi(\phi)$. Moreover,
shapes represented by (16) are convex for
$0\leqq\phi\leqq\phi_{\mathrm{c}}:=\operatorname{arccot}\left(-\frac{1}{2}\right)\doteq
2.03$ and concave for $\phi_{\mathrm{c}}<\phi\leqq\frac{3\pi}{4}$. The latter
are represented by the red strip in Fig. 4;

Figure 4: Configuration space with the admissible shapes described by (15).
The blue region below the graph of the function $\varpi(\phi)$ in (17)
represents all prolate shapes. The red strip for
$\phi_{\mathrm{c}}\leqq\phi\leqq\frac{3\pi}{4}$ represents the concave shapes
that we call _butterflies_ ; all shapes falling on the left of this strip are
convex. The circular disc is represented by the point $(\frac{\pi}{2},1)$,
marked by a red circle. We also call _tactoids_ the shapes for
$0\leqq\phi\leqq\frac{\pi}{16}$ (_genuine_ tactoids, only those for $\phi=0$,
marked by a red line), _discoids_ those for
$\frac{\pi}{16}\leqq\phi\leqq\frac{9\pi}{16}$, and _bâtonnet_ those for
$\frac{9\pi}{16}\leqq\phi\leqq\phi_{\mathrm{c}}$, see also Table 1, and Fig. 5
for a fuller gallery of shapes. The barriers marking transitions from one
family of shapes to another are represented by vertical dashed lines.

we call them _butterflies_ : their waist narrows as $\phi$ approaches the
boundary of $\mathsf{S}$ at $\phi=\frac{3\pi}{4}$, where it vanishes
altogether and the droplet splits in two.

$\mathscr{R}$ has pointed tips only whenever $R^{\prime}(\mu)$ is finite;
according to (16), the only value of $\phi$ that makes $b$ vanish is $\phi=0$.
We call _genuine_ these tactoids. For small enough values of $\phi$ the shape
represented by (15) via (16) cannot be visually distinguished from pointed
tactoids; we find the conventional barrier at $\phi=\frac{\pi}{16}$
appropriate to delimit the realm of tactoids (genuine or not). Further
increasing $\phi$, $\mathscr{R}$ has tips that look smoother, justifying our
calling them _discoids_. A conventional barrier is set at
$\phi=\frac{9\pi}{16}$ to mark where discoids evolve into little batons, for
which we use the French word _bâtonnet_. The names used to identify different
shapes of $\mathscr{R}$ and the corresponding strips in $\mathsf{S}$ where
they are found are recalled in Table 1.

Table 1: We identify four strips in configuration space $\mathsf{S}$, which correspond to four qualitatively different shapes for a region $\mathscr{R}$ represented by (15) via (16). Here $\phi_{\mathrm{c}}=\operatorname{arccot}\left(-\frac{1}{2}\right)\doteq 2.03$. Tactoids | Discoids | Bâtonnet | Butterflies
---|---|---|---
$0\leqq\phi<\dfrac{\pi}{16}$ | $\dfrac{\pi}{16}<\phi<\dfrac{9\pi}{16}$ | $\dfrac{9\pi}{16}<\phi<\phi_{\mathrm{c}}$ | $\phi_{\mathrm{c}}<\phi\leqq\dfrac{3\pi}{4}$

Fig. 5 illustrates a gallery of shapes for $\mathscr{R}$ obtained from (15)
and (16) for $\mu=1$ and a number of values of $\phi$ falling in the different
types listed in Table 1.

(a) $\phi=0$
genuine tactoid

(b) $\phi=\frac{\pi}{32}$
tactoid

(c) $\phi=\frac{\pi}{4}$
discoid

(d) $\phi=\frac{10\pi}{16}$
bâtonnet

(e) $\phi=\frac{11\pi}{16}$
butterfly

Figure 5: Gallery of shapes illustrating for $\mu=1$ the taxonomy introduced
in Table 1. The color coding of the shapes is the same used in Fig. 4.

##  III Optimal Shapes

This section is devoted to the study of the minimizers of the free-energy
functional in (II.3). Our major result will be the prediction of a shape
bistability, which has not yet been observed in this setting. Before
describing this phenomenon, however, we need to make sure that the droplets we
consider are not too small for the tangential anchoring condition in (5) to be
valid.

###  III.1 Admissible Drop Sizes

It is known [34] that drops sufficiently small in three-dimensional space tend
to break the director’s tangential anchoring on their boundary, favouring the
uniform alignment of $\bm{n}$ in their bulk. A simple heuristic argument tells
us that a similar breaking would also take place in the present two-
dimensional setting.

The elastic cost of the bulk deformation scales as $Kh$, where $K$ is a
typical elastic constant, while the surface anchoring energy scales as
$\gamma\omega hR_{\mathrm{e}}$, and so $\alpha$ (13) estimates the ratio of
the latter to the former. Thus, when $\alpha$ is sufficiently small, the bulk
energy becomes dominant and it is minimized by the uniform alignment of
$\bm{n}$, which breaks the tangential anchoring, undermining our analysis. In
Appendix B, we perform an energy comparison that provides an estimate for the
_safeguard_ vale $\alpha_{\mathrm{s}}$ of $\alpha$, above which tangential
anchoring is expected to remain unbroken. We obtained the following explicit
formula for $\alpha_{\mathrm{s}}$,

$\alpha_{\mathrm{s}}:=\frac{\frac{\pi}{4}\left(k_{3}-1-k_{3}\ln
2-\ln\varepsilon\right)+\varepsilon}{j(\omega)-\left(\frac{\pi}{2}-\varepsilon\right)-\frac{\omega}{2}\pi\varepsilon},$
(18)

where $j(\omega)$ is the function defined in (42). For $\omega=5$, which is a
choice supported by some evidence,666See, for example, [35] and [15].
$\alpha_{\mathrm{s}}\approx 0.2k_{3}-0.5(1+\ln\varepsilon)$. In particular,
for $k_{3}=1$ and $\varepsilon=10^{-3}$, $\alpha_{s}\approx 3$, which will be
our reference choice henceforth. Taking $K\sim 1$-$10\,\mathrm{pN}$ as typical
value for all elastic constants777This estimate is supported for example by
[36] for material such as DSCG, SSY, and PBG. and $\gamma\sim
10^{-5}\,\mathrm{J/m}^{2}$ as typical value for the surface tension of a
chromonic liquid crystal in contact with its melt,888Discordant estimates of
$\gamma$ have been given in the literature [15, 37, 38]. Here we take the
average order of magnitude found in these works (see also Sec. IV below for an
independent justification of this choice). by (13) taking $\alpha>3$ means
taking $R_{\mathrm{e}}\gtrsim 0.3$-$3\,\mu\mathrm{m}$, which provides a lower
bound on the admissible size of the droplets that can be treated within our
theory.

###  III.2 Shape Bistability

Finding analytically the minima of the _reduced_ free energy
$F_{\alpha}(\phi,\mu)$, the function defined on the configuration space
$\mathsf{S}$ by computing the functional $F\left[\mu;R\right]$ in (II.3) on
the special family of shapes in (15), is simply impracticable. Thus, for a
given choice of the elastic parameter $k_{3}$, we evaluated numerically
$F_{\alpha}$ and we sought its minimizers in $\mathsf{S}$ for increasing
$\alpha>\alpha_{\mathrm{s}}$.

We found out that there are two critical values of $\alpha$, $\alpha_{1}$ and
$\alpha_{2}>\alpha_{1}$, such that for either $\alpha<\alpha_{1}$ or
$\alpha>\alpha_{2}$, $F_{\alpha}$ attains a single (absolute) minimum in
$\mathsf{S}$, whereas it attains two (relative) minima for
$\alpha_{1}\leqq\alpha\leqq\alpha_{2}$. There is a third critical value,
$\alpha_{\mathrm{b}}\in(\alpha_{1},\alpha_{2})$, such that for
$\alpha=\alpha_{\mathrm{b}}$ the two minima of $F_{\alpha}$ are equal and its
absolute minimizer abruptly shifts from one point in $\mathsf{S}$ to another.
Were the points of $\mathsf{S}$ to represent the different phases of a
condensed system, this scenario would be described as a (first-order) _phase
transition_. In our setting, it more simply describes the (local) stability of
two equilibrium shapes for a squeezed drop: for
$\alpha_{1}<\alpha<\alpha_{2}$, both a tactoid and a discoid are local energy
minimzers, the global minimum shifting from the former to the latter at
$\alpha=\alpha_{\mathrm{b}}$. For $\alpha<\alpha_{1}$, the only equilibrium
shape is a tactoid, whereas it is a discoid for $\alpha>\alpha_{2}$. The
_shape bistability_ exhibited by this two-dimensional system will now be
documented in more details.

We start by representing the equilibrium landscape in the language of
bifurcation theory. Taking $\alpha$ as bifurcation parameter and $\phi$ as
equilibrium shape representative, in Fig. 6

Figure 6: Bifurcation diagram for the parameter $\phi$ that describes the
equilibrium profile (15) as a function of the bifurcation parameter $\alpha$.
For every $k_{3}$, there are three critical values of $\alpha$, namely,
$\alpha_{1}$, $\alpha_{\mathrm{b}}$, and $\alpha_{2}$. Solid lines represent
global minima, while broken lines represent local minima. Open circles mark
the appearance or disappearance of a local minimum at $\alpha=\alpha_{1}$ and
$\alpha=\alpha_{2}$. Two green lines delimit the coexistence interval
$[\alpha_{1},\alpha_{2}]$. At $\alpha=\alpha_{\mathrm{b}}$, both local minima
are also global minima; two black dots mark the jump of the global minimum
from one branch to the other. Here $k_{3}=1$, $\alpha_{1}=172$,
$\alpha_{\mathrm{b}}=221$, and $\alpha_{2}=305$. The droplet’s equilibrium
shape is drawn for the critical values $\alpha_{1}$ and $\alpha_{2}$.

we illustrate the minima of $F_{\alpha}$ for $k_{3}=1$: one falls in
$0<\phi<\frac{\pi}{16}$ (blue line), and so it is a tactoid, while the other
falls in $\frac{\pi}{16}<\phi<\frac{\pi}{2}$ (red line) and is a discoid.
Solid lines represent global minima, while broken lines represent local
minima. Two separate local minima are also global minima for
$\alpha=\alpha_{\mathrm{b}}$, where a perfect bistability is established
between the two equilibrium branches. The tacoidal branch can be further
continued, as it is locally stable, until $\alpha$ reaches the critical value
$\alpha_{2}$, where it ceases to exist altogether. Similarly, as soon as
$\alpha$ exceeds $\alpha_{1}$, the discoidal branch comes first into life as a
locally stable equilibrium, which then becomes globally stable for
$\alpha>\alpha_{\mathrm{b}}$. For $\alpha\in[\alpha_{1},\alpha_{2}]$, tactoids
and discoids coexist as optimal shapes; both are metastable, one or the other
is globally stable, according to whether $\alpha<\alpha_{\mathrm{b}}$ or
$\alpha>\alpha_{\mathrm{b}}$.

The dimensionless parameter $\alpha$ is ultimately related through (13) to the
amount of material trapped in the drop. So the coexistence interval
$[\alpha_{1},\alpha_{2}]$ corresponds to a window of areas $A_{0}$ for which
two different shapes could be observed, more likely (and frequently) the one
corresponding to the global minimum. If, ideally, one could gently pump
material into a tactoidal drop, so as to follow the blue branch in Fig. 6, the
drop would continue to grow as a tactoid until the critical volume corresponds
to $\alpha_{2}$, where a dynamical instability would presumably prompt the
transition towards a discoid. Conversely, if material could be gently removed
from the latter, it would keep its discoidal shape until the critical volume
corresponds to $\alpha_{1}$, where it would dynamically evolve into a tactoid.
The green lines in Fig. 6 delimit such a hysteresis loop.

Figure 7 illustrates the energy landscape for $\alpha<\alpha_{1}$;

(a) Graph of $F_{\alpha}$ against $\mathsf{S}$; the red dot designates its
single minimum.

(b) Contour plot of $F_{\alpha}$. The minimum is attained for $\phi\doteq
0.01$ and $\mu\doteq 1.11$ (red circle).

(c) Equilibrium tactoidal shape corresponding through (15) to the minimizer of
$F_{\alpha}$.

Figure 7: For $\alpha<\alpha_{1}$, the reduced free energy $F_{\alpha}$ is
convex on the configuration space $\mathsf{S}$ and attains a single minimum in
$0<\phi<\frac{\pi}{16}$. Here, $k_{3}=1$, $\alpha=170$, $\alpha_{1}=172$.

$F_{\alpha}$ is convex, and so it attains a single minimum, which falls in
$0<\phi<\frac{\pi}{16}$, representing a tactoid. The scene changes in Fig. 8,
where $\alpha=\alpha_{\mathrm{b}}$ and $F_{\alpha}$ attains two equal minima,
corresponding to a tactoid and a discoid.

(a) Graph of $F_{\alpha}$ against $\mathsf{S}$. $F_{\alpha}$ is no longer
convex: it attains two equal minima designated by red dots.

(b) Contour plot of $F_{\alpha}$. One minimum is attained for $\phi\doteq
0.025$ and $\mu\doteq 1.11$ (tactoid), while the other is attained for
$\phi\doteq 1.2$ and $\mu\doteq 1.04$ (doscoid); both are marked by red
circles.

(c) Tactoidal equilibrium shape.

(d) Discoidal equilibrium shape.

Figure 8: For $\alpha=\alpha_{\mathrm{b}}$, the reduced free energy
$F_{\alpha}$ has two (equal) global minima, one for $0<\phi<\frac{\pi}{16}$
(tactoid) and the other for $\frac{\pi}{16}<\phi<\frac{\pi}{2}$ (discoid).
Here, $k_{3}=1$ and $\alpha_{\mathrm{b}}=221$.

For $\alpha>\alpha_{2}$, $F_{\alpha}$ is again convex, with a single minimum
on a discoid,

(a) Graph of $F_{\alpha}$ against $\mathsf{S}$; the red dot designates its
single minimum.

(b) Contour plot of $F_{\alpha}$. The minimum is attained for $\phi\doteq
1.34$ and $\mu\doteq 1.02$ (red circle).

(c) Equilibrium discoidal shape corresponding through (15) to the minimizer of
$F_{\alpha}$.

Figure 9: For $\alpha>\alpha_{2}$, the reduced free energy $F_{\alpha}$ is
gain convex on the configuration space $\mathsf{S}$ and attains a single
minimum for $\frac{\pi}{16}<\phi<\frac{\pi}{2}$. Here, $k_{3}=1$,
$\alpha=307$, $\alpha_{2}=305$.

which gets closer and closer to the round disc, represented by the point
$(\frac{\pi}{2},1)$ in $\mathsf{S}$, as $\alpha$ grows indefinitely.

We have computed the three critical values, $\alpha_{1}$,
$\alpha_{\mathrm{b}}$, and $\alpha_{2}$ for several values of $k_{3}$. Fig. 10
shows that, to within a good approximation, they all grow linearly with
$k_{3}$.

Figure 10: Critical values of $\alpha$ for different values of $k_{3}$. To
within a good approximation, they all depend linearly on $k_{3}$. Red and blue
ranges refer to discoids and tactoids, respectively, according to the same
color coding adopted in Fig. 6.The broken lines represent the best linear fits
in (19).

The best linear fit is provided by the following functions,

$\displaystyle\alpha_{1}$ $\displaystyle\approx 157+15k_{3},$ (19a)
$\displaystyle\alpha_{\mathrm{b}}$ $\displaystyle\approx 205+16k_{3},$ (19b)
$\displaystyle\alpha_{2}$ $\displaystyle\approx 289+19k_{3}.$ (19c)

##  IV Comparison with experiments

A major motivation for this paper was offered by the experiments conducted in
[15] with water solutions of disodium cromoglycate (DSCG). Here we apply our
theory to interpret those experiments and show how to extract from them an
estimate for the isotropic surface tension $\gamma$ at the interface between
the nematic phase of a DSCG sol at a given concentration and its isotropic
melt.

Among many other things, Kim et al. [15] explored water solution of DSCG at
concentration $c=16\,\mathrm{wt\%}$ confined between two parallel glass plates
at a distance ranging in the interval $1$-$5\,\mu\mathrm{m}$; the plates were
spin-coated with a polymide layer, SE-$7511$, which exerts a degenerate
tangential anchoring on the nematic director. This system can be treated as
two-dimensional since the bounding plates suppress the out-of-plane
distortions in the observed samples, as required by our theory.

Upon quenching the system from the isotropic phase into the coexistence
regime, tactoidal droplets were observed, surrounded by the parent isotropic
phase. In particular (see, Fig. 6a of [15]), a tactoid with bipolar nematic
orientation was observed more closely, for which the area
$A_{0}=200\,\mu\mathrm{m}^{2}$ and the aspect ratio $\delta=1.3\pm 1$ were
measured at the temperature $T=37.5\,^{\circ}\mathrm{C}$, which we read off
from Fig. 2c of [15]. We wish to make use of these data to validate our
theory. To this end, we need to estimate the elastic constants $K_{11}$ and
$K_{33}$ at $T=37.5\,^{\circ}\mathrm{C}$.

Interpolating the curves representing in [36] the temperature dependence of
the elastic constants of the nematic phase of DSCG at $c=16\,\mathrm{wt\%}$,
we readily arrived at

$K_{11}=4\,\mathrm{pN},\quad K_{33}=9\,\mathrm{pN},\quad\hbox{at}\quad
T=37.5\,^{\circ}\\!\mathrm{C}\quad\hbox{and}\quad c=16\,\mathrm{wt\%},$ (20)

and so we take $k_{3}=2.25$. We can also obtain $\alpha$ from the measured
aspect ratio $\delta$. It follows from (15) that in our theory $\delta$ can be
given the form

$\delta:=\frac{\mu}{R(0)}=\frac{\mu^{2}\left(\frac{8}{3}\cos\phi+\pi\sin\phi\right)}{\pi(\cos\phi+\sin\phi)},$
(21)

where use has also been made of (16). The plot of $\delta$ computed on the
minima of $F_{\alpha}$ over $\mathsf{S}$ is drawn against $\alpha$ in Fig. 11
for $k_{3}\doteq 2.25$.

Figure 11: The droplet’s aspect ratio $\delta$ computed on the minima of
$F_{\alpha}$, plotted against $\alpha$ for $k_{3}=2.25$. The broken horizontal
line designates the level $\delta=1.3$, which is attained for $\alpha=17.2$.
Local and global minima are marked with the same symbols as in Fig. 6. The
critical values of $\alpha$ are $\alpha_{1}=192$, $\alpha_{2}=330$, and
$\alpha_{\mathrm{b}}=243$. Equilibrium tactoids are genuine for
$\alpha\lessapprox 137$.

We see that the value $\delta=1.3$ is attained by the tactoidal branch (as
expected) for $\alpha\doteq 17.2$; the corresponding coordinates in
$\mathsf{S}$ of the minimum of $F_{\alpha}$ are $\phi\doteq 0$ and $\mu\doteq
1.24$, and so the equilibrium shape is a genuine tactoid (with pointed tips).
For $k_{3}=2.25$, equilibrium tactoids cease to be genuine at $\alpha\approx
137$.

Since the equivalent radius $R_{\mathrm{e}}$ is known from the isoperimetric
constraint (7), $R_{\mathrm{e}}\approx 8\mu\mathrm{m}$, we readily obtain for
the long axis $l$ of the droplet $l=2R_{0}=2\mu R_{\mathrm{e}}\approx
20\,\mu\mathrm{m}$. The equilibrium tactoid predicted by our theory is plotted
in Fig. 12 against the observed shape.

Figure 12: The droplet’s equilibrium shape (in red) corresponding to
$\alpha=17.2$, $\phi=0$, and $\mu=1.24$ is contrasted against the shape
observed in [15] (see their Figs. 6a and 6b). Our estimate for the long axis
is $l\approx 20\,\mu\mathrm{m}$. On the left: PolScope image. On the right:
reconstructed shape.

Our expected value for $l$ also agrees quite well with the measurement
performed in [15] on the reconstructed shape, as does the _cusp angle_
$\tau_{\mathrm{N}}$ marked in Fig. 12. In our formalism, for a genuine
tactoid, the latter can be expressed as

$\tau_{\mathrm{N}}=-2\arctan(R^{\prime}(\mu))=2\arctan\left(\frac{3\pi}{4\mu^{2}}\right),$
(22)

which for $\mu=1.24$ delivers $\tau_{\mathrm{N}}\doteq 1.99$. The value of
$\tau_{\mathrm{N}}$ reported in [15] for the shape in Fig. 12 is
$\tau_{\mathrm{N}}=1.05\pm 0.05$, whereas a direct measurement on Fig. 12
would suggest that this value is indeed $\tau_{\mathrm{N}}/2$, in good
agreement with our theoretical prediction. Moreover, by use of equation (13),
we can give the following estimate for the surface tension at the
nematic/isotropic interface of an aqueous DSCG solution at
$16\,\mathrm{wt\%}$,

$\gamma\approx 0.89\times 10^{-5}\,\mathrm{J}/\mathrm{m}^{2}.$ (23)

Different, discordant estimates have been given in the literature for the
order of magnitude of $\gamma$. For example, in [37] they estimate $\gamma\sim
10^{-6}\,\mathrm{J}/\mathrm{m}^{2}$, whereas in [15] they give $\gamma\sim
10^{-4}\,\mathrm{J}/\mathrm{m}^{2}$, an estimate obtained by applying the
pendant drop technique [39, 40]. We trust that the measurements based on the
theory presented in this paper might be more accurate. The value in (23) is
closer to that found in [41] at the nematic-isotropic interface of 5CB (see
also [42, 43]).

The critical values of $\alpha$ corresponding to $k_{3}=2.25$ are
$\alpha_{1}=192$, $\alpha_{2}=330$, and $\alpha_{\mathrm{b}}=243$, while the
observed droplet shown in Fig. 12 corresponds to $\alpha=17.2$. According to
our theory, one would then expect coexistence of tactoids and discoids for
$R_{\mathrm{e}}$ in the range $90\,\mu\mathrm{m}\lessapprox
R_{\mathrm{e}}\lessapprox 153\,\mu\mathrm{m}$, that is, for area $A_{0}$ in
the range $0.3\,\mathrm{mm}^{2}\lessapprox A_{0}\lessapprox
0.7\,\mathrm{mm}^{2}$, a regime of large drops, for which no data are
available in [15].

##  V Conclusion

We studied chromonic droplets in two space dimensions. Our motivation was the
thorough experimental investigation in [15] on thin cells with degenerate
tangential anchoring for the nematic director. In particular, a solution of
DSCG in water was quenched in the temperature regime where nematic and
isotropic phases coexist in equilibrium, the former forming islands with a
tactoidal shape.

We introduced a wide class of two-dimensional shapes for the equilibrium
droplets, which includes tactoids, discoids, bâtonnet, and butterflies, the
latter of which are concave, whereas the former three are convex. Our analysis
revealed that upon increasing the droplet’s area a tactoidal equilibrium
branch gives way to a discoid branch, while neither bâtonnet nor butterflies
can ever be equilibrium shapes. Moreover, there is a regime of shape
coexistence, where a tactoid and a discoid are both local minima of the free
energy, the global minimum shifting from one to the other at a critical value
of the droplet’s area, where perfect bistability is established. A typical
bifurcation diagram with hysteresis describes the situation, reminiscent of a
first-order phase transition with super-heating and super-cooling temperatures
(replaced here by corresponding values of the area).

We put our theory to the test by interpreting experimental data provided by
[15]. We found a fairly good quantitative agreement between experiment and
theory, although further data should be collected to establish on firmer
grounds the degree of confidence of the theory. In particular, we could
extract from the data available in [15] an estimate for the isotropic
component $\gamma$ of the surface tension at the interface between coexisting
nematic and isotropic phases of DSCG in $16\,\%\mathrm{wt}$ aqueous sol. This
estimate seems to promise more accuracy than the rough evaluation of order of
magnitude available in the literature for this material and its chromonic
siblings. We hope that the theory proposed here could be used for a systematic
determination of $\gamma$ for different temperatures and concentrations.

The available data do not cover the range of predicted bistability. We
estimated the area that a droplet should reach to display an abrupt transition
from tactoid to discoid. It remains to be seen whether a controlled growth in
the droplet’s size can be realized to observe neatly this transition.

We have shown that the critical values of the area that delimit the shape
hysteresis are (increasing) linear functions of the ratio $K_{33}/K_{11}$
between bend and splay elastic constants. If this critical phenomenon could be
explored experimentally, our theory would also offer an independent way to
measure $K_{33}/K_{11}$. This might be especially welcome for non conventional
new lyotropic phases, such as chromonics.

##  Appendix A Retracted Tangential Field

In this Appendix, we justify the expression for the free-energy functional
(II.3) associated with the bipolar director field $\bm{n}$ defined as the unit
vector field tangent to the lines with given $t$ and varying $y$ (see Fig. 2).
A generic curve in that family is represented by the position vector

$\bm{p}_{t}(y):=g(t)R(y)\bm{e}_{x}+y\bm{e}_{y},\quad-R_{0}\leqq y\leqq R_{0},$
(24)

where $g$ is any function of class $C^{1}$ strictly increasing on $[0,1]$ and
such that $g(0)=0$ and $g(1)=1$. For given $t\in[0,1]$, the tangent vector
field $\bm{n}$ is given by

$\displaystyle\bm{n}=\frac{gR^{\prime}\bm{e}_{x}+\bm{e}_{y}}{\sqrt{1+(gR^{\prime})^{2}}},$
(25)

where a prime denotes differentiation. The unit vector field

$\bm{n}_{\perp}:=\bm{e}_{z}\times\bm{n}=\frac{-\bm{e}_{x}+gR^{\prime}\bm{e}_{y}}{\sqrt{1+(gR^{\prime})^{2}}}$
(26)

is everywhere orthogonal to $\bm{n}$ and such that
$\bm{n}_{\perp}|_{\partial\mathscr{R}}=-\bm{\nu}$, where $\bm{\nu}$ is the
outer unit normal to $\partial\mathscr{R}$ (see again Fig. 2).

Imagine now a smooth curve in $\mathscr{R}$ parameterized as
$\xi\mapsto(t(\xi),y(\xi))$; it follows from (24) and (25) that

$\dot{\bm{p}}=g^{\prime}R\dot{t}\bm{e}_{x}+\dot{y}(gR^{\prime}\bm{e}_{x}+\bm{e}_{y})=g^{\prime}R\dot{t}\bm{e}_{x}+\sqrt{1+(gR^{\prime})^{2}}\dot{y}\bm{n},$
(27)

where a superimposed dot denotes specifically differentiation with respect to
$\xi$. Thus, the elementary area $\operatorname{d}\\!A$ is

$\operatorname{d}\\!A=\operatorname{d}\\!t\operatorname{d}\\!yg^{\prime}R\sqrt{1+(gR^{\prime})^{2}}\bm{e}_{x}\times\bm{n}\cdot\bm{e}_{z}=g^{\prime}R\operatorname{d}\\!t\operatorname{d}\\!y,$
(28)

and the elementary length $\operatorname{d}\\!\ell_{t}$ on the retracted curve
$\partial\mathscr{R}_{t}$, for given $t$, is

$\operatorname{d}\\!\ell_{t}=\sqrt{1+(gR^{\prime})^{2}}\operatorname{d}\\!y.$
(29)

In particular, the area of $\mathscr{R}$ and the length of its boundary are
given by

$A(\mathscr{R})=2\int_{0}^{1}g^{\prime}\operatorname{d}\\!t\int_{-R_{0}}^{R_{0}}R\operatorname{d}\\!y=2\int_{-R_{0}}^{R_{0}}R\operatorname{d}\\!y$
(30)

and

$\ell(\partial\mathscr{R})=2\int_{-R_{0}}^{R_{0}}\sqrt{1+R^{\prime
2}}\operatorname{d}\\!y.$ (31)

Differentiating $\bm{n}$ in (25) along the smooth curve
$\xi\mapsto(t(\xi),y(\xi))$, we find that

$\dot{\bm{n}}=-\frac{g^{\prime}R^{\prime}\dot{t}+gR^{\prime\prime}\dot{y}}{1+(gR^{\prime})^{2}}\bm{n}_{\perp}.$
(32)

Assuming that $\bm{n}$ is differentiable in $\mathscr{R}$, $\dot{\bm{n}}$ and
$\dot{\bm{p}}$ must be related through

$\dot{\bm{n}}=(\nabla\bm{n})\dot{\bm{p}}.$ (33)

Since $\bm{n}$ is a unit vector field, $(\nabla\bm{n})^{\mathsf{T}}$
annihilates $\bm{n}$, and so there exists a vector
$\bm{a}=a_{1}\bm{n}+a_{2}\bm{n}_{\perp}$ such that

$\nabla\bm{n}=\bm{n}_{\perp}\otimes\bm{a}.$ (34)

To determine the scalar components $a_{i}$ of $\bm{a}$, we observe that, by
(34), (33) also reads as

$\dot{\bm{n}}=(\bm{a}\cdot\dot{\bm{p}})\bm{n}_{\perp}.$ (35)

Making use of (27) and both (25) and (26), we readily see that

$\bm{a}\cdot\dot{\bm{p}}=\frac{g^{\prime}R(gR^{\prime}a_{1}+a_{2})\dot{t}}{\sqrt{1+(gR^{\prime})^{2}}}+\sqrt{1+(gR^{\prime})^{2}}a_{1}\dot{y},$
(36)

and inserting this into (35) alongside with (32), we obtain an identity for
arbitrary $(\dot{t},\dot{y})$ only if

$a_{1}=-\frac{gR^{\prime\prime}}{[1+(gR^{\prime})^{2}]^{3/2}},\quad
a_{2}=\frac{R^{\prime}}{R}\frac{1}{\sqrt{1+(gR^{\prime})^{2}}}-\frac{g^{2}R^{\prime}R^{\prime\prime}}{[1+(gR^{\prime})^{2}]^{3/2}},$
(37)

which leads us to

$\nabla\bm{n}=-\frac{gR^{\prime\prime}}{[1+(gR^{\prime})^{2}]^{3/2}}\bm{n}_{\perp}\otimes\bm{n}+\left(\frac{R^{\prime}}{R}\frac{1}{\sqrt{1+(gR^{\prime})^{2}}}-\frac{g^{2}R^{\prime}R^{\prime\prime}}{[1+(gR^{\prime})^{2}]^{3/2}}\right)\bm{n}_{\perp}\otimes\bm{n}_{\perp}.$
(38)

The following expressions for the traditional measures of distortion then
follow from (38),

$\displaystyle\operatorname{div}\bm{n}$
$\displaystyle=\frac{R^{\prime}}{\sqrt{1+(gR^{\prime})^{2}}}\left(\frac{1}{R}-\frac{g^{2}R^{\prime\prime}}{1+(gR^{\prime})^{2}}\right),$
(39a) $\displaystyle\operatorname{curl}\bm{n}$
$\displaystyle=-\frac{gR^{\prime\prime}}{[1+(gR^{\prime})^{2}]^{3/2}}\bm{e}_{z},$
(39b) $\displaystyle\bm{n}$ $\displaystyle\cdot\operatorname{curl}\bm{n}=0,$
(39c) $\displaystyle\bm{n}\times\operatorname{curl}\bm{n}$
$\displaystyle=\frac{gR^{\prime\prime}}{[1+(gR^{\prime})^{2}]^{3/2}}\bm{n}_{\perp},$
(39d) $\displaystyle\operatorname{tr}(\nabla\bm{n})^{2}$
$\displaystyle-(\operatorname{div}\bm{n})^{2}=0.$ (39e)

We found it useful to rescale all lengths to the radius $R_{\mathrm{e}}$ of
the disc of area $A_{0}$. Letting $\mu$ as in (9) and using (6) and (30), we
arrive at the following reduced functional,

$\displaystyle F[\mu;R]:=\frac{\mathscr{F}[\mathscr{B}]}{K_{11}h}=$
$\displaystyle=\int_{-\mu}^{\mu}\operatorname{d}\\!y\int_{0}^{1}g^{\prime}\left[\frac{g^{4}RR^{\prime
2}R^{\prime\prime 2}}{\big{(}1+(gR^{\prime})^{2}\big{)}^{3}}+\frac{R^{\prime
2}}{R}\dfrac{1}{\big{(}1+(gR^{\prime})^{2}\big{)}}-2\frac{g^{2}R^{\prime
2}R^{\prime\prime}}{\big{(}1+(gR^{\prime})^{2}\big{)}^{2}}+k_{3}\frac{g^{2}RR^{\prime\prime
2}}{\big{(}1+(gR^{\prime})^{2}\big{)}^{3}}\right]\operatorname{d}\\!t+2\alpha\int_{-\mu}^{\mu}\sqrt{1+R^{\prime
2}}\operatorname{d}\\!y,$ (40)

where $k_{3}$ and $\alpha$ are as in (12) and in (13). The integration in $t$,
which delivers (II.3) in the main text, is independent of the specific
function $g$, provided it is monotonic and obeys the prescribed boundary
conditions.

##  Appendix B Preventing Anchoring Breaking

Here, we perform an energy comparison to identify the _safeguard_ value
$\alpha_{\mathrm{s}}$ of $\alpha$, that is, the lower bound that should be
exceeded for a drop to be bipolar at equilibrium. It is known that for
$\alpha$ sufficiently small the tangential anchoring favored by the
interfacial energy (4) is bound to be broken, so that the nematic alignment
becomes uniform throughout the droplet. Uniform and bipolar alignments will
indeed be the terms of comparison for our estimate of $\alpha_{\mathrm{s}}$.

For drops with uniform alignment, the total free energy reduces to (4),
subject to the area constraint (10). The optimal shape is delivered by the
classical Wulff’s construnction [44] (see also [18, p. 490]). Assuming that
$\bm{n}\equiv\bm{e}_{y}$ and that $\mathscr{R}$ is mirror-symmetric with
respect to both axes $\bm{e}_{x}$ and $\bm{e}_{y}$, one needs only determine
the equilibrium shape of $\mathscr{R}$ in the positive $(x,y)$ quadrant. We
let $y=y_{\mathrm{W}}(x)\geqq 0$ represent the profile of $\mathscr{R}$ and
define $\lambda>0$ by setting $y_{\mathrm{W}}(\lambda)=0$. Here, as in the
main text, all lengths are scaled to $R_{\mathrm{e}}$. The method illustrated
in [19, Ch.5] and [45] delivers Wulff’s shape through the following explicit
function

$y_{\mathrm{W}}\left(x\lambda^{-1}\right)\lambda^{-1}=\frac{1+\omega\left(1-\xi^{2}\right)-\xi
x\lambda^{-1}}{\sqrt{1-\xi^{2}}}.$ (41a) where $\xi$ is given in terms of $x$
by solving the algebraic equation
$\omega\xi^{3}+(1-\omega)\xi-x\lambda^{-1}=0$ (41b) and $\lambda$ is
determined by the isoperimetric constraint (10), which here reads as
$4\int_{0}^{\lambda}y_{\mathrm{W}}(x)\operatorname{d}\\!x=4\lambda^{2}\int_{0}^{1}y_{\mathrm{W}}\left(x\lambda^{-1}\right)\lambda^{-1}\operatorname{d}\\!\left(x\lambda^{-1}\right)=\pi.$
(41c)

Correspondingly, by computing the free energy in (4) on Wulff’s shape, we
obtain that

$\mathscr{F}_{\mathrm{W}}=4K_{11}h\alpha j(\omega),$ (42)

where the function $j$, defined by

$j(\omega):=\frac{1}{2}\sqrt{\frac{\pi}{\int_{0}^{1}y_{\mathrm{W}}\operatorname{d}\\!x}}\left(\int_{0}^{1}\sqrt{1+y_{\mathrm{W}}^{\prime
2}}\operatorname{d}\\!x+\omega\int_{0}^{1}\frac{1}{\sqrt{1+y_{\mathrm{W}}^{\prime
2}}}\operatorname{d}\\!x\right),$ (43)

is computed on the solution $y_{\mathrm{W}}$ of (41), renormalized so that
$y_{\mathrm{W}}(1)=0$. Figure 13 illustrates the Wulffian shape obtained with
this method for $\omega=5$ and the graph of $j$ for $0\leqq\omega\leqq 10$.

(a) Wulffian shape obtained by solving equations (41) for $\omega=5$. With
lengths scaled to $R_{\mathrm{e}}$, $\lambda=0.51$.

(b) Graph of the function $j$ in (43) plotted against $\omega$.

Figure 13: Shape and (scaled) energy for the fully aligned droplet.

We now compare this energy to that of a disc with an in-plane bipolar director
field whose integral lines are Apollonian circles passing through the poles
(see, for example, $\S$ 2 of [46]); their radius increases to $+\infty$ upon
approaching the $y$-axis, as shown in Fig. 14, which represents a quadrant of
the disc.

Figure 14: A quarter of a unit disc described in bipolar coordinates
$(\xi,\eta)$. Apollonian circles correspond to different values of $\eta$;
they are orthogonal to the coordinate lines with $\xi$ constant. As in Fig. 3,
also here $\varepsilon$ is out of scale.

Adapting the computations in bipolar coordinates $(\xi,\eta)$ of Williams [47]
to the present two-dimensional setting, we arrive at

$\displaystyle\mathscr{F}_{\mathrm{bip}}$
$\displaystyle:=K_{11}h\left\\{4\int_{\varepsilon}^{\frac{\pi}{2}}\operatorname{d}\\!\xi\int_{0}^{\frac{\pi}{2}}\operatorname{d}\\!\eta\left[\frac{1}{2}\frac{\cos^{2}\xi}{\sin\xi}\frac{1}{(1+\sin\xi\cos\eta)^{2}}+\frac{k_{3}}{2}\frac{\sin^{2}\eta\sin\xi}{(1+\sin\xi\cos\eta)^{2}}\right]+2\alpha\pi\right\\}$
$\displaystyle=4hK_{11}\left\\{\left[\frac{\pi}{4}\left(k_{3}-1-k_{3}\ln
2-\ln\varepsilon\right)+\varepsilon+\alpha\left(\frac{\pi}{2}-\varepsilon\right)\right]+\frac{\alpha\omega\pi\varepsilon}{2}\right\\},$
(44)

where the detect core has been identified with the region
$0\leqq\xi\leqq\varepsilon$. The estimate in (18) follows from requiring that
$\mathscr{F}_{\mathrm{bip}}<\mathscr{F}_{\mathrm{W}}$.

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