# Dynamics of elongation of nematic tactoids in an electric field

Mohammadamin Safdari  Roya Zandi  Department of Physics, University of
California, Riverside, California 92521, USA  Paul van der Schoot  Department
of Applied Physics and Science Education, Eindhoven University of Technology,
Eindhoven, The Netherlands

###### Abstract

Nematic tactoids are spindle-shaped droplets of a nematic phase nucleated in
the co-existing isotropic phase. According to equilibrium theory, their
internal structure and shape are controlled by a balance between the elastic
deformation of the director field, induced by the preferred anchoring of that
director field to the interface, and the interfacial free energy. Recent
experiments on tactoids of chitin nanocrystals dispersed in water show that
electrical fields can very strongly elongate tactoids, at least if the
tactoids are sufficiently large in volume. However, this observation
contradicts the predictions of equilibrium theory as well as findings from
Monte Carlo simulations that do not show this kind of extreme elongation to
take place at all. To explain this, we put forward a relaxational model based
on the Oseen-Frank free energy of elastic deformation of a director field
coupled to an anisotropic surface free energy. In our model, we use two
reaction coordinates to describe the director field and the extent of
elongation of the droplets, and evaluate the evolution of both as a function
of time following the switching on of an electric field. Depending on the
relative magnitude of the fundamental relaxation rates associated with the two
reaction coordinates, we find that the aspect ratio of the drops may develop a
large and very long-lived overshoot before eventually relaxing to the much
smaller equilibrium value. In that case, the response of the curvature of the
director field lags behind, explaining the experimental observations. Our
theory describes the experimental data reasonably well.

††preprint: APS/123-QED

##  I Introduction

Dispersions containing elongated colloidal particles have long been known to
form nematic liquid-crystalline phases if their volume or weight fractions
exceeds some critical value that actually can be surprisingly small [1]. As
the transition from the isotropic to a nematic state is discontinuous, the
coexistence of these two states occurs in a range of concentrations in between
the binodals associated with them [2]. The development of a macroscopic
nematic phase under conditions of coexistence with the isotropic phase is
rather slow, however, and involves the formation of nematic droplets called
tactoids in the isotropic parent phase [3, 4, 5, 6, 7]. Nematic tactoids have
an unusual, elongated shape reminiscent of a spindle, a rugby ball or an
American football, reflecting the underlying uniaxial symmetry of the nematic
phase. That the structure and properties of tactoids may be studied
experimentally is because it may take months to years before the nematic
droplets sediment and coalesce, and a single nematic phase presents itself.
The nematic phase is typically denser than the co-existing isotropic phase
[2].

Since the pioneering work of Zöcher on vanadium pentoxide sols in 1925 [8],
tactoids have been observed in a wide range of molecular, polymeric and
colloidal lyotropic liquid-crystalline fluids. These include dispersions of
different kinds of filamentous and rod-like virus particles [9, 10, 11, 12],
inorganic nanorods [13, 14], polypeptides [15], carbon nanotubes [16, 3],
F-actin [17], protein amyloids [18], chromonic liquid crystals [19] and
cellulose nanocrystals [20, 21, 22, 23]. The characteristic shapes and
director-field structures of nematic tacoids have been the subject of a large
number of theoretical and computer simulation studies [24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35]. From these studies we understand that the spindle
shape so typical of tactoids must be due to the competition between the
preference of rod-like particles for a planar anchoring of the director field
to the interface, and the resulting elastic deformation of the (quasi bipolar)
director field. Because the free energy associated with the elastic
deformation of the director field and the interfacial free energy depend
differently on the volume, both the shape and director field configuration
depend on the volume of a tactoid [26].

Experimental findings on the structure and shape of tactoids seem to be
reasonably well described by macroscopic theory, even if they are not very
large on the scale of the length of the colloidal particles. Indeed, the
application of macroscopic theory makes possible the extraction of information
on the elastic constants of the nematic and interfacial free energies between
the co-existing isotropic and nematic phases from polarization microscopic
images alone if the experimental data include a sufficiently wide range of
tactoid sizes [3, 28, 25, 26, 27, 36, 37, 38]. Interestingly, the experiments
on the aspect ratio (or length and breadth) as a function of the volume of
tactoids typically exhibit a significant amount of scattered data. Puech et
al. [16] argued that this must be due to thermal fluctuations on account of
the fact that the interfacial free energies of lyotropic nematics must be very
low. It so happens that for this type of colloidal system interfacial tensions
can be of the order of $\mu$N m-1 or even significantly below that [16, 39,
40, 4]. Incidentally, this also explains why tactoids require very little
energy to significantly deform, and why the shape and structure of tactoids
are so strongly affected by the contact with an adsorbing surface or the
presence of an externally applied electric, magnetic or flow field [41, 42,
37, 28].

It is also clear from the work of Jamali et al. on nematic tactoids found in
dispersions of carbon nanotubes in a superacid, however, that this variation
cannot be explained in terms of the magnitude of thermal (equilibrium)
fluctuations predicted by macroscopic theory [3]. This already suggests that
the equilibration of the shape and director field of tactoids might be quite
slow [35]. In fact, recent work by Mezzenga and collaborators on tactoids in
aqueous dispersions of amyloids and of cellulose nanocrystals supports this:
tactoids produced via nucleation and growth may have a different aspect ratio
and internal structure than those produced in a microfluidic device [7]. This
explains to some extent why experiments on tactoids in aqueous dispersions of
chitin in the presence of an applied AC electric field [37] show a large
disconnect with predictions from equilibrium theory and the results from Monte
Carlo simulations [36, 43]. Experimentally, tactoids are able to elongate by a
factor of up to about ten through their coupling to the electric field. In
contrast, in equilibrium theory and Monte Carlo simulations it is the director
field that responds to the external orienting field with the aspect ratio of
the tactoids showing much less sensitivity, see also Figure 1.

Figure 1: Polarization microscopic images of the evolution in time of tactoids
in an aqueous solution of chitin fibers, following the switching on and off of
an AC electric field of root-mean-square magnitude of $E=160$ V mm-1 and a
frequency of $300$ kHz. (a) to (e): The field is applied at time zero, and a
collection of tactoids imaged at 0, 120, 210, 640, and 1080 seconds. (f) to
(i): The field is switched off and the same tactoids are imaged after 230,
380, 530, and 820 seconds following the removal of the electric field. Figure
reproduced with permission from Ref. [37].

In Refs. [36, 43] this discrepancy was elucidated by suggesting that the
response of droplet shape and the director field occurs on different time
scales: Initially the anchoring enslaves the director field and forces it to
follow the droplet shape. Only after the droplet shape has relaxed, then the
director field starts relaxing. This eventually leads to the final relaxation
of the droplet shape. which in turn leads to the subsequent relaxation of the
shape of the drop to the actual equilibrium value in the very much later
stages of the process. Indeed, Mezzenga and collaborators recently also
suggested that tactoid equilibration might be kinetically controlled, leading
to very long-lived metastable states [18]. This, then, would also explain the
large scatter in tactoid shape and director field in dispersions that have
been left to equilibrate for a long time, in that full equilibrium might not
yet have been reached for the tactoids themselves. This is remarkable, given
that they typically measure from ten to a few hundreds of $\mu$m in length [3,
42].

In this article, we follow up on our previous work [36], where we investigated
the equilibrium shape and structure of nematic tactoids in an external
alignment field, and introduce a relaxational dynamics based on two reaction
coordinates. These reaction coordinates describe (i) the elongation of a
tactoid with a prescribed spindle shape, and (ii) the degree of deformation of
the director field where we prescribe the geometry of the deformation. The
prescribed droplet shape and director-field geometry allows us to
straightforwardly evaluate the well-known Oseen-Frank elastic free energy as
well as the surface free energies that we use as input for our dynamical
theory. For the surface free energy, we use the ansatz of Rapini and Papoular
[44]. By construction, the steady-state solution to our equations produces the
optimal aspect ratio and degree of curvature of the director field, given the
volume of the drop and the strength of the alignment field [36].

We find that the elongation of tactoids in an electric field is entirely a
kinetic effect. Depending on the ratio of the two fundamental relaxation rates
associated with the two reaction coordinates, we either obtain a monotonic
time evolution of both the aspect ratio and curvature of the director field,
or a monotonic response of the curvature of the director field and an
overshoot in the aspect ratio of the tactoids. The actual response times
depend strongly on the strength of the electric field and the volume of the
tactoids, appropriately scaled to the elastic constants and interfacial free
energies associated with the nematic. Curve-fitting to the experimental data
of Metselaars and collaborators [37] gives reasonable agreement, showing that
the relaxation of the bipolarness of the director field must be extremely
sluggish. The cause of this remains unclear and requires further study.

The remainder of this paper is structured as follows. In Section II we
describe the ingredients of our theory. Section III summarizes our main
findings based on a numerical evaluation of the theory. In Section IV we
compare our theory with experiments, and finally we present our conclusions
and discuss our results in the Section V.

##  II Theory

Relaxational or model A dynamics is based on a phenomenological description of
how a system relaxes to a state of equilibrium, described by an appropriate
free energy[45, 46, 47]. The free energy of the nematic droplet is the sum of
three components,

$F=F_{S}+F_{E}+F_{C},$ (1)

where $F_{S}$, $F_{E}$ and $F_{C}$ refer to the contributions of (i) the
interface between the nematic tactoid and the host isotropic phase it is in
contact with, (ii) the elastic deformation of the director field in the
tactoid and (iii) the interaction of the nematic with alignment field,
respectively.

Figure 2: Cross section (solid) and director field (dotted) of the structure
of a tactoid presumed in our calculations. The droplet is cylindrically
symmetric about its main axis. $R$ denotes the length of a tactoid and $r$ its
width. $\tilde{R}$ is the distance between the virtual boojums, which are the
focal points of the (extrapolated) bipolar director field. For $R=\tilde{R}$,
the virtual boojums become actual boojums, i.e., surface point defects. Also
indicated is the opening angle $\alpha$ of the spindle-shaped droplet. See
also the main text.

For the interfacial free energy, we use the well-known phenomenological
expression of Rapini and Papoular [48, 44],

$\displaystyle F_{S}=$
$\displaystyle\sigma\int\left[1+\omega(\vec{q}\cdot\vec{n})^{2}\right]\mathrm{d}A,$
(2)

that in the context of dispersions of rod-like particles has some merit [39].
Here, $\sigma$ denotes the interfacial tension between the nematic droplet and
isotropic medium for the case of perfect planar anchoring, $\vec{q}$ is the
local surface normal, $\vec{n}$ the director field at the surface $A$ and
$\omega>0$ a dimensionless anchoring strength that penalizes non-planar
anchoring of the director field to the interface. The integration is over the
entire surface $A$ of the droplet. For simplicity, we presume the interfacial
tension and anchoring strength to be independent of the strength of the
alignment field, even though the isotropic phase becomes paranematic in such a
field and we would expect the interfacial tension and anchoring strength to
respond to that [49]. As usual, we also ignore any curvature dependence of the
interfacial free energy and invoke the so-called capillarity approximation.

For the elastic deformation, we make use of the Oseen-Frank free energy [50],

$\displaystyle F_{E}=$
$\displaystyle\int\left(\frac{1}{2}K_{11}\left(\vec{\nabla}\cdot\vec{n}\right)^{2}+\frac{1}{2}K_{33}\left(\vec{n}\times\left(\vec{\nabla}\times\vec{n}\right)\right)^{2}\right.$
(3)
$\displaystyle-\left.K_{24}\vec{\nabla}\cdot\left(\vec{n}\vec{\nabla}\cdot\vec{n}+\vec{n}\times\left(\vec{\nabla}\times\vec{n}\right)\right)\right)\mathrm{d}V,$

where $K_{11}$, $K_{33}$, and $K_{24}$ represent the elastic modulus of splay,
bend, and saddle-splay terms respectively. In our description, we do not allow
for a twisted director field [27]. Integration is over the entire volume $V$
of the drop. We take the elastic moduli to be independent of the strength of
the field. Strictly speaking this cannot be true as their values depend on the
degree of nematic alignment of the particles [51]. For the kind of bipolar
director field that we shall be considering, the saddle-splay deformation
renormalizes the splay elastic contribution to the free energy, implying that
$K_{11}$ becomes $K_{11}-K_{24}$ and we can remove the saddle-splay term from
Eq. 3 [25, 26].

In the experiments of Metselaar and collaborators [37], the alignment field is
a high-frequency AC electric field that we treat as a static field $\vec{E}$
with a magnitude equal to the root-means-square value of that of the AC field.
At time zero we switch on the field, and are interested in the response of
tactoids to the switching on of the field. We expect the dielectric
susceptibility of the nematic phase to be anisotropic and reflect its uniaxial
symmetry [52]. Hence, the dielectric susceptibility of the nematic can be
described as a second-rank tensor with two principal susceptibilities
$\epsilon_{\parallel}$ parallel and $\epsilon_{\perp}$ and perpendicular to
the local director. Defining the susceptibility anisotropy as
$\Delta\epsilon\equiv\epsilon_{\parallel}-\epsilon_{\perp}\geq 0$, the
electric-field (or Coulomb) contribution to the free energy becomes

$F_{C}=-\dfrac{1}{8\pi}\Delta\epsilon\int\left(\vec{n}\cdot\vec{E}\right)^{2}\mathrm{d}V,$
(4)

apart from a constant term that we can ignore [36, 53]. The integration is
again over the entire volume $V$ of the tactoid. The anisotropy of the
dielectric susceptibility depends on the degree of nematic ordering and hence
also depends, in principle, on the strength of the electric field. This, we
also ignore.

Obviously, we do not know the shape and director-field configuration a priori.
As in our previous work, we follow Prinsen et al. and Kaznacheev et al. and
choose to prescribe the shape of the droplet and the geometry of the director
field [26, 28, 36]. For the shape we use a circle section of revolution, and
for the director field a bipolar director field, sketched in Fig. 2. The shape
and director field are then fully described by the length of the drop $R$, the
width of the drop $r$, and the distance between the focal points of the
bipolar director field $\tilde{R}$. The aspect ratio of a tactoid, $x\equiv
R/r$, is related to the opening angle $\alpha$, also indicated in Fig. 2, via
the relation $x=\cot{\alpha/2}$. The degree of bipolarness of the director
field is described by the quantity $y=\tilde{R}/R$. For a homogeneous director
field $y\rightarrow\infty$ whilst for a purely bipolar field $y\rightarrow 1$.
In the latter case the foci of the bipolar director field are located on the
poles of the tactoid and represent surface point defects called boojums [54].
We refer to our earlier work [36] and also to Fig. 2. Note that for any value
of the bipolarness $1<y<\infty$, the director field is quasi bipolar.

It turns out that the free energy can be entirely described in terms of the
variables $\alpha=2\cot^{-1}x$ and $y$, and in terms of four dimensionless
groups describing all the elastic and surface materials parameters of the
model. The volume of a droplet $V$ is fixed during an experiment and also
constitutes a model parameter that we make dimensionless by defining
$v\equiv\left(\sigma/(K_{11}-K_{24})\right)^{3}V$, recalling that $\sigma$
denotes the bare surface tension and $K_{11}$ and $K_{24}$ are the elastic
constants associated with a splay and saddle-splay deformation of the director
field. The free energy we also render dimensionless. A definition that proves
to be practical is $f\equiv\sigma F/\left(K_{11}-K_{24}\right)^{2}$.

With these definitions, the full expression of our free energy now becomes
[36]

$\displaystyle f(\alpha,y)=$
$\displaystyle\quad\mathrm{v}^{2/3}\phi^{-2/3}_{\mathrm{v}}(\alpha)(\phi_{\sigma}(\alpha)+\omega\phi_{\omega}(\alpha,y))$
(5)
$\displaystyle+\mathrm{v}^{1/3}\phi_{\mathrm{v}}^{-1/3}(\alpha)(\phi_{11}(\alpha,y)+\kappa\phi_{33}(\alpha,y))$
$\displaystyle-\Sigma\mathrm{v}\phi_{\mathrm{v}}^{-1}(\alpha)\phi_{C}(\alpha,y)$

The first term of Eq. 5 represents the contribution from the surface tension
and anchoring energy, with

$\phi_{\mathrm{v}}(\alpha)=\frac{7\pi}{3}+\frac{\pi}{2}\left(\frac{1-4\alpha\cot\alpha+3\cos
2\alpha}{\sin^{2}\alpha}\right),$ (6)
$\displaystyle\phi_{\sigma}(\alpha)=4\pi\left(\frac{1-\alpha\cot\alpha}{\sin\alpha}\right),$
(7)

and

$\displaystyle\phi_{\omega}(\alpha,y)$
$\displaystyle=\frac{\pi}{2}(y^{2}-1)^{2}\sin^{3}\alpha$ (8)
$\displaystyle\times\int_{0}^{\pi}\mathrm{d}\xi\left[\dfrac{\sin^{2}\xi\cos^{2}\xi}{N(y,\xi,\alpha)\left(1+\sin\xi\cos\alpha\right)^{3}}\right].$

Here,

$\displaystyle N(y,\xi,\eta)=$
$\displaystyle\left(\sin\xi\cos\eta+\frac{1}{2}Z(\xi,\eta)\left(y^{2}-1\right)\right)^{2}$
(9) $\displaystyle+y^{2}\sin^{2}\xi\sin^{2}\eta,$

and

$Z(\xi,\eta)=1+\sin\xi\cos\eta,$ (10)

where we set $\eta=\alpha$ in Eq. 9.

The second term of Eq. 5 is due to the Frank elastic energy, where the
dimensionless group $\kappa=K_{33}/(K_{11}-K_{24})$ acts as a measure for the
importance of the bend elastic deformation. Here,

$\displaystyle\phi_{11}(\alpha,y)=$ $\displaystyle
8\pi\int_{0}^{\pi}\mathrm{d}\xi\int_{0}^{\alpha}\mathrm{d}\eta\sin^{2}\xi\cos^{2}\xi\sin\eta$
(11)
$\displaystyle\times\frac{1}{N(y,\xi,\eta)\left(1+\sin\xi\cos\eta\right)^{3}},$

describes the contribution of the splay and saddle-splay deformations, and

$\displaystyle\phi_{33}(\alpha,y)=$ $\displaystyle
8\pi\int_{0}^{\pi}\mathrm{d}\xi\int_{0}^{\alpha}\mathrm{d}\eta\sin^{4}\xi\sin^{3}\eta$
(12)
$\displaystyle\times\frac{1}{N(y,\xi,\eta)\left(1+\sin\xi\cos\eta\right)^{3}},$

that of the bend deformation.

Finally, the third term describes the contribution from the interaction of the
nematic droplet with an electric field, with
$\Sigma\equiv\frac{1}{8\pi}\epsilon_{\mathrm{a}}E^{2}\sigma^{-2}(K_{11}-K_{24})$
a measure for its strength relative to the surface and elastic deformation
free energy cost, and

$\displaystyle\phi_{\mathrm{C}}(\alpha,y)=8\pi\int_{0}^{\pi}\mathrm{d}\xi\int_{0}^{\alpha}\mathrm{d}\eta\frac{\sin^{2}\xi\sin\eta}{(1+\sin\xi\cos\eta)^{3}}$
(13)
$\displaystyle\times\frac{\left(y^{2}Z^{2}+\sin^{2}\xi\sin^{2}\eta-\cos^{2}\xi\right)^{2}}{N(y,\xi,\eta)\left(1+\sin\xi\cos\eta\right)^{2}}.$

We have not been able to analytically evaluate the various integrals, except
when the tactoids are extremely elongated and the opening angle becomes very
small. In that case, asymptotic relations may be obtained as well as robust
scaling estimates [26, 36]. Since we do not wish to restrict ourselves to
large aspect ratios, we rely on a numerical evaluation of the integrals. For
this, we employ the Mathematica software package [55] and use the $NIntegrate$
function to compute integrals and the $D$ function to calculate derivatives.

Now that we have formulated our free energy in terms of the two reaction
coordinates $\alpha$ and $y$, we are able to formulate our relaxational theory
in terms of the generalized forces:

$\displaystyle\dfrac{\partial\alpha}{\partial
t}=-\tilde{\Gamma}_{\alpha}\dfrac{\partial F}{\partial\alpha},$ (14)
$\displaystyle\dfrac{\partial y}{\partial
t}=-\tilde{\Gamma}_{y}\dfrac{\partial F}{\partial y},$

where we choose to ignore the contribution of cross terms [56], the main
reason being to limit the number of adjustable parameters in our model. Here,
$\tilde{\Gamma}_{\alpha}$ and $\tilde{\Gamma}_{y}$ are fundamental relaxation
rates that have dimensions of J-1s-1, so reciprocal Joules per second. Because
$y$ describes the deformation of the director field relative to the length of
a tactoid this would arguably require the collective reorientation of the
colloidal particles in the nematic phase. Hence, we would expect
$\tilde{\Gamma}_{y}$ to be inversely proportional to a rotational viscosity of
the nematic phase. Relaxation of the aspect ratio of the droplets requires the
transport of material in both the isotropic and nematic phases. It would
therefore seem sensible to presume the relaxation rate
$\tilde{\Gamma}_{\alpha}$ to be some average of the relevant viscosities of
the two phases [57].

For dimensional reasons this then implies that both rates should also be
inversely proportional to some volume scale. Considering that the volume scale
must represent the volume in which the viscous dissipation takes place, we
conclude that both rates must be inversely proportional to the volume $V$ of a
tactoid. Hence, we write for our dynamical equations in terms of the
dimensionless free energy $f$ and tactoid volume $v$ as

$\displaystyle\dfrac{\partial\alpha}{\partial
t}=-\dfrac{\Gamma_{\alpha}}{v}\dfrac{\partial f}{\partial\alpha},$ (15)
$\displaystyle\dfrac{\partial y}{\partial
t}=-\dfrac{\Gamma_{y}}{v}\dfrac{\partial f}{\partial y},$

where $\Gamma_{\alpha}=\tilde{\Gamma}_{\alpha}V\sigma^{2}/(K_{11}-K_{24})$ and
$\Gamma_{y}=\tilde{\Gamma}_{y}V\sigma^{2}/(K_{11}-K_{24})$ are our scaled
fundamental rates with dimensions of reciprocal seconds that do no longer
depend on the volume of a drop. Our approach should be equivalent to but
extends that of Weirich et al. [38] and of Almohammadi et al. [35], who
balance the rate of change in mechanical energy with the rate of energy
dissipation for fixed values of our parameter $y$.

To numerically integrate the two rate equations described in Eq. 15, we apply
the (forward) Euler method function within the Mathematica package [55]. As
initial conditions, we use the solutions of $\partial
f/\partial\alpha=\partial f/\partial y=0$ in the absence of an external field,
so for the case $\Sigma=0$. A complete phase diagram describing the shape and
director field of tactoids as a function of the dimensionless tactoid volume
$v$ and the dimensionless electric field strength $\Sigma$ can be found in our
earlier work [36]. Notice that we can make time dimensionless by defining
$\tau=t\Gamma_{\alpha}$. In scaled time, our dynamical theory therefore
introduces a single additional dimensionless group, namely
$\gamma\equiv\Gamma_{y}/\Gamma_{\alpha}$. In our numerical evaluation of the
kinetic equations, we choose to make use of a time-adaptive approach in order
to efficiently deal with the fast and slow processes that turn out to
characterize the response of tactoids. This means that we dynamically adjust
the time step, referred to as $\mathrm{d}\tau$, during each evaluation step.
Specifically, we steadily increase $\mathrm{d}\tau$ by a factor of 1.01 and
assess the difference between the new values of $\alpha$ and $y$ relative to
the ones of the previous time step. If the difference is less than 0.001 for
$\alpha(t+\mathrm{d}\tau)-\alpha(t)<0.001\ \alpha(t)$ or
$y(t+\mathrm{d}\tau)-y(t)<0.001\ y(t)$, we continue with a new value of the
time step $d\tau$. Conversely, if one or both of the differences exceeds the
mentioned thresholds, we divided $\mathrm{d}\tau$ by a factor of 50 in each
subsequent time step. This adaptive approach is particularly useful in the
late stages of simulation when changes in $\alpha$ and $y$ become exceedingly
small, allowing us to expedite the numerical evaluation of the kinetic
equation.

Having presented the main ingredients of our theory, we next discuss our most
salient findings and compare our results with the experimental data of
Metselaar et al. [37].

##  III Results

To determine the conditions under which elongated tactoids can be observed,
even if only transiently, it is important to consider the following. As is now
well established, small nematic tactoids tend to have a uniform director
field, while for sufficiently large ones the director field is (for all
intents and purposes) bipolar [25, 26, 3]. Switching on an electric field only
acts to reorient tactoids with a uniform director field but does not affect
their elongation. Bipolar ones reorient and do not become more elongated if
their volume is smaller than some critical value [37]. If sufficiently large,
bipolar tactoids may elongate substantially under the action of the field but
only if somehow the director field remains bipolar and does somehow not
immediately respond to the electric field [36, 43]. This can only happen if
(i) the ratio of the two fundamental relaxation rates
$\gamma=\Gamma_{y}/\Gamma_{\alpha}$ is sufficiently small and (ii) the volume
of the drop is sufficiently large for the electric field to be able to deform
it. Making use of the scaling theory of our previous work [36], we deduce by
balancing the interfacial and Coulomb free energies that the latter happens if
$v\gg\Sigma^{-1/3}$. Note that, typically, the anchoring strength varies from
about $1.5$ to $6$ [3]. For the tactoids to be bipolar, we must in addition
insist that $v\gg\omega^{-5/2}$ [25]. The maximum aspect ratio we expect to
find, provided that these conditions are met, is
$R/r\sim\Sigma^{3/7}v^{1/7}\gg 1$ at the level of the scaling theory, so
ignoring any constants of proportionality. This shows that the field strength
more strongly impacts upon the maximum aspect ratio than the volume does. Our
numerical evaluation of the kinetic equations confirms this.

Now that we know under what conditions we might expect transients to arise, we
first explore how the ratio of fundamental relaxation rates $\gamma$
influences the aspect ratio $x$ and bipolarness $y$ of tactoids. Figure 3
shows the aspect ratio vs. time for different values of $\gamma$, and fixed
values of $v=10^{5}$, $\Sigma=65$, $\omega=1.3$, and $\kappa=20$. The latter
two values we obtain from the properties of chitin tactoids in the absence of
an electric field [36]. For these values of the parameters, we have
$v\Sigma^{3}\gg 1$, and based on the scaling estimate we expect to see an
overshoot of the aspect ratio of the order of 10 to 100 if $\gamma\ll 1$.
Figure 3 confirms this: the aspect ratio goes through a maximum before
reaching its equilibrium value in the late stages of the process, and the
maximum value reached increases as the magnitude of $\gamma$ decreases. Also,
the smaller the value of $\gamma$, the longer lived the high-aspect-ratio
states are. For late times, when the tactoids approach the true equilibrium
state, the value of aspect ratio is not so large and does not depend on the
ratio of $\gamma$. Hence, this confirms that large aspect ratios are possible
only for smaller values of $\gamma$ when the director field has not yet fully
equilibrated. Fig. 4, showing the degree of bipolarness $y$ as a function of
time, confirms this.

Fig. 4 shows that, not entirely unexpectedly, the smaller the value of
$\gamma$ the more slowly the changes in bipolarness occur. This prevents the
immediate alignment of particles and therefore that of the director field with
the applied external electric field. The particles remain essentially enslaved
to the surface anchoring until the internal dynamics allow them to break free
from this and relax to the equilibrium value. Fig. 4 shows that there is some
feedback between the two: whilst the bipolarness does increase monotonically
with time, it does seem to develop a weak “shoulder” around the region where
the aspect ratio of the droplets reaches its maximum. As time passes, the
bipolarness of the tactoid increases reaching its equilibrium value, which
leads to a reduction in the aspect ratio to a value consistent with our
previous equilibrium results [36]. This is the key reason for the presence of
a maximum value for the aspect ratio as a function of time.

Figure 3: Aspect ratio $x$ of a tactoid as a function of dimensionless time
$\tau$ for different values of the ratio $\gamma$ of the fundamental
relaxation rates associated with the response of the director field and that
of the aspect ratio. From top to bottom: $\gamma=$ 0.01 (blue), 0.05 (yellow),
0.1 (green), 0.5 (red), 1 (purple), 10 (brown), 50 (light blue). The
dimensionless volume of the droplet is $v=10^{5}$, the anchoring strength
$\omega=1.3$, the dimensionless strength of the electric field $\Sigma=65$,
and the ratio of the bend and splay elastic constants $\kappa=20$. Figure 4:
Bipolarness of a tactoid $y$ as a function of dimensionless time $\tau$ for
the different values of the ratio $\gamma$ of the fundamental relaxation rates
associated with the response of the director field and that of the aspect
ratio. From bottom to top: $\gamma=$ 0.01 (blue), 0.05 (yellow), 0.1 (green),
0.5 (red), 1 (purple), 10 (brown), 50 (light blue). The dimensionless volume
of the droplet is $v=10^{5}$, the anchoring strength $\omega=1.3$, the
dimensionless strength of the electric field $\Sigma=65$, and the ratio of the
bend and splay elastic constants $\kappa=20$.

We next investigate the impact of the strength of the external electric field
on the shape and director-field structure of nematic tactoids. Figure 5 shows
that the final, equilibrium value of the aspect ratio does not appreciably
depend on the electric field strength, at least not for the values of
$\Sigma=10,100,200,400,1000$ and $2000$ shown in the graph and the
dimensionless volume of the drop $v=10^{2}$. This is consistent with previous
equilibrium studies, showing that the final state is characterized by a more
or less uniform director field and an aspect ratio that approaches the value
of $2\sqrt{\omega}$ if the field is sufficiently strong [36, 43]. Still,
transients with a large aspect ratio do arise with a maximum that increases
with the strength of the electric field. We notice that the transients are
very long-lived and more so the larger the field strength. A careful review of
the plots of the maximum value of aspect ratio found in Fig. 5) shows that it
is proportional to $\Sigma^{0.42}$. The exponent is close to the prediction of
$3/7$ mentioned earlier. Fig. 5 also reveals that the time for a tactoid to
reach its maximum elongation becomes shorter as the strength of the electric
field becomes stronger. In fact, we find numerically that this time scales as
$\Sigma^{-0.77}$.

Figure 5: Aspect ratio $x$ of tactoid as a function of dimensionless time
$\tau$ for different values of the dimensionless electric field strength
$\Sigma=10,100,200,400,1000,2000$ (bottom to top). The dimensionless volume of
the droplet is $v=10^{2}$, the anchoring strength $\omega=1.3$, and the ratio
of the bend and splay elastic constants $\kappa=20$. The ratio of the
fundamental relaxation rates was set at a value of $\gamma=1/15$.

Let us now explore how the bipolarness $y$ of tactoids depends on the strength
of the electric field $\Sigma$. As shown in Fig. 6, the equilibrium value of
bipolarness increases with time, and more so the stronger the electric field.
This means that the virtual point defects (the focal points of the
extrapolated bipolar director field) move away from the poles of the tactoids
and the director field becomes increasingly more homogeneous. A careful
examination of our numerical results for late times shows that it increases as
$\Sigma^{0.48}$. This is consistent with the equilibrium theory of Safdari et
al. [36], which predicts that the bipolarness grows with the field strength as
$\Sigma^{0.5}$. Comparing Figs. 5 and 6 also shows that the relaxation of the
bipolarness to its equilibrium value is very much more sluggish than that of
the aspect ratio. In fact, it is more sluggish than what we would expect based
on the value of $\gamma$, which in the figure is equal to $1/15$. This is not
all that surprising given that any director field with bipolarness above a
value of, say, three is difficult to distinguish from a uniform director
field. This implies that any response of the aspect ratio must be very small
beyond that.

Figure 6: Bipolarness of tactoid $y$ as a function of the dimensionless time
$\tau$ for the different electric field strength
$\Sigma=10,100,200,400,1000,2000$ (bottom to top). The dimensionless volume of
the droplet is $v=10^{2}$, the anchoring strength $\omega=1.3$, and the ratio
of the bend and splay elastic constants $\kappa=20$. The ratio of the
fundamental relaxation rates was set at a value of $\gamma=1/15$.

Finally, we investigate the effect of the volume of a tactoid on both its
bipolarness and aspect ratio for a given external field and a given asymmetry
in the relaxation dynamics of the director field and the droplet shape. From
the earlier-mentioned scaling theory we expect that the director field is
essentially uniform and the drop does not respond to any alignment field other
than aligning along the field direction if the dimensionless volume $v$ is
smaller than $\omega^{-5/2}$. For our choice of $\omega=1.3$, $v$ needs to be
much larger than unity for it to have a noticeable degree of curvature of the
director field. For the external field to be able to straighten out the curved
director field, $v$ must be larger than about $\Sigma^{-5/4}$ according to the
scaling theory. If we set $\Sigma=65$, this implies that a tactoid responds to
that field as long the tactoid is bipolar, so $v\gg 1$. The maximum value of
the aspect ratio scales then with the tactoid volume as $v^{1/7}$, which
predicts a very weak dependence on the volume.

Figures 7 and 8 show how a variation of the dimensionless volume for
$v=1,310,2340$ and $5780$ affect the time-evolution of the aspect ratio and
bipolarness following the switching on of the electric field for $\Sigma=65$,
$\omega=1.3$, $\gamma=0.1$ and $\kappa=20$. The figure confirms once more that
the equilibrium value of the aspect ratio of a tactoid does not, as expected,
strongly depend on its volume [36, 43]. For $v=1$ the director field is almost
uniform with $y\approx 3$, so for this volume, the external field has very
little impact on both the aspect ratio and bipolarness. The larger tactoids
are all essentially bipolar at time zero, with a bipolarness $y$ close to
unity, and these do respond to the switching on of the field. The aspect ratio
of this larger droplet does exhibit transient overshoots, while their
bipolarness increases monotonically with time as the field straightens the
director field.

Figure 7 confirms that the maximum value of the aspect ratio increases only
weakly with the dimensionless volume that we varied over four decades in
magnitude. The timescale required to reach this maximum value decreases with
the volume of the drop but also not very strongly. The same can be said about
the relaxation of the director field. The equilibrium value of the bipolarness
does depend on the volume albeit not very sensitivly. According to the scaling
theory of Safdari et al., which is confirmed by numerical minimization of the
free energy [36], we expect $y$ to scale as $v^{1/6}$. The late-stage results
shown in Fig. 8 agree with this prediction, as they should.

Figure 7: Aspect ratio of tactoids $x$ as a function of time for the different
dimensionless volume $v=1,310,2340,5780,10000$ (bottom to top, central part).
The electric field strength is $\Sigma=65$, the anchoring strength
$\omega=1.3$, the ratio of the fundamental relaxation times $\gamma=1/15$ and
the ratio of the bend and splay elastic constants $\kappa=20$ Figure 8:
Bipolarness of tactoid $y$ as a function of the dimensionless time $\tau$ for
different dimensionless volume $v=1,310,2340,5780,10000$ (bottom to top on the
right). The electric field strength is $\Sigma=65$, the anchoring strength
$\omega=1.3$, the ratio of the fundamental relaxation times $\gamma=1/15$ and
the ratio of the bend and splay elastic constants $\kappa=20$.

After having investigated the response of the switching on of an electric
field, we ask ourselves the question what happens when the electric field is
turned off and the external field is removed? Specifically, we are interested
in understanding the dynamics of the change in aspect ratio during the
transition. Does it revert to its initial state swiftly? Or does it remain
relatively unchanged? This is a relevant question, because Metselaar et al.
find that switching off of the field prior to full relaxation in the presence
of the field the largest tactoids seem not to revert to the field-free aspect
ratios in roughly the same amount time as when the field was switched on [37].
See also Figure 1, noting that quantitative data are not available.

To address these questions, we turn on and next turn off the electric field
once the tactoids have elongated to a certain fraction of the maximum value
for $\Sigma=425$ and $v=5\cdot 10^{3}$. The results of our numerical
calculations are presented in Fig. 9. Consistent with the experiments of
Metselaar et al. [37], we find that the tactoids take more time to approach
their equilibrium configurations than the time required to elongate them. For
the case shown, the difference in time amounts to three or four orders of
magnitude. We notice that the state of elongation at which the field is
switched off does not seem to strongly influence the (dimensionless)
relaxation time, which for all cases shown in the figure is about 70,000. The
blue curve in Fig. 9 shows the situation when the electric field remains
turned on, showing a considerably slower convergence to the equilibrium state
in the presence of the field compared to the cases where the field was
switched off, reverting to the equilibrium state in the absence of an
alignment field. Both are much larger, though, than the time it takes to reach
peak elongation. This illustrates that there are many very divergent
timescales involved in the relaxation of tactoids, highlighting the non-linear
character of the kinetics at hand.

Figure 9: The evolution of the aspect ratio $x$ as a function of the
dimensionless time $\tau$. The blue curve shows the response to switching on
of the field at time zero. The curves in black, green, red, pink and orange
show what happens after shutting of the field. The dimensionless volume of the
droplet is $v=5\cdot 10^{3}$, the anchoring strength $\omega=1.3$, the
strength of the electric field is $\Sigma=425$, the ratio of the bend and
splay elastic constants $\kappa=20$, and the ratio of the fundamental
relaxation rates $\gamma=1/15$.

In the next section, we apply the theory developed here in order to interpret
the experimental results of Metselaar et al. [37].

##  IV Comparison with Experiment

In a series of measurements, Metselaar et al. recently measured the dimensions
of tactoids in different batches of an aqueous dispersion of chitin
nanocrystals as a function of time, following the application of a high-
frequency AC electric field [37]. Experiments were done for root-mean-square
field strengths in the range from 160 V/mm to 450 V/mm and frequencies ranging
from 300 to 700 kHz. Even though the findings between different experiments
differed quantitatively, qualitatively the results are consistent with each
other. Despite the small difference in the dielectric properties of the
coexisting isotropic and nematic phases, which are dominated by the
contribution from the aqueous solvent, it turns out that electric fields can
significantly elongate tactoids at least if these are of sufficiently large
volume. See also Figure 1. In our previous work [36], focusing on the
thermodynamic properties of tactoids in an electric field, we showed we could
only get reasonable agreement between theory and experiment for the maximum
elongation by invoking a restricted equilibrium. In this restricted
equilibrium, we fixed the bipolarness of the tactoids to the equilibrium
values obtained in the absence of the field.

Our aim here is to relax the restricted equilibrium, and adjust and calibrate
our relaxational model in order to achieve the closest possible alignment with
the experimental data. Given that the available data from different
experiments agree only qualitatively, we seek to reproduce trends rather than
achieve quantitative agreement. For this, we use model parameters obtained by
fitting the theory to the available experimental data on field-free aspect
ratio for a range of tactoid volumes varying four orders of magnitude [36]. In
our previous work, we found the values for the anchoring strength
$\omega=1.3$, the ratio of bend to splay elastic constants $\kappa=20$, and
the extrapolation length $\left(K_{11}-K_{24}\right)/\sigma=4$ $\mu$m. Hence,
in the conversion of dimensionless volume $v$ to dimension-bearing volume $V$
we multiply $v$ by $64$ to change the unit of our dimensionless volumes to
$\mu m^{3}$.

Figure 10: Time evolution of the aspect ratio of three droplets of different
dimensionless volumes $v$, indicated by different colors: green for $v=5780$,
orange for $v=2340$ and blue for $v=310$. Dots are experimental data points
extracted from Ref. [37] and the drawn curves display our curve fits. We set
the anchoring strength $\omega=1.3$, the ratio of bend to splay $\kappa=20$,
the coefficient of the electric field strength $\Sigma=190$ the ratio of
relxation rate $\gamma=0.5$ and $\Gamma_{\alpha}=1.6\times 10^{-4}s^{-1}$. The
$r^{2}$ values for the quality of the curve fits are 0.61, 0.91, and 0.94 for
blue, orange, and green respectively.

Fig. 10 shows the aspect ratio $x=R/r$ vs. the actual time $t$ in seconds for
three different tactoid volumes of $V\simeq 20\times 10^{3}$, $150\times
10^{3}$ and $370\times 10^{3}$ $\mu$m3. Shown are the ratios of lengths and
widths of the chitin tactoids obtained experimentally by Metselaar et al.
[37], together with the solid curves that are the result of our curve-fitted
numerical solutions to the kinetic equations. Each color corresponds to a
different volume. To find the best fit, we applied a hyperparameter grid
search (on the parameters $\Sigma$, $\gamma$ and $\Gamma_{\alpha}$) [58] and
minimize the so-called cost function,
$\sum(x_{\mathrm{the}}-x_{\mathrm{exp}})^{2}/N$ with N the number of data
points, a measure for the mean-square distance between the aspect ratio of the
theoretical prediction $x_{\mathrm{the}}$ and the experimental data points
$x_{\mathrm{exp}}$. The values for the various parameters that we find are
$\gamma=0.5$, $\Sigma=190$ and $\Gamma_{\alpha}=1.6\times 10^{-4}s^{-1}$.

We get reasonable agreement between theory and experiments, noting that we use
the same values of all of the parameters except the volume of the drops, with
coefficients of determination $r^{2}$ between $0.61$ and $0.95$. The curves
clearly show that whilst the initial response time goes down with increasing
droplet volume, final equilibration actually slows down and takes longer the
larger the tactoids. Our curve-fitting procedure produces maxima that are not
quite observed yet in the data, except perhaps for the smallest volume. It is
important to point out that setting $\gamma\ll 1$ produces curves for which
the maximum moves to much larger times. In fact, setting $\gamma=0$ and
suppressing the maximum entirely produces curve fits that have much smaller
values of $r^{2}$. In fact, a simple exponential relaxation, put forward in a
slightly different context in Refs. [38, 35], cannot describe the experiments
in any satisfactory way (results not shown). We expect that for times much
larger than, say, 7000 s, the experiments should show a downturn in the aspect
ratio.

Figure 11: Aspect ratio of tactoids $x$ as a function of their volume $V$ in
$\mu$m3 for different times. a) Experimental findings for chitin in water,
taken from [37]. b) Our theoretical predictions are based on the curve
fitting. Parameter values for the theoretical predictions: $\omega=1.3$,
$\kappa=20$, $\Sigma=500$, $\gamma=1/20$ and $\Gamma_{\alpha}=1.6\times
10^{-4}s^{-1}$. Results for the same moment in time are presented with the
same color. Blue circles: 120 seconds, red triangles: 210 s, green squares:
410 s, black diamonds: 640 s, purple triangles: 850 s, orange triangles: 1080
s. See also the main text.

In Figure 11a) and b) we compare for the time evolution of tactoids ranging
three orders in magnitude in size following the switching on of the electric
field. Figure 11a) shows the experimental findings for the aspect ratio $x$ of
the tactoids as a function volume, and Figure 11b) the results of our
numerical simulations, where we used the value of $\Sigma=500$ estimated from
the restricted model of Ref. [36]. The other parameters we set at
$\gamma=0.05$ and $\Gamma_{\alpha}=1.6\times 10^{-4}$ s-1 to obtain reasonable
agreement (by eye). Note that the data set of the results of this figure is
different from the one shown in Fig. 10. Different colors and symbols are used
to denote various points in time. Agreement is quantitative. The figures show
that for larger droplets, a gradual increase in elongation occurs over time
due to the coupling to the external field. In contrast, smaller droplets
exhibit minimal changes in their aspect ratios, because the Coulomb energy is
not strong enough to be able to affect any changes in the droplet shape as it
is strongly volume-dependent. The behavior predicted from our model closely
mirrors the trends observed in the experimental data points, indicating an
agreement between our simulations and the experimental observations.

##  V Conclusion

In this study, we investigate theoretically the behavior of spindle-shaped
nematic tactoid droplets in response to the switching on of an external
electric field. For this, we set up a relaxational kinetic model based on a
free energy landscape in terms of two reaction coordinates, extending our
earlier work on the equilibrium structure of tactoids in an electric (or a
magnetic) field [36]. These reaction coordinates are the aspect ratio and the
degree of curvature of the director field of the droplet, which is presumed to
be quasi bipolar and continuously interpolates between a uniform and a bipolar
director field. The free energy is determined by a combination of the usual
Frank-Oseen elasticity, a surface contribution that includes a preference for
planar anchoring to the interface, and the coupling to the field via the
dielectric anisotropy of the nematic phase.

We find that the elongation of the tactoids is purely a kinetic effect.
Indeed, it is not uncommon in nature for kinetic effects to transiently modify
the shape and symmetry of a structure before it attains its final equilibrium
form [59]. In this paper, we showed that the elongation of the tactoids is a
transient effect and that they are only able to elongate strongly under the
action of the electric field if the relaxation time of the degree of
bipolarness of the director field is large compared to the relaxation time of
the aspect ratio. Hence, the elongation of tactoids is only transient albeit
that it may be very long-lived. Eventually, the aspect ratio returns to a much
smaller equilibrium value predicted by equilibrium theory. Since equilibrium
theory cannot explain strongly elongated configurations of tactoids observed
in the experiments of Metselaar en collaborators [37], we propose that it must
be the approach to the transient state that the measurements have probed. This
means that significantly longer measurement times are required to observe the
return to equilibrium.

This conclusion is not as far-fetched as it may seem, because the aspect
ratios of the largest drops observed in the experiments seem not yet to have
reached the largest value at the latest times measured, as can in fact be
concluded from the data shown in Figs. 10 and 11. That the relaxational
dynamics of tactoids may exhibit a very long-time tail can also be deduced
from the large scatter in aspect ratio found experimentally in a host of
different systems [3, 37, 16, 17, 20, 21, 22, 23, 18]. Perhaps even more
convincingly, a recent study shows that the producing tactoids by means of
nucleation-and-growth or making use of a microfluidic device result in
different aspect ratios that seem stable for a very long period of time [7].

Our simple relaxational model captures the most salient features of the
experimental findings. One of these is the observation that relatively small
tactoids do not appreciably respond to an applied field, even if they are
bipolar. The reason for this is that anchoring, Frank elastic and Coulomb free
energies all scale with different powers of the volume of a tactoid [36]. We
find that the larger the volume or the field strength, the shorter it takes
for the aspect ratio to reach its largest value, but the longer it takes to
relax back to its equilibrium value. Another significant aspect of this
investigation is our finding that if the electric field is suddenly turned
off, tactoids revert to their initial aspect ratio on a considerably shorter
time scale than needed for the full equilibration to take place in response to
switching on of the field. This seems also to agree with the experimental
findings shown in Fig. 2, although no quantitative data are available as far
as we are aware.

It seems that the underlying fundamental relaxation time of the bipolarness of
the director field of tactoids is much lower than that of the aspect ratio of
the tactoids. This makes their aspect ratio become in a sense enslaved by the
anchoring of the director field to the interface with the isotropic host
suspension [37, 36]. The reason for this behavior remains unclear. Within our
model, it might reflect differences in the values of the viscosities that
describe the relaxations of the shape and director field of a tactoid. These
include five Leslie coefficients and the viscosity of the isotropic phase
[52]. On the other hand, it may also point at the existence of complex
interfacial relaxations not captured by current models. The issue in our view
merits further study.

###### Acknowledgements.

We thank Patrick Davidson and Ivan Drozov (Université Paris-Saclay) for
helpful discussions and sharing of experimental data. M.S. and R.Z.
acknowledge support from NSF DMR-2131963 and the University of California
Multicampus Research Programs and Initiatives (Grant No. M21PR3267).

## References

  * Dogic [2016] Z. Dogic, Filamentous phages as a model system in soft matter physics, Frontiers in microbiology 7, 1013 (2016).
  * van der Schoot [2022] P. van der Schoot,  _Molecular Theory of Nematic (and Other) Liquid Crystals: An Introduction_ (Springer Nature, 2022).
  * Jamali  _et al._ [2015] V. Jamali, N. Behabtu, B. Senyuk, J. A. Lee, I. I. Smalyukh, P. van der Schoot, and M. Pasquali, Experimental realization of crossover in shape and director field of nematic tactoids, Physical Review E 91, 042507 (2015).
  * van der Beek  _et al._ [2006] D. van der Beek, H. Reich, P. van der Schoot, M. Dijkstra, T. Schilling, R. Vink, M. Schmidt, R. van Roij, and H. Lekkerkerker, Isotropic-nematic interface and wetting in suspensions of colloidal platelets, Physical Review Letters 97, 087801 (2006).
  * van Bruggen  _et al._ [1996] M. P. van Bruggen, F. M. van der Kooij, and H. N. Lekkerkerker, Liquid crystal phase transitions in dispersions of rod-like colloidal particles, Journal of Physics: Condensed Matter 8, 9451 (1996).
  * Oakes  _et al._ [2007a] P. W. Oakes, J. Viamontes, and J. X. Tang, Growth of tactoidal droplets during the first-order isotropic to nematic phase transition of f-actin, Physical Review E 75, 061902 (2007a).
  * Almohammadi  _et al._ [2023] H. Almohammadi, S. Martinek, Y. Yuan, P. Fischer, and R. Mezzenga, Disentangling kinetics from thermodynamics in heterogeneous colloidal systems, Nature Communications 14, 607 (2023).
  * Zöcher [1925] H. Zöcher, Über freiwillige Strukturbildung in Solen. (Eine neue Art anisotrop flüssiger Medien.), Zeitschrift für anorganische und allgemeine Chemie 147, 91 (1925).
  * Bawden [1936] F. C. Bawden, Liquid crystalline substances from virus-infected plants, Nature 138, 1051 (1936).
  * Dogic [2003] Z. Dogic, Surface freezing and a two-step pathway of the isotropic-smectic phase transition in colloidal rods, Physical Review Letters 91, 165701 (2003).
  * Modlińska  _et al._ [2015] A. Modlińska, A. M. Alsayed, and T. Gibaud, Condensation and dissolution of nematic droplets in dispersions of colloidal rods with thermo–sensitive depletants, Scientific Reports 5, 18432 (2015).
  * Tarafder  _et al._ [2020] A. K. Tarafder, A. von Kügelgen, A. J. Mellul, U. Schulze, D. G. Aarts, and T. A. Bharat, Phage liquid crystalline droplets form occlusive sheaths that encapsulate and protect infectious rod-shaped bacteria, Proceedings of the National Academy of Sciences 117, 4724 (2020).
  * Coper and Freundlich [1937] K. Coper and H. Freundlich, The formation of tactoids in iron oxide sols, Transactions of the Faraday Society 33, 348 (1937).
  * Sonin  _et al._ [2017] A. Sonin, N. Churochkina, A. Kaznacheev, and A. Golovanov, Mineral liquid crystals, Colloid Journal 79, 421 (2017).
  * Robinson [1956] C. Robinson, Liquid-crystalline structures in solutions of a polypeptide, Transactions of the Faraday Society 52, 571 (1956).
  * Puech  _et al._ [2010] N. Puech, E. Grelet, P. Poulin, C. Blanc, and P. van der Schoot, Nematic droplets in aqueous dispersions of carbon nanotubes, Physical Review E 82, 020702 (2010).
  * Oakes  _et al._ [2007b] P. W. Oakes, J. Viamontes, and J. X. Tang, Growth of tactoidal droplets during the first-order isotropic to nematic phase transition of F-actin, Physical Review E 75, 061902 (2007b).
  * Bagnani  _et al._ [2018] M. Bagnani, G. Nystrom, C. De Michele, and R. Mezzenga, Amyloid fibrils length controls shape and structure of nematic and cholesteric tactoids, Acs Nano 13, 591 (2018).
  * Kim  _et al._ [2013] Y.-K. Kim, S. V. Shiyanovskii, and O. D. Lavrentovich, Morphogenesis of defects and tactoids during isotropic–nematic phase transition in self-assembled lyotropic chromonic liquid crystals, Journal of Physics: Condensed Matter 25, 404202 (2013).
  * Revol  _et al._ [1992] J.-F. Revol, H. Bradford, J. Giasson, R. Marchessault, and D. Gray, Helicoidal self-ordering of cellulose microfibrils in aqueous suspension, International Journal of Biological Macromolecules 14, 170 (1992).
  * Kitzerow [1994] H.-S. Kitzerow, Polymer-dispersed liquid crystals from the nematic curvilinear aligned phase to ferroelectric films, Liquid Crystals 16, 1 (1994).
  * Park  _et al._ [2014] J. H. Park, J. Noh, C. Schütz, G. Salazar-Alvarez, G. Scalia, L. Bergström, and J. P. F. Lagerwall, Macroscopic control of helix orientation in films dried from cholesteric liquid-crystalline cellulose nanocrystal suspensions, ChemPhysChem 15, 1477 (2014).
  * Wang  _et al._ [2016] P.-X. Wang, W. Y. Hamad, and M. J. MacLachlan, Structure and transformation of tactoids in cellulose nanocrystal suspensions, Nature Communications 7, 11515 (2016).
  * Kaznacheev  _et al._ [2002] A. V. Kaznacheev, M. M. Bogdanov, and S. A. Taraskin, The nature of prolate shape of tactoids in lyotropic inorganic liquid crystals, Journal of Experimental and Theoretical Physics 95, 57 (2002).
  * Prinsen and van der Schoot [2003] P. Prinsen and P. van der Schoot, Shape and director-field transformation of tactoids, Physical Review E 68, 021701 (2003).
  * Prinsen and van der Schoot [2004a] P. Prinsen and P. van der Schoot, Continuous director-field transformation of nematic tactoids, The European Physical Journal E 13, 35 (2004a).
  * Prinsen and van der Schoot [2004b] P. Prinsen and P. van der Schoot, Parity breaking in nematic tactoids, Journal of Physics: Condensed Matter 16, 8835 (2004b).
  * Kaznacheev  _et al._ [2003] A. Kaznacheev, M. Bogdanov, and A. Sonin, The influence of anchoring energy on the prolate shape of tactoids in lyotropic inorganic liquid crystals, Journal of Experimental and Theoretical Physics 97, 1159 (2003).
  * Lettinga  _et al._ [2005] M. P. Lettinga, K. Kang, A. Imhof, D. Derks, and J. K. G. Dhont, Kinetic pathways of the nematic–isotropic phase transition as studied by confocal microscopy on rod-like viruses, Journal of Physics: Condensed Matter 17, S3609 (2005).
  * Lettinga  _et al._ [2006] M. P. Lettinga, K. Kang, P. Holmqvist, A. Imhof, D. Derks, and J. K. G. Dhont, Nematic-isotropic spinodal decomposition kinetics of rodlike viruses, Physical Review E 73, 011412 (2006).
  * Williams [1986] R. D. Williams, Two transitions in tangentially anchored nematic droplets, Journal of Physics A: Mathematical and General 19, 3211 (1986).
  * van Bijnen  _et al._ [2012] R. M. W. van Bijnen, R. H. J. Otten, and P. van der Schoot, Texture and shape of two-dimensional domains of nematic liquid crystals, Physical Review E 86, 051703 (2012).
  * Otten and van der Schoot [2012] R. H. J. Otten and P. van der Schoot, Deformable homeotropic nematic droplets in a magnetic field, The Journal of Chemical Physics 137, 154901 (2012).
  * Everts  _et al._ [2016] J. C. Everts, M. T. J. J. M. Punter, S. Samin, P. van der Schoot, and R. van Roij, A landau-de gennes theory for hard colloidal rods: Defects and tactoids, The Journal of Chemical Physics 144, 194901 (2016).
  * Almohammadi  _et al._ [2022] H. Almohammadi, S. A. Khadem, M. Bagnani, A. D. Rey, and R. Mezzenga, Shape and structural relaxation of colloidal tactoids, Nature Communications 13, 2778 (2022).
  * Safdari  _et al._ [2021] M. Safdari, R. Zandi, and P. Van der Schoot, Effect of electric fields on the director field and shape of nematic tactoids, Physical Review E 103, 062703 (2021).
  * Metselaar  _et al._ [2017] L. Metselaar, I. Dozov, K. Antonova, E. Belamie, P. Davidson, J. M. Yeomans, and A. Doostmohammadi, Electric-field-induced shape transition of nematic tactoids, Physical Review E 96, 022706 (2017).
  * Weirich  _et al._ [2017] K. L. Weirich, S. Banerjee, K. Dasbiswas, T. A. Witten, S. Vaikuntanathan, and M. L. Gardel, Liquid behavior of cross-linked actin bundles, Proceedings of the National Academy of Sciences 114, 2131 (2017).
  * van der Schoot [1999] P. van der Schoot, Remarks on the interfacial tension in colloidal systems, The Journal of Physical Chemistry B 103, 8804 (1999).
  * Chen and Gray [2002] W. Chen and D. G. Gray, Interfacial tension between isotropic and anisotropic phases of a suspension of rodlike particles, Langmuir 18, 633 (2002).
  * Almohammadi  _et al._ [2020] H. Almohammadi, M. Bagnani, and R. Mezzenga, Flow-induced order–order transitions in amyloid fibril liquid crystalline tactoids, Nature Communications 11, 5416 (2020).
  * Jamali  _et al._ [2017] V. Jamali, E. G. Biggers, P. van der Schoot, and M. Pasquali, Line tension of twist-free carbon nanotube lyotropic liquid crystal microdroplets on solid surfaces, Langmuir 33, 9115 (2017).
  * Kuhnhold and van der Schoot [2022] A. Kuhnhold and P. van der Schoot, Structure of nematic tactoids of hard rods, The Journal of Chemical Physics 156, 104501 (2022).
  * Rapini and Papoular [1969] A. Rapini and M. Papoular, Distorsion d’une lamelle nématique sous champ magnétique conditions d’ancrage aux parois, Le Journal de Physique Colloques 30, C4 (1969).
  * Hohenberg and Halperin [1977] P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Reviews of Modern Physics 49, 435 (1977).
  * Clark  _et al._ [2023] A. B. Clark, M. Safdari, S. Zoorob, R. Zandi, and P. van der Schoot, Relaxational dynamics of the t-number conversion of virus capsids, The Journal of Chemical Physics 159 (2023).
  * Kra  _et al._ [2023] K. Kra, S. Li, L. Gargowitsch, J. Degrouard, J. Pérez, R. Zandi, S. Bressanelli, and G. Tresset, Energetics and kinetic assembly pathways of hepatitis b virus capsids in the presence of antivirals, ACS nano (2023).
  * Primicerio [1995] M. Primicerio, Eg virga, variational theories for liquid crystals”, Chapman and Hall, London 30, 736 (1995).
  * Finner  _et al._ [2020] S. P. Finner, I. Pihlajamaa, and P. van der Schoot, Geometric percolation of hard nanorods: The interplay of spontaneous and externally induced uniaxial particle alignment, The Journal of Chemical Physics 152, 064902 (2020).
  * Frank [1958] F. C. Frank, I. liquid crystals. on the theory of liquid crystals, Discussions of the Faraday Society 25, 19 (1958).
  * Odijk [1986] T. Odijk, Elastic constants of nematic solutions of rod-like and semi-flexible polymers, Liquid Crystals 1, 553 (1986).
  * De Gennes and Prost [1993] P. G. De Gennes and J. Prost,  _The physics of liquid crystals_ (Oxford University Press, 1993).
  * Toropova  _et al._ [2022] L. V. Toropova, E. V. Makoveeva, S. I. Osipov, A. P. Malygin, Y. Yang, and D. V. Alexandrov, Nucleation and growth of an ensemble of crystals during the intermediate stage of a phase transition in metastable liquids, Crystals 12, 895 (2022).
  * Volovik and Lavrentovich [1983] G. Volovik and O. Lavrentovich, Topological dynamics of defects: boojums in nematic drops, Zh. Eksp. Teor. Fiz. 85, 1997 (1983).
  * Wolfram [1999] S. Wolfram,  _The MATHEMATICA® book, version 4_ (Cambridge university press, 1999).
  * Doi [2011] M. Doi, Onsager’s variational principle in soft matter, J. Phys.: Condens. Matter 23, 284118 (2011).
  * Doi and Edwards [1988] M. Doi and S. F. Edwards,  _The theory of polymer dynamics_ (Oxford University Press, 1988).
  * Lerman [1980] P. Lerman, Fitting segmented regression models by grid search, Journal of the Royal Statistical Society Series C: Applied Statistics 29, 77 (1980).
  * Panahandeh  _et al._ [2022] S. Panahandeh, S. Li, B. Dragnea, and R. Zandi, Virus assembly pathways inside a host cell, ACS Nano 16, 317 (2022).