1. Introduction
Liquid crystals (LC) are fluids often made up of rod-like molecules that can be arranged in a variety of equilibrium phases, depending on their geometry and reciprocal interaction [
1,
2]. Of particular relevance is the cholesteric phase where the local molecular alignment is captured by a unit magnitude director field
displaying a natural twist deformation in the direction perpendicular to the molecules. Such liquid crystals can be found in a wide range of biological systems, such as bacterial flagella [
3] and DNA molecules in solution [
4], and have found increasing application in modern display devices, in which the cholesteric phase results from mixing a nematic liquid crystal (such as E7, Merck KGaA, Darmstadt, Germany) with a chiral dopant. In recent years, much interest has been focused on the realization of liquid crystal-particle composites, a new exciting class of soft material with tunable elastic and electro-optic properties, in which either colloidal particles or droplets of standard fluids are dispersed in an LC phase [
5]. The presence of these inclusions induces a disturbance in the director orientation, which typically aligns either tangentially or normally to the surface of the particles. This leads to the formation of a variety of topological defects that mediate long-range anisotropic particle-particle interactions and stabilize novel ordered structures, such as chain [
6,
7] and defect glass [
8] in nematics or colloidal crystals [
9] and isolated clusters [
10] in cholesterics, with potential use in photonics [
11] and in new devices [
12].
Recent theoretical and experimental works have investigated the typical defect structures observed at equilibrium when particles are dispersed in cholesteric liquid crystals (CLC) and have shown that these can be controlled by tuning the ratio between the particle size
R (typically its radius) and the cholesteric pitch
p. For instance, if strong perpendicular (homeotropic) anchoring is set on the surface of a spherical colloid, one observes either a planar Saturn ring (a circular defect line of half-integer charge located around the equator of the particle) if
, or a twisted Saturn rings, wrapping around the particle, if
[
13,
14,
15,
16]. If, on the other hand, the anchoring is tangential to the surface, the defect pattern ranges from the “boojum” (two surface defects of integer charge located on opposite sides) if
is small, to its twisted version if
is large enough [
15]. A further complication arises when, in place of a solid particle, one considers a deformable liquid droplet. Here, the strength of the interface anchoring relative to the elasticity of the LC becomes a crucial parameter to control the droplet shape (in addition to the director pattern near the surface). For instance, one observes either a spherical shape or a nutshell-like structure if the anchoring strength is respectively weak or strong [
17]. So far, most of the studies have focused on the equilibrium properties of these inclusions [
15], and much less is known when these systems are subject to external perturbations, such as electric and flow fields.
Liquid crystal have long been known as systems that exhibit a rich dynamical response such as shear banding [
18,
19,
20] and molecular tumbling [
1,
21,
22] when subject to shear flow [
1]. This is due to the complex coupling between hydrodynamics and the director field (known as backflow), a feature that is even more relevant in a CLC where the inherent three-dimensional twisted arrangement of the director field can give rise to striking effects, such as a significant increase in viscosity when subject to a shear flow in the direction of its helix (permeation) [
23,
24,
25]. Recent simulations on colloidal dispersions in CLC report, for instance, a violation of the Stokes law when a single particle is dragged parallel to the cholesteric helix [
13], or an induced rotation, either continuous or stepwise, when two particles, forming a dimer, are pulled through the cholesteric phase with a constant force along the helical axis [
26].
Here, we present a preliminary study on the effect that an imposed shear flow has on the rheology of an inverted cholesteric emulsion, described as a single isotropic (liquid crystal) droplet (in which molecules are randomly oriented) surrounded by a cholesteric liquid crystal. The study has been carried out by using a lattice Boltzmann approach [
27], a method that solves numerically the Beris–Edwards equations of cholesteric hydrodynamics [
28] coexisting with an isotropic phase and already tested for an inverted nematic emulsion in the presence of an electric field [
29] or a shear flow [
17]. The main strength of the algorithm is that backflow effects are automatically included and the dynamics of topological defects can be easily tracked. By varying the shear rate and the ratio between the elastic energy scales of the cholesteric in the bulk and at the droplet interface, we show that the presence of a liquid droplet strongly reduces the secondary flow usually observed in a pure cholesteric phase. Such an effect has been found more pronounced when strong perpendicular anchoring is set on the droplet interface and is also confirmed when the chirality degree of the cholesteric is increased, even though in this case, the system may temporarily enter the blue phase for high shear rates [
30].
The paper is organized as follows. In
Section 2, we describe the numerical model of a 2D droplet of isotropic fluid suspended in a CLC, in particular its equilibrium phase behavior, encoded by a Landau–de Gennes free energy, and its hydrodynamics, captured by the Beris–Edwards equations of motion for CLCs coupled to Cahn–Hilliard dynamics. In
Section 3, we first discuss the equilibrium properties of a cholesteric sample and then those of a single droplet of Newtonian fluid dispersed into it, both in the absence of anchoring and with homeotropic anchoring. We next present the main results of the paper, namely the effect that a symmetric shear flow has on the droplet-CLC system when different values of interface anchoring and chirality are considered. Finally, the last section is dedicated to discussing the results and conclusions.
2. Model and Methods
We consider an isolated droplet of a Newtonian isotropic fluid dispersed in a medium of cholesteric liquid crystal. This setup is usually referred as inverted cholesteric emulsion [
7,
17,
31] to be distinguished from a direct emulsion, where a liquid crystal droplet is immersed in an isotropic fluid. The hydrodynamics of such a system can be described by using an extended version of the Beris–Edwards theory [
28] for chiral fluids, already adopted in previous works on liquid crystals [
32,
33,
34,
35]. Here, we briefly recap the model.
The equilibrium properties are captured by a Landau–de Gennes [
1] free-energy functional
, where the free-energy density
f is given by the sum of the following terms:
while the second contribution
is added in the presence of a bounding surface. Here,
is a scalar order parameter related to the concentration of the cholesteric phase relative to the isotropic phase, while
is a tensor order parameter that, within the Beris–Edwards theory [
28], describes the cholesteric phase. It is defined as
, where
is the director field (Greek subscripts denote Cartesian coordinates) describing the local orientation of the molecules (in the uniaxial approximation
), and
q is the local degree of order, proportional to the largest eigenvalue of
(
). The first term of Equation (
1) describes the bulk properties of the mixture and is given by:
where
a and
are two positive phenomenological constants controlling the interface width
of the droplet, which, for a binary fluid without liquid crystal, goes as
. Equation (
2) is borrowed from binary fluid mixtures and enables the formation of two phases: the isotropic one (inside the droplet where
) and the cholesteric one (outside the droplet where
), separated by an interface whose energetic cost is gauged by the gradient term. The second term is the cholesteric liquid crystal free-energy density given by:
The terms multiplied by the positive constant
stem from a truncated polynomial expansion up to the fourth order in
[
1] and describe the bulk properties of a uniaxial nematic liquid crystal with an isotropic-to-nematic transition at
. Here,
plays the role of an effective temperature. For a nematogen without chirality (
), the phase is isotropic if
; otherwise, it is cholesteric. By following previous studies [
17,
29,
36], we set
, where
and
control the boundary of the coexistence region. The remaining terms of Equation (
3), multiplied by the constant
K, take into account the elastic energy due to the local deformations of the cholesteric arrangement and enter the free energy through first order gradient contributions, except a gradient-free term included to have a positive elastic free energy. The parameter
determines the pitch length
of the cholesteric, and
is the Levi–Civita antisymmetric tensor. Here, we consider the “one elastic constant” approximation, an approach usually adopted when investigating liquid crystals as it considerably simplifies theoretical calculations [
1]. The energetic cost due to the anchoring of the liquid crystal at the droplet interface is included through:
where the constant
L controls the strength of the anchoring. If it is negative, the liquid crystals are homeotropic (perpendicular) to the interface, whereas if positive, the liquid crystal lies tangentially. Finally, if confining walls are included, a further term needs to be considered in the free energy functional. For homeotropic anchoring, which is the case considered here, one has:
where
W controls the strength of the anchoring at the walls and
sets the preferred configuration of the tensor order parameter at the surface.
The thermodynamic state of our mixture is specified by two dimensionless quantities, the reduced temperature and the reduced chirality, given by:
The former multiplies the quadratic terms of the dimensionless bulk free energy and vanishes at the spinodal point of a nematic, while the latter multiplies the gradient terms and gauges the amount of twist accumulated in the system [
30]. Such parameters have been used, for instance, in [
37,
38] for the numerical calculation of the phase diagram of cholesteric liquid crystals. Note, in particular, that according to Equation (7), the knowledge of
is not sufficient information (except if
) to correctly assess the value of the chirality (hence, the phase of the liquid crystal), as this is also affected by other thermodynamic parameters (
A,
K and
).
The dynamics of the system is governed by a set of balance equations, the first of which is the equation of the tensor order parameter
:
where
is the velocity of the fluid, and the term on the left-hand side is the material derivative, describing the rate of change of
advected by the flow. The derivative includes the tensor
since the order parameter can be rotated and stretched by local velocity gradients
and is given by:
Here,
and
are the symmetric and antisymmetric parts of the velocity gradient tensor,
denotes the tensorial trace and
is the unit matrix. The constant
depends on the molecular details of the liquid crystal and controls the dynamics of the director field under shear flow. Indeed, after imposing a homogeneous shear on a nematic liquid crystal, at steady state, the director will align along the flow gradient at an angle
fulfilling the relation
. Real solutions are obtained when
. Throughout our simulations, we have set
. Finally, on the right-hand side of Equation (
8),
is the collective rotational diffusion constant and
is the molecular field given by:
The time evolution of the concentration field
is governed by a convection-diffusion equation:
where
M is the mobility and
is the chemical potential, while the force balance is ensured by the incompressible Navier–Stokes equation,
where the total stress tensor
is the sum of four contributions:
The first term
is the ideal background pressure, and
T is the temperature. The second one is the viscous stress tensor:
where
is the isotropic shear viscosity. The third one is the elastic stress due to the liquid crystalline order:
and the last term is the interfacial stress between the isotropic and liquid crystal phase:
The total isotropic pressure of the system is the sum of the ideal background pressure
plus the isotropic term of Equation (
16) (constant in our simulations, except nearby the defects and close to the droplet interface where it augments) and of Equation (
17). Both
and
take into account the non-Newtonian fluid effects. In fact, if
, they reduce, respectively, to the scalar pressure and to the interfacial stress of a binary Newtonian fluid mixture.
Equations (
8) and (
11)–(13) are solved numerically by means of a hybrid lattice Boltzmann method [
36,
39], which uses a combination of a standard lattice Boltzmann approach to solve the Navier–Stokes and the continuity equation and a finite-difference scheme to solve Equations (
8) and (
11).
All simulations are performed on a quasi-two-dimensional rectangular box of size in which the cholesteric liquid crystal and the droplet are embedded. We choose a longer size along the y-direction in order to minimize the interference between periodic images of the droplet moving with the flow when a shear flow is applied. The droplet, in particular, is modeled as a circular isotropic (liquid-crystalline) region with radius lattice sites, initially placed at the center of the sample and surrounded by the liquid crystal phase. This quasi-2D setup is chosen to allow an out-of-plane component of the vector fields along the x-direction that accommodates the three-dimensional twisted arrangement of the cholesteric and, in the presence of a shear flow, captures the secondary flow appearing perpendicular to the direction of the shearing, along the x-direction. The entire system is periodic along the y-direction and is sandwiched between two flat walls parallel to the -plane and lying at and . The walls can be either at rest or moving along the y-direction with velocity and (with at and , respectively. There, we impose neutral wetting conditions (i.e., , where is a unit vector perpendicular to the wall) for the concentration, no-slip conditions (the fluid moves with the same velocity of the walls) for the fluid velocity and homeotropic conditions (i.e., the director is perpendicular to the walls) for the liquid crystal order parameter.
Initial conditions of
and
in the bulk are as follows. They are both set to zero inside the droplet, while outside,
is fixed to a constant value (
), and the components of
are given by:
where
,
, to accommodate a cholesteric liquid crystal with the helical axis parallel to the
y-direction. Such components stem from assuming that the ground state of the director field is given by
, where
is a unit vector oriented along the Cartesian axis. The parameter
controls the number
N of
twists that the liquid crystal displays in a cell of length
. Indeed, one can define
in terms of the linear size
, as
, to compare the pitch length with the cell size.
Starting from these initial conditions, the system is first allowed to relax to its equilibrium state, and afterwards, both walls are sheared along opposite directions at constant speed. Typical parameters used in our simulations are
,
,
,
,
,
,
,
and
. Finally, one has to choose the values of
and
in order to have the cholesteric phase outside the droplet [
37,
38]. We set
(i.e.,
) and
(which controls the chirality
; see Equation (7)) equal to
or
, in order to have
or
on a lattice of linear size
. If, for instance,
, one has
, well inside the cholesteric phase according to the phase diagram of [
37,
38].
In order to map such numerical values to typical physical ones, we consider a droplet of size 1–10
m immersed in a cholesteric liquid crystal of elastic constant roughly 10 pN and a viscosity of 1 Poise. This corresponds to a lattice space
m, and the time step is
s (they are both equal to one in our simulations). Finally, the anchoring coefficient
L correspond to values
Jm
[
29,
40].
4. Conclusions
To summarize, we have studied, by numerical simulations, the rheological response of an inverted cholesteric droplet sandwiched between two planar walls under a symmetric shear flow. We have shown that the dynamics is affected by the shear rate, the strength of the interface anchoring and the chirality of the cholesteric. In particular, the presence of a relatively intense secondary flow (emerging out of the plane of the cholesteric) suggests that the system is essentially non-Newtonian, an effect generally weakened if an isotropic droplet is included and further reduced if strong interface anchoring is imposed.
If the droplet is absent, a moderate shear flow drives the sample towards a steady state in which the director field is almost everywhere tilted along the direction imposed by the shear itself, except in regions where the S-like -charge region is sustained by disclinations pinned at the walls. Importantly, increasing the chirality favors a non-Newtonian response of the cholesteric witnessed by a sustained secondary flow. On the other hand, augmenting the shear-rate determines a decrease of the transition temperature, which leads the system to either the nematic state or, if the chirality is higher, to a blue-phase-like state. The dynamics is significantly different if an isotropic droplet is embedded in the sample. If the interface anchoring is weak, we generally find a reduction of the secondary flow with respect to the droplet-free cholesteric sample, an effect even more pronounced if the interface anchoring is strong. We ascribe the latter effect to the resistance encountered by the flow to overcome a larger elastic deformation of the director field, in particular near the droplet interface where two fully in-plane topological defects form. These results are overall confirmed when the chirality is increased, although here an intense secondary flow, comparable to that observed in the droplet-free sample, is found for the low shear-rate and weak anchoring. This highlights the fact that the rheological response displayed by an inverted cholesteric droplet has a complex landscape where a key role is played by at least three quantities: shear rate, elasticity and chirality.
Our results represent a first step in the study of the rheology of an inverted cholesteric droplet and is of possible interest for designing CLC-based devices built from an emulsion. This opens up several directions for future research. It would be worth investigating the case in which tangential anchoring is considered (both at the droplet interface and at the walls) in a cholesteric sample whose axis is perpendicular to the walls. Besides yielding to the formation of different topological defects (such as a twisted Saturn ring in 3D), such a configuration is expected to display a rich dynamical behavior like that observed with homeotropic interface anchoring, which strongly depend on the cholesteric pitch (or the chirality). This can pave the way to extend the study to include several droplets, either in a monodisperse setup or in the more intriguing polydisperse one where different droplet interface anchoring sets may be considered. One can envisage, for instance, the design of a new soft material made of highly-packed isotropic droplets, the resistance to deformation of which could be sustained by the liquid crystal dispersed in between. Finally, although three-dimensional simulations can be computationally demanding, it would be surely of great interest to investigate the physics of more realistic systems in order, for instance, to minimize finite size effects. However, we note that quasi-two-dimensional liquid crystal devices could be experimentally realized, such as that described in [
45], where a smectic-C film surrounding droplets, obtained by nucleation, is proposed.