Defect structures in nematic liquid crystals around charged particles.

Abstract

We numerically study the orientation deformations in nematic liquid crystals around charged particles. We set up a Ginzburg-Landau theory with inhomogeneous electric field. If the dielectric anisotropy epsilon 1 is positive, Saturn-ring defects are formed around the particles. For epsilon 1< 0 , novel "ansa" defects appear, which are disclination lines with their ends on the particle surface. We find unique defect structures around two charged particles. To lower the free energy, oppositely charged particle pairs tend to be aligned in the parallel direction for epsilon 1> 0 and in the perpendicular plane for epsilon 1< 0 with respect to the background director. For identically charged pairs the preferred directions for epsilon 1> 0 and epsilon 1< 0 are exchanged. We also examine competition between the charge-induced anchoring and the short-range anchoring. If the short-range anchoring is sufficiently strong, it can be effective in the vicinity of the surface, while the director orientation is governed by the long-range electrostatic interaction far from the surface.

Full text

Page 1

EPJ manuscript No.
(will be inserted by the editor)
Defect structures in nematic liquid crystals around charged
particles
Keisuke Tojo1, Akira Furukawa2, Takeaki Araki1, and Akira Onuki1
1Department of Physics, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan
2Institute of Industrial Science, University of Tokyo, Meguro-ku, Tokyo 153-8505, Japan
Received: August 11, 2009
Abstract. We numerically study the orientation deformations in nematic liquid crystals around charged
particles. We set up a Ginzburg-Landau theory with inhomogeneous electric field. If the dielectric
anisotropy ε1 is positive, Saturn ring defects are formed around the particles. For ε1 < 0, novel “ansa”
defects appear, which are disclination lines with their ends on the particle surface. We find unique defect
structures around two charged particles. To lower the free energy, oppositely charged particle pairs tend
to be aligned in the parallel direction for ε1 > 0 and in the perpendicular plane for ε1 < 0 with respect
to the background director . For identically charged pairs the preferred directions for ε1 > 0 and ε1 < 0
are exchanged. We also examie competition between the charge-induced anchoring and the short-range an-
choring. If the short-range anchoring is sufficiently strong, it can be effective in the vicinity of the surface,
while the director orientation is governed by the long-range electrostatic interaction far from the surface.
PACS. 61.30.Dk Continuum models and theories of liquid crystal structure – 61.30.Jf Defects in liquid
crystals – 77.84.Nh Liquids, emulsions, and suspensions; liquid crystals – 61.30.Gd Orientational order of
liquid crystals; electric and magnetic field effects on order
1 Introduction
A variety of mesoscopic structures have been found in liq-
uid crystals around inclusions such as colloids and water
droplets [1,2,3]. In nematics, inclusions distort the ori-
entation order over long distances, inducing topological
defects [4,5,6,7,8,9,10,11,12,13]. We mention the forma-
tion of structures or phases, such as string-like aggregates
[2,12,14,15], soft solids supported by a jammed cellular
network of particles [16], and a transparent phase includ-
ing microemulsions [17,18]. The origin of the long-range
distortions has been ascribed to the anchoring of the liq-
uid crystal molecules on the inclusion surface [4,5,6,7,8,9,
10,11,12,13,19,20]. It arises from the short-range molec-
ular interactions between the liquid crystal molecules and
the surface molecules. In the Ginzburg-Landau-de Gennes
theory, we have a surface free energy depending on the ori-
entation of liquid crystal molecules on the surface.
In this paper, we are interested in another anchoring
mechanism. That is, electrically charged inclusions align
the liquid crystal molecules in their vicinity to lower the
electrostatic energy [21,22,23], which can be relevant for
ions and charged particles. In fact, de Gennes [24,25] at-
tributed the origin of the small size of the ion mobility
in nematics to a long-range deformation of the orienta-
tion order around ions. However, the effect of charges in
Send offprint requests to:
liquid crystals remains complicated and has rarely been
studied, despite its obvious fundamental and technologi-
cal importance. It is of great interest how the electric-field
anchoring mechanism works and how it is different from
the usual short-range anchoring mechanism.
The electric field and the liquid crystal orientation are
coupled because the dielectric tensor εij depends on the
local orientation tensor Qij(see equation (14)). The align-
ment along a homogeneous electric field is well-known [25],
but the alignment in an inhomogeneous electric field has
not yet been well studied. When the dielectric tensor is
inhomegeneous, it is a difficult task to solve the Pois-
son equation and seek the electric potential Φ. We here
perform numerical simulations placing charged particles
in liquid crystals in a three-dimensional cell. We use the
Ginzburg-Landau-de Gennes scheme in terms of the ori-
entation tensor Qij[23,25,26,27]. A similar approach has
recently been used to calculate the polarization and com-
position deformations around charged particles in elec-
trolytes [28]. It is worth noting that hydration of water
molecules around ions is analogous to the orientation an-
choring of liquid crystal molecules around charged parti-
cles, as pointed out by de Gennes [24,25].
In Section 2, we will present a Ginzburg-Landau-de
Gennes theory for liquid crystals containing charged par-
ticles. In particular, we will give two general forms of the
electrostatic free energy for the fixed-charge and fixed-
arXiv:0908.1455v1 [cond-mat.stat-mech] 11 Aug 2009

Page 2

2Keisuke Tojo et al.: Defect structures in nematic liquid crystals around charged particles
potential cases (which can be used for any dielectric fluids
containg charges). In Section 3, we will explain the nu-
merical method adopted in this work. in Section 4, we will
present numerical results of equilibrium configurations of
the orientation order around charged particles. We will
also examine competition of the short-range and electric-
field anchoring mechanisms. In Section 5, a summary and
critical remarks will be given.
2 Theoretical Background
We consider a liquid crystal system in a cubic box and
place one or two charged spherical particles with radius R
inside the box. The particle positions are written as Rn
(n = 1,2). The liquid crystal order is described in terms of
the symmetric orientational tensor Qij(r) with the trace-
less condition Qii= 0 [25]. We place one or two charged
particles with radius R considerably longer than the ra-
dius of the solvent molecules. In this work the Boltzmann
constant is set equal to unity and the temperature T rep-
resents the thermal energy of a liquid crystal molecule.
2.1 Model
We are interested in the equilibrium liquid crystal orienta-
tion around the particles, which minimizes the sum of the
Landau-de Gennes free energy, the short-range anchoring
energy, and the electrostatic energy. Thus the total free
energy of the liquid crystal containing charged particles
consists of four parts as [21,22,23]
F = F0+ Fg+ Fa+ Fe.
(1)
The first term is of the Landau-de Gennes form,
??
where we introduce
F0=
dr
?A
2J2−B
3J3+C
4J2
2
?
,
(2)
J2= Q2
ij,J3= QijQjkQki.
(3)
Hereafter repeated indices are implicitly summed over.
The coefficient A is dependent on the temperature T,
while the coefficients B and C are positive constants as-
sumed to be independent of T. The second term is the
gradient free energy in the one-constant approximation,
??
where ∇k= ∂/∂xk(xk= x,y,z) are the space derivatives
and L is a positive constant. The space integrals??dr in
the particles |r − Rn| > R. It is convenient to define the
length,
d = T/L,
Fg=L
2
dr(∇kQij)2,
(4)
equations (2) and (4) are to be performed only outside
(5)
Fig. 1. A capacitor and an inhomogeneous fluid containing a
net charge Qin in the fixed-charge case (a) and in the fixed-
potential case (b). The charge and potential of the lower plate
are Qb and Φb, while those of the upper plate are Qt and Φt.
which is the typical molecular size of liquid crystal. The
term Farepresents the short-range anchoring free energy.
It is expressed as the integral on the particle surfaces,
?
where da is the surface element, ν is the outward normal
unit vector to the surface, and w represents the strength
of the anchoring. For the uniaxial form Qij = S(ninj−
δij/3), we have Fa = wS?da[1/3 − (ν · n)2]. Thus, for
homeotropic and planar anchoring, respectively.
We explain the electrostatic part Fe, which depends
on the experimental method. As a generalization of the
theory by one of the present authors [21], we allow that
the fluid region can contain a net charge Qin =
where ρ = ρ(r) is the charge density inside the fluid. As
in Figure 1, we insert the fluid between parallel metallic
plates in the region 0 < z < H. The surface charge and
the potential of the lower plate at z = 0 are Qband Φb,
while those of the upper plate at z = H are Qtand Φt.
We require the overall charge neutrality condition,
Fa= −w
daνiνjQij,
(6)
neutral particles, positive and negative values of w lead to
?drρ,
Qin+ Qb+ Qt= 0,
(7)
since the eletric field in the metal plates should vanish. In
terms of Q ≡ (Qt− Qb)/2, we may set
Qb= −Q − Qin/2,
(i) In (a) in Figure 1, Q can be fixed and can be a control
parameter, where the potential difference,
Qt= Q − Qin/2.
(8)
V = Φt− Φb,
(9)
depends on the fluid inhomogeneity induced by the chaged
particles. Here the electrostatic energy of the surface charges
of the plates is fixed, the appropriate form of Feis
1
8π
?
Fe=
?
drD · E
drρ
2
=
?
Φ −Φt+ Φb
2
?
+QV
2
,
(10)

Page 3

Keisuke Tojo et al.: Defect structures in nematic liquid crystals around charged particles3
where Ei= −∂Φ/∂xiis the electric field and Di= εijEj
is the electric induction with εijbeing the dielecric tensor.
Here we superimpose small variations δQb, δQt, δρ, and
δεijon Qb, Qt, ρ, and εij, respectively. We use the relation
?drE · δD/4π = ΦbδQb+ ΦtδQt+?drΦδρ. We then
?
−1
8π
obtain the incremental change of Feas
δFe= V δQ +
drδρ
?
Φ −Φt+ Φb
2
?
?
drδεijEiEj.
(11)
(ii) On the other hand, in (b) in Figure 1, the potential
difference V can be fixed and can be a control parameter
with Q being dependent on the fluid inhomogeneity. The
appropriate form of Feis
1
8π
?
where the second line follows from the second line of equa-
tion (10). This is the Legendre transformation of the elec-
trostatic free energy in the fixed-charge case. Here we use
the same notation Fe in the two cases. Then the incre-
mental change of Fereads
?
−1
8π
Fe=
?
drD · E − V Q
?
=
drρ
?
Φ −Φb+ Φt
2
?
−D · E
8π
?
,
(12)
δFe= −QδV +
drδρ
?
Φ −Φt+ Φb
2
?
?
drδεijEiEj.
(13)
where the first term on the right hand side is different from
that in equation (11). It is worth noting that the second
line of equation (12) yields the frequently used expression
Fe= −?drD · E/8π in the fixed-potential condition for
for example).
The potential Φ satisfies the Poisson equation,
dielectric fluids without charge (ρ = 0) (see reference[24],
∇i(εij∇jΦ) = −4πρ.
(14)
We assume the linear form of the dielectric tensor,
εij(r) = ε0δij+ ε1Qij(r),
(15)
in the liquid crystal region (the particle exterior)1. Defin-
ing Φ in the whole space, we may solve equation (14) by
setting εij(r) = εpδijin the particle interior. Then the in-
tegrals in equations (10) and (12) are over the whole cell
region. We then have δFe/δQij = −ε1EiEj/8π both at
fixed Q and at fixed V .
1In the nematic state we have ε?= ε0+ 2Sε1/3 along the
director n and ε?= ε0−Sε1/3 in the perpendicular directions
[24], where the amplitude S is given in equation (20).
2.2 Equilibrium conditions
In our numerical work we will adopt the geometry (b) in
Figure 1 and set V = 0. The charge density ρ is fixed. We
define the tensor, hij≡ δF/δQij+λδij, where λ is chosen
such that hijbecomes traceless. Some calculations give
hij= (A + CJ2)Qij− B
− L∇2Qij−ε1
?
QikQkj−1
EiEj−1
3J2δij
?
?
8π
?
3E2δij
.
(16)
In equilibrium, minimization of F yields
hij= 0,
(17)
in the particle exterior. The boundary condition of Qijon
the particle surface is given by
Lν · ∇Qij+ w(νiνj− δij/3) = 0.
Obviously, the defect structure is independent of the sign
of the particle charge, since Qijis coupled to the bilinear
terms of E in equation (16).
For B > 0 uniaxial states are selected in the bulk re-
gion below the isotropic-nematic transition A < At[25],
where Qij= S(ninj− δij/3) and
At= B2/27C.
(18)
(19)
Substituting the uniaxial form into the first line of equa-
tion (16), we obtain 2CS2−BS +3A = 0, which is solved
to give
S = B/4C + [(B/4C)2− 3A/2C]1/2.
Just below the transition we have S = St≡ B/3C. How-
ever, it is known that the liquid crystal order is consider-
ably biaxial inside defect cores [8,23,26]. See Figure 3 of
Ref.[23] for the biaxiality of the Saturn ring core (where
the spatial mesh size is finer than in this work). Note that
Qijcan generally be expressed as
(20)
Qij= S1(ninj− δij/3) + S2(mimj− ?i?j),
where n, m, and ? constitute three orthogonal unit vec-
tors. Inside defect cores, the amplitude S2of biaxial order
is of the same order as the amplitude S1(= S in this work)
of uniaxial order. Outside the defect cores, S2nearly van-
ishes and the orientation order becomes uniaxial.
In addition, the polarization vector of the liquid crystal
is given by Pi= χijEjin terms of the susceptibility tensor
χij. From εijEj= Ei+ 4πPi, we have
(21)
χij= (εij− δij)/4π.
(22)
This tensor should be positive-definite in equilibrium to
ensure the thermodynamic stability in the (paraelectric)
nematic phase [23]. For the special form (15) this require-
ment becomes
ε0− 1 + ε1qα> 0,
where qα(α = 1,2,3) are the eigenvalues of Qij.
(23)

Page 4

4Keisuke Tojo et al.: Defect structures in nematic liquid crystals around charged particles
2.3 Electric field effect near the surface
Let us consider the electric field effect near a particle sur-
face. For simplicity we assume |ε1| <∼ε0. Then the sur-
face electric field Esis estimated to be of order eZ/ε0R2,
where Ze is the particle charge (with e being the elemen-
tary charge). (i) Far above the transition A ? Atin the
isotropic phase, we neglect the terms proportional to B,
C, and L in equation (17) to obtain Qij∼= ε1E2(xixj/r2−
δij/3)/8πA, which grows as A is decreased as a pretransi-
tional effect. (ii) Just below the transition, a nonlinear de-
formation occurs for |ε1|E2
is easily realized for small B. (iii) In the nematic phase
far below the transition, strong nonlinear deformations of
Qijare induced on the surface for R < ? with [23]
s/8π >
∼AtSt= B3/81C2, which
? = |Z|(|ε1|?Bd/12πε0S)1/2,
(24)
where d is defined by equation (5) and
?B= e2/ε0T
(25)
is the Bjerrum length. This criterion arises from the bal-
ance of the gradient term (∼ LSR−2) and the electrostatic
term (∼ ε1E2
more, for sufficiently large ?/R, a defect is formed around
the particle, where the distance from the surface is of order
? − R.
It is important to clarify the condition of defect for-
mation in real systems. Let us assume ε0∼ 2, |ε1| ∼ ε0,
S ∼ 1, d ∼ 2nm, and ?B∼ 24nm. Then ? ∼ |Z|nm. Thus,
the relation R < ? holds for microscopic ions, though our
coarse-grained model is inaccurate on the angstrom scale.
See the remark (3) in the last section for a comment on
ions in liquid crystal. We may also consider a large particle
with a constant surface charge density
s/8π ∝ R−4) in hijin equation (17). Further-
σ = Z/4πR2.
(26)
It may be difficult to induce sufficient ionization on col-
loidal surfaces in liquid crystal solvents. One method of
realizing charged surfaces will be to attach ionic surfac-
tant molecules on colloidal surfaces. For such a particle,
the condition of defect formation becomes R ? Rc, where
Rc= (3ε0S/4π|ε1|?Bd)1/2σ−1.
Using the above parameter values, we have Rc∼ 0.1σ−1nm
(with σ in units of nm−2). For example, if σ = 0.0624nm−2
or eσ = 1µC/cm2, we obtain Rc= 1.6nm. Here the elec-
tric field at the surface is eσ/4πε0 ∼ 100V/µm, which
is strong enough to align the director field. Electric field
applied macroscopically is typically of order 1V/µm [30,
31].
(27)
3 Simulation method
We give our simulation method in the Landau-de Gennes
scheme under the condition of V = 0. For simplicity, we
Fig. 2. Derivative ∂J2/∂r in units of d−1and gradient free
energy density fg = L(∇kQij)2/2 in units of Td−3vs normal-
ized distance (r−R)/R from the surface of a charged spherical
particle. The path starts from a surface position and passes
through a Saturn ring (see Figure 3).
impose the periodic boundary condition in the xy plane.
We suppose nanoscale particles confined between a thin
layer.
In the previous section we have assumed sharp bound-
aries between the particles and the liquid crystal region.
However, precise simulations are not easy in the presence
of sharp curved boundaries on a cubic lattice, unless the
mesh size is very small. In this work, to overcome this dif-
ficulty, we employed the smooth particle method. That is,
we introduce diffusive particle profiles by [9,11,12,19,20]
?R − |Rn− r|
where the surface is treated to be diffuse with thickness
d = T/L in equation (5), Rnrepresents the particle cen-
ter, and R is the particle radius.
In terms of φn(r), the overall particle and charge dis-
tributions are expressed as
?
ρ(r) =e
v
n
φn(r) =1
2tanh
d
?
+1
2,
(28)
φ(r) =
n
φn(r),
(29)
?
Znφn(r),
(30)
where Zne are the particle charges and v = 4πR3/3 is
the particle volume. The charge distribution is assumed
to be homogeneous inside the particles. In F0in equation
(2) and Fgin equation (4), the space integrals outside the
particles??dr should be redefined as
dr(···) =
??
?
dr[1 − φ(r)](···).
(31)
The surface integral in equation (6) is also redefined as
?
Then, the short-range anchoring free energy (6) is rewrit-
ten as
?
da(···) =
?
dr|∇φ|(···).
(32)
Fa= −wdrQij(∇iφ)(∇jφ)/|∇φ|.
(33)

Page 5

Keisuke Tojo et al.: Defect structures in nematic liquid crystals around charged particles5
The dielectric tensor is space-dependent as
εij(r) = [ε0+ (εp− ε0)φ]δij+ ε1(1 − φ)Qij,
where εpis the dielectric constant inside the particles.
To seek Qij satisfying equations (17) and (18), we
treated Qij(r,t) as a time-dependent tensor variable obey-
ing the evolution equation,
(34)
1
ζ
∂
∂tQij(r,t) = −δF
= −(1 − φ)hij− L(∇kφ)(∇kQij)
+
|∇φ|
where ζ is a constant kinetic coefficient. In the first line,
the functional derivative is taken both inside and outside
the particles with the redefinitions (29)-(34), with λ ensur-
ing Qii= 0. In the second line, hijis defined in equation
(16) and ∇kφ arises from the factor 1−φ in equation (31).
On a cubic 64 × 64 × 64 lattice, we integrated the above
equation for Qij. Space and time are measured in units of
d and
τ = d2/ζL,
δQij
+ λδij
w
?
∇iφ∇jφ − |∇φ|2δij
3
?
,
(35)
(36)
respectively. The space mesh size is d and the time mesh
size is ∆t = 0.01τ in the integration. The cell interior is in
the region 0 ? x,y,z ? 64d. We solved the Poisson equa-
tion (14) at each integration step using a Crank-Nicolson
method [23].
As the boundary conditions of Qij at z = 0 and 64d,
we assume the homeotropic anchoring ni= δizfor ε1> 0
and the parallel alignment ni = δix for ε1 < 0, where
n = (nx,ny,nz) is the director with i = x,y,z. Those of
Qij in the x and y directions are the periodic boundary
conditions. The potential Φ vanishes at z = 0 and 64d
and is periodic in the xy plane. Note that the electric field
at z = 0 and 64d is along the z axis, so the electrostatic
energy is lowest for the selected director alignments both
for ε1 > 0 and ε1 < 0. In order to approach a steady
state, we performed the integration until |dF/dt| became
less than 10−5T/τ.
In our steady states thus attained, we confirmed that
both equations (17) and (18) excellently hold in the bulk
liquid crystal region and near the particle surfaces, re-
spectively. Mathematically, they should hold in the thin-
interface limit d ? R, where −∇φ∼= δ(r − R)ν around
a spherical surface with ν being the normal unit vector.
In Figure 2, we show our numerical result of the deriva-
tive ∂J2/∂r = 2Qij∂Qij/∂r and the gradient free energy
density fg= L(∇kQij)2/2 around a particle surrounded
by a Saturn ring defect for w = 0. See the next section
for details of the calculation and Figure 3 for its 3D pic-
ture. We can see that ∂J2/∂r is nearly equal to zero at
the surface and exhibits double peaks around the Saturn
ring position. The boundary condition ν · ∇Qij = 0 in
equation (14) is thus nearly satisfied even in the presence
of a defect in our diffuse interface model.
Fig. 3.
particle for (a) Z = 60, (b) Z = 100 and (c) Z = 160 in a
nematic solvent with ε1 = 1.8ε0. Short lines (in blue) represent
the director n = (nx,ny,nz) and cylinders (in green) in (b) and
(c) contain a Saturn ring.
(color online) Orientational field around a charged
4 Numerical results
In our simulations, we set
A = −15T/d3,
?B= 12d,
B = |A|/2,
εp= ε0.
C = 3B,
For example, for ε0 = 2.3 and T = 300K, we have d =
2nm, ?B = 24nm, and L = 2pN. The nematic order pa-
rameter S in equation (20) is calculated as S = 0.75. We
show simulations results, where the charge number per
particle is Z = 30, 50, 60, 80, 100, and 160. If it is 100
and the radius R is 25nm, the surface electric field Es
becomes 100 V/µm. We also set εp= ε0. In the case of
one particle, the interior dielectric constant εp does not
affect the exterior electric potential and is irrelevant. In
the case of two particles, we also performed simulation
with εp= 2ε0in the examples in figures 6 and 8, but no
marked difference was found.
In Subsections 4.1 and 4.2, we will neglect the short-
range anchoring interaction and set w = 0, focusing on
the electric field effect on the director field. In Subsec-
tion 4.3, we will include the short-range anchoring in-
teraction around a charged particle. In our Landau-de
Gennes scheme, the orientation order is almost uniaxial
outside the defect cores both for ε1 > 0 and ε1 < 0.
Thus we will display the director n around the parti-
cles. Tube-like surfaces in Figures 4-10 will be those where
fgd3/T = (d∇kQij)2/2 = 0.2. This threshold is so high
such that the resultant tubes enclose defects.
In addition, we confirmed that the eigenvalues of χij
in equation (22) were kept to be positive everywhere in
the system. For example, in the uniaxial state with ε1=
1.8ε0and S = 0.75, the eigenvalues of χij, are given by
χ?∼= 0.27 and χ⊥∼= 0.024.
4.1 A single particle in nematic liquid
We fisrt consider a single charged particle for the two
cases, ε1> 0 and ε1< 0. Its charge number Z is in the
range [60,160]. The orientation tensor Qijis independent
of the sign of Z.