Biaxial torus around nematic point defects.

Abstract

We study the biaxial structure of both line and point defects in a nematic liquid crystal confined within a capillary tube whose lateral boundary enforces homeotropic anchoring. According to Landau-de Gennes theory the local order in the material is described by a second-order tensor Q, which encompasses both uniaxial and biaxial states. Our study is both analytical and numerical. We show that the core of a line defect with topological charge M=1 is uniaxial in the axial direction. At the lateral boundary, the uniaxial ordering along the radial direction is reached in two qualitatively different ways, depending on the sign of the order parameter on the axis. The point defects with charge M=+/-1 exhibit a uniaxial ring in the plane orthogonal to the cylinder axis. This ring is in turn surrounded by a torus on which the degree of biaxiality attains its maximum. The typical lengths that characterize the structure of these defects depend both on the cylinder radius and the biaxial correlation length. It seems that the core of the point defect does not depend on the far nematic director field in the bulk limit.

Full text

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Biaxial torus around nematic point defects
S. Kralj,1,2E. G. Virga,3and S. Zˇumer2,4
1Department of Physics, Faculty of Education, University of Maribor, Koros ˇka 160, 2000 Maribor, Slovenia
2Institute Joz ˇef Stefan, Jamova 39, 1000 Ljubljana, Slovenia
3Department of Mathematics, INFM Research Unit, University of Pavia, via Ferrata 1, 27100 Pavia, Italy
4Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia
?Received 14 January 1999?
We study the biaxial structure of both line and point defects in a nematic liquid crystal confined within a
capillary tube whose lateral boundary enforces homeotropic anchoring. According to Landau–de Gennes
theory the local order in the material is described by a second-order tensor Q, which encompasses both uniaxial
and biaxial states. Our study is both analytical and numerical. We show that the core of a line defect with
topological charge M?1 is uniaxial in the axial direction. At the lateral boundary, the uniaxial ordering along
the radial direction is reached in two qualitatively different ways, depending on the sign of the order parameter
on the axis. The point defects with charge M??1 exhibit a uniaxial ring in the plane orthogonal to the
cylinder axis. This ring is in turn surrounded by a torus on which the degree of biaxiality attains its maximum.
The typical lengths that characterize the structure of these defects depend both on the cylinder radius and the
biaxial correlation length. It seems that the core of the point defect does not depend on the far nematic director
field in the bulk limit. ?S1063-651X?99?07408-5?
PACS number?s?: 61.30.Cz, 61.30.Jf
I. INTRODUCTION
Generally, in ordered media defects take different names
in different contexts: so they are called dislocations, discli-
nations, singularities, or domain walls. The study of these
defects is traditionally one of the most important fields of
physics ?1?. This is due to their presence in connection with
diverse physical phenomena, where their contribution is cru-
cial and often exhibits a universal behavior. They appear as a
consequence of the universal concept of broken symmetry,
either associated with a phase transition, or due to the topo-
logical characteristics of the confining boundary ?2?. Defects
in ordered media are singular regions exhibiting order pa-
rameter configurations that cannot be transformed into a ho-
mogeneous ground state via continuous transformations. In
general, at a defect site some continuum field describing a
defectless state of the system is not uniquely defined. The
core of the singularity is indeed the region in space where a
finer description of the states experienced by the system is
needed to remedy such an apparent failure of the continuum
theory: it mostly consists of a different phase with higher
energy than the surrounding. The linear dimension of the
core is roughly given by the correlation length of the relevant
order parameter field employed to explore it.
In this respect, the majority of experiments were carried
out in various liquid crystal phases ?3?. This is because of the
rich variety of qualitatively different defects exhibited by
these fluids: they are reminiscent of singularities in other
condensed media ?1,4,5? and physical fields, such as cosmol-
ogy ?6,7?, which are often less accessible experimentally. On
the other hand, it is relatively easy and inexpensive to pre-
pare adequate liquid crystal samples. The morphology of a
defect can be controlled by choosing a suitable liquid crystal
phase, a confining geometry, and a surface anchoring condi-
tion. On the micron length scale defect structures can easily
be observed optically, due to the optical anisotropy of these
molecules. Another advantage with liquid crystals is that
they reach equilibrium structures on experimentally acces-
sible time scales.
The nematic phase, on which we focus in this contribu-
tion, exhibits in general both point and line defects that are
conventionally classified through their topological charge M
?also called the disclination index? ?1,4?. This is defined with
respect to the surrounding nematic director field n, which is
singular exactly at the defect: around the defect the director
rotates by the angle 2M?. For point defects the strength ?M?
is an integer, while for line defects it can also be half an
integer, by the head-tail invariance of the nematic director.
The excess free energy associated with defects is in most
cases roughly proportional to M2?8?. Consequently, defects
with ?M??1 appear only rarely. Here, following the
Landau–de Gennes theory, we employ a second-order tensor
Q to describe the local molecular order: it encompasses
within the same setting both uniaxial and biaxial states. A
defect for n is generally not so for Q, and so Q is fit to
explore the biaxial structure of the uniaxial defects.
There have been various studies devoted to the structure
of both point and line defects in nematic liquid crystals; the
following lists of references, though far from being exhaus-
tive, witness the interest attracted, respectively, by these
types of defects: ?9–21?, ?2,10,21–23?. Nevertheless, despite
this endeavor, several issues remained open. Among them
are the detailed analysis of the biaxial structure of defects
and the effects of confinement on their characteristic fea-
tures. These questions have been answered only in part in
?9,16,18,19,21,22?. Here we further explore the effect of cy-
lindrical confinement on the biaxial structures of both line
and point defects in uniaxial nematic liquid crystals.
The plan of the paper is the following. In Sec. II we
introduce the mathematical model employed throughout the
paper. In Sec. III we illustrate our main results, which are
then discussed in the last section.
PHYSICAL REVIEW E AUGUST 1999VOLUME 60, NUMBER 2
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II. MODEL
A. Free energy
The local ordering of a nematic liquid crystal can be de-
scribed by a tensor order parameter ?16,24?:
Q??
i?1
3
siei?ei,
?1?
where the orthogonal unit vectors eiare clearly the eigenvec-
tors of Q, and siare the corresponding eigenvalues, that is,
Qei?siei. By Eq. ?1?, the tensor Q is symmetric because
QT?Q; it is further required to be traceless:
trQ??
i?1
3
si?0.
?2?
Consequently, Q is in general defined by five independent
parameters. Three of them determine the orientation of the
eigenvectors, and the remaining two the eigenvalues.
In the uniaxial ordering two eigenvalues are equal, and so
only three independent parameters are needed to describe a
nematic configuration. Q can then be given the form
Q?s?n?n?1
3I?,
?3?
where the scalar s is the uniaxial order parameter, and the
unit vector n is the nematic director pointing along the local
optic axis. In Eq. ?3? s can have either sign: when it is posi-
tive the ensemble of molecules represented by Q tends to be
aligned along n, whereas when s is negative it tends to lie in
the plane orthogonal to n.
In practice, various perturbations can make a confined
liquid crystal exhibit weakly biaxial states, especially in the
vicinity of a defect for n. Thus, the representation for Q in
Eq. ?3? is no longer valid throughout the region occupied by
the material, and use has to be made of the complete repre-
sentation in Eq. ?1?. In a real sample the state represented by
Q changes from point to point, and so Q is to be regarded as
a tensor field. Wherever Q?0, the nematic order is locally
lost and the fluid becomes isotropic. A convenient quantity
to measure the degree of biaxiality is the parameter ?2de-
fined by ?25?
?2?1?6?trQ3?2
?trQ2?3,
?4?
which ranges in the interval ?0,1?. In all uniaxial states ?2
?0, and a state with maximal biaxiality would correspond to
?2?1.
The free-energy density f of a nematic liquid crystal can
be expressed as the sum of two terms:
f?fe?fb.
They are, respectively, the elastic and the bulk free-energy
densities. The former depends on the distortion in space of
the tensor field Q; within a simplified model it can be given
the form
fe?L?“Q?2,
?5?
where L is an elastic constant which does not depend on the
temperature. This description corresponds to the approxima-
tion with equal Frank elastic constants for uniaxial nematics
?26?. The bulk free-energy density fbis a potential that pro-
motes the uniaxial order in an undistorted nematic liquid
crystal. It is conventionally described by an expansion in Q
up to the fourth order ?8?:
fb?A?T?T*?trQ2?B trQ3?C?trQ2?2.
Here A, B, C are positive material constants, T is the
temperature, and T*is the nematic supercooling temp-
erature. For T?T*, the potential fbattains a local mini-
mum at the isotropic phase, whereas for T?T*
local minimum ceases to exist. The material constants in
Eq. ?6? are chosen so that the minimizer of fbis a uniaxial
order tensor like that in Eq. ?3?. Within this model the
isotropic-nematic phase transition in the bulk occurs at the
temperature TIN?T*?B2/24AC. Moreover, the equilibrium
value of the uniaxial scalar order parameter seqin Eq. ?3?
alsodependsonthe
??9?8?(T?T*)/(TIN?T*)??/4,
??(3A/2C)(TIN?T*)?B/4C.
Here we will only be concerned with strong anchoring
conditions, and so we need not consider any contribution to
the free energy from the boundary.
?6?
this
temperature:
seq(T)ªs0?3
s0ªseq(TIN)where
B. Lyuksyutov constraint
In most cases the I-N phase transition is weakly first or-
der, so reflecting relatively small values of the material con-
stant B. Deep in the nematic phase B is approximately an
order of magnitude smaller than both A(T*?T) and C.
Thus, for weak elastic distortions a good approximation for
both f and fbis the following ?27?:
f?fb?A?T?T*?trQ2?C?trQ2?2.
This function attains its minimum for a value of trQ2that
can alternatively be expressed in terms of the equilibrium
value of s within this approximation:
trQ2?A?T*?T?
2C
?
2seq
3
2
.
?7?
In our model this is to be regarded as a constraint for Q.
Thus, we assume that the orientational order of a liquid crys-
tal responds to local distortions in a way that leaves trQ2
unchanged, even for strong distortions. Two facts concerning
this constraint are worth noting.
?i? Within this approximation the liquid crystal cannot
melt locally and become isotropic, because Q cannot vanish.
From a physical point of view, this scenario is plausible in
the deep nematic phase, where melting becomes exceedingly
costly.
?ii? In Eq. ?6? only the cubic term makes the uniaxial
states preferred to the biaxial ones. Thus, we will keep this
term in as a perturbation to the free-energy density on the
states that minimize the leading terms in fb.
Henceforth we take the constraint in Eq. ?7? as valid. Con-
sequently, only one parameter is needed to determine all ei-
genvalues of Q.
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C. Parametrization
We study a nematic liquid crystal confined within an in-
finite cylindrical cavity with radius R. The cylindrical coor-
dinates are represented by ?r,?,z?, and the corresponding
unit vectors along the coordinate axes are er, e?, and ez.
We confine attention to distortions where the eigenvectors of
Q can be expressed as
e1?cos?er?sin?ez,
e2??sin?er?cos?ez,
e3?e?,
?8?
where ? is an angle ranging in the whole real axis. Thus, in
going from a point within the cylinder to the next the eigen-
vectors of Q can only rotate around the e?axis. This ex-
cludes, for example, any distortion twisted along the axis of
the cylinder.
It is easily checked that both constraints Eqs. ?2? and ?7?
are identically satisfied when the eigenvalues of Q are given
the following representation in terms of a single angle ?:
s1?2
3seqcos?,
s2??2
3seqcos????
3?.
3?,
s3??2
3seqcos????
?9?
Moreover, the degree of biaxiality defined in Eq. ?4? can be
expressed as a function of ?:
?2?1?16cos2? cos2????
3?cos2????
3?,
?10?
which is periodic with period ?/3. For i??1,2,3?, the con-
figurations with ??(i?1)2?/3?? correspond to uniaxial
states with negative order parameter and nematic director
along ei, while the configurations with ??(i?1)2?/3 cor-
respond to uniaxial states with opposite order parameter, but
respectively the same director. It easily follows from Eq.
?10? that these are the only zeros of ?2in ???,??. The
states with other values of ? reflect biaxial molecular distri-
butions. The degree of biaxiality attains its maximum for ?
?(j?1)?/6, j??1,...,6?. The essential features of this rep-
resentation for Q are illustrated in Fig. 1.
Through Eqs. ?8? and ?9? we describe all biaxial structures
admissible in our model by use of only two parameters,
namely, ? and ?. The tensor Q delivered by Eq. ?1? can then
be regarded as a function of ? and ?, which, however, is not
injective. There are indeed transformations in the parameters
? and ? that leave Q unchanged. Two of them are immedi-
ate consequences of the parametrization itself: they are em-
bodied by the identities
Q??,???Q??,??2k??,
Q??,???Q???k?,??,
valid for all relative integers k. Changing ? into ??2k? or
? into ??k? does not affect Q, because its eigenvalues
remain the same and its eigenvectors just get reversed. Be-
sides these trivial transformations, there is another which is
not so, that is,
??,??˜????
2,2?
3???.
?11?
This exchanges s1and s2, maps e1into e2, and e2into
?e1, while leaving both s3and e3unchanged: by Eq. ?1?, it
has no effect on Q. The identity
Q??,???Q????
2,2?
3???
will play a central role in the following. In particular, it en-
sures that both ? and ? can suffer a jump without causing
any discontinuity in Q. We shall exploit this indeterminacy
to represent a continuous field Q through discontinuous
fields ? and ?, whenever this does not cause a divergence in
the free-energy functional.
D. Scaling
In the strong anchoring limit the only relevant character-
istic length entering the model is the biaxial correlation
length ?see Appendix?
?bª?
2L
3Bseq.
?12?
It can easily be expressed in terms of the uniaxial correlation
length ?nª?L/A(TIN?T*) at the I-N phase transition as
3?seq?TIN?
?b??n
seq?T?.
?13?
For later use, we measure the free energy F in terms of
F0ªRLseq
and the order parameter in terms of s0
?seq(TIN): thus, in the following F˜F0F and s˜s0s. We
furtherintroducethe reduced
?T*) /(TIN?T*) and measure all lengths relative to the
cylinder radius R, so that r˜Rr, z˜Rz, ?b˜R?b, ?n
˜R?n, “˜(1/R)“; in these units R?1, seq
convenience, we also define the excess free energy as ?F
ªF?Fbulk, where Fbulkdenotes the free energy of a bulk-
undistorted nematic.
2
temperature
?ª(T
2??1
2?. For
FIG. 1. The nematic states described by the order parameter ?.
Full lines: uniaxial states with a positive eigenvalue and nematic
director along ei(i?1, ??0; i?2, ??2?/3; i?3, ???2?/3).
Dashed lines: uniaxial states with a negative eigenvalue and nem-
atic director along ei(i?1, ???; i?2, ????/3; i?3, ?
??/3). Dotted lines: states with maximal degree of biaxiality.
1860PRE 60
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In terms of these definitions one obtains the following
dimensionless expression for the excess free energy of the
nematic liquid crystal stored in a cylinder with length h:
?F?16?
3?
0
1
r dr?
?h/2
?h/2
dz?
1
?b
2?b??e?,
?14?
where
?bª1
18?1?cos3??,
?15?
?eª1
4?“??2?sin2????
3??“??2
?sin2????/3?cos2?
r2
?sin2? sin2?
r2
.
?16?
The corresponding Euler-Lagrange equations are
?2 sin?2????
?2??
1
?b
2
sin3?
33???“??2
?2?sin?2????/3??cos2??sin2? sin2??
r2
?0,
?17?
and
sin2????
3??2???“?•“??sin?2????
3??
?sin2?sin2????/3??sin2?
2r2
?0.
?18?
In these equations “??(??/?r)er?(??/?z)ez and ?2?
??2?/?r2?(1/r)(??/?r)??2?/?z2, for ? either ? or ?.
E. Boundary conditions
We assume that the lateral boundary of the cylinder en-
forces the strong homeotropic anchoring condition so that
the nematic order is uniaxial along erwith positive order
parameter:
Q?r?1?Qr?ªseq?er?er?1
3I?.
?19?
This state can be described by the pair ??,????0,0?, which
by Eq. ?11? is completely equivalent to the pair ??/2,2?/3?.
It follows from Eq. ?14? that for the integral to converge,
only two pairs ??,?? are admissible on the cylinder axis,
namely, ?0,2?/3? and ?0,??/3?. They correspond to the fol-
lowing tensors:
Qz?ªQ?0,2?/3??seq?ez?ez?1
Qz?ªQ?0,??/3???Qz?,
3I?,
?20?
?21?
which represent uniaxial states with nematic director along ez
and opposite scalar order parameters. Clearly, by Eq. ?11? the
same states are also represented by the pairs ??/2,0? and
??/2,??, respectively. Besides Eq. ?19?, Q will also be sub-
ject to either Q?r?0?Qz?or Q?r?0?Qz?.
The equilibrium nematic structures subject to these
boundary conditions were obtained numerically from the
above Euler-Lagrange equations by using the over-relaxation
method ?28?.
III. RESULTS
A. Line defects
We first restrict attention to distortions where Q only var-
ies with the r coordinate. With the terminology introduced in
?29?, possible minimizers for this one-dimensional problem
are the escaped radial structure ?ER? and two qualitatively
different planar radial solutions ?PR? with either positive or
negative scalar order parameter at r?0 ?respectively, de-
noted by PR?and PR?). In the genuinely uniaxial descrip-
tion, both PR structures would exhibit a line defect with
strength 1 along the axis ?1?.
It is also expedient recalling the uniaxial ER structure
?30? for the role it plays in our study of point defects. In this
solution both boundary conditions Qz?and Qr?are met by
simply rotating the eigenvectors of Q while r spans the in-
terval ?0,1?. In our setting, it is represented by the pair of
functions ??ER,?ER?, where
?ER?r?ª?
2?2 arctanr,
?ER?0.
?22?
The corresponding excess free energy is ?FER?8?h in di-
mensionless units.
On the contrary, in the PR solutions the eigenvectors re-
main fixed relative to the frame ?er, e?, ez?, so that ??0
for both: the uniaxial states at r?0 and r?1 are connected
through an exchange between the eigenvalues of Q ?22?;
here this is described as a change in the angle ?, which starts
from 0 at r?1 and reaches either 2?/3 or ??/3 at r?0, in
the solutions PR?and PR?, respectively. We denote by
?PR?and ?PR?the functions of r that describe these solu-
tions: their graphs are shown in Fig. 2?a?.
While ?PR?ranges in the interval ???/3,0?, thus cross-
ing no uniaxial state for all 0?r?1, ?PR?crosses at r
?runthe uniaxial state with ???/3, which has negative
order parameter and nematic director along e?. Figure 2?b?
illustrates the degree of biaxiality of both solutions for
R/?b?10.
In the PR?solution the nematic ordering attains the maxi-
mal biaxiality at rb1and rb2, where 0?rb1?run?rb2?1.
For the PR?solution, however, there is a single value of
r with maximal biaxiality; this will be denoted by rb. The
influence of the confinement on these parameters is shown in
Fig. 3. For R??bthey all come close to 2?b, yielding the
size of the defect core in the bulk.
When R/?b?0 there are analytic expressions for both PR
solutions. Although this limit is unphysical the solutions ex-
hibit the general features recalled above. In this limit and for
??0, the free energy in Eq. ?14? reduces to
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?FPR?16?h
3?
0
1?
1
4??2?sin2?
r2 ?r dr,
?23?
where ?ª???/3, and a prime denotes differentiation with
respect to r. The corresponding Euler-Lagrange equation
reads as
?r?????2 sin2?
r
.
?24?
After multiplying both sides of Eq. ?24? by r?? we arrive at
a first integral in the form
1
4?r???2?c?sin2?,
?25?
where c is an integration constant. Inserting this expression
for ?? into Eq. ?23? and requiring ?FPR?? , one obtains
c?0. A further integration of Eq. ?25?, subject to the bound-
ary conditions ?(1)?2?/3 or ?(1)???/3, corresponding
to the PR?or the PR?solution, respectively, leads to
?PR??r??arccos?
?PR??r??arccos?
3r4?1
3r4?1???
3?r4???
3,
?26?
3?r4
3,
for which ?FPR??8?h and ?FPR??(8?/3)h in dimen-
sionless units.
The excess free energies for the ER and the PR solutions
are plotted in Fig. 4 as functions of R/?b. One sees that the
PR?solution is always the most energetic among them.
There is a critical value of the ratio R/?b, close to 10.7,
marking a transition ?31? between the ER and the PR?solu-
tion: below this value the former stores more energy than the
latter. It is known from ?32? that this transition has indeed a
more complex structure: when the ER solution loses stabil-
ity, there is a range of values for R/?bwhere the least ener-
getic solution is planar polar with line defects ?33? ?PPLD?,
that is, a solution with two biaxial escapes along the axis of
the cylinder, resembling the uniaxial disclination with
strength
sumed in our parametrization, and so it escapes our model,
which instead captures the transition to the PR?solution,
actually prevailing over the PPLD solution for R/?bsuffi-
ciently small. Henceforth we take R/?b?10, so that the ER
solution is the absolute minimizer of the free-energy func-
tional.
To find out in which regime the Lyuksyutov constraint is
acceptable, we compare the biaxial PR?solution to the
uniaxial PRusolution, which requires melting at the cylinder
axis. To determine the PRusolution, we set n?erand allow
1
2. The PPLD solution breaks the symmetry pre-
FIG. 2. Spatial structure of the PR?and PR?solutions. ?a? ?
??(r) for different values of R/?b; ??0. At r?0, ??2?/3 in
the PR?solution and ????/3 in the PR?solution. The curves
labeled with ?i?, ?ii?, and ?iii? correspond to (R/?b)2?1350, 135,
and 0, respectively. ?b? ?2??2(r) for (R/?b)2?135.
FIG. 3. The influence of confinement on the characteristic
lengths rcfor the PR structures. For the PR?solution, rcis either
rb1, run, or rb2, while for the PR?solution it is just rb.
FIG. 4. The excess free energy ?F for the ER, PR?, and PR?
structures normalized to the excess free energy ?FERfor the ER
structure, for different values of R/?b.
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for spatial variations of s in Eq. ?3?: it is easily shown that in
dimensionless units the free energy is then expressed by
r dr?
?n
FPRu?4?h
3?
0
1
1
2??s2?2s3?s4??s?2?3s2
r2?.
?27?
The corresponding Euler-Lagrange equation is
s??s?
r?3s
r2?
1
?n
2??s?3s2?2s3??0.
?28?
Figure 5 shows the excess free energy of both the PR?and
the PRu
solution as a function of the ratio ?(?)
ª?b(?)/?(?)??2??4??9
tween the nematic biaxial and uniaxial correlation lengths
?see Appendix?. The ‘‘deep nematic’’ phase corresponds to
the regime where ?(?)?1. Just below the I-N phase transi-
tion, for which ?(1)?1
There exists a critical temperature below which the crossover
to the PR?solution ?obtained within the Lyuksyutov con-
straint? takes place. In reality, the value of trQ2drops at the
defect core of the PR?solution ?17,18?, so pushing the criti-
cal temperature towards higher values.
2?3
2?9?8?/3?3??9?8? be-
3, the isotropic solution is preferred.
B. Point defects
Here we focus on the biaxial structure of point defects
with strength ?M??1 , where Q also depends on the z coor-
dinate. The free energy of the ER solution is invariant under
the transformation that reverses the sign of ?ERin Eq. ?22?.
Thus, domains with opposite ER structures are equally likely
to arise in an infinitely long cylinder. Wherever two such
domains join together, a point defect with topological charge
?1 appears on the cylinder axis ?29,34?. The resulting struc-
ture, which is often referred to as ERPD ?escaped radial with
point defects?, is metastable and tends to relax towards the
topologically equivalent ER structure. An ERPD structure is
generally produced on cooling the liquid crystal from its iso-
tropic phase. Each cross section through a defect exhibits a
distortion resembling a PR solution, since ? has opposite
signs on the two sides of the section, and so must vanish on
it. Thus, this section plays the role of a domain wall. Only
the PR?solution can be accommodated in it, because, unlike
the PR?solution, on the cylinder axis it matches both ER
domains, which are uniaxial with positive order parameter. It
is remarkable that the PR?solution, which would never be
energetically preferred in the absence of point defects, is
indeed relevant to their biaxial structure.
In this study we restrict attention to a single defect of
either sign: this effectively amounts to assuming that the
distance between two adjacent defects is larger than 2R, so
that their mutual attraction becomes negligible ?34?. Within
our model the equilibrium biaxial structure of a defect is
described by the functions ???(r,z) and ???(r,z): they
are represented in Figs. 6, 7 together with the degree of bi-
FIG. 5. The excess free energy ?F for both the PR?and the
PRustructures normalized to the excess free energy ?FERfor the
ER structure, for different values of the ratio ?(?)ª?b(?)/?(?). ?i?
(R/?b)2?135, ?ii? (R/?b)2?1350.
FIG. 6. The graphs of the functions ?a? ???(r,z), ?b? ?
??(r,z), and ?c? ?2??2(r,z) for (R/?b)2?135, and different val-
ues of z: ?i? z?0, ?ii? z??z, ?iii? z?2?z, ?iv? z?3?z, where
?z/R?0.05. The center of the defect core is at (r,z)?(0,0). The
graphs for z?0 in both ?a? and ?b? reflect the transformation
??,??˜????/2,2?/3??? for r?rundescribed in the text.
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1863BIAXIAL TORUS AROUND NEMATIC POINT DEFECTS

Page 7

axiality ?2. The plane at z?0 exhibits a structure similar to
the one shown in Fig. 2?a?, where the nematic state at r?0 is
described by the pair ??,????0,2?/3?. On the other hand,
just above this plane, but still at r?0, the ER structure pre-
scribes the pair ??/2,0?, which by the transformation in Eq.
?11? corresponds to the same state. This discontinuity would,
however, cause a divergence in the free-energy functional,
because it also involves ?. In our calculations we avoided
such a divergence by expressing both the fields ? and ? and
their gradients in the Euler-Lagrange equations through one
and the same representation. We privileged the ‘‘perspec-
tive’’ of the ER solution, and so at z?0 we switched from
the pair ??,?? for r?runto the pair ????/2,2?/3??? for
r?run, with runthe point where ???/3: as in ?35?, the
discontinuity in ? at this point does not make the free-energy
functional infinite. The value of ? at r?runremains arbi-
trary, reflecting the degeneracy of the eigenvalues of Q in the
(r,z) plane.
The resulting structure is characterized by the following
qualitative features: some are already evident from Figs.
6?c?, 7, which show the graph of the function ?2??2(r,z).
The symmetry plane at z?0 exhibits a uniaxial ring with
radius r?runsurrounded by biaxial zones with maximal bi-
axiality at the rings r?rb1and r?rb2, as in the PR?solu-
tion studied above: ?2(rb1,0)??2(rb2,0)?1 with 0?rb1
?run?rb2, though these values of r are different from those
for the genuine PR?solution. Just above ?or below? z?0 the
uniaxial ring disappears. Farther away from this plane, both
radia rb1and rb2survive, but they vary with z: they approach
each other, and eventually merge at z??zb1. For ?z??zb1
the function ?2never reaches 1, and it exhibits a single
maximum at r?rb(z), which monotonically decreases with
z. The value of ?2„rb(z),z… drops to
In other words, the uniaxial ring with negative order pa-
rameter lying in the symmetry plane is surrounded by a bi-
axial torus, where the degree of biaxiality attains its maxi-
mum. For R large enough ?‘‘saturated’’ regime?, the torus
cross section is circular with radius rt?0.8?b, as shown in
Fig. 8?a?. In this regime rb1?4.2?b, run?5.0?b, rb2
?5.8?b, zb1?0.8?b, zb2?1.6?b. For smaller values of R
the torus cross section becomes prolate along the z direction.
The way the above characteristic lengths depend on the con-
finement resembles the one obtained for line defects. Some
1
2at z??zb2.
preliminary results are shown in Fig. 8?b?. We plan to
present a detailed study focused on the confining effect of
the geometry on the biaxial torus elsewhere.
IV. CONCLUSIONS
The main topic of this contribution is the detailed biaxial
structure of nematic line and point defects with strength
?M??1 in a cylindrical cavity enforcing homeotropic an-
choring. We mainly built on the work by Lavrentovich and
co-workers ?12,36?, Penzenstadler and Trebin ?9?, Gartland
and co-workers ?17–19?, and Rosso and Virga ?16?.
We limited attention to the nematic low temperature re-
gime, where a uniaxial liquid crystal responds to distortions
by entering biaxial states, rather than melting. We treated
only structures with no twist deformation exhibiting cylindri-
cal symmetry. Twisted structures might appear for relatively
low values of the twist Frank elastic constant relative to the
bend and splay constants ?8?.
Within this framework we obtained two qualitatively dif-
ferent line defect core structures, referred to as PR?and
PR?, that basically differ by the sign of the uniaxial order
parameter s at the very defect core (s?0 in PR?, and s
?0 in PR?). Our results confirm calculations of Sigillo
et al. ?23? based on a molecular approach. In the PR?struc-
ture the line defect is surrounded by a cylinder with radius rb
exhibiting maximal degree of biaxiality. The PR?structure,
which is more energetic, can be characterized by three co-
axial cylinders with radia rb1?run?rb2displaying, in the
order, maximal biaxiality (rb1, rb2), and negative uniaxial
ordering in the azimuthal direction (run). For equal Frank
FIG. 7. Two-dimensional plot of ?2for R/?b?100 in the vari-
ables uªlnr ?for r?10?3) and z. It clearly indicates the cross
section of the torus exhibiting a maximal degree of biaxiality.
FIG. 8. ?a? The cross section of the biaxial torus for R/?b
?11.5. ?b? Influence of the confinement on the aspect ratio of the
cross section. The torus width is 2rt, while its thickness is 2rz.
Circles mark calculated points.
1864PRE 60
S. KRALJ, E. G. VIRGA, AND S. ZˇUMER

Page 8

elastic constants, rb1?1.7?b, run?rb?2.3?b, rb2?3.7?bin
the regime R?Rsat?5.5?b, where for typical liquid crystals
?b?10 nm. Below this regime, confinement effects become
significant,pushing characteristic
roughly linearly with R. Note that a homeotropic anchoring
with strength W prevails only if RW/K?1, where K is a
typical nematic Frank elastic constant. For K?5?1012J/m
and R?Rsat?55 nm, this requires W?10?4J/m2for finite
size effects to be observed. In addition, the radius must be
small enough (R?4.4?b?45 nm) for a PR?core structure
to correspond to the absolute minimizer. For typical liquid
crystals these limits are rather hard to reach experimentally.
It is also to be noted that in the limit for small values of R the
validity of the Lyuksyutov constraint is questionable, as is
the validity of the elastic approach. Thus, in this regime the
results are more of an academic interest.
We explored in detail the biaxial structure of a point de-
fect displayingcylindrical
?9,16,17? have shown that this structure is in most cases
stable relative to the uniaxial solution. The center of the core
is uniaxial along the symmetry axis, with positive order pa-
rameter; it is surrounded by a uniaxial ring with negative
degree of order and radius run. The azimuthal plane through
the center of the defect has the PR?structure typical of a line
defect: this structure can thus be characterized by the radia
run, rb1, and rb2, which are, however, approximately twice
as large as for the line defect. The ring with r?runis in the
center of the torus displaying maximal biaxiality. In the satu-
rated regime (R?10?b) the torus cross section is circular
with diameter 2rt?rb2?rb1. For equal elastic nematic con-
stants, we obtained rb1?4.2?b, run?5.0?b, rb2?5.8?b.
The position of the uniaxial ring is similar to the one retrace-
able in Gartland’s pioneering simulations ?37?. Below the
saturated regime the torus cross section takes a shape pro-
lated in the direction of the cylinder axis.
Our preliminary results indicate that in the limit R/?b
˜? the core structure of point defects is exactly the same
for both spherical and cylindrical confinements. This sug-
gests that the core structure does not depend on the far di-
rector field, if the confining cavity is large enough compared
to ?b. We will focus elsewhere on this universal feature ?38?.
radia towardszero,
symmetry.Recent results
ACKNOWLEDGMENTS
We thank both E.C. Gartland and A.M. Sonnet for stimu-
lating discussions and comments. This work was supported
by the Ministry of Science and Technology of Slovenia
?Grant No. Y1-0595, Y2-7609? and the European Commu-
nity ?INCO Copernicus Project IC15-CT96-0744?.
APPENDIX: CORRELATION LENGTHS
To obtain estimates for both the uniaxial and biaxial cor-
relation lengths, we express the tensor Q in the Cartesian
coordinate system with ?e1,e2,e3?: e1is in the direction of
the x axis, along which variations in space are only allowed.
The corresponding dimensionless free-energy density is
3?
?n
f?2
1
2??s2?2s3cos3??s4??s2?
d?
dx?
2
??
ds
dx?
2?.
?A1?
We first take the uniaxial case ?i.e., ??0), and locally ?at
x?0) we perturb the order parameter by ?s0from its equi-
librium value s(?)?seq, where s(?)?0 for ??1 and s(?)
?(3??9?8?)/4 for ??1. We let s?seq??s(x) and ex-
pand Eq. ?A1? about equilibrium, up to the second order in
?s. The solution to the corresponding Euler-Lagrange equa-
tion reads as ?s??s0e?x/?(?), where ?(?) is the nematic
correlation length at the reduced temperature ?:
?A2?
??????
?n
??
for ??1
?n
?2
??4??
9
2?
3
2?9?8?
for ??1.
?A3?
The quantity ?ndefines the value of the correlation length at
the I-N phase transition, i.e., ?n??(1).
We next limit attention to the nematic phase ?i.e., ??1),
set s(x)?seq, and perturb locally ? from its equilibrium
value (??0). The same approximation described above now
leads to defining the biaxial correlation length as
?bª
?n
3?seq
.
?A4?
Taking into account that 24A(TIN?T*)C/B2?1 and that
here s is measured in terms of seq(TIN), we then easily re-
trace the expressions in Eqs. ?12? and ?13?.
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