The Geometry of the Cholesteric Phase
Daniel A. Beller,1 Thomas Machon,2 Simon Čopar,1, 3, 4 Daniel M. Sussman,1
Gareth P. Alexander,2 Randall D. Kamien,1 and Ricardo A. Mosna1, 5
1
Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104-6396, USA
Department of Physics and Centre for Complexity Science, University of Warwick, Coventry CV4 7AL, UK
3
Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia
4
Jozef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia
5
Departamento de Matemática Aplicada, Universidade Estadual de Campinas, 13083-859, Campinas, SP, Brazil
(Dated: June 13, 2014)
arXiv:1406.3304v1 [cond-mat.soft] 12 Jun 2014
2
We propose a construction of a cholesteric pitch axis for an arbitrary nematic director field as
an eigenvalue problem. Our definition leads to a Frenet-Serret description of an orthonormal triad
determined by this axis, the director, and the mutually perpendicular direction. With this tool we
are able to compare defect structures in cholesterics, biaxial nematics, and smectics. Though they
all have similar ground state manifolds, the defect structures are different and cannot be, in general,
translated from one phase to the other.
The standard lore in liquid crystal physics is that the
cholesteric and the smectic are essentially the same beast:
the Caille instability in smectics [1] is the same as the
Lubensky instability in cholesterics [2], both examples of
the Peierls instability. Fluctuations around the ground
states of each are dominated by a single Goldstone mode
with an unusual fluctuation spectrum that distinguishes
modes along the layer normal or pitch axis from modes
along other directions. Since these modes are also the
coördinates on the ground state manifold (GSM), one is
tempted to identify defects in smectics with defects in
cholesterics, with the cholesteric pitch axis P analogous
to the smectic layer normal N.
On the other hand, the cholesteric phase is biaxial
[3, 4]: intuitively, a cholesteric is defined by a unit director field n and a derived pitch axis P that describes the
direction along which n twists, whose precise definition
we will discuss below. At each point, a local orthonormal
triad is specified by n and P, both defined up to sign,
along with their cross product n×P ≡ n
/ (“not n”). From
this point of view, one might attempt to identify defects
in the cholesteric with defects in the biaxial phase characterized by the standard homotopy theory of topological defects [5–8], in particular via the non-Abelian fundamental group π1 [SO(3)/D2d ] = {±1, ±σx , ±σy , ±σz },
the unit quaternions. This has been the motivation behind a classification of cholesteric defects as singularities
in two of three triad directions, termed the λ±1 , τ ±1 ,
and χ±1 singularities, in correspondence with ±σx , ±σy ,
and ±σz singularities of the biaxial nematic phase [9].
For example, just as σx σy = σz in the quaternion group,
composing a λ with a τ yields a χ in the cholesteric.
Meanwhile, the cholesteric-smectic analogy suggests that
disclinations in the layer normal N are to be identified
with the λ and τ defects where P is not defined, so that
the decomposition of a smectic dislocation into a disclination pair [10] is also analogized to the decomposition
of a χ into a λ-τ pair.
In order to explore the similarities and differences between the topology of defects in smectics, cholesterics,
and biaxial nematics, we will develop the geometric tools
necessary to map between these three systems. We find
that though there is a natural mapping between the
ground states of these three systems, the descriptions of
their defects are all different. In particular, we demonstrate the failure of this natural mapping in three example textures: the χ2 to λ2 defect transformation, the
oblique helicoidal cholesteric texture, and the screw dislocation. Our result highlights the well-known problem
with classifying topological defects in translationally ordered media [5, 10, 11] and confronts the usual analogies
between our triptych of phases.
There are crucial differences between the three systems, illustrated most clearly in the symmetries of their
ground states. A smectic ground state with layer normal
ẑ and layer spacing a is invariant under a translation by
a along ẑ; in other words, a continuum of ground states
is available under translations modulo a along ẑ. This
discrete symmetry is associated with the existence of dislocations in the smectic phase. The smectic ground state
also possesses a continuous symmetry under arbitrary rotations about ẑ. In contrast, the biaxial nematic ground
state with director n parallel to ẑ possesses a continuous
symmetry under arbitrary translations along ẑ, but a discrete symmetry under rotations by π about ẑ. The latter
discrete symmetry is related to the biaxial singularities
not found in uniaxial nematics. Along with uniaxial nematics, smectics and biaxial nematics are invariant under
π rotations around x̂ and ŷ.
From the point of view of symmetries the cholesteric
is intermediate between the smectic and biaxial nematic
phases. A cholesteric ground state with pitch axis ẑ and
helical wave number q0 has a continuous screw symmetry:
rotations by an angle α about ẑ composed with translations by α/q0 along ẑ. However, a translation by π/q0
along ẑ, and a rotation by π about ẑ, are each (separately) discrete symmetry operations of the cholesteric
ground state (as are rotations by π about x̂ or ŷ). In
other words, the cholesteric ground state has the discrete symmetries of both the smectic and the biaxial ne-
2
matic ground states, which might lead us to expect that
cholesteric defects will bear some resemblance to both
biaxial nematic defects and smectic defects. However,
broken translational invariance leads to a breakdown of
the homotopy theory of defects [5, 10, 11] and so symmetries are not the whole story.
We note also that smectics can have point defects, for
instance when the layers are concentric spheres around
a single point, whereas the biaxial nematic cannot support such defects as π2 [SO(3)/D2d ] = 0. This makes the
interpretation of the cholesteric even more subtle. From
one point of view the cholesteric configuration is completely specified by a director field everywhere in space
and is thermodynamically equivalent to a uniaxial nematic, which has ground state manifold RP 2 and can
support point defects. However, as discussed in [12] if we
include the pitch axis then a simple point defect in the
director field necessarily sprouts line defects in the pitch
axis. From this point of view the connection between
the cholesteric and biaxial nematic may seem promising.
However, there is a weakness in this link as well, which
will appear when we give an explicit definition of the P.
To proceed any further, let us now define the unit
cholesteric pitch axis P. With the orthonormal triad of
the biaxial in mind, we require P · n = 0 and define the
direction “not n” as n
/ = n × P; the cholesteric triad is
then {n, P, n}.
/ Intuitively, P is a direction about which
n twists, meaning that the directional derivative (P·∇)n
has no component along P. As n is a unit vector, this
directional derivative is also perpendicular to n, so we
can define P by the equation (P · ∇)n = −q n,
/ or the
equivalent eigenvalue problem
Pi Cij = qPj
(1)
where q is the local eigenvalue and Cij = nk ǫℓjk ∂i nℓ is
precisely the same chirality tensor defined in [13]. Note,
in addition, that Tr C = −n · ∇×n. This approach is
similar to, but more restricted than, the definition proposed in [14] that only holds for the cholesteric ground
state texture. Our construction generates a triad defined
up to sign so the GSM is SO(3)/D2d .
Whence the weak link in the analogy between biaxial nematics and cholesterics? Note that in the biaxial
nematic, the local triad can undergo a global rotation
that may change the energy but not the topology, in precise analogy with the two-dimensional uniaxial nematic;
a +1 disclination, for example, can be purely bend (azimuthal) or purely splay (radial). If the bend and splay
elastic constants differ, the energies will differ but the
topology will not. However, this internal rotation is not
a symmetry of the cholesteric since the pitch axis, defined
through Cij , involves spatial derivatives. As a result, rotations of the triad also require a rotation of the system,
i.e. only the trivial rotation of the whole sample leaves
the topology invariant. Though it is tempting to view
the cholesteric’s λ, τ , and χ defects in analogy with the
σx , σy , and σz defects of the biaxial nematic, the correspondence is not precise: Homotopies between biaxial
nematic triad configurations, which establish the topological equivalence of two configurations, are not in general available to cholesteric triads, where P is determined
by spatial derivatives of n and cannot rotate freely.
As a result, it is not possible to fully preserve the algebraic topology of the biaxial triad in the cholesteric.
For example, in the biaxial nematic σi2 = −1 for all
i = x, y, z. This implies that there is a single homotopy class for defects with 2π winding, independent of the
triad direction around which the rotation is performed.
Thus, a pair of σx singularities could be brought together
and, through a continuous family of biaxial nematic configurations, could be transformed into a pair of σz singularities. However, this equivalence spectacularly fails
in the cholesteric. Let us see what happens when we
try to transform a χ2 defect configuration, with 2π rotation about P, nχ2 = r̂ sin(q0 z) + φ̂ cos(q0 z) (in cylindrical coördinates), into a λ2 defect configuration, with
2π rotation about n, nλ2 = ẑ cos(q0 r) + φ̂ sin(q0 r). In
the nχ2 configuration, P is vertical and n is in a planar
+1 disclination configuration, rotating between the radial and azimuthal geometries as a function of z. The
nλ2 configuration is the classic double-twist configuration, with a radial P. Consider the interpolation
nχ2 cos Θ(r) + nλ2 sin Θ(r)
n= p
1 + nχ2 · nλ2 sin 2Θ(r)
(2)
where Θ(r) is a sigmoidal function ranging from
0 to π/2.
To be concrete
we pick Θ(r) =
π (1 − e−r )/(1 + e−(r−r0 ) ) /2 with r0 = 4π/q0 . We discover that this transformation from nχ2 to nλ2 gives rise
to a series of λ±1 ring defects, with alternating winding
sign, encircling the central χ2 defect at radii close to r0
(Fig. 1a), an iconic structure considered long ago [15].
Thus the straightforward homotopy between σx2 and σz2
defects in the biaxial nematic does not carry over to a
homotopy between the supposedly analogous λ2 and χ2
defect configurations. Instead, the transformation is mediated by the appearance of another set of defects. We
will see below that these λ±1 ring defects are made necessary by smectic-like features of the cholesteric phase.
Just as the analogy between cholesteric and biaxial
nematic ground states leads us astray in the study of
defects, the intuitive correspondence between cholesteric
and smectic ground states fails when extended to general
configurations. We can make a straightforward identification of the cholesteric pitch axis P with a smectic layer
normal N. With P = ẑ, we could draw a level set wherever the cholesteric’s director n points in a particular
direction, say x̂, and identify these level sets with smectic layers. Indeed, when we view a cholesteric with P in
the plane of the image through crossed polarizers we see
alternating bright and dark stripes; when the cholesteric
has distortions we may see a distorted pattern of stripes,
as in the famous “fingerprint” patterns. The problem
with thinking of cholesterics as a “layered” phase, however, is that the pitch axis need not be normal to any
3
set of layers. Nothing requires the pitch axis to satisfy
the Frobenius integrability condition [16], P · ∇ × P = 0,
which is necessary for P to be normal to a family of surfaces defined by level sets of some phase field φ(x).
As an example, consider the recently observed oblique
helicoidal cholesteric phase [17], with director field
n = [cos β cos kz, cos β sin kz, sin β] for some tilt angle β. Using our construction we find that P =
[− sin β cos kz, − sin β sin kz, cos β] with eigenvalue q =
k cos2 β. Moreover, P · (∇×P) = −k sin2 β and so this
conical texture, existing as an excited cholesteric state,
cannot be foliated, so there is no corresponding smectic
structure whose layer normal N agrees with P. There
is, however, a linear combination of P and n
/ whose twist
does vanish everywhere, providing a suitable ∇φ direction. We will return to this example below after discussing the physical interpretation of φ in a cholesteric.
Herein lies the central difficulty in connecting the defect
classification between cholesterics, smectics, and the biaxial nematic. In order to identify cholesteric χ disclinations with smectic dislocations, it is necessary to foliate
space; this is impossible with our choice of pitch axis, but
our pitch axis is necessary to establish a triad to connect
to the GSM of the biaxial nematic.
Not all uniaxial nematic configurations can be described as cholesteric, for instance the achiral nematic
ground state. The breakdown of the cholesteric description of nematics is also encoded in the tensor Cij . Because the tensor is not symmetric, its eigenvalues need
not be real; when they are complex no pitch axis can be
identified in the director field by our prescription [18].
Consider the characteristic polynomial for Cij
− q 3 + q 2 Tr C − qΓ(C) + det C = 0
(3)
where 2Γ(C) = ∇ · [n (∇ · n) − (n · ∇) n], the saddlesplay. Since n is a null right eigenvector of Cij , one
eigenvalue vanishes. Reality of the remaining two requires that R = (Tr C)2 − 4Γ(C) ≥ 0 where we dub R
the “cholestericity” – when it is positive we have two distinct pitch axes [19], when it is negative there are no real
values of q and there are no pitch axes, and when R = 0
we only have one pitch and one or two pitch directions,
since we are not guaranteed a full basis when C 6= C T .
When the saddle-splay vanishes the two eigenvalues are
−n·(∇×n) and 0, and so the pitch axis is the vector corresponding to the only nonzero eigenvalue. On the other
hand, when n is a surface normal the twist vanishes, and
so Cij is traceless. In that case, whenever the surface has
negative Gaussian curvature [20] there are two equal and
opposite eigenvalue “chiralities” q2 = −q1 . This is the
origin of the directionally dependent chirality discussed
by Efrati and Irvine [13] in saddle surfaces. A corollary
of this is that surfaces with positive Gaussian curvature
cannot have a real “pitch eigenvalue.” Only at points
where R = 0 can the pitch axis degenerate to the circle
of directions perpendicular to n. Therefore, R vanishes
around λ±1 and τ ±1 defect lines where P is ill-defined,
just as the nematic degree of order S vanishes around
nematic defects where n is ill-defined. For example, the
λ±1 rings that appeared above in the transition between
nχ2 and nλ2 appear where R = 0 as displayed in Fig. 1a.
Of the two pitch axes identified by Cij when R > 0,
the physical pitch axis depends on the boundary conditions. We assume that at infinity (or much closer) we
have a standard cholesteric ground state with vanishing
saddle-splay and thus one pitch axis. As long as R > 0
we can follow the pitch axis at infinity into the bulk of the
sample. Regions where there is no pitch axis or with degenerate pitch axes are defects in the cholesteric. Though
allowed in principle from the structure of the eigenvalue
problem, we do not know if it is physically possible for
there to be regions in which R = 0 and there is only one
pitch axis. We leave this for future study.
Fortuitously our definition of the pitch axis, used to
describe the rotation of n, specifies the rotation of the
entire cholesteric triad as we move along P. This will
have important consequences for our study of topological
defects. Note that in addition to (P · ∇) n = −q n,
/ P·
n = 0 and P2 = 1, which imply (P · ∇) P = κ/
n where
κ is a function of location. We can see that κ is the
curvature of the integral curves of P. In the manner of
the Frenet-Serret apparatus [20], we have:
0 κ 0 P
P
/
/ = −κ 0 q n
(4)
(P · ∇) n
n
0 −q 0
n
Thus, the torsion of integral curves of P equals the local
cholesteric wavenumber, provided that κ does not vanish. This approach leads to an alternate construction of
the “mean torsion” [16, 21]: the two nontrivial eigenvalues of Cij are the possible torsions of the two integral
curves with binormal n and 12 Tr C is their average, the
mean torsion. When Cij has two eigenvalues q1 6= q2
we denote the two eigenvectors P(1) and P(2) , respec(i)
tively, and n
/ ≡ n × P(i) . Because n · P(i) = 0 we have
(1)
P(2) = P(1) cos δ + n
/ sin δ with P(1) · P(2) = cos δ. It
follows from (4) that ∇ · n = n
/ · (/
n · ∇)n = (q1 − q2 ) cot δ;
when there is splay the pitch axes are not orthogonal.
Following streamlines of the pitch axis, we integrate
the local q to determine the total rotation of n about
P between two points. In the ground state the angle
made by the director at z = z0 with the director at z =
0 is simply qz when P = ẑ. We could define a phase
field φ(x) = qz for the ground state that measures the
total rotation of n about P from a particular reference
height such as z = 0. For a general, distorted cholesteric,
the Frenet-Serret apparatus enables us to define a phase
field φ(x) generally, with the change in φ along integral
curves of P recording the integrated rotation of n about
P. Suppose that a particular pitch axis integral curve
P passes through a point x after traversing arc length s
from a reference
point where φ = 0. Then the phase field
R
is φ(x) = P q(x)ds and ∇φ · P = q.
We will show that the general cholesteric configuration
does not admit a smectic phase field whose gradient is everywhere parallel to P. However, the phase field φ can be
4
defined locally for all cholesterics, so a layered structure
can be found lurking inside any cholesteric configuration,
even if the level sets do not correspond to obvious features
of a micrograph or illustration. On a simply connected
domain without defects, this phase field is also globally
defined and the phase field modulo 2π defines layers of
the cholesteric texture. However, in the presence of line
defects the absolute phase depends on the homotopy class
of the integral curve from the reference point. Unlike the
phase of the density fluctuations in smectics, the pitch
streamlines need not generate holonomy in multiples of
2π. Thus, even with the phase field, a cholesteric in general cannot be decorated with continuous layers, though
it is possible in the absence of line defects. In the oblique
helicoidal configuration we considered above, one can find
a direction P̃ that is both perpendicular to n and satisfies P̃ · (∇ × P̃) = 0, so that there exists a phase field
φ whose gradient points along P̃. Writing this direction
as P̃ ∝ P + fn,
/ we are led to a first order Ricatti equation for f that can be solved in smooth regions of the
cholesteric. Though P̃ can be foliated, we know of no
general argument that relates P̃ to ∇φ/|∇φ|.
Do dislocations in a smectic correspond to χ defects
in cholesterics? Note that the ground states of the
cholesteric and smectic share a discrete symmetry under
translations by π/q along the P direction. This discrete
symmetry makes possible the dislocation defects suffered
by smectics. Thus we might expect dislocation defects
in the level sets of φ in cholesterics, as well. As in smectics, we would expect such dislocations in φ to be realized
as disclination dipoles, with ±π winding of ∇φ. If the
eigenvalue q is not allowed to vanish except at singular
lines, then a nontrivial winding in ∇φ implies a nontrivial winding in P, since the dot product of the two vectors
never changes sign. Therefore, defects in ∇φ are also defects in P, either λ or τ singularities. Prohibiting q from
vanishing requires us to move only through cholesteric
textures, analogous to requiring that a biaxial nematic
be nowhere uniaxial [22], the “hard” biaxial phase.
Returning to the interpolated field in (2), we can understand the necessity of λ±1 defects in the P field in
light of the constructed φ field. At small r, where P = ẑ,
the pitch axis does not twist and so φ = q0 z plus an
arbitrary uniform shift; the level sets are planar and horizontal. At large r, where P = r̂, the pitch axis also
does not twist and so φ = q0 r plus an arbitrary uniform
shift; the levels sets are cylinders concentric about r = 0.
The situation is illustrated schematically in Fig. 1b. In
the biaxial nematic, the analogous transformation could
be accomplished by a uniform “escape up” or “escape
down” of one of the triad vectors with decreasing r, and
the same rotation of another triad vector with increasing
r. However, for a cholesteric, such a resolution is incompatible with the existence of a φ field that maintains
nonzero layer spacing (except possibly at line defects).
Instead, the layers formed by level sets of φ must suffer
dislocations, realized as disclination dipoles, line defect
pairs with ±π winding, as illustrated schematically in
(a)
5
R
0
(b)
(c)
FIG. 1. The cholesteric interpolation of Equation (2). (a) A
χ2 line (green) at r = 0 surrounded by λ (blue) and λ−1 (red)
rings near r = r0 . The grey scale is a density plot of R in a
portion of the xz plane; zeros (black) indicate defects in P.
Orange rods show P and its nontrivial winding around the
defect rings; purple rods show n, which is nonsingular. (b)
Level sets of the cholesteric φ field, in the small and large r
limits. (c) Filling space between the two boundaries requires
dislocations in the layers, realized as ±π disclination pairs.
φ ∈ Z are in black; half-layers are in gray.
Fig. 1c. Under our assumptions, these defects in ∇φ are
also defects in P. Since n is nonsingular, the defects are
λ±1 rings. Although the cholesteric configuration might
not geometrically admit a foliation everywhere perpendicular to P, its topology is nonetheless limited by the
topologies available to a family of layers. This example
illustrates that the algebra of cholesteric defects differs
from that of biaxial nematics.
Are all smectic textures subsets of the cholesteric textures with the added constraint that P ∝ ∇φ? As long
as the pitch axis remains straight, one can make this
identification using φ provided that the distortions are
purely z-dependent. Similar identifications are possible for spherically symmetric (hedgehog) and cylindrically symmetric (jelly roll) configurations. The Volterra
construction is then used to create dislocations in the
cholesteric in analogy with those in a smectic. However,
is it possible to apply the Volterra construction while
preserving the cholesteric structure implied in (1)? In
general, the answer is no: The topology of a screw defect in a smectic cannot be constructed in a cholesteric.
This implies that the identification between smectics and
cholesterics fails in the context of topological defects.
To see this, consider the change in angle of the director field along an infinitesimal shift ǫ along the pitch
axis, ∆θ = −ǫ n
/ · (P · ∇) n = ǫq where ∆θ and q will,
in general, depend upon which integral curve of P we
are considering. We see that q = dθ/dǫ. In this context
a Lagrangian set of coördinates is natural – each point
5
in space is labeled by the arclength along each integral
curve, and the integral curve is labeled by its coördinates
at one of the two-dimensional boundaries. However, if
the pitch axis is supposed to be perpendicular to the
“smectic layers” defined by the same value of rotation
θ around each integral curve, then P ∝ ∇θ and we
must implicitly switch between Lagrangian coördinates
and the Eulerian coördinates in space in order to form
the gradient. Accordingly, P = ∇θ/|∇θ| and q = ±|∇θ|
depending on the sign of twist, held constant throughout
the sample as before. Consider now a standard smectic
screw dislocation in θ with layer spacing a and Burgers scalar b, level sets of θ(x) = [z − b arg(x
p + iy)]/a
2
2
2 2
with unit normal
p N(x) = [by/r , −bx/r , 1]/ 1 + b /r
2
2
where r =
x + y . One can verify that the unit
speed curves R(s) = [A cos(γs), −A sin(γs),
√ Bγs] satisfy P = Ṙ = N [R(s)] where γ = 1/ A2 + B 2 and
B = A2 /b. Finally, as r → ∞ the √torsion of this
curve is q = b/(b2 + A2 ) while |∇θ| = b2 + A2 /(Aa).
We can consider ever larger helical paths by letting A
grow, and find that q → 0 as A → ∞ – the integral
curves become straight lines. At the same time, however, |∇θ| → 1/a. These two boundary conditions are
incompatible, q 6= |∇θ|, and prevent us from decorating
the smectic screw dislocation with a director field. Can
we embellish the integral curves to maintain the torsion?
Recall that there is a family of helices [20] that maintain
constant torsion while their curvature vanishes. However, such a family of concentric curves will have singleor double- twist in the core – a vector field that cannot
be foliated since N · (∇ × N) 6= 0.
To summarize, we have proposed a definition of the
pitch axis in the cholesteric that leads to a rich geometry
of an orthonormal triad consisting of the pitch axis P,
the director field n, and the perpendicular direction n.
/
Our construction explicitly demonstrates the differences
between the algebra of the topological defects in smectics,
cholesterics, and biaxial nematics – phases with apparently similar or identical ground state manifolds. The
issue is not the ground states or even the states with
topological defects; for homotopy theory to apply it must
be possible to smoothly distort the director complexion
between any two representatives of the same class. Because of the restrictions on the pitch axis in cholesterics
and the layer normal in smectics, algebraic “equivalence
classes” are not homotopic – the adventure begins.
We acknowledge stimulating discussions with B.G.
Chen and V. Vitelli and support from the UK EPSRC
and NSF Grant DMR12-62047. D.A.B. was supported by
NSF Graduate Research Fellowship DGE-1321851. T.M.
was supported by a University of Warwick Chancellor’s
International Scholarship. S.Č. was partially supported
by Slovenian Research Agency contract P1-0099. R.D.K.
was partially supported by a Simons Investigator grant
from the Simons Foundation. R.A.M. was supported
from FAPESP grant 2013/09357-9.
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