The Geometry of the Cholesteric Phase

Daniel A. Beller,1 Thomas Machon,2 Simon Copar,1, 3, 4 Daniel M. Sussman,1

Gareth P. Alexander,2 Randall D. Kamien,1 and Ricardo A. Mosna1, 5

1Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104-6396, USA2Department of Physics and Centre for Complexity Science, University of Warwick, Coventry CV4 7AL, UK

3Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia4Jozef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia

5Departamento de Matematica Aplicada, Universidade Estadual de Campinas, 13083-859, Campinas, SP, Brazil(Dated: October 8, 2018)

We propose a construction of a cholesteric pitch axis for an arbitrary nematic director field asan eigenvalue problem. Our definition leads to a Frenet-Serret description of an orthonormal triaddetermined by this axis, the director, and the mutually perpendicular direction. With this tool weare able to compare defect structures in cholesterics, biaxial nematics, and smectics. Though theyall 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 thecholesteric and the smectic are essentially the same beast:the Caille instability in smectics [1] is the same as theLubensky instability in cholesterics [2], both examples ofthe Peierls instability. Fluctuations around the groundstates of each are dominated by a single Goldstone modewith an unusual fluctuation spectrum that distinguishesmodes along the layer normal or pitch axis from modesalong other directions. Since these modes are also thecoordinates on the ground state manifold (GSM), one istempted to identify defects in smectics with defects incholesterics, with the cholesteric pitch axis P analogousto the smectic layer normal N.

On the other hand, the cholesteric phase is biaxial[3, 4]: intuitively, a cholesteric is defined by a unit direc-tor field n and a derived pitch axis P that describes thedirection along which n twists, whose precise definitionwe will discuss below. At each point, a local orthonormaltriad is specified by n and P, both defined up to sign,along with their cross product n×P ≡ /n (“not n”). Fromthis point of view, one might attempt to identify defectsin the cholesteric with defects in the biaxial phase char-acterized by the standard homotopy theory of topologi-cal defects [5–8], in particular via the non-Abelian fun-damental group π1[SO(3)/D2d] = {±1,±σx,±σy,±σz},the unit quaternions. This has been the motivation be-hind a classification of cholesteric defects as singularitiesin 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 σxσy = σz in the quaternion group,composing a λ with a τ yields a χ in the cholesteric.Meanwhile, the cholesteric-smectic analogy suggests thatdisclinations in the layer normal N are to be identifiedwith the λ and τ defects where P is not defined, so thatthe decomposition of a smectic dislocation into a discli-nation pair [10] is also analogized to the decompositionof a χ into a λ-τ pair.

In order to explore the similarities and differences be-tween the topology of defects in smectics, cholesterics,

and biaxial nematics, we will develop the geometric toolsnecessary to map between these three systems. We findthat though there is a natural mapping between theground states of these three systems, the descriptions oftheir defects are all different. In particular, we demon-strate the failure of this natural mapping in three ex-ample textures: the χ2 to λ2 defect transformation, theoblique helicoidal cholesteric texture, and the screw dis-location. Our result highlights the well-known problemwith classifying topological defects in translationally or-dered media [5, 10, 11] and confronts the usual analogiesbetween our triptych of phases.

There are crucial differences between the three sys-tems, illustrated most clearly in the symmetries of theirground states. A smectic ground state with layer normalz and layer spacing a is invariant under a translation bya along z; in other words, a continuum of ground statesis available under translations modulo a along z. Thisdiscrete symmetry is associated with the existence of dis-locations in the smectic phase. The smectic ground statealso possesses a continuous symmetry under arbitrary ro-tations about z. In contrast, the biaxial nematic groundstate with director n parallel to z possesses a continuoussymmetry under arbitrary translations along z, but a dis-crete symmetry under rotations by π about z. The latterdiscrete symmetry is related to the biaxial singularitiesnot found in uniaxial nematics. Along with uniaxial ne-matics, smectics and biaxial nematics are invariant underπ rotations around x and y.

From the point of view of symmetries the cholestericis intermediate between the smectic and biaxial nematicphases. A cholesteric ground state with pitch axis z andhelical wave number q0 has a continuous screw symmetry:rotations by an angle α about z composed with transla-tions by α/q0 along z. However, a translation by π/q0along z, and a rotation by π about z, are each (sepa-rately) discrete symmetry operations of the cholestericground state (as are rotations by π about x or y). Inother words, the cholesteric ground state has the dis-crete symmetries of both the smectic and the biaxial ne-

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matic ground states, which might lead us to expect thatcholesteric defects will bear some resemblance to bothbiaxial nematic defects and smectic defects. However,broken translational invariance leads to a breakdown ofthe homotopy theory of defects [5, 10, 11] and so sym-metries are not the whole story.

We note also that smectics can have point defects, forinstance when the layers are concentric spheres arounda single point, whereas the biaxial nematic cannot sup-port such defects as π2[SO(3)/D2d] = 0. This makes theinterpretation of the cholesteric even more subtle. Fromone point of view the cholesteric configuration is com-pletely specified by a director field everywhere in spaceand is thermodynamically equivalent to a uniaxial ne-matic, which has ground state manifold RP 2 and cansupport point defects. However, as discussed in [12] if weinclude the pitch axis then a simple point defect in thedirector field necessarily sprouts line defects in the pitchaxis. From this point of view the connection betweenthe cholesteric and biaxial nematic may seem promising.However, there is a weakness in this link as well, whichwill appear when we give an explicit definition of the P.

To proceed any further, let us now define the unitcholesteric pitch axis P. With the orthonormal triad ofthe biaxial in mind, we require P · n = 0 and define thedirection “not n” as /n = n×P; the cholesteric triad isthen {n,P, /n}. Intuitively, P is a direction about whichn twists, meaning that the directional derivative (P·∇)nhas no component along P. As n is a unit vector, thisdirectional derivative is also perpendicular to n, so wecan define P by the equation (P ·∇)n = −q /n, or theequivalent eigenvalue problem

PiCij = qPj (1)

where q is the local eigenvalue and Cij = nkε`jk∂in` isprecisely the same chirality tensor defined in [13]. Note,in addition, that Tr C = −n ·∇×n. This approach issimilar to, but more restricted than, the definition pro-posed in [14] that only holds for the cholesteric groundstate texture. Our construction generates a triad definedup to sign so the GSM is SO(3)/D2d.

Whence the weak link in the analogy between biax-ial nematics and cholesterics? Note that in the biaxialnematic, the local triad can undergo a global rotationthat may change the energy but not the topology, in pre-cise analogy with the two-dimensional uniaxial nematic;a +1 disclination, for example, can be purely bend (az-imuthal) or purely splay (radial). If the bend and splayelastic constants differ, the energies will differ but thetopology will not. However, this internal rotation is nota symmetry of the cholesteric since the pitch axis, definedthrough Cij , involves spatial derivatives. As a result, ro-tations of the triad also require a rotation of the system,i.e. only the trivial rotation of the whole sample leavesthe topology invariant. Though it is tempting to viewthe cholesteric’s λ, τ , and χ defects in analogy with theσx, σy, and σz defects of the biaxial nematic, the cor-respondence is not precise: Homotopies between biaxial

nematic triad configurations, which establish the topo-logical equivalence of two configurations, are not in gen-eral available to cholesteric triads, where P is determinedby spatial derivatives of n and cannot rotate freely.

As a result, it is not possible to fully preserve the al-gebraic topology of the biaxial triad in the cholesteric.For example, in the biaxial nematic σ2

i = −1 for alli = x, y, z. This implies that there is a single homo-topy class for defects with 2π winding, independent of thetriad direction around which the rotation is performed.Thus, a pair of σx singularities could be brought togetherand, through a continuous family of biaxial nematic con-figurations, could be transformed into a pair of σz sin-gularities. However, this equivalence spectacularly failsin the cholesteric. Let us see what happens when wetry to transform a χ2 defect configuration, with 2π ro-

tation about P, nχ2 = r sin(q0z) + φ cos(q0z) (in cylin-drical coordinates), into a λ2 defect configuration, with

2π rotation about n, nλ2 = z cos(q0r) + φ sin(q0r). Inthe nχ2 configuration, P is vertical and n is in a planar+1 disclination configuration, rotating between the ra-dial and azimuthal geometries as a function of z. Thenλ2 configuration is the classic double-twist configura-tion, with a radial P. Consider the interpolation

n =nχ2 cos Θ(r) + nλ2 sin Θ(r)√

1 + nχ2 · nλ2 sin 2Θ(r)(2)

where Θ(r) is a sigmoidal function ranging from0 to π/2. To be concrete we pick Θ(r) =π[(1− e−r)/(1 + e−(r−r0))

]/2 with r0 = 4π/q0. We dis-

cover that this transformation from nχ2 to nλ2 gives riseto a series of λ±1 ring defects, with alternating windingsign, encircling the central χ2 defect at radii close to r0(Fig. 1a), an iconic structure considered long ago [15].Thus the straightforward homotopy between σ2

x and σ2z

defects in the biaxial nematic does not carry over to ahomotopy between the supposedly analogous λ2 and χ2

defect configurations. Instead, the transformation is me-diated by the appearance of another set of defects. Wewill see below that these λ±1 ring defects are made nec-essary by smectic-like features of the cholesteric phase.

Just as the analogy between cholesteric and biaxialnematic ground states leads us astray in the study ofdefects, the intuitive correspondence between cholestericand smectic ground states fails when extended to generalconfigurations. We can make a straightforward identifi-cation of the cholesteric pitch axis P with a smectic layernormal N. With P = z, we could draw a level set wher-ever the cholesteric’s director n points in a particulardirection, say x, and identify these level sets with smec-tic layers. Indeed, when we view a cholesteric with P inthe plane of the image through crossed polarizers we seealternating bright and dark stripes; when the cholesterichas distortions we may see a distorted pattern of stripes,as in the famous “fingerprint” patterns. The problemwith thinking of cholesterics as a “layered” phase, how-ever, is that the pitch axis need not be normal to any

3

set of layers. Nothing requires the pitch axis to satisfythe Frobenius integrability condition [16], P ·∇×P = 0,which is necessary for P to be normal to a family of sur-faces defined by level sets of some phase field φ(x).

As an example, consider the recently observed obliquehelicoidal cholesteric phase [17], with director fieldn = [cosβ cos kz, cosβ sin kz, sinβ] for some tilt an-gle β. Using our construction we find that P =[− sinβ cos kz,− sinβ sin kz, cosβ] with eigenvalue q =k cos2 β. Moreover, P · (∇×P) = −k sin2 β and so thisconical texture, existing as an excited cholesteric state,cannot be foliated, so there is no corresponding smecticstructure whose layer normal N agrees with P. Thereis, however, a linear combination of P and /n whose twistdoes vanish everywhere, providing a suitable ∇φ direc-tion. We will return to this example below after dis-cussing the physical interpretation of φ in a cholesteric.Herein lies the central difficulty in connecting the defectclassification between cholesterics, smectics, and the bi-axial nematic. In order to identify cholesteric χ disclina-tions with smectic dislocations, it is necessary to foliatespace; this is impossible with our choice of pitch axis, butour pitch axis is necessary to establish a triad to connectto the GSM of the biaxial nematic.

Not all uniaxial nematic configurations can be de-scribed as cholesteric, for instance the achiral nematicground state. The breakdown of the cholesteric descrip-tion of nematics is also encoded in the tensor Cij . Be-cause the tensor is not symmetric, its eigenvalues neednot be real; when they are complex no pitch axis can beidentified in the director field by our prescription [18].Consider the characteristic polynomial for Cij

− q3 + q2 TrC − qΓ(C) + detC = 0 (3)

where 2Γ(C) = ∇ · [n (∇ · n)− (n ·∇)n], the saddle-splay. Since n is a null right eigenvector of Cij , oneeigenvalue vanishes. Reality of the remaining two re-quires that R = (TrC)2 − 4Γ(C) ≥ 0 where we dub Rthe “cholestericity” – when it is positive we have two dis-tinct pitch axes [19], when it is negative there are no realvalues of q and there are no pitch axes, and when R = 0we only have one pitch and one or two pitch directions,since we are not guaranteed a full basis when C 6= CT .When the saddle-splay vanishes the two eigenvalues are−n·(∇×n) and 0, and so the pitch axis is the vector cor-responding to the only nonzero eigenvalue. On the otherhand, when n is a surface normal the twist vanishes, andso Cij is traceless. In that case, whenever the surface hasnegative Gaussian curvature [20] there are two equal andopposite eigenvalue “chiralities” q2 = −q1. This is theorigin of the directionally dependent chirality discussedby Efrati and Irvine [13] in saddle surfaces. A corollaryof this is that surfaces with positive Gaussian curvaturecannot have a real “pitch eigenvalue.” Only at pointswhere R = 0 can the pitch axis degenerate to the circleof directions perpendicular to n. Therefore, R vanishesaround 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, theλ±1 rings that appeared above in the transition betweennχ2 and nλ2 appear whereR = 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 condi-tions. We assume that at infinity (or much closer) wehave a standard cholesteric ground state with vanishingsaddle-splay and thus one pitch axis. As long as R > 0we can follow the pitch axis at infinity into the bulk of thesample. Regions where there is no pitch axis or with de-generate pitch axes are defects in the cholesteric. Thoughallowed in principle from the structure of the eigenvalueproblem, we do not know if it is physically possible forthere to be regions in which R = 0 and there is only onepitch axis. We leave this for future study.

Fortuitously our definition of the pitch axis, used todescribe the rotation of n, specifies the rotation of theentire cholesteric triad as we move along P. This willhave important consequences for our study of topologicaldefects. Note that in addition to (P ·∇)n = −q /n, P ·n = 0 and P2 = 1, which imply (P ·∇)P = κ/n whereκ is a function of location. We can see that κ is thecurvature of the integral curves of P. In the manner ofthe Frenet-Serret apparatus [20], we have:

(P ·∇)

P/nn

=

0 κ 0−κ 0 q0 −q 0

P/nn

(4)

Thus, the torsion of integral curves of P equals the localcholesteric wavenumber, provided that κ does not van-ish. This approach leads to an alternate construction ofthe “mean torsion” [16, 21]: the two nontrivial eigenval-ues of Cij are the possible torsions of the two integralcurves with binormal n and 1

2Tr C is their average, themean torsion. When Cij has two eigenvalues q1 6= q2we denote the two eigenvectors P(1) and P(2), respec-

tively, and /n(i) ≡ n×P(i). Because n ·P(i) = 0 we have

P(2) = P(1) cos δ + /n(1)

sin δ with P(1) · P(2) = cos δ. Itfollows 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 integratethe local q to determine the total rotation of n aboutP between two points. In the ground state the anglemade by the director at z = z0 with the director at z =0 is simply qz when P = z. We could define a phasefield φ(x) = qz for the ground state that measures thetotal rotation of n about P from a particular referenceheight such as z = 0. For a general, distorted cholesteric,the Frenet-Serret apparatus enables us to define a phasefield φ(x) generally, with the change in φ along integralcurves of P recording the integrated rotation of n aboutP. Suppose that a particular pitch axis integral curveP passes through a point x after traversing arc length sfrom a reference point where φ = 0. Then the phase fieldis φ(x) =

∫P q(x)ds and ∇φ ·P = q.

We will show that the general cholesteric configurationdoes not admit a smectic phase field whose gradient is ev-erywhere parallel to P. However, the phase field φ can be

4

defined locally for all cholesterics, so a layered structurecan be found lurking inside any cholesteric configuration,even if the level sets do not correspond to obvious featuresof a micrograph or illustration. On a simply connecteddomain without defects, this phase field is also globallydefined and the phase field modulo 2π defines layers ofthe cholesteric texture. However, in the presence of linedefects the absolute phase depends on the homotopy classof the integral curve from the reference point. Unlike thephase of the density fluctuations in smectics, the pitchstreamlines need not generate holonomy in multiples of2π. Thus, even with the phase field, a cholesteric in gen-eral cannot be decorated with continuous layers, thoughit is possible in the absence of line defects. In the obliquehelicoidal configuration we considered above, one can finda direction P that is both perpendicular to n and satis-fies P · (∇ × P) = 0, so that there exists a phase field

φ whose gradient points along P. Writing this directionas P ∝ P + f/n, we are led to a first order Ricatti equa-tion for f that can be solved in smooth regions of thecholesteric. Though P can be foliated, we know of nogeneral argument that relates P to ∇φ/|∇φ|.

Do dislocations in a smectic correspond to χ defectsin cholesterics? Note that the ground states of thecholesteric and smectic share a discrete symmetry undertranslations by π/q along the P direction. This discretesymmetry makes possible the dislocation defects sufferedby smectics. Thus we might expect dislocation defectsin the level sets of φ in cholesterics, as well. As in smec-tics, we would expect such dislocations in φ to be realizedas disclination dipoles, with ±π winding of ∇φ. If theeigenvalue q is not allowed to vanish except at singularlines, then a nontrivial winding in ∇φ implies a nontriv-ial winding in P, since the dot product of the two vectorsnever changes sign. Therefore, defects in ∇φ are also de-fects in P, either λ or τ singularities. Prohibiting q fromvanishing requires us to move only through cholesterictextures, analogous to requiring that a biaxial nematicbe nowhere uniaxial [22], the “hard” biaxial phase.

Returning to the interpolated field in (2), we can un-derstand the necessity of λ±1 defects in the P field inlight of the constructed φ field. At small r, where P = z,the pitch axis does not twist and so φ = q0z plus anarbitrary uniform shift; the level sets are planar and hor-izontal. At large r, where P = r, the pitch axis alsodoes not twist and so φ = q0r plus an arbitrary uniformshift; the levels sets are cylinders concentric about r = 0.The situation is illustrated schematically in Fig. 1b. Inthe biaxial nematic, the analogous transformation couldbe accomplished by a uniform “escape up” or “escapedown” of one of the triad vectors with decreasing r, andthe same rotation of another triad vector with increasingr. However, for a cholesteric, such a resolution is in-compatible with the existence of a φ field that maintainsnonzero layer spacing (except possibly at line defects).Instead, the layers formed by level sets of φ must sufferdislocations, realized as disclination dipoles, line defectpairs with ±π winding, as illustrated schematically in

R

0

5(a)

(b) (c)

FIG. 1. The cholesteric interpolation of Equation (2). (a) Aχ2 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 aportion of the xz plane; zeros (black) indicate defects in P.Orange rods show P and its nontrivial winding around thedefect rings; purple rods show n, which is nonsingular. (b)Level sets of the cholesteric φ field, in the small and large rlimits. (c) Filling space between the two boundaries requiresdislocations 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 arealso defects in P. Since n is nonsingular, the defects areλ±1 rings. Although the cholesteric configuration mightnot geometrically admit a foliation everywhere perpen-dicular to P, its topology is nonetheless limited by thetopologies available to a family of layers. This exampleillustrates that the algebra of cholesteric defects differsfrom that of biaxial nematics.

Are all smectic textures subsets of the cholesteric tex-tures with the added constraint that P ∝ ∇φ? As longas the pitch axis remains straight, one can make thisidentification using φ provided that the distortions arepurely z-dependent. Similar identifications are possi-ble for spherically symmetric (hedgehog) and cylindri-cally symmetric (jelly roll) configurations. The Volterraconstruction is then used to create dislocations in thecholesteric in analogy with those in a smectic. However,is it possible to apply the Volterra construction whilepreserving the cholesteric structure implied in (1)? Ingeneral, the answer is no: The topology of a screw de-fect in a smectic cannot be constructed in a cholesteric.This implies that the identification between smectics andcholesterics fails in the context of topological defects.

To see this, consider the change in angle of the di-rector field along an infinitesimal shift ε along the pitchaxis, ∆θ = −ε /n · (P ·∇)n = εq where ∆θ and q will,in general, depend upon which integral curve of P weare considering. We see that q = dθ/dε. In this contexta Lagrangian set of coordinates is natural – each point

5

in space is labeled by the arclength along each integralcurve, and the integral curve is labeled by its coordinatesat one of the two-dimensional boundaries. However, ifthe pitch axis is supposed to be perpendicular to the“smectic layers” defined by the same value of rotationθ around each integral curve, then P ∝ ∇θ and wemust implicitly switch between Lagrangian coordinatesand the Eulerian coordinates in space in order to formthe gradient. Accordingly, P = ∇θ/|∇θ| and q = ±|∇θ|depending on the sign of twist, held constant throughoutthe sample as before. Consider now a standard smecticscrew dislocation in θ with layer spacing a and Burg-ers scalar b, level sets of θ(x) = [z − b arg(x + iy)]/a

with unit normal N(x) = [by/r2,−bx/r2, 1]/√

1 + b2/r2

where r =√x2 + y2. One can verify that the unit

speed curves R(s) = [A cos(γs),−A sin(γs), Bγs] sat-

isfy P = R = N [R(s)] where γ = 1/√A2 +B2 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 Agrow, and find that q → 0 as A → ∞ – the integralcurves become straight lines. At the same time, how-ever, |∇θ| → 1/a. These two boundary conditions areincompatible, q 6= |∇θ|, and prevent us from decoratingthe smectic screw dislocation with a director field. Canwe 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. How-ever, such a family of concentric curves will have single-or double- twist in the core – a vector field that cannotbe foliated since N · (∇×N) 6= 0.

To summarize, we have proposed a definition of thepitch axis in the cholesteric that leads to a rich geometryof an orthonormal triad consisting of the pitch axis P,the director field n, and the perpendicular direction /n.Our construction explicitly demonstrates the differencesbetween the algebra of the topological defects in smectics,cholesterics, and biaxial nematics – phases with appar-ently similar or identical ground state manifolds. Theissue is not the ground states or even the states withtopological defects; for homotopy theory to apply it mustbe possible to smoothly distort the director complexionbetween any two representatives of the same class. Be-cause of the restrictions on the pitch axis in cholestericsand the layer normal in smectics, algebraic “equivalenceclasses” are not homotopic – the adventure begins.

We acknowledge stimulating discussions with B.G.Chen and V. Vitelli and support from the UK EPSRCand NSF Grant DMR12-62047. D.A.B. was supported byNSF Graduate Research Fellowship DGE-1321851. T.M.was supported by a University of Warwick Chancellor’sInternational Scholarship. S.C. was partially supportedby Slovenian Research Agency contract P1-0099. R.D.K.was partially supported by a Simons Investigator grantfrom the Simons Foundation. R.A.M. was supportedfrom FAPESP grant 2013/09357-9.

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