Hydrodynamic instabilities in active cholesteric liquid crystals

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by  Carl A. Whitfield

EPJ manuscript No.
(will be inserted by the editor)
Hydrodynamic instabilities in active cholesteric liquid crystals
Carl A. Whitfield1, Tapan Chandra Adhyapak2, Adriano Tiribocchi3, Gareth P. Alexander1,4, Davide Marenduzzo5,
and Sriram Ramaswamy6,7a
1 Department of Physics, University of Warwick, Coventry CV4 7AL, UK
2 Institut fu¨r Physik, Johannes Gutenberg-Universita¨t Mainz, Staudingerweg 7-9, 55128 Mainz, Germany
3 Dipartimento di Fisica e Astronomia, Universita` di Padova, Via Marzolo 8, I-35131 Padova, Italy
7 4 Centre for Complexity Science, University of Warwick, Coventry CV4 7AL, UK
1
5 SUPA, School of Physics and Astronomy, University of Edinburgh, JCMB Kings Buildings, Mayfield Road, Edinburgh EH9
0
3JZ, Scotland
2
6 TIFRCentreforInterdisciplinarySciences,TataInstituteofFundamentalResearch,21BrundavanColony,Narsingi,Hyder-

n abad 500 075 India
a 7 Department of Physics, Indian Institute of Science, Bangalore 560 012 India
J

8
Received: date / Revised version: date
1

] Abstract. Wedescribethebasicpropertiesandconsequencesofintroducingactivestresses,withprincipal
t direction along the local director, in cholesteric liquid crystals. The helical ground state is found to be
f
o linearly unstable to extensile stresses, without threshold in the limit of infinite system size, whereas con-
s tractilestressesarehydrodynamicallyscreenedbythecholestericelasticitytogiveafinitethreshold.This
.
t is confirmed numerically and the non-linear consequences of instability, in both extensile and contractile
a
cases,arestudied.Wealsoconsiderthestressesassociatedtodefectsinthecholestericpitch(λlines)and
m
show how the geometry near to the defect generates threshold-less flows reminiscent of those for defects
- inactivenematics.Atlargeextensileactivityλlinesarespontaneouslycreatedandcanformsteadystate
d
patterns sustained by constant active flows.
n
o
c
[ 1 Introduction and Phenomenology symmetries different from simple nematic liquid crystals,

or polar fluids, remain largely unexplored.
1
v Activeliquidcrystalshavecometorepresentanarchetype Chiralityisubiquitousinnature,fromthehelicalstruc-
2 foractivematter,offeringaframeworkfororganisingideas tureofDNA[23],tobacterialflagellaandtheirrotarymo-
2 about biological processes and biologically inspired ma- tors [24,25], to selective reflection and structural colour
0
terials. Starting with studies of polar flocks [1,2,3], the in both plants and animals [26,27]. Although there has
5
field has grown to encompass bacterial swarms and grow- been some consideration of active stresses that are chi-
0
. ing colonies [4,5,6], systems of self-propelled rods [7,8], ral [28,29], such as those arising from torque-dipoles [30],
1 the cell cytoskeleton and suspensions of the biopolymers active materials that have a chiral structure have not yet
0
thatconstituteit[9,10,11,12],amongmanyotherbiologi- been studied. There is ample motivation to do so: First,
7
cally inspired systems. From the phenomenology of liquid from the structural point of view, it is noteworthy that
1
: crystals certain generic traits of active matter have been cholesterictexturesarefoundinawiderangeofbiological
v identified,suchasspontaneousflowtransitionsandhydro- systems. Cross-sectional cuts through many fibrous tis-
i
X dynamic instabilities, giant number fluctuations and the sues,suchasdinoflagellatechromosomes,thecarapacesof
role of defects in active turbulence [13,14]. From these a insectsandcrustaceans,fisheggshellsandcompactbones,
r
a picture of living matter, viewed as a material, is emerg- display a distinctive series of arced fibrils [31,32,33,34].
ing. However, from the perspective of condensed matter Thesearethehallmarkofathree-dimensionalhelicalstack-
physics all of these studies represent only a small part ingofstraightfilamentsidenticaltothestructureofcholesteric
of the broad scope offered by active matter phases, cor- liquidcrystals.Moresignificantly,polymersextractedand
responding largely to the two-dimensional nematic and purified from such materials, as well as numerous other
polar-ordered[15,16,17,18]liquidcrystalphases.Recently, biopolymers, including actin, cellulose, chitin, collagen,
active smectic phases have also been studied [19,20,21], microtubules, nucleic acids, polypeptides and polysaccha-
and there has been a parallel development of active mat- rides all show cholesteric phases in solution [31,35,36,37,
terwithoutorientationalorder[22],butactivephaseswith 38,39,40,41,42,43,44], and the same cholesteric ordering
is found in colloidal suspensions of rod-like viruses, such
a Present and permanent address. as the fd virus [45,46,47]. Further examples of cholesteric
2 Carl A. Whitfield et al.: Hydrodynamic instabilities in active cholesteric liquid crystals
orderinbiologicalmaterialsincludethepackingofDNAin (a) (b)
phagecapsids[48,49],theoriginofiridescenceofScarabaei-
dae beetle exoskeletons [36,50,51], the natural colour of
flower petals [52] and Pollia fruit [53], and silk spinning
processes [54].
Given these examples, it is natural to expect that ac-
tive stresses in cholesteric phases should have broad rele-
vance to a diverse range of biological and biologically in-
spiredmaterials.Ourmotivationsarealsosparkedbypos-
Pitch
sibilities of experimental realisation of active cholesterics.
All the natural systems that we have already mentioned
dohavesuchprospects:forinstance,F-actinsolutionsac-
tually exist in a cholesteric, rather than a nematic phase,
Fig. 1. (a)Projectionsofacholestericdirectorfield,equation
asaconsequenceofthedoublehelical,hencechiral,nature (1), onto successive horizontal slices. The right-handed twist
of an actin fibre. Another appealing experimental candi- in the z direction results in a texture of periodic arcs on an
dateforanactivechiralgelisasolutionofDNAmolecules arbitrary slice (top slice). (b) Sketch of the Helfrich-Hurault
interactingwithDNAorRNApolymerases,whichleadto instability of the cholesteric ‘layers’ for a passive cholesteric
relative DNA-enzyme motion and may exert non-thermal under mechanical strain applied at the walls in the vertical
active stresses on the polymers. That macroscopic chiral- direction.
ity is important here is clear from looking at the passive
counterpart of this system, namely a concentrated DNA
and is shown in schematic form in figure 1. In terms of
solution without polymerases: such solutions have long
their fundamental description, nematics and cholesterics
been known to exhibit cholesteric or blue phases in differ-
differ only in the presence of the chiral coupling constant,
ent salt conditions or concentrations [39,40,55]. It is also
q , in the Frank free energy [56,57]. At length scales that
worth noting that many passive cholesterics are formed 0
are short compared to the pitch, π/q , the two materi-
by adding small amounts of a chiral dopant to nematic 0
als are effectively alike. Nonetheless, the bulk properties
materials. This suggests that current experimental reali-
andcharacteristicsofcholestericsdiffersubstantiallyfrom
sations of active nematics [11,12] could be converted to
those of nematics. At wavenumbers k (cid:28) q the struc-
cholesterics by addition of suitable dopant, for instance a 0
ture has a one-dimensional periodicity analogous to the
biopolymer known to form cholesteric mesophases, or fd
density modulation of smectics and hence its elasticity
virus.
is the same [58]. However, in significant ways cholester-
Here, we study active cholesterics from the framework
ics and smectics are also fundamentally unalike. In the
ofactiveliquidcrystals,combiningtheforce-dipolestresses
deepest sense this is because – even if we ignore fluctu-
alreadyestablishedforactivematerialswiththecholesteric
ations – the cholesteric state does not break translation
ordering that comes from a passive chiral nematic. One
invariance: the pitch axis is a continuous screw axis, an
can expect chirality to lead to important effects in active
arbitrary translation along which can be compensated by
materials,similartothesituationinpassivecholestericliq-
arotationaboutit.Inthecholesterictheone-dimensional
uid crystals, whose hydrodynamics and physics are much
periodic modulation carries the director through a con-
different from those of nematics [56]. With respect to the
tinuousfamilyofsymmetry-equivalentdirections;nosuch
fewexistingtheoreticalworksonchiralactivity[28,29]our
symmetryrelatesthedifferentdensitiesencounteredinthe
paperprovidesasystematicanalysisoftheactiveinstabil-
smectic mass-density wave. Whereas the layer normal in
ities of the cholesteric ground state, as well as the active
a smectic is parallel to the molecular alignment, the pitch
stresses and flows generated by defects in the cholesteric
axis is orthogonal to the director in a cholesteric. Geom-
order,knownasλlines.Wedosobymeansoflinearstabil-
etry aids greatly in the description of liquid crystalline
ityanalysisanddirectnumericalsolutionofthenon-linear
structures and their distortions. Several combinations of
equations of motion. Furthermore, we provide a detailed
these with activity can be identified and it is useful to
comparisonbetweenactivecholestericsandthebehaviour
consider each separately. Active stresses in liquid crystals
alreadyestablishedforactivenematicandsmecticphases.
drive fluid flows in response to distortions in the director,
Wefirstgiveaqualitative,phenomenologicalsummary
with splay distortions generating flows locally parallel to
ofactivecholestericsandlistourmainresults:Likenemat-
the director and bend distortions creating perpendicular
ics,cholestericsaredescribedbyaunitmagnitudedirector
flows.
field, nˆ, corresponding to the local molecular alignment.
The main results of our work are: (i) It is well estab-
Unlike nematics this is not uniform in the ground state,
lished that contractile stresses drive an instability of the
but is a linear function of position, the director rotating
nematicstatetosplaydeformationsandextensilestresses
at a uniform rate, q , about a spontaneously chosen di-
0 yield instability to bend [13]. In passive liquid crystals,
rectioncalledthepitchaxis.Denotingbyz thisdirection,
Lubenskyhasshownthatsplaydistortionsinthedirector
the cholesteric ground state corresponds to the director
are not true hydrodynamic modes in a cholesteric but are
screenedatlengthscalesoforderthecholestericpitch[59].
nˆ =cosq zeˆ +sinq zeˆ ≡nˆ , (1) Thusonemayexpectthatthecontractileinstabilityofac-
0 x 0 y 0
Carl A. Whitfield et al.: Hydrodynamic instabilities in active cholesteric liquid crystals 3
tive nematics will be suppressed in a cholesteric until the namicsteadystate[11,67,68,69].Thefundamentaldefects
activityexceedsathresholdoforderKq2andindeedthisis incholestericorderarecalledλlines;theycanbethought
0
what we find. The Stokesian hydrodynamic deformations ofasdefectsinthepitch,ordislocationsinthecholesteric
ofacholestericinvolvebenddistortionsofthedirectorand layers. The layer undulation instability ultimately gives
may be characterised either as a splay of the pitch direc- way to the formation of pairs of λ± lines, reminiscent to
tion,orabendofthecholesteric‘layers’.Asthesearetrue the defect formation process in active nematics. A local
hydrodynamic modes, extensile active materials are still analysis of the director field around a λ line shows that
linearlyunstabletothem.This‘layerundulation’instabil- they do not self-propel, as defects in active nematics do,
ityisanalogousincharactertotheHelfrich-Huraulteffect but they still generate active flows in the cholesteric. We
(figure 1(b)) observed in passive cholesterics and smec- findthatextensileactivecholestericscanformsteadystate
tics in response to an applied electric field [60,61], or an defectpatternsinbothbulkandconfinedgeometries.The
extensional strain [62]. The same layer undulation insta- resulting structure is a lattice of λ lines mediated by re-
bility also occurs in active smectics [19], but for contrac- gions of cholesteric order, with a defect density that in-
tileratherthanextensilematerials.Thedifferencederives creases with activity. Ultimately, this ordered state is de-
from the contrasting alignment of the director relative to stroyed at larger activity with a transition to a state of
the one-dimensional periodicity and emphasises that the active turbulence.
active properties of cholesterics and smectics differ signif-
icantly even though their passive elasticity is the same.
(ii) The direction associated to bend distortions is al- 2 Hydrodynamics of Active Liquid Crystals
ways orthogonal to the local director field. In a nematic
all such orthogonal directions are equivalent, but this is The equilibrium elasticity of cholesteric liquid crystals is
not so in a cholesteric: one of the orthogonal directions given by the Frank free energy for the director field
corresponds to the cholesteric pitch axis, the direction
along which the alignment is rotating. In the hydrody- F =(cid:90) d3r(cid:26)K1(cid:0)∇·nˆ(cid:1)2+ K2(cid:0)nˆ ·∇×nˆ +q (cid:1)2
namic mode associated to extensile instability, the bend 2 2 0
(2)
dflioswtosritniotnhiissddirierecctitoend.aItloisngkntohwenptihtcahtsauxcihs flaonwdsgaerneepralutegs- + K3(cid:0)(nˆ ·∇)nˆ(cid:1)2(cid:27),
2
like, rather than Poiseuille-like, in a passive cholesteric,
and are resisted by an effective viscosity that can exceed where q is a chiral coupling constant that vanishes in a
0
thatinnematicsbyfiveorsixordersofmagnitude[63,59, nematic. The free energy is minimised by a director with
64]. constant right-handed twist, as in (1).
(iii) In cholesterics, the case in which bend distor- In addition to the director field, the hydrodynamic
tionsareperpendiculartothepitchaxisdeservesseparate variables in an active cholesteric are the fluid mass den-
study, both for geometric reasons and because the active sity, ρ, and momentum density, g = ρv, as well as the
flows that are generated lie within the cholesteric layers, concentration, c, of active particles [13,70]. Their equa-
where the effective viscosity is much lower. Such distor- tions of motion are constructed by retaining, at leading
tions and flows arise most simply in response to a uni- orders in gradients, all terms allowed by symmetries and
form conical tilt of the cholesteric director into the pitch conservationlawsandnotnecessarilyderivablefromafree
direction. This situation can be described in a quasi-one- energy. We will assume that the mass density and con-
dimensionalsettingidenticaltothatstudiedextensivelyin centration of active particles are both homogeneous and
the context of spontaneous flow transitions in active ne- constant, which implies that the fluid velocity is incom-
matics[65]andpolargels[66].Inthissenseitmaybecon- pressible, ∇·v = 0. In the viscous regime appropriate to
sidered the natural cholesteric analogue of those studies represent active matter the continuity of the momentum
and the analysis is broadly the same, although the struc- density reduces to the Stokes equation, ∇·σ = 0, where
ture of the director distortions and flows are markedly the stress tensor has components
different, again emphasising the contrasting character of
ν(cid:0) (cid:1)
activecholestericscomparedtootherformsofactivemat- σ =−Pδ +2ηu + n h +h n
ij ij ij 2 i j i j
ter.Theone-dimensionalsettingallowsforexactsolutions, (3)
1(cid:0) (cid:1)
beyond just linear analysis, although the restriction to + n h −h n −Φ ∂ n −ζn n .
2 i j i j ik j k i j
anassumedone-dimensionalvariationalsoartificiallysup-
presses the basic layer undulation instability so that such
Here, P is the pressure, η the viscosity and u =(∂ v +
ij i j
analysis gives only partial, qualitative insight. However,
∂ v )/2thesymmetricpartofthevelocitygradientsofthe
j i
this is no different to the situation in active nematics,
fluid. h=−δF/δnˆ is the molecular field conjugate to the
where such quasi-one-dimensional studies are still highly
liquid crystal director, and ν is a flow alignment parame-
instructive [65,66,14].
ter. The liquid crystal is flow aligning if |ν| ≥ 1 and flow
(iv) Defects play a fundamental role in the non-linear tumbling if |ν|<1 [56]. We choose ν <−1 corresponding
description of materials, both passive and active. In ac- toaflow-aligningactivecholestericwithrod-likeparticles
tive nematics, active stresses nucleate defects and drive (e.g. F-actin filaments or DNA molecules). The “Erick-
their self-propulsion, eventually creating a turbulent dy- sen stress” term contains Φ =[∂f/∂(∂ n )](δ −n n )
ij i k jk j k
4 Carl A. Whitfield et al.: Hydrodynamic instabilities in active cholesteric liquid crystals
where f is the free energy density (such that F =(cid:82) d3rf isconvenienttosolvetheStokesequationintheCartesian
inequation(2)).Thefinaltermistheactivestress,which basis. The active contribution to the Stokes equation is
correspondstoaforcedipolealignedalongthelocaldirec-
(cid:26)
tor field of the liquid crystal [70]. The phenomenological fa ≈−ζ ik (δθ +δθ )eˆ
coefficient ζ is proportional to the concentration of active n 2 x n+1 n−1 z
particles and is positive in extensile materials and nega- (cid:104) (cid:105)
+ k (δφ −δφ )+i(k +nq )(δθ +δθ ) eˆ
tiveincontractileones.Theactivestressultimatelyarises x n+2 n−2 z 0 n+1 n−1 x
through an off-diagonal piece in the matrix of kinetic co- (cid:104) (cid:105) (cid:27)
efficients linking stress and fuel consumption, viewed as + ikx(δφn+2+δφn−2)−(kz+nq0)(δθn+1−δθn−1) eˆy ,
‘fluxes’, to strain-rate and chemical-potential imbalance,
(9)
viewed as ‘forces’ [14].
Finally,thedynamicequationfortherelaxationofthe and interestingly does not depend on δφ but rather the
n
director field is modesinneighbouringBrillouinzones.Similarly,thepas-
sivedistortioncontributiontotheStokesequationfd can
1 n
∂tni+vj∂jni+ωijnj =−νuijnj + γhi, (4) be written in terms of the molecular field in equations,
and is given in Appendix A. The stability of the initial
where ωij =(∂ivj−∂jvi)/2 is the antisymmetric, or rota- perturbations δφn and δθn is determined by equation (4)
tional, part of the velocity gradients, and is the same as and is given by
inapassiveliquidcrystal.Theparameterγ isarotational ik (cid:20) (cid:16) (cid:17)
viscosity which sets the timescale for reorientation due to −iωδφ =−q v(z)+ x iν v(x) −v(x)
n 0 n 4 n+2 n−2
the molecular field.
(10)
(cid:16) (cid:17)(cid:21) h⊥0
+2v(y)−ν v(y) +v(y) + n
n n+2 n−2 γ
3 Linear Instabilities in an Active Cholesteric ν+1(cid:26)(cid:104) (cid:105)
−iωδθ =− k +(n+1)q (iv(x) −v(y) )
n 4 z 0 n+1 n+1
3.1 Generic hydrodynamic instability: pitch-splay
(cid:104) (cid:105) (cid:27)
+ k +(n−1)q (iv(x) +v(y) )
The hydrodynamics of cholesterics is subtle, as not all z 0 n−1 n−1
director deformations correspond to true hydrodynamic ν−1 (cid:16) (cid:17) hz
modes. We follow the framework of Lubensky [59] for − ik v(z) +v(z) + n .
4 x n+1 n−1 γ
the hydrodynamics of passive liquid crystals to calculate
(11)
hydrodynamic instabilities in active cholesterics in the
Stokesianregime.Agenericperturbationofthecholesteric
In general ω is the solution to an infinite-dimensional
ground state (1) can be written
eigenvalue problem [59] where each mode n is coupled to
(cid:2) (cid:3) neighbouringmodesupton±6.Inthehydrodynamiclimit
nˆ =cosδθ cos(q z+δφ)eˆ +sin(q z+δφ)eˆ
0 x 0 y (5) k (cid:28) q0 the lowest energy modes decouple at order k0 to
+sinδθeˆ , a5×5eigenvalueprobleminvolvingthemodesδφ ,δθ ,
z 0 ±1
and δφ , given explicitly in Appendix A. We look first
≈nˆ +δφnˆ +δθeˆ , (6) ±2
0 ⊥0 z
for hydrodynamic modes of the passive case which corre-
wherenˆ =−sinq zeˆ +cosq zeˆ ,andδφ=δφ(x,z,t) spond to eigenvalue solutions of ω → 0 as ζ,k → 0. We
⊥0 0 x 0 y
andδθ =δθ(x,z,t)aresmallfluctuations.Withoutlossof find that these exist in two cases, when kx =0 or kz =0.
generalityweconsiderthemodewithwavevector(kx,0,kz) When kx = 0 the matrix is diagonalised and the sys-
lying in the x-z plane, which, due to the symmetry of the tem becomes quasi-one-dimensional. In this case the only
cholesteric ground state, can be written as a sum over all hydrodynamic mode has eigenfunction δφ0 and is purely
Brillouin zones diffusive with ω = −iK2kz2/γ. This mode remains unaf-
fected by non-zero activity and so is not generically un-
(cid:88)
δφ(x,z,t)= δφ ei[kxx+(kz+nq0)z−ωt] (7) stable in an active cholesteric.
n
n When kz =0 the problem can be further reduced to a
δθ(x,z,t)=(cid:88)δθ ei[kxx+(kz+nq0)z−ωt] (8) systemofthreecoupledequationsforδφ0 andδθ±1 inthe
n
longwavelengthlimitk (cid:28)q .Inthepassivecase(ζ =0)
n x 0
we find that the lowest energy mode is diffusive and has
where k ∈ [−π/q ,π/q ] is in the first Brillouin zone. corresponding director perturbation:
z 0 0
Henceforth all terms with subscript n are understood to
denotethenthcomponentoftheinfinitesumoverallBril- δφ=δφ0ei(kxx−ωt) (12)
louin zones. ik
The molecular field hn (given in Appendix A) is cou- δθ =− qxδφ0cos(q0z)ei(kxx−ωt) (13)
0
pledtothemodesδφ andδθ .Indeed,thesymmetry
n±2 n±1
of the cholesteric basis means that even n modes of δφ which agrees with the analysis of the static equations by
n
are coupled to odd n modes of δθ and vice versa [59]. It Lubensky [59]. Including a small activity |ζ| (cid:28) Kq2 this
n 0
Carl A. Whitfield et al.: Hydrodynamic instabilities in active cholesteric liquid crystals 5
3.2 Suppression of splay instability and spontaneous
flow transition
Contractile active nematics are linearly unstable to splay
distortions in the director field [13,70,73]. For a director
lying in the xy-plane these splay distortions may be writ-
ten in the form
δn=δθei(kz−ωt)eˆ . (15)
z
When k (cid:29) q the cholesteric ground state (1) locally re-
0
sembles a nematic, on length scales of this perturbation,
and the distortion retains its splay-like character. How-
Fig. 2. Sketchofthepitch-splay,orlayerundulation,instabil- ever, for k ∼ q the cholesteric order “screens” the insta-
0
ity in extensile cholesterics. Black lines show the projection of bility and produces a finite threshold for its onset.
thetwisteddirectorfieldontotheplanecontainingthesplayed
Substituting (15) into the force balance equations we
pitch axis (red lines). The blue arrows show the active flow
find that the fluid velocity is
direction which acts to increase the distortion and drives the
instability. v =δθei(kz−ωt)(cid:20)ζ+ 1+ν(cid:0)K k2+K q2(cid:1)(cid:21)
2 1 3 0
(16)
q nˆ −iknˆ
mode becomes generically unstable to extensile activity × 0 ⊥0 0.
η(k2−q2)
with 0
Substituting this into (4) and keeping terms to first order
ω ≈ i(cid:20)ζ − 3K3k2(cid:21) 8η+γ(1+ν)2 . (14) in the perturbations we find the dispersion relation
0 η 2 8 8η+2γ(1+ν)2
ω =−iζ1+ν −i(cid:0)K k2+K q2(cid:1)(cid:20)1 + (1+ν)2(cid:21). (17)
2η 1 3 0 γ 4η
A sketch of this perturbation mode and the accompa-
nying active flow is shown in figure 2. As shown in [59] Theactivetermisidenticaltothatforasplayedperturba-
this mode corresponds to a pure splay deformation of the tioninanactivenematic[73],andagainislongrange.The
pitch,orabendofthedirectorfieldthatisdirectedalong passivesplaycontributionproportionaltoK alsoremains
1
the pitch axis. When the activity is extensile, this defor- unchanged, and in the nematic limit q → 0 we recover
0
mation mode gives rise to active flows along the pitch the familiar splay instability of active gels. The new con-
axis, parallel to the bend direction, which destabilise the tribution arising from the cholesteric order is the passive
cholesteric order.This hydrodynamic pitch-splay modein bend term proportional to K q2. We see that this is al-
3 0
a cholesteric is analogous to the layer undulation mode waysnegativeandimaginary,andhenceactstoscreenthe
of smectic A materials. Indeed, their passive elasticity is nematic splay mode. Instability only sets in when the ac-
the same [58]. However, the director distortions that un- tivityexceedsathresholdoforderK q2.Asinthenematic
3 0
derlie the cholesteric pitch-splay mode are of a different case [66,65], for flow-aligning, rod-like particles (ν <−1)
type than those associated to the layer undulations of a the instability occurs for extensile activity (ζ >0).
smectic. As we have described, in a cholesteric the di- Thislinearanalysisisconfirmedbyfullnumericalsolu-
rector distortions are of bend type as this leads only to tionofthehydrodynamicequationsforsuchasplaymode
splay deformations of the pitch axis. In contrast, the di- in an active cholesteric, as we show in figure 3. As there
rector distortions of smectic layer undulations are splay is only z-dependence of both the perturbation and the
deformations, since the director corresponds to the layer cholestericgroundstate,thesimulationscanbeperformed
normal. It is for this reason that the instability of active in a quasi-one-dimensional setting, although we defer de-
cholesterics occurs for extensile materials, in contrast to tails of the numerical method until § 4. This set up is,
thesituationinactivesmectics,whereitisthecontractile therefore,entirelyanalogoustothatconsideredinthecon-
material that is unstable [19]. textofspontaneousflowtransitionsinactivenematics[66,
Modes which are destabilised by contractile activity 65].Alsointhecholestericcase,thereisaspontaneousflow
(ζ <0) all have finite stability thresholds in the limit k = transition above the threshold for linear instability, how-
0 of order Kq2. Thus one expects contractile cholester- ever its character is rather different. Figure 4 shows the
0
ics to be stable unless the magnitude of the activity ex- results of numerical simulations of this transition in an
ceeds(approximately)Kq2.Thisappliestoω whichgives extensile cholesteric confined between parallel plates with
0 0
an instability for large enough contractile activity, but tangential anchoring conditions, as well as the analogous
in general this may not by the lowest instability thresh- results for a nematic [65], reproduced for comparison. In
old mode. For example, the offset family of perturbations the nematic, the spontaneously flowing state is charac-
δθ ,δφ alsopredictfiniteinstabilitythresholdsinthe terised by director splay distortions localised around the
2n 2n−1
longwavelengthlimit.Inthefollowingsectionweconsider mid-planeofthecell,andalsoattheboundariestoaccom-
a special case of these in the quasi-one dimensional limit. modatethetangentialanchoring.Thebenddistortionsare
6 Carl A. Whitfield et al.: Hydrodynamic instabilities in active cholesteric liquid crystals
(a) (b)
z x
n v
z z
(c) (d)
e e
d d
u u
nit nit
g g
a a
M M
y d
a n
pl e
S B
z z
Fig. 4. Profiles of (a) director tilt, n , and (b) velocity along the x-direction, v , for three different steady states found for
z x
extensile active cholesterics (ζ =0.025). Panels (c) and (d) show the magnitude of splay and bend deformations, respectively,
for each of these steady states. q =0 is the nematic case, shown for reference. The legend on the right also shows schematics
0
of the steady state director field in each case.
uniformlysmallbycomparison.Theflowispredominantly
along the director field, in conformity with the splay na-
π ture of the distortions driving them.
16 By contrast, in the cholesteric the spontaneous flow
Stable transition leads to an approximately uniform conical tilt
ofthedirectoralongthepitchaxisthroughoutthebulkof
q Unstable
0 the cell, vanishing only near the boundaries to satisfy the
π tangentialanchoringconditions.Thisleadstoaheliconical
32 director profile in which the bend distortion has constant
magnitude and is directed orthogonal to the pitch axis.
There is no splay, except close to the boundaries where it
is due to the surface anchoring. The spontaneous flow is
predominantly along the direction of the bend distortion
andhenceparalleltothevectornˆ .Throughoutthebulk
⊥0
0.01 0.02 0.03 0.04 of the cell it has approximately constant magnitude, but
ζ
0 helically varying direction. Although we do not show it,
simulationswithafullyperiodiccholesterictextureandno
Fig. 3. Quasi-one-dimensional numerical results for extensile
boundaryconditionsyieldaspontaneousflowtransitionto
active cholesterics with three values of the pitch. The active
a perfect heliconical director with a constant conical tilt
gel is sandwiched between two infinite plates, parallel to the
angle.
xy plane, lying at z =0 and z =L . Planar boundary condi-
z
tionsaresetonbothwalls.Thelineistheanalyticalprediction These simulations suggest a simple one-dimensional
from § 3.2 for k =π/L , demonstrating the q2 dependence of analysisoftheflowingstateanalogoustothatofVoituriez
z 0
theactivitythresholdforthismode.Thediscrepancyatsmall et al. [66] for polar gels, but for the heliconical director
valuesofq isdueto thefixedanchoringboundaryconditions field
0
used in the simulations.
nˆ =cosθnˆ +sinθeˆ , (18)
0 z
withtheconicalangleθaconstantparametertobesolved
for. There is no splay in this texture, but the bend vector
Carl A. Whitfield et al.: Hydrodynamic instabilities in active cholesteric liquid crystals 7
is non-zero We use our method to simulate an active cholesteric,
eitherinaquasi-one-dimensionalgeometry(Figures3and
(nˆ ·∇)nˆ =q0sinθcosθnˆ⊥0, (19) 4), or in a quasi-two-dimensional geometry (xz-plane).
The typical system size is 64 (quasi-1D geometry) or 64×
and of constant magnitude, oriented perpendicularly to
64 (quasi-2D geometry) lattice sites with either periodic
boththedirectorfieldandthepitchaxis.Theactivestress
boundary conditions or homogeneous (planar) strong an-
associated to it drives a flow in the same direction
choring of the director field on both the upper and the
(cid:18) ζ 1+νcos2θ lower wall bounding the simulation domain. The initial
v =−q0sinθcosθ ηq2 + 2η conditionistheequilibriumhelix(1),forasuitablechoice
0 (20) of q , with a small deformation in the midplane and no
(cid:19) 0
(cid:2) (cid:3) flow. In our calculations we have chosen α = −0.1, β =
× K +(K −K )cos2θ nˆ .
2 3 2 ⊥0
0.1, γ = 1, ν = −1.1 (corresponding to a rod-like, flow-
aligningliquidcrystal),andη =5/3asinpreviousnumer-
Finally, the angle θ is determined by the director relax-
ical works [75,76]. We have also assumed the one elastic
ation equation (4). In addition to the passive solution
constant approximation, setting K = K = K = K =
1 2 3
θ = 0, the activity allows for a solution with non-zero
0.01.Inthecontractilecase,forinstance,thesevaluescan
conical tilt, given implicitly by
be mapped as done in [77] onto an actomyosin gel with
ζ (cid:18) 1 (cid:0)1+νcos2θ(cid:1)(cid:19) effective elastic constant equal to 1 nN, γ =1kPa·s, and
− = + η = 1.67kPa/s. When using walls, the value of the sur-
2ηq02 γ(cid:0)1+νcos2θ(cid:1) 4η (21) faceanchoringwaschosensoastobealwaysinthestrong
×(cid:2)K +(K −K )cos2θ(cid:3). anchoring limit.
2 3 2
We conclude that in this simplified one-dimensional In the case of extensile activity, there is a continuous
picture, the cholesteric is primarily unstable to bend de- flow transition with the activity ζ. Figure 5 shows suc-
formations resulting from a constant tilt of the director cessive steady states for increasing extensile activities. As
along the pitch axis. Thus the splay instability of an ac- predicted in § 3.1 for an infinite cholesteric, we observe
tivenematicisreplacedbyabendinstabilityintheactive a pitch-splay instability as the director field bends and
cholesteric. The bend vector twists with the director field is advected along the direction of positive curvature as
and thus generates flows of constant magnitude along di- sketched in figure 2 for the linear instability. The bifur-
rections perpendicular to the pitch axis that rotate along cation to spontaneous flow is supercritical, corresponding
itinahelicalfashion.Inviewofitsnature,werefertothis toacontinuousnonequilibriumtransition,asfoundin[78]
remarkable active flowing state as a “sliding cholesteric”. foractivenematics.Thetransitionthresholdissystemsize
dependent and goes to zero in an infinite system (corre-
sponding to k → 0 in the linear stability condition (14)).
4 Numerical Simulations As the activity is increased the resulting bend deforma-
tions become more pronounced and vortices begin to ap-
To study the nature of the state that develops from the pearintheflow(figures5(a)-(c)).Atlargeractivitiesstill,
cholesteric ground state in response to the fundamental λlines(defectsinthepitchaxis)arecreatedandseparated
hydrodynamic pitch-splay instability, we solve the non- into distinct λ± pairs akin to the dynamics of defects in
linear equations of motion numerically. More specifically, two-dimensional active nematics. Initially the dynamics
we use a hybrid lattice-Boltzmann (LB) algorithm, previ- of this system is unsteady and resembles the turbulence
ouslyusedforpassiveoractivenematicliquidcrystals[74, of active nematics [68], with λ lines being both sponta-
75,76],tosolve(4),andthefollowingNavier-Stokesequa- neously created and annihilated. However, at long times
tion, the system can reach a steady state defect configuration
with a fixed director field, accompanied by steady active
ρ(∂ +v ∂ )v =∂ (σ )+η∂ (∂ v +∂ v ), (22) flows. This state is discussed further in § 5.
t j j i j ij j j i i j
where all terms were defined previously (see Section § 2). Inthecaseofcontractileactivitywedonotexpectthe
The molecular field hi is defined in terms of the following hydrodynamic modes of the cholesteric liquid crystal to
free energy, be unstable from the linear calculation in § 3.1. This is
F =(cid:90) d3r(cid:26)αn2+ β(cid:0)n2(cid:1)2+ K1(cid:0)∇·n(cid:1)2 rsetaflbecleteudpbtyo rtehleatsivimeluylahtiigohnscoanstrtahceticlehoalcetsitveirtiycvtaelxuteusr.eAins
2 4 2 instability occurs at ζ = −0.01 which is approximately
(23)
+ K2(cid:0)n·∇×n+q (cid:1)2+ K3(cid:0)(n·∇)n(cid:1)2(cid:27). two orders of magnitude larger than the threshold in the
2 0 2 extensile case. Figure 6 shows that above this threshold,
thecholestericorderisdestroyedandtheresultingdirector
We note that in our simulations n is not a unit vector; field and flow field resembles that of a contractile active
this constraint is though enforced softly through the bulk nematic at large activity [79]. Note that if ν > −1, devi-
free energy which is minimised by n2 =−α/β (we choose ations of the director along z can become unstable (from
α=−β, see below). the linear prediction of § 3.2) and the director field can
8 Carl A. Whitfield et al.: Hydrodynamic instabilities in active cholesteric liquid crystals
(I) (II)
(a)
(b)
(c)
(d)
Fig. 5. Simulation results for an extensile active cholesteric in a quasi-two-dimensional geometry. The snapshots correspond
to director field profile (I) and the corresponding velocity profile (II) for ζ equal to: 0.00009 (a), 0.00025 (b), 0.001 (c), 0.005
(d). In (a)-(c) these profiles are steady states of the system, whereas (d) is a representative snapshot of the time-dependent
asymptotic state. All results are obtained for q =π/16.
0
Carl A. Whitfield et al.: Hydrodynamic instabilities in active cholesteric liquid crystals 9
(I) (II)
(a)
(b)
(c)
(d)
Fig. 6. Simulation results for a contractile active cholesteric in a quasi-two-dimensional geometry. The snapshots correspond
to director field profile (I) and the corresponding velocity profile (II) for ζ equal to: −0.01 (a), −0.015 (b), −0.03 (c), −0.05
(d). In all (a)-(d) the snapshots are representative of a time-dependent asymptotic (statistically steady) state. All results are
obtained for q =π/16.
0
10 Carl A. Whitfield et al.: Hydrodynamic instabilities in active cholesteric liquid crystals
(I) (II)
(a)
(b)
(c)
Fig. 7. Results of an extensile active cholesteric confined in a quasi-two-dimensional geometry with flat walls. Homogeneous
anchoringofthedirectorissetonboththeupperandlowersurfaces.Thesnapshotscorrespondtodirectorfieldprofile(I)and
the corresponding velocity profile (II) for ζ equal to: 0.001 (a), 0.0025 (b), 0.005 (c). In (a) and (b) these profiles are steady
states of the system, whereas (c) is a snapshot of the time-dependent asymptotic state. All results are obtained for q =π/16.
0
transiently display a heliconical steady state above the fi- roll structure of the velocity profile, clearly visible in col-
nite threshold. umn (II) of figure 7. This director pattern is accompa-
nied by the generation of symmetric λ line pairs, similar
Finally, the effect of confinement is studied by con- to the defect lattices observed in the periodic case and
sidering a cell geometry in which the active cholesteric also to those formed in passive cholesterics undergoing a
is sandwiched between two non-slip walls at the top and Helfrich-Hurault instability [80]. An increase in the activ-
bottomofthesimulation.Figure7showsthedirector(col- itydecreasesthelengthscaleofthedefectseparationalong
umn(I))andcorrespondingvelocityfield(column(II))for the x-direction, sustained by shorter wavelength rolls in
threevaluesofactivityforwhichthesystemisintheactive the flow field (figure 7(b)). These ordered structures are
state. Interestingly, for ζ = 0.001 we observe a Rayleigh- destroyed for higher values of activity when the system
Benard-like pattern of the director field with the typical enters the chaotic state.
undulationsseenintheperiodiccase(seefigure5(c))now
squeezed between two flat walls. This pattern requires a