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Lattice Boltzmann simulations of liquid crystalline fluids:

active gels and blue phases

M. E. Cates1, O. Henrich1, D. Marenduzzo1, K. Stratford2

1SUPA, School of Physics and Astronomy, and 2EPCC,

University of Edinburgh, JCMB Kings Buildings,Mayfield Road, Edinburgh EH9 3JZ, Scotland

October 29, 2018

Lattice Boltzmann simulations have become amethod of choice to solve the hydrodynamicequations of motion of a number of complex flu-ids. Here we review some recent applications oflattice Boltzmann to study the hydrodynamicsof liquid crystalline materials. In particular, wefocus on the study of (a) the exotic blue phasesof cholesteric liquid crystals, and (b) active gels– a model system for actin plus myosin solutionsor bacterial suspensions. In both cases latticeBoltzmann studies have proved useful to pro-vide new insights into these complex materials.

1 Introduction

In recent years, the lattice Boltzmann (LB) al-gorithm [1] has emerged as a powerful methodto study fluid dynamics. Due to its conceptualsimplicity and codability (particularly on paral-lel computers), LB provides an attractive alter-native to other methods such as finite elementsalgorithms. In the last couple of decades, inparticular, the LB method has increasingly beenapplied to the hydrodynamics of complex fluids,such as binary fluids, colloidal suspensions andliquid crystals [2, 3, 4, 5, 6, 7, 8].In applying LB to complex fluids, one often aimsat solving two coupled sets of partial differentialequations. One set describes the evolution ofthe order parameter (e.g., composition for bi-nary mixtures, or orientational order in liquidcrystals), whereas the other set describes con-servation of mass and momentum (via the con-

tinuity and Navier-Stokes equations for the ve-locity field). In liquid crystals, which are thefocus of this article, one typically considers theBeris-Edwards model [9], which is defined start-ing from a free energy expressed in terms ofa tensorial order parameter, Q, whose largesteigenvalue describes locally the strength of lo-cal molecular alignment (nematic order), andwhose corresponding eigenvector defines the di-rector n along which this alignment prevails.(See the Appendix for the precise form of thefree energy adopted.) Note that simpler descrip-tions, in which the magnitude of the ordering isassumed constant and only the director varies,are unsuitable for describing blue phases. Thisis because such phases contain defect lines atwhich the ordering drops locally [10].The equation of motion for Q is then [9]:

DtQ = ΓH. (1)

The left member of equation 1 is a “materialderivative” describing the time evolution of theorder parameter advected with the velocity u

of a fluid element. For fluids of rod-like parti-cles such as liquid crystals, flow gradients maylead to local rotations and these are also takeninto account by the material derivative. In Eq.1, Γ is a collective rotational diffusion constant,which sets the time scale for the relaxation oforientational order (usually in the millisecondrange for small-molecule liquid crystals), and H

is the “molecular field”, which provides the forcefor this relaxation motion. The molecular field

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involves the derivative of the free energy with re-spect to the order parameter (see the Appendixfor the specific form of H).As stated previously, the fluid velocity obeys thecontinuity equation, ∂tρ = −∇.u ≡ ∂αuα whereρ is the fluid density, which in practical casesthis can be taken as constant (so that the fluidis incompressible). However, a feature of LBis that slight fluid compressibility is maintainedin the algorithm; this makes all the dynamicsfully local (the sound-speed is finite) rather thanhaving to solve each timestep for a pressure fieldthat responds instantaneously to distant events.(Locality is important to efficient parallelizationstrategies, and maintains near-linear scaling ofthe computational cost with the size of the sys-tem under investigation [11].)The fluid velocity also obeys the Navier-Stokesequation, which for effectively incompressiblefluids reads

ρ [∂t + uβ∂β ]uα = ∂βΠαβ (2)

+ η∂β (∂αuβ + ∂βuα) ,

where η is the viscosity, Cartesian componentsare denoted by Greek indices, and Παβ is a ther-modynamic stress tensor. This tensor, like H,is found by differentiation of the free energy,and its divergence ∂βΠαβ represents an effectivebody force, acting on the fluid, arising from theresponse to deformation of the order parameterfield. (Appendix A gives the explicit form.) Inthe case of active fluids, the forces do not solelystem from a free energy but include terms aris-ing from the dissipative conversion of chemicalenergy into motion. This creates an additionalcontribution to Παβ .Eqs. 1 and 2 are very complex to solve due totheir inherent nonlinearities, and only limitedprogress is possible with analytical techniques.In contrast, LB offers an ideal method to solvenumerically these equations, allowing their fulldynamics to be addressed not only in one di-mension but also in 2D and 3D flows. Suchstudies can not only test approximate analyticsolutions where these exist, but also lead to im-proved insight into the nonlinear physics con-tained in the underlying models.For purely Newtonian fluids, the LB algorithmproceeds by the introduction of “mesoscopic”

velocity distribution functions f(ci;x), propor-tional to the density of (notional) fluid particlessitting at a lattice node at x having a certain ve-locity ci chosen from a discrete set [1]. (Thesediscrete velocities correspond to moving by onelattice site in a timestep.) The distributionfunctions evolve according to an appropriate lo-cal dynamics, which recovers the Navier-Stokesand continuity equations upon coarse graining.(The density ρ then equates to

∑

i f(ci) and theaverage fluid velocity u becomes the first mo-ment,

∑

i f(ci)ci, of the distribution.)

Historically, the first LB approach to study com-plex fluids [3, 5] consisted of one extra set of dis-tribution functions for each new order parame-ter component entering the equations of motion.More recently, however, it has become apparentthat this “full LB” approach had the drawbackof requiring a large memory to store all the dis-tribution functions for the whole lattice, on topof being rather cumbersome from a theoreticalviewpoint. At the same time, the new genera-tion of LB studies for complex fluids has shownthe need to address ever larger systems, to makebest use of the potentially very good scalabilityof parallel LB codes.

As a result, new hybrid algorithms have beencoded and deployed, for both binary fluids andliquid crystals [12, 13], where the LB algorithmis used to solve the Navier-Stokes equation (Eq.2), and is coupled to a standard finite-differencesolver for the order parameter dynamics (Eq.1). At each timestep, the fluid velocity foundby LB is used to calculate the advective deriva-tive in Eq. 1, while the order parameter foundfrom that equation is used to compute the forc-ing term, ∂βΠαβ , in Eq. 2. By this division oflabour, LB is only used to handle the momen-tum and mass transport – the problem for whichit was originally devised.

In what follows, we review recent applications offull and hybrid LB algorithms to the study of ac-tive liquid crystalline fluids and the blue phasesof cholesteric liquid crystals. In both cases, aswe shall see, numerical simulations have provedextremely helpful in providing a link betweentheoretical predictions and experimental obser-vations.

While in this work we focus on a “hydrody-

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namic” description of liquid crystalline fluidsvia continuum models, it is important to notethat there is a variety of other coarse grainedmodels and methods to study liquid crystals.Most relevant to our topics here, liquid crys-talline molecules may be individually modelledvia e.g. soft spherocylinders, interacting withGay-Berne potentials (see e.g. [14, 15] for a se-ries of recent examples) and a related approachhas also managed to stabilise blue phases [16],although the length scale accessible with thismore detailed approach is typically significantlysmaller than the ones which can be studied withan LB continuum description.

2 Active fluid simulations

Active fluids have become a highly topical re-search area at the interface of soft matter andbiological physics. Generally speaking, an ac-tive particle “absorbs energy from its surround-ings or from an internal fuel tank and dissipatesit in the process of carrying out internal move-ments”. (The quote is from a well-known pa-per of the Bangalore group, which helped pi-oneer the use of continuum models of activematter [17].) This definition applies to bacte-ria, swimming algae and other microorganisms,as well as to living cells or cell extracts. How-ever active materials may also be non-biological,and a synthetic example is a shaken granularfluid [18]. As the definition clearly implies, ac-tive materials may remain far from thermody-namic equilibrium even in steady state, due totheir continuous energy intake, and this renderstheir properties of particular interest to physi-cists.

We consider active fluids comprising a con-centrated suspension of such active particles.Paradigmatic examples of active fluids are bac-terial suspensions and solutions of cytoskeletalgel components (actin fibers or microtubules)with molecular motors (myosin or kinesin). Theterm “active gels” is also widely used for thesematerials; but while all are non-Newtonian, notall of them are strongly viscoelastic. Experi-mental studies with active fluids have uncov-ered a wide range of intriguing and non-trivial

physical properties. For instance, microscopystudies of droplets of Salmonella [19] and of B.subtilis [20, 21, 22] reveal that when the bacte-ria are concentrated enough (more than about20-30% in volume fraction), long range corre-lations arise, creating eye-catching patterns offlow involving long-lived vortices (see Fig. 1).These resemble the turbulent flow of fluids athigh Reynold numbers, although remarkably inthis case, the Reynolds number is very small –effectively zero [23]. (In this respect, the result-ing “bacterial turbulence” resembles elastic tur-bulence in polymer solutions, which is attainedpast a critical value of the Wiessenberg num-ber [24, 25].)

The equations of motion of an active liquid crys-talline fluid have been written down, either onthe basis of symmetry [17, 26], or via a coarsegraining of an underlying microscopic model ofstiff cytoskeletal gels and molecular motors [27].There are three main differences between theseequations and the equations of motion of a pas-sive liquid crystal. The first one is the presenceof an active term in the stress tensor, whosedivergence acts as a force in the Navier-Stokesequation (Eq. 2). This extra active term hasbeen first shown in [28] to be proportional toan activity constant, ζ , which to lower order islinear in the energy uptake of the fluid (e.g.,via ATP hydrolysis), times the local order pa-rameter Q. A second term arises from activ-ity also, but the form of this can be absorbedinto the free energy (see Appendix) and we setthis to zero. A third and final difference canarise in cases where an oriented swarm of swim-ming particles have a collective mean velocityrelative to the surrounding fluid, causing a ‘self-advection’ effect. Such materials are called ‘po-lar’ and are distinct from the ‘apolar’ case whichdescribes either a swarm in which equal numberof particles swim forward and backward alongthe director axis, or particles which are non-motile but nonetheless exert active forces on thefluid. (Motile and non-motile active particlesare sometimes called “movers” and “shakers”respectively.) More details on the exact formsof these active terms, and of the equations ofmotion used to describe polar systems, are givenin the Appendix.

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The sign of ζ – hence of the active apolar con-tribution to the stress tensor – is of vital im-portance for the hydrodynamics and the rhe-ology of active fluids [6, 7, 17]. A positive ζcorresponds to a suspension of extensile activeparticle, or “pushers” [29], which exert forcesalong the molecular axis away from the cen-tre of mass and towards the surrounding fluid.A negative ζ corresponds to a fluid of contrac-tile active particles, or “pullers” [29], for whichthe force dipoles is exerted axially towards thecentre of mass. Examples of contractile fluidsare suspensions of the biflagellated alga Chlamy-

domonas and actomyosin gels (or more generalsuspensions of non-permanently cross-linked cy-toskeletal gels and molecular motors). On theother hand, the majority of bacteria are thoughtto be extensile [29]. However, we are currentlylacking quantitative experiments which measurethe velocity field around active particles, whichcould lead to estimates of the values of ζ in thesevarious cases.

Alongside the two distinctions already made(apolar versus polar, and contractile versus ex-tensile), all rodlike molecules, including passiveones, fall into two further categories, known as“flow aligning” and “flow tumbling”. With-out activity, the former exhibit stable flow inwhich the director is inclined to the flow direc-tion at a certain angle (the Leslie angle) whereasthe latter undergo continuous director evolu-tion, which is frequently chaotic [30]. To avoidthe double complexity created by the flow tum-bling instability on top of activity, we addresshere only the flow-aligning case.

Early theoretical work determined the linearstability of these systems and it was foundthat an infinite sample of active material withnonzero Q (whether polar or apolar) is hy-drodynamically unstable to order parameterfluctuations, and that this instability is con-nected to the generation of a spontaneous fluidflow [17, 26, 28]. (For extensile – but not con-tractile – particles, this instability is presenteven if the flow velocity is constrained to bea function of one spatial coordinate only.) Itwas also realised that the introduction of bound-aries together with suitable “anchoring condi-tions” (fixing the molecular orientation) would

lead to the stabilisation of the non-flowing or-dered phase of uniform Q. (Stability is restoredfor small values of ζ , lying below a threshold ζcwhich decreases with system size, L, as 1/L2.)However, within these analytical approaches, itwas not possible to determine the ultimate flowpattern resulting from this hydrodynamic insta-bility.

The simulations reported in [6, 7, 13] gave there-fore the first quantitative predictions for theresulting spontaneous flow patterns in unsta-ble active fluids. We focus first on a quasi-1Dslab geometry with planar anchoring along theboundary. Here it was found that for an apo-lar fluid, upon increasing the activity, the sys-tem organises into a spontaneous Poiseuille flow(with a smoothly varying flow velocity, maximalat the centre of the slab). This spontaneouslybreaks symmetry and causes a net mass fluxalong the slab axis. For stronger activity and/orlarger system sizes, one can also find sponta-neously “shear-banded” flows in which succes-sive layers of material have very different shearrates.

In a 2D (thin film) flow geometry, the pat-terns differed significantly from the 1D case andwere in good qualitative agreement with obser-vations of flow patterns in, for instance, con-centrated B. subtilus and Salmonella suspen-sions [19, 20, 21, 22]. Fig. 2 shows some of thepatterns which were found in 2D. In the apolarcase, for values of ζ just above the threshold, theinstability first evolves into steady state “con-vective” rolls (Figs. 1a-b) [13]. In some caseswe also observed active bands or a successionof rolls (Fig. 2c), whereas deeper in the ac-tive phase we found that an initial array of rollsbreaks up into what looks chaotic flow (“bacte-rial turbulence”) at low Reynolds number (Fig.2d-f). The equations of motion for polar activesuspensions (see Appendix) can be treated withthe same method. This is somewhat similar,in that above a threshold there is a transitionto a spontaneously flowing state. In this case,however, we do not observe rolls or bands andthe system jumps directly into the “turbulent”flowing state (see Fig. 2f-h, with 2h showingtrajectories of tracer particles which highlightthe chaotic appearance of the flow).

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LB simulations have also proved helpful to char-acterise the rheology of active fluids. In Ref. [27]it was suggested that the viscosity of a con-tractile active fluid should diverge at the pas-sive isotropic-nematic transition in 2D. Simula-tions have confirmed and generalised this to thecase of a 3D order parameter (albeit constrainedto undergo a 1D flow) where it was clarifiedthat the divergence there occurs at the spinodalpoint (which unlike the 2D case is different fromthe isotropic-nematic transition point). LB sim-ulations also showed that upon increasing thedensity towards this spinodal point, one shouldobserve a decrease in the viscosity for extensilefluids.

The non-linear rheology should show even morestriking behaviours [7]. Contractile fluidsshould strongly shear thicken for small forcing,and the extent of thickening should depend onthe distance from the isotropic-nematic tran-sition. Within a bulk nematic phase, a for-mal yield stress (a stress threshold below whichthere is no flow in steady state) is predicted,whereas for larger forcing, these materials ex-hibit shear thinning and approach the unen-hanced (passive) viscosity at very large shearrates.

In contrast, isotropic suspensions of concen-trated extensile particles (bacteria) should startfrom a low viscosity and thicken to again ap-proach the passive behaviour upon increasingthe shear rate. Strikingly, a bulk oriented (ne-matic) phase of extensile active particles shouldshow a zero effective viscosity for shear ratesbelow some critical value. This is because the3D ordered system in the absence of stress (butconstrained to have 1D flow) spontaneously or-ganizes into two shear bands with flow in oppo-site directions, so that there is zero net velocityof the fluid. A finite relative velocity of the con-fining walls can now be accommodated, still atzero stress, by adjusting the relative amounts ofthe two bands. (Since there is now a finite shearrate and no shear stress, the viscosity is formallyzero.) Finally, our studies suggested that whensubjected to shear flow, extensile fluids shouldform shear bands more readily than passive onesclose to the isotropic-nematic transition, whilecontractile fluids should shear-band less readily.

These rheological predictions remain provi-sional, assuming as they do a 1D flow profileand the results may well be modified when thisassumption is relaxed. (Such work is now under-way in our group.) Nonetheless, the recent ad-vances in experimental techniques, which havemade it possible to grow thin concentrated filmsof, e.g. B. subtilis, should render several ofour results, such as the 2D predictions of Fig.2,testable in the near future.

3 Blue phase simulations

The “blue phases” (BPs) of chiral nemato-genic molecules offer spectacular examples offunctional soft matter; each comprises a self-sustained network of disclinations [10] embed-ded within a nematic matrix. (A disclination isa topological defect line, defined such that thenematic director rotates through a half-turn ontreversing any circuit that encloses this line.)

At high temperature, a fluid of chiral ne-matogenic molecules remains in the “isotropicphase”, with no preferred orientation of themolecular axes. Upon cooling down the sam-ple, the molecules become oriented (Q is fi-nite), but due to molecular chirality the direc-tor field n rotates with spatial position, describ-ing everywhere a helix with a well-defined axis.(This is called the cholesteric phase.) However,very close to the transition, it is more advan-tageous locally for the director field to rotatein a helical fashion about any axis perpendic-ular to a straight line – this complicated pat-tern was named a “double twist cylinder” [10].Mathematically, it is impossible to patch to-gether such double twist cylinders without cre-ating defects in between, and this frustrationgives rise to the disclination network observedin the blue phases, and responsible for manyof their remarkable physical properties. Moststriking among these are their optical proper-ties (such phases can be made in all colours,not just blue) which stem from the presence ofa lattice of disclinations with a unit cell whosesize is comparable to the wavelength of visiblelight. In BP I and II (the two most common)this lattice has long range cubic order; BP III

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however is certainly not cubic, and probably notordered.

BPs have a fascinating scientific history. Theywere first reported in the late 19th century, by ascientist named Reinitzer, and then long forgot-ten, until some new experimental interest arosein the 1960s and 1970s. Initial theories of BPsonly came out in the 1980s (see e.g. [31]), whenthe concept of double twist was first proposed.At this stage BPs were widely considered tobe of purely academic interest, mainly becausethey were only stable in a very narrow temper-ature range (about 1 K) close to the isotropic-cholesteric transition. In 1983, a world-leadingexpert on liquid crystals, F. C. Frank, said:“They [Blue phases] are totally useless, I think,except for one important intellectual use, that ofproviding tangible examples of topological odd-ities, and so helping to bring topology into thepublic domain of science, from being the pri-vate preserve of a few abstract mathematiciansand particle theorists.” [10]. In the first decadeof the 21st century, this view rather suddenlychanged, following advances in fabrication thatenormously increased the stability range of BPs,up to about 50 K [32, 33]. In May 2008 Sam-sung presented the first blue-phase based liquidcrystal display at the annual SID InternationalSymposium. This new display is able to oper-ate at a high frame frequency (240 Hertz), doesnot require costly alignment treatment at theboundaries of the liquid crystal, and may oneday supersede current LC (twisted nematic) dis-play technologies.

Traditional theories of BPs (see e.g. [34])were based on semi-analytical approximations.While they were extremely useful to gain a qual-itative understanding of the physics of BPs,these approaches had to rely on severe assump-tions for progress to be possible. For instance,when estimating the phase diagram, the Q ten-sor was approximated by a truncated Fourierseries. Computational constraints only allowedvery few harmonics to be considered. As a re-sult, when important experimental observationswere mispredicted by the theory, it was not clearwhether this was a drawback of the underlyingfree energy functional (the Landau-de Gennesfree energy, see Appendix), or simply an arti-

fact of the simplifications which were employed.These early theoretical works left unexplainedthe detailed shape of the phase diagram – BPIand BPII appeared in the wrong order on in-creasing the chirality κ. (This parameter isproportional to the inverse helical pitch of thecholesteric; see Appendix.) With an electricfield applied, the analytic theories were also un-able to account for the anomalous field-induceddistortion (electrostriction) of BPI, nor to ex-plain why a new phase, named BPX, should bestable at all. Finally, most theories assume cu-bic symmetry, so cannot describe BPIII, or the“blue fog”, which is thought to comprise a net-work of disclination lines without long-range or-der.

In recent years, LB simulations have been re-markably useful in filling most of these concep-tual theoretical gaps, and have significantly ex-tended our quantitative understanding of thephysics of blue phases. Firstly, in Refs. [35, 36],Eq. 1 was solved by non-hybrid LB in theabsence of fluid flow (u = 0) with a set ofinitial conditions suggested from analytical ex-pressions for the infinite chirality limit of BPI,BPII and O5. (O5 is another disclination lattice,which was proposed as stable by early theories,but not observed in experiments.) This pro-cedure amounts to a free energy minimisationwith a topological constraint; that is, the purelyrelaxational dynamics of Eq. 1 was shown tomaintain the point group of the disclination net-work chosen initially. Therefore, this approachcan be used to map out the full phase diagram asa function of chirality, κ, and a reduced temper-ature parameter τ (see Appendix for the math-ematical definition of these paramaters). Thisapproach has the advantage of not making anyapproximations beyond those implicit in the se-lected (Landau-de Gennes) free energy.

These LB simulations showed that the phase di-agram predicted by the continuum theory is ac-tually in good qualitative agreement with theexperiments: BPI and BPII show up in theright order on increasing chirality, and O5 isrelegated to unphysical regions in the phasediagram [35]. The structure of these phasesis shown in Fig.3a,b where a surface is drawnaround each defect line at a certain contour of

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the ordering strength. (This creates a renderingof each disclination as a fattened tube.) The nu-merical phase diagram (Fig.3c) is in good agree-ment with the experimental one (Fig.3d). Aninteresting calculation in [38] has shown thateven including an extra set of spherical har-monic in the analytical scheme of Ref. [34] isunable to reproduce the details shown by thenumerics. This is an example in which simula-tions are extremely important to accurately findwhat the predictions of a given theory are.

Similarly, LB simulations were performed in thepresence of an electric field in Ref. [36]. Itwas found that under a small electric field theunit cell of BPII tended to elongate along thefield direction, and shrink perpendicularly to it,whereas BPI displayed an opposite behaviour,once more in agreement with experiments. Anintermediate field also turned the disclinationnetwork of BPI into a different structure (whichit is tempting to identify with the experimen-tally observed BPX). Therefore, also when anelectric field is present, the Landau-de Gennesfree energy works remarkably well, although an-alytically tractable approximations of the re-sulting equations are not adequate to captureits predictions.

As well as allowing one to find equilibrium statesunder different fields and thermodynamic con-ditions, LB of course comes into its own for dy-namical problems in which fluid flow cannot beignored. For instance, the existence of a discli-nation network affects the response of a BP toan imposed Poiseuille flow [39]. Here, LB workhas shown that flow can lead to the unzipping ofdisclinations of integer topological charge. (Ona circuit around such a defect line, the direc-tor rotates through a whole number of turns,rather than the half-turn around a standarddisclination; these higher-order lines can formmetastable networks in the absence of an ap-plied flow.) Flow also can cause the bendingand twisting of the BPI and BPII disclinationnetworks. This bending and twisting lead to anelastic component in the rheological behaviour,and as a result the simulations predict “perme-ative” flows, in which the molecules comprisingthe blue phase flow through a static disclinationpattern whose geometry is hardly perturbed by

the underlying molecular transport. The samekind of flow also occurs when cholesterics aresheared by small forcing along the direction oftheir helical axis [40]. Typically, BPs also dis-play significantly shear thinning behaviour, asa strong enough flow disrupts the disclinationnetwork in a manner that reduces the elasticstress.Until very recently, LB work on BPs was lim-ited to one unit cell of the disclination lat-tice, within which several disclination cores arepresent and require a fine enough discretisationto be correctly resolved. However, supercom-puters now allow supra-unit cell simulations ofBPs, the first account of which we have recentlygiven in Ref. [41], where we studied the do-main growth dynamics of a BPII domain insidea cholesteric or isotropic “slab”, in a parameterregion in which the BP is the thermodynam-ically stable state. The simulations give evi-dence of an intriguing domain growth kinetics.For small values of the chirality, the growth isslow and the resulting blue phase has no or fewdefects. When the chirality exceeds a certainthreshold, however, the advancing disclinationnetwork changes its symmetry and reconstructinto a new hexagonal phase, which is so far un-documented in experiments. (This process isshown in Fig. 3e,f.) It would be of interest todetermine whether this new BP is a metastablestructure found due to the geometry we have fo-cussed on (we considered just one planar slice ofunit cells) or has a wider physical meaning. Inall cases, these simulations are encouraging asthey suggest that large scale simulations of BPsare within reach. This is of course of interestto the modelling of real devices, which can bemanufactured in the micron scale which we canconsider computationally.

4 Conclusions and future

prospects

We hope that this selection of results has shownthat LB simulations of liquid crystals are poten-tially extremely powerful in gaining new insightsinto the physics which is contained in the hy-drodynamic equations of motion of liquid crys-

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talline fluids. For both active fluids and bluephases, it would have been very difficult to com-pare theory and experiments – even at a qual-itative level – without using these simulations.Although such comparisons remain in their in-fancy for active nematics, the spontaneous flowpatterns in (for instance) concentrated bacte-rial suspensions, simulated via the continuumequations of motion of an active liquid crystal,are qualitatively comparable with the patternsobserved in the experiments. In the case ofblue phases, such comparisons are more clearcut. Here it was thought that the classic theorybased on a Landau-de Gennes free energy wasmissing some physics because (for example) thephase diagram was poorly predicted and anoma-lous electrostriction in BPI was not found. Re-markably, LB has shown that this was a draw-back of the approximations used to make ana-lytical progress, and not of the original theory,which is qualitatively and semi-quantitativelyaccurate.

The fields we have covered in this short revieware, of course, still full of open questions, andwe hope that future LB simulations will play animportant role in clarifying some of these.

In active fluids, it will be important tocharacterise the flow patterns in fully three-dimensional active suspensions, and also to ex-tend the treatment we have covered here to thecase in which there are density fluctuations orinhomogeneity in the fluid. Another related is-sue would be to study “active emulsions”, inwhich droplets of active gels are suspended inan aqueous passive medium, possibly enclosedby an elastic membrane. Ultimately, it wouldbe very exciting if continuum theories like theone we solved numerically may be applied to,for instance, suspensions of cell extracts in anextracellular matrix. From the theory point ofview, it appears that an urgent issue is to clar-ify to what extent active fluids faithfully rep-resent concentrated suspensions of motile parti-cles, or swimmers, by, for instance, comparingthe results of continuum simulations to thoseof more microscopic models with fully resolvedswimmers, which can also be treated via LB(though of a different kind than the one pre-sented here) [29, 42, 43].

Large scale simulations of blue phases will alsobe likely to be important in the future. Froman application point of view, the exciting po-tential of BP devices can ultimately be fullyexploited if we manage to reach a quantitativeunderstanding of their thermodynamics, theirswitching dynamics, and the role of flow. Supra-unit cell simulations are needed to this end, be-cause the field leads to unit cell deformationsand may cause full scale reconstruction of thedisclination network. From a more fundamen-tal point of view, we do not have a satisfactoryunderstanding of non-cubic blue phases. Mostnotably, the structure of BPIII – the “blue fog”– is still not understood to date, and we hopethat large scale simulations of amorphous discli-nation networks may shed some light on thiselusive problem.

We are grateful to G. P. Alexander, S. M.Fielding, A. N. Morozov, E. Orlandini and J.M. Yeomans for useful discussions. We ac-knowledge EPSRC grants EP/E045316/1 andEP/E030173/1 for funding, and computer timeon Hector funded by EP/F054750/1. MECholds a Royal Society Research Professorship.

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10

Fig. 1 (a) Turbulence in a sessile droplet ofB. subtilis, viewed from below a petridish. The horizontal line is the edgeof the droplet (picture taken fromFig. 3 of Ref. [20]). The scale baris 35 µm. (b) Flow pattern in a sim-ilar bacterial droplet – the arrow atthe right stands for a speed of 35µm/s (picture taken from Fig. 4 ofRef. [20]).

Fig. 2 Selected results from active fluidsimulations. We only plot the ve-locity field, resulting from LB so-lutions of Eq. 2. The top row (a-c) shows stationary states obtainedfor apolar extensile fluids with mod-erate activity: it can be seen thatthe spontaneous flow has the shapeof rolls (a,b) or of bands, in generaltilted (c). The middle row show non-stationary “turbulent” solutions forlarger values of the activity. Thebottom row shows solution of theequations of motion of polar activegels. In (g) there is no self-advectionterm, so that the fluid is equivalentto an apolar gel, whereas in (h-i)this term is switched on. In (i) weplot the trajectories of 3 tracer par-ticles, which show the “turbulent”nature of the flow. Parameters com-mon to the apolar runs in (a)-(f) are:γ = 3 (ensuring that we work in theordered phase), ξ = 0.7 (which to-gether with our choice of γ selectsflow-aligning liquid crystals), Γ =0.33775, K = 0.08, η = 0.57. Theactivity parameter ζ was 0.001 (a),0.002 (b), 0.01 (c), and 0.04 (d-f).Parameters for the polar runs in (g-i) are ζ = 0.02, λ = 1.1, K = 0.04,Γ = 0.3, η = 1.67, and w = 0 (g), orw = 0.01 (h,i).

Fig. 3 The top row shows the disclinationlattices of BPI (a) and BPII (b),as obtained from LB simulations –2 unit cells in each directions areshown. The second row shows a

computational (c, within the oneelastic constant approximation), andan experimental (d) typical phase di-agram for blue phases (picture takenfrom Fig. 3 of Ref. [37]). The keyfeature is that LB simulations pre-dict the correct order of appearanceof BPI and BPII upon increasingthe chirality. Note that BPIII isnot a cubic phase hence is not in-cluded in the theoretical phase di-agram. The bottom two rows (e-f) show dynamical states obtainedwhen a BPII domain grows insidean initially cholesteric matrix (seeRef. [41]). The reduced tempera-ture was τ = 0 and the chirality wasκ = 2 (e) or κ = 1 (f).

11

(a) (b)

Figure 1:

12

0 0.2 0.4 0.6 0.8 10

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Apolar active gels: stationary states

Apolar active gels: "turbulent" states

Polar active gels

Figure 2:

13

Figure 3:

14

Appendix: Hydrodynamic equations

of motion for active and passive liquid

crystalline fluids

In this Appendix we review the equations of mo-tion for (active and passive) liquid crystallinefluids, which we solve by lattice Boltzmann sim-ulations. These are the equations used to gen-erate the results reviewed in our work.We first describe the thermodynamics of a liquidcrystalline fluid in the absence of active stresses.This covers cholesterics and blue phases (andalso active gels within a passive phase). We em-ploy a Landau-de Gennes free energy F , whosedensity we indicate by f . The free energy den-sity can be written as a sum of two contribu-tions, f1 and f2. The first is a bulk contribu-tion,

f1 =A0

2(1−

γ

3)Q2

αβ −A0γ

3QαβQβγQγα

+A0γ

4(Q2

αβ)2,

while the second is a distortion term. Fornonchiral liquid crystals, we take the (standard)one elastic constant approximation [9]

f2 =K

2(∂γQαβ)

2 .

Where A0 is a constant, γ controls the magni-tude of order (it may be viewed as an effectivetemperature or concentration for thermotropicand lyotropic liquid crystals respectively), whileK is an elastic constant. To describe cholester-ics, we employ the slightly generalised distortionfree energy, which is again standard [10]:

f2 =K

2

[

(∂βQαβ)2 + (ǫαγδ∂γQδβ + 2q0Qαβ)

2]

.

Here and in what follows Greek indices denotecartesian components and summation over re-peated indices is implied.For blue phases, it is customary to identify theposition in thermodynamic parameter space viathe chirality, κ, and the reduced temperature,τ . These may be defined in terms of previousquantities via [38, 36]:

κ =

√

√

√

√

108Kq20A0γ

τ = 27

(

1− γ/3

γ

)

.

(Note that the reduced temperature was defined

in older literature as τ = 271−γ/3γ

+ κ2.)When needed, the anchoring of the director fieldon the boundary surfaces (Fig. 2) to a chosenunit vector n0 is ensured by adding a surfaceterm in the free energy density

fs =1

2W0(Qαβ −Q0

αβ)2

Q0αβ = S0(n

0αn

0β − δαβ/3)

The parameter W0 controls the strength of theanchoring, while S0 determines the degree of thesurface order. If the surface order is to equal thebulk order, S0 should be set equal to q, the or-der parameter in the bulk (3/2 times the largesteigenvalue of the Q tensor). W0 is large (stronganchoring) in what follows.The equation of motion for Q is taken to be [9]

(∂t + ~u · ∇)Q− S(W,Q) = ΓH+ λ̃Q

where Γ is a collective rotational diffusion con-stant, and λ̃ is an activity parameter which forsimplicity we set to zero in our simulations.(The resulting term can anyway be absorbedinto a shift of A0 and/or γ in the free energy.)The first term on the left-hand side of the equa-tion above is the material derivative describingthe usual time dependence of a scalar quantityadvected by a fluid with velocity ~u. This is mod-ified for rod-like molecules by a second term

S(W,Q) = (ξD+ ω)(Q+ I/3)

+ (Q+ I/3)(ξD− ω)

− 2ξ(Q+ I/3)Tr(QW)

where Tr denotes the tensorial trace, whileD = (W + WT )/2 and ω = (W − WT )/2are the symmetric part and the anti-symmetricpart respectively of the velocity gradient tensorWαβ = ∂βuα. The constant ξ depends on themolecular details of a given liquid crystal, anddetermines, together with γ, whether a liquidcrystal is flow aligning or flow tumbling (we re-strict to the former case in this work). The firstterm on the right-hand side of the order param-eter evolution equation describes the relaxation

15

of the order parameter towards the minimum ofthe free energy. The molecular field H whichprovides the driving motion is given by

H = −δF

δQ+ (I/3)Tr

δF

δQ.

The fluid velocity, ~u, obeys the continuity equa-tion and the Navier-Stokes equation, whose in-compressible limit is Eq. 2 in which Παβ =Πpassive

αβ +Πactiveαβ . The stress tensor Πpassive

αβ nec-essary to describe ordinary LC hydrodynamicsis (up to an isotropic pressure term) given by:

Πpassiveαβ = 2ξ(Qαβ +

1

3δαβ)QγǫHγǫ

− ξHαγ(Qγβ +1

3δγβ)

− ξ(Qαγ +1

3δαγ)Hγβ

− ∂αQγνδF

δ∂βQγν

+ QαγHγβ −HαγQγβ.

whereas the active term is given by, in leadingorder

Πactiveαβ = −ζQαβ

where ζ is an activity constant [17]. Note thatwith the sign convention chosen here ζ > 0 cor-responds to extensile rods and ζ < 0 to contrac-tile ones [17].We can also use a variant of these equations tostudy polar active gels. The order parameter isthis time a vector Pα (with variable magnitude)as there is no longer head-tail symmetry in thesystem. The equations of motion we used inour LB simulations (reported in Fig. 2, bottomrow), are a simplified version of those presentedin [44]. The equation governing the evolution ofthe vectorial order parameter is

[∂t + (uβ + wPβ) ∂β)]Pα =

λDαβPβ − ωαβPβ + Γ′hα.

In this equation, w is another active term, dueto swimming, which causes self-advection of theorder parameter, while λ is a material depen-dent constant – positive for rod-like molecules.If |λ| > 1 the liquid crystalline passive phaseis flow-aligning, otherwise it is flow-tumbling.

The “molecular field” is now given by hα =−δFpol/δpα where Fpol is the free energy for apolar active nematic, whose density is (see alsoRef. [44] where a more general form is used):

f =a

2|P|2 +

b

4|P|4 +K (∂αPβ)

2 (3)

where K is an elastic constant, b > 0, and wechose a = −b to ensure that the minimum ofthe free energy is with |P| = 1. (Note that inthis model P this time denotes a vector ratherthan a tensor).The Navier-Stokes equation is as in the tensorialmodel, but the stress tensor this time is

1

2(Pαhβ − Pβhα)−

λ

2(Pαhβ + Pβhα)− ζPαPβ.

As mentioned in the text, the active term in thestress tensor, proportional to ζ , has thereforethe same form for apolar and polar active gels.A few limits of the two theories consideredabove are worth noting. The tensorial modelwe have written down is equal to, for ζ = 0, theBeris-Edwards model for liquid crystal hydrody-namics. Analogously, for ζ = w = 0 the polarmodel reduces to the Leslie-Ericksen model ofnematodynamics. For w = 0, and a sample ofuniaxial active liquid crystals, with a spatiallyuniform degree of orientational order, the ten-sorial model may be mapped onto the vectorialone (see [13] for a proof of this).

16