rheology of active polar emulsions: from linear to

This journal is©The Royal Society of Chemistry 2019 Soft Matter, 2019, 15, 8251--8265 | 8251

Cite this: SoftMatter, 2019,

15, 8251

Rheology of active polar emulsions: from linear tounidirectional and inviscid flow, and intermittentviscosity†

G. Negro,a L. N. Carenza, *a A. Lamura,b A. Tiribocchi cd and G. Gonnellaa

The rheological behaviour of an emulsion made of an active polar component and an isotropic passive

fluid is studied by lattice Boltzmann methods. Different flow regimes are found by varying the values of

the shear rate and extensile activity (occurring, e.g., in microtubule-motor suspensions). By increasing

the activity, a first transition occurs from the linear flow regime to spontaneous persistent unidirectional

macro-scale flow, followed by another transition either to a (low shear) intermittent flow regime with

the coexistence of states with positive, negative, and vanishing apparent viscosity, or to a (high shear)

symmetric shear thinning regime. The different behaviours can be explained in terms of the dynamics of

the polarization field close to the walls. A maximum entropy production principle selects the most likely

states in the intermittent regime.

1 Introduction

Active gels1–3 are a new class of complex fluids with strikingphysical properties and many possible innovative applications.4–8

As other kinds of active systems,9–17 they are maintained in theirdriven state – far from thermodynamic equilibrium – by energysupplied directly and independently at the level of individualconstituents. Examples are suspensions of biological filaments,such as actomyosin and microtubule bundles, activated withmotor proteins18–21 and bacterial cultures.22,23 The constituentsof these systems have the natural tendency to assemble and align,thus developing structures with typical polar or nematic order.The combination of this property with self-motility capacity is atthe origin of a wealth of interesting phenomena, not observable inthe absence of activity,24 including spontaneous flow,1,25,26 activeturbulence at low Reynolds numbers,22,27,28 and unusual rheo-logical properties.3,29 Most of these behaviours were found insingle component fluids, while mixtures of active and passivecomponents have not been investigated too much so far.30

Complex rheological behaviours in active matter depend onthe interplay between the external forcing and the circulatingflow induced by active agents. For instance, the swimming

mechanism of pusher microswimmers, like E. Coli, producesa far flow field characterized by quadrupolar symmetry, inwhich fluid is expelled along the fore–aft axes of the swimmerand drawn transversely, thus leading to extensile flow patterns.These enforce the applied flow, in the case of flow-aligningswimmers, causing shear thinning.31 This may lead to theoccurrence of a superfluidic regime with vanishing (apparent)shear viscosity, which was speculated in ref. 32 for the case ofactive liquid crystals close to the isotropic–nematic transition.Experiments33,34 and further theories35 confirmed that extensileactive components are able to lower the viscosity of thin filmsuspensions. An effective inviscid flow was observed in ref. 36 andmore recently in ref. 37, when the concentration and activityof E. Coli are sufficiently large to support coherent collectiveswimming. A related feature in extensile gels is the appearanceof persistent uni-directional flows in experiments on bacterialsuspensions38 and ATP-driven gels.39 Recently, numerical simula-tions in quasi-1d geometries and linear analysis of active polarliquid crystal models have shown the occurrence of vanishing andeven negative viscosity states.40 However, a complete characteriza-tion of the rheology of fully 2d active compounds has never beenaccomplished so far, despite being fundamental to unveildynamical mechanisms leading to the complex propertiespresented. We also mention that puller swimmers – exertinga contractile force dipole on the surrounding fluid – stillgenerate a quadrupolar far flow field, but this time the fluidis expelled transversely to their body. This explains the shearthickening behaviour observed in experiments performed onsuspensions of C. reinhardtii41 – a species of micro-alga thatpropels itself by means of two flagella producing contractile

a Dipartimento di Fisica, Universita degli Studi di Bari and INFN, Sezione di Bari,

via Amendola 173, Bari, I-70126, Italy. E-mail: [email protected] Istituto Applicazioni Calcolo, CNR, Via Amendola 122/D, I-70126 Bari, Italyc Center for Life Nano Science@La Sapienza, Istituto Italiano di Tecnologia,

00161 Rome, Italyd Istituto per le Applicazioni del Calcolo CNR, via dei Taurini 19, 00185 Rome, Italy

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c9sm01288e

Received 28th June 2019,Accepted 12th September 2019

DOI: 10.1039/c9sm01288e

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8252 | Soft Matter, 2019, 15, 8251--8265 This journal is©The Royal Society of Chemistry 2019

movements – and in numerical studies on contractile gels.26,35,42

In this paper we will focus on extensile systems.We study, by extensive lattice Boltzmann simulations, the

rheology of a 2d emulsion, made of an active fluid component(a polar gel) and an isotropic passive fluid, under simple shearflows. The system that we consider43,44 has the property thata tunable amount of active material can be homogeneouslydispersed in an emulsion. With respect to the single-componentactive gel theory, this approach has the great advantage that thetypical length-scale of active injection can be kept under controlby setting opportunely the parameters of the model. Thiswould also elucidate the mechanisms for the rich rheologicalbehaviour previously described. Experimental realization canbe achieved by either confining activated cellular extracts in anemulsion of water-in-oil20,21 or dispersing bacteria in water,ensuring microphase separation through depletion forces anda suitable surfactant.45

In Section 2 we describe the model for active emulsions andthe lattice Boltzmann method implemented for the numericalstudy. Moreover, we introduce our main adimensional parameters,which are related to the strength of both active injection and shearflow. In Section 3 we describe the numerical results for the flowbehaviour in a region of the parameter space characterized by adouble transition, first from linear velocity profiles to unidirectedmotion and then, by increasing the activity, to symmetric shearthinning behaviour. We describe the relevance of the role ofthe polarization and active stress in the proximity of external walls.In Section 4 intermittent flow – occurring in a region of theparameter space at low shear rate and sufficiently high viscosity –will be analyzed, also in terms of the entropy production rateevaluated for the different coexisting flow states. Finally, in Section 5we show the phase diagram summarizing results for the differentbehaviours we have found and we draw some conclusions.

2 Model and numerical methods

We outline here the hydrodynamic model and the numericalmethod used to conduct our study. We consider a fluid in 2dcomprising a mixture of active material and an isotropic solventwith total mass density r. The physics of the resulting compositematerial can be described by using an extended version of thewell-established active gel theory.1,3,24,46–48 The hydrodynamicvariables are the density of the fluid r, its velocity v, the concen-tration of the active material f, and the polarization P, whichaccounts for the average orientation of the active constituents.The dynamical equations ruling the evolution of the system in theincompressible limit are:

r@

@tþ v � r

� �v ¼ r � ~stot; (1)

@f@tþr � fvð Þ ¼Mr2m; (2)

@P

@tþ v � rð ÞP ¼ �~O � Pþ x ~D � P� 1

Gh: (3)

The first one is the Navier–Stokes equation, where ~stot is thetotal stress tensor.49 Eqn (2) and (3) govern the time evolutionof the concentration of the active material and of the polariza-tion field, respectively. By the assumption that the amount ofthe active component is locally conserved, the evolution of theconcentration field, eqn (2), can be written as a convection–diffusion equation, where M is the mobility, and m = dF/df thechemical potential, with F a suitable free energy functional,encoding the equilibrium properties of the system, to bedefined later. The dynamics of the polarization field followsan advection–relaxation equation, eqn (3), borrowed from polarliquid crystal theory. Here G is the rotational viscosity, and x isa constant controlling the aspect ratio of active particles(positive for rod-like particles and negative for disk-like ones)and their response to external flows (|x| 4 1 for flow-aligningparticles and |x| o 1 for flow-tumbling ones), while h = dF/dP isthe molecular field. D = (W + WT)/2 and ~O = (W � W T)/2respectively represent the symmetric and the anti-symmetricparts of the velocity gradient tensor Wab = qbva, where the Greekindexes denote Cartesian components. These contributions arein addition to the material derivative, as the liquid crystal canbe rotated or aligned by the fluid.49

The stress tensor ~stot considered in the Navier–Stokes equation,eqn (1), can be split into an equilibrium/passive and a non-equilibrium/active part:

~stot = ~spass + ~sact. (4)

The passive term accounts for the viscous dissipation andreactive phenomena – the elastic response of the binary fluidand of the liquid crystal – and is given by four contributions:

~spass = ~shydro + ~svisc + ~spol + ~sbm. (5)

The first term is the hydrodynamic pressure contribution givenby shydro

ab = �pdab. The second term is the viscous stress, writtenas svisc

ab = Z0(qavb + qbva), where Z0 is the shear viscosity. Thesecond term is the polar elastic stress, analogous to the oneused in nematic liquid crystal hydrodynamics:49

spolab ¼1

2Pahb � Pbha� �

� x2Pahb þ Pbha� �

� kP@aPg@bPg; (6)

where kP is the elastic constant of the liquid crystal, in thesingle constant approximation.50 The third term on the right-hand side of eqn (5) is borrowed from binary mixture theoriesand it includes the interfacial contribution between the twophases:

sbmab ¼ f � fdFdf

� �dab �

dF@ @bf� � @af; (7)

where we have denoted with f the free energy density. The activecontribution to the stress tensor, the only one not stemmingfrom the free energy, is given by25,48

sactab ¼ �zf PaPb �1

3jPj2dab

� �; (8)

where z is the activity strength, positive for extensile systems(pushers) and negative for contractile ones (pullers). The active

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stress drives the system out of equilibrium, by injecting energyon the typical length scales of deformation of the polarizationpattern.

By assuming that local equilibrium is satisfied even in thepresence of activity, the thermodynamic forces in eqn (1)–(3)can be deduced by the following free-energy functional, couplingthe Landau–Brazovskii model51,52 to the distortion free-energy of apolar system:

F ½f;P� ¼ðdr

~a

4fcr4f2 f� f0ð Þ2þ kf

2rfj j2þ c

2r2f� �2�

�a2

ðf� fcrÞfcr

Pj j2þ a4Pj j4þ kP

2ðrPÞ2 þ bP � rf

�:

(9)

This is a generalization of the free energy functional foractive binary mixtures defined in ref. 48. The first term allowsfor the segregation of the two phases when the bulk energydensity a 4 0, so that free energy has two minima at f = 0and f0. The second and third terms determine the interfacialtension. Notice that here a negative value of kf favours theformation of interfaces while a positive value of c is used toguarantee thermodynamic stability.51 Lowering kf from positiveto negative values leads the system to move from a pure ferro-magnetic phase to configurations where interfaces betweencomponents are favoured.53–57 In the Appendix A Dimensionalnumbers we will show that, for a symmetric composition of themixture, the system sets into a lamellar phase modulated at

wavenumber k ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijkfj=2c

p, when a o kf

2/4c + b2/kP (see panels(a) and (b) of Fig. 1 for the typical lamellar morphology).

The bulk properties of the polar liquid crystal are insteadcontrolled by the |P|2 and |P|4 terms, multiplied by the positiveconstant a. The choice of fcr has been made in order to breakthe symmetry between the two phases and confine the polar-ization field in the active phase, f4 fcr. The term proportionalto (rP)2 describes the energy cost due to elastic deformation inthe liquid crystalline phase, connected to the theory throughthe elastic constant kP (a more general treatment can be foundin ref. 58). Finally, the last term takes into account the orienta-tion of the polarization at the interface of the fluid. If b a 0, Ppreferentially points perpendicularly to the interface (normalanchoring) and towards the passive (active) phase if b4 0 (bo 0).A plot of the typical polarization field orientation to the interfacesin the lamellar phase is shown with white arrows in panel (e) ofFig. 1. This allows for the confinement of the active behaviour onsmall scales in a low-Reynolds number environment, offering away to control the typical length-scales of energy injection – anactual challenge for experiments in the field.21 The model exhibitswide phenomenology on varying the activity parameter z,43 whichhas been summarized at the beginning of Section 3.

2.1 Numerical method and parameters

The equations of motion of the polar active emulsion, eqn (1)–(3),are solved by means of a hybrid lattice Boltzmann (LB) scheme,which combines a predictor–corrector LB treatment for the Navier–Stokes equation59 with a finite-difference predictor–corrector

algorithm to solve the order parameter dynamics, implementing afirst-order upwind scheme, for the convection term, and fourth-order accurate stencil for the computation of space derivatives.

In this approach the evolution of the fluid is described interms of a set of distribution functions fi(ra,t) (with index ilabelling different lattice directions, thus ranging from 1 to 9)defined on each lattice site ra. Their evolution follows adiscretized predictor–corrector version of the Boltzmann equationin the BGK approximation:

fi ra þ xiDtð Þ � fi ra; tð Þ ¼ �Dt2

Cð fi; ra; tÞ þ C fi�; ra þ xiDt; tð Þ½ �:

(10)

Here {xi} is a set of discrete velocities, which for the d2Q9 modelare x0 = (0,0), x1,2 = (�u,0), x3,4 = (0,�u), x5,6 = (�u,�u), and x7,8 =(�u,�u), where u is the lattice speed. The distribution functionsf * are first-order estimations of fi(ra+ xiDt) obtained by settingfi* � fi in eqn (10), and C( f,ra,t) = �( fi � f eq

i )/t + Fi is thecollisional operator in the BGK approximation expressed in termsof the equilibrium distribution functions f eq

i and supplementedwith an extra forcing term for the treatment of the anti-symmetricpart of the stress tensor. The density and momentum of the fluidare defined in terms of the distribution functions as follows:X

i

fi ¼ rXi

fixi ¼ rv: (11)

The same relations also hold for the equilibrium distributionfunctions, thus ensuring mass and momentum conservation.In order to correctly reproduce the Navier–Stokes equation weimpose the following condition on the second moment of theequilibrium distribution functions:X

i

fixi � xi ¼ rv� v� ~sbin � ~spols ; (12)

and on the force term:Xi

Fi ¼ 0;

Xi

Fixi ¼ r � ~spola þ ~sact� �

;

Xi

Fixi � xi ¼ 0;

(13)

where we denote with ~spols and ~spol

a the symmetric and anti-symmetric part of the polar stress tensor, respectively. Theequilibrium distribution functions are expanded up to secondorder in the velocities:

f eqi = Ai + Bi(x�v) + Ci|v|2 + Di(x�v)2 + Gi: (x # x).

(14)

Here coefficients Ai, Bi, Ci, Di, and Gi are to be determinedimposing the conditions in eqn (11) and (12). In the continuumlimit the Navier–Stokes equation is restored if Z0 = t/3.

We made use of a parallel approach implementing MessagePassage Interface (MPI) to parallelize the code. We divided thecomputational domains into slices, and assigned each of them toa particular task in the MPI communicator. Non-local operations

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(such as derivatives), have been treated through the ghost-cellapproach.60

Simulations have been performed on a square lattice of sizeL = 256. The concentration f ranges from fC 0 (passive phase)to f C 2 (active phase). Unless otherwise stated, the parametervalues are a = 4 10�3, kf = �6 10�3, c = 10�2, a = 10�3,kP = 10�2, b = 0.01, G = 1, x = 1.1, f0 = 2.0, and Z0 = 1.67. Allquantities in the text are reported in lattice units. We initializedthe system starting from a uniform phase, with f(r) = hfi +df(r), where hfi = fcr is the conserved (area) averaged value ofthe concentration field and df is a small perturbation field

favouring phase separation and ranging within �fcr

10;fcr

10

. The

initial condition for the polarization field is completely random,its orientation being randomly distributed in the plane, while itsintensity is randomly chosen within [0, 1].

Our choice of parameters is such that the Schmidt number

(Sc ¼ Z0rD

where D = 2Ma/fcr4 is the diffusion constant) is

fixed at values typical for liquids, B2000, where, in the absenceof activity, lamellae show low resistance to the flow and caneasily order. It was shown in ref. 57 that at smaller Sc lamellardomains hardly align to the flow and may eventually undergopearling instability, persistent even in the long dynamics.

We considered flow in a channel with no-slip boundaryconditions at the top and the bottom walls ( y = 0 and y = L),implemented by bounce-back boundary conditions for thedistribution functions,61 and periodic boundary conditions inthe y direction. The flow is driven by moving walls, with velocityvw for the top wall and �vw for the bottom wall, so that the

shear rate is given by _g ¼ 2vw

L.

Moreover neutral wetting boundary conditions wereenforced by requiring on the wall sites that the following relationshold:

r>m = 0, r>(r2f) = 0, (15)

Fig. 1 Linear velocity profiles and lamellar phase. Concentration contour plots at Eract = 0.057 and Er = 0.0030 and Eract = 0.058 and Er = 0.0075 arerespectively shown in panels (a) and (d). Here (and in the rest of the paper) red regions correspond to the active phase (fE 2.0), while blue regions to thepassive fluid (f E 0.0). Panels (b) and (e) show a zoom of the regions highlighted by the black squares in panels (a) and (d). Black and white vectorsrespectively denote velocity and polarization fields in panel (e), while only the velocity field has been plotted in panel (b). Panels (c) and (f) show theaveraged velocity profile and the time evolution of the shear stress (red and orange curves correspond to the cases considered in panel (a) and (d)).The black arrow in panel (f) points to a jump in the relaxation of sxy, due to the annihilation of two dislocations. Note that in the present figure, as well as inthe following, the colors of the velocity and the stress profiles have been chosen in relation to the colors of the corresponding region of the phasediagram of Fig. 8. In panels (c–f) the profiles have been plotted in such a way to make reference to the red/orange region of the linear profiles.

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where r> denotes the partial derivative computed normally tothe walls and directed towards the bulk of the system. Here thefirst condition ensures density conservation, and the seconddetermines the wetting to be neutral. In the case of bacterialswimmers, it is commonly observed that, close to the boundaries,they orient along the wall direction.62 In actomyosin solutions, theactin filaments can also be assumed to be anchored parallel tothe walls due to focal adhesion.63 Therefore, suitable boundaryconditions for the polarization P are a strong anchoring conditionwith P aligned parallel to the walls

P>|walls = 0, r>P8|walls = 0, (16)

where P> and P8 denote, respectively, the normal and tangentialcomponents of the polarization field with respect to the walls.In order to compare external and active forcing in our system, wemake use of the Ericksen number Er and the active Ericksennumber Eract as relevant adimensional quantities. The former isoften used in the study of liquid crystals to describe the deforma-tion of the orientational order parameter field under flow and it isdefined as the ratio of the viscous stress to the elastic stress. Inparticular in lamellar systems a suitable choice is given by:

Er ¼ Z0 _gB; (17)

where B is the lamellar compression modulus, namely the energycost for the variation of the lamellar width l = 2p/k per unitlength, whose expression in terms of the parameters of the modelis explicitly derived in Appendix A. The active Ericksen number,suggested by Giomi for the first time in ref. 64, is, in turn, definedas the ratio between the module of the activity parameter z andthe compression modulus:

Eract ¼jzjB: (18)

3 Linear flow and symmetry breakingtransition

Before presenting specific result cases we summarize themorphological phenomenology arising in an active extensilepolar lamellar system on varying the activity parameter (z 4 0)in the absence of any external forcing. Bonelli et al.43 showedthat the shear-free system is characterized by a transition atEract E 0.11 from the lamellar phase to an emulsion withmoving active droplets. The bending instability, typical ofextensile gels,25 favors this rearrangement. For Eract \ 1 thesystem enters into a totally mixed phase, characterized by chaoticvelocity patterns.28 The following Sections will be devoted topresenting the different behaviours of the sheared system onvarying the intensity of both active and external forcing.

3.1 Linear velocity profiles and lamellar phase

The scenario just described is strongly influenced by an exter-nal shear flow. Due to the tendency of lamellae to align with theflow, an applied shear, even small, is found to counter activity-induced bending, thus extending the range of stability of

lamellar order towards larger Eract (Eract t 0.18) with respectto its unsheared counterpart. Under this threshold and for avast range of shear rates, the system sets into a lamellar phase,as shown in panels (a) and (d) of Fig. 1. The region with theseproperties is red in Fig. 8, where flow regimes found byscanning the Er�Eract plane are summarized. At small shear,the relaxation dynamics leads to the formation of long-liveddislocations in the lamellar pattern, such as the one high-lighted by the black box in panel (a) of Fig. 1. Panel (b) of thesame figure shows the detail of the velocity field in theneighborhood of the dislocation. If _g is weak enough, defectsare capable of consistently altering the velocity pattern, sincedislocations develop flows transversal to the direction oflamellar-alignment, thus leading to permanent shear bendingsin the velocity profile (as shown by the red line in the inset ofFig. 1c). At greater values of shear rate, the superimposed flowis strong enough to eliminate dislocations (see for examplepanel (d) in Fig. 1 and Movie S1, ESI†), eventually leading to theformation of disruptions that have much less effect on the flowthan dislocations, as confirmed by the linear behaviour of thecorresponding velocity profile. In this regime lamellae areglobally aligned to the flow, while the polarization field, homeo-tropically anchored to the interfaces (panel (e) of Fig. 1), ispointing towards the passive phase (blue regions in the zoom ofpanel (e)). Panel (f) compares the time evolution of shear stress forthe two cases considered. The dynamics at high shear leads tosmoother and faster relaxation towards lower values of sxy. Whenthe imposed shear is weaker, oscillations or jumps, such as theone marked by an arrow at t = 6 106 in panel (f) of Fig. 1, are dueto the annihilation of two dislocations.

3.2 Unidirectional motion

The behaviour becomes more complex when the activity isincreased. The combination of activity and shear has dramaticconsequences. The system undergoes a morphological transi-tion from the lamellar phase towards an emulsion of activematerial in a passive background, a behaviour also found byBonelli et al.43 at lower active dopings. Fig. 2 shows two casesat Er = 0.031 and Eract = 0.29 (top row) and Er = 0.062 andEract = 0.29 (bottom row) characterized by the formation of athick layer of material close to one boundary (see the dynamicsin Movie S2, ESI†), and small features on Brazovskii lengthscale l = 2p/k coexisting with larger aggregates of activematerial elsewhere. Such symmetry breaking is mediated andsustained by the formation of these large active domains wherebending polarization instabilities, typical of extensile systems,act as a source of vorticity (see panel (b)). Big active domains aremostly advected by the intense flow close to the walls, as shownby purple velocity streamlines, differently from what happensin shear-free systems where polarization bending results in therotational motion of the bigger active droplets. Fig. 2c showsthe related x-averaged velocity profile hvxi (grey curve): insteadof the linear behaviour of Fig. 1c, one observes banded flowswith the higher gradient corresponding to the wall with theactive layer. Similar cases occur at different Er and Eract withdeposition of active material randomly on the top or bottom wall.

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The streamlines of v (in panels (b) and (e) of Fig. 2) show that theinversion of the fluid velocity takes place at the interface of theactive layer. The top–bottom symmetry breaking leads to a net fluxof matter in the flow direction and has been named unidirectionalmotion. Cases exhibiting such a property have been plotted ingrey in Fig. 8. Moreover, as the shear is increased, the positionof flow inversion migrates towards the bulk of the system(see the dark grey profile at the larger Er in panel (c) and thecorresponding vx contour plot in panel (f)). Unidirectional flow(grey region in Fig. 8) may occur with a vanishing gradient ofhvxi almost everywhere, and in this case it will be calledsuperfluidic32,65 (see the light grey profile in Fig. 2c).

What are the mechanisms for the observed velocity profiles?And how to explain the flow symmetry breaking transition? Dueto complexity of the system we can only partially answer thesequestions. The velocity behaviour is strictly related to that ofpolarization close to the walls. Thick layers such as the one inFig. 2a, d and in Fig. 3 are characterized by the bendingof polarization due to competition between strong parallelanchoring to the walls and perpendicular orientation to domaininterfaces, leading to a negative active shear stress contribution.This is clearly shown in Fig. 3, where the white/black regions

correspond to positive/negative values and correspond to activedomains, while beige ones are associated with the isotropicbackground and correspond to almost null values. Moreover,topological defects in the active layer are strongly inhibited byelastic energy, as suggested by the uniform polarization pattern inthe black bottom layer.

Within the active layer sactxy z2f0 Peq

� �2sin 2y, where y denotes

the local orientation of polarization with respect to theimposed velocity (0 r y r p), thus generating an activeforce density in the flow direction ( f act

8 = @>sactxy , where @>

denotes the derivative in the direction normal to the walls).This can either reinforce the imposed flow if the polarizationfield is oriented as vw (since @> sin 2y 4 0), or lead to areduction of the fluid velocity if opposite (since @> sin 2y o 0).However, between the two possible orientations, the onereinforcing the flow does not appear in the cases discussed sofar. In order to clarify this point we define the average polarizationPw on each wall – it can be calculated as Pw B hP8|wallsi, wherehere h�i stands for the average over a few layers close to the wallsites. For the unidirectional motion case, Pw is null at the wallwhere the thick active layer is absent, so that we denote such apolarization state as 0. If the polarization is opposite to vw, as in

Fig. 2 Unidirectional motion. (a) Concentration contour plot at Eract = 0.29 and Er = 0.031. (b) Contour plot of the vorticity o = (qyvx � qxvy)/2 (blackcorresponds to o = 0 and red to o = 10�2) in the region framed with the black box in panel (a); purple lines represent velocity streamlines while thepolarization field is plotted in white (for graphical clarity, only in the bottom active layer). (d) Contour plot of the concentration field at Eract = 0.29 andEr = 0.062 and (e) zoom of the region framed with the black box in panel (d), with black lines representing v streamlines and P plotted in white. (f) Contourplot of the velocity field in the flow direction for the same case shown in panel (d), with a few isolines plotted in white. Panel (c) shows hvxi for the twocases considered and also for the case at Eract = 0.172 and Er = 0.0075.

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the bottom/top of panel (b)/(e) of Fig. 2, we will indicate it with �.Following the notation just introduced, we will refer to the globalstates shown in Fig. 2 with 0�, independently of the top–bottomasymmetry.

3.3 Symmetric shear thinning profiles

By further increasing both the activity and the shear rate, phasedemixing is more pronounced (Fig. 4a) with the formation of anemulsion of amorphous active domains in a passive matrix.Active layers form on both walls so that symmetry is restoredalso at the level of the velocity profiles hvxi, with gradients inthe bulk of the system lower than the imposed one, as shown inpanel (c) of Fig. 4 (thick lines). Symmetric cases exhibiting such

phenomenology have been plotted in green in Fig. 8. Underthese conditions, it may happen that the velocity gradient in thebulk of the system is either vanishing everywhere or, eventually,opposite to the one externally imposed (negative viscosity),despite the fact that such states are found to be unstable inthe long term (see Section 4). To explain the flow propertiespresented, we analyse the active shear stress profiles hsact

xy iaveraged in the flow direction (see the thin lines in panel (c)).This confirms that the active stress is considerably differentfrom zero only in the layer close to both walls, where it assumesnegative values – thus leading to the sharp decrease of theintensity of the flow in the same region – while it is approxi-mately null in the bulk. A contour plot of the active stress is alsoshown in panel (b) of Fig. 4, clearly showing that the Pw

polarization state at the boundaries is in a – configuration.Such behaviour is also accompanied by shear thinning,

typical of extensile fluids as Eract is increased. This is analyzedin Fig. 5 where the ratio between the apparent viscosity Z = hstot

xy i/_g(where hstot

xy i denotes the time average of the total stress tensor)and the shear viscosity Z0 has been plotted versus Eract. We variedEr in the range Er \ 0.05, where the viscosity states are foundto be stable for any value of Eract. The viscosity mainly dependson the intensity of the active doping, while no substantialdependence is found on the shear rate if Eract t 0.6. Thissuggests that activity, inducing shear thinning, is a parametercapable of controlling the rheological property of extensilesuspensions.

3.4 Activity quench

We further analyzed the nature of the transition between thesymmetric configurations at higher activity (green region inFig. 8) and states with unidirectional flow at weaker activity,starting from a stationary state at Eract = 0.57 and quenchingthe activity to Eract = 0.28 at fixed Er = 0.06. Panels (a)–(c) in Fig. 6show the quenching dynamics: starting from the symmetric

Fig. 4 Symmetric shear thinning profiles. Concentration (a) and active shear stress (b) contour plot for the case at Eract = 0.70 and Er = 0.04. The colourcode is the same as Fig. 3. Red arrows denote the direction of the moving walls, and cyan ones the direction of the polarization, averaged within theactive layers close to the walls. (c) Velocity hvxi (thick lines) and active shear stress profiles hsact

xy i (thin) averaged along the flow direction, for the casecorresponding to panels (a) and (b) and analogous ones.

Fig. 3 Active shear stress close to a wall at Er = 0.0074 and Eract = 0.172.The polarization field P (cyan arrows) exhibits a splayed profile under themutual effect of strong anchoring to the wall (tangential) and to theinterface (homeotropic), while the red arrow shows the direction ofthe imposed velocity vw. The angle y denotes the local orientation of thepolarization, sketched by the magnified reference cyan arrow, with respectto the flow. White lines trace the interface (f = fcr) between active andpassive phases. Passive (beige) regions are almost stress free, whilenegative stress in the boundary layer (black) corresponds to a net forceopposite to the flow direction. Droplets have quadrupolar structures.

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configuration of panel (a) at Eract = 0.57, amorphous active domainsprogressively stretch in the flow direction and cluster on the bottomboundary (see panel (b)). This accompanies the melting of the activelayer close to the upper wall, finally generating the asymmetricconfiguration of panel (c) characterized by unidirectional flow. Theevolution of the velocity profiles is shown in panel (d). Panels (e)–(g)show the results of a similar experiment: this time we quenched theactive parameter so as to move from the grey region (Eract = 0.22)with unidirectional motion to the red one (Eract = 0.057) with linear

profiles, in Fig. 8, while keeping the shear rate fixed at Er = 0.01.After the quench droplets are no longer stable, since the activedoping is not strong enough to maintain the bending instability. Alamellar phase progressively grows from the bottom of the system,where most of the active material was initially found, towards theupper wall. The final configuration is characterized by a symmetric,defect-free lamellar configuration, with linear velocity profile, whoseevolution can be appreciated by looking at panel (h).

4 Intermittent flow

Surprisingly, yet another behaviour appears in a vast region of theparameter plane at small shear, Er t 0.04, and sufficiently largeactivity, Eract \ 0.4 (blue in Fig. 8). Under this condition, thesystem is symmetric and exhibits an intermittent flow regime.This is related, once again, to the polarization state at theboundaries. Indeed, this time we find (unstable) configurationswhere Pw is oriented in the same direction as the imposed velocity,so we will denote such states as +. The dynamics of the system ischaracterised by jumps between ++, �+ and �� states.

Such intermittent behaviour is reflected in the evolution ofthe area averaged stress, as shown in Fig. 7a. Elastic contributionsare on average constant, while active stress fluctuates aroundpositive, negative or vanishing values. These are found to belargely determined by the portion of the system closer to theboundaries (see Fig. 7c). For each wall the sign of active stresscoincides with the one of Pw, so that positive and negative total

Fig. 5 Shear thinning in extensile mixtures. Ratio between apparentviscosity Z (measured as hstot

xy i/ _g) and shear viscosity Z0, while varying theactivity for some values of the Ericksen number.

Fig. 6 Activity quench. Panels (a)–(c) show the evolution of the system after Eract has been quenched from 0.57 to 0.28, thus moving from the greenregion of Fig. 8 to the grey one, at Er = 0.06. The colors of the profiles have been chosen in accordance with the color of the corresponding region in thephase diagram of Fig. 8. The labels (0 or �) at the top and at the bottom of the initial and the final panel denote the state of the polarization at theboundary. The evolution of the velocity profiles is also shown in panel (d). Panels (e)–(g) show the same for the quench from the grey region (Eract = 0.22)towards the red region (Eract = 0.057) of Fig. 8 at Er = 0.01, with the evolution of the velocity profiles in panel (h). Contour plots correspond to the first,third and last velocity profiles in panels (d) and (h).

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stress correspond respectively to ++ and �� Pw states, whiletotal zero active stress comes from opposite Pw contributions(�+ states). Viscosity jumps (Fig. 7a) correspond to the inversionof the polarization on one of the two walls during evolution. Theinversion of polarization on the boundaries, generally preventedby elastic effects and strong anchoring, can occur as a result ofcatastrophic events, such as collision of big domains with theactive layer (see Appendix B, and Movies S3 and S4, ESI†). ��states typically live longer than the others, as can be appreciatedlooking at the pdf of the total shear stress reported in Fig. 7b. Thestatistics of viscosity states has been constructed considering 40different runs for the same couple of parameters Eract = 1.14 andEr = 0.0075. The typical length of each run is about 107 latticeBoltzmann iterations. The data for the shear stress have beensampled under stationary conditions and then fitted with the sumof three normal distributions, centred at values marked withdotted lines in panel (b). These mean values are consistentwith the average of the stress in the different states of Fig. 7a,and their probability of occurrence will correspond to the amountof time that the system spends in each of them.

To go deeper into the characterization of this behaviour wemeasured the entropy production during the system timeevolution. Generally, in non-equilibrium systems, the entropydensity S(r,t) obeys the continuity equation

@tS + r�(Sv) = s, (19)

where s is the rate of entropy production per unit volume,subject to the condition s Z 0. This can be written in terms ofgeneralized fluxes Ji and forces Xi, as66

Ts = JiXi, (20)

with T the temperature – fixed in our simulations (T = 0.5) sincewe are neglecting heat transfer. Thermodynamic forces arechosen as follows:

Xv = (rv)S � D, (21)

XP ¼ �dFdP� h; (22)

Xf ¼ rdFdf� rm; (23)

where F is given by eqn (9), and D is the strain rate tensor.Moreover, in our model the following linear phenomenologicalrelations between forces and fluxes hold:1

Jv = ~stot, (24)

JP ¼ @tP þ ðv � rÞP þ ~O � P ¼ 1

Ghþ x ~D � P ; (25)

Jf = �Mrm, (26)

where we denote with ~O the vorticity tensor, that is the anti-symmetric part of the gradient velocity tensor. By substitutingthese relations into eqn (20) we find that

Ts ¼ 2Z ~D : ~Dþ 1

Gh � hþMðrfÞ2; (27)

where we retained only those terms even under time reversalsymmetry. We here identify three contributions: the first one isthe entropy production due to viscous effects, 2ZD:D, while the

Fig. 7 Entropy production in multistable states. (a) Time evolution of stress contributions at Eract = 1.14 and Er = 0.00075. The total stress is plotted intransparent yellow, while the active one is the dark blue curve underneath. Panel (b) shows the pdf of the total stress (data from 40 independent runs,fitted by 3 normal distributions peaking at sxy = �0.0068, 0.0009, and 0.0086). (c) Total stress close to the bottom and top walls, respectively computedas stwxy ¼

ÐdxÐ LyLy�ldysxy and sbwxy ¼

ÐdxÐ l0dysxy; where l is the width of the layer. Here l = 15. Nevertheless, as long as l o Ly/2, the results remain unaltered.

(d) Entropy production. Negative viscosity states, corresponding to – configurations of polarization at the boundaries, live longer than others. Entropyproduction s(t) assumes greater values corresponding to these regions. Each contribution to s(t) has been measured by integrating in the computationaldomain the terms on the right-hand side of eqn (27).

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second1

Gh � h

� �and the third (M(rf)2) ones are respectively

the molecular and the chemical terms, accounting for theentropy produced during the relaxation dynamics of P and f.This expression is quadratic in the thermodynamic forces andsatisfies all required conditions, among which is invarianceunder Galilean transformations as well as the second principleof thermodynamics. In the stationary regime, entropy productionmust be equal to the energy injected into the system, due to thework of the walls and active pumping. We emphasize that activityz is a reactive parameter and therefore does not appear in theentropy production formula.67 However the activity influences thedynamics, acting as a velocity source through the active stress,thus contributing indirectly to dissipation.

Panels (a) and (d) compare the behaviour of the stress andentropy production. The evolution of the stress is characterizedby two regions of stability of negative viscosity – whose lifetimeis O(107) LB iterations – which respectively occur from t E 107

to 2.5 107, and from t E 3.2 107 to 4 107. These stateshave also the highest probability, as can be appreciated lookingat the pdf of the total stress in panel (b) of Fig. 7.

In panel (d) the different entropy production contributions areshown. We first notice that the contribution due to diffusion/chemical effects (yellow line) is almost null. In addition, the viscousdissipation (blue line) is always greater than the contribution due tothe molecular field (violet line). This suggests that the hydro-dynamics of the system – driven by active injection and externalforcing – is mainly countered by viscous dissipation phenomena.Moreover, the total entropy production oscillates around two differ-ent values and jumps during the time evolution, with the highestvalue corresponding to the negative viscosity states. This behaviourwith prevalence of � polarization states is typical of all the inter-mittent cases (blue region in Fig. 8), and is compatible with amaximum entropy production principle (MaxEPP). Various varia-tional principles, related to entropy production rates, have been putforward to quantitatively select the most probable state in multi-stable systems. While much effort has been spent in the searchfor a general principle and recent progress has been made68 on atheoretical derivation of such a principle, questions about howthis should be interpreted and applied have not been answered,especially for systems evolving far from thermodynamic equilibrium.In particular, a MaxEPP has been implemented as a selection criteriato study systems characterized by multiple non-equilibriumstationary states68 and very recently the Schlogl model69 – whichis a simple, analytically solvable, one-dimensional bistable chemicalmodel – has been used to demonstrate that the steady state withthe highest entropy production is favoured.70 In the system hereconsidered, we found that entropy production s(t) is higher for themost likely – states, suggesting that MaxEPP may act as a thermo-dynamic principle in selecting non-equilibrium states.

5 Overview, phase diagram andconclusions

The various behaviours found in the active polar emulsion undershear have been summarized in Fig. 8, on varying Er and Eract.

At small Eract, the system arranges in a lamellar configuration.In this range, at small shear rates, a few persistent defects (disloca-tions) give rise to slight deformations in the velocity profiles (dashed-red region in Fig. 8). With increasing Er, the dislocations are washedout by the flow and the system enters the region of linear velocityprofiles (red region in Fig. 8). As the activity is increased, themorphology is characterized by a transition towards asymmetricconfigurations, with the formation of a thick layer of material closeto one boundary, thus generating a non-vanishing flux of matter – abehaviour known as unidirectional flow. Such a regime is stable fora broad range of Er at intermediate active dopings (grey region inFig. 8). By further increasing the activity, at high shear rate, phasedemixing is more pronounced and this has important consequenceson the flow. Active layers form on both walls and symmetry isrestored. Under these conditions, it may happen that the velocitygradient in the bulk of the system is either vanishing everywhereor, eventually, opposite to the one externally imposed (negativeviscosity), despite the fact that such states are found to be unstablein the long term. The region with stable symmetric profiles is greenin Fig. 8. Cases in the blue region are instead characterized by thejumping dynamics described in the previous section.

We also remark that the results presented in this paperstrictly hold for bidimensional geometries – as often happensin experimental realizations of active systems, where bacteriaand cytoskeletal suspensions are usually confined at a water–oilinterface.7,27 In full 3d environments – where both vortex stretchingof the flow field and twisting of the polarization field are allowed –the proliferation of degrees of freedom may strongly affect thebehaviour of the system, which is indeed different even inthe absence of any internal and/or external forcing.71,72

In conclusion, we showed how the competition betweenexternally imposed shear and local energy injection results ina wealth of different rheological behaviours, which can beexplained in terms of specific dynamical mechanisms. As anexample, jumps between velocity profiles with positive andnegative gradients are due to collisions between large activedroplets or domains and active layers coagulated on the movingwalls. The generalized active gel model proposed in this work

Fig. 8 Flow regimes in the Er–Eract plane, as described in the text. Hollowmarks denote simulation points.

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has allowed us to perform a full 2d analysis, by keeping undercontrol the time evolution of the important variables, such asthe local concentration and the orientation of the active constituents.Thus we confirmed, by varying both external and internalforcing, the existence of superfluidic and negative viscositystates found experimentally in bacterial suspensions36,37 whosefirst numerical confirmation by means of quasi-1d simulationswas furnished in ref. 40. Moreover, we also found that amaximum entropy production principle holds in selecting themost probable state in the intermittent viscosity regime.

Since most of the observed behaviours mainly arise due tothe elastic properties of the order parameter, they are expectedto stay valid also in nematic systems. This because the dynamicsof polar systems differs from their nematic counterpart mainlydue to the allowed topological defects. We hope that this studycan stimulate the design of new active materials and devices withpioneering applications.

Conflicts of interest

There are no conflicts of interest to declare.

Appendix A: adimensional numbers

In this Appendix we will furnish a derivation for the compressionmodulus B that we made use of to define the adimensionalEricksen number, Er, and the active Ericksen number Eract.

In the following we consider the elastic coefficients for a binarymixture in the lamellar phase, in which one of the components is anisotropic fluid and the other is a polar liquid crystal. One of them,the compression modulus, is used to define the adimensionalEricksen number Er and its active counterpart Eract, in terms ofthe model parameters. The analytical treatment generalizes thatgiven in ref. 57 for a simple lamellar fluid.

It is first convenient to rewrite the Landau–Brazovskii free-energy functional of eqn (9) in a more symmetric form, in termsof the field c = f � fcr, as

F ½c;P� ¼ðdr

~a

2c2 þ b

2c4 þ kf

2rcj j2þc

2ðr2cÞ2

�a2c Pj j2þa

4Pj j4þkP

2ðrPÞ2 þ bP � rc

:

(28)

Eqn (9) of the main text can be obtained with a = �a/fcr2,

b = a/fcr4 and f0 = 2fcr. At equilibrium, the chemical potential

m and the molecular field h must vanish:

m � dFdc¼ ~acþ bc3 � kfr2cþ cr4c� aP� br � P ¼ 0;

(29)

h � dFdP¼ �acPþ aP2P� kPr2Pþ brc ¼ 0: (30)

We then take the single mode approximation,57 exact if con-sidering only gradient terms in the above expressions,

c = ~csin(ky), (31)

where the amplitude ~c and the wavenumber k of the modula-tion have to be computed. Our simulations confirm that, aslong as the bulk parameters are small if compared to the elasticones, the concentration field is modulated in a sinusoidalfashion. By substituting eqn (31) into eqn (30), and neglectingnon linear contributions, we find that Py must satisfy

@y2Py ¼ �

bkkP

~c sinðkyÞ; (32)

whose periodic solutions, with lamellar width l = 2p/k, aregiven by

Py ¼ C þ~P

kcosðkyÞ; (33)

where C is a constant and

P = b ~c/kP. (34)

In order to find the coefficients C and ~c and the wavenumber k,we substitute the profiles of c and Py in eqn (28), integrate overthe lamellar wavelength l and minimize with respect to k,which is found to be

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijkfj=2c

p, the same as for the polarization-

free case. Thus we rewrite the free-energy density as

f ¼ 1

4ð~a� af � aPÞ~c2 þ 3

8bþ 4ac2b4

kf2kP4

� �~c4

þ aC2 C2 � 6cb2~c2

kfkP2

!#;

(35)

where we defined af = kf2/4c and aP = b2/kP. Minimization of f

with respect to C gives C = 0. Then, by further minimizing f withrespect to ~c, we find

~c ¼ �2 kf�� kP2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaf þ aP � ~a

3 kf2kP4bþ 4ac2b4� �

s: (36)

This result shows that lamellar ordering occurs for a o acr =af + aP. Thus, polarization enlarges the range of stability of thelamellar phase with respect to the Brazovskii theory, wherelamellar ordering occurs if a o af.

We now introduce the elastic coefficients related to the free-energy cost of deviations from the harmonically modulatedprofile. We perturb equilibrium profiles by introducing a layerperturbation field W(x,y), in terms of which the perturbatedprofiles become

c(x,y) = ~c sin[k(y � W(x,y))], (37)

Pðx; yÞ ¼~P

kcos kðy� Wðx; yÞÞ½ �: (38)

The field W is chosen so that its amplitude is much smallerthan the lamellar width l, but its typical variation length scaleis much longer.

Because of the slowly-varying behaviour of W, we flush outhigh-frequency modes to obtain a coarse-grained description of

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the model solely in terms of the layer displacement. Thisimplies57 that the following normalization condition holds:

1

LxLy

ðdr sin2 kðy� Wðx; yÞ½ �

cg

¼ 1; (39)

where Lx and Ly are the linear dimensions of the system and theboundary conditions are assumed to be periodic in bothdirections. By substituting eqn (37) and (38) and their deriva-tives in eqn (28) and by retaining only the elastic contributions,namely the gradient terms, we find the following coarse-grained free-energy functional:

Fcg W½ � ¼ðdr

~c2

2kfk2 þ 2ck4 þ b2

kP

� �ð@xWÞ2

þ kfk2 þ 6ck4 þ b2

kP

� �@yW� �2þck2 r2W

� �2

�ðdr

S2@xWð Þ2þB

2ð@yWÞ2 þ

U2r2W� �2

(40)

where we have used eqn (39), and eqn (34) to get rid of P. In thisexpression we identify three contributions, a combination ofthe three following effects: (i) stretching/shrinking of thelamellar surface in the layer direction (x in this section), (ii)compression/expansion of the lamellar layers in the gradientdirection (y) and (iii) bending of the layers.

The coefficient of the derivative along the gradient directionis half of the compression modulus B and it gives the energy

penalty per unit surface due to a change in the layer width.Its explicit expression is then given by

B ¼ b2

kPþ kf

2

c

� �~c2: (41)

Analogously we can define the surface tension S as theenergy penalty per unit surface due to the stretching of thelayer as half of the coefficient in the layer direction

S ¼ b2

kP� kf

2

2c

� �~c2: (42)

It is worth noticing that, on one hand, the liquid crystalnetwork makes the lamellar structure stiffer, since thecompression modulus is strengthened with respect to thepolarization-free model, while on the other hand, it counter-balances the negative surface tension of the lamellar phase.Finally, the coefficient of the laplacian term is half of thecurvature modulus, which gives the energetic cost associatedwith infinitesimal bending of a layer. It can be written as

U ¼ kf

2~c2: (43)

Appendix B: polarization flip

In this section we illustrate the mechanism at the origin of theflip of polarization at the boundary, giving rise to intermittentviscosity behaviour, characterizing the blue region of the phasediagram presented in Fig. 8. This phenomenon is driven by the

Fig. 9 Polarization flip. Contour plots of concentration field f for z = 0.005 (Eract = 0.57) and _g = 3.12 10�5 (Er = 0.003). Panels (a) and (f) show theinitial and final state, respectively, of the polarization at the bottom wall. Panels (b–e) show snapshots of the collision of a droplet with the bottom wall,which causes a local rearrangement of the polarization state. The red contour in panel (b) encloses a +1 defect in the polarization field pattern.

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collision of active domains against the boundary layers.In Fig. 9 we show, for the case at Eract = 0.57 and Er = 0.003,a series of snapshots of a change of polarization on the bottomwall from the antiparallel alignment of P with respect to theimposed velocity (� state) to a parallel configuration (+ state).The overall dynamics of the event can be better appreciated bylooking at the attached Movies S3 and 4 (ESI†). Panel (a) ofFig. 9 shows the configuration at time t = 52 380, before theevent. Panel (b) shows a zoom at the same time of a collidingactive domain, characterized by a typical vortical pattern in thepolarization field, with a +1 defect, highlighted by a red squarein panel (b). As the droplet marges with the layer, strong elasticinteractions produce bendings of the liquid crystal network(see panel (c) at time t = 53 580). In panel (d) at a closesubsequent time t = 56 580, the system is still found in ahomogeneous –Pw state on the wall, despite the fact that thepolarization on the top part of the layer is now directed inthe same direction as the imposed flow. The +1 defecthas disappeared due to the interaction with the wall, thusgenerating complex rearrangement dynamics. As a result ofthe advection and the elastic deformations, the polarizationflap, directed along the flow direction and highlighted by theblack box in panel (d), is pushed on the walls and adheres on it,resulting in a local change of the polarization state (see panel(e) at time t = 58 286). After a similar event, not shown in Fig. 9,the overall hydrodynamic state changes, thus leading to a finalhomogeneous + state, shown in panel (f). We observe that theflip dynamics generally takes place on a time-scale of orderBO(105) timesteps, much shorter than the lifetime of viscositystates, which is found to be BO(107).

Acknowledgements

Simulations were performed at Bari ReCaS e-Infrastructurefunded by MIUR through the program PON Research and Com-petitiveness 2007-2013 Call 254 Action I. A. T. acknowledgesfunding from the European Research Council under the EuropeanUnion’s Horizon 2020 Framework Programme (No. FP/2014-2020)ERC Grant Agreement No. 739964 (COPMAT). We thank DavideMarenduzzo and Ilario Favuzzi for the useful discussions.

Notes and references

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rheology of active polar emulsions: from linear to

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