Faculty of mathematics and physics

Seminar - 1st year

Active turbulence

Author: Ziga KrajnikMentor: prof. dr. Miha Ravnik

May 10, 2018

Abstract

A wide range of concentrated systems of active monomers often exhibit a disordered steady stateknown as active turbulence, owing to its apparent similarity with inertial turbulence. The systemis driven out of equilibrium by energy injection at monomer scale. We introduce a continuumdescription of active turbulence based on the equations of nematic liquid crystals, augmented by anadditional active term, which we derive microscopically. Turbulent flow fields lead to the formationof topological defects whose creation and annihilation we describe as a part of an ongoing defectcycle.

1 Introduction

Active matter is a general term encompasing a plethora of mainly biological systems consisting of agentsthat consume energy and transform it into kinetic energy by self-propelled motion. Due to their biologi-cal origins most monomers have a preffered direction, which is often head-tail symetric. This anisotropyis reminiscent of that found in nematic liquid crystals. A prototypical example of active matter withnematic symetry is a mixture of microtubules and kinesin as seen in Figure 1. Depending on monomerconcentration and activity such a system can exhibit a multitude of self organized phases among whichare flocking, jamming and active turbulence [1], [10].

Figure 1: A typical active nematic liquid crystal. Adapted from [7].

Active turbulence is a term used to describe a wide range of seemingly irregular flow patterns in activesystems suchs as those present in swimming bacteria [1], flocks of birds [1], cell suspensions [1] [10], mi-cotubule and kinesin motor suspensions [1]. These are called turbulent due to their striking resemblenceto inertial turbulence. Despite their apparent dissimilarity and disparity of scale all of these active sys-tems share a non isotropic spatial micro structure and are driven out of equilibrium by energy injectionat ”monomer” level. Due to its ubiquitousness, active turbulence might be a universal phenomenon,independant of the specific dynamics of the system in question. In the present seminar active turbulencewill be examined in a nematic liquid crystal setting, which can be wieved as an effective coarse-grainedtheory for a diverse set of systems [1] as can be seen in Figures 2a and 2b.

(a) Liquid-crystaline ordering in a bacterial flock (b) Orientational ordering in a school of fish

Figure 2: Examples of liquid-crystaline order at two widely different length scales. Adapted from [1].

The two main ingrediants of active turbulence are anisotropy resulting in monomer aligning which isquantified by a tensor order paramter Q and activity, which governs the rate of energy injection intothe system. As with inertial turbulence the starting point for describing active turbulence is the Navier-Stokes equation. Due to the non trivial structure of the nematic the Navier-Stokes equation has tobe generalized to account for the effect of the Q tensor, which captures the orientational order of thesystem. A Landau-de Gennes expansion [5] is used for the bulk free energy of the system together with

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a harmonic elasticity theory [5] of the nematic. An aditional equation for the evolution of Q is requiredwhich, together with the Navier-Stokes equation yields the Beris-Edwards equations [6]. An extra termdescribing the activity of the monomers completes the basic set of equations of nematohydrodynamicsused in modelling active turbulence. Solving the equations of nematohydrodynamics, one finds, in theturbulent regime (such as those in Figures 3a, 3b), topological defects and disclinations in the directorfield[11], in addition to the seemingly chaotic flow patterns characteristic of turbulence. Defect dynamicstogether with their creation and annihilation contribute to active turbulence [13].

(a) Active turbulence in a microtubule bundle andkinesin suspension.

(b) Bacteriall turbulence of Bacillus subtilis. Thescale bar measures 35µm.

Figure 3: Examples of active turbulence. Adapted from [1] and [2]

It should be emphasized that this continuum description is not the only possible theoretical model ofactive turbulence. Other models include a discrete self-propelled rod model or various generalization ofthe Vicsek model [3] and its continuum counterpart, the Toner-Tu model [4].

In the present seminar we firstly introduce the requisite hydrodynamic equations for a continuum model ofsufficient complexity to exhibit active turbulence. To acomplish this we analyze how monomer anisotropyaffects the systems free energy and its consequences for nematohydrodynamics. In the second part ofthe seminar we compare the resulting model to experiments and observe the formation of topologicaldefects. Lastly we briefly examine the processes of defect generation and annihilation.

2 Nematohydrodynamics

Nematohydrodynamics is the extension of classical hydrodynamics to systems with nontrivial monomerstructure (with head-tail symetry), which manifests itself as orientational order and represents an ad-ditonal degree of freedom that must be accounted for. The coupling of velocity and orientational orderleads to backflow in which a reverse flow is generated by an elastic stress caused by the original flow.

The continuum description of an incompressible fluid with no external forcing can be written as

(∂t + vj∂j)vi = ∂jΠij , (1)

∂jvj = 0, (2)

where Π is the stress tensor and the second equations specifies that we are dealing with an incompressiblefluid. In the case of a simple Newtonian fluid the stress tensor takes the form

Πsimpleij = −pδij + η(∂jvi + ∂ivj), (3)

where p is the pressure and η the viscosity.For an active nematic liquid crystal the Navier-Stokes equation still holds with a modified stress tensor,which accounts for the effects arising from elasticity and activity.

Π = Πsimple + Πelastic + Πactive (4)

To specifiy and understand these extra terms we must first quantify the orientational order of a nematicliquid crystal.

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2.1 Q tensor

The orientational order of the nematic liquid crystal can be described with a coarse-grained local averagedirection described by the director n, which specifies the direction, but bears no information about thedegree of the ordering. To rectify this we introduce a a scalar order parameter S as

S = 〈P2(cos θ)〉, (5)

where P2 is the second Legendre polynomial and θ is the angle between the molecule and director.Averaging is performed over all molecular orientations.Nematic molecules are head-tail symetric, which implies that n and −n are equivalent. To account forthis degeneracy we introduce the tensor Qij as

Qij =3

2S(ninj −

1

3δij), (6)

where S is the aforementioned scalar order parameter. Such an expression accounts only for a uniaxialnematic and we have neglected the possible biaxial ordering of nematic monomers as it does not impactthe current discussion in a meaningful way.

2.2 Nematic free energy

Using the above defined tensor order parameter we can write the free energy density of the nematic f ,from which we can derive the elastic contributions to the stress tensor. The free energy is a sum of twocontributions, a bulk free energy fbulk and an elastic free energy felastic.

f = fbulk + felastic (7)

For a liquid crystal near the isotropic-nematic transition we can write the bulk free energy as a Landau-deGennes expansion in powers of invariants of Qij as

f =a

2QijQji −

b

3QijQjkQki +

c

4(QijQji)

2, (8)

where a, b, c, are material constants.The elastic free energy arises due to spatial inhomogeniety of Qij and can thus be written as a sum ofderivates of Qij , which obey the symetries of the nematic liquid crystal

felastic =L1

2(∂kQij)

2+L2

2(∂jQij)(∂kQik) +

L3

2Qij(∂iQkl)(∂jQkl), (9)

where Li are three independant elastic constants, corresponding to the three independant deformationsof twist, splay and bend.

2.3 The Beris-Edwards equations

Using the above free energy, one can derive [6] the generalized Navier-Stokes equations for nematic liquidcrystals, the Beris-Edwards equations for incompressible flow

(∂t + vk∂k)vi = ∂jΠij , (10)

(∂t + vk∂k)Qij − Sij = ΓHij , (11)

∂jvj = 0 (12)

Equation 11 desribes the temporal evolution of Qij . Because nematic molecules are anisotropic thematerial derivative is supplemented with a generalized advection term Sij which desribes the rotation ofmonomers in the flow

Sij = (λEij + Ωij)(Qkj +1

3δkj) + (Qik +

1

3δik)(λEkj − Ωkj)− 2λ(Qij +

1

3δij)(Qkl∂kvl), (13)

Eij =1

2(∂jvi + ∂ivj), (14)

Ωij =1

2(∂jvi − ∂ivj), (15)

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where λ is called the tumbling parameter and quantifies the monomer anisotropy. For elongated, rod-likemonomers λ is positive, whereas for squashed, disk-like monomers λ is negative.The right hand side of equation 11 includes the molecular field Hij

Hij = − δf

δQij+δij3tr(

δf

δQkl), (16)

which ensures relaxation of Qij towards an equilibrium state. The prefactor Γ is the rotational diffusivityof nematic molecules.

As has been mentioned, equation 10 is the Navier-Stokes equation, where the stress tensor incorporatesthe nematic and active contributions

Πij = Πsimpleij + Πelasic

ij + Πactiveij . (17)

Using the free energy functional 7 one can derive the elastic stress tensor as

Πelasticij = −pδij+2λ(Qij+

1

3δij)(QklHlk)−λHik(Qkj+

1

3δkj)−λ(Qik+

1

3δik)Hkj−∂iQkl

δf

δ∂jQlk+QikHkj−HikQkj ,

(18)where λ is once again the tumbling parameter.

In contrast to the convoluted elastic stress the active stress is comparatively simple

Πactiveij = ζQij , (19)

where ζ is the activity. The microscopic origin of the active stress is discussed below.

2.4 Active stress

To understand active stress, we take a look at a microswimmer with an axis of symetry in a simpleNewtonian fluid. At the micrometer length scale of, for example bacteria, the Reynolds number tendstoward 0. This implies that inertial effects can be neglected and the Navier-Stokes equation reduces tothe linear Stokes equation

∂ip = η∂j∂jvi + fi, (20)

where fi now denotes the force distribution.Since the Stokes equation is linear it is possible to write the velocity field using a convolution of the forcedistribution and a Greens function G for the Stokes equation

vi(~r) =

∫Gij(~r − ~r′)fj(~r′)d3r′. (21)

It is possible to calculate the Greens function explicitly, but the following argument is independant ofthe exact form of the Greens function. (As long as the dipolar component of G is non zero.)Performing a multipolar expansion of the velocity for a localised force distribution one gets

vi(~r) = Gij(~r)Fj +∂Gij

∂r′kDjk +

1

2

∂2Gij

∂r′k∂r′l

Qjkl + . . . (22)

where

Fj =

∫fj(~r

′)d3r′, (23)

Djk =

∫fj(~r

′)r′kd3r′. (24)

Now for the crux of the argument. Since we are interested in the low (zero) Reynolds limit where inertiais negligable, the net force on the microswimmer must vanish

Fj =

∫fj(~r

′)d3r′ = 0. (25)

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Figure 4: (A) Eperimental measurement of the velocity field of E. coli. (B) Theoretical dipolar field. (C)The difference between the measured and an ideal dipole flow field. One can observe that the near fieldis not dipolar. (D) Force measurement indicates a dipolar far field with characteristic r−2 dependance.Adapted from [8].

Hence, the leading term in the far-field velocity will, in general, be dipolar, as can be seen in Figure 4.

Writting the traceless part of the dipolar term as a sum of a symetric and antisymetric tensor onegets

Djk −1

3Dllδjk = Sjk + Tjk, (26)

where Tjk is the antisymetric part od Djk, asociated with the torque, that will, due to the symetry ofthe microswimmer, vanish.The symetric part Sjk represents the stress due to the dipolar field of the microswimmer

Sjk =1

2

∫(fjr

′k + fkr

′j)d

3r′ − 1

3δjk

∫flrld

3r′. (27)

Passing to the continuum description of a suspension of microswimmers, the active stress will be theaverage dipolar stress of a microswimmer per unit volume

Πactivejk = 〈Sjk〉, (28)

where the average runs over the orientations of the microswimmers.As an idealized microswimmer will produce a pair of forces acting parallel to the director and we averageover a region where the director is constant we have

fj = ±fnj (29)

r′j = ±dnj , (30)

where fd is the dipole moment of the microswimmer. Plugging equations 29 and 30 into equation 28,we get

Πactivejk = 〈fd(

1

2njnk +

1

2nknj −

1

3nlnlδjk)〉 = fd〈(njnk −

1

3)〉 = ζQjk. (31)

We can see that ζ is a measurement of activity, being proportional to the dipole moment of a swimmer.Qualitatively active stress will respond to any local variation in orientaional order Q by generating acorresponding flow and thus potentially driving Q yet further from equilibrium. This effect can, atsufficiently high activities, lead to various hydrodynamical instabilities.

3 Active turbulence

Having introduced the continuum model of active nematics, we are now in a position to look at theexperiments on active turbulence and compare them with the numerical predictions of the continuummodel.

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A typical experimental setup consists of a dense suspension of bacteria or a mixture of microtubules,kinesin and ATP which provides the energy. Experiments are often done in thin films or at an oil-waterinterface to obtain approximate 2D behaviour.In order to better visualize the results and appreciate the turbulent character of the flow fields, it ishelpful to introduce vorticity ~ω as

ωi = εijk∂jvk. (32)

Looking at the comparison of experimental and theoretical prediction in Figure 5a we find the two tobe in good qualitative agreement. Both exibit highly irregular flow patterns and the formation of areasof concentrated vorticity, which correspond to vortices and are characteristic of turbulent flow. One canobserve that the vortices seem to be of rougly equal size, indicating that they are not scale independantas in inertial turbulence.

(a) (A) Concentrated B. Subtilis suspension. (B)Flow and vorticity fields of the enlarged portionof the snapshot. (D) Simulation of continuum the-ory corresponding to the experimental results. Thescale bar is 50 µm. Adapted from [10]

(b) Flow and vorticity fields at high (a) and low(c) activity. Director field at high (b) and low (d)activity. Blue and red dots indicate defects of thedirector field. Adapted from [11]

Another peculiarity of active turbulence in nematics is the formation, at sufficiently high activities [11],of disclinations and eventually topological defects in the director field, as can be seen in Figure 5b. Dueto the disorganized and seemingly chaotic flow fields present in the turbulent regime, there is a continualinterplay of defects, disclinations and velocity that augments turbulent behaviour. We shall endeavor toillustrate the role of defects in the following section.

4 Topological defects

Topological defects are singularities of the director field, that cannot continously relax into a homogenousnematic state. For the sake of brevity and simplicity we shall confine ourselves mainly to the descriptionof topological defects in two dimensions.Twodimensional topological defects can be characterized by their winding number m, given by

m =1

2π

∮dθ, (33)

where θ is the angle of rotation of the director as it travels along the curve looping around the defect.Since n and −n are equivalent the total angle must be an integer multiple of π and consequently m canbe integer or half integer.It can be shown (see [12]) that the free energy asociated with a topological defects scales quadraticallywith its charge

fdefect ∝ m2. (34)

Thus unless stronger defects are in some way stabilized, only ± 12 defects will be present in equilibrium.

Their flow and vorticity fields are shown in Figure 6.

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Figure 6: Flow and vorticity field of a + 12 (left) and − 1

2 (right) defect. White lines represent the directorfield. Adapted from [2].

Topological defects will, similarly to charged particles, expirience a force due to other defects. By analogyto electrodynamics the winding number m may be though of as a charge. The force will be repulsivefor defects with the same charge and attractive for defects with charges of opposite sign. When twooppositely charged defects come together, they annihilate, transfering their energy to the flow field anddestroying the charge they contained.Another insight can be gleaned from Figure 6. Observing the symetry of the flow field for a − 1

2 defectwe can see that it will be stationary, barring the pressence of some external flow. On the other hand, a+ 1

2 defect will move of its own accord along the symetry axis, due to active flow generated by non zerodivergence of the Q tensor[13].

4.1 Creation and annihilation of defect pairs

As we have already seen active turbulence will generate topological defect at sufficiently high activity.This observation is not surpising if we think about the underlying hydrodynamic theory.Increasing the the activity leads to greater and greater energy densities in the medium, which in turnleads to hydrodynamic instabilities and the nematic starts to bend. If activity is sufficiently high theresulting kink might grow to the point it becomes unstable and separetes from the underyling nematic.An example of such a procces is illustrated in Figure 7.

Figure 7: Eperimental observation of the production of a ± 12 defect pair in a kinesin and microtubule

bundle suspension. Blue and red arrows point to a − 12 and a + 1

2 defect respectively. Adapted from [13].

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Going from left to right, we can observe the start of a bend instability in microtubule bundle. This in-stability elongates and its tip becomes more pointed, resuling in a non zero divergence of Q. Afterwardsthe resulting kink separates yet further from the base bundle and eventually breaks off, thus completingthe formation of a defect pair. The lower frames show an idealised director picture of such a process.

In a steady turbulent state the rate of defect formation and annihilation must balance, so let us turn ourattention to the annihilation of defect pairs.When two defect with opposite charge come sufficiently close together they atract and annihilate, releas-ing their energy into the surrounding medium. While the strength of attraction between two defects ischanged by the presence of activity [13], the idea remains unchanged.An example of defect anihilation is presented in Figure 8.

Figure 8: A schlieren image of defect annihilation in an active system. Adapted from [13].

Figure 9: A schematic representation of defect dynamics.Adapted from [7].

After the two defects collide (b) a pair ofbands is formed (c), which is further de-formed due to activity (d) and eventuallyproduces new defect pairs (e), (f).We have thus come full circle in the processof defect dynamics. Hydrodynamical insta-bilites, strenghtened by activity, lead to thecreation of defect pairs, which in time an-nihilate, producing yet further instabilitesand thus perpetuating the turbulent cycleas depicted in Figure 9.

5 Conclusion

Active turbulence is an intriguing phenom-ena arising as a consequence of a complexinterplay of energy injection at the microscale and orientational ordering. It is aninherently emergent phenomenon, havingno counterpart at the level of an isolatedmonomer. We have introduced active tur-bulence in a nematic liquid crystal settingby generalizing the Navier-Stokes equationto anisotropic fluids. Anisotropy enters viathe tensor order parameter Q, the temporal

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evolution of which is approximately givenby the Beris-Edwards equations. Activity is accounted for by a term in the stress tensor proportional tothe divergence of Q. We have explained the microscopic origin of this term as the average of a multitudeof microswimmer, whose far field velovity is dipolar due to the low Reynolds limit.

Comparing the numerical simulation of the continuum theory with experiments good qualitative agree-ment is observed. They both exibit the formation of areas of concentrated vorticity, a characteristic ofturbulent flows. Moreover, both lead to the formation of topological defect in the director field, whosedynamics prove crucial to understanding the turbulent regime.We have described the mechanism whereby pairs of ± 1

2 defects are produced and stressed the role ofactivity in amplifying hydrodynamical instabilities. The produced defects annihilate and, aided by ac-tivity, introduce new instabilites from which defect pairs can develop, thus arriving at the beginning ofthe defect cycle. It is as of yet unclear whether active turbulence is an universal phenomenon, but thewide scope of systems in which it can be observed, as well as some recent work [14] indicate it might be.

References

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[2] Luca Giomi, Geometry and Topology of Turbulence in Active Nematics, Physical Review, X 5, 031003(2015)

[3] T. Vicsek, A. Czirok, E. B. Jacob, I. Cohen, O. Shochet, Novel Type of Phase Transition in a Systemof Self-Driven Particles, Phys. Rev. Lett. 75, 1226, 7 August 1995

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[8] K. Drescher, J. Dunkel, L. H. Cisneros, S Ganguly, R. E. Goldstein Fluid dynamics and noise inbacterial cell–cell and cell–surface scattering, 10940–10945, PNAS, July 5, 2011, vol. 108, no. 27

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[10] H. H. Wensink, J. Dunkel, S. Heidenreich, K. Drescher, R. E. Goldstein, H. Lowen, J. M. Yeomans,Meso-scale turbulence in living fluids,14308–14313, PNAS, September 4, 2012, vol. 109, no. 36

[11] S. P. Thampi, R. Golestanian, J. M. Yeomans, Vorticity, defects and correlations in active turbulence,Phil. Trans. R. Soc. A 372: 20130366

[12] P.G. de Gennes, J. Prost, The Physics of Liquid Crystals, Second Edition, Clarendon Press, 1993

[13] L. Giomi, M. J. Bowick, X. Ma, M. C. Marchetti, Defect Annihilation and Proliferation in ActiveNematics, arXiv:1303.4720v2

[14] A. Doostmohammadi, T. N. Shendruk, K. Thijssen, J. M. Yeomans, Onset of meso-scale turbulencein active nematics, Nature Communications, 8:15326, DOI: 10.1038/ncomms15326

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