The role of activity versus elasticity on active nematic liquid crystalsN. M. Silvestre and M. M. Telo da Gama

Departamento de Física da Faculdade de Ciências andCentro de Física Teórica e Computacional, Universidade de Lisboa,

Avenida Professor Gama Pinto, 2, P-1649-003 Lisboa, Portugal.

J.M. Yeomans

The Rudolf Peierls Centre for Theoretical Physics, University of Oxford1 Keble Road, Oxford, OX1 3NP, United Kingdom.

Motivation● Cell extracts and bacterial suspensions are active gels. Systems of microscopic

swimmers that have ordering tendencies, and that are driven by a continuous energy burn, e.g. from chemical reactions, driving them out of thermodynamic equilibrium even in steady state.

● Active gels may exhibit polar correlation or nematic correlation, depending on the specific features of hydrodynamic interactions between swimmers.

● Studies on polar active gels have revealed an interesting phase diagram where the interplay between the active and the elastic forces leads the system into some non-trivial liquid crystal configuration.

● To our knowledge there has been no similar study for active nematic liquid crystals.

Conclusions● Collective microscopic-swimming is strongly dependent on how active are the

swimming constituents.● The interplay between active forces and elastic forces results in a variety of

hydrodynamical states.● When active forces are stronger than elastic forces, the active gel evolves towards a

turbulent phase. In this phase, the system continuously creates and annihilates topological defects.

● As elastic forces get stronger than active forces, the system evolves towards a stripe phase. This phase has already been observed for polar active gels.

● When elastic forces completely dominate over active forces, the system evolves towards a uniformly aligned phase, such as in passive nematic liquid crystals.

References[1] R. Voituriez, J.F. Joanny, and J. Prost, Prys. Rev. Lett. 96, 028102 (2006).[1] Y. Hatwalne, S. Ramaswamy, M. Rao, and R.A. Simha, Phys. Rev. Lett. 92, 118101 (2004).[2] A. N. Beris and B. J. Edwards, Thermodynamics of Flowing Systems. Oxford University Press,

Oxford (1994).[3] D. Marenduzzo, E. Orlandini, M.E. Cates, and J.M. Yeomans, Phys. Rev. E 76, 031921 (2007).

AcknowledgementsNMS acknowledges the financial support of Foundation for Science and Technology (FCT) through Grant No. SFRH/BPD/40327/2007, and of CFTC/FFCUL.

Modelling active nematic liquid crystals● Tensor order parameter:

● Landau – de Gennes free energy density:

Equation of motion for tensor order parameter [2]:

● Navier-Stokes for incompressible fluids:

● Stress-tensor:

● Beris-Edwards stress tensor [2]:

● Active stress tensor [1]:

● Extensile swimmers:

● Contractile swimmers:

● Equations were numerically solved using the Hybrid Lattice Boltzmann Method for

nematic liquid crystals [3].

● Periodic boundary conditions were considered.

● For simplicity, we consider only the effect of activity on the stress tensor, ζ. The

active parameter λ is known to affect only the nematic-isotropic transition.

Flow phases

● When active forces are stronger than aligning (elastic) forces, defects are continuously created and annihilated,and the velocity field has a turbulent pattern.

● Increasing the strength of aligning (elastic) forces diminishes the number of eddies present in the flow field and increases the jet streams. These in turn locally deform the orientational field.

● In some situations the eddies are stretched and the jet streams organize into undulated stripes.

● If the aligning (elastic) forces are much larger than the active forces, the system assumes a flowing state organized in stripes with different flow orientations that, in turn, induces the orientational field to exhibit a stripe configuration.

Velocity field (left) and director field (right) for LB time t=500000, elastic constant L=0.001, and active stress ζ=0.001.

Velocity field (left) and director field (right) for LB time t=500000, elastic constant L=0.01, and active stress ζ=0.001.

Velocity field (left) and director field (right) for LB time t=500000, elastic constant L=0.1, and active stress ζ=0.01.

Velocity field (left) and director field (right) for LB time t=500000, elastic constant L=0.1, and active stress ζ=0.001.

(A)

(B)

(C)

(B)

(D)

Velocity structure functions exponents α as a function of activity stress ζ, for several values of the elastic constant L=0.001, 0.01, and 0.1.

Integral scale and structure functions exponents● A measure of the extent of region over which velocities are correlatedis given by the integral scale:

● In the presence of turbulent flowsthe integral scale gives the maximum size of eddies. As activity increases the size of eddies decreases resulting in a more turbulent flow.

● Velocity structure functions:

● According to Kolmogorov theory of turbulence velocity structure functions should have a power law

valid for .

● For active nematic liquid crystals the exponents are functions of the activity and of the elastic constant

Integral scale as a function of activity stress ζ, for several values of the elastic constant L=0.001, 0.01, and 0.1.

(A)(A)

(A)

(B)

(D)Uniform

(C) (C)

(B)

(B)(A)/(B)