1. The role of activity versus elasticity on active nematic liquid crystals
N. M. Silvestre and M. M. Telo da Gama
Departamento de Física da Faculdade de Ciências and
Centro 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 Oxford
1 Keble Road, Oxford, OX1 3NP, United Kingdom.
Motivation Modelling active nematic liquid crystals
● Cell extracts and bacterial suspensions are active gels. Systems of microscopic ●
Tensor order parameter:
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 ● Landau – de Gennes free energy density:
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- Equation of motion for tensor order parameter [2]:
trivial liquid crystal configuration.
● To our knowledge there has been no similar study for active nematic liquid crystals. ● Navier-Stokes for incompressible fluids:
● Stress-tensor:
Flow phases
(A)
● Beris-Edwards stress tensor [2]:
● Active stress tensor [1]:
● When active forces are stronger than
aligning (elastic) forces, defects are ● Extensile swimmers:
continuously created and annihilated,
and the velocity field has a turbulent ● Contractile swimmers:
pattern. ● Equations were numerically solved using the Hybrid Lattice Boltzmann Method for
Velocity field (left) and director field (right) for LB nematic liquid crystals [3].
time t=500000, elastic constant L=0.001, and active
stress ζ=0.001. ● Periodic boundary conditions were considered.
(B) ● For simplicity, we consider only the effect of activity on the stress tensor, ζ. The
● Increasing the strength of aligning
(elastic) forces diminishes the number active parameter λ is known to affect only the nematic-isotropic transition.
of eddies present in the flow field and
increases the jet streams. These in
turn locally deform the orientational Integral scale and structure functions exponents
field. ● A measure of the extent of region
over which velocities are correlated Uniform
Velocity field (left) and director field (right) for LB is given by the integral scale: (D)
time t=500000, elastic constant L=0.01, and active
stress ζ=0.001.
(C) (C)
(C) (B)
(A)/(B)
● In the presence of turbulent flows
In some situations the eddies are (B) (A)
the integral scale gives the
●
stretched and the jet streams maximum size of eddies. As activity (A) (B)
organize into undulated stripes. increases the size of eddies (A)
decreases resulting in a more
turbulent flow. Integral scale as a function of activity stress ζ, for several
values of the elastic constant L=0.001, 0.01, and 0.1.
Velocity field (left) and director field (right) for LB time
t=500000, elastic constant L=0.1, and active stress
ζ=0.01. ● Velocity structure functions:
(D) ● If the aligning (elastic) forces are
much larger than the active forces, ● According to Kolmogorov theory of
the system assumes a flowing state turbulence velocity structure
organized in stripes with different flow functions should have a power law
orientations that, in turn, induces the
orientational field to exhibit a stripe
configuration.
valid for .
Velocity field (left) and director field (right) for LB time
t=500000, elastic constant L=0.1, and active stress ● For active nematic liquid crystals
ζ=0.001. the exponents are functions of the
Velocity structure functions exponents α as a function of
activity stress ζ, for several values of the elastic constant
activity and of the elastic constant
L=0.001, 0.01, and 0.1.
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
References
hydrodynamical states. [1] R. Voituriez, J.F. Joanny, and J. Prost, Prys. Rev. Lett. 96, 028102 (2006).
● When active forces are stronger than elastic forces, the active gel evolves towards a [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,
turbulent phase. In this phase, the system continuously creates and annihilates
Oxford (1994).
topological defects. [3] D. Marenduzzo, E. Orlandini, M.E. Cates, and J.M. Yeomans, Phys. Rev. E 76, 031921 (2007).
● 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 Acknowledgements
towards a uniformly aligned phase, such as in passive nematic liquid crystals. NMS acknowledges the financial support of Foundation for Science and Technology (FCT) through
Grant No. SFRH/BPD/40327/2007, and of CFTC/FFCUL.