Motility of active fluid
drops on surfaces
Diana Khoromskaia & Gareth P. Alexander
CoSyDy Meeting
Phase Transitions and Scale Invariance in Biology
Imperial College London
28th September 2015
Active fluids confined to drops
Active fluids confined to droplets exhibit many biomimetic behaviours:
complex large-scale flows
self-driven motility
[H. Wioland et al., PRL 2013]
[T. Sanchez et al., Nature 2012]
How does activity interact with geometrical confinement to create
these behaviours, which are not seen in bulk active systems?
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Active fluid drops on substrates
[E.L. Barnhart et al., PLOS Biol. 2011]
[K. Keren et al., Nat. Cell Biol. 2009]
[E. Tjhung et al., Nat. Comm. 2015]
For a three-dimensional active drop on a planar substrate so far only
symmetrical spreading and stationary shapes have been investigated [Joanny and Ramaswamy, J. Fluid. Mech. 2012].
We study theoretically how such drops can become motile as a result of imposed orientation profiles with topological defects, and how the shape of the drop and friction at the substrate affect motility.
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Model of a drop on a surface
The orientation field satisfies tangential anchoring at both bounding surfaces
S(x,t)
(a)
z
y
h(x,y,t)
Activity drives large-scale flows in the drop through active stress tensor
x
P(x,y,t)
(b)
z
u(x,t)
y
The flow of active fluid is given by the
Stokes equation and obeys continuity:
x
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Thin film approximation
Assuming a thin droplet, we can expand the equations in a small parameter
Diana Khoromskaia
•
to retain effects of activity we scale •
surface tension is neglected, since otherwise
•
boundary conditions:
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Solution for the flow field
S(x,t)
(a)
z
The horizontal and vertical flow components,
and ,
y
are mainly determined by the effective active force
h(x,y,t)
x
P(x,y,t)
(b)
z
u(x,t)
y
x
splay
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shape
bend
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Example: aster defect
(a)
(b)
(c)
(d)
0.5
y
0.0
-0.5
-0.5
0.0
x
-0.5
0.5
(e)
0.0
x
0.5
(f)
-0.5
0.0
x
-0.5
0.5
(g)
0.0
x
0.5
(h)
0.025
0.5
0.020
y
0.015
0.0
0.010
-0.5
0.005
-0.5
0.0
x
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0.5
-0.5
0.0
x
0.5
-0.5
0.0
x
0.5
Motility of active fluid drops on surfaces
-0.5
0.0
x
0.5
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Example: vortex defect
(a)
(b)
(c)
(d)
0.5
y
0.0
-0.5
-0.5
0.0
x
-0.5
0.5
(e)
0.0
x
0.5
(f)
-0.5
0.0
x
-0.5
0.5
(g)
0.0
x
0.5
(h)
0.025
0.5
y
0.020
0.015
0.0
0.010
-0.5
0.005
-0.5
0.0
x
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0.5
-0.5
0.0
x
0.5
-0.5
0.0
x
0.5
Motility of active fluid drops on surfaces
-0.5
0.0
x
0.5
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Motility of the drop
defect posi+on
ξ=0.3
r0/rd
●
slip length
ξ=0
ξ=0.1
ξ=0.2
vcm
0.02
●
0.01
●
-1.5
0.03
-1.0
-0.5
0
■
-0.01
■
-0.02
As a measure for the motion of the drop we use
the centre-of-mass velocity in the x-direction
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Effect of surface friction on motility
top view
0.4
z
(a)
0.2
0.0
slip length
0.4
z
-0.5
0.5
x
(b)
0.2
0.0
0.4
z
0.0
-0.5
0.0
0.5
-0.5
0.0
0.5
x
The friction-dependent prefactor determines the amount of rotation in the flow: rolling vs. sliding
(c)
0.2
0.0
x
0
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1
2
3
4
5
changes sign if
6
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Stationary shapes
For an axisymmetric drop with an aster defect we
calculate the stationary shape profile analytically from
1.4
hf = 0.5
hf = 1.0
hf = 1.3
1.2
1.0
h 0.8
0.6
0.4
0.2
0.2
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0.4
0.6
0.8
r
1.0
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1.4
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Shape deformations
For an asymmetric profile, e.g. aster or vortex defect placed at
the boundary, we can look at the deviation
from the spherical cap after a small time step
(a)
(b)
0.5
y
0.0
-0.5
-0.5
x
- 0.15 - 0.10 - 0.05
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0.5
0.0
0
0.05
-0.5
- 0.10 - 0.05
0.5
0.0
x
0
0.05
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Conclusion and outlook
Conclusion
•
In the scope of a thin film approximation, we derived exact expressions for the flow field in a drop of active fluid driven by a given polarisation profile
•
We identified two key requirements for self-propulsion of an active drop on
a planar substrate: asymmetrically bent or splayed orientation field, e.g.
induced by a topological defect in the interior of the drop, and sufficient
surface friction provided by the substrate
Outlook
•
Investigate how micro-patterned substrates could induce drop motion
•
Study active flows and shape deformations induced by topological defects on a thin spherical shell of active liquid crystal
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Thank you for your attention!
I thank my supervisor Gareth P. Alexander for help and guidance.
D. Khoromskaia and G. P. Alexander, Motility of active fluid drops on surfaces. [arXiv:1508.05242]
(accepted for publication in Phys. Rev. E (2015))
D.Khoromskaia@warwick.ac.uk warwick.ac.uk/dianakhoromskaia @diana_khorom
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