Active matter -
overview
18.354 L23
dunkel@math.mit.edu
Active matter
lds Numbers in Biology
Typical Reynolds numbers
number is dimensionless group that characterizes the ratio o
fined as
⇥U L
UL
Re =
=
µ
density of the medium the organism is moving through; µ is t
; is the kinematic viscosity; U is a characteristic velocity of
stic length scale. When we discuss swimming biological organ
eatures that are moving through water (or through a fluid with
hose of water). This means that the material properties µ and
ber is roughly determined by the size of the organism.
e characteristic size of the organism and the characteristic sw
rule-of-thumb, the characteristic locomotion velocity, U , in bi
y U L/second e.g. for people L 1 m and we move
at U 1
dunkel@math.mit.edu
Birds
Fish
Bacteria
Berg lab, Harvard
Bacteria
Wioland et al (2013) PRL
Vortex life time ~ minutes
Motor-driven filaments
Drosophila embryo, Goldstein lab, Cambridge
Motor-driven filaments
Dogic lab (Brandeis) Nature 2012
Amoeba
Physarum
Tero et al, Science 2010
Physarum
Tero et Tero
al,etScience
2010
al (2010) Science
Questions
•
universal aspects of collective motion & selforganization ?
•
•
•
biological functions ?
•
effects of boundary conditions ?
information transport ?
mathematical description? (microscopically,
macroscopically, …)
lds Numbers in Biology
Typical Reynolds numbers
number is dimensionless group that characterizes the ratio o
fined as
⇥U L
UL
Re =
=
µ
density of the medium the organism is moving through; µ is t
; is the kinematic viscosity; U is a characteristic velocity of
stic length scale. When we discuss swimming biological organ
eatures that are moving through water (or through a fluid with
hose of water). This means that the material properties µ and
ber is roughly determined by the size of the organism.
e characteristic size of the organism and the characteristic sw
rule-of-thumb, the characteristic locomotion velocity, U , in bi
y U L/second e.g. for people L 1 m and we move
at U 1
dunkel@math.mit.edu
Bacterial motors
movie: V. Kantsler
~20 parts
20 nm
Berg (1999) Physics Today
source: wiki
Chen et al (2011) EMBO Journal
dunkel@math.mit.edu
E. coli (non-tumling)
non-tumbling HCB 437
10 ㎛
Drescher et al (2011) PNAS
Results
w
of itswe
“puller”
image.
d. field
Atwalls,
distances
rfocused
<6µ
mon
theadipole
overestimates
flow field. (E)
Experimentally
measured
flow to
planemodel
50 µm
from thethe
topbacterial
and bottom
tancesurfaces
2 µm parallel
to sample
the wall. chamber,
(F) Best fitand
force-dipole
model,
(G) residual
field.
Notewhere
the
Bacterial
cell
body,
of the
recorded
∼ 2 and
terabytes
of flow
4
flowmovie
fieldResults
ofdata.
an E. In
coli
“pusher”
decays
much faster,
when
a bacterium
swims close
thecule
surface,
fortothe
length
of t
theflow
mea
this
data we
identified
∼ 10437)
rare
events when
(non-tumbling
HCB
achieved
by
fitting
theByTo
measured
andminisbest-fit
force
decays
cellsBacterial
swam in the
for > surfaces.
1.5 s.
tracking
labeled,
n
flowfocal
fieldplane
far from
resolve the
the
at
variable
locatio
fluidcule
tracers
infield
eachcreated
of the
rare
events, relating
their
the
decays
ofposition
thetracked
flow speed
uof of
with
flow
by
individual
bacteria,
we
gfp- sion
fluc
m surfaces.
To
resolve
the
minisfield
(r
>
8
µm).
and labeled,
velocity to
the position E.
andcoli
orientation
ofthe
the cell
bacterium,
field
dis
non-tumbling
as
they
swam
through
a (Fig.
suspenof
body
(Fig.
1D)
illustr
walls,
we
dividual
bacteria,
we
tracked
gfpthe
measured
and
best-fit
force
dipole
field
1C).
The
the
specific
fitting
and performing an ensemble average over all tracers, we reHoweveo1
sionswam
ofdecays
fluorescent
tracer
particles.
For
measurements
farcharacteristic
from
surfaces
field
displays
the
dipole
length
ℓ=
of
the
flow
speed
u
with
distance
r
from
the
center
i
as
they
through
a
suspensolved the time-averaged flow field in the E. coli swimming
he minismeasure
walls,
we
focused
on
a
plane
50
µm
from
the
top
and
bottom
value
of
F
is cons
movie
dat
However,
the
force
dipole
flow
sig
oftothe
cell
(Fig.
1D) illustrate
that
flow
down
0.1%
of body
the mean
swimming
speed V0 =
22 ±the
5 measured
ticles.
For
measurements
far
from
ckedplane
gfpcell
bod
surfaces
of
the
sample
chamber,
and recorded
terabytes
ofresistive
2 ∼ 2their
and
force
µm/s.
As
E.
coli
rotate
about
their
swimming
direction,
cells
swam
field
displays
the
characteristic
1/r
decay
of
a
force
dipole.
measured
flow
to
the
side
of
thel
4
a 50
suspenµmmovie
fromdata.
the top
and
bottom
for
the
In
this
data
we
identified
∼
10
rare
events
when
note
that
in
the
b
time-averaged
flow field
in
three
dimensions
isbody,
cylindrically
fluid
trace
However,
the
force
dipole
flow
significantly
overestimates
the
cell
where
the
flow
magnit
far from
achieved
ber,
and
recorded
2 terabytes
of forall>components
cells
swam
in∼the
focal plane
1.5 s. Byoftracking
thebehind
µm
the ce
symmetric.
Our
measurements
capture
this
measured
flow
to
the
side
of
the
cell
body,
and
behind
the
4
and
veloci
didentified
bottom
for the
length
ofposition
the
flagellar
bun
fluid
dragatonvaria
the
10symmetric
rare
events
when
fluid∼
tracers
in each
of the
rare
relating
their
cylindrically
flow,
except
theevents,
azimuthal
flow
due
to
cellof body,
where
the
floworientation
magnitude
isof nearly
and
perfo
abytes
field
(r f
achieved
by
twoconstant
opposite
and
velocity
tocell
the
position
and
ofu(r)
thefitting
bacterium,
the of
rotation
the
about
its the
body
axis.
The topology
ne
for
> 1.5
s.
By
tracking
for the
length
of the
flagellar
bundle.
The
force
dipole
fitthe
was
solved
the
nts
when
spec
the
measured
flow
field
(Fig.
1A)
is
the
same
as
that
of
a
and
performing
an
ensemble
average
over
all
tracers,
we
reat
variable
locations
along
the
sw
are events, relating
their
position
Bacterial
flow
fiel
achieved
by1B),
fitting
two
opposite
force
monopoles
(Stokeslets)
dipole
le
plane
dow
kingforce
the
dipole
flowtime-averaged
(Fig.
defined
by
solved
the
flow
field
in
the
E.
coli
swimming
field
(r
>
8
µm).
From
the
best
and
orientation
of
the
bacterium,
dipole
flow
descri
variable
locations
alongswimming
the swimming
direction
thevalue
far As
of
position
µm/s.
plane at
down
to 0.1%
of the mean
speedfitting
V0 = 22routines
± to
5 good
the
specific
and
fi
with
accura
hall
i weFrom
e
average
over
tracers,
refield
(r
>
8
µm).
the
best
fit,
which
is
insensitive
to
and
resi
A
r
ℓF
acterium,
µm/s.
As E.
coli
about
their ,swimming
direction,
theirapproximatio
time-avera
ˆ 2rotate
this
u(r)
=
3(r̂.
d)
−
1
r̂,
A
=
r̂
=
,
[
1
]
dipole
length
ℓ
=
1.9
µm
and
dip
ow
field
in the
E. coli
swimming
the
routines
and fitting
regions,
we
obtain
the
note
tha2
|r|2 specific
8πηdimensions
|r| is
s, we
retime-averaged
flowfitting
field in
three
cylindrically
symmetric
a
wall.
Focusing
value
F is Fconsistent
with
opt
dipole
length
1.9±µm
and
dipole
force
= of
0.42
This
µm beh
mean
swimming
speed
V0 ℓ==22
5 capture
symmetric.
Our
measurements
allof
components
thispN.
wimming
and
applying
the
cylindrica
and
resistive
force
theory
calculat
fluid
dra
of
Fforce,
is consistent
with
optical
trapforce
measurements
[45]
where
F isvalue
the dipole
ℓ the
distance
separating
the
ut
direction,
their
symmetric
flow,
except
the
azimuthal
flow due
to the
= their
22
±cylindrically
5swimming
resulted
in
a sligh
rotati
note
thatThe
in
the
best
fit,
the
cell
and
resistive
force
theory
calculations
[46].
It is
interesting
to
η the
viscosity
the
fluid,
dˆ the
orientation
vector
theof
flow
field
struc
the
rotation
ofisof
the
cell
about
itsunit
body
axis.
topology
on, pair,
their
three
dimensions
cylindrically
the
measu
(swimming
direction)
the best
bacterium,
and
rbehind
the
distance
surfaces,
the
note
thatflow
inofthe
fit, 1A)
theµm
cell
drag
Stokeslet
isfrom
0.1
the measured
field
(Fig.
is the
same
as that
oflocated
aof
the
center
the
cell
ndrically
nts
capture
all
components
of
this
force
dipo
Bacteria
vector
relative
to
the
center
of
the
dipole.
Yet
there
are
some
ity
of
a
no-slip
sur
µm
behind
the
center
of
the
cell
body,
possibly
reflecting
the
force
dipole
flow
(Fig.
1B),
defined
by
fluid drag on the flagellar bundle
tsexcept
of1.this
flow
due
to
Fig. differences
Averagethe
flow fieldazimuthal
created by a single freely-swimming
bacterium.
(A) Experimentally measured flow field far from a surface. Stream lines indicate local
direction
of fl
dipole
Drescher
et aldrag
(2011)
PNAS
close
to
the
cell
body
as
shown
by
the
residual
of
outward
streamlin
fluid
on
the
flagellar
bundle.
flow. (B) Best fit force-dipole model, and (C) residual flow field, obtained by subtracting the best-fit dipole from the experimentally measured field. The presence of the flagella
E.coli
w due to
‘pusher’ dipole
Hydrodynamic scattering
Assuming that r̂ is uniformly distributed on a sphere, and dˆ
uniformly distributed on a circle in the tangential plane at
radius r, we obtain
A
dipole flow
v
2 2
2 3
2
2A τ
r
⟨|∆θ(τ, r)| ⟩H = (Γ + 1)
.
[ 22 ]
6
5
A r
vorticity
=⇤ v⇥ 3
Equating this expression with rrotational diffusion (see
Eq. [ 17 ]) yields the effective hydrodynamic horizon
encounter time
HD rotation
⇥/V
rH
»
2
3
2A τ
(Γ2 + 1)
≃
20
⇥| ⇥| ⇤ (⇤ D
)2r
–1/6
A.
r3
⇥2
[ 23 ]
Note that, due to the τ 1/6
- dependence, the result is rather
2
⇥| ⇥| ⇤ value
Dr used for τ and, similarly,
rotational
diffusion
insensitive
to the particular
to changes in the other parameters. Adopting τ = a/V0
⇥1/6
and inserting experimentally
values (a, ℓ, F, V, Dr )
2measured
A
balance
as given in the mainrtext,
we obtain rH ≃ 3.3 µm for E. coli.
H
r
Equation [ 23 ] can be viewedDas
an upper bound, as the dipolar flow model overestimates u for r < 6 µm (see Fig. 1D in
Drescher et al (2011) PNAS
the
main text).
fluorescent polystyrene beads, which remained uniformly dis-
left to right and cRL for cross
E. coli
Rectification of prokaryotic locomotion
VOL. 189, 2007
FUNNEL WALL CONCENTRATES SWIMMING BACTERIA
8
V . 189, 2007
CONCENTRATES SWIMMING BACTERIA
8705
FIG. 2. Distribution
of bacteria in a structureFUNNEL
withWALL
a funnel
wall. (A) Uniform distribution
after injectio
80 min. (C) Ratios of densities in the left and right compartments versus time. The blue circles are experim
a fit of equation A2 from the Appendix.
OL
FIG. 1. Microstructures with funnel walls. (A) Schematic drawing of the interaction of bacteria with the funnel opening. Bacteria on the left
side may (trace 1) or may not (trace 2) get through the gap, depending on the angle of attack. On the right, all bacteria colliding with the wall are
diverted away from the gap (traces 3 and 4). (B) Scanning electron micrograph of the device. (C) Distribution of incoming and outgoing angles
FIG.
1. Microstructures
with
walls.
Schematic drawing of the interaction of bacteria with the funnel
for bacteria
colliding with a wall.
Datafunnel
were taken
for 70(A)
events.
opening. Bacteria on the
Galadja et al (2009) side may (trace 1) or may not (trace 2) get through the gap, depending on the angle of attack. On the right, all bacteria colliding with the wall
J Bacteriology
diverted away from the gap (traces 3 and 4). (B) Scanning electron micrograph of
the device. (C) Distribution of incoming and outgoing an
tributed during a 24-h period, and thus this population imbalRESULTS AND DISCUSSION
for bacteria colliding with a wall. Data were taken for 70 events.
Austin lab, Princeton, 2009
Our swimmers were green fluorescent protein (GFP)-expressing motile (E. coli) bacteria (strain RP 437/pGFP!2).
They were initially uniformly spread in both compartments
ANDbacteria
DISCUSSION
filled with LB RESULTS
medium. Individual
were tracked as
ance occurs only if the objects actively swim, as opposed to
spreading due to diffusion (data not shown). Since bacteria
communicate with each other (1) and (moreover) move towards one another tributed
(9, 10), it isduring
possibleathat
suchperiod,
quorum- and
24-h
thus this population imb
Chlamydomonas
PRL 105, 168101 (2010)
Movie: Jeff Guasto (TUFTS)
‘puller’
PHYSICAL
FIG. 4 (color online). Time- and azimuthally-averaged fl
Drescher et al PRL 2010
from!velocity vectors (blue [dark
gray]). The spiraling near
Guasto et al PRL 2010
velocity field. A color scheme indicates flow speed magnitu
model: flagellar thrust is distributed among two Stokesle
arrows), whose sum balances drag on the cell body (cen
separate colors in the inset, compared to results from the
size ~ 20µm speed ~ 100µm/s
beat frequency ~30 Hz
Mechanical control of algal
locomotion
PRL 105, 168101 (2010)
PHYSICAL REVI
FIG. 4 (color online). Time- and azimuthally-averaged flow field o
from velocity vectors (blue [dark gray]). The spiraling near elliptic po
velocity field. A color scheme indicates flow speed magnitudes. (b) St
model: flagellar thrust is distributed among two Stokeslets placed
arrows), whose sum balances drag on the cell body (central red ar
separate colors in the inset, compared to results from the three-Stok
10㎛
flow may be important [30]. We are currently investigating
whether similar conclusions hold for the flow field around
bacteria, the prototypical ‘‘pusher’’ microorganisms.
We thank K. C. Leptos for suggesting the use of autofluorescence to track Chlamydomonas cells, S. B. Dalziel,
V. Kantsler, and T. J. Pedley for discussions, D. Page-Croft
and N. Price for technical assistance, and acknowledge
support from the EPSRC, the BBSRC, the Marie-Curie
Program (M. P.), and the Schlumberger Chair Fund.
Kantsler, Dunkel, Polin, Goldstein (2012) PNAS
[1] L. Turner, W. S. Ryu, and H. C. Berg, J. Bacteriol. 182,
2793 (2000).
Surface scattering laws
Kantsler, Dunkel, Polin, Goldstein (2012) PNAS
Control of algal locomotion
2h
2 mm
Kantsler, Dunkel, Polin, Goldstein (2012) PNAS
Sperm near surfaces
A
C
B
Fig. 1. Surface scattering of bull spermatozoa is governed by ciliary contact interactions, as evident from the scattering sequences of individual cells at two
temperature values: (A) T = 10 °C and (B) T = 29 °C. The background has been subtracted from the micrographs to enhance the visibility of the cilia. The cyanPolin, Goldstein
(2012)
PNAS
colored line indicates the corner-shapedKantsler,
boundary Dunkel,
of the microfluidic
channels (see
Movies
S1 and S2 for raw imaging data). The horizontal dotted line in the last
Sperm
Surface + shear flow
!
Kantsler et al 2014 (submitted)
Rheotaxis facilitates upstream navigation
!
Kantsler et al 2014 (submitted)
Viscosity & shear dependence
long distance navigation by rheotaxis ?
⇣? describes
u(C(s)) chirality-induced
Ċ(s) · [I t(s)t(s)] deviations from exact (31)
he second term
anti-alignment,
Assuming that the tip R(t) of the helix performs a quasi-2D motion along the surface
rsal velocity
component,
as observed
in ratio
the
experiments.
ential
and perpendicular
drag
coefficients.
The
drag
we are
interested
in
obtaining
simplified
e↵ective equations for the mean drag velocity R
the orientation
Ṅ (t) duehere,
to the is
action
of the flow gradient
the rigid helical curve
del of a rigid conicalinhelix,
as discussed
a relatively
crudeon
approximation
to C
t
⇣
?
such
can be derived from resistive force theory (RFT). (32)
 =equations
,
erm cell, for it neglects
dynamical
aspects of the flagellar beat (exact wave form
From⇣||Eq. (26), the velocity of some point s 2 [0, S] on the helix can be decomposed as
ects due to translation and rotation of the cell’s head. Notwithstanding, it is pl
ods, takes
values  force
' 1.4 theory
1.7 for realistic flagella. Combining
the
(31)
with
Ċ(s)
= RFT
Ṙ + Ṙansatz
U+
ṘN · Ĉ ✏ .
N · Ĉ ✏ =
Resistive
es
larger
than
the
typical
beat
period,
Eqs.
(36)
and
(39)
provide
a
useful coarse-gr
rque conditions of the over-damped Stokes-regime
shear flow
profile u,the
RFTmain
assumessymmetries
that the force line-density
(force per unit le
ng near a surface,
asGiven
thethe
model
captures
of the problem.
Z
2D minimal model
S
0 = Fi =
ds
0
0 = ⌧i =
Z
S
ds
0
dĈ(s)
fi (s),
ds
f (s) = ⇣||
nh
u(C(s))
nh
⇣? u(C(s))
i
o
Ċ(s) (33)
· t(s) t(s) +
i
o
Ċ(s) · [I t(s)t(s)]
Minimal model
dĈ(s)
✏ijk [Cj (s) Xj⇤ ]fk (s),
(34)
ds ⇣|| and ⇣? are tangential and perpendicular drag coefficients. The drag ratio
where
arise
the minimal
quasi-2D
model
ourexact
simulations.
⇣? results Assuming a
er of rotation,
yields a 6⇥6-linear
system
which implemented
could be solved to in
obtain
 =RFT,
⇣
||
ghethe
y-axis
(Fig. 1B,
Main
Text), and
Eqs.
and
(39)
imply
the
following
minimal
resulting
expressions
are very
complicated
do (36)
not o↵er
much
insight.
Fortunately,
it
+ some
approximations
noise
gives
to
leading
order
e analytical
formulas
for which
U andequals
Ṅ
, that
capture
theYtakes
essential
parts
of
a sperm
with
position
R(t)
=+
(X(t),
(t))values
and
(t)by=flagella.
(Nx (t),
Ny (t
2 for
rigid
rods,
orientation
'of1.4their1.7dynamics,
for N
realistic
Combining
cases U
ṘN · Ĉ (translation-dominated
regime) and
U ⌧ Ṙ
Ĉ (rotation-dominated
the zero-force and zero-torque
conditions
ofNthe
over-damped Stokes-regime
Z
S
at steric interactions between flagellum and channel wall compensate drag
forces
dĈ(s)in vertical
0
=
F
=
ds
fi (s),
Ṙ = V N
U ey , are non-zero. Consideringi the 0translation-dominated
he (x, y)-components
of +
the velocity
ds
✓ in the (x,
◆y)-directions,
✓ F12= 0 and
◆ F2Z= 0, can be solved for
zero-force conditions (34)
Nxangular
Ny distribution, we
Nxfind for
1 ✏ ⌧ 1 Sand1/2
dĈ(s) leading
aging over
with
a
uniform
0 = ⌧i+
= (2D)
ds  '(I1 to✏ijk
X ⇤ ]fk (s),
Ṅ = ˙ ↵
+ ˙
N[CN
) · ⇠(t).
j (s)
Ny2
1
Nx Ny
0
ds
j
✓ ◆
✓
◆
1
0 with 2X ⇤ denoting the2 center
0 of rotation, yields a 6⇥6-linear system which could be solved t
U' ✏ ˙ S
✏ ( 1) ˙ S
,
self-swimming
±1 defines
the expressions
flow direction,
˙ >(35)0 and
is the
shear
1 speed,
Nthe
for3U and Ṅ=
. However,
are very complicated
do not
o↵er m
2
x Nresulting
y
he
! simple
is possible
to obtain
analytical
formulas
for U and
Ṅ , that
capture the essential
experienced
by
the
cell,
and
2
{0,
±1}
the
beat
chirality.
The
dimensionless
oximate length of the flagellum.
The
first
term
is
the
mean
drag
on
the
geometric
center
of regime) and geom
Kantsler
et
al
2014
(submitted)
focussing on the two limit cases U
ṘN · Ĉ (translation-dominated
U ⌧
code
details
of
the
shape
of
the
flagellar
beat,
and
the
coefficient
D
determines
second is an orientation-dependent
regime). drag contribution due to chirality . For passive chiral
2D minimal model
!
Kantsler et al 2014 (submitted)
Collective motion
Broken reflection-symmetry at surfaces
Sea urchin sperm
near surface (dilute)
in bulk (dilute)
F I GURE 1
Dark- f i e l d mi crographs o f l ive sperm o f Tr i pneus t es suspended in natura l seawa ter conta i n i ng
0.2 mM EDTA and ad j usted to pH 8 .3 ( refer red to as standard seawa ter ) . The mi crograph , wh i ch was
taken a f ew seconds af ter mov i ng this f ield into the l ight beam, shows some sperm in l i ght - i nduced
qu i escence , and some that are sw i mm i ng. Among those sw i mm i ng , mos t show l i t t le asymme t ry as i nd i cated
by the near st ra ightness of the i r pa ths . Exposure : 1 s. X 380 .
similar for bacteria (E. coli):
ght absorpt i on proper t i es o f the g l ass. By us i ng
ser i es of f i l ters (Ze i ss : UG5 , BG3 , BG12) , the
t i ve wave l ength i n l i ght - i nduced s topp i ng has
Gibbons (1980) JCB
Di Luzio et al (2005) Nature
Howeve r , we canno t be cer ta i n on th i s po i nt because i t is di f f i cul t to make an accura t e es t i ma t e
of the percent age of qu i escent spe rm when th i s is
arxiv: 1208.4464
2d Swift-Hohenberg model
tropic fourth-order model for a non-conserved scalar or !
reflection-symmetry
⇤(t, x), given by
b
=
2
⇤
⇤,
(1)
2
2
2
0
⇤t ⇥ =2 U (⇥) + 0 ⇥ ⇥
2 (⇥ ) ⇥
he time derivative, and ⇤ = ⌅2 is the d-dimensional
ved from a Landau-potental U (⇤)
a
b
c
U (⇤) = ⇤ 2 + ⇤ 3 + ⇤ 4 ,
2
3
4
0
a>0
(2)
e rhs. of Equation (1) can also be obtained by variational
ed energy functional. In the context of active suspensions,
x) =fluctuations,
⇥ v
tify local (t,
energy
local alignment, phase
will assume throughout that the system is confined to
d
of volume
(3)
a<0
arxiv: 1208.4464
2d Swift-Hohenberg model
tropic fourth-order model for a non-conserved scalar or !
reflection-symmetry
⇤(t, x), given by
b
=
2
⇤
⇤,
(1)
2
2
2
0
⇤t ⇥ =2 U (⇥) + 0 ⇥ ⇥
2 (⇥ ) ⇥
he time derivative, and ⇤ = ⌅2 is the d-dimensional
ved from a Landau-potental U (⇤)
a
b
c
U (⇤) = ⇤ 2 + ⇤ 3 + ⇤ 4 ,
2
3
4
(2)
e rhs. of Equation (1) can also be obtained by variational
ed energy functional. In the context of active suspensions,
x) =fluctuations,
⇥ v
tify local (t,
energy
local alignment, phase
will assume throughout that the system is confined to
d
of volume
(3)
0
arxiv: 1208.4464
2d Swift-Hohenberg model
tropic fourth-order model for a non-conserved scalarbroken
or
reflection-symmetry
⇤(t, x), given by
2
b(1)=
⇤
⇤,
2
2 2
0
2
⇤t ⇥ = U (⇥) + 0 ⇥ ⇥
2 (⇥ ) ⇥
he time derivative, and ⇤ = ⌅2 is the d-dimensional
ved from a Landau-potental U (⇤)
a
b
c
U (⇤) = ⇤ 2 + ⇤ 3 + ⇤ 4 ,
2
3
4
0
a>0
(2)
e rhs. of Equation (1) can also be obtained by variational
ed energy functional. In the context of active suspensions,
x) =fluctuations,
⇥ v
tify local (t,
energy
local alignment, phase
will assume throughout that the system is confined to
d
of volume
−1
(3)
0
a<0
b<0
1
trajectory of a swimming cell can
a pre
arxiv:exhibit
1208.4464
2d Swift-Hohenberg
example,model
the bacteria Escherichia coli [40] an
broken
tropic fourth-order model for
a
non-conserved
scalar
or
⌃ ij denotes the Cartesian components of the Levi-Civ
reflection-symmetry
⇤(t, x), given by
a summation convention
for equal indices throughout.
2
b(1)=
⇤
⇤,
2
2 2
0
2
⇤t ⇥ = U (⇥) + 0 ⇥ ⇥
2 (⇥ ) ⇥
he time derivative, and ⇤ = ⌅2 is the d-dimensional
ved from a Landau-potental U (⇤)
a
b
c
U (⇤) = ⇤ 2 + ⇤ 3 + ⇤ 4 ,
2
3
4
(2)
e rhs. of Equation (1) can also be obtained by variational
ed energy functional. In the context of active suspensions,
x) =fluctuations,
⇥ v
tify local (t,
energy
local alignment, phase
will assume throughout that the system is confined to
d
of volume
(3)
0
eferred handedness [40, 48, 49, 50]. For
trajectory of a swimming cell can
a pre
arxiv:exhibit
1208.4464
nd Caulobacter
[48] have been observed
2d Swift-Hohenberg
example,model
the bacteria Escherichia coli [40] an
broken vita tensor, ⌅i = ⌅/⌅xi for i = 1, 2, and we use
⌃ ij denotes the Cartesian components of the Levi-Civ
reflection-symmetry
a summation convention
for equal indices throughout.
Sea urchin sperm cells
near surface (high concentration)
Riedel et al (2007) Science
b=0
Active polar fluids !
(things with a head and tail)
Results
w
of itswe
“puller”
image.
d. field
Atwalls,
distances
rfocused
<6µ
mon
theadipole
overestimates
flow field. (E)
Experimentally
measured
flow to
planemodel
50 µm
from thethe
topbacterial
and bottom
tancesurfaces
2 µm parallel
to sample
the wall. chamber,
(F) Best fitand
force-dipole
model,
(G) residual
field.
Notewhere
the
Bacterial
cell
body,
of the
recorded
∼ 2 and
terabytes
of flow
4
flowmovie
fieldResults
ofdata.
an E. In
coli
“pusher”
decays
much faster,
when
a bacterium
swims close
thecule
surface,
fortothe
length
of t
theflow
mea
this
data we
identified
∼ 10437)
rare
events when
(non-tumbling
HCB
achieved
by
fitting
theByTo
measured
andminisbest-fit
force
decays
cellsBacterial
swam in the
for > surfaces.
1.5 s.
tracking
labeled,
n
flowfocal
fieldplane
far from
resolve the
the
at
variable
locatio
fluidcule
tracers
infield
eachcreated
of the
rare
events, relating
their
the
decays
ofposition
thetracked
flow speed
uof of
with
flow
by
individual
bacteria,
we
gfp- sion
fluc
m surfaces.
To
resolve
the
minisfield
(r
>
8
µm).
and labeled,
velocity to
the position E.
andcoli
orientation
ofthe
the cell
bacterium,
field
dis
non-tumbling
as
they
swam
through
a (Fig.
suspenof
body
(Fig.
1D)
illustr
walls,
we
dividual
bacteria,
we
tracked
gfpthe
measured
and
best-fit
force
dipole
field
1C).
The
the
specific
fitting
and performing an ensemble average over all tracers, we reHoweveo1
sionswam
ofdecays
fluorescent
tracer
particles.
For
measurements
farcharacteristic
from
surfaces
field
displays
the
dipole
length
ℓ=
of
the
flow
speed
u
with
distance
r
from
the
center
i
as
they
through
a
suspensolved the time-averaged flow field in the E. coli swimming
he minismeasure
walls,
we
focused
on
a
plane
50
µm
from
the
top
and
bottom
value
of
F
is cons
movie
dat
However,
the
force
dipole
flow
sig
oftothe
cell
(Fig.
1D) illustrate
that
flow
down
0.1%
of body
the mean
swimming
speed V0 =
22 ±the
5 measured
ticles.
For
measurements
far
from
ckedplane
gfpcell
bod
surfaces
of
the
sample
chamber,
and recorded
terabytes
ofresistive
2 ∼ 2their
and
force
µm/s.
As
E.
coli
rotate
about
their
swimming
direction,
cells
swam
field
displays
the
characteristic
1/r
decay
of
a
force
dipole.
measured
flow
to
the
side
of
thel
4
a 50
suspenµmmovie
fromdata.
the top
and
bottom
for
the
In
this
data
we
identified
∼
10
rare
events
when
note
that
in
the
b
time-averaged
flow field
in
three
dimensions
isbody,
cylindrically
fluid
trace
However,
the
force
dipole
flow
significantly
overestimates
the
cell
where
the
flow
magnit
far from
achieved
ber,
and
recorded
2 terabytes
of forall>components
cells
swam
in∼the
focal plane
1.5 s. Byoftracking
thebehind
µm
the ce
symmetric.
Our
measurements
capture
this
measured
flow
to
the
side
of
the
cell
body,
and
behind
the
4
and
veloci
didentified
bottom
for the
length
ofposition
the
flagellar
bun
fluid
dragatonvaria
the
10symmetric
rare
events
when
fluid∼
tracers
in each
of the
rare
relating
their
cylindrically
flow,
except
theevents,
azimuthal
flow
due
to
cellof body,
where
the
floworientation
magnitude
isof nearly
and
perfo
abytes
field
(r f
achieved
by
twoconstant
opposite
and
velocity
tocell
the
position
and
ofu(r)
thefitting
bacterium,
the of
rotation
the
about
its the
body
axis.
The topology
ne
for
> 1.5
s.
By
tracking
for the
length
of the
flagellar
bundle.
The
force
dipole
fitthe
was
solved
the
nts
when
spec
the
measured
flow
field
(Fig.
1A)
is
the
same
as
that
of
a
and
performing
an
ensemble
average
over
all
tracers,
we
reat
variable
locations
along
the
sw
are events, relating
their
position
Bacterial
flow
fiel
achieved
by1B),
fitting
two
opposite
force
monopoles
(Stokeslets)
dipole
le
plane
dow
kingforce
the
dipole
flowtime-averaged
(Fig.
defined
by
solved
the
flow
field
in
the
E.
coli
swimming
field
(r
>
8
µm).
From
the
best
and
orientation
of
the
bacterium,
dipole
flow
descri
variable
locations
alongswimming
the swimming
direction
thevalue
far As
of
position
µm/s.
plane at
down
to 0.1%
of the mean
speedfitting
V0 = 22routines
± to
5 good
the
specific
and
fi
with
accura
hall
i weFrom
e
average
over
tracers,
refield
(r
>
8
µm).
the
best
fit,
which
is
insensitive
to
and
resi
A
r
ℓF
acterium,
µm/s.
As E.
coli
about
their ,swimming
direction,
theirapproximatio
time-avera
ˆ 2rotate
this
u(r)
=
3(r̂.
d)
−
1
r̂,
A
=
r̂
=
,
[
1
]
dipole
length
ℓ
=
1.9
µm
and
dip
ow
field
in the
E. coli
swimming
the
routines
and fitting
regions,
we
obtain
the
note
tha2
|r|2 specific
8πηdimensions
|r| is
s, we
retime-averaged
flowfitting
field in
three
cylindrically
symmetric
a
wall.
Focusing
value
F is Fconsistent
with
opt
dipole
length
1.9±µm
and
dipole
force
= of
0.42
This
µm beh
mean
swimming
speed
V0 ℓ==22
5 capture
symmetric.
Our
measurements
allof
components
thispN.
wimming
and
applying
the
cylindrica
and
resistive
force
theory
calculat
fluid
dra
of
Fforce,
is consistent
with
optical
trapforce
measurements
[45]
where
F isvalue
the dipole
ℓ the
distance
separating
the
ut
direction,
their
symmetric
flow,
except
the
azimuthal
flow due
to the
= their
22
±cylindrically
5swimming
resulted
in
a sligh
rotati
note
thatThe
in
the
best
fit,
the
cell
and
resistive
force
theory
calculations
[46].
It is
interesting
to
η the
viscosity
the
fluid,
dˆ the
orientation
vector
theof
flow
field
struc
the
rotation
ofisof
the
cell
about
itsunit
body
axis.
topology
on, pair,
their
three
dimensions
cylindrically
the
measu
(swimming
direction)
the best
bacterium,
and
rbehind
the
distance
surfaces,
the
note
thatflow
inofthe
fit, 1A)
theµm
cell
drag
Stokeslet
isfrom
0.1
the measured
field
(Fig.
is the
same
as that
oflocated
aof
the
center
the
cell
ndrically
nts
capture
all
components
of
this
force
dipo
Bacteria
vector
relative
to
the
center
of
the
dipole.
Yet
there
are
some
ity
of
a
no-slip
sur
µm
behind
the
center
of
the
cell
body,
possibly
reflecting
the
force
dipole
flow
(Fig.
1B),
defined
by
fluid drag on the flagellar bundle
tsexcept
of1.this
flow
due
to
Fig. differences
Averagethe
flow fieldazimuthal
created by a single freely-swimming
bacterium.
(A) Experimentally measured flow field far from a surface. Stream lines indicate local
direction
of fl
dipole
Drescher
et aldrag
(2011)
PNAS
close
to
the
cell
body
as
shown
by
the
residual
of
outward
streamlin
fluid
on
the
flagellar
bundle.
flow. (B) Best fit force-dipole model, and (C) residual flow field, obtained by subtracting the best-fit dipole from the experimentally measured field. The presence of the flagella
E.coli
w due to
‘pusher’ dipole
Active polar fluids
B. subtilis
tracers
bright field
fluorescence
Wensink et al PNAS 2012
Dunkel et al PRL 2013
Bacterial ‘turbulence’
PIV
Vortex diameter ~ 70µm
Vortex life time ~ 1 sec
tracers
fluorescence
Dunkel et al PRL 2013
Minimal continuum theory for
bacterial velocity field
incompressibility
polar alignment
nematic stresses
E=(
0
2
†
†
⇥
)(⇥
v
+
⇥v
)
2
vortices
PNAS 2012
New J Phys 2013
PRL 2013
Isotropic fixed-point
Polar fixed-point
Eigenvalues of
determine stability
experiment
quasi-2D slice
theory
vs.
2D slice
from 3D simulation
Dunkel et al PRL 2013
7 movie segments (40 fps, each 50 s long)
to 7 di⌅erent activity levels.
Global bacterial flows were quantified
Velocity correlations
kinetic energy E (t) = ⌃(v + v )/2⌥ an
xy
2
x
2
y
strophy ⇤z (t) = ⌃⇥z2 /2⌥, where ⇥z = ⇤x v
vertical component of vorticity and ⌃ · ⌥ is
age. While Exy and ⇤z fluctuate, their
(E xy , ⇤z ) are approximately constant d
time interval used in the data analysis (Fi
two orders of magnitude in energy (Fig. ?
the linear scaling ⇤z = E xy /⇥2 , with ⇥
roughly one half of the typical vortex rad
Vortex diameter ~ 70µm
Vortex life time ~ seconds
Dunkel et al PRL 2013
Continuum theory for
bacterial velocity field
incompressibility
non-conservative
E=(
0
2
†
†
⇥
)(⇥
v
+
⇥v
)
2
the creation of a vortex state.
Conserved dynamics ?
II.
HYDRODYNAMIC EQUATIONS
Flow equations
0 = r·v
@t v + (v · r)v = rp + r ·
(1a)
(1b)
Equatio
6th orde
requires
adopt p
2⇥6 BC
(fixes 2
with stress tensor
= [
0
2 (r
2
)+
4 (r
) ](r> v + rv > ) (1c)
2 2
In component notation
0 = @i vi
Interpretation:
@t vj + vk @k veffective
@j pflow-field
+ @k kj
j =
with
(2a)
for (2b)
passive solvent + active component
that creates non-local stresses
kj
=[
0
2 @nn
+
4 @nn @mm ](@k vj
+ @j vk )
(2c)
where @kk = @k @k . Take divergence of (2b) to obtain
That is,
(vx , vy )
S-type
W-typ
Here,
tively.
Conserved dynamics ?
II.
HYDRODYNAMIC EQUATIONS
Flow equations
0 = r·v
@t v + (v · r)v = rp + r ·
(1a)
(1b)
Equatio
6th orde
requires
adopt p
2⇥6 BC
(fixes 2
with stress tensor
= [
0
2 (r
2
)+
4 (r
) ](r> v + rv > ) (1c)
2 2
In component notation
with
growth rate wHkL
1.0
kj
0 = @i vi
0.5
@t vj + vk @k vj = @j p + @k
kj
0.0
-0.5
=[
0
-1.0
-2
2 @nn
-1
+
4 @nn @mm ](@k vj
0
1
wavenumber k
(2a)
(2b)
linear stability
+ @j vk )
That is,
(vx , vy )
S-type
W-typ
Here,
tively.
(2c)
2
where @kk = @k @k . Take divergence of (2b) to obtain
In 2D
the creation of a vortex state.
p(t, x, ±H/2) ⌘ P.
Conserved dynamics ?
II.
(5b)
HYDRODYNAMIC EQUATIONS
Equation (2b) for the velocity field v = (vx , vy ) is of
6th orderFlow
in two
spatial coordinates (x, y) and therefore
equations
requires 2 ⇥ 2 ⇥ 6 BCs in total. Consistent with (5a), we
0 = r(which
·v
adopt periodic BCs in x-direction
leaves us(1a)
with
@t v + (v
· r)v = conditions
rp + r · at y = ±H/2
(1b)
2⇥6 BCs) and no-slip
boundary
(fixes 2 BCs per field component),
Equatio
6th orde
requires
adopt p
2⇥6 BC
(fixes 2
with stress tensor
2 2 L,>y),
(6a) That is,
= [v(t,
(r2 ⌘
) + v(t,
](r v + rv > ) (1c)
0 x,2y)
4 (rx)+
v(t, x, ±H/2)
(6b) (vx , vy )
In component
notation = (±V, 0).
S-type
6th order PDE
W-typ
0 = 2@i⇥
vi 4 more BCs for(2a)
That is, we still need to specify
v=
Here,
@t vj + We
vk @kwill
vj =consider
@j p + @two
(vx , vy ) at y = ±H/2.
k kj classes.(2b)
tively.
S-type: First and second-order derivatives of vanish.
with
W-type: Second and fourth-order derivatives vanish.
[ 0 stand
+ ‘strong’
+ @j vk ) respec(2c)
Here, S and
and
kj = W
2 @nnfor
4 @nn @mm ](@
k vj‘weak’,
tively. where @ = @ @ . Take divergence of (2b) to obtain
kk
k k
r
the w
ity oc
We ho
like re
by pla
Mean field prediction for
shear flow between two plates
the wi
H = 12
0.2
E.
H
=
13.77
FIG. 1 LEFT: Shear force on the upper wire Fx+ ity occ
.4
0.1 on the
Hdistance
= 13.9 between the wires H and theWe ho
depends
To
parameter 2 . White +
spots indicate regions
.0 swimmer
Fx Flow
(H, velocity
7.5) profile,
0.0to the
7.5
close
RIGHT:
2 =singularity.
like re
2
⇥10
separations. The inset
.6 u(y), at three di↵erent wire
2.20
0.1
shows how shear force2.25
varies with separation at by pla
2.30 2 .
.2
constant
2.35
0.2
11 13 15 17
.8
1
0
1
y second derivatives to zero,
S-type: Setting first and
2
u(y)
10
.8
u0 (±H/2) = u00 (±H/2) = 0,
+
force on the
upper
wire
Fx
+
periodic
BCs
in
other
directions
is less restrictive and the non-zero coefficients C
(25)
2
and C3
E.
C1 =
Active nematics
Dogic lab (Brandeis) Nature 2012
autonomous motility, which are not observed in their passive
ana- relative po
1
77
Massachusetts
Avenue
E17-412,
Cambridge,
M
Department
of Mat
Active
nematics
logues. Taken together, these observations exemplify
how assemmicrotubul
(Dated:
October
23,
2013)
BASICS
77biomimetic
Massachusetts
Av
blages of animate microscopic objects exhibit
collective
between
mi
Active nematics
PACS
numbers:1
a
Jörn Dunkel1, ⇤
b
d
In 2d,
the
symmetric
order-parameter
tensor
Q(t
Department
of Mathematics, Massachusetts
In
+
+
PACS numbers:
with
77 Massachusetts Avenue E17-412, Cambridge,
PEG
Depletion
force
BASICS
Qij = Qji ,
(Dated: October
23,can
2013)
This
be
Tr Q = 0,
BASICS
numbers:
In 2d, thePACS
symmetric
can
be order-parameter
represented as tensor Q(t, x, y)
Microtubules
with
✓
◆
In
2d,
the
symmetric
order-parameter
tens
Motor
Time
+
µ
Kinesin clusters
force
with
Q
=
.(1)
where
the bu
Q = QjiBASICS
,
Tr Q = 0,
This can
c ij
µ
Dogic lab (Brandeis) Nature 2012
S=
> 0,0, we
Q
=
Q
,
Tr
Q
ij
ji
can be represented as
whereas n =
Defining
In 2d, the
symmetric
order-parameter
tensor
Q(t,
x,
y)
✓
◆
no head or tail canQ-tensor
order-parameter
be represented
as
p
µ
th
Q=
.
2+µ
2 ,(2) ◆
✓
=
µ
µ
Q=
Qij = Qji ,
Tr Q = 0,
(1) . where the
Defining
the eigenvalues of Q are given by µ
S > 0, we
n be represented as pDefining
We startn
whereas
±
=✓ 2 + ◆
µ2 ,
(3)
⇤ =±
energy dens
p
µ
i
i
j ij
Dt
Dt
approach
each derivaother along
þ
v
!
r
indicates
the
material
where
D=Dt
¼
@
P
H
Y
S
I
C
A
L Rweek
E V Iend
E
t
PRL
110,
228101
(2013)
DQij
We
have
integrated
num
P
H
Y
S
I
C
A
L
R
E
V
I
E
W
L
E
T
T
E
R
S
&1
þ
v
!
r
indic
where
D=Dt
¼
@
31 MAY 2
10, 228101 (2013)
t diffusion
¼ %Suij þtive,
Qik !D
&¼
!ikDQ0kj
þþ
& DH
; is(1c)
'
Q
the
anisotropic
kj ij
ijij
ij
1
Dt
of
uniform
P H Ytive,
S I fluid
Cconfiguration
Edensity
V
E
case of an incompressible
ofLconstant
D
¼
D0 'R
D
Q!,
isacctc
PRL
110,
228101
(2013)
pffiffiffiffiffiffiffiffiffiffiffiffi
ijA
ij þ
1I
ij W
tensor,
is the!,where
viscosity,
isaequations
the
pressure,
and
according
square-root
law,
xðtÞ two
/% tis
an incompressible fluid of
constant#density
&the
t, with
r ! v ¼p
0, to
the
are given
by
the
a
locity,
with
disclinati
tensor,
#
is
the
viscosity,
p
is
þnematic
v ! rbyindicates
the parameter.
material
derivawhere
D=Dt
¼ @tare
r!v
¼ 0, the
equations
given
the annihilation
time.
More
precise
calculations
ht
sho
¼
ð@
v
þ
@
v
Þ=2
alignment
Here
u
ijthe
i j of
ja isquareHe
x axis
Dc
nematic
alignment
parameter.
2 @ friction
de
shown
that
the
effective
is
itself
a
function
of
¼
@
½D
@
c
þ
"
c
Q
%;
(1a)
tive,
D
¼
D
'
þ
D
Q
is
the
anisotropic
diffusion
accord
an0 2incompressible
fluid
ofÞ=2
constant
!, rate
i
ij j
1 density
j ij
ij
ij and
1 !
ij
Dc case of
Dt
¼
ð@
v
&
@
v
are
the
symmetrized
of
x
ð0Þ
¼
ð)L=4;
0Þ.
The
in
do
ij
i(1a)
j
j i separation
)
logx=a,
although
defect
[29,30],
(ð@
¼i v
(j0 &
¼
@
v
Þ=2
are
and
!
¼
@i ½Dij @#j cis
þ the
"1 c viscosity,
@j Qij %;
ij
j
i
tensor,
p
is
the
pressure,
and
%
is
the
Dvi given
Dt where r ! v ¼ 0,
2v &
thetensor
equations
by@substantial
thesche
a
Th
finite
differences
does
inmolecular
the overall
pictu
!are
¼not
#rimply
@the
(1b)
strain
and
the
vorticity,
The
i respectively.
i p þ tensor
j $ij ;changes
strain
and
the
vorticity,
re
Dt
þ @j vmodel
alignment parameter. Here uij ¼ ð@
vi nematic
ap
i vj simple
i Þ=2 predicts
This
thatof
thethe
defect
and
antidef
To
render
Eqs.
(1)
dimen
¼ #r2 vi & @i p þ @j $ij ;field Hij embodies
(1b) the
shown
relaxational
dynamics
nematic
DQ
ij
relaxationa
field
Dt and !ij ¼ ð@i vj & @j vi Þ=2 are the symmetrized
ofik !kjH
approach
each
other
along
¼ %Surate
&ij!embodies
Qkj þ &&1the
Htrajectories.
(1c)
ij þ Q
iksymmetric
ij ;
Dc
by
the
approximate
length
Dt
co
2
Qij strain tensor and the phase
(with
&
a
rotational
viscosity)
and
can
be
obtained
defect
phase
(with
&
a
rotational
visco
We
have
integrated
numerically
Eqs.
(1)
for
an ini
&1
vorticity,
respectively.
The
molecular
¼
@
½D
@
c
þ
"
c
@
Q
%;
(1a)
¼ %Suij þ Qik !kj & !iik Qkjijþ &j Hij ; 1(1c) j ij
loco
stress
bymaterial
the
elastic
stress
t field H embodies
configuration
of
uniform
concentration
and
zero
flow
þ
v
!
r
indicates
the
derivawhere
D=Dt
¼
@
Dt
from
the
variation
of
the
Landau-de
Gennes
free
energy
of
a
t
from
the
variation
of
the
Landau-d
the relaxational dynamics of the nematic
2 then
does
ij
time
by )diffusion
¼ #‘
=K
tive, D
D1disclinations
Qij is and
the anisotropic
locity,
with
two
of
charge
)1=2
located
ij ¼ D
0 'ij þ
xH
two-dimensional
nematic
[21],
two-dimensional
nematic
[21],
H
¼
&'F='Q
,
with
þ
v
!
r
indicates
the
material
derivaD=Dt
¼
@
Dv
)
phase t(with i& a rotational
viscosity) and
can
be
obtained
tensor,
# is
viscosity,
p is viscous
theL pressure,
and
is the
2
the
x the
axis
of a ijsquare
+ L and
boxij
at %initial
positi
This
s
elastic
stress.
the
!
¼
#r
v
&
@
p
þ
@
$
;
(1b)
D
'
þ
D
Q
is
the
anisotropic
diffusion
ij ¼
0
ij
1
ij
i
i
j
ij
þ @j vi Þ=2 by us
nematic
parameter.
Here
uij ¼!ð@i v
from the variation of the Landau-de Gennes
free
of a0Þ. The
j performed
x)alignment
ð0Þenergy
¼ ð)L=4;
integration
is
!and
"of2 ¼ "
To
simplicity,
we
let
"appro
Dt
Z
# is the viscosity,
p is the pressure, and
% is the
Z
1
¼finite
ð@i,vwith
& @j vi Þ=2
are
the symmetrized
rate
!the
1
1
1
ij
j
differences
scheme
described
in
Refs.
[11,1
two-dimensional
nematic
[21],
H
¼
&'F='Q
?
2
ij v Þ=2
by
?ij
2 F=K ¼
2 dA
2
ðcjrQj
& c 2ÞtrQ
þ
boundary
conditions
are
c alignmentDQ
parameter. Here uF=K
ij ¼ ð@¼
i vj þ @dA
j i strain
ðc
&
c
cðtrQ
;
ÞtrQ
þ
Þ
þ
tensor
and
the
vorticity,
respectively.
The
molecular
To render Eqs. (1)
dimensionless,
we
normalize
dista
ij
42
str
We
&1
p
4
4
vi Þ=2¼
the symmetrized
rate offield
!are%Su
"
embodies
the
relaxational
dynamics
of
the
nematic
H
allowed
to
evolve
until
th
j ¼ ð@i vj & @jZ
þ
Q
!
&
!
Q
þ
&
H
;
(1c)
ij
ij
ik 1 kj
ij of the active rods i‘ ¼ 1=an
byikthe kj
approximate length
1respectively.
ij
? The2 molecular
2 2(with1& a rotational
2
ensorF=K
and the
vorticity,
phase
viscosity)
and
can
be
obtained
Dt
config
snapshot
of
the
order
param
jrQj
¼ dA ðc & c ÞtrQ þ cðtrQ Þstress
;
þ by
(2)
vis
the elastic stress of the nematic phase $ ¼ K‘
variation
of the Landau-de
Gennes free energy of a
4 dynamics of the nematic
4 from the
2
2
j embodies the relaxational
sim
and time by ) ¼ #‘ =K
representing
the
ratio
betw
the
beginning
of
the
relax
locity,
two-dimensional nematic
[21],
H
¼
&'F='Q
,
with
ij
ij
withwhere
& a rotational
viscosity)
and
can
be
obtained
where
K
is derivaan elastic
constant
w
bo
(2) stress.
viscous
and elastic
In
these
dimensionless
units,
þ
v
!
r
indicates
the
material
D=Dt
¼
@
i
extensile
system,
with
"
¼
t
where
K
is
an
elastic
constant
with
dimensions
of
energy,
2
2
2
?
e variation of the Landau-de Gennes free energy of a
!
"
all
the
x
Z
S 1and
=2,
and
is 22j=2.
the Perio
criti
simplicity,
let?"2 ¼
¼2 "
take
"1c1¼ j"
1 we trQ
2þ D 2Q
?
2 2the
(see
also
supplemental
tive,
D
¼
D
'
is
the
anisotropic
diffusion
mensional
nematic
[21],
H
¼
&'F='Q
,
with
trQ
¼
S
=2,
and
c
is
the
critical
concentration
for
the
sna
ðc
&
c
cðtrQ
jrQj
F=K
¼
dA
;
ÞtrQ
þ
Þ
þ
ij
0 ij
ij dimensions
boundary
are 4assumed, transition,
and
the xdefects
where K isij an elastic
constant 1ijwith
of energy,
4conditions
2
isotropic-nematic
so
ð0Þ
the
)
In annihilate.
passive nematic
liqui
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
!
"
2
2
?
allowed
to
evolve
until
they
Figure
1
show
ZtrQ
isotropic-nematic
transition,
so
that,
at
equilibrium,
S
¼
tensor,
#
is
the
viscosity,
p
is
the
pressure,
and
%
is
the
(2)
¼
S
=2,
and
c
is
the
critical
concentration
for
the
?
1
1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
exo
2
1 &parameter
cknown
=c. Finally,
the
stress
ten
that
the
dynamics
the
fin
snapshot
of
the
order
and
flow
field
shortly
a
cðtrQ2 Þ2 þ
jrQj
¼ dA ðc & c? ÞtrQ2 þ p
;
ij
?
r
a
(se
4
4
2
isotropic-nematic
transition,
atwhere
equilibrium,
Sð@
¼the
1&
c so
=cthat,
. Finally,
the
stress
¼
$
þ
$ofbackflow,
is due
the to
¼tensor
þ
@so-called
vfor
nematic
alignment
parameter.
Here
u
ijthe
K isbeginning
an
constant
with
dimensions
the
of
relaxation
both
aenergy,
contractile
ijelastic
i vof
j$
jelastic
i Þ=2
ij
ij
the
th
sum
stress
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
ij
To
ren
Giomi
et
al
PRL
2012
2
2
?
(2)
?
r
a
rthe¼ ab
trQ ¼
S =2,
and
cto
is nematic
the critical
concentration
for
extensile
system,
with
"ientation
¼ )0:2
inof
the
units
defined
1 & c!=c.¼
Finally,
the
stress
tensor
$are
$
þsymmetrized
$
is
the
sum
of
the
elastic
due
elasticity,
$
the
nematic
ord
kn
ij ¼stress
ij
ij
ð@
v
&
@
v
Þ=2
the
rate
of
and
þ
Q
H
&
H
&%SH
ijQkj , wh
ij
i j
j i
kj [31]). ikS
isotropic-nematic
soij that,
atikequilibrium,
¼
(see also thetransition,
supplemental
movie
S1
Active nematics
PRL 110, 228101 (
case of an incomp
where r ! v ¼ 0, th
r
Dc
¼ @ ½D
Dt
Dv
!
¼ #r v
Dt
DQ
¼ %Su
Dt
r
by the
Active nematics
REVIEW LETTERS
PRL 110, 228101 (2013)
week ending
31 MAY 2013
PHYSICAL
FIG. 2 (color online). Defect pair production in an active
case
of an incompressible
fluid ofandconstant
densi
suspension of microtubules not
and consistent
kinesin (top) with
the
same
where
r ! vobserved
¼ 0, in
theourequations
aresetup
given
by
phenomenon
numerical simulation
of an extenexperimental
sile nematic fluid with " ¼ 100 and ! ¼ "0:5. The experimental picture
T. Sanchez et al.,
Dcis reprinted with permission from
2@ Q
Nature (London)
491,
431
(2012).
Copyright
¼ @Giomi
½D
@
c
þ
"
c
i
ij etj al PRL12012j 2012,
ij %; Macmillan.
Dt
Note that the potential cannot contain odd-power terms
since Tr Q2k+1 = 0 in 2D. Consider the corresponding
field equation
Alternative approach
To obtain closed equation, we must express
v = (vk ) in
F
@t Qdiscuss
@k Qpossible
(12)
ij + vk two
ij =
terms of Q. We
choices
Qij
vk = Dvelocity
@n Qnk and
where v is the advection
(16a)
3 Qjk )
vk = D @n (Qnj
(16b)
3
F 2
a @ Tr Q2
b @ Tr Q2 4
=
+
where
1the constant D is a response
Qij 2
2 @Qij
4 @Qij1 coefficient with units
length 0 /time. Note the 2crucial 0 di↵erence between
the
2
@(rQ)
@(rrQ)
2
4-1 assumes
-1
two closure
conditions:
Eq.
(16a)
that active
@k
+ @k @n
. LC
(13)
-2
2 Q@(@
2-2 flow@(@
Qijopposite
)
k @nin
configurations
andk QijQ) create
fields
-3
-3
-3 -2 which
-1
0
1 may
2
3 also be
-3 -2 -1
1
2
3
directions.
By0 contrast,
Eq. (16b)
Using
written asD<0 extensile
D>0 contractile
k
2
gridsp
Note
that
the
potential
cannot
contain
odd-power
terms
situations because they have identical spectra (although
work
2k+1
since
Tr Q
= 0 in 2D.are
Consider
the
corresponding
eigenvectors
swapped).the corresponding
100 ⇥
field
equation
Adopting
(16a), which distinguishes between contracshoul
Alternative approach
tile and extensile nematics, the equations of motions take
F
the form
could
@t Qijequation,
+ vk @k Qwe
To obtain closed
must express v = (vk )(12)
inIt s
ij =
Q
nates
ij
3choices
terms
of
Q.
We
discuss
two
possible
@t Qij + D(@n Qnk )@k Qij = aQij bQ ij +
(18a)
T =
2 @k @k Qvelocity
ij
4 @k @
k @n @n Qij
where v is the advection
and
vk = D @n Qnk
4 Qjk )
or, equivalently,
vk2 = bD@@Tr
n (Q
F
a @ Tr Q
Qnj
=
+
(16a)
(16b)
3
Q
2
@Q
4
@Q
ij
ij
ij
@
Q
+
D[(r
·
Q)
·
r]Q
=
aQ
bQ
+
(18b)
t
where the constant D is a response coefficient
with unitsDen
2
2
2
2 2
@(rQ)
@(rrQ)
r
Q
(r
)
Q
length2 /time.
Note
the
crucial
di↵erence
between
the
2
4
2
4
unit
t
@
+
@
@
.
(13)
k
k
n
two closure2conditions:
Eq.
(16a)
assumes
LC
norm
@(@k Qij )
2
@(@kthat
@n Qactive
ij )
Dynamics
conservesQthe
Tr Q ⌘ flow
0. As
indicated
configurations
andtrace,
Q create
fields
in opposite
vecto
above,
the velocity
breaksEq.
Q !(16b)
Q symmetry,
butalso be
directions.
By term
contrast,
which may
Using
could be replaced by (D, Q) ! ( D, Q) symmetry.
written as
In principle one could include
additional terms on
k
@t Qij + D(@n Qnk )@k Qij =
aQij
bQ
ij
+
(18a)
Alternative approach
2 @k @k Qij
4 @k @k @n @n Qij
T =1
or, equivalently,
@t Q + D[(r · Q) · r]Q =
bQ3 +
aQ
2r
2
Q
4 (r
(18b)
2 2
) Q
Deno
unit ta
normal
vector
growth rate wHkL
Dynamics conserves the trace, Tr Q ⌘ 0. As indicated
1.0
above, the velocity term breaks Q ! Q symmetry, but
could be replaced0.5by (D, Q) ! ( D, Q) symmetry. linear In principle one could include additional termsstability
on
the rhs., such as0.0
correctly symmetrized combinations of
and Ga
vorticity coupling
terms
involving
!
Q
where
!
=
ik kj
ik
-0.5
@i vk @k vi with vk being expressed in terms of QChaos
via thepossible?
adopted closure-1.0
condition
(this could
lead
-2
-1
0
1 to less2 isotropic
structures).
wavenumber k
Exam
Prelim. simulation results
D<0 extensile
Prelim. simulation results
D>0 contractile