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Boulder 2009

Active and Driven Soft MatterM. Cristina MarchettiSyracuse University

Boulder School 2009

•Aphrodite Ahmadi (SU → SUNY Cortland)•Shiladitya Banerjee (SU)•Aparna Baskaran (SU → Brandeis)•Luca Giomi (SU → Brandeis/Harvard)•Tannie Liverpool (Bristol, UK)•Gulmammad Mammadov (SU)•Shraddha Mishra (SU)

Boulder 2009

Lectures 1 & 2

Lecture 1 - Introduction• What are active systems?• Hydrodynamics: symmetries, conservation laws and brokensymmetries• Tutorial: phenomenological hydrodynamics of a fluid

Lecture 2 - Hydrodynamics of “living liquid crystals”• Tutorial: hydrodynamics of nematic liquid crystals• Active Hydrodynamics: active stresses, nematic vs polar order• Spontaneous flow in active films• Asters, vortices and spirals

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Lectures 3 & 4

Lecture 3 - From microdynamics to hydrodynamics: SProds on a substrate

• Tutorial: Langevin, Fokker-Planck and Smoluchowskidynamics• Hydrodynamics of SP hard rods• Instabilities, persistent diffusion, and enhanced order

Lecture 4 - Hydrodynamics of bacterial suspensions• The simplest "swimmer”: pullers vs pushers• Hydrodynamic interactions• Instabilities and other results• Outlook

Boulder 2009

Today’s Lecture

A tour through living and non-living active systems

Hydrodynamics as an effective field theory:symmetries, conservation laws and brokensymmetries

Tutorial: hydrodynamics as a generic field theory -isotropic fluid

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Boulder 2009

What is active matter? Collection of interacting self-driven units with emergent behaviorat large scales

Energy input that maintains thesystem out of equilibrium is on eachunit, not at boundaries

Robust behavior in the presenceof noise (generally nonthermal)

Orientable units → states withorientational order

Boulder 2009

Examples from Living & Non-Living World

Bacterial suspensions

In vitro mixtures of cytoskeletal filaments and motor proteins

Cell cytoskeleton

Layers of vibrated granular rods

Nanoparticles propelled by catalytically self-generated forces

Collections of robots

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Vortex patterns on all scales

E-coli from Wiebel’s lab,Wisconsin-Madison

A layer of vibrated granular rods(L=4.6mm), Narayan et al.Science 2007

20µm

10µm

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Cell cytoskeleton: polymer network that serves as skeletonand muscle to the cell - controls motility & mechanical properties

nucleus

actin cytoskeleton

200nm

Borisy &Svitkina

200nm

Fragment of lamellipodia canmove on their own in theabsence of cell body.(Verkhovsky et al. 1996)

fish keratocyte

V. Small,Vienna

Actinlamellipodium

Microtubules (MT)

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Boulder 2009

In vivo: active cytoskeleton controls cell motility

V. Small, IMBA, Vienna.

mousemelanoma(20min)

troutkeratocyte(4min)v=15µm/min

mousefibroblasts(3h)

chickfibroblasts(2h)

Boulder 2009

Ingredients of cytoskeleton:filaments & crosslinkers

Crosslinkers: motor proteins,stationary linkers, [capping proteins],[severing proteins]

Stiff filamentary proteins:•F-actin: d ~ 6nm, lp ~ 10µm•MT : d ~ 20nm, lp ~ 1mmPolarity & Treadmilling

lpd

-+Motor proteins hydrolyze ATP to “walk”in a given direction on polar filaments:

kinesins and dyneins on MT myosin on F-actin

Motor clusters crosslink filaments anddrive filament sliding & contraction

→ Active crosslinkersmyosin II

200nm

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Boulder 2009

In vitro: activity drives pattern formationextracts of cytoskeletal filaments and motor proteins

T. Surrey et al, Science 292, 5519 (2001)

Microtubules & kinesins

Backouche et al., Phys. Biol.3, 264 (2006)

Actin, myosin II & fascin

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Long-Lived Giant Number Fluctuationsin a Swarming Granular NematicVijay Narayan, Sriram Ramaswamy, Narayanan MenonScience 317, 105 (2007)

Fluctuations in # particles expected tosatisfy central limit theorem

!N = N2" N

2

~ N #!N

N~

1

N

Active rod nematic !N ~ N !

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Collective behavior of active matter

•Emergent behavior qualitatively different from that of the individualconstituents•Onset of directed large-scale motion (relation to cell motility?)•Athermal order-disorder transitions: LRO in 2d

•Pattern formation on various scales

•Novel correlations (giant number fluctuations seen in vibratedlayers of granular rods)

•Unusual mechanical and rheological properties: cells stiffen whenstresses and adapt their mechanical properties to the environment

→ “living liquid crystals”

Boulder 2009

Today’s Lecture

A tour through living and non-living active systems

Hydrodynamics as an effective field theory:symmetries, conservation laws and brokensymmetries

Tutorial: hydrodynamics as a generic field theory -isotropic fluid

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Boulder 2009

Goal of Theoretical WorkUse the tools and ideas of (soft) condensed matter physics(hydrodynamics and broken symmetries) to develop an effectivetheory describing the behavior of active matter:

Characterization of phases & ordered structures

Role of noise

Rheological and mechanical properties

What is the role of physical interactions vs biochemical orchemotactic ones?

Can we think of motility as a “material property” and of the onsetof motion as a nonequilibrium phase transition?

Understanding the microscopic mechanisms responsible forcollective behavior

Boulder 2009

Rule-based models (Vicsek, 1995)

Numerical simulations

Symmetry-based phenomenological hydrodynamics•Toner & Tu (1998)•Ramaswamy et al (2002)•Kruse, Joanny, Julicher, Prost & Sekimoto (2004)

Theoretical work can be grouped into three classes:

Microscopic derivation of hydrodynamics for specificmodels•Liverpool & MCM (2003)•Baskaran & MCM (2008)•Aranson et al. (2005)

Lectures 1 & 2

Lectures 3 & 4

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Coarse-graining and hydrodynamicsCoarse-grained description of phases and dynamics of systems of manyinteracting degrees of freedom (liquids, solids, liquid crystals) in terms of fewmacroscopic fields:

cf. field theories in particles physics, hard CM, cosmology

Two Routes:Phenomenology: based on symmetries, generic, but with undetermined parametersDerivation from microscopic models: approximations needed, model-dependentexpression for various parametersInterplay between the two approaches crucial for full understanding

First step: identify the hydrodynamic fields as “slow variables”, withrelaxation rates ω(k)→0 when k →0 (Martin, Parodi, Pershan, PRA 6, 2401 (1972))

Two classes of hydrodynamic fields: densities of conserved variables: # particles, momentum , energy broken symmetry fields in systems with spatial order

!r

i,!p

i{ } ! ",!g,# , ...{ }

~1023

<10

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Conserved fieldsExample: colloidal fluid = large (~µm) particlesin a solvent→ only one conserved field: density ρ(r,t)

t j

j D !

"!

"

# = $% &

= $ %

! !

!!

The decay of fluctuations in timeis controlled by a conservationlaw

2 D

t

!"!"

#$ = %

# diffusion

Fluctuations form equilibrium δρ(r,t)= δρ(r,t) - ρ0occur at finite T with probability ~ exp(-F/kT)

!"(!r ,t) = !"

k(t)e

i!k #!r

!k

$

!"k(t) = !"

ke%& (k )t

& (k) = Dk2

hydrodynamic mode

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Broken symmetry fields → example ofNematic Liquid Crystals

Liquid of rod-like molecules: upon increasing density (orlowering T) the molecules order in a phase with long-rangeorientational order (Onsager 1949)

Isotropic liquid:orientational symmetry

Nematic liquid:broken orientational

symmetry

n2

1 1~

vN

exL b

! !

να

Lb

Competition of translational & rotational entropy

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Entropy at work

Elephant seals on the Californiacoast - courtesy of Arshad Kudrolli

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Ordered state

director n (|n|=1) is brokensymmetry fieldn

-n

order parameter: 1

3ˆ ˆ( )ij i j ijQ ! !

!

" " #=< $ >%

Fluctuations δnk=nk-n0 of wavelength λ=2π/k cost no energy when k→0and decay via a “slow” mode δnk(t) ~ exp[- ω(k)t], with ω(k → 0)=0.

The broken symmetry fields are hydrodynamic fields.

!̂"

rods have no head/tail → cos 0!

!

"< >=#!

"

1

3( )jij i j in nSQ !"=

n → -n is a symmetry of ordered state

Boulder 2009

Today’s Lecture

A tour through living and non-living active systems

Hydrodynamics as an effective field theory:symmetries, conservation laws and brokensymmetries

Tutorial: hydrodynamics as a generic field theory -isotropic fluid

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Boulder 2009

Constructing a hydrodynamic theory1. Identify hydrodynamic variables (conserved densities and

broken symmetry fields). The hydrodynamic equations mustcontain only these variables and their gradients.

2. Identify symmetries.

3. Identify conservation laws.

4. Construct constitutive equations for the fluxes as functions ofthe hydrodynamic fields and their gradients:a) Rule out any term explicitly forbidden by symmetry or conservation

laws

b) Since one is interested in large scales and slow variation, keepterms to leading order in the gradients of the hydrodynamic fields

Boulder 2009

Tutorial 1: isotropic fluid1. Conserved densities:

2. Symmetries:

3. Conservation laws:

4. Constitutive equation for the fluxes: !t" = #

!$ %!g !t gi = #! j& ij

!g = !

!v

density ρ momentum(energy)

rotational invariance, space & time translationalinvariance ,Galileian invariance

density flux !g = !

!v

momentum flux " ij = " ij(!,!g,#,$!,...)

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Entropy production: fluxes & forcesAssume local thermodynamics and calculate rate of entropyproduction (isothermal processes)

F =g

2

2!+ f

0(!)

"#$

%&'!

r

( "F =d

dt(U ) TS) = )T "S

R ! TdS

dt= "

!g " #

!v( ) $!%µ

m+ & ij " #viv j " ' ij p( )(iv j

)*+

,-.!

r

/

πij

ρ

ForceFluxHyd. field

!g

!!µ

!g !

ivj

R = ! flux " force#!r

$

Boulder 2009

Constitutive Eqs & Hydrodynamics

R ! TdS

dt= "

!g " #

!v( ) $!%µ

m+ & ij " #viv j " ' ij p( )(iv j

)*+

,-.!

r

/ = " flux 0 force1!r

/

Reversible process R=0

Irreversible process R>0

! ij = "#viv j " $ ij $ ij stress tensor

$ ij

r= " p% ij $ ij

d= 2&(uij "

1

3% ijukk ) +&b% ijukk

flux=(flux)rev+ (flux)diss

!g = !

!v

!gd= 0

Crucial role of parity under time reversal

uij=1

2!ivj+ !

jvi( )

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Hydrodynamic Equations

!t" = #

!$ % ("

!v)

"(!t+!v %!$)!v = #

!$p +&$2!

v+(&b# 2

3&)!$(!$ %!v) Navier-Stokes

equation

In the context of active systems two approximationswill often be relevant:

• incompressible fluid

• flow at low Reynolds number ! = constant "

!# $!v=0

! "

t

!v +!v #!$!v( ) =

!$ #"% &

!$ #"%

Boulder 2009

Low Reynolds number hydrodynamics

! ,"

! "

t

!v+!v #!$!v( ) = %

!$p +& $2!

v L

!v

Navier-Stokes eqn of fluid flow:

! "t

!v +!v #!$!v( )

accelaration/inertial forces

~!v2 /L

" #$$ %$$=!$ #&% = &

!$p + ' $2!

v

viscous force

~'v/L2

'

Re =inertial force

viscous force!v! L

"

•Re<<1 Stokes flow:•Re>>1 turbulent flow

To keep in mind: yet we saw that bacterialsuspensions (Re<<1) and overdamped systems(Re=0) exhibit swirling “turbulent” motion

bacteria*:L ~ micronsv ~ 10 microns/secRe~10-4-10-5

*E. M. Purcell, Am. J. Phys.45, 3 (1977)

! ,"

!v

!! ""# = 0

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Boulder 2009

Summary A tour through living and non-living active systems

Hydrodynamics as an effective field theory: symmetries,conservation laws and broken symmetries

Tutorial: hydrodynamics as a generic field theory - isotropicfluid

Next Hydrodynamics of active systems