Rods, Rotations, Gels

2/23/06 Texas Colloquium

Rods, Rotations, Gels

Soft-Matter Experiment and Theory from Penn.


The Penn Team• Theory

– TCL– Randy Kamien– Andy Lau– Fangfu Ye– Ranjan

Mukhopadhyay– David Lacoste– Leo Radzihovsky– X.J. Xing

• Experiment– Arjun Yodh– Dennis Discher– Jerry Gollub– Mohammad Islam– Ahmed Alsayed– Zvonimir Dogic– Jian Zhang– Mauricio Nobili– Yilong Han– Paul Dalhaimer– Paul Janmey


Outline• Some rods • Rotational and Translational

Diffusion of a rod - 100th Anniversary of Einstein’s 1906 paper

• Chiral granular gas • Semi-flexible Polymers in a nematic

solvent• Nematic Phase• Carbon Nanotube Nematic Gels


Comments

• Experimental advances in microsocpy and imaging: real space visualization of fluctuating phenomena in colloidal systems

• Wonderful “playground” to for interaction between theory and experiment to test what we know and to discover new effects

• Simple theories - great experiments


Rods

fd Virusfd Virus

100 nm – 10,000 nm100 nm – 10,000 nm

~1 nm~1 nm

900 nm900 nm

1 1 mm

PMMA Ellipsoid

Carbon Nanotube


Einstein – Brownian Motion1. “On the Movement of Small Particles Suspended in Stationary Liquid Required by the Molecular Kinetic Theory of Heat”, Annalen der Physik 17, 549 (1905);

( )

1

2

; (6 )

2 ;B

x f a

x Dt D k T

ph -= G G=

D = = G

&

a = particle radius; = viscosity; f = force


Langevin Oscillator Dynamics2

212 2p

H kxm

p mv mx

= +

= = &

1

22 2 2

;

( ) ( ') 2 ( ')

2| ( ) |

x kx

t t T t t

T kx

k k

x g

x x d

ww

-= - G + G=

= G -

G=

+G

&

Ignore inertial terms:

( ) ( ') 2 ( ')

6

p kx v

t t T t t

a

g V

V V g d

g ph

= - - +

= -

=

&

( )2 2| |2

Bd k Tx x

kw

wp

= =ò

Dynamics must retrieve equilibrium static fluctuations: sets scale of noise fluctuations to be 2T

P. Langevin, Comptes Rendues 146, 530 (1908) Uhlenbeck and Ornstein, Phys. Rev. 36, 823 (1930)


Diffusion with no potential

( )

2

2

0:

[ ( ) (0)]

21

2 2

i tB

B

k x

x t x

d k Te

k T t Dt

w

x

wp w

-

® =

-

G= -

= G =

ò

&

AVG

16 6

Bk T RTD

a N aph ph= = Measurement of D

gives Avogrado’s number

2

( , )

1exp

4 4

P x t

xDt Dtp

=

æ ö÷ç- ÷ç ÷çè ø

Gaussian probability distribution

R = Gas constant


Density Diffusion

( )

( )

( ) ( )

( )21 2 1 2

2

2

( , ) ( )

; ( ) ( ) ( ') '

( , ) ( )

( ) ( )

1( ) ( ) ( )

2

( , ) ( , )

( , ) ( , )

t t

t

x

t t t t

xt t

t x x

t

n x t x x t

x x t t x t t dt

n x t t x x t t

x x t x x x t

x x t t t dtdt

n x t nv D n x t

n t n D n t

aa

a a a a a

aa

a a aa a

a a aa

d

x x

d

d d

d x x

+D

+D +D

= -

= +D = +

+D = - +D

= - - ¶ -

+ ¶ -

¶ = - ¶ + ¶

¶ = - Ñ × + Ñ

å

ò

å

å å

å ò ò

r v r

&

&


J. Perrin Expts. (1908)

NAVG=7.05 x 1023

( )

20

3/ 2

( , )

exp44

n r t

N rDtDtp

=

æ ö÷ç- ÷ç ÷çè ø


Rotational Diffusion“On the Theory of Brownian Motion,” ibid. 19, 371

(1906): Idea of Brownian motion of arbitrary variable, application torotational diffusion of a spherical particle.

= torque

( ) 13

2

; 8

( ) 2 ;B

a

D t D k T

q q

q q q

q t ph

q

-= G G =

D = = G

&

P. Zeeman and Houdyk, Proc. Acad. Amsterdam, 28, 52 (1925)W. Gerlach, Naturwiss 15,15 (1927)G.E. Uhlenbeck and S. Goudsmit, Phys. Rev. 34,145 (1929)F. Perrin, Ann. de Physique 12, 169 (1929) W.A. Furry, “Isotropic Brownian Motion”, Phys. Rev. 107, 7 (1957)


Diffusion of Anisotropic Particles

1. Brownian motion of an anisotropic particle: F. Perrin, J. de Phys. et Rad. V, 497 (1934); VII, 1 (1936).

Interaction of Rotational and Translational Diffusion: Stephen Prager, J. of Chem. Physics 23, 12 (1955)


Diffusion of a rodHan, Alsayed,Nobili,Zhang,TCL,Yodh

Texas Colloquium2/23/06

Defining trajectories

x0

x1

x2

x3

x4

x5

x6

x7

2

1

34

5

6

7

Can extract lab- and body-frame displacements, trajectories at fixed initial angle and averaged over initial angles.


Rotational Langevin (2d)

[ ]2 2

2

2

( ) ( ') 2 ( ')

( ) (0) [ ] 2

[ ]

2cos[ ] exp[ ]

B

HI p

H

t t k T t t

t D t

Dt

n n D t

q q q q

q q q

q q q

q

q

q

q q xq

q x q xq

x x d

q q q

q

q

¶= - - G +G =

¶¶

= - G + ® =¶

= G -

- º D =

D=

D = -

&& & &

& &


Translation and Rotation

Anisotropic friction coefficients ;a bx f x f^ ^= - G = - GP P& &

0

( )

( ) ( ) ( ) [ ( ) ( )]

( ) ( ') 2 ( ( )) ( ')

i ij ij

aij i j ij i jb

i j ijB

Hx

x

n n n n

t t k T t t tq

q x

q q q d q q

x x q d

¶= - G +

¶

G = G +G -

= G -

&

Lab-frame equations

.In lab-frame, noise depends on angle: expect anisotropic crossover

F. Perrin, J. de Phys. et Rad. V, 497 (1934); VII, 1 (1936).


Body-frame equations

[ ] [ ]2 2

cos sin

sin cos

( )

; ( ) ( ') 2

0

0

2 ; 2

i nin

t i i i i ijB

a

ijb

a b

x x

y y

x t x

x t t k T

x D t y D t

d q q d

d q q d

d

x x x

æ ö æ öæ ö÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç=÷ ÷ ÷ç ç ç÷ ÷ ÷-÷ ÷ ÷ç ç çè ø è øè ø

=

¶ = = G

æ öG ÷ç ÷çG = ÷ç ÷G ÷çè ø

D = D =

å

%

%

% %

% % % %%

%

% %

x yBody frame : and are

are independent; simple

Langevin equations with

constant diffusion.

% %

x2

x1

1

2dx~

dy~


Anisotropic Crossover[ ]

00

0 0

40 0

1( , ) ( ) ( )

2cos2 sin2

( )sin2 cos22

ij i j

ij

D t x t x tt

DD t

t

qq

q qd t

q q

é ù= D Dë û

æ öD ÷ç ÷ç= + ÷ç ÷- ÷çè ø

( )/ 2

(1 )( )

a b

a b

nD t

n

D D D

D D D

et

nD

q

q

t-

= +

D = -

-=

1. Diffusion tensor averaged over angles is isotropic.

2. Lab-frame diffusion is anisotropic at short times and isotropic at long times

3. Body-frame diffusion tensor is constant and anistropic at all times

[ ] [ ]

0 0

2 2

1( , )

2

2 ; 2

ij ij ij

a b

D d D t D

x D t y D t

q q dp

= =

D = D =

ò% %


Lab- and body-frame diffusion


Gaussian Body Frame Statistics

;

( ) ( ') 2

t i i

i i ijB

x

t t k T

x

x x

¶ =

= G

%%

% % %

21( ) exp

4 4i

ii i

xP x

Dt Dtp

æ ö÷ç ÷= -ç ÷ç ÷çè ø

%%


Non-Gaussian lab-frame statistics[ ] [ ]

( )0

24 2(4)

2 212 4 4

204 16 4

( ) 3 ( )

[3( ( ) ( ))

( ( ) ( ) 3 ( ))cos4 ]

C x t x t

D t t t

t t t

q

q q

q q

t t t t

t t t t t q

= D - D

= D - -

+ - -

[ ]

( )( )

0

(4)

22

2

2

( )( )

3 ( )

2

a b

a b

C tp t

x t

D D

D D

q=

D

-®

+

Fixed q0: Gaussian at short times – same as body frame; Gaussian at long times – central limit theorem.Average q0: nonGaussian at short times, Gaussian at long times


Non-Gaussian Distribution.

Probability distribution at fixed angle and small t is Gaussian. The average over initial angles is not


Rattleback gasChiral Rattlebacks spin in a preferred direction; Achiral ones do not.

Chiral wires spin in a preferred direction on a vibrating substrate

Tsai, J.-C., Ye, Fangfu , Rodriguez, Juan , Gollub, J.P., and Lubensky, T.C., A Chiral Granular Gas, Phys. Rev. Lett. 94, 214301 (2005).


Rattleback gas II


Spin angular momentum dynamics

( )

Spin angular momentum

Spin angular frequency

( , ) ( )

l nI

l t p t

HI p

qa aa

q q q q

d

q q xq

= W=

W=

= -

¶= - - G +G =

¶

åx x x

&& & &

2

2

( ) ( ( )

( )

t i j ij

j j

l n D p t

n D l

lv D

q a qa aa

q

q d

W

¶ = - GW+¶ ¶ -

= - GW+ Ñ

= - ¶ - GW+ Ñ W

å x x


Rattleback gas III

2( ) ( )t j jl lv Dw tW

W¶ = - ¶ - G W- GW- + Ñ W+

2 12( ) ( )v

t i j i j i i i ij jg gv p v vh e w¶ = - ¶ - ¶ + Ñ - G + ¶ GW-

( ) / 2

i i

z

l I

g vr

w

= W=

= =

W=

= Ñ ´ =v

Spin angular momentum

Center-of-mass mometum

Spin angular frequency

CM angular frequency

Substrate friction Spin-vorticity couplingVibrational torque

1( ) ( ) ( )/ ; rR R v R R v vw -W = = ¶ = - lB.C.:


Nematic phase1 13 3( )

ij i j ij i j ijQ S nn d nn d= - = -

{ ( )

( )[ ]

( )[ ] }

2312 1

2

2

2

3

F d x K

K

K

n

n n

n n

= Ñ ×

+ ×Ñ ´

+ ´ Ñ ´

ò

Order Parameter

Splay Twist

Bend

)( nn n

)( nn

n =Frank director

Frank free energy


Isotropic-to-Nematic Transition

4 1I N

D LL D

whenf-

= ?

D - rod diameterL – rod length

-

-I N

I N

rod concentration

at phase transitionorder parameter S :

orientational distribution functions

increasingconcentration

212 sin ( )(3cos 1)S dq qf q q= -ò

Onsager Ann. N. Y. Acad. Sci. 51, 627 (1949)


Semi-flexible biopolymers

2 – 30 micron length 7-8 nm in diameter

~ 16 micron persistence length

ActinWormlike Micelle( polybutadiene-polyethyleneoxide )

10 – 50 micron length~ 15 nm in diameter~ 500 nm persistence

length

DNA

16 micron length 2 nm in diameter40 nm persistence

length

Neurofilament

5 - 20 micron length 12 nm in diameter

~ 220 nm persistence length


Semi-flexible Polymer in Aligning Field

( )d

sdsR

t=2

| ( ) | 1;

( ) ( ( ), 1 | ( ) | )

s

s s s

t

t t t^ ^

=

= -

R(s)

t(s)

R

h=kBTG/


Fluctuations2

2

22

| |2

| |2

BzP

BP

k T dH ds l

ds

k T dds l

ds

tt

tt^

^

é ùæ öê ú÷ç= - G÷ç ÷ê úè øë ûé ùæ öê ú÷ç» +G÷ç ÷çê úè øë û

ò

ò/

/

=

p B

B

p

l k T

h k T

l

Persistence length

Alignment parameter

Odijk deflection length

k

l

= =

G= =

=G

| |/( ) (0)2

zx x

p

t z t el

ll -=/ 2( ) (0) ps ls et t -á × ñ= = 0

> 0


Polymers in fd-virus suspensionsActin16 m

WormlikeMicelle500 nm

DNA50 nm

Neurofilament200 nm

Isotropic Nematic

10 m 10 m

Hairpindefects

Actin in Nematic Fd

Dogic Z, Zhang J, Lau AWC, Aranda-Espinoza H, Dalhaimer P, Discher DE, Janmey PA, Kamien RD, Lubensky TC, Yodh AG, Phys. Rev. Lett. 92 (12): 2004


Tangent-tangent correlations

actin in nematic phase

/ 2( ) (0) ps ls et t -á × ñ=Orientational correlationsdecay exponentiallylp – persistence length of actin in isotropic phase

Isotropic phase – quasi 2D

t(s’) t(s’+s)

h(s)


Polymer in nematic solvent

( )2

2 23

0 0

1/ ( ) (0, )

2 2 2

L Lpl d

F T dz dz z z K d xdz

tt n nd^

^

æ ö G÷ é ùç= + - + Ñ÷ç ë û÷çè øò ò ò

pl

K

Persistence length

Coupling constant

Elastic constant

=

G=

=

Bending energy Coupling energy Elastic energy

Pl

Odijk lengthl = =G

2

/

2

2 2 2 2 2 20

1 cos( / )lo

( ) (

g(1 / )4 (1 )[1 log(1

)2

/ )]

xz

px

xz D xdx

K x

t z

x

z

x

z e

x

t

D

l

lp l a

l -

¥

¢+

++ + +

+

=

+ò

l

Documents

Rods, Rotations, Gels