Soft Motors: molecular to macroscopic perspectives · PDF filetorque results from ... • steady unidirectional rotation • orientational ratchet / Landauer’s blowtorch mechanism

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Soft Motors:molecular to macroscopic

perspectives

Peter Palffy-Muhoray Liquid Crystal Institute Kent State University

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Outline: microscopic• What is a Motor?

• Molecular Motors 1. Anomalous Photoalignment– ideas of Landauer, Prost & Astumian

• Molecular Motors 2. Yokoyama - Tabe Experiment– Lehman effect

• Molecular Motors 3. ATP synthase– Boyer & Walker

• Molecular Motors 4. Myosin and Actin– muscles

• Molecular Motors 5. LC elastomers– elongation/contraction

• Broken symmetry– broken symmetry drives motors

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Outline: macroscopic

• molecular & macroscopic length scales

• shape change

• motors based on elongation

• motors based on bend

• summary

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what is a motor?

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• what is a motor?– anything that produces or imparts motion (dictionary.com)

– a machine that supplies motive power for a vehicle or other device (O.E.D.)

Motor

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Motor• what is a motor?

– a device which uses energy (but not momentum!) to cause motion

• how does motion come about?• car convinces the road to exert a force on it, and push it

forward

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Motor• what is a motor?


• motor causes one part of a system to exert a force/torque on another, causing motion

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• what is a motor?


• motor causes one part of a system to exert a force/torque on another, causing motion

Motor

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Molecular Motors 1.Anomalous Photoalignment

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Direct optical torque

laserray

table

_hp = λ τ = ×optvol D E

angular momentum transfer:

torque results fromchange in extrinsicangular momentumof light.

τ τ+ = 0el optvol vol


Indirect optical torque

⇒ ×1

addition of 1% of anthroquinone dye reduction of threshold intensity by ~ 100!

laser

- dye

τ τ

τ τ

+ =

+ =+1100

without dye: 0with dye: ?? 0

el optvol vol

el optvol el

Torque on nematic causing elastic deformationCANNOT come from light! Source of torque??

1. I. Janossy, A.DD. Lloyd and B.S. Wherret, Mol. Cryst.Liq. Cryst. 179, 1, 1990.I. Janossy, Phys. Rev. E 49, 2957, 1994.


• light causes the director to reorient, against a restoring elastic torque, essentially without the transfer of angular momentum.

• light causes rotation without exerting a torque!

Puzzle 1:

???

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Prost, Astumian

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The translational ratchet*

U

xU=0

xU

x

* R.D. Astumian et al: Phys.Rev.Lett.72, 1766, 1994. J. Prost et al: Phys. Rev. Lett. 72, 2652, 1994.

Particles in asymmetric periodic potential:

When potential is turned OFF, particles diffuse, no net current

When potential is turned ON,particles move towards minimum,giving rise to current.

J


Simple model:• dye

• anisotropic molecule with orientation

• in ground state does not interact with nematic

• in excited state becomes 'nematic' molecule

• probability of excitation depends on

• lifetime of excited state is

• optical field• transfers energy, but no momentum

to the sample

dl

2ˆ( )dl Eot

k

E


Torque on liquid crystal:

n

ld^

n

ld^

Average orientation of excited dye molecule is NOT along nematic director

interaction between liquid crystal and excited dye gives rise to torque!


Rotating dye gives rise to shear flow

n

ld^

flow

n

ld^

flow

dye: - rotation source of vorticityliquid crystal: - aligned by nematic field of dye

⇒

viscous shear carries angular momentum from cell walls to dye

angular momentum current: dye nematic cell walls dyemolecular elastic shear


Summary of Puzzle 1

– Janossy effect1 (source provides energy but not torque yet causes rotation) ⇒ orientational ratchet2

– dye is light-driven rotor with continuous rotation

– energy from optical field drives shear flow

– viscous stress carries angular momentum from dye to cell walls

1. T. Kosa and I. Janossy, Opt.Lett. 17, 1183, 1992; Opt. Lett. 20, 1231, 1995. 2. W. E and P. Palffy-Muhoray, Mol.Cryst.Liq.Cryst. 320, 193, 1998


Puzzle 2. Photoinduced Twist


Photoinduced Twist

laser OFF

nematic

photosensitive

alignment layer

strong anchoring

dye


Photoinduced Twist

laser ON

nematic

photosensitivealignment layer

strong anchoring

E

polarized

light

dye

θ


Origin of torque

• before deformation takes place, polarization is parallel to the director, hence optical torque D x E = 0.

• optical field stabilizes configuration!

• source of torque?

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Landauer

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Landauer’s Blowtorch

x

T(x)x

U(x)

R. Landauer, J. Stat. Phys. 53, 233 (1988)

particles in periodic potential

‘hot’ molecules are more excited & diffuse over barrier

steady current: system is molecular motor (pump) ∴

one part of system pushes on other;no momentum transfer from outside.

( )U x

( )T xx

x

M. Büttiker, Z. Phys. B 68, 161 (1987).

J


Indirect optical torque: Brownian ratchet

x

T(x)x

U(x)dye molecules in nematic field

dye molecules parallel to pump polarization are excited & diffuse over barrier

steady rotation: dye is light driven molecular motor ∴

torque results from rotation of dye;no momentum transfer from light.

P. Palffy-Muhoray, T. Kosa and Weinan E, Appl. Phys. A 75, 294 (2002)

( )U θ

( )I θθ

θ


Schematic of Brownian motor:

dye in trans-state starts to rotate towards director

it exerts torque on director & vice versa

as it becomes parallel to pump polarization

it is excited into cis-state, undergoes diffusion

relaxes into trans-state, rotates towards director...


Dynamics• dye: Fokker-Planck

• transition rates

t

c

2t

2c

2U / 2

2U / 2

( )t

( )t

ˆ ˆ = e ( )ˆ ˆ = e ( )

t t t t t t t c c

c c c c c c c t t

kTt to t d

kTc co c d

D M U f f

D M U f f

f f e

f f e

ρ ρ ρ ρ ρ

ρ ρ ρ ρ ρ

ν

ν

∂= ∇ +∇⋅ ∇ − +

∂∂

= ∇ +∇⋅ ∇ − +∂

+ ⋅

+ ⋅

l E

l E


Dynamics• liquid crystal

– bulk:

– surface:

2ˆ ˆ ˆ ˆ=K ( ) It

γ ∂ ∇ −∂n n nn

t c

ˆ ˆ ˆˆ ˆ={K( )

ˆ ˆ < > < >}( ) ˆ ˆ

s

t cd d

tU Ud d I

γ

ρ ρ

∂∇ ⋅ + ×∇×

∂∂ ∂

− − −∂ ∂

n N n N n

nnn n


Simulations

• optical field drives orientational current

• dye exerts steady torque on nematic

0.0

0.5

1.0di

rect

or a

ngle

(rad

ians

)

0.0 5000.0 10000.0time

Field is turned off

0.0

0.5

1.0di

rect

or a

ngle

(rad

ians

)

0.0 5000.0 10000.0time

Field is turned off

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Summary of Anomalous Photoalignment

• light-driven molecular motors

• dye* molecules are rotors in nematic potential

• energy (but not momentum) from light

• steady unidirectional rotation

• orientational ratchet / Landauer’s blowtorch mechanism

• work: steady torque on director by viscous stress + dissipation

• achiral

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Molecular Motors 2. Yokoyama - Tabe Experiment

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• chiral Langmuir monolayer

• substrate: glycerol + H2O~3 nm

Yokoyama – Tabe Experiment

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Collective Molecular Rotor/Pump

Yuka Tabe and H. Yokoyama, Nature Materials, 2, 806(2003).

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Chiral LC molecules on glycerol

FELIX013 on pure glycerol (SmC* phase)

clockwise rotationCS4001 on pure

glycerol (SmCA* phase)

counter-clockwise rotation

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Chirality inversion

100µm

(S)-OPOB

(R)-OPOBon pure glycerol

on pure glycerol

CW

CCW

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Precession speed linearly depends on ∆PW/P0

∆Pw/P0=Pv-Ps

Pv: actual water vapor pressure

Ps: saturated water vapor pressure

(R)-OPOB(S)-OPOBCW

CCW

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Onsager Reciprocal Relations

υm: molecular volume of water in vaporγ: rotational viscosity of LC directorη: water mobility through the filmb: Lehmann coefficient (∝ chirality strength)

PJt

ST m∆⋅+∂∂

=•

υφτOnsager Relations:

Entropy Production:

P b dtd

m ∆+= υτγ

φ 1

PbJ m∆+= υη

τ 1

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zc

cc

∂∂

×+−=∂∂ PF

tν

δδγ

∫ ∑

+−∇=

= yxii

ucKdrF,

422 ||4

||2

||2

ccτLehman term

Shibata & Mikhailov, Europhys. Lett., 73 (2006)H. Brand et al., Phys. Rev. Lett. 96 (2006) Tsori & de Gennes, Euro.Phys. J. E 14 (2004)T. Okuzono, Kyoto Workshop (2005)

Simulated by T. Okuzono

Phenomenological Theory of Lehmann effect

Equation of motion :

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Summary of Yokoyama – Tabe experiment• substrate & monolayer system is motor

• energy stored in chemical potential of H2O

• vapor current drives rotation of chiral molecules

• work: produces circulating flow, dissipation

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Molecular Motors 3.ATP Synthase

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F1FO ATP Synthase

• enzyme that synthesizes adenosine-5'-triphosphate (ATP)

from

adenosine diphosphate (ADP) + phosphate

• reaction:

ADP + Pi → ATP

• ATP:"molecular unit of currency" of intracellular energy transfer


F1FO ATP Synthase• energy from glucose sets up

H+ gradient• H+ current drives motor• stator opens & closes,

producing ATP

http://upload.wikimedia.org/wikipedia/commons/3/37/ATPsynthase_labelled.png


F1FO ATP Synthase


F1FO ATP Synthase• normal human produces

~100lbs of APT/day

• motor can run backwards

Boyer & Walker, Nobel Prize in Chemistry, 1997



Why does rotor turn?• sequential protonation &

deprotonation of side groupscoupled with rotation

• chemical potential differenceof H+ drives current, whichdrags surface of C-rotor.

• pot. diff. →current →

rotation.A. Aksimentiev et al.Biophysical Journal 86 ,1332 (2004)


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Modeling

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A. Aksimentiev et al.Biophysical Journal 86 ,1332 (2004)

• system of Langevin equations

• Lennard-Jones + hydrophobic +screened Coulomb

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• proposed sequence

based on simulations

Modeling

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A. Aksimentiev et al. Biophysical Journal 86 ,1332 (2004)

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Summary of ATP Synthase• rotary molecular motor

• rotation driven by proton current

• energy from stored electrochemical potential

• work: creation of ATP + dissipation

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Molecular Motors 4.Actin & Myosin

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Biological transport• all organisms (eukaryotic cells of yeasts, plants, animals) contain

‘motor proteins’

• two well known examples:

– kinesins• two active heads• hydrolyses ATP• moves along microtubules

– myosins• two active heads• hydrolyses ATP• moves along actin fibers


Muscle force and movement

F=5.3pN /cross-bridgestep-size=5.5nm1 step/ATPase reaction

myosin filament length=1.55µm

C.J. Pennycuick, Newton Rules Biology (Oxford, 1995)

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Summary of Actin &Myosin

• processive motor (unidirectional translation)

• hand-over-hand motion

• driven by chemical energy of ATP

• work: translation of cargo against force, dissipation

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Molecular Motors 5.LC elastomers

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director

If orientational order increases,

expansion contraction

If orientational order decreases,

expansioncontraction

LC elastomer network changes shape

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Summary of LC elastomers

• elongational motor:

– change in orientational order produces stress

– stress produces change in shape

• driven by heat, light, or impurities

• work: motion against force, dissipation

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Broken Symmetry

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Proper- and pseudo-tensors

tensorrank

0 1 2 3 4

proper

pseudo

π x

( )×A B C

E DP, , ,

, ,

,,

H M B

αβδ Qαβ

αβΓ

αβεαβµ

ijkε

dαβγ

scalar vector

etc.

changes sign on inversion

does not change sign on inversion handedness =pseudoscalar c*!

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Symmetry and motors

• symmetry of the system determines motion of motor!

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Translational motion

My rugs are alive and moving!?

I have wall to wall carpets in my apartment. I have placed large rugs on top of the carpet. Without damaging the carpet or the rug, how can I stop them from wrinkling and moving? Is it caused by static electricity?

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http://global.ard.yahoo.com/SIG=15pdj7aqe/M=650008.12773057.13811805.8356343/D=know/S=2115300224:HEAD/Y=YAHOO/EXP=1303359552/L=5_0nqEWTZOOmFvjoTa8ZBwO3g3voU02vlB8ADIyB/B=Rl5KeEwNPLA-/J=1303352352127986/K=j4iIFF_nUC9tmxsn_pqtzw/A=5856910/R=11/SIG=10qrh7h9e/*http:/answers.yahoo.com



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Translational motion



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tilted strands




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Translation



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tilted strands




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Translation



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tilted strands




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Translation



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tilted strands




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Translation



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tilted strands




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Translation



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tilted strands




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Translation



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tilted strands




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Symmetry and motors

• symmetry of the system determines motion of motor!

• broken left-right symmetry: – proper vector exists in system – translation in direction of vector (carpet, kinesin, myosin)

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Symmetry and motors• symmetry of the system determines motion of motor!

• broken translational symmetry: – proper vector exists in system – translation in direction of vector (carpet, kinesin, myosin)[Q: is there a vector in Landauer's blow torch?]

• broken chiral symmetry – pseudoscalar c* exists in system

• broken chiral + broken translational symmetry– pseudoscalar + proper vector = pseudovector– rotation (Yokoyama – Tabe, ATP Synthase, etc.)

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Symmetry and motors• if system has proper dyad (director)

– strain (elongation)

• if system has proper dyad and gradient (vector), – proper vector (curvature, bend)

• if system has proper dyad, gradient & pseudoscalar– pseudovector (twist)

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Symmetry & deformation/motionSymmetry can construct deformation/motion material

vector translation in kinesin, actin

dyad uniaxial strain nematic networks

c* - -

c* pseudovector rotation Lehman, ATP, Tabe

c* pseudovector twist LCE twist

proper vector bend LCE photoactuation

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Connecting molecular & macrosopic scales• cumulative effect of many molecular motors →

macroscopic response– rotating dye system, Yokoyama-Tabe expt:

• distributed local vorticity → bulk torque & bulk flow

– ATP synthase:• produces bulk ATP

– myosin & actin• muscle fibers form long muscles, these contract

– effective shape change• contraction, elongation and bend and twist

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Smooth shape change: elongation

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• liquid crystal elastomers:– coupling of orientational order and strain– large mechanical response to excitations

• elongation due to temperature change

• motor based on elongation

H. Finkelmann, nematic LCE

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Another motor based on elongation • rubber band heat engine

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Another motor based on elongation

• collagen contracts in salt water

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Aharon Katzir-Katchalsky 1913-1972

http://en.wikipedia.org/wiki/File:Collagentriplehelix.png

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Another motor based on elongation

• a modified version

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M. V. Sussman and A. Katchalsky, Science, 167, 45 (1970)

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Dunking bird of the first kind• heat engine

– Carnot efficiency limit

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Dunking bird of the second kind• not a heat engine

• isothermal

• involves shape change

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N. Abraham, P. Palffy-Muhoray., Am.J. Phys. 72, 782 (2004)

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Bend Motors• samples of liquid crystal elastomers with azo dye

bend on exposure to light

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Yanlei Yu, Makoto Nakano, Tomiki Ikeda, Nature 425, 125 (2003)

timescale: 10 s

LC + diacrylate network+ functionalizedazo-chromophore

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Water vapor sensitive network

K. Harris, C. Bastiaansen, D. Broer, J. MEM Syst., 16, 480 (2007)

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Bend Motors

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sample: nematic elastomer EC4OCH3+ 0.1% dissolvedDisperse Orange 1 azo dye 5mm

300 mµ

Response time: 70ms

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• if sample bends both sides are illuminated, producing oscillations

Photoinduced oscillations

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nematic azo-elastomer

S. Serak, N. Tabiryan, R. Vergara, T. White, R. Vaia, T. Bunning, Soft Matter 6, 779–783 (2010)

90o>

max. frequency = 270Hz

sample size: 5 mm 0.8 mm 0.05 mm

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Bend Motors

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• floating nematic LCE sample illuminated from above

Laser beam

container

waterElastomer

M. Chamacho-Lopez, H. Finkelmann, P. Palffy-Muhoray, M. Shelley, Nature Mat. 3, 307, (2004)

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Bend Motors

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• robotic arm

CLCP Layers (Homogeneous)

CLCP Layers(Homeotropic) PE Film

T. Ikeda, Chuo University (priv. comm.)

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Bend Motor: focus of current interest

• azo-dye doped liquid crystal elastomer

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M. Yamada, M. Kondo, J. Mamiya, Y. Yu, M. Kinoshita, C. Barrett, T. Ikeda, Angew. Chem. 47, 4986 (2008)

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Bend Motor: focus of current interest• shape-memory Ni-Ti alloy,

– two phases:

• orthorhombic martensite (cold)

• cubic austenite (hot)

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Existing rotary bend motors

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LCE bends on exposure to light

wire becomes stiff when heated

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• bend vs. compression:– if ends are brought closer, will filament compress or bend?

Model

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d R

L

L L−∆

21 ( )24

L LL R∆

21/ ( )2 12b

d wdl ER

= 21/ ( )2c

Ll E wdL∆

=2 2~ ( ) ( )c

b

L Ld R

Bend is much cheaper energetically than stretch;→ length is constant.

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Model

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1s

3s

4s

2s

xsos

2θ

3θ1θ4θ

2R1R

arbitrary shape

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Model

– arc length along filament

– position vector of point on filament

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1s

3s

4s

2s

xsos

2θ

3θ1θ4θ

2R1R

s

( )sR z

x

y

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Model

– bend energy density:

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2 21 1/2b

cl EIR R

= =

1 2

1

3 4

2 3 4

22 2

21

22 2 2

22 1

1 ˆ( ) ( ( ))

1 1ˆ( ) ( ( ) ( ) ,

o

o

s s

s s

s s s

s s s

c ds c dsR s s

c ds c ds c dsR s s R

κ κ

κ κ κ

∂ ∂= − + − × +

∂ ∂

∂ ∂− + − × + −

∂ ∂

∫ ∫

∫ ∫ ∫

R R z

R R z

1s

3s

4s

2s

xsos

2θ

3θ1θ4θ

2R1R

stiffness natural curvaturefn. of positionfn. of position

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Model

– constraints:

• filament length is constant

• displacement in x-dir. is fixed

• displacement in y-dir. is fixed

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1s

3s

4s

2s

xsos

2θ

3θ1θ4θ

2R1R

2

11 1 2 2 1 2

ˆ ˆ(1 cos ) (1 cos ) ,s

sR ds R L R Rθ θ− − ⋅ + − = + +∫ t x

4

31 4 2 3 1 2

ˆ ˆ(1 cos ) (1 cos ) .s

sR ds R L R Rθ θ− + ⋅ + − = + +∫ t x

2

11 1 2 2

ˆ ˆsin sin 0,s

sR ds Rθ θ+ ⋅ − =∫ t y

4

31 4 2 3

ˆ ˆsin sin 0s

sR ds Rθ θ+ ⋅ − =∫ t y

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Model: Energy with constraints

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1 2 3

1 2

4

3 4

2

1

2

1

2 2 2

01 2

12 2

1

211 1 2 1 2

1 2

211 1 2

1 2

1 1ˆ ˆ ˆ( ) ( ( )) ( )

1ˆ ˆ ˆ( ( )) ( )

ˆ ˆ{ (1 cos ) (1 cos( )) ( )}

ˆ ˆ{ sin sin(

s s s

s s

s

s s

s xx s

s xy s

c ds c ds c dsR R

c ds c dsR

s ssf R ds R L R RR R

s ssf R ds RR R

κ κ κ

κ κ

′

′

= − + − + −

+ − + −

−+ − − ⋅ + − − + +

−+ + ⋅ −

∫ ∫ ∫

∫ ∫

∫

∫

t t z

t t z

t x

t y

×

×

4

3

4

3

342 1 2 1 2

1 2

342 1 2

1 2

)}

1 ˆ ˆ{ (1 cos( )) (1 cos( )) ( )}

1 ˆ ˆ{ sin( ) sin( )}

s xx s

s xy s

s ssf R ds R L R RR R

s ssf R ds RR R

−−+ − + ⋅ + − − + +

−−+ + ⋅ −

∫

∫

t x

t y

1s

3s

4s

2s

xsos

2θ

3θ1θ4θ

2R1R

Lagrange multipliers are components of forces in top and bottom filaments.

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Model: Energy minimization

• ODEs are obtained by minimizing w.r.t.

• these govern the shape of the filaments on top and bottom

– BC for is obtained by minimizing w.r.t.

– this gives continuous tangent and curvature at contact pts.

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ˆ (cos ,sin )θ θ=t

1 1 1 22( ( ) ) sin cos 0, if ( , )x yc c c f f s s sθ θ κ θ θ′′ ′ ′ ′− + + + + = ∈

1s

3s

4s

2s

xsos

2θ

3θ1θ4θ

2R1R

2 2 3 42( ( ) ) sin cos 0, if ( , ).x yc c c f f s s sθ θ κ θ θ′′ ′ ′ ′− + + − + = ∈

θ 1 2 3 4, , ,s s s s

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Numerical Solution1. Assign initial values for and and solve ODE

with BCs.

2. Assign initial values for and and solve ODE

with BCs.

3. Vary the forces to satisfy constraints (root finding)

4. Vary and go to 1. until

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1 2, , xs s s 1 1,x yf f

1 12( ( ) ) sin cos 0x yc c c f fθ θ κ θ θ′′ ′ ′ ′− + + + + =

3 4,s s 2 2,x yf f

2 22( ( ) ) sin cos 0x yc c c f fθ θ κ θ θ′′ ′ ′ ′− + + − + =

1 1 2 2, , ,x y x yf f f f

1 2 3 4, , , , xs s s s s 0, 1,...4, .i

i xs∂

= =∂

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Numerical Solution: varying the curvature

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local spontaneous curvature

κ

s

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Numerical Solution: varying the stiffness

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local stiffening

E

s

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Calculating torques

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– to calculate the net torque, we consider part of system

– look at forces

– look at point torques

1xf1yf

2 yf

2xf

1τ

2τ

ˆ ˆ ˆ2 ( ( ))cτ κ′= −t t z×

ODE is precisely the condition that the torque on wheelremain constant, regardless of where the filament is cut!

1 1 10 02 ( ) sin ( ) cos ( ) , (upper part)

s s

x yc f s ds f s ds Kθ κ θ θ′ + − − =∫ ∫

Note: integrating the ODE w.r.t. to s gives

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What happens on the pulleys?

• Problem: elastic beam on rigid cylinder

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Interesting observation– straight beam touches cylinder along one line

– curved beam touches cylinder over extended area

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F F Transition?

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Experiment

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Determination of point of contact

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Data

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0 10 20 30 40 50

-80

-60

-40

-20

0

20

40

60

80Co

ntac

t Ang

le (d

eg.)

Force (N)

Contact angle vs. total force

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Modelling: as before

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NOT a pitchfork bifurcation!

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Force distribution on pulleys

• when force is applied to filament,

– point contact if – extended contact if

• torque balance:

• Volterra integral equation of first kind.

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force density on pulley

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Force distribution on pulleys

• When force is applied to filament,

– point contact if – extended contact if

• Solution for force density:

where

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F

cF F≤

cF F> cEIFLR

=

( ) ( ( ) cos sin )c c cFfR

θ δ θ θ θ θ= − +

sin ( )cc

F F LF R

θ −=

R

F Fθ

L

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Force distribution

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How does rotation come about?

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• away from the small wheel, – longer lever arm on one side

• near small wheel– greater point torque on one side

• curvature is the same, – but stiffer on one side

– have pontaneous curvature on one side

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• away from the small wheel, – longer lever arm on one side

• near small wheel– greater point torque on one side

• curvature is the same, – but stiffer on one side

– have pontaneous curvature on one side

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Motion due to point torques• how can a point torque in filament cause translation?

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Motion due to point torques• how can a point torque in filament cause bulk translation?

– net force is zero,

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1F

2F

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– net force is zero, but torque causes constraint force to appear

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1F

2F

3F

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– net force along

– translation

• angular momentum transport!

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1F

2F

3F

1F

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– net force along

– translation

• no rotation without friction between filament & wheel!

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1F

2F

3F

1F

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• point torque in filament causes constraint force F to appear*

• reaction force -F appears;exerts torque on hand

• angular momentum currentflows into bend motor from hand

• * via stress transport:

Angular momentum current

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v

-F2

2( )E tρ ∂

∇ ∇ =∂σσ

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Summary• detailed model for bend motors driven by

– local spontaneous curvature caused by light (LCE-Ikeda)– local stiffening caused by heat (Nitinol - Wang)

• corrects/completes existing qualitative explanations

• prediction: – LCE motor with perp. alignment runs the other way:

confirmed!

• understand angular momentum transport

• efficiency calculation: remains to be done

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Angular momentum

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NN

stimulus is applied

filament bends

torque appears

angular momentum flows into motor

rotation starts

angular momentum is conserved &earth speeds up

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Conclusions• variety of molecular -scale mechanisms can be exploited

– rotary (dyes, Yokoyama-Tabe)– need better molec. motors with processive translation – elongation (LC elastomers/networks)

• symmetry plays important role in determining motion

• macroscopic phenomena are cumulative effects of molecular motors

• understand bend motors well

• many exciting possibilties to explore!

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Documents

Soft Motors: molecular to macroscopic perspectives · PDF filetorque results from ... • steady unidirectional rotation • orientational ratchet / Landauer’s blowtorch mechanism