Soft Active Materials

Zhigang SuoHarvard University

James F. Bell1914-1995

The James F. Bell Memorial Lecture in Continuum MechanicsDepartment of Mechanical Engineering, Johns Hopkins University5 November 2009 1

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Professors Sharp and Bell (1983)

Professor Bell (about 1990)

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elastomer = networkgel = network + solvent

reversible

solvent

networkgel

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Super absorbent diaper

Sodium polyacrylate: polyelectrolyte

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Gels regulate flow in plants

Missy Holbrook

Zwieniecki, Melcher, Holbrook,Hydrogel control of xylem hydraulic resistance in plants Science 291, 1095 (2001)

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Self-regulating fluidics

David Beebe

•pH•Salt•Temperature•light

Responsive toPhysiological variables:

•Many stimuli cause deformation.•Deformation regulates flow.

Beebe, Moore, Bauer, Yu, Liu, Devadoss, Jo, Nature 404, 588 (2000)

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octopus

Mäthger, Hanlon, Kuzirian – Marine Biological Laboratory, Woods Hole MA

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Squid changes color

Expand pigmented sacs by contracting muscles

Mathger, Denton, Marshall, Hanlon, J. R. Soc. Interface 6, S149 (2009)

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Adaptive OpticsAh, optics!

•Many stimuli cause deformation.•Deformation affects optics.

Wilbur, Jackman, Whitesides, Cheung, Lee, PrentissElastomeric opticsChem. Mater. 1996, 8, 1380-1385

Aschwanden, StemmerOptics letters 31, 2610 (2006)

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Soft Active Materials (SAM)

Soft: large deformation in response to small forces(rubbers, gels,…)

Active: large deformation in response to diverse stimuli(electric field, temperature, pH, salt,…)

A stimulus causes deformation. Deformation provides a function.

StimulipH, E, T, C…

Functionsoptics, flow…

SAMdeforms

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Very well, but how does stimulus X cause deformation?

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Dielectric elastomer

Reference State Current State

Φ

l

a Q+

Q−Compliant Electrode

Dielectric Elastomer

L

A

Pelrine, Kornbluh, Pei, Joseph High-speed electrically actuated elastomers with strain greater than 100%. Science 287, 836 (2000).

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Parallel-plate capacitor P

Φ

Q+

Q−batterya

electrode

lvacuum

electrode

P force

ED 0ε=

ε0, permittivity of vacuum

202

1Eεσ = Maxwell stress

lE

Φ=

a

QD =

a

P=σ

Electric field

Electric displacement field

stress field

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Field equations in vacuum, Maxwell (1873)

ii x

E∂Φ∂

−=0εq

x

E

i

i =∂∂

Electrostatic field

ii qEF =A field of forces maintain equilibrium of a field of charges

⎟⎠⎞

⎜⎝⎛ −

∂∂

= ijkkijj

i EEEEx

F δεε20

0

ijkkijij EEEE δεεσ20

0 −=

Q−

20

2E

εσ =

P

E

Q+ Maxwell stressP

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James Clerk Maxwell (1831-1879)

“I have not been able to make the next step, namely, to account by mechanical considerations for these stresses in the dielectric. I therefore leave the theory at this point…”

A Treatise on Electricity & Magnetism (1873), Article 111

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Trouble with Maxwell stress in dielectrics

+ + + + + + + + +

- - - - - - - - - - - - - - - - - - - - - - - -

+ + + + + + + + +

+-

Maxwell stressElectrostriction2

33 2Eεσ −=

±

Our complaints:

•In general, ε varies with deformation.•In general, E2 dependence has no special significance.•Wrong sign?

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Trouble with electric force in dielectrics

In a vacuum, external force is needed to maintain equilibrium of charges

+Q +Q

P Pii qEF =

+Q +Q

In a solid dielectric,force between charges is NOT an operational concept

ii qEF =

The Feynman Lectures on PhysicsVolume II, p.10-8 (1964)

“It is a difficult matter, generally speaking, to make a unique distinction between the electrical forces and mechanical forces due to solid material itself. Fortunately, no one ever really needs to know the answer to the question proposed. He may sometimes want to know how much strain there is going to be in a solid, and that can be worked out. But it is much more complicated than the simple result we got for liquids.”

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All troubles are gone if we use measurable quantities

Current StateReference State

L

A

P

l

a Q+

Q−

Φ

QlPF δδδ Φ+=equilibrate elastomer and loads

Nominal

Ll /=λ

APs /=

LE /~

Φ=

AQD /~=

aP /=σ

lE /Φ=

aQD /=

( )ALFW /=

DEsW~~δδλδ +=

LA

Q

AL

lP

AL

F δδδ Φ+=divide by volume

name quantitiesTrue

( )λλ∂

∂=

DWs

~, ( )

D

DWE ~

~,~

∂∂

=λ

equations of state

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Fine, Zhigang, but what is ?( )DW~

,λ

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Dielectric constant is insensitive to stretch

VHB 4910

Kofod, Sommer-Larsen, Kornbluh, PelrineJournal of Intelligent Material Systems and Structures 14, 787 (2003).

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Ideal dielectric elastomer

( ) ( )ε

λλλλ2

,~

,,2

2121

DWDW stretch +=

Elasticity Polarization

a

QD =

A

QD =~

21

~

λλD

D =

1321 =λλλ

Dielectric behavior is liquid-like, unaffected by deformation.

For an ideal dielectric elastomer, electromechanical coupling is purely a geometric effect:

incompressibility

1

11

1λ2λ

3λ

Zhao, Hong, Suo, Physical Review B 76, 134113 (2007)

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Ideal dielectric elastomer

2P 1P

22Lλ

33Lλ

11LλΦ

Q

2P 1P

22Lλ

33Lλ

11LλΦ

Q( ) ( ) 22

21

22

22

122

2121 2

~3

2

~,, −−−− +−++= λλ

ελλλλµλλ D

DW

( )1

211

~,,

λλλ

∂∂

=DW

s( )

2

212

~,,

λλλ

∂∂

=DW

s

In terms of nominal quantities

In terms of true quantities

( ) 222

21

211 Eελλλµσ −−= −− ( ) 22

22

1222 Eελλλµσ −−= −−

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Young man, don’t be too fast. The theory sounds fine, but can you relate the theory to experiments?

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Two modes of failureElectromechanical instabilitySoft material

Φ

Q

l

Q

Φ

polarizing thinning

Q+

Q−l

Q

Φ

polarizing

Electrical breakdownHard material

Stark & Garton, Nature 176, 1225 (1955)

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A dilemma

•To deform appreciably without electrical breakdown, the elastomer must be soft.

•But a soft elastomer is susceptible to electromechanical instability.

How large can deformation of actuation be?

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Electromechanical instability limits deformation of actuation

ΦQ+

Q−

1λ

λ1

26.12 3/1 ≈=cλ

Experiment: Pelrine, Kornbluh, Joseph, Sensors & Actuators A 64, 77 (1998)

Theory: Zhao & Suo Appl. Phys Lett. 91, 061921 (2007)

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Pre-stretch increases deformation of actuation

2P 1P

22Lλ

33Lλ

11LλΦ

Q

2P 1P

22Lλ

33Lλ

11LλΦ

Q

Theory: Zhao, SuoAPL 91, 061921 (2007)

Experiment: Pelrine, Kornbluh, Pei, JosephScience 287, 836 (2000).

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Coexistent states stiffening

Φ

l

Q+

Q−

Interpretation: Zhao, Hong, SuoPhysical Review B 76, 134113 (2007)

Coexistent states: flat and wrinkledObservation: Plante, Dubowsky, Int. J. Solids and Structures 43, 7727 (2006)

Qthick thin

Φ

thinningpolarizing

Cross sectionTop view

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When stretched, a polymer stiffens

n links

nb

n

λ

Langevin

Gauss

releasedb

P

P

stretched

P

fully stretched

bnPP

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Stiffening stabilizes elastomer

( ) ( )( ) ⎥⎦⎤

⎢⎣⎡ +−+++−++ ...9

20

13

2

1 223

22

21

23

22

21 λλλλλλµ

n( )3

223

22

21 −++ λλλµ

LangevinGauss

Arruda, Boyce, J. Mech. Phys. Solids 41, 389 (1993)

Q

Φ

Φ

l

Q+

Q−

Zhao, Hong, SuoPhysical Review B 76, 134113 (2007)

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Giant deformation of actuation?Φ

λ

λ

E

BE

1

1

ΦQ+

Q−

λ

λ

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Interpenetrating networks

•Short chains provide a safety net.•Long chains fill the space.

Experiment: Ha, Yuan, Pei, PelrineAdv. Mater. 18, 887 (2006).

Theory: Suo, Zhu. Unpublished

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3D inhomogeneous field

dAdVqdAxTdVxBWdV iiii ∫∫∫∫∫ Φ+Φ++= δωδδδδ

Φ

P

Qδ

lδ

Need to specify a material model

( )DF~

,W( )DW~

,λ

( ) KKiKiK DEFsW~~~

, δδδ +=DF

( )iK

iK FW

s∂

∂=

DF~

, ( )K

KD

WE ~

~,~

∂∂

=DF

Condition of equilibrium

PDEs

( )K

K Xt

E∂Φ∂

−=,~ X( )

K

iiK X

txF

∂∂

=,X

( ) ( ) 0,,

=+∂

∂tB

X

tsi

K

iK XX ( ) ( )tq

X

tD

K

K ,,

~X

X=

∂∂

( )K

KD

WE ~

~,~

∂∂

=DF( )

iKiK F

Ws

∂∂

=DF~

,

Toupin (1956), Eringen (1963), Tiersten (1971), Goulbourne, Mockensturm and Frecker (2005), Dorfmann & Ogden (2005), McMeeking & Landis (2005)…

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Suo, Zhao, Greene,J. Mech. Phys. Solids 56, 467 (2008)

The nominal vs the true

Reference State Current State

P

Φ

l

a Q+

Q−L

A

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( ) iKjK

ij sF

Fdet=σ

( ) KiK

i DF

D~

det F=

KiiK EEF~

=

APs /=

LE /~ Φ=

AQD /~ =

lE /Φ=

aQD /=

aP /=σ

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Ideal dielectric elastomersLiquid-like dielectric behavior, unaffected by deformation

( ) ( )ε2

~,

2DWW s += FDF

Stretch Polarization

( ) KiK

i DF

D~

det F=

Ideal electromechanical coupling is purely a geometric effect:

( ) ( )iK

iK F

Ws

∂∂

=DF

DF~

,~,( ) ( )

K

KD

WE ~

~,~

,~

∂∂

=DF

DF

( )( )

⎟⎠⎞

⎜⎝⎛ −+

∂∂

= ijkkji

jK

siKij EEEE

F

WF δεσ2

1

det

F

Fii ED ε=

Zhao, Hong, Suo, Physical Review B 76, 134113 (2007)

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Programmable deformation

Design:Kofod, Wirges, Paajanen, BauerAPL 90, 081916 (2007)

Simulation:Zhao, SuoAPL 93, 251902 (2008)

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Electrostriction

+ + + + + + + + +

- - - - - - - - - - - - - - - - - - - - - - - -

+ + + + + + + + +

+-

Maxwell stressElectrostriction2

33 2Eεσ −=

±

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Non-ideal dielectric elastomer

( )[ ]31 321 −+++= λλλεε c

Die

lect

ric c

onst

ant

Area ratio

c=-0.062

deformation affects dielectric constant

Wissler, Mazza, Sens. Actuators, A 138, 384 (2007).

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Quasi-linear dielectric elastomer

( )( )31 321 −+++= λλλεε c

( ) ( )ε

λλλµλλ2

32

~,,

223

22

2121

DDW +−++=

( )1

211

~,,

λλλ

∂∂

=DW

s

Zhao, Suo, JAP 104, 123530 (2008).

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Energy harvesting

Generate electricity from walking Generate electricity from waves

Pelrine, Kornbluh…

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Maximal energy that can be converted by a dielectric elastomer

R

LT

EB

E=0

EMI

6 J/g, Elastomer (theory)0.4 J/g, Elastomer (experiment)0.01 J/g ceramics

Really that large? Someone should do more experiments.

Koh, Zhao, Suo, Appl. Phys. Lett. 94, 262902 (2009)

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Summary• Convergence of the two worlds (soft biology, hard engineering).

• Soft active materials (SAMs) have many functions (soft robots, adaptive optics, self-regulating fluidics, programmable haptics, oil recovery, energy harvesting, drug delivery, tissue regeneration, low-cost diagnosis…).

• Mechanics of SAMs is interesting and challenging (mechanics, electricity, chemistry; large deformation, mass transport, instability).

• The field is wide open (fabrication, computation, application).

3 lectures on SAMs: http://imechanica.org/node/3215•Dielectric elastomers•Gels•Polyelectrolytes