Zhigang Suo
Harvard University
Work with
X. Zhao, W. Hong, J. Zhou, W. Greene
1

L
A
Dielectric
Elastomer
Compliant
Electrode
Reference State l a
+
Q
−
Q
Current State
Φ
Pelrine, Kornbluh, Pei, Joseph
High-speed electrically actuated elastomers with strain greater than 100%.
Science 287, 836 (2000).
2

•Large deformation
•Compact
•Lightweight
•Low cost
•Low-temperature fabrication
Kofoda, Wirges, Paajanen, Bauer
APL 90 , 081916, 2007
3

P
−
Q
E
+
Q
σ =
ε
0
2
E
2
σ ij
= ε
0
E j
E i
−
ε
0
2
E k
E k
δ ij
P
A field of forces needed to maintain equilibrium of a field of charges
F
= i qE i
Electrostatic field
E i
= −
∂ Φ
∂ x i
∂
E i
∂ x i
= q
ε
0
F i
=
∂
∂ x j
ε
0
E j
E i
−
ε
0
2
E k
E k
δ ij
4

Include Maxwell stress in a free-body diagram
Φ
1
2
ε
0
E
2 h ρ gh
“Free-body” diagram
ρ gh
=
1
2
( ε − ε
0
)
E
2
1
2
ε
E
2
5

Trouble with Maxwell stress in dielectrics
±
- - - - - - - - - - - -
+ + + + + + + + +
Maxwell stress
σ
33
= −
ε
2
E
2
- - - - - - - - - - - -
+
-
+ + + + + + + + +
Electrostriction
Our complaints:
•In general,
ε varies with deformation.
•In general, E 2 dependence has no special significance.
•Wrong sign of the Maxwell stress?
In solid, Maxwell stress is not even wrong; it’s a bad idea.
Suo, Zhao, Greene, JMPS (2007)
6

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
7

E i
= −
∂ Φ
∂ x i
+Q +Q
In a vacuum , force is needed to maintain equilibrium of charges
Define electric field by E = F/Q
F
= i qE i
∂
E i
∂ x i
= q
ε
0
+Q
+Q
In a dielectric , force between charges is
NOT an operational concept
F
= i qE i
Historical work
•Toupin (1956)
•Eringen (1963)
•Tiersten (1971)
……
Recent work
•Dorfmann, Ogden (2005)
•Landis, McMeeking (2005)
• Suo, Zhao, Greene (2007)
……
8

Volume II, p.10-8 (1964)
“What does happen in a solid? This is a very difficult problem which has not been solved, because it is, in a sense, indeterminate. If you put charges inside a dielectric solid, there are many kinds of pressures and strains. You cannot deal with virtual work without including also the mechanical energy required to compress the solid, and 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.”
9

All troubles are gone if we use measurable quantities
L
Reference State
A l
Current State a
+
Q
Φ
−
Q
P
Battery does work
Φ δ
Q
λ = l / L s
=
P / A
~
E
= Φ
/ L
~
D
=
Q / A
Weight does work P
δ l
For elastic dielectric , work fully converts to free energy:
δ
U
=
P
δ l
+ Φ δ
Q
Material laws
δ
U
AL
=
P
δ l
+
AL
Φ δ
Q
LA
Suo, Zhao, Greene, JMPS (2007) s
=
∂
W
( )
∂ λ
δ
W
= s
δλ +
~
E
δ
~
D
~
E
=
∂
W
∂
λ
~
D
,
10

• Extend the theory to 3D.
• Construct free-energy function W.
• Study interesting phenomena.
• Add other effects (stimuli-responsive gels).
11

Linear PDEs
Suo, Zhao, Greene, JMPS (2007)
A field of weights F iK
A field of batteries
~
E
K
δ
Q
Φ
=
∂ x i
∂
X
K
,
= −
∂ Φ
∂
X
K
,
P δ l
∫ s iK
∂ ξ i
∂
X
K dV
= ∫ ~ b i
ξ i dV
+ ∫ t
~ i
ξ i dA
∂ s iK
∂
X
X ,
K
~ b i
X ,
=
0
( s
− iK
( ) − s
+ iK
( ) )
N
K
( ) = t
~ i
( )
∫
⎛
⎜⎜
−
∂ η
∂
X
K
⎞
⎟⎟
~
D
K dV
= ∫ η q dV
+ ∫ η ω dA
∂
~
D
K
∂
X
X ,
K
=
X ,
(
~
D
K
+
( )
−
~
D
K
−
( )
)
N
K
( ) ( )

Elastic dielectric , defined by a free energy function W
Free energy of the system
δ
W
=
∂
W
∂
F iK
δ
F iK
+
∂
W
∂
( )
~
D
K
,
δ ~
D
K
δ
Q
Φ
Free energy of dielectric
δ
G
= ∫ δ
WdV
Potential energy Potential energy
− ∫ of weights
~ b i
δ x i dV
− ∫ t
~ i
δ x i dA
− ∫ of batteries
Φ δ q dV
− ∫ Φ δ ω dA
P
A little algebra
δ
G
= ∫
⎣
⎢
⎡ ∂
W
∂
F iK
− s iK
⎦
⎥
⎤
δ
F iK dV
+ ∫
⎣
⎢
⎡ ∂
W
∂
F
~
( )
D
K
,
−
~
E
K
⎦
⎥
⎤
δ
~
D
K dV
δ
G
=
0
δ
F
δ
~
D
Material laws s iK
=
∂
W
∂
F iK
,
~
E
K
=
∂
W
∂
F
~
D
( )
K
,
13
δ l

L
Reference State Current State
A l a
+
Q
−
Q
P
Φ
~
E
= Φ
/ L
~
D
=
Q / A
( E
= Φ
/ l )
( D
=
Q / a )
Nominal electric field and nominal electric displacement are work-conjugate
Battery does work
Φ δ
Q
=
( ) ( )
=
( ) ~
E
δ
~
D
True electric field and true electric displacement are NOT work-conjugate
Battery does work
Φ δ
Q
=
( ) ( ) ( )
E
δ
D
+
EDl
δ a
14

L
Reference State
A s
=
P / A
~
E
= Φ
/ L
~
D
=
Q / A
σ =
P / a
E
= Φ
/ l
D
=
Q / a l
Current State
Φ a
+
Q
−
Q
P
σ ij
=
F det jK
( ) s iK
E i
=
H iK
~
E
K
D i
=
F iK det
( )
~
D
K
( H
=
F
−
1
)
15

Dielectric constant is insensitive to stretch
Kofod, Sommer-Larsen, Kornbluh, Pelrine
Journal of Intelligent Material Systems and Structures 14 , 787-793 (2003).
16

Zhao, Hong, Suo, Physical Review B 76, 134113 (2007).
W
( )
=
W s
( )
+
D
2
2
ε
Stretch Polarization
W s
=
µ
2
(
λ
2
1
+ λ
2
2
+ λ
2
3
−
3
)
D i
=
F iK det
( )
~
D
K
~
E
K
D i
= ε
E i
=
∂
W
∂
F
~
D
( )
K
,
σ ij
=
F det iK s iK
∂
W s
∂
F jK
=
∂
W
∂
F iK
+ ε
E i
E j
−
1
2
E k
E k
δ ij
⎞
⎠ 17

l l
Φ
+
Q
−
Q
W
( )
=
µ
2
(
λ +
2
λ −
1 −
3
)
+
λ
2
~
D 2
ε s
=
∂
W
( )
∂ λ
,
~
D
=
0
Q
λ =
⎛
⎝
1
+
~
D
2
εµ
⎞
⎟⎟
−
1 / 3
Φ
~
E
=
∂
W
∂
( )
~
D
,
~
~ D
E
=
ε
λ
2
µ
~
E
/
ε
=
~
D
εµ
⎛
⎜⎜ 1
+
~
D
2
εµ
⎞
⎟⎟
−
2 / 3
Q
~
E c
~
µ
ε
~
10
6
10
−
10
N
F
/ m
/ m
=
10
8
V / m
Stark & Garton, Nature 176, 1225 (1955).
Zhao, Suo, APL 91, 061921 (2007) 18

Experiment : Pelrine, Kornbluh, Pei, Joseph
Science 287, 836 (2000).
Theory : Zhao, Suo
APL 91, 061921 (2007)
19

l
Φ
Φ
+
Q
−
Q thick thin
Q
Top view
Cross section
Coexistent states: flat and wrinkled
Experiment : Plante, Dubowsky,
Int. J. Solids and Structures 43 , 7727 (2006).
Theory : Zhao, Hong, Suo
Physical Review B 76, 134113 (2007)..
20

W
( )
=
W s
( )
+
D
2
2
ε
Stretch Polarization
Stiffening as each polymer chain approaches its fully stretched length
(e.g., Arruda-Boyce model)
W s
= µ
⎡
⎢⎣
1
2
(
I
−
3
)
+
1
20 n
(
I
2
−
9
)
+
...
⎤
⎥⎦
I
= λ
2
1
+ λ
2
2
+ λ
2
3
µ
: small-strain shear modulus n: number of monomers per chain
21

l
+
Q
−
Q
Φ
Φ
Q
22
Zhao, Hong, Suo, Physical Review B 76, 134113 (2007).

Thick State
Transition
Thin State
Thin State
Transition
Thick State
23
Zhou, Hong, Zhao, Zhang, Suo, IJSS, 2007

collapsed reversible
Gel
•long polymers (cross-linked but flexible)
•small molecules (mobile) swollen
Stimuli
•temperature
•electric field
•light
•ions
•enzymes
24
Ono et al, Nature Materials, 2007

Contact lenses
Artificial tissues
Gates in microfluidics
Drug delivery
25

• A nonlinear field theory. No Maxwell stress. No electric body force.
• Effect of electric field on deformation is a part of material law.
• Ideal dielectric elastomers: Maxwell stress emerges.
• Electromechanical instability: large deformation and electric field.
• Add other effects (solvent, ions, enzymes…)
26