Electric Charge / Deformation
and Polarization
Matt Pharr
ES 241
5/21/09
Electric Charge
Total charge is conserved
QA= 0
B
A
+QB= 0
Qnet= 0
10 protons, 10 electrons
QA= +4 A
Net charge = 10 - 10 = 0
+QB= -4
SI Units: 1 Coulomb =
Qnet= 0
= 6.242 * 1018 elementary charges
B
Capacitor
F’(Q’)
F  F Q 
F Q 
F 
Q
Q
Φ
SI Units: 1 Volt
= 1.602e-19 Joule
Q '
F’’(Q’’)
F’’(Q’’)
Q 'Q ' '  const  Q '  Q ' '
Fcomp  F ' Q '  F ' ' Q ' '
F ' Q ' 
F ' ' Q ' '
Fcomp 
Q '
Q ' '  0
Q '
Q ' '
For
F ' ' Q ' '  F ' Q '

: Q '  0
Q ' '
Q '
Measurement of Electric Potential
V?
R
I
Current measured with galvanometer
Ohm’s Law: V = IR gives the potential
A Capacitor, a Weight, and a Battery
F  F l , Q 
F l , Q 
F l , Q 
F 
l 
Q
l
Q
Mechanical work
Electric work
In equilibrium
P
Pl
 Q
F  Pl   Q
F l , Q 
F l , Q 
, 
l
Q
F(l,Q) and Stress
a
l
Experimental Relation
Q


a
l
Φ
Recall:
-Q
+Q
oil
Electric field E 
Electric displacement D 

l
Q
a
Stress field   P
a
F l , Q 
lQ 2

 F l , Q  
Q
2a
F l , Q  Q 2
P

l
2a
Maxwell Stress
z 
E 2
2
Deformable Dielectrics
Reference State
Current State
A
a
L
l
Q
P
Stretch   l
L
Nominal ~ 
electric field E  L
Nominal electric ~ Q
D
displacement
A
Nominal stress
P
s
A
Nominal freeF
W
energy density
AL

Q
F  Pl   Q
F P l  Q


AL A L L A
~ ~
W  s   E D

~
W  , D
s



~
~ W  , D
E
~
D

Definition of Stress
Nominal stress s 
P
A
No weight, no stress???
How is there deformation due to voltage change?
Analogous to thermal expansion
T
σ
T
Small α
Very stiff
Stress-free
deformation
σ
Stress
generated due
to constraint
3D Homogeneous Deformation

~
W 1 , 2 , 3 , D

F  P1l1  P2l2  P3l3  Q
F
P1l1 P2l2 P3l3  Q




AL A L
AL
AL
LA
~ ~
W  s11  s22  s33  ED




~
W 1 , 3 , 3 , D
s1 
1
~
W 1 , 3 , 3 , D
s3 
3

~
W 1 , 3 , 3 , D
s2 
2


~
~ W 1 , 2 , 3 , D
E
~
D

Field Theory Recovers
Maxwell Stresses in a Vacuum
Electric energy per F  0 E 2
~ 1

 W 1 , 1 , 1 , D    0 E 2123
current volume l1l2l3
2
2
D
Recall
~
D
12
D  0E
~
D 23
~
 W 1 , 1 , 1 , D 
2 012


1
1   2    0 E 2
2
1
 3  0E 2
2
Maxwell Stresses
Q
E
Q
P

0
2
E2
 ij   0 E j Ei 
0
2
Ek Ek ij
Ideal Dielectric Elastomers
Elastomer Structure
Incompressibility
123  1

~
 W 1 , 2 , D

Stretching
Polarization
E 2
~
W 1 , 2 , D  Ws 1 , 2  
2
~ 2
1  D 
~


W 1 , 2 , D  Ws 1 , 2  
2  12 




Ws 1 , 2 
 1   3  1
 E 2
1
 2   3  2
Ws 1 , 2 
 E 2 ,
2
D  E
Electrostriction
Well below extension limit
Close to extension limit
Low cross-link density
High cross-link density
Polarization unaffected by
deformation
Deformation affects
polarization
Deformation Affects Polarization
A model: quasi-linear dielectrics
   1, 2 
 1 , 2  2
~
W 1 , 2 , D  Ws 1 , 2  
E
2


Quasi-linear dielectrics
W  ,  
1 
 1   3  1 s 1 2  E 2 
1E 2
1
 2   3  2
2 1
Ws 1 , 2 
1 
 E 2 
2 E 2
2
2 2
Ideal dielectric elastomer
Ws 1 , 2 
 1   3  1
 E 2
1
 2   3  2
Ws 1 , 2 
 E 2
2
Pull-in Instability
Q
a
l
Q

Experimental
Observation for oil
Q


a
l
P
As l , Q 
This can lead to electrical breakdown
Pull-in Instability
Exercise: Find critical electric field for instability
subject to a biaxial force in the plane of membrane
3 L3
1L1
2 L2

Q
Assume ideal dielectric
elastomer and incompressibility
~ 2

1
D 
~


W 1 , 2 , D  Ws 1 , 2  
2  12 


P1
P2
Choose a free energy of stretching function:
Neo-Hookean law

Ws 1 , 2   12  22  32  3
2
Pull-in Instability
In equilibrium




~
~2
W 1 , 2 D
D
s1 
  1  1 3 22  
13 22
~
~
1

~ W 1 , 2 , D D 2 2
E
 1 2
~
~
~2

D
W 1 , 2 , D
D  3 2
 3 2
s2 
  2  2 1  
2 1
2



For equal biaxial stress, s1 = s2 = s and λ1 = λ2 = λ
~
E
~
D4

~ 2 5
D

5
, s       
2
Combining these two equations gives the following
~
E
  s  3  2 8 
       
 

Pull-in Instability
~
E
~
E
  s  3  2 8 
       
 

~
dE
0
reaches a peak when
d
If s/μ = 0

d   2
8

     0  c  21 / 3  1. 26

d  


~
 Ec  0 . 69
~

106 N / m
 108V / m
10
10 F / m
If s/μ = 1

~
c  1.7463 Ec  0.56

Larger stretch
before breakdown
Solder Bumps
e-
Solder: Relation to Class
• Multiple forces
•
•
•
•
Chemical potential
Electric current
Package Warpage
Temperature gradient
• Multiple phases
Ideas from Paper Covered in Class
Kinetic laws – chemical potential, diffusion flux
Principle of virtual work – work conjugates
Traction
Deformation Rate
Eulerian vs. Lagrangian