Electric Charge / Deformation and Polarization

Matt Pharr

ES 241

5/21/09

Electric Charge

10 protons, 10 electrons

Net charge = 10 - 10 = 0

Total charge is conserved

A B

SI Units: 1 Coulomb =

= 6.242 * 1018 elementary charges

A B

QA= 0

+QB= 0

Qnet= 0

QA= +4

+QB= -4

Qnet= 0

F’(Q’) F’’(Q’’)F’(Q’) F’’(Q’’)

Capacitor

QQ

QFF

QFF

Φ

'Q

0':

'

''

''

''''For

0''''

'''''

'

''

''''''

''''''

QQ

QF

Q

QF

QQ

QFQ

Q

QFF

QFQFF

QQconstQQ

comp

comp

SI Units: 1 Volt

= 1.602e-19 Joule

Measurement of Electric Potential

I

RV?

Current measured with galvanometer

Ohm’s Law: V = IR gives the potential

A Capacitor, a Weight, and a Battery

QQ

QlFl

l

QlFF

QlFF

,,

,

Mechanical work lPElectric work Q

In equilibrium QlPF

Q

QlF

l

QlFP

,,

,

F(l,Q) and Stress

la

Q

Experimental Relation

a

Q

l

QlFP

2

, 2

a

lQQlF

Q

QlF

2,

, 2

Recall:

oil

Φl

a

+Q

-Q

lE

aQ

D

aP

Electric field

Electric displacement

Stress field

Maxwell Stress

2

2Ez

Deformable Dielectrics

A

Reference State

P

l

a Q

Q

Current State

LE

~

A

QD ~

A

Ps

Nominal electric field

Nominal electric displacement

Nominal stress

L

lStretch

Nominal free-energy density AL

FW

QlPF

AL

Q

LA

lP

AL

F

DEsW~~

DW

s~

, D

DWE ~

~,~

L

Definition of Stress

A

Ps Nominal stress

T

No weight, no stress???

Analogous to thermal expansionStress-free deformation

Small α

Very stiff

T

σ σ

Stress generated due to constraint

How is there deformation due to voltage change?

3D Homogeneous Deformation

DEsssW~~

332211

QlPlPlPF 332211

AL

Q

LA

lP

LA

lP

LA

lP

AL

F 332211

1

3311

~,,,

DW

s

D

DWE ~

~,,,~ 321

3

3313

~,,,

DW

s

2

3312

~,,,

DW

s

DW~

,,, 321

Field Theory Recovers Maxwell Stresses in a Vacuum

3212

0111

20

321 2

1~,,,

2

EDWE

lll

F

ED 0

Electric energy per current volume

Recall21

~

D

D

210

32

111 2

~~

,,,

DDW 2

03

2021

2

12

1

E

E

ijkkijij EEEE 2

00

Q

Q

20

2E

P

E

Maxwell Stresses

Ideal Dielectric ElastomersElastomer Structure

1321 Incompressibility

DW~

,, 21

2

,~

,,2

2121

EWDW s

Stretching ization

2

212121

~

2

1,

~,,

D

WDW s

Polar-

EDE

W

EW

s

s

,,

,

2

2

21232

2

1

21131

Electrostriction

Well below extension limit

Low cross-link density

Polarization unaffected by deformation

Close to extension limit

High cross-link density

Deformation affects polarization

Deformation Affects Polarization

A model: quasi-linear dielectrics

21,

2212121 2

,,

~,, EWDW s

2

2

21232

2

1

21131

,

,

EW

EW

s

s

22

2

2

2

21232

21

1

2

1

21131

2

1,

2

1,

EEW

EEW

s

s

Ideal dielectric elastomerQuasi-linear dielectrics

Pull-in Instability

la

Q

P

l

a Q

Q

Experimental Observation for oil

Q,As l 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

2P 1P

22L

33L

11L

Q

2P 1P

22L

33L

11L

Q

Assume ideal dielectric elastomer and incompressibility

2

212121

~

2

1,

~,,

D

WDW s

Choose a free energy of stretching function: Neo-Hookean law

32

, 23

22

2121 sW

Pull-in Instability

22

31

22

23

111

211

~~,

DDWs

21

32

22

13

222

212

~~,,

DDW

s

22

21

21

~

~

~,,~

D

D

DWE

For equal biaxial stress, s1 = s2 = s and λ1 = λ2 = λ

2

~,

~~ 52

54

D

sD

E

Combining these two equations gives the following

823~

s

E

In equilibrium

Pull-in Instability

E~

reaches a peak when 0~

dEd

If s/μ = 0

mVmF

mNE

d

d

c

c

/10/10

/10~69.0

~

26.120

810

6

3/182

If s/μ = 1

56.0

~7463.1 cc E

Larger stretch

before breakdown

823~

s

E

Solder Bumps

e-

• Multiple forces• Chemical potential• Electric current• Package Warpage• Temperature gradient

• Multiple phases

Solder: Relation to Class

Ideas from Paper Covered in Class

Kinetic laws – chemical potential, diffusion flux

Principle of virtual work – work conjugates

Traction

Deformation Rate

Eulerian vs. Lagrangian