Yuhang Hu

Advisor: Zhigang Suo

May 21, 2009

Based on Zhigang’s notes, ucsb talk and an on going paper by Zhigang, Wei and Xuanhe

2

Introduction (microstructure and applications)A field theory of gels coupling large deformation and electrochemistry of ions and solventHomogeneous field in the interior of a gel and solutionInhomogeneous field near interface between gel and external solution

3

NetworkNetwork SolventSolvent Fixed ionsFixed ions Mobile ionsMobile ions

+

http://en.wikipedia.org/wiki/Polyelectrolyte

strong polyelectrolyte

+ HCOOCOOHor

weak polyelectrolyte

hydrogel

solution

polyelectrolyte

++- -

+-+-+-+-

Negatively charged proteins

4

Repulsion retain water during compression. And thus maintain small friction.

5

6

7

battery, pump,

ele

ctro

ns,

q Ions, M

standard electrolyte

electrolyte

electrode

MqF

0 Mezq

MezF

MF

ez

neutrality

equilibrium

work done by the pump, M work done by the battery, q Helmholtz free energy, F

Electrochemical potential: Mechanical work done by bringing one ion from a standard state to a specified concentration and electric potential

Gibbs (1878)

8

QMlF a,,Helmholtz free energy of the gel

lδP work done by the weight

aa Mδμ work done by the pumps

QδΦ work done by the battery

Applicable to a single macromolecule, a cell or a large system

l

9

deformation gradientdeformation gradient

Reference state(Dry state)

Current state

K

iiK X

txtF

,X

,XX

x(X, t)Marker

nominal nominal concentrationsconcentrations

state reference in volume(ions) molecules of#

,a

tCa X

nominal electric nominal electric fieldfield

t,Φ X

x(X+dX, t)

td ,Φ XX

K

K Xt

tE

,Φ,

~ XX

ground ,,,E

~,Fˆˆ 21 CCWW Free-energy densityFree-energy density

10

dAξTdVξBdVXξ

s iiiiK

iiK

in volume

on interface

Define the stress siK, such that

holds for any test function i (X)

Apply divergence theorem, one obtains that

tTtNtsts iKiKiK ,X,X,X,X

0,,

tB

Xts

iK

iK XX

XXB dVt,

XXT dAt,

11

dAdVQdVDX K

K

~

in volume

on interface

Define the electric displacement , such that

holds for any test function (X).

Apply divergence theorem, one obtains that

KD~

tQX

tD

K

K ,,

~X

X

ttNtDtD KKK ,Ω,,~

,~

XXXX

++-+

+-

-

XXQ dVt,

XX dAt,

12

in volume

on interface

trX

tXItCtC a

K

Ka

aa ,X,

,X,X

0

titNtItI aK

aK

aK ,X,X,X,X

Number of ions is conserved:

XX dVt,ar XX dAtia ,

dAidVtrdVX

IdVtCtC aa

K

aK

aa ,X,X,X 0

The above two equations is equivalent to

holds for any test function X

in volume

on interface

aaCezCzqQ 00

aaiez

by batteryby pumps

fixed mobile

13

14

Work done by the Work done by the weightsweights

Work done by the Work done by the batteriesbatteries

Work done by the pumpsWork done by the pumps

dAxTdVxB iiii

dAidVr aaaa

dAδωqdVδ ΦΦ

a field of weights, pumps and batteries

15

dVCC

CCWD

D

CCWF

FCCW

Wa

aaK

KiK

iK

,,,~

,~~

,,,~

,,,,~

, 212121 DFDFDF

work done by weights

Free energy change of the composite Free energy change of the composite system:system:

work done by pumpswork done by batteries

a

aa

a

aaii

ii dAidVrdA

tdV

tq

dAtx

TdVtx

BdVt

WdV

tG

Free energy density change of the gel Free energy density change of the gel element:element:

ThermodynamicThermodynamics:s:

0~

~~

a

aK

bb

K

aaaa

aaaa

KK

K

iKiKK

iK

dVIezC

WX

dAiezC

W

dVrezC

W

dVt

DE

D

WdV

tF

sNFW

tG

a

aa

CCCW

ezμ

,,,~

,Φ

21DF

iK

iK FCCW

s

,,,~

, 21DF

K

KD

CCWE ~

,,,~

,~ 21

DF

Local Equilibrium:

Kinetic law: K

aaK X

tMtJ

,

,X

X

16

0~

~~

a

aK

bb

K

aaaa

aaaa

KK

K

iKiKK

iK

dVIezC

WX

dAiezC

W

dVrezC

W

dVt

DE

D

WdV

tF

sNFW

tG

17

K

iiK X

txtF

,

,X

X K

K Xt

tE

,,

~ XX

0,,

tB

Xts

iK

iK XX tQ

XtD

K

K ,,

~X

X

iKiK F

Ws

K

KD

WE ~~

a

aa

C

Wez

,,,~

, 21 CCWW DF

fixfixaa CezCezqQ

tTtNtsts iKiKiK ,,,, XXXX ttNtDtD KKK ,Ω,,~

,~

XXXX

K

aaK X

tCMtJ

,

,~

,,X

DFX

aaiez

trX

tXItCtC a

K

Ka

aa ,X,

,X,X

0

titNtItI aK

aK

aK ,X,X,X,X

state reference in volume

(ions) molecules of#,

atCa X

18

Free energy of Free energy of stretchingstretching

D~

,F,,FD~

,,F pnis

solsa WCCWCWWCW 21

0

FF detlog2321 iKiKs FFNkTW

ss

sss

sol vCv

vCvC

CkTCW11

log Free energy of mixingFree energy of mixing

Free-energy functionFree-energy function

Free energy of polarizationFree energy of polarization LKiLiK

p DDFF

εW

~~det2

1~,

FDF (Ideal dielectric material)

microscopic effect Swelling increases entropy by mixing solvent and polymers, but decreases entropy by straightening the polymers. Redistributing mobile ions increases entropy by mixing, but increase polarization energy

sbbs

bb

ion cvCC

CkTCCW 10

21 log,, Free energy of dissolving ionsFree energy of dissolving ions

Flory-Rhner

19

+ =

Vdry + Vsol = Vgel

Assumptions:Assumptions: Individual solvent molecule and polymer are incompressible. Gel has no voids. An ion occupies a same volume in the solvent or in the gel

Fdet1 a

aaCv

va – volume per particle of species a

++-

-

++-

-

20

bbss

bbb v

cCv

CkTze

0

log

F

Fdet

21

det

~~

iKiKnMnJMKiJMJ

iK

siK HHFFF

FDD

FW

s

JiKiJ

K DFF

E~

det~

F

ions

s

sbs

b

ssssss

sss v

C

C

CvCvCv

CvkT

2

11

1

1log

solvent

21

In liquid far from interface between gel and solventIn liquid near interfaceIn gel far from interfaceIn gel near interface

22

0211

ijmmjiijij DDDD

0

ii

DE

0log0

bb

bbb v

c

ckTze

ssbs vkTcvckTv 02

In equilibrium

kTvze

ccbb

bb ΠΦexp0

-

- --

--

--

-

-

-

-

-

-

-

-

-

-

+

+

+

+

++

+

+

+

+

++

+

+

+

++

+

+

-

-

-

-

-

--

--

-

-

-

GelExternal solution

-

- --

--

--

-

-

-

-

-

-

-

-

-

-

+

+

+

+

++

+

+

+

+

++

+

+

+

++

+

+

-

-

-

-

-

--

--

-

-

-

GelExternal solution

Infinity in liquid

00 cc

Π=E=D=0

&

State equations

bbss

bbb v

cCvC

kTzeμ ΠlogΦ0

F

FdetΠ

21

det

~~

iKiKnMnJMKiJMJ

iK

siK HHFFδF

FεDD

FW

s

s

sbs

b

ssssss

sss v

CC

Cv

χCvCv

CvkTμ Π

111

1log 2

JiKiJ

K Dε

FFE

~det

~F

23

d

Lx

kTxe

D2tanhlog0Φsgn2

Φ

022 ce

kTLD

Debye length:

DLxx exp0When |Φ| << kT/e

DLx

kTεce

εD

σ expΦ2

2

200

22

22Stress near the interface Negative surface tension!

kTeec

dx

d

sinh

2 02

2

10 cv

0211

ijmmjiijij DDDD

2

2

dx

dcceQ

0log0

v

c

ckTze

ssbs vkTcvckTv 02

x0

LD

solution+

+

++

+

+

__

_

_

_

_

__

+

+

gel

10 cv22σ

22σ

00 cc infinity

General solution:

fast decay electric field in liquid near interface

24

0log0

v

cCv

CkTze

ss

0detΠ

FF

iKiK

siK H

FW

s

ss

sbs

b

ssssss

sss vkTcv

C

C

CvCvCv

CvkT 02

211

1

1log

constantInfinity in liquid

00 cc fixCCC

infinity in gel

0log0

v

cCv

CkTze

ss

C

C

Cs

-

- --

--

--

-

-

-

-

-

-

-

-

-

-

+

+

+

+

++

+

+

+

+

++

+

+

+

++

+

+

-

-

-

-

-

--

--

-

-

-

GelExternal solution

-

- --

--

--

-

-

-

-

-

-

-

-

-

-

+

+

+

+

++

+

+

+

+

++

+

+

+

++

+

+

-

-

-

-

-

--

--

-

-

-

GelExternal solution

fixCCC

321 13 ssCv

Electric field vanishes and electric charge neutralgel swells uniformly

incompressibility

25

10-6

10-5

10-4

10-3

10-2

35

40

45

50

55

60

65

70

0.005

vc0

vCs +

1

Nv = 0.001 = 0.1

0.002

vC0 = 0.01

vC*s + 1

nonionic gel

Sw

ellin

g ra

tio

Concentration of ions in external solution

Concentration of fixed ions

26

02

12

1

22

22

1111

dXd

NkTs

2

2

1

22

0 dX

dXCXCCeXQ

0log0

sCvc

CkTe

022

11

1

1log ckTvv

Cv

CCv

CvCvCv

CvkT ss

ssssssss

sss

x0

LD

solution+

+

++

+

+

__

_

_

_

_

__

+

+

gel

10 cv

22s

22s

00 cc infinity

1221 ssCv

Inhomogeneous field 132

incompressibility

Deep in gel, electric filed vanishes and no net chargeExternal solution

0

dXd

kTe

eLkT

dXd

D 20

sinh2

0

0, f

+

27

x0

LD

solution+

+

++

+

+

__

_

_

_

_

__

+

+

gel

10 cv

22s

22s

00 cc infinity

-5 -4 -3 -2 -1 0-2

-1.5

-1

-0.5

0

x/LD

e/k

T

vc0 = 10-5

10-410-3

-5 -4 -3 -2 -1 00

2

4

6

8x 10

-5

x/LD

vs22

/kT

vc0 = 10-5

10-4

10-3

-5 -4 -3 -2 -1 0

-1

-0.5

0x 10

-3

x/LD

vQ/e

10-5

10-4

vc0 = 10-3

Nv = 0.001 = 0.1vC

0 = 0.002

-5 -4 -3 -2 -1 0

39

40

41

42

x/LD

vCs

vc0 = 10-5

10-4

10-3

28

Hydrogel : poroelasticity

Li battery : field theory coupling large deformation and electrochemistry of ions and solvent

Electron

PositiveElectrode

NegativeElectrode

Electrolyte

Li-ion

A discharging Li-ion cell.

Load

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