11

Soft Active Materials

Zhigang Suo

Harvard University

III. Polyelectrolytes

2

Polyelectrolyte: polymer with electrolyte group

In a solvent, the electrolyte dissociates.

solvent

fixed charge

counterionCa

Clco-ion

counterion

covalent bond

ionic bond

polymer chain

3

Polyelectrolyte gel

-

- --

--

--

-

-

-

-

-

-

-

-

-

-

+

+

+

+

++

+

+

+

+

++

+

+

+

++

+

+

-

-

-

-

-

--

--

-

-

-

GelExternal solution

Fixed charges

Counterions

Co-ions

-

+

-

A network of polymers Solvent (e.g., water)

Crosslinks

4

Many tissues are polyelectrolyte gels

Structure and function:•Cartilage is polyelectrolyte. •Does this fact have anything to do with low friction of a joint?

5

Plant cell walls are polyelectrolytes

Pectin

6

Gels regulate flow in plants

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

Missy Holbrook

7

Osmosis in a polyelectrolyte gel

mem

bran

e

solution

solvent

Osmosis balanced by gravity

•Network tatters fixed charges.•To maintain electroneutrality, fixed charges are balanced by counterions. •Counterions cause osmosis

crosslink

-

-

-

-

-

- -

-

--

+

++

+

+

++

++

fixed charge

counterion

water

network

Osmosis balanced by elasticity

8

Super absorbent diaper

Sodium polyacrylate: polyelectrolyte

Masuda, Trends in the development of superabsorbent polymers for diapers, pp. 88-89, Superabsorbent polymers: science and technology (1994).

9

Ion exchanger

2NaXCaCaXNa2

10

pH-sensitive gel

HRCOORCOOHfixed mobile

K

RCOOHHRCOO

11

pH-sensitive hydrogel

HlogpH 10

12

pH-sensitive valves in microfluidics

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

Deformation and force regulated by physiological variables (pH, salt, temperature…)Mechanics and chemistry

David Beebe

13

Adaptive optics with soft materials

Dong, Agarwal, Beebe, Jiang, Nature 442, 551 (2006)

14

Electric potential Electrochemical potential

Gibbs (1878)

electrode

battery, pump,

ele

ctro

ns,

q

Ions, M

MqF

0 Mezq

MezF

MF

ez

electroneutrality

equilibrium

Material A

Material B

electrode

•Battery does work on electrons, q•Pump does work on ions, M•Helmholtz free energy of system, F

15

Lithium-ion battery

Electrolyte

Li-ion

LoadNegativeelectrode

Positive electrode

Wang, Wu, Wang, Chen, 2005. Journal of Power Sources 140, 125

Chan et al., 2008. Nature Nanotechnology 3, 31

electron

LiFePO4

SiliconJohn B. Goodenough

16

3 ways of doing work to a gel

a

aa

iiii

dVC

dAqdV

dAxTdVxB

state reference in volume species of ions of#

,a

tCa X

gel

apump

Pweight δx

δMa

a) reference state

b) current state

dry polymer

X = (X1, X2, X3)

x(X, t)

batteryΦ

δQq

a

Hong, Zhao, Suo. J. Mech. Phys. Solids 58 (2010) 558-577.

17

A field theory of thermodynamic equilibriumDeformation gradient

K

iiK X

txtF

,

,X

X

K

K Xt

tE

,

,~ X

X 0,

,

tBX

tsi

K

iK XX

tQX

tD

K

K ,,

~X

X

Equivalent conditions of equilibrium

iKiK F

Ws

K

KD

WE ~~

Gauss’s law

aaa

C

Wez

,,,~

, 21 CCWW DF

tCa ,X

fixfixaa CezCezqQ

Concentrations

Material model:

constanta

a

aaiiii dVCdAqdVdAxTdVxBWdV Equilibrium:

Hong, Zhao, Suo. J. Mech. Phys. Solids 58 (2010) 558-577.

18

microscopic processes

•Swelling increases entropy by mixing solvent and polymers.•Swelling decreases entropy by straightening the polymers.•Redistributing mobile ions increases entropy by mixing.•Neutrality reduces electrostatic energy.

-

-M

-

-

X

X

M

gel external solution

19

Free-energy function

DFFDF~

,,,~

,, 21pms

a WCCWWCW

FF detlog2321 iKiKs FFNkTW

Flory, Rehner, J. Chem. Phys., 11, 521 (1943)

sb

bs

bb

ss

ssa

m cvCC

CkTvCvχ

vCvC

CkTCW 1log11

log0

Free energy of mixing

Free-energy function

Free energy of polarization LKiLiK

p DDFF

εW

~~det2

1~,

FDF

Free energy of stretching

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

20

Swelling regulated by concentration of ions

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

21

Constrained swelling

Concentration of ions in external solution

10-6

10-5

10-4

10-3

10-2

30

32

34

36

38

40

42

0.005

vc0

vCs +

1

Nv = 0.001 = 0.1

0.002

vC0 = 0.01

vC*s + 1

Concentration of fixed ions

Sw

ellin

g ra

tio

gel

External solution

5.10

22

Debye length

x0

-

-

-

-

-

-

-

+

+

+

+

+

+

+

+0

22 ce

kTLD

0

gel external solution

23

-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

In gel, near interface

x0

-

-

-

-

-

-

-

+

+

+

+

+

+

+

+

00 cc infinity

10 cv22s

22s

liquidgel

24

pH-sensitive hydrogel

HRCOORCOOHfixed mobile

aK

RCOOHHRCOO

25

ionsol

NkT

11

321

1

2

321321321

111log

Ssol v

kT

HH// cccc 1

HH//

cccc

cccccckTion HH

Acccc

H

KNc

ccA

AH

A H

321 vf

ccAAH

Donnan equilibrium

Electroneutrality

Constant of acidic dissociation

Conservation of particles

1

11

12

3

Marcombe, Cai, Hong, Zhao, Lapusta, Suo, Soft Matter 6, 784 (2010)Ricka, Tanaka,Macromolecules17, 2916 (1984).

Osmosis is balanced by elasticity

26

Free swelling

27

Constrained swelling

28

Finite element method

SSCCCWW )()(ˆHH

ccWW ,,ˆˆH

F

dAxTdVxBdVW iiii ˆ

dAxTdVxBnnnnWdV iiiiSS HH

H

,,, CCCWW F

HH// cccc

1

HH//

cccc

aAKN

cccvf

cccc

H

1HH

det/ F

Condition of equilibrium

Legendre transformation

Shengqiang Cai

Marcombe, Cai, Hong, Zhao, Lapusta, Suo, Soft Matter 6, 784 (2010)ABAQUS UMAT posted online.

29

pH-sensitive valve

DB

A

gel

pillar

Channel wall

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.40

1

2

x 10-4

x/L

pvs/K

T

A/D=0.1

32.0DB

35.0DB

36.0DB

4.0DB

pH = 2

pH = 6

30

Summary

• Couple large deformation and electrochemistry.

• Swelling regulated by concentration of ions.

• Swelling regulated by pH.