Soft Active Materials
III. Polyelectrolytes
Zhigang Suo
Harvard University
1
Polyelectrolyte: polymer with electrolyte group
polymer chain
covalent bond
ionic bond
In a solvent, the electrolyte dissociates.
solvent
C l
co-ion
fixed charge

Ca
counterion
counterion
2
Polyelectrolyte gel
A network of polymers
+
+
+
-
+
+
-
+
+
-
+
-
+
-
+
Gel
+
+
-
Fixed charges
+ Counterions
+
-
-
-
-
-
-
+
-
-
+
-
-
+
+
-
-
-
-
-
-
-
-
-
Crosslinks
-
-
+
Solvent (e.g., water)
+
Co-ions
-
External
solution
3
Many tissues are polyelectrolyte gels
Structure and function:
•Cartilage is polyelectrolyte.
•Does this fact have anything to
do with low friction of a joint?
4
Plant cell walls are polyelectrolytes
Pectin
5
Gels regulate flow in plants
Missy Holbrook
Zwieniecki, Melcher, Holbrook,
Hydrogel control of xylem hydraulic resistance in plants
Science 291, 1095 (2001)
6
Osmosis in a polyelectrolyte gel
fixed charge
water
crosslink
-
solution
-
+
+
-
+
-
+
solvent
+
+ +
+
+ -
network
-
-
membrane
counterion
Osmosis balanced by gravity
•Network tatters fixed charges.
•To maintain electroneutrality, fixed charges
are balanced by counterions.
•Counterions cause osmosis
Osmosis balanced by elasticity
7
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).
8
Ion exchanger
XNa2  Ca  XCa 2 Na
9
pH-sensitive gel
RCOOH RCOO  H
fixed
mobile
R COO H   K


R COOH
10
pH-sensitive hydrogel
 
pH  l og10 H
11
pH-sensitive valves
in microfluidics
David Beebe
Deformation and force regulated by physiological variables (pH, salt, temperature…)
Mechanics and chemistry
12
Beebe, Moore, Bauer, Yu, Liu, Devadoss, Jo, Nature 404, 588 (2000)
Adaptive optics with soft materials
13
Dong, Agarwal, Beebe, Jiang, Nature 442, 551 (2006)
Electric potential
Electrochemical potential
Gibbs (1878)
Material B
•Battery does work on electrons, Fdq
•Pump does work on ions, mdM
•Helmholtz free energy of system, F
electrons, dq
electrode
battery, F
pump, m
electrode
Material A
equilibrium
dF  Fdq  mdM
electroneutrality
dq  ezdM  0
dF   ezF  m dM
F
m  ezF 
M
14
Lithium-ion battery
electron
Positive
electrode

Load
Negative
electrode
John B. Goodenough
Silicon
Li-ion

Electrolyte
LiFePO4
Chan et al., 2008. Nature Nanotechnology 3, 31
15
Wang, Wu, Wang, Chen, 2005. Journal of Power Sources 140, 125
dry
polymer
X = (X1, X2, X 3)
3 ways of doing work to a gel
b) current state
x(X, t)
pump
ma
ma
 B dx dV   T dx dA
i
δM
a
gel
i
i
i
 FdqdV   FddA
dq
δQ
 m  dC
a
battery
Φ
a
dV
a
P
weight δx
C a X,t  
# of ions of species a
volume in reference state
16
Hong, Zhao, Suo. J. Mech. Phys. Solids 58 (2010) 558-577.
A field theory of thermodynamic equilibrium
Deformation gradient
FiK X , t  
Gauss’s law
x i X , t 
X K
Material model:
Equilibrium:
Concentrations
C a X,t 
~
DK X,t 
 QX,t 
X K

Q  q   ez aC a  ez fixC fix

~
W  W F, D, C 1 , C 2 ,
a
a
d
WdV

B
d
x
dV

T
d
x
dA

F
d
qdV

F
d
dA

m
d
C

i
i
i
i





 dV
a
Equivalent conditions of equilibrium
siK 
W
FiK
siK X , t 
 Bi X , t   0
X K
W
~
EK  ~
DK
m a  ez a F 
~
FX, t 
E K X , t   
X K
Hong, Zhao, Suo. J. Mech. Phys. Solids 58 (2010) 558-577.
W
C a
m a  constant
17
microscopic processes
-
-
M
-
-
gel
X
M
X
external solution
•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.
18
Free-energy function
Free-energy function


 
~
~
W F, C a , D  Ws F   Wm C 1 , C 2 ,  Wp F, D
 Free energy of stretching
Ws F   21 NkTFiK FiK  3  2 logdet F 
 Free energy of mixing

 s
vC s
χ v 
Cb
b



Wm C   kT  C log


kT
C
log

1

s
s 
s b

1

vC
1

vC
vC
c
b s
0




 Free energy of polarization
~
1 FiK FiL ~ ~
W p F, D 
DK DL
2ε det F
a
 
19
Flory, Rehner, J. Chem. Phys., 11, 521 (1943)
Zhao, Hong, Suo, Physical Review B 76, 134113 (2007)
Swelling regulated by concentration of ions
70
Nv = 0.001
 = 0.1
vC = 0.01
60
vCs + 1
Swelling ratio
65
0
Concentration
of fixed ions
55
0.005
50
45
0.002
40
nonionic gel
s
*
vC + 1
35
10
-6
10
-5
10
vc
-4
10
-3
10
-2
0
Concentration of ions in external solution
20
Constrained swelling
42
Nv = 0.001
 = 0.1
gel
0  1.5
vC = 0.01
38
vCs + 1
Swelling ratio
40
0
36
External solution
Concentration
of fixed ions
0.005
34
32
0.002
30
vC + 1
s
*
10
-6
10
-5
10
vc
-4
10
-3
10
-2
0
Concentration of ions in external solution
21
Debye length
+
LD 
-
kT 
2e 2 c0
+
-
-
+
+
-
0
x
+
+
+
+
0
F  
22
gel
external solution
In gel, near interface
0
10-3
10-4
vCs
eF/kT
-0.5
42
-1
-5
-2
-5
-4
-3
-2
-1
0
10-4
39
10-3
-3
-2
x/LD
-5
-3
-4
8
-3
10-5
10-4
22
-0.5
Nv = 0.001
 = 0.1
vC0 = 0.002
-4
-1
0
-
+
+
-3
vc 0 = 10-5
-
4
gel
+
10
-1
0
x
liquid +
+
-3
0
-5
0
infinity
c0  c0
-4
2
-2
x/LD
x 10
6
vs /kT
vQ/e
vc 0 = 10
-5
+
-
-5
x 10
-1
-
40
x/LD
0
+
41
vc 0 = 10
-1.5
-
vc 0 = 10-5
v  c0  1
s22
10
-4
-3
-2
x/LD
-1
+
0
s22
23
pH-sensitive hydrogel
R COO H   K

RCOOH RCOO  H
fixed
mobile

R COOH
a
24
Osmosis is balanced by elasticity
1 
 sol
NkT 
1
 1 
2 3 
1

   sol   ion 

3
1

kT  
1 
1




l og 1 

vS   1 2 3  1 2 3 1 2 3 2 
1
1
2
1
 ion  kTcH  c  c  cH  c  c 

Donnan equilibrium
Electroneutrality

c / c  cH / cH
c / c  cH / cH


c  cH  c  c A

Conservation of particles


1

c AH  c A 
Constant of acidic dissociation


c A cH

c AH

f
v1 2 3
 N AK
25
Ricka, Tanaka,Macromolecules17, 2916 (1984).
Marcombe, Cai, Hong, Zhao, Lapusta, Suo, Soft Matter 6, 784 (2010)
Free swelling
26
Constrained swelling
27
Finite element method
Condition of equilibrium
Legendre transformation
 dWdV  m dn
S


 mH dnH  m dn  m dn  Bidx i dV  Tidx i dA

S

ˆ  W  (m  m )C  (m  m )C  m C
W




S S
H
H


 dWˆ dV   B dx dV   T dx dA
i
i
i
i
Shengqiang Cai
W  W F, C  , C  , CH


c / c  cH / cH



1
ˆ W
ˆ F, c , c 
W

H

c / c  cH / cH


cH cH  c  c 


 f / vdet F 
1
 cH  c  c 
 N A Ka

Marcombe, Cai, Hong, Zhao, Lapusta, Suo, Soft Matter 6, 784 (2010)
ABAQUS UMAT posted online.
28
pH-sensitive valve
-4
pH = 2
x 10
Channel wall
A/D=0.1
B
 0.4
D
B
D
A
pillar
2
B
 0.36
D
s
pv /KT
gel
pH = 6
B
 0.35
D
1
B
 0.32
D
0
-0.4
-0.3
-0.2
-0.1
0
x/L
0.1
0.2
0.3
0.4
29
Summary
• Couple large deformation and
electrochemistry.
• Swelling regulated by concentration of
ions.
• Swelling regulated by pH.
30