entration of ions in external solution
20
Constrained swelling
42 40
Nv = 0.001 = 0.1
gel
Swelling ratio
0 1.5
vC = 0.01
0
38
vCs + 1
36 0.005 34 32 30 0.002 vC + 1 10
-6
Concentration of fixed ions
External solution
s *
10
-5
10 vc
0
-4
10
-3
10
-2
Concentration of ions in external solution
21
Debye length
+ LD kT 2e 2 c0
+
+
0 + x
microscopic processes
-
-
M
X
M X
gel
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.
-
-
+ -
-
Fixed charges
+
+ Counterions
+
- Co-ions
+
Gel
-
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?
a
24
Osmosis is balanced by elasticity
1
NkT 1 1 2 3 1
+ + + +
0
F
22
gel
external solution
In gel, near interface
0 -0.5
eF/kT
10-3 10-4
vCs
42 41 40 39
vc 0 = 10-5
-
s22
+ -
v c0 1
-1 -1.5 -2 -5 vc 0 = 10 -4
-3
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)
8
Ion exchanger
XNa Ca XCa 2 Na 2
9
pH-sensitive gel
RCOOH RCOO H
fixed
mobile
R COO H K

R COOH
10
pH-sensitive hydrogel
pH l og10 H
70 65 Nv = 0.001 = 0.1 vC = 0.01
0
Swelling ratio
60
vCs + 1
55 0.005 50 45 40 vC + 1 35 10
-6
Concentration of fixed ions
0.002
s *
nonionic gel
10
-5
10 vc
0
-4
10
6
Osmosis in a polyelectrolyte gel
water fixed charge crosslink
solution
solvent network + + +
+
+ + + + + -
-
counterion membrane •Network tatters fixed charges. •To maintain electroneutrality, fixed charges are balanced by counterions. •Counterions cause osmosis Osmosis balanced by elasticity
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
-5
10-4 10-3 -3 -2 x/LD
-1 0
+ -
-3 x/LD
-2
-1
0
-5
-4
-5
+
0
+
infinity c0 c0 x
0
x 10
8
-3
x 10
vc 0 = 10
vQ/e
10-5
vs22/kT
6 4 2
-0.5 Nv = 0.001 = 0.1 vC0 = 0.002 -4 -3 x/LD -2
3 ways of doing work to a gel
b) current state
dry polymer
X = (X1, X2, X 3)
x(X , t)
pump δM gel
a
ma
ma
B dx dV T dx dA
i i i i
dQ q δ
FdqdV FddA m dC
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
1 NkTF F 3 2 log det F Ws F 2 iK iK
7
Osmosis balanced by gravity
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).
electrons, dq
dF Fdq mdM
dq ezdM 0
battery, F
dF ezF m dM
F m ezF M
electrode Material A
14
Lithium-ion battery
electron Positive electrode
Deformation gradient Gauss’s law Concentrations
C a X,t
FiK X , t
x i X , t X K
~ DK X,t QX,t X K
~ W W F, D, C 1 , C 2 ,
Q q ez aC a ez fixC fix
10-4
vc 0 = 10-5
-4 -3
gel -
liquid + + +
10
-1 -5
+
0
-1
0
0 -5
10 -4 -3 -2 x/LD -1
s22
23
pH-sensitive hydrogel
RCOOH RCOO H
R COO H K

fixed
mobile
R COOH
a a a
battery Φ P weight δx
C a X,t
dV
# 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
C l
co-ion fixed charge