智能软材料-水凝胶
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Soft Active Materials
III. Polyelectrolytes
Zhigang Suo Harvard University
1
Polyelectrolyte: polymer with electrolyte group
polymer chain
covalent bond
ionic bond In a solvent, Hale Waihona Puke Baiduhe electrolyte dissociates. solvent
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
-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
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
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
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
a
24
Osmosis is balanced by elasticity
1
NkT 1 1 2 3 1
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
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)
siK
W FiK
W ~ EK ~ DK
m a ez a F
W C a
siK X , t Bi X , t 0 X K
~ FX, t E K X , t X K
m a constant
17
Hong, Zhao, Suo. J. Mech. Phys. Solids 58 (2010) 558-577.
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
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
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
+ + + +
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
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
11
pH-sensitive valves in microfluidics
David Beebe
Deformation and force regulated by physiological variables (pH, salt, temperature…) Mechanics and chemistry
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.
Free energy of mixing Free energy of polarization
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
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).
Material model:
a
Equilibrium:
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
Equivalent conditions of equilibrium
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
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
-
-
+ -
-
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?
-3
10
-2
Concentration 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
Electric potential Electrochemical potential
Gibbs (1878)
Material B electrode •Battery does work on electrons, Fdq •Pump does work on ions, mdM •Helmholtz free energy of system, F equilibrium electroneutrality pump, m
a
~ 1 FiK FiL ~ ~ W p F, D DK DL 2ε det F
19
Flory, Rehner, J. Chem. Phys., 11, 521 (1943) Zhao, Hong, Suo, Physical Review B 76, 134113 (2007)
Swelling regulated by concentration of ions
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
C l
co-ion fixed charge
Ca
counterion
2
counterion
Polyelectrolyte gel
A network of polymers
+
Solvent (e.g., water)
+
+
-
+ +
+ -
+ + + + + + + -
Crosslinks
-
+
+ -
III. Polyelectrolytes
Zhigang Suo Harvard University
1
Polyelectrolyte: polymer with electrolyte group
polymer chain
covalent bond
ionic bond In a solvent, Hale Waihona Puke Baiduhe electrolyte dissociates. solvent
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
-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
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
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
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
a
24
Osmosis is balanced by elasticity
1
NkT 1 1 2 3 1
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
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)
siK
W FiK
W ~ EK ~ DK
m a ez a F
W C a
siK X , t Bi X , t 0 X K
~ FX, t E K X , t X K
m a constant
17
Hong, Zhao, Suo. J. Mech. Phys. Solids 58 (2010) 558-577.
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
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
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
+ + + +
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
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
11
pH-sensitive valves in microfluidics
David Beebe
Deformation and force regulated by physiological variables (pH, salt, temperature…) Mechanics and chemistry
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.
Free energy of mixing Free energy of polarization
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
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).
Material model:
a
Equilibrium:
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
Equivalent conditions of equilibrium
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
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
-
-
+ -
-
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?
-3
10
-2
Concentration 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
Electric potential Electrochemical potential
Gibbs (1878)
Material B electrode •Battery does work on electrons, Fdq •Pump does work on ions, mdM •Helmholtz free energy of system, F equilibrium electroneutrality pump, m
a
~ 1 FiK FiL ~ ~ W p F, D DK DL 2ε det F
19
Flory, Rehner, J. Chem. Phys., 11, 521 (1943) Zhao, Hong, Suo, Physical Review B 76, 134113 (2007)
Swelling regulated by concentration of ions
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
C l
co-ion fixed charge
Ca
counterion
2
counterion
Polyelectrolyte gel
A network of polymers
+
Solvent (e.g., water)
+
+
-
+ +
+ -
+ + + + + + + -
Crosslinks
-
+
+ -