11 soft active materials zhigang suo harvard university iii. polyelectrolytes
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Soft Active Materials
Zhigang Suo
Harvard University
III. Polyelectrolytes
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Polyelectrolyte: polymer with electrolyte group
In a solvent, the electrolyte dissociates.
solvent
fixed charge
counterionCa
Clco-ion
counterion
covalent bond
ionic bond
polymer chain
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Polyelectrolyte gel
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GelExternal solution
Fixed charges
Counterions
Co-ions
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+
-
A network of polymers Solvent (e.g., water)
Crosslinks
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Many tissues are polyelectrolyte gels
Structure and function:•Cartilage is polyelectrolyte. •Does this fact have anything to do with low friction of a joint?
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Plant cell walls are polyelectrolytes
Pectin
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Gels regulate flow in plants
Zwieniecki, Melcher, Holbrook,Hydrogel control of xylem hydraulic resistance in plants Science 291, 1095 (2001)
Missy Holbrook
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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
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-
--
+
++
+
+
++
++
fixed charge
counterion
water
network
Osmosis balanced by elasticity
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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).
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Ion exchanger
2NaXCaCaXNa2
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pH-sensitive gel
HRCOORCOOHfixed mobile
K
RCOOHHRCOO
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pH-sensitive hydrogel
HlogpH 10
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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
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Adaptive optics with soft materials
Dong, Agarwal, Beebe, Jiang, Nature 442, 551 (2006)
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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
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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
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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.
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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.
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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
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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)
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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
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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
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Debye length
x0
-
-
-
-
-
-
-
+
+
+
+
+
+
+
+0
22 ce
kTLD
0
gel external solution
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-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
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pH-sensitive hydrogel
HRCOORCOOHfixed mobile
aK
RCOOHHRCOO
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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
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Free swelling
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Constrained swelling
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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.
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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
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Summary
• Couple large deformation and electrochemistry.
• Swelling regulated by concentration of ions.
• Swelling regulated by pH.