Mechanics and electrochemistry of
polyelectrolyte gels
Zhigang Suo
School of engineering and Applied Sciences
Harvard University
Work with W. Hong and X. Zhao
1
Polyelectrolyte: polymer with electrolyte group
http://en.wikipedia.org/wiki/Polyelectrolyte
covalent bond
ionic bond
In a solvent, the electrolyte dissociates.
solvent
Cl −
co-ion
fixed ion
Ca
++
counterion
counterion
2
gel = network + solvent
solvent
reversible
gel
network
Solid-like: long polymers crosslink by strong bonds. Retain shape
Liquid like: polymers and solvent aggregate by weak bonds. Enable transport
Ono, Sugimoto, Shinkai, Sada, Nature Materials 6, 429, 2007
3
Polyelectrolyte gels
+
-
-
+
+
+
-
-
-
+
+
-
+
+
-
+
-
+
-
+
Gel
+
-
-
-
-
•Network
•Solvent
•Fixed ions
•Mobile ions
+
+
-
-
+
-
-
+
+
-
-
-
-
-
-
-
+
-
+
-
External
solution
4
Ion exchange resin
XNa 2 + Ca + + → XCa + 2Na +
5
pH-sensitive valves in microfluidics
6
Beebe, Moore, Bauer, Yu, Liu, Devadoss, Jo, Nature 404, 588 (2000)
Many tissues are polyelectrolyte gels
Structure and function:
•Cartilage is polyelectrolyte.
•Does this fact have anything to
do with low friction of a joint?
7
Time-dependent process
Shape change: short-range motion of solvent molecules, fast
Volume change: long-range motion of solvent molecules, slow
8
Hong, Zhao, Zhou, Suo, JMPS 56, 1779 (2008)
Electrochemical potential
Work done to pump an ion from one electrolyte to another
electrolyte
•work done by the battery, Φδq
•work done by the pump, µδM
•Helmholtz free energy, F
electrons, δq
equilibrium
battery, Φ
neutrality
δF = Φδq + µδM
δq + ezδM = 0
Ions, δM
δF = (− ezΦ + µ )δM
pump, µ
µ = ezΦ +
∂F
∂M
electrode
standard electrolyte
Gibbs (1878)
9
3D inhomogeneous state of equilibrium
a
a
δ
WdV
=
B
δ
x
dV
+
T
δ
x
dA
+
Φ
δ
qdV
+
Φ
δω
dA
+
µ
δ
C
∑
∫
∫ i i
∫ i i
∫
∫
∫ dV
a
x(X, t)
pump
µa µ a
δMa
gel
δq
δQ
battery
Φ
W=
free energy of gel
volume in reference state
C a (X,t ) =
# of ions of species a
volume in reference state
P
weight δx
3 ways of doing work to a gel
10
A field of markers: stretch
L
λ=
l
Reference state
X
l
L
Current state
x(X, t)
X+dX
x(X+dX, t)
FiK
∂x i (X ,t )
=
∂X K
x i (X + dX ,t ) − x i (X ,t )
dX K
11
A field of batteries: electric field
Reference state
L
Current state
l
X
x(X, t)
Φ
~ Φ
E=
L
x(X+dX, t)
X+dX
Φ(X ,t )
Φ (X + dX ,t )
Φ (X + dX ,t ) − Φ (X ,t )
dX K
ground
∂Φ (X ,t )
~
=
−
EK
∂X K
12
A field of weights: stress
Define the stress siK, such that
∫ siK
∂ξ i
dV = ∫ Bi ξ i dV + ∫ Ti ξ i dA
∂X K
holds for any test function ξi (X)
Apply divergence theorem, one obtains that
∂siK (X ,t )
+ Bi (X ,t ) = 0
∂X K
in volume
(s
−
iK
(X ,t ) − siK+ (X ,t ))N K (X ,t ) = Ti (X ,t )
on interface
13
A field of charges: electric displacement
Q = q + ∑ ez aC a + ez fix C fix
++
- +
Define the electric displacement D~K, such that
−∫
∂ζ ~
DK dV = ∫ ζQdV + ∫ ζωdA
∂X K
holds for any test function ζ(X).
Apply divergence theorem, one obtains that
~
∂DK (X , t )
= Q (X , t )
∂X K
in volume
(
)
~+
~−
DK (X , t ) − DK (X , t ) N K (X ,t ) = ω (X , t )
on interface
14
Equilibrium conditions
a
a
δ
WdV
=
B
δ
x
dV
+
T
δ
x
dA
+
Φ
δ
qdV
+
Φ
δω
dA
+
µ
δ
C
∑ ∫ dV
∫
∫ i i
∫ i i
∫
∫
a
(
)
~ 1 2
W
=
W
F
,
D
, C , C ,L
Material model:
siK =
∂W
∂FiK
∂W
~
EK = ~
∂DK
µ a = ez a Φ +
∂W
∂C a
15
Sum up: a field theory
A field of batteries
A field of markers
FiK (X , t ) =
∂x i (X , t )
∂X K
~
∂Φ (X , t )
E K (X , t ) = −
∂X K
A field of weights
A field of charges
∂siK (X , t )
+ Bi (X , t ) = 0
∂X K
C a (X,t )
Equations of state
siK =
∂W
∂FiK
~
∂DK (X ,t )
= Q (X ,t )
∂X K
Q = q + ∑ ez a C a + ez fix C fix
(
)
~
W = W F, D, C 1 , C 2 ,L
∂W
~
=
EK
~
∂DK
µ a = ez a Φ +
∂W
∂C a
16
Osmosis
N
p = kT
V
17
Incompressibility of particles
+
Vdry +
=
+ - +
Vsol =
Vgel
+- +
1 + ∑ vaC a = det F
a
va – volume per particle of species a
Assumptions:
– Individual solvent molecule and polymer are incompressible.
– Gel has no voids. (a gel is different from a sponge.)
– An ion occupies a same volume in the solvent and in the gel
18
Osmotic pressure
Enforce incompressibility as a constraint by introducing a Lagrange multiplier Π
(
) (
~
W = W F, C a , D + Π 1 + ∑ v a C a − det F
siK =
)
∂W
− Π H iK det F
∂FiK
µs =
µb =
∂W
s
+
Π
v
∂C s
solvent
∂W
b
b
+
Φ
+
Π
e
z
v
∂C b
negligible
∂W
~
EK = ~
∂DK
ions
19
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.
•Unbalanced changed causes electrostatic energy.
20
Free-energy function
Free-energy function
(
)
(
~
~
W F , C a , D = Ws (F ) + Wm (C 1 , C 2 ,L) + W p 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
log
1
kT
C
+
−
−
∑
s
s ⎟
s b
⎜
1
1
vC
vC
vC
c
+
+
b≠ s
0
⎠
⎝
⎠
⎝
„ Free energy of polarization
~
1 FiK FiL ~ ~
DK DL
W p F, D =
2ε det F
a
( )
21
Flory, Rehner, J. Chem. Phys., 11, 521 (1943)
Zhao, Hong, Suo, Physical Review B 76, 134113 (2007)
Equations of state
siK
~ ~
∂Ws (F ) D J D M ⎛
1
⎞
=
+
⎜ FiJ δ MK − FnJ FnM H iK ⎟ − Π H iK det F
ε det F ⎝
2
∂FiK
⎠
⎡
vsC s
1
χ
+
+
µ = kT ⎢ log
1 + vsC s 1 + vsC s 1 + vsC s
⎢⎣
s
(
)
2
Cb ⎤
− ∑ s ⎥ + Π v s solvent
b≠ s C ⎥
⎦
Cb
µ = eΦz + kT log s s b + Π vb ions
v C c0
b
b
F F ~
~
E K = iJ iK D J
ε det F
22
Free swelling
Infinity in liquid
-
-
X−
-
Φ = constant
M+
C + = C − + C −fix
X−
gel
external solution
∂Ws (F )
− ΠHiK det F = 0
∂FiK
Π
⎡
vsC s
χ
1
s
+
+
µ = kT ⎢ log
1 + vsC s 1 + vsC s 1 + vsC s
⎢⎣
(
µ + = eΦz + + kT log
+
C
+
+
Π
v
=0
s s +
v C c0
C−
µ = eΦz + kT log s s − + Π v − = 0
v C c0
−
M+
-
infinity in gel
siK =
c0± = c0
−
C ⎤
− ∑ s ⎥ + Π v s = −2kTc0 v s
b≠ s C ⎥
⎦
b
)
2
Cs
Φ
C+
C−
23
Swelling ratio
70
Nv = 0.001
χ = 0.1
vC0 = 0.01
60
Concentration
of fixed ions
55
0.005
s
vC + 1
Swelling ratio
65
50
45
0.002
40
nonionic gel
s
vC* + 1
35
10
-6
10
-5
10
-4
10
-3
10
-2
vc0
Concentration of ions in external solution
24
Double layer: electric potential due to free swelling
0
0.002 →
-1
-2
eΦ(-∞)/kT
← 0.005
-3
← vC0 = 0.01
-4
Concentration
of fixed ions
-5
-6
-7
10
-6
10
-5
10
-4
10
-3
10
Φ
-2
vc0
0
Concentration of ions in external solution
Φ (− ∞ )
25
gel
external solution
In external solution, near interface
+
+
-
-
c0± = c0
+
+
-
infinity
0
x
+
+
+
+
Φ(x ) = Φ(0)exp (− x LD )
0
Φ (− ∞ )
LD =
gel
external solution
kTε
2e 2 c0
Debye length
26
In gel, near interface
0
10-3
10-4
-5
-1
-5
vc 0 = 10
-1.5
+
40
-2
-5
-4
-3
-2
-1
0
-5
-4
x/LD
-3
8
-3
vc 0 = 10
10-5
-5
Nv = 0.001
χ = 0.1
vC0 = 0.002
-4
+
-1
0
-
-3
vc 0 = 10-5
-
4
10-4
2
x/LD
-1
0
0
-5
+
+
10
-4
-3
-2
-1
0
x
gel liquid +
-
-3
-2
infinity
c0± = c0
+
+
x 10
6
10-4
-0.5
-1
10-3
-3
-2
x/LD
-5
x 10
vs22/kT
0
-
10-4
39
vQ/e
-
vc 0 = 10
v ± c0 << 1
s22
41
vCs
eΦ/kT
-0.5
42
+
0
x/LD
s22
27
A thin layer of gel
0
40.5
vc 0 = 10-5
-4
40
-0.5
vCs
eΦ/kT
10
39.5
-1
vc 0 = 10
-1.5
0.5
x/LD
-4
vQ/e
2 LD
liquid
2
vc 0 = 10-5
0
-8
-3
10
-10
1
x 10
10-3
10-4
-2
Nv = 0.001
χ = 0.1
vC0 = 0.002
-4
0
0.5
x/LD
0
10-4
-12
10-3
-5
x 10
-6
gel
38.5
-4
x
c0± = c0
1
vs22/kT
0
0
10-4
39
-5
0.5
x/LD
1
-6
0
vc 0 = 10-5
0.5
x/LD
1
28
pH-sensitive gel
RCOOH ⇔ RCOO − + H +
fixed
mobile
29
Kim et al, J. App. Polymer. Sci., 91, 3613 (2004)
Weak electrolyte:
RCOOH ⇔ RCOO − + H +
new variable
+
H
δC H = δC ext
+ δC RCOO
Conservation of H+
a
(
+
−
−
~
W = W F,D, C s , C H , C Cl , C RCOO
Helmholtz free energy
∂W
∂C
−
∫ δWdV = ∫ Biδx i dV + ∫ Tiδx i dA + ∫ ΦδqdV + ∫ ΦδωdA + ∑a µ ∫ δC ext dV
Equilibrium condition
+
−
Q = q + ∑ ez a C a − eC RCOO
Conservation of charge
µ H = eΦ +
+
H+
−
µ Cl = −eΦ +
∂W
∂C
Cl −
∂W
∂C
H
+
+
∂W
∂C
RCOO −
a
)
=0
30
Free-energy function
(
)
(
)
(
)
(
+
−
−
+
−
−
~
~
W F, D, C s , C H , C Cl , C RCOO = Ws (F ) + Wm C s , C H , C Cl + W p F, D + Wd C RCOO
Free energy of dissociation
(
Wd C
RCOO
−
)
RCOO −
RCOO −
⎡
⎛
C
C
−
−
= kT ⎢C RCOO ln RCOOH + C 0RCOOH − C RCOO ln⎜ 1 − RCOOH
⎜ C0
⎢
C0
⎝
⎣
(
)
Entropy of mixing between RCOO- and RCOOH
In equilibrium
ln
C
v
s
H+
+
C s c0H
+ ln
C
C
RCOO −
C 0RCOOH
RCOO −
=
−C
RCOO −
K d C 0RCOOH
CH
Kd + s
C
+
−
γ
kT
⎞⎤
⎟⎥ − γC RCOO −
⎟⎥
⎠⎦
Enthalpy of
dissociation
=0
+
⎛ γ ⎞
K d = v s c0H exp⎜ −
⎟
kT
⎝
⎠
31
)
Summary
• Couple mechanics and electrochemistry.
• Maxwell stress emerges under ideal dielectric
assumption
• Free swelling.
• Double layer at interface.
• Future work: relate to experiments; analyze
interesting phenomena.
32