Yuhang Hu
Advisor: Zhigang Suo
May 21, 2009
Based on Zhigang’s notes, ucsb talk and an on going paper by Zhigang, Wei and Xuanhe
Introduction (microstructure and applications)
A field theory of gels coupling large deformation
and electrochemistry of ions and solvent
Homogeneous field in the interior of a gel and
solution
Inhomogeneous field near interface between gel
and external solution
2
solution
+
--+
hydrogel
+
+
+
+
- +
-
polyelectrolyte
+
or
strong polyelectrolyte
COOH  COO   H
weak polyelectrolyte
Network
Solvent
Fixed ions
Mobile ions
http://en.wikipedia.org/wiki/Polyelectrolyte
3
Negatively charged proteins
Repulsion retain water
during compression. And
thus maintain small friction.
4
5
6
Electrochemical potential: Mechanical work done by bringing one ion
from a standard state to a specified concentration and electric potential
electrons, dq
electrolyte
battery, F
work done by the pump, mdM
work done by the battery, Fdq
Helmholtz free energy, F
equilibrium
neutrality
pump, m
dF  Fdq  mdM
dq  ezdM  0
dF   ezF  m dM
electrode
standard electrolyte
m  ezF 
Gibbs (1878)
F
M
7
μaδM a work done by the pumps
Pδl work done by the weight
ΦδQ work done by the battery
l
Helmholtz free energy of the gel
F l, M a , Q 
Applicable to a single macromolecule, a cell or a large system
8
x(X, t)
Marker

deformation gradient
FiK X, t  
X

nominal concentrations
C a X, t  
Reference state
(Dry state)
x i X, t 
X K
#of a molecules (ions)
volume in reference state
Current state
x(X+dX, t)
ΦX,t 
ΦX  dX,t 
ground

nominal electric field
~
ΦX, t 
EK X, t   
X K


~ 1 2
ˆ W
ˆ F, E
Free-energy density W
, C , C ,
9
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 i (X)
B X , t dV  X 
T  X , t dA X 
Apply divergence theorem, one obtains that
in volume
siK X,t 
 Bi X,t   0
X K
on interface
s

iK
X,t   siK X,t N K X,t   Ti X,t 
10
electric displacement
Define the electric displacement D~K , such that


 ~
DK dV  QdV  dA
X K


Q X , t dV  X 
++
- + X , t dAX  +
holds for any test function (X).
Apply divergence theorem, one obtains that
in volume
on interface
~
DK X, t 
 Q X, t 
X K
D~

K
X,t   D~K X,t N K X,t   ΩX,t 
11
conservation of ions
Number of ions is conserved:
I a K  X , t 
C X, t   C X, t0  
 r a X, t 
X K
a
in volume
on interface
I
a
K
a
X, t   I Ka X, t  N K X, t   i a X, t 
r a  X , t dV  X 
i a  X , t dA X 
The above two equations is equivalent to
 C X, t   C X, t dV   I
a
a
0
a
K

dV   r a X, t dV   i adA
X K
holds for any test function   X 
12
fixed
in volume
mobile
Q  q  z 0C 0   ezaC a
by pumps
by battery
on interface
     ezai a
13
a field of weights, pumps and batteries

Work done by the weights
 B dx dV   T dx dA
i

i
i
i
Work done by the batteries
 ΦδqdV   ΦδωdA

Work done by the pumps
m a  dr a dV  m a  di a dA
14
Free energy density change of the gel element:




~ 1 2
~ 1 2
 W F, D
,
C
,
C
,


W
F
,
D
, C , C , ~
dW  
dFiK 
dDK 
~

F
DK
iK



a


~

W F, D, C 1 , C 2 ,
a
d
C
dV

C a

Free energy change of the composite system:
dxi
dxi
dG
dW
dq
d
a
a
a
a
dV

dV

B
dV

T
dA

F
dV

F
 dt
 dt
 i dt
 i dt
 dt
 dt dA  a m  dr dV  a m  di dA
work done by weights
work done by batteries
Thermodynamics:
work done by pumps
  W

a
a
a
  a  ez F  m dr dV 

  C

  W

a
a a
  a  ez F  m di dA   0

  C



  W
b 
a

 b  ez F dI K dV 
 X K  C



~
 W ~  dD
 W
 dF
dG
 
N K  siK  iK dV    ~  E K  K dV  
 D
 dt
dt
 FiK
 dt
a
 K



15
  W

a
a
a

ez
F

m
d
r
dV




a

  C

  W

a
a a
  a  ez F  m di dA   0

  C



  W
b 
a

 b  ez F dI K dV 
 X K  C



~
 W ~  dD
 W
 dF
dG
 
N K  siK  iK dV    ~  E K  K dV  
 D
 dt
dt
 FiK
 dt
a
 K



Local Equilibrium:




~
W F, D, C 1 , C 2 ,
siK 
FiK
~
~
W F, D, C 1 , C 2 ,
EK 
~
DK


~ 1 2

W
F
,
D
, C , C ,
μa  ez aΦ 
C a
Kinetic law:
J Ka
m a X , t 
X,t    M
X K
16
FiK X,t  
s

iK
x i X,t 
X K
siK X, t 
 Bi X, t   0
X K
X,t   siK X,t N K X,t   Ti X,t 
~
FX, t 
E K X, t   
X K
~
DK X ,t 
 Q X ,t 
X K
D~

K
X,t   D~K X,t N K X,t   ΩX,t 
Q q
siK
W

FiK



C a X, t  
C a X, t   C a X, t0  
I
ez aC a  ez fix C fix
~
W  W F, D, C 1 , C 2 ,
W
~
W
EK  ~
m a  ez a F  a
DK
C
#of a molecules (ions)
volume in reference state
a
K
I a K  X , t 
 r a X, t 
X K
X, t   I Ka X, t  N K X, t   i a X, t 
  
ez i
a a
~
m a X , t 
J Ka X , t    M F, D, C
X K


17
Free-energy function W F, C a , D   Ws F   Ws ol C s   Wi 0n C 1 , C 2 ,  Wp F, D 
~
~
microscopic effect
 Swelling increases entropy by mixing solvent and polymers,
but decreases entropy by straightening the polymers.
 Redistributing mobile ions increases entropy by mixing,
but increase polarization energy


Free energy of stretching
Free energy of mixing
Ws F   21 NkTFiK FiK  3  2 log det F 
Wsol

Free energy of dissolving ions

Free energy of polarization
Flory-Rhner
 s
vC s
 v 

C  kT  C log

s
s 
1

vC
1

vC


 
s
Wion


Cb

C , C ,  kT  C  log

1
s b
vC
c
b s
0



 
1
2

~
1 FiK FiL ~ ~
W p F, D 
DK D L
2ε det F
b
(Ideal dielectric material)
18
+
+- +
=
+ - +
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.
An ion occupies a same volume in the solvent or in the gel
19
siK
~ ~
Ws F  DJ D M 
1



 FiJ d MK  FnJ FnM H iK   H iK det F
FiK
 det F 
2


vsC s
1

m  kT log


1  vsC s 1  vsC s 1  vsC s

s

Cb
m  eFz  kT log s s b  vb
v C c0
b
b

2
Cb 
  s   v s
b s C 

solvent
ions
F F ~
~
E K  iJ iK DJ
 det F
20
In liquid far from interface between gel and solvent
In liquid near interface
In gel far from interface
In gel near interface
21
-- +
+
+
+ +
+
- + ++
+
- - - - +- +
+
+
+
+
+
+
+
-+
- +
+
+
- - + +-- + -+
- +
-
+
-
-
+
-
+
siK
FiK

ε det F 
iJ
MK
+
State equations
~ ~
Ws F  DJ DM 
1


F δ  F
– liquid electrolyte
Gel
+-
--
+
+ +
-
-
-
Gel

nJ FnM HiK   ΠHiK det F
2

--
External
solution
+
-
+
Infinity in liquid
c0  c0
-
External
In
solution
equilibrium
1

1
 ij  d ij   Di D j  Dm Dmd ij   0

2


v sC s
1
χ
Cb 
μ s  kT  log



 Πv s

2
s s
s s
s
s
s
1  v C 1  v C 1  v C  bs C 

m s  kTvc b  vs  2kTc0vs
Cb
μ  eΦz  kT log s s b  Πvb
v C c0
m b  eFz b  kT log
b
b
~
F F ~
EK  iJ iK DJ
ε det F
Ei 
Di

cb
 v b  0
b
c0
0
c
b

eΦz
 c0b exp 


 Πvb 

kT

b
&
Π=E=D=0
22
σ22
_
+
_
+
_
_
_
_
+
0+
_
gel
v  c0  1
_
+
+
1
m   eFz   kT log

c
 v   0

c0

d 2F
Q  e c c  
dx 2
+ solution
σ22


v  c0  1
LD
When |Φ| << kT/e

m s  kTvc b  vs  2kTc0vs
infinity
c0  c0
x
+
General solution:
1
 ij  d ij   Di D j  Dm Dmd ij   0

2


 x

eΦ  x 
 2 sgn Φ0 log  tanh
 d 
kT
 2 LD


d 2F 2ec0
eF

sinh

kT
dx 2
Debye length: LD 
kT
2e 2c0
Fx   F0exp x LD  fast decay electric field in liquid near interface
2
2
2
Stress near the interface σ22   D   2e 2c0Φ0 exp  x  Negative surface tension!


ε
ε kT
 LD 
23
+
+
C   C   C fix
- -
-
+
+
+
+
-
+
-
+
siK 
2
++
-
- +- - +
+-
--
+-
--
-
-
Gel
3
+
+
+
-
-
- +free -swelling
- -
-
-
-+
--
+ +
-
+
+
+
+
+
+
-
-
External
External
incompressibility
solution
solution
s s
3
v C   1


vsC s
1

m  kT log


1  vsC s 1  vsC s 1  vsC s

C


m  eFz  kT log s s   v   0
v C c0

C
m  eFz  kT log s s   v  0
v C c0

Infinity in liquid
c0  c0
-
-
--
-
Ws F 
 ΠHiK det F  0
FiK
s

-
-
Electric field vanishes and electric charge
Gelneutral
gel swells uniformly       
1
+
-
-
-+
-
+
+
-
+
-
+
--
+
-
-- -
+ +
-
-
infinity in gel
F  constant
- +
-
+
Cb 

  v s  2kTc0 v s
s
b s C 

 
2
Cs
F
C
C
C   C   C fix
24
- free swelling
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
25
s22
_
+
_
+
_
_
_
_
_
gel
v  c0  1
_
+
0+
+
x
+
+
+ solution
s22
LD
Inhomogeneous field 2  3  1
infinity
c0  c0
v s C s  1 22  1
incompressibility
2

1 
  dF 
 0
s11  NkT  1    22  22 
1 
2  1 dX 


vsC s
1

m  kT log


s s
s s
1v C
1v C

1  vsC s
s


2

v C C

vsC s
  v

s
 2kTv s c0
C
m  eF  kT log
0
vc0C s



22 d 2F
QX   e C 0  C X   C X    1
 dX 2


f F, F   0
Deep in gel, electric filed vanishes and no net charge
+
External solution
dF
dX

0
2kT
 eF0 
sinh 

LD e
 2kT 
dF
dX
0

26
0
10-3
10-4
+
+
_
_
_
_
_
gel
_
s22
LD
+
0+
+
+
vCs
vc 0 = 10-5
infinity
c0  c0
x
-2
-5
-4
-3
-2
-1
0
vc 0 = 10-5
41
40
10-4
39
10-3
-3
-2
x/LD
-5
-4
x/LD
-3
0
8
vc 0 = 10-3
+
+ solution
-5
10
Nv = 0.001
 = 0.1
vC0 = 0.002
-3
4
10-4
2
-2
x/LD
vc 0 = 10-5
6
10-4
-4
0
x 10
-5
-0.5
-1
-1
-5
x 10
22
_
-1
-1.5
vQ/e
_
v  c0  1
vs /kT
s22
eF/kT
-0.5
42
-1
0
0
-5
10-3
-4
-3
-2
-1
x/LD
27
0
Hydrogel : poroelasticity
Li battery : field theory coupling large deformation and electrochemistry of
ions and solvent
Electron
Positive
Electrode
Load

Negative
Electrode
Li-ion

Electrolyte
A discharging Li-ion cell.
28
29