Download A theoretical prediction of the ions in amphoteric polymer gel

Materials Science and Engineering A285 (2000) 314 – 325
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A theoretical prediction of the ions distribution in an amphoteric
polymer gel
Hirohisa Tamagawa *, Minoru Taya
Department of Mechanical Engineering, Uni6ersity of Washington, Box 352600, Seattle, WA 98195, USA
Abstract
In order to obtain the optimum design for the realization of a high performance gel actuator, the ion distribution profiles in
the gel are obtained theoretically by solving the Poisson – Bolzmann equation. Regardless of the types of ions (mobile cation,
immobile cation, mobile anion and immobile anion), ion concentration is found to change abruptly at the electrode – gel interface.
Based on this result, we found that gel could be deformed only in this interface region. Then we concluded that: (i) the use of an
amphoteric gel rather than a cationic or an anionic gel; (ii) applying high voltage to gel; and (iii) the import of the electrode–gel
interface as many as possible are promising strategies for the design of practical use gel actuator. © 2000 Elsevier Science S.A.
All rights reserved.
Keywords: Polymer gel; Ion distribution; Poisson–Boltzmann equation; The electrode – gel interface; Hydrogen bonding
1. Introduction
Application of the polymer to the actuator such as an
artificial muscle was pioneered by Kachalsky et al. [1].
They built a mechanochemical turbine made of collagen
in LiBr solution. In 1970s, the phase transition of
polymer gel characterized by its abrupt high volume
change was found by Tanaka [2]. Then the polymer gels
attracted a broad attention. Especially, this material
was considered to be promising material to imitate the
muscle. Therefore, the intensive investigations on polymer gel properties have been performed since then,
which clarified that the phase transition could be induced by a number of different types of environmental
stimuli such as type of solvents, pH temperature, electric field etc. [2–13].
Ionic gels show the higher order of volume change
ratio rather than neutral gels on account of the existence of ions in the gel network, and some ionic gels
shows thousand times of volume change ratio. The
deformation of such polymer gels is extremely high,
compared with other materials such as piezoelectric
materials. This unique property is considered to be
applicable to a numerous kinds of industrial products
such as drug delivery devices, soft but largely de* Corresponding author.
formable actuators etc.
Our research project aims at application of electrically driven polymer gel to the high deformation actuator, since the electrical actuation is a quite convenient
way as compared with solution exchange, pH change
and etc. As described above ionic gels exhibit a larger
deformation compared with neutral gels. Therefore we
are going to design ionic polymer gel for our purpose.
Tanaka et al. reported that partially hydrolyzed acrylamide could be deformed significantly by applying
electric field, since acrylamide groups were converted to
acrylic acid groups [13]. However, the gel actuators for
practical use have not been synthesized successfully yet.
Mainly two problems remain to be overcome; first, slow
response of the polymer gels to the environmental
stimuli, second, fragile structure of gels. The former
problem can be overcome to some extent by the scaling
down of unit gel size, since the volume change heavily
depends on the diffusion of the solvent. The latter can
also be overcome to some extent by adding high
amount of crosslinkers. However, users need right size
actuators, and adding crosslinkers results in slow response time. For the first step, we focus on the former
problem as to how the response time can be improved.
For the actuation of an ionic gel, the distributions of
ions contained in a gel network are expected to play a
key role. Thus, it is quite important to obtain the ion
0921-5093/00/$ - see front matter © 2000 Elsevier Science S.A. All rights reserved.
PII: S 0 9 2 1 - 5 0 9 3 ( 0 0 ) 0 0 6 8 2 - 1
H. Tamagawa, M. Taya / Materials Science and Engineering A285 (2000) 314–325
Fig. 1. Homogeneously and heterogeneously deformable gel actuator.
(a) Homogeneously deformable gel is deformed the same length lg,
towards up and down by applying electric field; (b) heterogeneously
deformable gel is deformed only towards down by applying electric
field.
Fig. 2. Cationic gel. Electric field attracts the mobile anions towards
right side, which results in a heterogeneous ion distribution.
Fig. 3. Amphoteric gel. The same amount of mobile cations and the
anions are attracted towards opposite direction, which results in a
homogeneous ion distribution in terms of macroscopic view even
under electric field.
Fig. 4. The coordinate system set to a cylindrical polymer gel, which
is laid between the electrodes and its expected potential behavior.
315
distributions theoretically in order to obtain the required gels. And as an actuator material, homogeneously
deformable
gel
is
preferable
than
heterogeneously deformable one (Fig. 1). Cationic or
anionic gel may display the heteregeneous deformation,
since its electronic structure is heterogeneous. For example, cationic gel contains the immobile cations fixed
on the polymer network and the mobile anions as
shown in Fig. 2. Without electric field, both cations and
anions distribute homogeneously. But by applying electric field, the mobile anions are attracted towards one
side, which results in the heterogeneous ion distribution. This phenomenon must cause a heterogeneous gel
deformation. This is also the case with an anionic gel.
Thus, amphoteric polymer gel is preferable than a
cationic or an anionic gel. Ion distributions in an
amphoteric gel is expected to be symmetric. Namely,
even the mobile anions are attracted towards one side
by electric field, the same amount of cations are expected to be attracted towards the other side, and then
the symmetric ion distribution is expected from the
macroscopic view (Fig. 3). This phenomenon must
cause a homogeneous deformation. Therefore, in this
paper, we show the analytical model for the ions distribution in an amphoteric polymer gel, and suggest a
promising design for the high performance gel actuator.
2. Analytical model for gel potential
The derivation method of the potential in an amphoteric polymer gel is explained. The coordinate system is
set to the cylindrical polymer gel, and for simplicity, the
potential behavior is supposed to be anti-symmetric in
relationship to z =0 as shown in Fig. 4. Since the
potential behavior is anti-symmetric, only the derivation methods of the potentials in region I, II and III are
explained. It is not necessary to derive the potential of
region IV and V.
If the concentration of the mobile anion at z=0 is
[MA](0), the anion concentration at a given point,
[MA](z), based on Boltzmann distribution is given by
Eq. (1) [14–16].
[MA](z)= [MA](0)exp +
qf(z)
kT
(1)
where q, f(z), k and T are elementary charge (1.6022×
10 − 19 C), potential at z, Boltzmann constant (1.3807×
10-23 J K − 1) and absolute temperature (room
temperature 298.15 K).
For our experimental work, the applied potential,
DV, (DV = f + L/2 − f − L/2, where f + L/2 and f − L/2 are
the potentials at the positive and the negative electrodes, respectively (Fig. 4) is expected to be rather
high. Namely, the anion concentration at z= +L/2,
[MA](+ L/2), given by Eq. (1) becomes unimaginably
H. Tamagawa, M. Taya / Materials Science and Engineering A285 (2000) 314–325
316
By use of Eqs. (3) and (4), the mobile ion concentrations at a given point are expressed by Eqs. (5) and (7),
where [MC](z) is the mobile cation concentration at a
given point.
[MC](z)=[S]w
Fig. 5. The hydrolyzed cation and anion.
exp[−(qf+L/2/kT)]
exp[−(qf+L/2)/(kT)]+exp[+(qf + L/2)/(kT)]
q(f(z) −f + L/2)
kT
exp −
= C− exp −
n
(5)
q(f(z)− f + L/2)
kT
n
(6)
where
C− [S]w
exp[− (qf + L/2/kT)]
exp[− (qf + L/2)/kT]+ exp[+ (qf + L/2)/kT]
(7)
By the same procedure, [MA] is given by Eq. (8).
Fig. 6. The hydrolyzed ions on the electrode surface.
[MA](z) =[S]w
high, even if [MA](0) is quite low. Therefore, another
method rather than using Eq. (1) should be taken up to
obtain the practical concentration value, which is explained below.
Both mobile cation and anion shapes are assumed to
be spherical. They must be hydrolyzed as shown in Fig.
5. If the water molecule is spherical in shape, its diameter is about 3 A, . If the diameter of both ions are
defined as di, the hydrolyzed ion diameter, dh, is 6+di
A, . These hydrolyzed ions are supposed to be located on
the electrode surface as shown in Fig. 6, each of them
occupies the volume of d 3h A, 3. Then the total mobile ion
(both cation and anion) concentration at z= + L/2,
[S]w is given by Eq. (2).
[S]w =
1/Vi
1
1
=
= 3
NA ViNA d hNA
(2)
where Vi and NA are mobile ion volume and Avogadro
number (6.02×1023 mol − 1), respectively.
Since the distribution of the mobile ions obey the
Boltzmann distribution, [S]w must distribute as exp(−
qf + L/2/kT):exp(+ qf + L/2/kT)
for
[MC](+ L/
2):[MA](+L/2). Then Eqs. (3) and (4) are obtained.
[MC](+ L/2)=
exp +
exp[+(qf + L + 2/kT)]
exp[− (qf + L/2)/kT] + exp[+ (qf + L/2)/kT]
q(f(z)− f + L/2)
kT
= C+ exp +
n
qf(z)− f + L/2
kT
(3)
[MC]( +L/2) =
[S]w
exp[ + (qf + L/2/kT)]
exp[(− qf + L/2)/(kT)]+ exp[( + qf + L/2)/(kT)
(4)
n
(9)
where
C+ [S]w
exp[+ (qf + L/2/kT)]
exp[− (qf + L/2)/kT]+ exp[+ (qf + L/2)/kT]
(10)
Then the total charge density, r at the given point is
given by Eq. (11).
r(z)= q([MC](z)− [MA](z)+ [IC](z)− [IA](z))
(11)
In order to obtain the potential behavior of the ionic
solution system, usually the Poisson–Boltzmann equation is employed [14–34]. The Poisson–Boltzmann
equation for this gel system shown in Fig. 7 is given by
Eq. (12).
d2f
r(z)
q
=−
= − ([MC](z)−[MA](z)+[IC](z)
dz 2
o
o
− [IA](z))
exp[ − (qf + L/2/kT)]
[S]w
exp[(− qf + L/2)/(kT)]+ exp[( + qf + L/2)/kT]
(8)
(12)
2.1. Region III
First of all, the potential behavior in the proximity of
the positive electrode is calculated. In this region
f(z)−f + L/2 must be small, namely − kT/qBf(z)−
f + L/2 B + kT/q. Therefore, Taylor expansion is appli-
H. Tamagawa, M. Taya / Materials Science and Engineering A285 (2000) 314–325
cable to Eqs. (6) and (9). Then Eqs. (13) and (14) are
obtained.
[MC](z)=C− 1−
q(f(z) −f + L/2)
kT
[MA](z)= C+ 1+
q(f(z) −f + L/2)
kT
(13)
(14)
In respect to ‘IC’ (concentration of the immobile action
fixed on the polymer network), ‘IA’ concentration of
the immobile anion fixed on the polymer network),
‘IAMC’ concentration of the functional atomic group
fixed on the polymer network, which consists of ‘IA’
and ‘MC’) and ‘ICMA’ (concentration of the functional
atomic group fixed on the polymer network, which
consists of ‘IC’ and ‘MA’), the equations of the chemical reactions, Eqs. (15) and (16) are obtained.
K
[IAMC](z)l [IA](z) +[MC](z)
K
[ICMA](z)l [IC](z) + [MA](z)
(15)
[IA](z)[MC](z)
[IAMC](z)
(17)
[IC](z)[MA](z)
[ICMA](z)
(18)
Total concentration of the fixed functional groups,
[FG]= [IC](z)+ [ICMA](z)= [IA](z)+ [IAMC](z), is
independent of the portions of the gel. For example,
Fig. 8 shows two different portions of a same gel. As to
Fig. 8a, there are four ‘ICMA’ and four ‘IAMC’
namely, total number of fixed functional groups are
four, respectively. As to Fig. 8b, there are three
‘ICMA’, one ‘IAMC’, one ‘IC’, three ‘IA’, one ‘MA’
and three ‘MC’ namely, total number of the fixed
functional groups, 3(ICMA)+ 1(IAMC), 1(IC)+3(IA),
are four, respectively. Total concentration of the fixed
functional groups never varies with the change of the
part of gel. Thus, Eqs. (19) and (20) hold.
[FG]= [IC](z)+ [ICMA](z)= constant
(19)
[FG]=[IA](z)+[IAMC](z)= constant
(20)
By use of Eqs. (18), (19) and (21) is obtained
(16)
where K is dissociation constant. For simplicity, both
Ks of Eqs. (15) and (16) are assumed to be the same.
Then Eqs. (17) and (18) are obtained.
K=
K=
317
K=
[IC](z)[MA](z)
[FG]− [IC](z)
(21)
Eq. (22) is obtained by solving Eq. (21) with respect to
[IC](z).
[IC](z)=
K[FG]
K+ [MA](z)
(22)
Substitution of Eq. (9) for Eq. (22) results in Eq. (23).
[IC](z)=
K[FG]
K+ C+ exp[+ q(f(z)− f + L/2)/(kT)]
(23)
:
K[FG]
K+ C (1+q(f(z)− f + L/2)/(kT))
(24)
=
K[FG]
(K+C+)+ {[C+q(f(z)− f + L/2)]/(kT)}
(25)
+
1
K[FG]
+
+
+
K+ C 1+[(C )/(K + C )]{[q(f(z)− f + L/2)]/
(kT)}
(26)
=
Fig. 7. The polymer gel network. ICMA, IAMC, IC and MC are
immobile atomic groups fixed on the polymer network. MC and MA
are mobile ions.
=
C+ q(f(z)− f + L/2)
K[FG]
1+
K+ C+
kT
K+ C+
:
K[FG]
C+ q(f(z)− f + L/2)
1−
+
K+ C
K+ C+
kT
−1
(27)
(28)
By use of Eqs. (17) and (20), Eq. (29) is obtained.
K=
[IA](z)[MC](z)
[FG]− [IA](z)
(29)
Eq. (30) is obtained by solving Eq. (29) with respect to
[IA](z).
Fig. 8. (a) Shows a small part of the gel system which contain only
the immobile functional atomic groups fixed on the gel network; (b)
shows the other part of the gel system which contain mobile cations
(MC) and anions (MA) as well as the fixed functional atomic groups.
[IA](z)=
K[FG](z)
K+ [MC](z)
(30)
Substitution of Eq. (6) for Eq. (30) results in Eq. (31).
H. Tamagawa, M. Taya / Materials Science and Engineering A285 (2000) 314–325
318
[IA](z)=
K[FG]
K+ C− exp{ − [q(f(z) − f + L/2)]/(kT)}
(31)
:
K[FG]
K+ C−{1− [q(f(z) −f + L/2)/(kT)]}
(32)
=
K[FG]
(K+C−)− [C−q(f(z) −f + L/2)/kT]
(33)
=
1
K[FG]
−
−
−
K +C 1− [C /(K + C )]{q[f(z) −f + L/2]/kT}
=
:
−
K[FG]
C
q[f(z) −f + L/2]
1−
−
−
K+ C
kT
K+ C
−
(34)
−1
(35)
q[f(z) −f + L/2]
C
K[FG]
1+
K+ C−
kT
K+ C−
q −
K[FG]
K[FG] qf + L/2
C − C+ +
−
+
+
kT
o
K+C
K+C−
−
C− + C+ +
A
B −
1+
=−
q[f(z) −f + L/2]
kT
−
K[FG]
C− q[f(z) −f + L/2]
1+
−
K +C−
K+C
kT
(37)
K[FG]q
C+
C−
+
f(z) +C− − C+
+ 2
kT
(K+C )
(K + C−)2
q
qf
K[FG]
f(z) +(C− +C+) + L/2 +
kT
kT
K+ C+
− (C− + C+)
K[FG] K[FG]qf + L/2
C−
C+
−
+
+
−
+ 2
K+C
kT
(K + C )
(K + C−)2
(38)
q
K[FG]C+ K[FG]C−
=−
(C− + C+) +
+
f(z)
kT
(K + C+)2 (K + C−)2
+ C− − C+ +
K[FG]
K[FG]
−
K+ C+ K + C−
+
= Af + B
−
(41)
+
K[FG]
K[FG]
q −
C − C+ +
−
+
K+C
K+C−
o
(42)
qf + L/2 −
K[FG]C+ K[FG]C−
C + C+ +
+
kT
(K+C+)2 (K+C−)2
df
]0
dz
„
Ca = o
(39)
S
L
(45)
Q=CaDV =o
S
(f
− f − L/2)
L + L/2
(46)
The surface charge density, s, is given by Eq. (47).
Q
f + L/2 − f − L/2
s= = o
L
S
(47)
s can be rewritten by Eq. (48).
)
df
dz z = + L/2
(48)
By use of Eq. (44), Eq. (49) is obtained.
)
df
2
= Af +
L/2 + 2Bf + L/2 + C
dz z = + L/2
By the use of Eqs. (12) and (39) Poisson – Boltzmann
equation, Eq. (40), is obtained.
ƒ
r(z)
q2
K[FG]C+
d2f(z)
=−
=
(C− +C+) +
2
dz
o
okT
(K + C+)2
−
K[FG]C
+
f(z)
(K + C−)2
By use of Eqs. (48) and (49), Eq. (50) is obtained.
(44)
where S is electrode surface area.
The surface charge on the positive electrode, Q, is
given by Eq. (46).
s= o
qf
K[FG]C
K[FG]C
+ + L/2 C− +C+ +
+
+ 2
kT
(K + C )
(K + C−)2
(43)
The surface charge density of the positive electrode is
obtained in order to determine the value of C. Both the
positive and the negative electrodes are supposed to
constitute a condenser. Then its capacitance, Ca, is
given by Eq. (45).
K[FG]
C+ q[f(z) −f + L/2]
1
−
K+C+
K +C+
kT
2
df
= Af 2 + 2Bf + C
dz
+
(40)
K[FG]C+ K[FG]C−
q2
(C− + C+)+
+
okT
(K+C+)2 (K+C−)2
(36)
q[f(z) −f + L/2]
−C+
kT
d2f 1 d df
=
dz 2 2 df dz
r = q([MC](z)− [MA](z) + [IC](z) − [IA](z))
where o is dielectric constant of water (80×8.85×
10 − 12 F m − 1).
Eq. (40) is solved by the following procedure.
The charge density, r(z), is given by Eq. (11). By use of
Eqs. (13), (14), (28) and (36), Eq. (37) is obtained.
= C− 1−
K[FG]C+ K[FG]C−
+
(K+C+)2 (K+C−)2
2
s= o
Af +
L/2 + 2Bf + L/2 + C
(49)
(50)
Eq. (51) is obtained by use of Eqs. (47) and (49).
H. Tamagawa, M. Taya / Materials Science and Engineering A285 (2000) 314–325
o
f + L/2 −f − L/2
2
=o
Af +
L/2 +2Bf + L/2 +C
L
f + L/2 − f − L/2
U
L
(51)
2
=Af
2
+ L/2
(52)
+2Bf + L/2 +C
UE+ − 2B −2Af =2
A(Af 2 + 2Bf + C)
UF+ − 2Af
(64)
=2
A(Af 2 + 2Bf + C) (F+ E+ − 2B)
(65)
UF+2 − 4AF+f+ 4A 2f 2 = 4A 2f 2 + 8ABf +4AC
(66)
Then C is given by Eq. (53).
f
−f − L/2
ƒC = + L/2
L
2
−Af
2
+ L/2
(53)
−2Bf + L/2
Eq. (53) is solved to obtain the analytical expression of
f.
df
= Af 2 +2Bf + C
dz
&
U
df
Af + 2Bf +C
2
Uz =
1
A
(54)
&
= dz
(55)
)
)
log 2Af + 2B + 2
A(Af 2 +2Bf +C) + D
(56)
)
L
1
D= −
log 2Af + L/2 +2B
2 A
2
+ 2
A(Af +
L/2 +2Bf + L/2 +C)
1
A
)
log
)
if
Uf =
)
2
(58)
)
2Af + 2B +2
A(Af 2 +2Bf + C)
+2Bf + L/2 + C)
(59)
2Af +2B +2
A(Af 2 +2Bf + C)
2
2Af + L/2 + 2B + 2
A(Af +
L/2 +2Bf + L/2 +C)
2
2Af + L/2 + 2B +2
A(Af +
L/2 + 2Bf + L/2 +C)
f=
(69)
(70)
(71)
\0
(60)
2
2Af + L/2 +2B+2
A(Af +
L/2 +2Bf + L/2 +C)
By use of Eqs. (23), (31) and (73) is obtained.
[IC](z)− [IA](z)
K[FG]
K+ {[S]w/(exp[−(qf + L/2)/kT]+ exp[+ (qf + L/2)/
kT])exp[+ (qf(z))/kT]
=
K[FG]
K+ {[S]w/(exp[−(qf + L/2)/kT]+ exp[+ (qf + L/2)/
kT])exp[− (qf(z))/kT]
(73)
−
=
K[FG]
K+ G exp[+ (qf(z))/(kT)]
−
(61)
(62)
where E+
(63)
K[FG]
K+ G exp[−(qf(z))/(kT)]
(74)
where G
[S]w
exp[− (qf + L/2)/(kT)]+ exp[+ (qf + L/2)/(kT)]
:−
e A[z − (L/2)](2Af + L/2 +2B
(72)
2.2. Region II
2Af + 2B +2
A(Af +2Bf + C)
2
+2
A(Af +
L/2 +2Bf + L/2 +C))
F−2 − 4AC
= f−
III(z)
8AB+ 4AF−
2
+ 2
A(Af +
L/2 + 2Bf + L/2 + C))
2
UE+ = 2Af + 2B +2
A(Af 2 +2Bf + C)
B0
where
e A[z − (L/2)] =
+
(68)
By the same procedure described for case (i), Eq. (70) is
obtained.
F− E− − 2B
)+L
2
2Af+L/2+2B+2
A(Af +
L/2+2Bf+L/2+C)
2Af + L/2 +2B +2
A(Af
2Af + 2B +2
A(Af 2 + 2Bf + C)
(57)
2Af+2B+2
A(Af 2+2Bf+C
2
+ L/2
F+2 − 4AC
= f+
III(z)
8AB+ 4AF+
(67)
and
E− − e A[z − (L/2)](2Af + L/2 + 2B
Ulog(e A[z − (L/2)])
= log
UF+2 − 4AF+f= 8ABf +4AC
if
Since f = f + L/2 at z = +L/2, Eq. (57) is obtained.
ƒz =
319
K[FG]
K+ G exp[− (qf(z))/(kT)]
(75)
(76)
By use of Eqs. (5), (8) and (75), Eq. (77) is obtained.
H. Tamagawa, M. Taya / Materials Science and Engineering A285 (2000) 314–325
320
[MC](z)−[MA](z)= G exp −
qf
qf
−G exp +
kT
kT
(77)
: − G exp +
H= Af 2III + BfIII + C−
(78)
f=
q[FG] 2
z + Hz+J
2o
J is given by Eq. (93).
r(z)
q
df
=−
= − ([MC](z) −[MA](z) + [IC](z)
dz 2
o
o
JfIII −
2
− [MA](z))
=
q K[FG]+ KG exp( +qf/kT) +G 2
o K+G exp[(− (qf(z))/(kT))]
(81)
:
q K[FG]+ KG exp( +qf/kT)
o K+ G exp[(−(qf(z))/(kT))]
(82)
if K= 10 − 4 M (10 − 1 mol m − 3)
:
q K[FG]+KG exp( +qf/kT)
o
K
(83)
=
qf
q
[FG]+G exp +
kT
o
(84)
if [FG] =1000 mol m − 3 (1 mol l − 1) and dh = 10 A,
(Fig. 2), G B[FG].
q
: [FG]
o
(85)
d2f q
= [FG]
dz 2 o
(86)
df q[FG]
=
z+ H
dz
o
(87)
Since (df/dz)z = zIII, Eq. (88), is obtained, by use of Eq.
(44), where zIII is the coordinate value of z at the
boundary between Region III and II.
)
df
=
Af 2III +BfIII +C
dz z = z III
(88)
kT
q
(89)
zIII, Eq. (90), is obtained, by use of Eq. (58).
A
) 2Af
III
=[S]w
exp[−(qf+L/2)/(kT)]
exp[(−(qf+L/2)/(kT))]+exp[(+(qf+L/2)/(kT))]
q(f(z)− f + L/2)
− [S]w
kT
exp[+ qf + L/2/kT]
exp[(−(qf + L/2))/(kT)]+ exp[(+ (qf + L/2))/(kT)]
exp −
exp +
=
q(f(z)− f + L/2)
kT
(94)
[S]w
exp[(− (qf + L/2)/(kT))]+exp[(+ (qf + L/2)/(kT))]
exp −
=−
qf + L/2
qf
− exp + + L/2
kT
kT
qf(z)
qf(z)
− 1−
kT
kT
2Gq
f(z)
kT
(95)
(96)
(97)
By the use of Eqs. (23) and (31), Eq. (98) is obtained.
Eq. (86) can be solved easily.
log
[MC](z)− [MA](z)
: G 1−
Consequently, Eq. (86) is obtained.
1
(93)
By use of Eqs. (5) and (8), Eq. (94) is obtained
(80)
=
q[FG] 2
z III − HzIII
2o
2.3. Region I
qf
−G exp +
kT
where fIII = f + L/2 −
(92)
(79)
q
K[FG]
−
o
K+ G exp[( −(qf(z))/(kT))]
(91)
Solving Eq. (87) results in the expression of f.
qf
kT
The charge density is given by Eq. (11). Then Poisson–
Bolzmann equation is obtained.
=−
q[FG]
zIII
o
+ B +2
A(Af 2III + BfIII + C)
2
+L/2
2Af+L/2+B+2
A(Af
+Bf+L/2+C
[IC](z)− [IA](z)
=
K[FG]
K+ C exp[+ [q(f(z)− f + L/2)]/kT]
+
−
K[FG]
K+ C−exp[− [q(f(z)− f + L/2)]/kT]
K[FG]
K+ [S]w{[exp(+qf + L/2/kT)]/[exp( − qf + L/2/kT)+
exp(+ qf + L/2/kT)]}exp[+ q(f(z)− f + L/2)/kT]
=
K[FG]
K+ [S]w{[exp(−qf + L/2/kT)]/[exp( − qf + L/2/kT)+
exp(+ qf + L/2/kT)]}exp{− q[f(z)− f + L/2]/kT} (99)
−
)+L
2
(90)
(98)
K[FG]
K+{[S]w/[exp(−qf(z)/kT)+exp(+ qf(z)/
kT)]}exp[+ qf(z)/kT]
=
H. Tamagawa, M. Taya / Materials Science and Engineering A285 (2000) 314–325
K[FG]
K+ {[S]w[exp(−qf + L/2/kT)+ exp( + qf + L/2/
kT)]}exp[+qf(z)/kT]
(100)
−
=
K[FG]
K+ G exp[(−(qf(z))/kT)]
1
K+G exp[(+(qf(z))/kT)]
(102)
= K[FG]
G{exp[− qf(z)/kT]−exp[ + qf(z)/kT]}
2
K + KG[(exp−(qf(z))/(kT)) + exp( + (qf(z))/
: K[FG]
=
− 2G(qf(z))/kT)
K 2 +2KG
=−
(120)
M=
q[FG]
2o
kq
(121)
(N= J− fII)
(122)
− H+
H 2 − 4MN
2M
(zII ] 0)
(123)
By use of the Eqs. (117) and (118), Eq. (124) is obtained
and solved with respect to M+.
kT
= M+(e + kzII − e − kzII)
q
UM+ =
q(e
kT
− e − kzII)
+ kzII
(124)
(125)
Consequently, Eq. (126) is obtained.
ƒf(z)=
(110)
=
(111)
kT
(e + kz − e − kz)
q(e + kzII − e − kzII)
kT e + kz − e − kz
q e + kzII − e − kzII
(126)
(127)
Since k, z and kzII are quite small. Taylor expansion is
applicable to Eq. (127).
The solution is given by Eq. (112).
'
(118)
UMz 2II + HzII + N=0
(109)
2G(K+ 2G +2[FG])
f
o(K+2G)kT
− −kz
kT
= fII
q
(119)
(108)
q 2Gq(K+ 2G + 2[FG])
=
f
o
(K+2G)kT
+M e
Eq. (118) is the definitions of f(zII), where zII is the
coordinate value of z at the boundary between Region
II and I.
(107)
q
2Gq
2G[FG]q
−
−
f
o
kT (K + 2G)kT
f(z)=M e
(117)
UzII =
−[IA](z))
+ +kz
ƒf(z)= M+(e + kz − e − kz)
q[FG] 2
z II + HzII + (J− fII)= 0
2o
df(z)2
r(z)
q
=−
= − ([MC](z) −[MA](z) +[IC](z)
2
o
o
dz
=q2
(116)
(106)
2Gq[FG]
f(z)
(K+ 2G)kT
UM− = − M+
zII is given by the following procedure. By use of Eq.
(92), Eq. (119) is obtained.
The charge density is given by Eq. (11). Then the
Poisson–Bolzmann equation obtained.
=−
(115)
(103)
(105)
−2Gqf(z)K[FG]
K(K+ 2G)kT
UM+(e + kz + e − kz)= − M−(e + kz + e − kz)
f(zII)=
G{1−[qf(z)/kT] −1 −[qf(z)/kT]}
K 2 + KG{[1−(qf(z))/kT]+ [1 +(qf(z))/kT]}
(104)
= K[FG]
(114)
(101)
1
K+G exp[(− (qf(z))/kT)]
(kT))]+G 2]
(113)
ƒM+e + kz + M−e − kz = − (M+e − kz + M−e + kz)
= K[FG]
−
Since f(z) is anti-symmetric in relationship to z =0,
the boundary condition, given by Eq. (113) is obtained.
f(z)= − f(− z)
K[FG]
K+ G exp[(+(qf(z))/kT)]
−
321
2G(K+2G+2[FG])
o(K+2G)kT
(112)
:
kT 1+ kz− (1−kz)
q 1+ kzII − (1− kzII)
(128)
:
kT 2kz
q 2kzII
(129)
322
H. Tamagawa, M. Taya / Materials Science and Engineering A285 (2000) 314–325
Fig. 9. The potential and ion concentration profiles in case where DV=1.0. (a) – (c), (d) – (f) and (g) – (i) show the profiles in case of gel length
L =10, 15 and 20 mm, respectively.
=
kT
z
qzII
(130)
3. Results and discussions
Figs. 9 and 10 show the z dependence of potential
and each ion concentration. Through total amount of
MC and MA should be same as that of IC and IA,
these results do not fulfill that requirement. Therefore,
these results are not so accurate quantitatively. However, based on these results, we cold qualitatively predict the optimum design for the electrically driven gel
actuators. Hereafter imagine that IC, MA, IA and MC
−
−
are CH2NH+
and H+, respectively.
3 , OH , COO
Then chemical reactions are represented by Eqs. (131)
and (132).
COOH =COO− +H+
(131)
−
CH2NH2 + H2O =CH2NH+
3 +OH
(132)
Figs. 9 and 10 suggest that the higher applied
voltage, DV, induces the more abrupt drop of
−
[CH2NH+
3 ] and [COO ] in the proximity of positive
and negative electrode surfaces, respectively. On the
other hand, the more abrupt soaring of [H+] and
[OH−] is induced in the proximity of negative and
positive electrode surfaces, respectively. These results
are interpreted that the higher DV gives rise to the
stronger attraction of the more plenty of H+ and OH−
toward the negative and positive electrodes, respectively. Then the soaring of [H+] and [OH−] makes both
the chemical reactions of Eqs. (131) and (132) proceed
left hand side strongly which results in the decrease of
−
[CH2NH+
3 ] and [COO ]. These results are summarized in Table 1.
Hydrogen bonding is one of the key elements, which
induce the phase transition [9]. For example, Fig. 11a
show the gel network which contains COOH and
CONH2 atomic groups. In this caseCOOH is ionized
into COO− + H+ and the gel is in the swelling status.
However, if COOH atomic groups. In this case,
COOH is ionized into COO− + H+ and the gel is in
the swelling status. However, if COOH is produced by
the association of COO and H+, the gel shrinks on
account of the formation of hydrogen bonding through
COOH as shown in Fig. 11b. Hydrogen bonding play
a part as a bridge between the gel network. Then gel
shrinkage is realized. As described above, there are
spates of COOH at the negative electrode surface
region. Thus shrinkage of the gel at the negative electrode surface region is expected due to the hydrogen
bonding (Fig. 12a). Since CH2NH2 also forms hydrogen bonding, the same phenomenon must be observed
in the positive electrode surface region (Fig. 12b).
To sum up, increase of [H+] and [OH] give rise to the
increase of [COOH] and [CH2NH2], which results in
H. Tamagawa, M. Taya / Materials Science and Engineering A285 (2000) 314–325
323
Fig. 10. The potential and ion concentration profiles in case where DV= 1.5. (a) – (c), (d) – (f) and (g) – (i) show the profiles in case of gel length
L =10, 15 and 20 mm, respectively.
the occurrence of gel shrinkage due to the formation of
hydrogen bonding. If DV is higher then large amount of
hydrogen bonding are expected. And since higher DV
gives rise to higher [H+] and [OH−] as described earlier, higher DV must induce the more effective
shrinkage.
Amphoteric gels will deform highly than cationic or
anionic gels. Though both cationic and anionic gels
must be deformed only at positive or negative electrode
surface region (Fig. 12), amphoteric gels must be deformed at both positive and negative electrode regions,
which results in two times larger deformation as compared with cationic and anionic gels. Amphoteric gels
are also expected to exhibit quicker response time compared with cationic and anionic gels due to the same
reason for the occurrence of larger deformation.
Contrary to the explanations above about the result
shown in Figs. 9 and 10, these results may be explained
that the lower DV gives rise to the increase of [H+] and
[OH−] in the broader region compared with higher DV
case. This explanation results in the increase of
[COOH] and [CH2NH2] in the broader region compared with higher DV case. If this is true, the occurrence of gel shrinkage due to the formation of hydrogen
bonding is observed in quite a wide range in case a
quite lower DV is applied. However, this explanation
conflicts with the experimental evidences as well as our
intuitions. It is already observed experimentally that
some hydrophilic gel shrinks under electric field in an
Table 1
Ion concentrations
The region in the proximity of the positi6e electorde surface
[CH2NH2]
[CH2NH+
[OH−]
3 ]
high
Low
High
The region in the proximity of the negati6e electrode surface
[COOH]
[COO−]
[H+]
High
Low
High
Fig. 11. Molecular structure of the gel when it swells and shrinks. (a)
Molecular structure of gel network in the swelling status; (b) molecular structure of gel network in the shrinking status.
324
H. Tamagawa, M. Taya / Materials Science and Engineering A285 (2000) 314–325
Fig. 13. Design of a gel actuator L − L1 L −L2. (a) Non-effective
gel actuator; (b) effective gel actuator.
Fig. 12. Shrinkage of gel due to the hydrogen bonding, (a) due to
formation of the hydrogen bonding that COOH plays a part,
negative electrode surface region of gel shrinks; (b) due to
formation of hydrogen bonding that CH2NH2 plays a part,
positive electrode surface region of the gel shrinks.
the
the
the
the
Table 2
Comparison of ion concentrations between the cases DV=1.0 and
1.5 V
DV= 1.0 V
The region in the proximity of the positi6e electrode surface
[CH2NH+
[OH−]
[CH2NH2]
3 ]
High
Low
High
The region in the proximity of the negati6e electrode surface
[COOH]
[COO−]
[H+]
High
Low
High
DV= 1.5 V
The region in the proximity of the positi6e electrode surface
[CH2NH2]
[CH2NH+
[OH−]
3 ]
Extremely high
Extremely low
Extremely high
The region in the proximity of the negati6e electrode surface
[COOH]
[COO−]
[H+]
Extremely high
Extremely low
Extremely high
aqueous solution, and it remains in swelling state under
no electric field [3]. Therefore, even if lower DV realizes
a broad region containing high amount of CH2NH2
and COOH at the positive and the negative electrode
surface regions, respectively, it is thought that
[CH2NH2] and [COOH] are not sufficiently high to
form enough amount of hydrogen bonding to induce
gel shrinkage. These results are summarized in Table 2.
As it is easily understood, abrupt change of ion
concentration is observed only in the interface between
electrode surface and gel terminal. The interface is
supposed to be quite important to realize a deformation
of gel. In order to obtain high performance gel actuator, it is necessary to import the interface as much as
possible (Fig. 13) in addition to using an amphoteric gel
and applying high voltage.
4. Conclusion
We obtained the potential and ion concentration
behavior theoretically, and predicted the expected properties of gel actuator. It was pointed out that the
variation of the mobile ion distribution gives rise to the
formation and the breaking of the hydrogen bonding.
Namely if we can build an actuator by use of gel whose
volume change is dominated by hydrogen bonding, we
can deform it deliberately by adjusting the applied
voltage. In addition to the use of such a kind of gel the
importance of three factors were suggested: (i) use of an
amphoteric gel; (ii) import of as many electrode–gel
interfaces as possible; and (iii) applying high voltage.
However, still serious problems remain to be overcome.
Applying high voltage induces the decomposition of
water solvent. Gel structure is quite weak. Response
time is still slow for the practical use actuator and so
on. For our next tasks, we investigate these factors to
improve the gel properties.
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