9583
J. Phys. Chem. 1994, 98, 9583-9587
Electrokinetic Modeling of the Contractile Phenomena of Polyelectrolyte Gels. One-Dimensional Capillary Model J. P. Gong, T. Nitta, and Y. Osada* Division of Biological Sciences, Graduate School of Science, Hokkaido University, Sapporo 060, Japan Received: December 29, 1993; In Final Form: July 5, 1994@
When a water-swollen polyelectrolyte gel is interposed between a pair of electrodes and DC current is applied, the gel undergoes electrically-induced chemomechanical contraction and concomitant water exudation in the air. In order to clarify the mechanism, a one-dimensional electrokinetic model of the contractile phenomenon of polymer gels under an electric field was postulated based on Poisson-Boltzmann and Navier- Stokes equations. This model well characterized the nature of the contraction profiles and theoretically predicted that the rate of contraction linearly depends on the electric field, and the contraction efficiency inversely depends on the cross-linking density. The model also showed a good agreement with experimental data and qualitatively demonstrated that the electrically-induced contraction of the gel is dominated by electrokinetic processes of hydrated ions and water in the polymer gels.
I
1. Introduction
Anode
When a water-swollen cross-linked polyelectrolyte gel is inserted between a pair of electrodes and dc voltage is applied by using an apparatus shown in Figure 1, it undergoes anisotropic contraction and concomitant fluid (water) Several explanations have been proposed to describe the mechanism of the gel contraction under the electric field such as local pH change near the volume collapse by phase transition,8 and others. We have reported in earlier papers that the electricallyinduced contraction of the gel is caused by a transport of hydrated ions and water in the network and.demonstrated several experimental facts supporting an interpretation that the contractile behaviors observed are essentially electrochemical p h e n ~ m e n a . ~ ?They ' ~ are (1) the absolute absence of the contraction for neutral (noncharged) hydrogel, (2) slight swelling near the cathode and extensive contraction near the anode for the polyanionic gel and the reverse for the cationic gel (refers to Figure 2), and (3) the rate of contraction being proportional to the electrical current. The most striking support for the presumed electrokinetic nature would be the direct observation of migration of water and of the charged ions toward the electrode bearing a charge opposite in sign to that borne by the polymer n e t ~ o r k .The ~ observed contractile behaviors of the hydrogels led us to consider the possibility that the external electric field undergoes direct interaction with macroions and counterions in the gel to transport water and ions by electroosmosis and electrophoresis (electrokinetic phenomenon). In other words, application of an electric field causes a pumping of largely mobile counterions and macronetwork ions together with the surrounding water in opposite directions until mobile ions reach the electrode, whereupon the velocity of migration and the velocity of gel contraction are governed by both the quantity of mobile ions and the electric field. The objective of this paper is to theoretically establish the mechanism of the contractile phenomenon and postulate a model expressing the total contraction process. Modeling of the contractile behavior of the gel based on the electrokinetic
* Author to whom correspondence should be addressed. @
Abstract published in Advance ACS Abstracts, August 15, 1994.
,
/Platinum
mesh electrode
DC 1OV
Meter
7 7 Balance
Figure 1. Experimental setup for measuring the gel contraction. Size of the gel: 10 x 10 x 10 mm3. (Anode)
/--7l
Omin
2min
6min r
Time Figure 2. Schematic view of shape change of gel with time under the electric field.
explanation has been made by combining the Poisson-Boltzmann equation with Navier-Stokes equation. A cylindrical symmetry has been used in the simulation to calculate the electrostatic potential of a polyelectrolyte solution. By comparing it with experimental results, our theory proved that the contractile behavior is essentially associated with electrokinetic phenomenon, i.e., transportation process of counterions (electrophoretic process) and water molecules (electroosmotic process) in the water-swollen cross-linked network. 2. Theory
As shown schematically in Figure 2, an extensive contraction occurs at one electrode of the polymer gel due to water supply
0022-3654/94/2098-9583$04.50/0 0 1994 American Chemical Society
Gong et al.
9584 J. Phys. Chem., Vol. 98, No. 38, 1994
axis, we get a frictional force
Ff = q2nr[dv(r)/dr], - q2n(r
+ dr)[dv(r)/dr](,+,,,
(1)
due to the difference in velocity between the volume element and its outside and inside adjacent volume elements. Here, q is the viscosity of solution. The electrical force on the same volume element is
F, = 2 n r d r g(r)E
(2)
where e ( r ) is the local charge density and E is the electric field. Both of the forces are expressed per unit length. Since the electric force should be balanced by the frictional force, a steady velocity of motion is given
Macro-ion Figure 3. Schematic model of the rodlike macroion chains. As described in the text, only the macroions parallel to the electric field
are considered (one-dimensional model). toward the other electrode, we have assumed that this contraction is governed by a migration of charged counterions toward the oppositely-signed electrode (electrophoresis) together with the water. When an outer electrical field is applied across the gel, both the macroion and the counterions experience electrical forces in opposite directions. We consider here that the macroions are a stationary phase since they are chemically fixed to the polymer network, while the counterions are mobile and are capable of migrating along the electric field and dragging water molecules with them. We have supposed that the macroions are evenly distributed along the extended macromoleculeswith an infinitely long chain compared with its radius. We have considered only the macroions located along the chains parallel to the electric field applied and neglected the effects of cross-linking points and the chains located perpendicular to the electric field. Under such a simplification, the problem of water transportation under the electric field in the polymer network becomes simply a problem of the transportation along the rodlike polymer chain. This problem is analogous to that of liquid flow through a capillary under the influence of electric field and we used a rodlike model developed by A. Katchalsky" in order to estimate the electrostatic potential distribution of the network based on the assumption that the polyelectrolyte chains are fully ionized and take relatively extended rodlike conformations due to strong intramolecular electrostatic repulsion. According to Debye-Hiickel theory, the electrostatic potential of the network can be obtained by solving the PoissonBoltzmann equation. The counterion distribution can be derived from the Poisson equation. Preliminary calculation has shown that the intermolecular electrostatic repulsion between two macromolecular chains can be neglected when the gel is highly swollen. We consider here a tube composed of two concentric cylinders of inner radius ri and outer radius r,, which corresponds to the radius of macroion and the distance to the midpoint of two adjacent macroions, respectively (Figure 3). When an electrical field is applied across this tube, water molecules in the tube start to move. The frictional force resists this movement and a steady state is quickly obtained. If we suppose that the flow is laminar, the water flow velocity v(r) is a function of r, the distance from the cylindrical axis due to the cylindrical symmetry. The velocity will be zero at r = ri, at the surface of macroion chain and a maximum at r = r,, which is the midpoint of two adjacent macroions. Considering a volume element contained between two concentric cylinders, a distance d r apart, with the inner cylinder located r from the
2 n r d r e ( r ) E = q2nr[dv(r)/dr],- q2n(r -I- dr)[d~(r)dr](,,~,
(3) using Taylor's expansion, we have [dV(r)dr](,+d,
= [dv(r)/dr],
-t [d2v(r)/dr*],d r
(4)
If eq 4 is substituted into eq 3 and terms in (dr)2 are neglected, eq 3 becomes 2 n r d r &-)E = -q2n{r[d2v(r)/dr*] d r
+ [dv(r)/dr] dr}
(5)
Dividing each side by 2nr dr, we get
e(r)E = -q[d2v(r)/dr2
+ (l/r) dv(r)/dr]
(6) The right term of eq 6 is the Laplace operator over v(r) in cylindrical coordinates. Therefore, e(r)E = -qAv(r) (7) Equation 7 is the simplified form of Navier-Stokes equation,'* which is a general equation describing fluid flow in hydrodynamics. The charge density, e ( r ) , is connected with the electrostatic potential V(r) by Poisson's equation
Aq(r) = -Q(r)/e Here,
E
(8)
is the dielectric constant of water. Therefore
A V ( 4 = [q/(€E)I'4V(4 (9) If we assume that q , E , and E are constant with r and denote Y = V ( r ) - [q/(EE)IV(r)
(10)
Eq 9 becomes Ay=O (1 1) The general solution of eq 11 with the cylindrical symmetry is
y = c1 In r Here
c1
and
c2
+ c2
(12)
are integration constants. Thus, we get
v(r) = (eE/q)[V(r)- c1 In r - c2]
(13)
Combining with boundary conditions of v(ri) = 0, dv(r,)/dr = 0 the integration constants can be determined c1 = 0 and c2 = V(ri)
(*:
#(I,)
(14)
= 0)
so
v(r) = (eE/v) [ V ( r ) - V@Jl (15) The electrostatic potential V ( r ) is given by the PoissonBoltzmann equation A+(r) = -(en/€) exp[-etp(r)/kTl (16) Here, e is the charge of an electron, k the Boltzmann constant,
Contractile Phenomena of Polyelectrolyte Gels
J. Phys. Chem., Vol. 98, No. 38, 1994 9585
and T the temperature. The counterion concentration, n, with units of l/m3, can be expressed by
thus, the rate of water transportation to that of charge u(r) is
= dw/dQ = l/@(r) (27) Taking an average u(r) over one macroion, we get the gel U(T)
n = d[.x(r: - rf)b]
(17)
Here, a is the degree of ionization and b is the distance between ionizable groups on the polyanion. A. Katchalsky et al. successfully solved eq 16 with the rodlike model:" +(I)
= (kT/e) l n { ( ~ ~ r ~ / 2sin2[-tan-' /3~)
p + p ln(r/ro)]} (18)
where K~
(19)
= ne2/rkT
A = (1 + P2)/[1- p cot@)]
(20)
A = ae2/4ncbkT (21) is the value corresponding to the Debye parameter of low molecular electrolyte solution, il is a dimensionless quantity expressing the ionic strength of the polymer network, and p is an intermediate parameter defined by eq 20, which can be numerically solved by using Newton's method. The quantity s is given as K
s = ln(ri/ro) = In q-'l2
(22)
since (ri/ro)2= (nr,b/nrob)2= Vd/V, = l/q, where 4 is the degree of swelling and v d and V, are gel volumes in the dry and wet state, respectively. From Poisson's equation (8), the distribution and density of counterions may then be derived from the potential
e ( r ) = (2kT@12/e)/{r2 sin2[-tan-'
p + p ln(r/ro)]}
p ln(r/r,)]/sin[-tan-'
W/W, = 1 - JQuaVdQ/LoS = 1 - uavQ/L0S
(29)
When a constant voltage U is applied across the gel, E would change during the contraction process with time because the distance between two electrodes changes accordingly. Thus, both E and v(r) should be a function of t. By denoting E at time t as E(t), v(r) as v(r,t), and vav(t) as the average of v(r,t) over r, we get
E(t) = U/L(t)= U/[Lo= J'vav(f) dt]
(30)
Here, L(t) is the length of gel at t. From eq 15, we have
v(r9t>= [W)/VI [+(r) - +(ri>l = v(~,o)E(~)/E(o) (3 1)
(23)
fi + p + p ln(ri/ro)]}2 (24)
In order to obtain the feature of the contraction process, we have empirically made the following assumptions: (1) The gel under contractionwas divided into two regions, the region which has been initially contracted (shadowed in Figure 2 ) and the region which contracted later. (2) Only consider the uncontracted region, neglecting the region already contracted. (3) The uncontracted part of the gel does not change its initial degree of swelling in the course of the contraction. The above assumptions were made in order to ensure that the contraction of a gel occurs only in the direction of the electric field E, i.e., in the direction of water transportation, keeping the cross-sectional area S of the gel constant during the contraction. The amount of water passing through the one-dimensional capillary has also been assumed to be the same as that leaving the gel. Thus, the amount of water released from the gel should be equal to the amount of water transported through the crosssectional area S of the gel. The volume of water dw transported through a cross-sectional element dS = 2 m dr per unit time is dw = v(r) dS = 2nw(r) d r (25) The rate of charge dQ flowing through the cross-sectional element dS is dQ = @(r)v(r)dS = 2 n r ~ ( r ) v ( r d) r
uav can also be expressed in g/C by multiplying by a factor of lo6, because the specific gravity of water is lo3 kg/m3. It is clear from eqs 27 and 28 that the contraction efficiency is determined by the nature of the gel and is independent of the electric field E. Under a certain temperature, it is only a function of degree of swelling 4 , which is inversely proportional to the cross-linking density. According to approximation (2), uav is constant during the contraction process. If we denote W and W, as the weight of gel at t and t = 0, respectively, and & as the initial length of the gel, the dependence of relative weight W/W, on the amount of charge Q flow through the gel is
Therefore
Using the expression for tp(r), v(r) can easily be obtained v(r) = (rEkT/ey) ln{(r/ri) sin[-tan-'
contraction efficiency in m3/C
(26)
vav(t) = vav(O)E(t)/E(O)
(32)
Combining eqs 30 and 32, we have vav(t)[Lo - J'vav(t) dt1= UV,,(O)/E(O)
(33)
Differentiating eq 33 with t and combining eq 30, we get { [ v a v ( t ) ~ -dvav(tYdt ~~ = E(OY[UV~,(O)I
(34)
Integrating eq 34 and using the relation E(0) = U/&, we find vav(t) = V,,(O)[ 1-2va,(o)t/Lol-'~2
(35)
The time dependence of relative weight W/W, is
W/W, = L(t)/Lo= 1 - Jrvav(t) dt/Lo
(36)
3. Experimental Materials. Poly(2-(acrylamido)-2-methylpropanesulfonic acid) (PAMPS) gel was prepared by radical polymerization using sodium persulfate as a radical initiator, and N,N-methylenebis(acrylamide) (MBAA) was used as a cross-linking agent. Polymerization was carried out at 50 "C under a N2 atomsphere for a prolonged period of time in a test tube 5 mm in diameter and 100 mm long. In order to remove any unreacted monomer and initiator, the PAMPS gel was immersed in a large amount of purified water for at least 1 week until the gel reached equilibrium state. Measurements. Before using for measurement, the degree of swelling ( 4 ) was calculated by weighing a PAMPS gel both in the dry and in the swollen state. Dry gel was obtained by evacuation until it reached a constant weight.
Gong et al.
9586 J. Phys. Chem., Vol. 98, No. 38, 1994 Figure 1 shows the experimental setup for measuring the gel contraction. A piece of cubic gel 10 mm in length was inserted between a pair of platinum-mesh electrodes and an electric field was applied by using a potentiostat (Hokuto-Denko HA-501) as a dc power supply. Contraction was followed by measuring the weight change of the PAMPS gel with time. The exuded water was automatically in situ pumped out by using a slightly evacuated pipet. During the course of experiment, the quantity of electricity was measured by using a coulomb meter (Hokuto-Denko HF-20 1).
1.0 0.8
0.2
0.0 0
The PAMPS gel is a strong acid polymer with fully ionized sulfonic groups as macroions and H+ as counterions. When an electric field is applied, hydrated H+ ions (more exactly H30+) migrate toward the cathode and are reduced, liberating H2. The water migrates together with H+ ions and exits the gel near the cathode. The overall electrode reactions are expressed as follows:
H,O
cathode
1.2
1.6
2
1
0
Time / min
4. Results and Discussion
anode
0.8
0.4
-
2H’
2H’
+ 2e- + 1/20,
+ 2e- - H,
The oxygen and hydrogen liberated can be observed throughout the course of the contraction. We have verified that the amount of gases released was consistent with the theoretical value derived by Faraday’s law according to the above equations. The overall population of ions should not significantly change during the contraction, because at every moment, the amount of H+ generation and consumption should be balanced at both electrodes. The local pH change near the cathode does not bring about observable change in volume due to the fully-ionized nature of the gel. The numerical simulation of contraction of PAMPS gel has m, been made by using following constants: ri = 6.08 x b = 2.55 x m, a = 1, = 78 (6 = E&O, EO is the dielectric constant in vacuum), q = 7.973 x s N m-2, = 0.01 m, e = 1.6 x C, T = 300 K. While 10 V was applied across the gel, our previous experimental results10 showed that a significant potential drop as large as 4.3 V occurs at the gel-electrode interfaces due to the formation of a Helmholtz bilayer, and we used 5.7 V as the effective voltage for the calculation of contraction. Figure 4 is theoretically (part a) and experimentally (part b) obtained time profiles of relative weight change of PAMPS gels (W/W,) with various degrees of swelling q. Both figures show that the rate of contraction increases with increase in the degree of swelling q. The theoretical results are convexly curved, indicating that the acceleration in contraction takes place and this acceleration has actually been observed in the experiments. The acceleration is apparently due to decrease in the electrode distance with time accompanied with the contraction of the gel. The experimental data in Figure 4b show that there exists a W/Ww value at which the gel practically stops its contraction. The higher the degree of swelling of the gel, the lower the value of WIW,. We do not have an exact interpretation explaining it, but it may be associated with the presence of so-called “bound water”. As is well established, the thermal movement of water molecules located adjacent to the macroions is locally restricted and the ‘H NMR spin-lattice relaxation timeI3 is also low due to strong attractive interaction with macrocharges located nearby. This suggests that the water molecules located within a certain distance from the macroions hardly move out from the potential energy valley made by the charged network. On the other hand, free water existing far from the ionic “atmosphere” can easily migrate under the electric force. From the results shown in
I 0.8
.
3*o.6 3 0.4 0.2
0 0
2
4
6
8
Time I min
Figure 4. Time profiles of the relative weight change of the gel for
various degree of swelling q: (a, top) simulation, (b, bottom), experiment. Applied voltage, 10 V; an effective voltage of 5.7 V was used in the simulation: 0, q = 25; 0,q = 70; M, q = 100; A, q = 200; 0, q = 256; A, q = 512; 0 , q = 750.
Figure 4b, one can evaluate the volume ratio of bound water to polymer ionic groups to obtain 10-15, which is equivalent to 148 water molecules per sulfonate moiety. This result is roughly equal to the amount of bound water observed in various synthetic and natural gels.14 Thus, our result suggests that the water molecules absorbed around the macroions within a distance of (3-4)ri are not displaced by an electric field. According to our electrokinetic contraction model described before, the amount of relative weight change is dependent only on the amount of charge transported through the gel (Le., on the quantity of electricity) and is independent of electric field (eq 29). Figure 5 shows the simulational and experimental results of the dependence of relative weight change on the quantity of electricity. The experimental results shows that WIW, is almost linearly dependent on Q, agreeing with the simulation. Figure 6 shows the dependence of the efficiency of contraction expressed in change of weight per coulomb (g/C) on the degree of swelling q. Figure 6b was replotted from the linear region of Figure 5b. Both the simulational and the experimental results show that the efficiency of the contraction almost linearly increases with the increase in the degree of swelling q, regardless of the value of the electric field applied. This result indicates that the contraction efficiency inversely depends on the crosslinking density of the gel. Here it should be noted that the values obtained by simulation are several times larger than those of experimental data. The main reason for this should be the simplification of the problem to a one-dimensional model, whereupon the effects of crosslinking points, and of the macromolecular ions located perpendicular to the electric field, have not been taken into account. At present, we cannot quantitatively evaluate how significantly the cross-linking points affect the electrophoretic and electroosmotic transport processes of counterions and water. In the case of a one-dimensional model, the outer electric field induces the migration of counterions along the capillary-like “channels” formed parallel to the direction of electric field, and this does not cause any unnecessary rearrangement of the counterion
Contractile Phenomena of Polyelectrolyte Gels 1.o
0.8
.
0.6
3* 3 0.4 0.2 0.0
0
1
2
3
4
5
Quantity of electricity / C
J. Phys. Chem., Vol. 98, No. 38, 1994 9587 and viscosity q for free water. As discussed earlier, this is not true in the case of the gel and both E and 7 values are considered to be enhanced in the highly-charged network. Although there exists disagreement on the rate of contraction between our one-dimensional analysis and the experimental results, the simulational results qualitatively demonstrate the electrokinetic nature of the contraction. Our simulation reasonably predicted that the time profile of contraction strongly depends on the electric field and on the degree of swelling of the gel and that the contraction efficiency only depends on the degree of swelling and not on the electric field.
1.0
Glossary 0.8
b (m)
3 3 0.6
.
3
0.4
0.2 0.0
0
10
20
30
40
Quantity of electricity / C
Figure 5. Relative weight change of the gel as a function of the quantity of electric flow through the gel: (a, top) simulation, (b, bottom) experiment. Symbols are the same as those in Figure 4.
--
4.0
100 200 300 400 500 600 700 800
0
Degree of Swelling 0
. 3
0.6
-
0.5
-
.-W0
0.4
-
0.3
-
0.2
-
00
g
e 'Z
distance between ionizable groups on the polymer chain charge of an electron applied electric field at t = 0 and t, respectively Boltzmann's constant length of the polymer gel at t = 0 and t, respectively counterion concentration in the gel degree of swelling of the gel amount of the charge flow through S radius of the macroion distance to the midpoint of two adjacent macroions distance from the axis of the macroion cross-sectional area of the polymer gel absolute temperature local gel contraction efficiency dc voltage applied across the gel local water flow velocity at t = 0 and t , respectively average values of v(r), v(r,t), and u(r) over r gel volumes in the dry and the wet state, respectively amount of water transported through S weight of gel at t = 0 and t, respectively degree of ionization dielectric constant of water viscosity of water local charge density in the gel local electrostatic potential in the gel
Gu M
0.1
References and Notes
6 0
0
100 200 300 400 500 600 700 800
Degree of Swelling
Figure 6. Dependence of the efficiency of the contraction on the degree of swelling q: (a, top) simulation, (b, bottom) experiment.
distribution. However, counterions of a real system, particularly those distributed around the macroions located perpendicular t o the electric field, make rearrangement of their distribution when they get electrostatic force from an outside electric field. This means that microions are forced to pass an extra potential barrier made by the charged network. This restricts electrophoretic and electroosmotic migration of ions as well as water and lowers the rate of contraction as compared with model prediction. In addition, in our simulation, we used dielectric constant E
(1) (2) (3) (4)
Osada, Y.; Hasebe, M. Chem. Lett. 1985, 1285. Osada, Y. Adv. Polym. Sci. 1987, 82, 1. Osada, Y.; Gong, J. P. Prog. Polym. Sci. 1993, 18, 187. Osada, Y.; Okuzaki, H.; Hori, H. Nature 1992, 355, 242. (5) Osada, Y.; Ross-Murphy, S. B. Sci. Am. 1993, 268, 82. (6) DeRossi; Tsukuba. Proc. Polym. Gel Symp. 1991, 33. (7) Doi, M.; Matsumoto, M.; Hirose, Y. Macromolecules 1992, 25, 5504. (8) Tanaka, T.; Nishio, I. Sun, S. T.; Nishio, S. V. Science 1982, 218, 467. (9) Osada, Y.; Kishi, R. J . Chem. Soc., Faraday Trans. 1989,85,655. (10) Kishi, R.; Hasebe, M.; Hara, M.; Osada, Y. Polym. Adv. Technol. 1990, 1, 19. (11) Lifson, S.; Katchalsky, A. J. Polym. Sci. 1954, 13, 43. (12) Tanford, C. Physical Chemistry of Macromolecules; John Wiley & Sons: New York; Univ. of Tokyo Press: Tokyo, 1969. (13) Yasunaga, H.; Ando, I. Polym. Gels Networks 1993, 1, 83. (14) Hatakeyama, T.; Yamauchi, A.; Hatakeyama, H. Eur. Polym. J. 1984, 20, 66.
