11092
J. Phys. Chem. 1996, 100, 11092-11097
Motion of Polymer Gels by Spreading Organic Fluid on Water J. P. Gong, S. Matsumoto, M. Uchida, N. Isogai, and Y. Osada* DiVision of Biological Sciences, Graduate School of Science, Hokkaido UniVersity, Sapporo 060, Japan ReceiVed: February 8, 1996X
We have constructed a new class of polymer gel that undergoes translational and rotational motions in water. These polymer gels consist of cross-linked hydrophobic-hydrophilic copolymers, which swell in watermiscible organic fluid but collapse in water. When these gels swollen in organic solvent are placed on supporting water, they start to do motion. The mode of motion depends on the shape of the gel: a disk- or spherical shaped gel exhibits translational motion while a triangular or a cubic one exhibits rotation. The velocity and duration of gel motion are strongly associated with its size and chemical structure. The driving force of the motion is originated from the surface spreading of organic solvent, which is pumped out of the gel because of the high osmotic pressure and hydrostatic pressure in the gel.
I. Introduction
II. Experimental Section
The isothermal transformation from chemical to mechanical power is called “chemomechanicals” and is found in various kinds of motilities of living systems. We have already reported on a polymer gel that moves by the molecular assembly reaction of surfactant molecules, where an electric field controlled the direction and equilibrium states of the surfactant binding.1-3 However, there should be an inherent economy if one can utilize a purely chemical osmotic pressure in order to convert it directly into mechanical work. When a drop of a liquid is placed over the planar surface of a liquid that solubilizes the first one, the spreading of the drop and the motion of the free surface outside the drop are observed.4-8 The rate of the spreading is determined mainly by the Harkins spreading coefficient, and the motion of the free surface is mainly associated with the variation of surface tension along this free surface (Marangoni effect). Therefore, if there exists any materials that are able to float on the supporting liquid surface and release continuously another liquid capable of spreading along the surface, the material should spontaneously move as a result of reaction. Such a motion due to the surface tension is very simple in principle and could possibly be one of the important mechanisms of the mass transportation in various living systems. In the previous paper, we have briefly reported that the crosslinked hydrophobic-hydrophilic amphoteric copolymer gels swollen in organic solvent undergo spontaneous motion when placed over water.9 These copolymers are composed of acrylic acid (AA) and acrylate monomers with a long alkyl side chain such as n-stearyl acrylate (SA) or 12-acryloyl dodecanoic acid (ADA). In this paper, we have systematically investigated the behavior of the motion using various kinds of cross-linked hydrophobichydrophilic copolymer gels. It was found that the motion is originated from the spreading of the water-soluble organic solvent in the gel to the water surface, whereupon the osmotic and hydrostatic pressures generated in the gel keep pumping the solvent for a prolonged period of time. The velocity and duration of gel motion are strongly associated with the shape, size, and chemical structure of the gel. The mechanism and process of the motion are described.
Gel Preparation. The amphoteric polymer gels are made of cross-linked copolymers of AA and hydrophobic acrylates such as SA,10-12 ADA, and acryloyl hexadecanoic acid (AHA).13 These amphoteric copolymer gels were obtained by radical copolymerization of various monomer mixtures in ethanol in the presence of a cross-linking agent of N,N′-methylenebis(acrylamide) (MBAA). The reaction conditions of the gel preparation were the following: total monomer concentration in ethanol, 3.0 mol/dm3; MBAA, 3.0 × 10-2 mol/dm3; temperature, 58 °C. All the monomers were polymerized for 24 h and formed a chemically cross-linked polymer gel. After polymerization, the gel was immersed in pure ethanol to remove unreacted substances. Detailed purification procedures of the gels were described elsewhere.10-13 The molar fraction of the hydrophobic monomer was denoted as F. The degree of crosslinkage (DCL) was simply calculated as a molar ratio of the cross-linking agent to the monomer. Degree of swelling, which is denoted as q, was calculated by the ratio of the weight of the gel in swollen and in dry states. Poly(SA) gel (PSA) was prepared by a radical polymerization in toluene in the presence of a cross-linking agent 1,6hexanediodiacrylate (2.66 mol %). The monomer concentration was 1.75 mol/L, and polymerization was carried out at 58 °C for 20 h to give the chemically cross-linked polymer gel. After polymerization, the gel was immersed in a large amount of tetrahydrofuran (THF) for 3 days to remove the monomer, uncross-linked polymer, and initiator and then was immersed in fresh THF for a week until it reached equilibrium. The spherical PSA micro gels with diameters of 30-200 µm were synthesized by the suspension polymerization technique. The monomer, cross-linking agent, and initiator were dispersed in a mixture of 10 mL of toluene and laurylbenzene sulfonic acid sodium salt (Nakarai Chemical Co. Ltd.) under nitrogen atmosphere. The solution was mixed with 150 mL water and was stirred by a mechanical agitator at 300 rpm. The polymerization was carried out at 58 °C for 20 h. After polymerization, the product was washed with a large amount of ethanol several times to remove unreacted substances and was dried in vacuo. Measurements. The motion of the gels was recorded by a video camera and analyzed with an image data analyzer (Olympus-Avia XL500). Motions of microgels were monitored by a video camera through an optical microscope.
X
Abstract published in AdVance ACS Abstracts, June 1, 1996.
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© 1996 American Chemical Society
Motion of Polymer Gels
J. Phys. Chem., Vol. 100, No. 26, 1996 11093
Figure 2. Time profiles of the rotational velocity of a cubic-shaped PSA gel. The swelling solvent was THF, qorg ) 6, length was 5 mm, and the weight was 123 mg. Closed circles are for PSA in pure water, and open circles are for PSA in 5 N NaCl solution.
(a)
Figure 1. Motions of PSA gels: (a) translational motion of a diskshaped gel; (b) rotational motion of a cubic-shaped gel. Photographs were taken by multiexposure with a stroboscope flashing at 10 Hz for 0.9 s.
III. Results The amphoteric polymer gels used in this paper are PSA, PSA microgel, poly(AA-co-SA), poly(AA-co-ADA), and poly(AA-co-AHA) with various monomer compositions. When placed in water, these gels shrink because of formation of crystalline lamella layers with a thickness of ca. 5 nm. The long alkyl side groups are in a tail-to-tail alignment arranged perpendicularly to the main chains.10 However, they swell easily in water-soluble organic fluids such as ethanol or THF by as much as 5-20 times in volume and change to the amorphous state.10-12 If a piece of these gels is swelled in organic solvent and placed in water, it instantaneously starts to move.9 The motion could be observed as long as the gel floats on the supporting water surface. Once the gel totally sinks into the bulk water, it stops the motion, indicating that the driving force arises from the water-air interface. The mode, duration, and velocity of the motion are associated with the size, shape, solvent, and chemical nature of the gel. Figure 1 shows examples of the progressive motions photographed by multiexposuring with a programmed stroboscope flasher (10 Hz) for 0.9 s. A disk-shaped gel undergoes a translational motion, occasionally accompanied by rotational motion (Figure 1a). Triangular and cubic gels show a rapid rotation with occasional change in the direction (Figure 1b). Figure 2 shows the prolonged time dependencies of the velocity of rotation exhibited by a cubic-shaped PSA gel swollen in THF. The closed circles represent the motion in pure water, and the open circles are in 5 N NaCl solution. All points shown in the figure are averages over 1 s. The velocity of rotation is 150-200 rpm at the beginning, which decreased slowly with time, but still shows 30 rpm after 10 min. Note that the gel in NaCl solution rotates more quickly than in water.
(b)
Figure 3. Short time profiles of the motion for poly(ADA-co-AA) gels where F ) 0.25, the swelling solvent is ethanol, and qorg ) 12: (a) rotational velocity of a cubic-shaped gel 2 mm in length; (b) translational velocity of a disk-shaped gel 4 mm in diameter and 2 mm thick.
To observe the instantaneous behavior of the gel, motions within several seconds were analyzed using an imaging analyzer. An image time interval of 1/30 s was used for the analyses. Parts a and b of Figure 3 are the initial rotational and translational velocities of cubic-shaped and disk-shaped poly(ADA-co-AA) gels swollen in ethanol, respectively. Velocities of rotational and translational motion fluctuate, but one can see that the values are in a range of 400-500 rpm and 1-4 cm/s, respectively. When these gels are allowed to swell in ethanol or THF and placed in water, they undergo volume collapse to form a hydrophobic dense surface skin layer and continue to release the organic solvent through it. The amount of THF released from the PSA gel was evaluated from the change in the weight of the gel with time, assuming no water can penetrate into the gel. As shown in Figure 4, the amount of THF released decreases with the time, which might be associated with a gradual decrease in the rotational velocity in Figure 2.
11094 J. Phys. Chem., Vol. 100, No. 26, 1996
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(a)
Figure 4. Time profile of the amount of THF released from the PSA gel during the motion. The values were divided by the total weight of THF contained in the gel before being placed on water. The PSA gel is 5 mm in length and weighs 123 mg. qorg ) 6, and the total weight of THF is 0.1 g.
Figure 5. Size dependencies of the duration of rotation of various gels: (O) triangular poly(ADA-co-AA) gel, F ) 0.25, q ) 12, swelling solvent is ethanol; (b) triangular poly(ADA-co-AA) gel, F ) 0.25, q ) 11, swelling solvent is methanol; (4) triangular poly(SA-co-AA) gel, F ) 0.25, q ) 16, swelling solvent is ethanol; (0) cubic-shaped PSA gel, q ) 6, swelling solvent is THF.
The duration of the motion was found to increase with an increase in the weight. The observed size dependence of the duration held for various polymers with different chemical compositions and for various swelling organic solvents. The durations for PSA gel swollen in THF and poly(ADA-co-AA) gel and poly(SA-co-AA) gel swollen in ethanol and methanol are shown in Figure 5. One should note that the rotation has a longer duration than that of translational motion. Since the amount of organic solvent contained in the gel is proportional to its size as expressed by the following equation
(
Worg ) W 1 -
)
1 qorg
(1)
the results in Figure 5 indicate that the gel behaves like a solvent reservoir that supplies the motion energy. Here, W and Worg are the weights of the gel and the organic solvent contained in the gel, respectively, and qorg is the degree of swelling of the gel in the organic solvent. The size effect of the microgel on the velocity of motion has also been investigated. Figure 6a shows the change in the velocity of translational motion when the spherical gel particles (open circle) or disk-shaped gels (closed circle) with various diameters were used. It is seen that the velocity of the motion gradually increases with an increase in size. However, if the velocity is divided by the cross-sectional area of the gel in order
(b)
Figure 6. (a) Translational velocities of spherical gel particles (O) and disk-shaped gels (b) with different sizes. Velocities were averaged values over the initial 2-3 s for each sample. (b) Relative velocity calculated by dividing the cross-sectional area of the PSA gel. The swelling solvent was THF.
to evaluate the relative velocity against its body, we found that these microparticles with ca. 200 µm diameters are able to move their bodies ca. 2 orders of magnitude more quickly than gels with millimeter-sized diameters (Figure 6b). IV. Discussion The process of a prolonged motion of the gel on the surface of water seems to be composed of two steps: one is the releasing of organic solvent from the gel by osmotic pressure and hydrostatic pressure; the other is the spreading of the organic solvent on the surface of the water and imparting motion to the gel as schematically illustrated in Figure 7. 1. Solvent Releasing (Figure 7a). As reported earlier,10-12 stearyl groups in the gels can form a tail-to-tail structure aligned perpendicularly to the main chain in water. A similar tail-totail alignment can also be formed by the long alkyl side chains of the hexadecanoic and dodecanoic acid residues of AHA13 and ADA gels. The diffraction peaks of the wet poly(AA-coSA) gels (F ) 0.50), for example, correspond to a spacing of 0.41 nm and are characteristic of hexagonal packing of long alkyl side groups. It was also found that the lamella distance (d2) of the swollen gels is almost 0.8 nm larger than that in their dry state. This experimental result and that in Figure 4 suggest that the gel, behaving as a perm selective membrane, allows the organic solvent to diffuse out but does not allow the water to come in. Therefore, when the amorphous gel highly swollen in organic solvent is placed on water, it quickly forms a crystalline skin layer and an osmotic pressure is produced between the inside and outside of the gel. Since the formation
Motion of Polymer Gels
J. Phys. Chem., Vol. 100, No. 26, 1996 11095
Figure 8. Young’s modulus of the PSA gel as a function of THF concentrations.
Here, Mw and Forg are the molecular weight and the density of the organic solvent, respectively. The hydrostatic pressure difference ∆P produced between the inside and outside of the gel can be evaluated as follows:
(
∆P ) E λ +
Figure 7. Schematic illustration of the mechanism for the motion of the gel: (a) releasing process of organic solvent; (b) surface spreading process of organic solvent.
of the surface skin layer produces a large hydrostatic pressure difference between the inside and outside of the gel, it is possible to eject the organic solvent from the gel. The collapse of the gel starts from the outer surface of the gel and develops into its inner part as schematically shown in Figure 7a. By virtue of this, the skin layer keeps “pumping out” the organic solvent for a prolonged period of time. Here, an important fact is that the crystalline skin layer formed at the gel-water interface can act as a “molecular orifice” through which the organic solvent in the gel flows out. The flux of the ejection of the organic solvent J can be expressed as follows:14
J ) Lp(∆π + ∆P)
(2)
where ∆π and ∆P are the osmotic and hydrostatic pressure differences inside and outside the gel and Lp is the ideal membrane permeability. The osmotic pressure appearing at the gel-water interface is expressed as
∆π ) RT∆Corg
(3)
where ∆Corg is the concentration difference of the organic solvent inside and outside the gel, which practically equals the concentration in the gel, since the outside concentration is negligible. Therefore, we have ∆Corg as follows:
∆Corg )
(qorg - 1)Forg qorgMw
(4)
)
1 λ2
(5)
where E is Young’s modulus and λ is the elongation ratio of the skin layer. The above formula was obtained under the assumption that the gel skin layer can behave like a rubber and can induce an inward pressure when being expanded. The elongation ratio of the skin layer λ can be estimated from
λ)
( ) qorg qw
1/3
(6)
Here, qw is the degree of swelling of the gel in water. We have found that a PSA gel with a cross-linking density of 2.66 mol % undergoes the amorphous-crystalline transition in a 50 v/v % THF-water mixture and shows an abrupt transition in the Young’s modulus of the gel (Figure 8). Using E ) 108 dyn/cm2 for the crystalline state from Figure 8, qw ) 1 in water, qorg ) 6 in THF solution, Mw ) 72, Forg ) 889 kg/m3, and T ) 298 K, we obtain ∆π ) 2.5 × 108 dyn/cm2 and ∆P ) 2.1 × 108 dyn/cm2. This means that the hydrostatic pressure can seriously contribute to the ejection of the organic solvent as much as the osmotic pressure does. 2. Solvent Spreading (Figure 7b). Once the organic solvent is ejected out of the gel, it rises up onto the water surface because of its lower density and rapidly spreads on it because of the large difference in their surface tensions. The spreading coefficient, which is the free energy change for the spreading of a liquid B over liquid A, can be written as5
∆SB/A ) γA - γB - γAB
(7)
Here, γA and γB are the surface tension of water and organic solvent, respectively, and γAB is their interface tension. For ethanol, for example, SB/A ) 50.4 dyn/cm. Numerous determinations of spreading rates of “oils” on water have been performed by means of a cinematographic technique.15 The results showed a first period during which the velocity decreases rapidly followed by a period of nearly constant spreading velocity. After a certain distance, the spreading velocity fell off, mainly as a result of the depletion
11096 J. Phys. Chem., Vol. 100, No. 26, 1996
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Figure 9. Driving torque applied to the cubic PSA gel: (O) ideal driving torque Nmax evaluated from Figure 4; (b) the actual torque N calculated from Figure 2.
of the spreading liquid. The initial spreading velocities V0 at the air-water interface are, for instance, 53 cm/s for ethanol and 33.6 cm/s for acetone.16 It is believed that the more hydrophobic the “oil”, the faster is its spreading speed. The surface spreading of the organic solvent imparts the reaction force to the gel and makes the gel do motion. The mode of the motion is associated with whether or not the net reaction force of organic solvent ejection passes through the mass center of the gel. Since the counteraction force acting on the gel is perpendicular to the surface of the gel, the spherical and disk-shaped gels always get a force that passes through the mass center, thus causing a translational motion. On the other hand, gels of other shapes may get a reaction force that do not pass the mass center. In such cases, a moment is generated and rotation occurs. The driving force of the gel motion should equal the reaction force F imparted to the gel, which is given by the momentum conservation law:
F ) V0 dm/dt
(8)
Here, dm/dt is the releasing rate of the organic solvent. The gel gets a maximum driving force when the organic solvent is released from one “orifice”. For the cubic PSA gel, the ideal maximum torque Nmax can be evaluated assuming this “orifice” is located at one of the cubic corners:
(dmdt )L2
Nmax ) V0
(9)
where L is the length of the cubic. Using constants of V0 ) 40 cm/s, L ) 0.5 cm, and the data of dm/dt, which is calculated by differentiating the data in Figure 4 with time, we obtain the maximum driving torque for the cubic PSA gel. The result is shown in Figure 9(open circle). On the other hand, we can estimate the driving torque N, which actually makes the gel do motion on water. According to the motion equation ((A-10) in the Appendix), a cubic-shaped gel undergoing a rotation with an angular velocity of ω should get a driving torque N as follows:
N ) (3/16)FL5ω2 + 6πηL3ω + (1/6)FgL5 dω/dt (10) where F and η are the density and viscosity of the supporting water solution, respectively, and Fg is the density of the gel. Substituting the data of motion in water in Figure 2 into eq 10 and using constants Fg ≈ F ) 1 kg/m3 and η ) 10-3 Ns/m2, we obtain the time profile of N and, the results are shown in Figure 9 (closed circle). It is seen that the driving torque increases at
the very beginning and then via a maximum after 10 s decreases with the elapse of time. The releasing of the organic solvent should occur randomly in its direction, and consequently, the net driving torque is strongly suppressed because of the mutual concellation of the force. This is why the actual driving torque calculated from the rotational velocity is 1 order of magnitude lower than the maximum driving force evaluated from the releasing rate of the organic solvent. This also suggests that there might exist a releasing “orifice” that is dominating the total direction of the motion. To confirm the described assumption of the mechanism, we attempted to make the motion on salt or surfactant solution to vary the surface tension. As shown in Figure 2 (open circles), the velocity of rotation on 5 N NaCl solution is larger than that on pure water because of increased surface tension of the salt solution. Conversely, if a drop of lauryl alcohol, which is not miscible with water, is added, the gel instantaneously stopped the motion because of the decrease in the surface tension. 3. Energy Efficiency. The energy efficiency of the gel motion largely depends on how to define the energy consumed by the gel. If we suppose that the spreading of the organic solvent is a two-dimensional isothermal expansion, we have the following relation for the work done by the spreading of the organic solvent on the water surface, Es:
Es) nRT
(11)
Here, n is the molar number of organic solvent making a two-dimensional spreading and R and T are the gas constant and temperature, respectively. E, the work done by the gel through rotation can be expressed by the following formula (see Appendix, eq A-13):
E ) (3/16)FL5∫0 ω3 dt + 6πηL3∫0 ω2 dt ∞
∞
(12)
The energy efficiency ηeff of the gel motion is equal to the ratio of the work done by the gel motion, E, to the work done by the spreading of the organic solvent Es:
ηeff ) E/Es ) E/nRT
(13)
For a PSA gel swollen in THF with L ) 5 × 10-3 m and qorg ) 6, we have n ) 1.4 × 10-3 mol. Using R ) 8.3 J/K and T ) 298 K, we have Es ) 3.5 J. On the other hand, using the ω-t relation in Figure 2a (closed circle) and F ) 103 kg/m3, η ) 10-3 Ns/m2 for water, we have E ) 5.4 × 10-7 J or ηeff ) 1.5 × 10-5%, which is relatively small in comparison with the values for mechanical engines. However, we should emphasize that the prolonged gel motion obtained by the surface spreading of the organic solvent has several advantages and unique characteristics. They produce no noise and no unnecessary exhaust products like combustion or other chemical reactions. The motion is obtained only by the dilution of organic fluid, which can be recovered by using separation technologies such as distillation or membranes. Acknowledgment. The authors express their sincere thanks to Professor D. W. Grainger (Colorado State University) for his kind and valuable suggestions. One of authors (J. P. Gong) thanks the Hayashi Memorial Foundation for Female Natural Scientists for financial support. This research was supported in part by a Grant-in-Aid for the experimental research project “Synthesis and Applications of the Shape Memory Gels” from the Ministry of Education, Science and Culture (07555591), Japan. The authors also acknowledge the Agency of Science
Motion of Polymer Gels
J. Phys. Chem., Vol. 100, No. 26, 1996 11097 Np, the total torque the gel gets, is
Np ) 4∫0
π/4
dNp ) (3/16)FL5ω2
(A-7)
We assume that the frictional resistance Ff of the cubic gel is approximately equal to that of a spherical one with a radius of L. Therefore, according to Stokes’ law,
Ff ) 6πηL2ω Figure 10. Cubic-shaped gel in rotation used for the analysis of motion.
Here, η is the viscosity of water. Therefore, the torque Nf due to the frictional resistance is
and Technology, Minister of International Trade and Industry (MITI) for the financial support. Appendix We consider the rotational motion of a cubic-shaped gel L in length. The rotational axis is vertical to the paper and through the mass center of the gel (Figure 10). The gel should get three forces during motion: the driving force from the surface spreading of the organic solvent, the water pressure resistance, and the frictional resistance due to motion. If we denote N, Np, and Nf as torques from the surface spreading, the water pressure resistance, and the frictional resistance, respectively, the motion equation of the gel is
N - Np - Nf ) I
dω dt
(A-1)
Nf ) FfL ) 6πηL3ω
N ) (3/16)FL5ω2 + 6πηL3ω + (1/6)FgL5 dω/dt
(A-2)
Pp ) Npω ) (3/16)FL5ω3
(A-11)
and of friction Pf is
(A-12)
The work done by the gel E is
E ) ∫0 (Pp + Pf) dt ∞
ds⊥ ) L dx sin θ
(A-3)
where θ ) ∠boc and x ) bc ) (L/2)tan θ. Therefore, the element ds gets a pressure resistance force of dFp:
sin θ dFp ) F ds⊥ V2 ) FL(L/2)3ω2 4 dθ cos θ
(A-4)
since
Lω 2 cos θ
(A-5)
where F is the density of water. dNp, the torque that the element ds gets, is
dNp ) dFp ob ) FL(L/2)4ω2
(A-10)
The work done in unit time in overcoming the resistance force of water pressure Pp is
Pf ) Nfω ) 6πηL3ω2
is the moment of inertia of the cubic gel with respect to the rotational axis, ω the rotational velocity, and Fg the density of the gel. Although a Reynolds number of about 40 has been estimated from the motion of the gel, it is considered that the frictional resistance cannot be neglected in comparing with the pressure resistance, since the gel does rotational motion. We simply use Newton’s theory to estimate the water pressure resistance to a cubic gel doing rotation. As shown in Figure 10, the gel surface element perpendicular to the velocity V is
V)
(A-9)
By substituting eqs A-7, A-9, and A-2 into eq A-1, we obtain the driving torque (moment of force) that the gel gets at time t:
Here,
I ) (1/6)FgL5
(A-8)
sin θ dθ cos5 θ
(A-6)
) (3/16)FL5∫0 ω3 dt + 6πηL3∫0 ω2 dt ∞
∞
(A-13)
References and Notes (1) Osada, Y.; Okuzaki, H.; Hori, H. Nature 1992, 355, 242. (2) Okuzaki, H.; Osada, Y. Macromolecules 1994, 27, 502. (3) Gong, J. P.; Osada, Y. J. Phys. Chem. 1995, 99, 10971. (4) Harkins, W. D.; Feldman, A. J. Am. Chem. Soc. 1922, 44, 2665. (5) Adamson, A. W. Physical Chemistry of Surfaces, 5th ed.; John Wiley & Sons, Inc.: New York, 1990; Chapter 4, p 116. (6) O’Brien, R. N.; Feher, A. I.; Leja, J. J. Colloid Interface Sci. 1976, 56, 474. (7) Saylor, J. E.; Barnes, G. T. J. Colloid Interface Sci. 1971, 35, 143. (8) Troian, S. M.; Wu, X. L.; Safran, S. A. Phys. ReV. Lett. 1989, 62, 1496. (9) Osada, Y.; Gong, J. P.; Uchida, M.; Isogai, N. Jpn. J. Appl. Phys. 1995, 34, L511. (10) Matsuda, A.; Sato, J.; Yasunaga, H.; Osada, Y. Macromolecules 1994, 27, 7695. (11) Tanaka, Y.; Kagami, Y.; Matsuda, A.; Osada, Y. Macromolecules 1995, 28, 2574. (12) Osada, Y.; Matsuda, A. Nature 1995, 376, 6537. (13) Uchida, M.; Kurosawa, M.; Osada, Y. Macromolecules 1995, 28, 4583. (14) Lakshminarayanaiah, N. Transport Phenomena in Membranes; Academic Press: New York, 1969; Chapter 6, p 314. (15) Davies, J. T.; Rideal, E. K. Interfacial Phenomena; Academic Press: New York and London, 1963; pp 25-28. (16) Suciu, D. G.; Smigelschi, O.; Ruckenstein, E. J. Colloid Interface Sci. 1970, 33, 520.
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