Flower Micelle of Amphiphilic Random Copolymers in Aqueous Media

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J. Phys. Chem. B 2010, 114, 11403–11408

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Flower Micelle of Amphiphilic Random Copolymers in Aqueous Media Yukio Tominaga,† Mari Mizuse,‡ Akihito Hashidzume,‡ Yotaro Morishima,§ and Takahiro Sato*,‡ Genomic Science Laboratories, Dainippon Sumitomo Pharma Co., Ltd., 33-94 Enoki, Suita, Osaka 564-0053, Japan, Department of Macromolecular Science, Graduate School of Science, Osaka UniVersity, 1-1 Machikaneyama-cho, Toyonaka, Osaka 560-0043, Japan, and Faculty of Engineering, Fukui UniVersity of Technology, 6-3-1 Gakuen, Fukui, Fukui 910-8505, Japan ReceiVed: May 22, 2010; ReVised Manuscript ReceiVed: July 22, 2010

The structure of the flower micelle formed by an amphiphilic random copolymer, sodium (2-acrylamido)2-methylpropanesulfonate and N-dodecylmethacrylamide p(AMPS/C12), in 0.05 M aqueous NaCl was investigated by fully atomistic molecular dynamics simulation as well as by light scattering, and the results were compared with the flower micelle model of the minimum loop size, recently proposed by Kawata et al. [Macromolecules 2007, 40, 1174-1180]. After a sufficiently long simulation time, simulated p(AMPS/C12) chain with the degree of polymerization of 200 and C12 content of 50 mol % formed a unicore micelle, of which radius of gyration was much smaller than the AMPS homopolymer with the same degree of polymerization. The simulated micellar structure was analyzed in terms of density distribution functions for dodecyl groups, the main chain, and sulfonate groups as functions of the radial distance r from the center of mass of dodecyl groups. Only dodecyl groups exist at r j 1.5 nm, and the main chain and sulfonate groups distribute in the range of r between 1.5 and 3.5 nm, but there were dodecyl groups coexisting with the main chain and sulfonate groups beyond r ) 1.5 nm. All these structural features, as well as hydrodynamic radius data for p(AMPS/C12) with C12 contents higher than ca. 20 mol % obtained by light scattering, agreed with the predictions of the flower micelle model of the minimum loop size. Introduction

SCHEME 1: Chemical Structure of p(AMPS/C12)

With increasing consciousness of ecology, environmentally benign systems based on macromolecules in aqueous media have attracted growing interest from researchers. With a goal to control properties of such aqueous systems, a wide variety of water-soluble polymers have been investigated by many research groups as potentially important materials. Among these, amphiphilic polyelectrolytes (APEs) that bear hydrophobic and electrolyte units on the same polymer chain have received particular attention in the past two or more decades because they exhibit versatile and yet characteristic properties in aqueous media.1 When APEs are dissolved in water, hydrophobic units associate with each other while charged units repel each other, forming various types of self-assemblies. The APEs selfassembling phenomena somewhat resemble the formation of a higher order structure of proteins, thus providing us with a simple model for the self-assemblies of biologically important macromolecules. In addition, APEs are practically important materials for commercial products, such as emulsifiers, solubilizers, suspensifiers, rheology modifiers, and thickeners. These APE-based products are used in various areas, including paints and coatings, drug delivery systems, cosmetics, and personal care items.2 APEs can be categorized into two classes: one is block APEs, and the other is statistical or random APEs. Block APEs usually form multimolecular micelles with a core-shell structure of well-defined size and shape.3 Statistical or random APEs form * Corresponding author: Fax: +81-6-6850-5461. E-mail: tsato@ chem.sci.osaka-u.ac.jp. † Dainippon Sumitomo Pharma Co., Ltd. ‡ Osaka University. § Fukui University of Technology.

a single- or multimolecular micelle in aqueous media, but its structure is much less defined because of the complexity of electrostatic and hydrophobic interactions occurring concurrently along and among the copolymer chains.4 Challenging experimental and theoretical studies were carried out for amphiphilic random or segmental copolymers by several researchers,5-15 but they were unable to draw a general picture of the micelle structure formed by such copolymers. Recently, Kawata et al.16 have investigated the structure of micelles formed from some statistical APEs bearing dodecyl groups by static and dynamic light scattering combined with time-dependent fluorescence quenching measurements. Hydrodynamic radii of the statistical APEs forming unicore micelles in aqueous salt solution were favorably compared with a flower micelle model of the minimum loop size, determined by the copolymer chain stiffness. In order to elucidate the detailed structure of micelles formed by statistical or random APEs, we have carried out, in the present study, fully atomistic molecular dynamics (MD) simulations on a copolymer of sodium (2-acrylamido)-2-methylpropanesulfonate (AMPS) and N-dodecylmethacrylamide (C12) [p(AMPS/C12); cf. Scheme 1] with equimolar composition (x

10.1021/jp104711q  2010 American Chemical Society Published on Web 08/12/2010

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Tominaga et al.

Figure 1. Schematic diagram of the flower micelle model, where all the hydrophobic groups are included in a single core (a) and with loops of the minimum size (b).

) 0.5) in aqueous salt solution. The micellar structure was characterized in terms of the radius of gyration and density distribution functions of dodecyl groups, the main chain, and sulfonate groups as well as of water molecules surrounding the micelle. In addition to the MD simulation, we have also investigated the C12 content x dependence of the micelle formation for p(AMPS/C12) in aqueous salt solution by light scattering and fluorescence emitted from pyrene added to the polymer solution to examine the consistency among the results of the MD simulation, light scattering, fluorescence, and the flower micelle model of the minimum loop size. In what follows, we first review the flower micelle model and then present results of the MD simulation, light scattering, and fluorescence. We will show that the flower micelle model is favorably compared with those results.

However, the above discussion has completely neglected the intrinsic stiffness of copolymer main chain. According to the wormlike chain model,17 the chain stiffness is expressed in terms of the persistence length q: it takes zero for the completely flexible chain (i.e., the Gaussian chain) and tends to infinity for the rigid rod. Yamakawa and Stockmayer18 calculated the ring closure probability of the wormlike chain and found it sharply increases from zero only when the chain contour length exceeds a critical chain contour length lmin. Furthermore, they calculated the rigid path of the minimum loop with the contour length lmin. According to their results, lmin and the distance dloop between the loop chain end and midpoint for the rigid path are approximately written as

lmin ) 1.6q,

(

(1)

where x and N0,1 are the content of the hydrophobic monomer units and the degree of polymerization, respectively, of the copolymer chain, and υC12 is the molecular volume of the hydrophobic dodecyl group (xmN0,1 is the number of dodecyl groups per micelle). The average number of monomer units between neighboring hydrophobic monomer units along the copolymer chain is given by ∑ig0i(1 - x)ix ) (1 - x)/x, and the average loop size must be smaller than lcontour/2 ) l(1 - x)/2x with l being the contour length per monomer unit. As a result, the radius of the whole micelle Rmicelle should fulfill the condition

Rmicelle < Rcore + l(1 - x)/2x (cf. Figure 1a).

)

N0,1l 4π + 1 mυC12 Rcore3 ) λ 3 lmin

(2)

(1′)

where we assumed that λ dodecyl groups are included in the core at each root of the loop (λ ) 3 in Figure 1b), and the radius of the flower micelle Rmicelle should be calculated by

Rmicelle ) Rcore + dloop

When an amphiphilic random copolymer is dissolved in water or aqueous salt solution, hydrophobic groups, say dodecyl groups, attaching to each copolymer chain tend to self-assemble to form hydrophobic cores, and the copolymer main chain is forced to take the loop conformation. As the result, the amphiphilic random copolymer forms a flower micelle (cf. Figure 1). In what follows, the flower micelle is assumed to consist of m copolymer chains (m is referred to as the aggregation number). If all the hydrophobic dodecyl groups in the flower micelle would be included in a single core, the radius Rcore of the hydrophobic core should be calculated by

(3)

If the flower micelle of the copolymer chains has these minimum loops, each chain forms N0,1l/lmin loops, and only dodecyl groups attaching to the roots of the loops can be included in the hydrophobic core. Thus, eq 1 should be replaced by

Flower Micelle Model16

4π R 3 ) xmN0,1υC12 3 core

dloop ) 0.62q

(2′)

along with eqs 1′ and 3. The average dimension of polymer chains is usually represented by the radius of gyration 〈S2〉1/2 which can be measured by light scattering.19 Assuming scattering powers of all the atoms constituting the chain to be identical, we can define 〈S2〉1/2 as the mean-square root of the distance of the atoms from the center of mass of the chain. If the hydrophobic core is assumed to be a sphere of a uniform density and minimum loops to stem from the spherical surface, we have 〈S2〉1/2 ) 3 n R 2 + nloop(Rcore2 + 0.4235Rcorelmin + 0.0605lmin2) 5 core core ncore + nloop

[

]

1/2

(4)

where ncore and nloop are the numbers of atoms constituting the hydrophobic core and all loop chains. Methods MD Simulation. For a random copolymer chain of p(AMPS/ C12) with N0,1 ) 200 and x ) 0.5, the following three-step MD simulations were implemented: (i) high temperature (400 K) implicit water MD simulations, (ii) medium temperature (350 K) explicit water MD simulations, and (iii) room temperature (300 K) explicit water MD simulations. The initial condition of the high temperature implicit water MD simulations was alltrans conformations of the backbone and dodecyl chains, but five different initial momenta for each constituent atom were

Flower Micelle of an Amphiphilic Random Copolymer

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TABLE 1: Molecular Characteristics of p(AMPS/C12) Samples Used x

N0,1

Mz,1/Mw,1

0.06 0.07 0.12 0.15 0.19 0.31 0.42

120 1100 260 110 95 51 61

1.4 2.0 1.4 1.5 1.7 1.5 1.6

chosen to check the attainment of the equilibrium conformation after sufficient simulation times. After 50 ns, four 400 K simulations among the five provided unicore micellar conformations with almost identical radii of gyration 〈S2〉1/2 and total potential energy Etot, but one simulation gave a double-core micellar conformation with slightly larger 〈S2〉1/2 and Etot than the other four simulations. This simulation was continued up to 100 ns. The final conformation was a unicore micelle, and 〈S2〉1/2 and Etot reduced to almost the same values as the other four simulations. We judged these unicore micelles to be equilibrated conformations, and one of the conformations was chosen as the starting conformation for the following explicit water MD simulations. The explicit water simulation system contains a fully ionized p(AMPS/C12) chain with a random sequence of 50 mol % C12, 118 sodium ions, 18 chloride ions, and 19 946 TIP3P water molecules (see the System Preparation and MD Simulation Protocol sections in Supporting Information). The molar concentration of sodium chloride corresponds to 0.05 M. The global micellar structures did not essentially change during the 350 and 300 K simulations. Light Scattering and Fluorescence Experiments. The micellar structure of real p(AMPS/C12) samples in 0.05 M aqueous NaCl was also studied by light scattering. The weightaverage aggregation number mw and the hydrodynamic radius RH of the micelle were determined by this experiment. Furthermore, a small amount of pyrene was added to aqueous p(AMPS/ C12) solutions, and the number of hydrophobic cores per micelle was estimated by the time-resolved fluorescence experiment emitted from pyrene. Detailed methods of both experiments are described elsewhere.16 Random copolymer samples used for both experiments were made from sodium 2-(acrylamido)-2-methylpropane-sulfonate (AMPS) and dodecylmethacrylamide (C12) using AIBN as the radical initiator.16 For each copolymer sample, the C12 content (the mole fraction) x was determined in D2O by 1H NMR, and the degree of polymerization N0,1 and the ratio of the z-average to weight-average molecular weights Mz,1/Mw,1 were estimated in methanol containing 0.1 M LiClO4 (where copolymer chains are molecularly dispersed) by sedimentation equilibrium.16 The results are listed in Table 1. Results and Discussion MD Simulation. Figure 2 shows an example of the snapshot of equilibrated p(AMPS/C12) chains with N0,1 ) 200 and x ) 0.5 in 0.05 M aqueous NaCl, obtained by the explicit water MD simulation at 300 K. The copolymer chain shrinks due to the self-association of dodecyl groups to take a globular conformation of almost spherical shape, just like globular proteins. The average chain dimension is quantitatively expressed in terms of the radius of gyration 〈S2〉1/2 averaged over a long time range after equilibration. (Here, 〈S2〉1/2 was calculated by summing up all the atoms constituting the chain with equal

Figure 2. Snapshot of an equilibrated chain of p(AMPS/C12) with x ) 0.5 obtained by the MD simulation.

Figure 3. Comparison of the radius of gyration for p(AMPS/C12) with x ) 0.5 estimated from the 300 K explicit water MD simulations (the filled circle) with those for the AMPS homopolymer obtained by light scattering20 (unfilled circles) and also with the theoretical result for the flower micelle model of the minimum loop size calculated by eq 4 (the solid curve). Dotted curve: theoretical values calculated by the wormlike chain model with the persistence length q ) 3.0 nm.20

weight.) Figure 3 compares 〈S2〉1/2 estimated from the simulations (the filled circle) with experimental 〈S2〉1/2 for AMPS homopolymer (pAMPS) under the same solvent condition, obtained from light scattering measurements by Yashiro et al.20 (unfilled circles). The original authors analyzed their pAMPS data with the perturbed wormlike chain model to estimate the persistence length q to be 3.0 nm. The dotted curve in Figure 3 indicates theoretical values of the wormlike chain model. The simulation result deviates downward from the theoretical dotted curve. Now, we compare the simulation result for p(AMPS/C12) with the flower micelle model of the minimum loop size. To calculate 〈S2〉1/2 using eq 4 for this model, we need four molecular parameters: q, l, υC12, and λ. We may choose q ) 3.0 nm for the AMPS homopolymer in 0.05 M aqueous NaCl,21 and l ) 0.25 nm for conventional vinyl polymers; the minimum loop consists of 19 () 1.6q/l) monomer units. Furthermore, the molecular volume υC12 of the dodecyl group is given to be 0.35 nm3.22 Thus, the only adjustable parameter is λ. According to Kawata et al.,16 we have chosen λ ) 4.5. The solid curve in Figure 3 indicates theoretical values of 〈S2〉1/2 for the flower micelle model calculated by eq 4 using the parameters above. The filled circle is close to the theoretical curve, demonstrating a good agreement between the simulation result and the flower micelle model. Using the simulation result, we also calculated density distribution functions F(r) for carbon atoms in dodecyl groups,

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Figure 4. Density distribution functions F(r) of carbon atoms in dodecyl groups (the blue solid curve), carbon atoms in the polymer main chain (the green dotted curve), sulfur atoms in sulfonate groups (the magenta dashed curve), and oxygen atoms in water molecules (the red dot-dash curve) for p(AMPS/C12) with x ) 0.5 in 0.05 M aqueous NaCl at 300 K, averaged over 60-160 ns. Here, r is the radial distance from the center of mass of the dodecyl groups.

carbon atoms in the main chain, sulfur atoms in sulfonate groups, and oxygen atoms in water molecules, where r is the radial distance from the center of mass of dodecyl groups. As shown in Figure 4, only dodecyl groups exist at r j 1.5 nm, and the main chain and sulfonate groups distribute in the range of r between 1.5 and 3.5 nm. The density of water molecules starts increasing at r ) 1.5 nm and approaches the bulk water density at r J 3.5 nm. Therefore, we can say that 1.5 nm is the radius of the hydrophobic core and 3.5 nm is the radius of the whole micelle. In Figure 4, two vertical segments indicate Rcore and Rmicelle for the flower micelle model of the minimum loop size, calculated from eqs 1′ and 2′ with the same parameters as in Figure 4. The flower micelle model precisely predicts the hydrophobic core and whole micellar sizes. Furthermore, it should be noticed that dodecyl carbon atoms coexist with mainchain carbon atoms, sulfonate groups, and water molecules in a region of r between 1.5 and 2.5 nm. This means that not all dodecyl groups are included in the hydrophobic core region, being consistent with the flower micelle model of the minimum loop size. Using the flower micelle model, the number of dodecyl groups inside the core is calculated to be 52 [) λ(N0,1l/lmin + 1)]. On the other hand, MD simulations for the spherical micelle of sodium dodecyl sulfate (SDS), performed by several authors,24 mostly assumed the aggregation number to be 60 and yielded the hydrophobic core domain boundary of ca. 1.5 nm. This agrees with Rcore of the p(AMPS/C12) micelle. However, the density distribution of dodecyl groups in the SDS micelle vanishes at r ≈ 2.0 nm, which is appreciably narrower than F(r) of dodecyl groups in the p(AMPS/C12) micelle. This difference in the dodecyl group distribution is owing to the existence of dodecyl groups on loop chains and outside the hydrophobic core in the AMPS/C12 copolymer micelle. Khokhlov et al.25 obtained flowerlike micelles by computer simulations of amphiphilic copolymers with specially designed sequences. Their copolymers have short-range sequence correlations, being in contrast with random copolymers, and such sequences are crucial to the micellar structure they obtained. On the other hand, our AMPS/C12 copolymer has a random sequence and seems to form the flower micelle by a different origin. Hydrophobes of our copolymer are relatively long dodecyl groups, while Khokhlov et al.’s copolymer models have spherical hydrophobes. The shape of the hydrophobe may play

Tominaga et al.

Figure 5. Hydrophobe content dependences of the weight-average aggregation number mw (filled circles) and the number of hydrophobic cores nc (unfilled circles) of the micelle formed by p(AMPS/C12) with different x in 0.05 M aqueous NaCl at 25 °C.

Figure 6. Hydrodynamic radii RH of p(AMPS/C12) (black filled circles), pAMPS (black unfilled circles),26 and amphiphilic random and alternating copolymers listed in Scheme 2 (red triangles and diamonds)16,27,28 in 0.05 M aqueous NaCl at 25 °C, plotted against the number of monomer units constituting the aggregate or polymer chain: blue squares, RH of globular proteins in aqueous solution with the ionic strength of 0.1-0.5 M; black solid curve, theoretical values of Rmicelle for the flower micelle model calculated by eq 2′; black dotted curve, theoretical values for the wormlike cylinder model with q ) 3.0 nm;29 blue dot-dashed line, eye guide with the slope of 1/3.

an important role in the hydrophobic core formation of the flower micelle. Light Scattering and Fluorescence Experiments. Figure 5 shows the hydrophobe content dependences of the weightaverage aggregation number mw and the number of hydrophobic cores nc of the micelle formed by p(AMPS/C12) in 0.05 M aqueous NaCl at 25 °C determined by light scattering and fluorescence from pyrene added to the copolymer solution. (Error bars for mw were estimated from uncertainties in the concentration extrapolation, while those for nc from results at different pyrene concentrations.) Furthermore, hydrodynamic radii RH of the p(AMPS/C12) samples with different x in the same solvent condition are shown in Figure 6 by filled circles as a function of the degree of polymerization N0,1 or the number of monomer units constituting the micelle mwN0,1. The same figure includes RH data (unfilled circles) for the AMPS homopolymer in the same solvent condition, reported by Yashiro and Norisuye,26 and the theoretical solid curve of Rmicelle for the flower micelle model of the minimum loop size, calculated by eq 2′ along with eqs 1′ and 3 using the same parameters as in Figure 3. It can be seen from Figures 5 and 6 that p(AMPS/ C12) transforms from a molecularly dispersed random coil to

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SCHEME 2: Chemical Structures of Amphiphilic Random and Alternating Copolymers Investigated Previously

the flower micelle with aggregation number of ca. 2, with increasing x. Figure 6 includes also previous results (red symbols) of RH for various amphiphilic random and alternating copolymers bearing the dodecyl hydrophobe with x ≈ 0.2-0.5,16,27,28 listed in Scheme 2. Even if the hydrophobic monomer unit is changed from the (meth)acrylamides to vinyl ether, and the electrolyte group is changed from sulfonate, carboxylate, to different amino acids, all the data points, including p(AMPS/C12) with x J 0.2, follow the solid curve for the flower micelle model of the minimum loop size. Thus, the micellar global conformation is independent of those chemical structures of amphiphilic copolymers. Furthermore, the conformation is also independent of the content and sequence manner of the hydrophobic monomer unit at 0.2 j x e 0.5, which results from the fact that the loop size is determined by the main-chain stiffness. Globular proteins are amphiphilic copolymers, and the hydrophobic and electrostatic interactions play important roles in determining their third-order structures in aqueous solution. Amino acid groups in the polypeptide main chain can form hydrogen bonding that contributes to the formation of an R-helix or β-structure, but this is not the case of the main chain of vinyl polymers. Moreover, globular proteins are made from 20 different kinds of amino acids whereas the copolymers in Schemes 1 and 2 are prepared from only two kinds of monomers. To compare the global conformation of the amphiphilic vinyl copolymers in Schemes 1 and 2 with that of globular proteins, Figure 6 also plots RH data for globular proteins (ribonuclease, lysozyme, β-lactogloblin, ovalbumin, serum albumin, and hemoglobin)30 by blue squares. Here, the degree of polymerization N0,1 in the abscissa was multiplied by 3/2 for the proteins to compare RH between the proteins and vinyl copolymers at the same number of main-chain atoms (the amino acid residue consists of CCN three main-chain atoms). The data points for globular proteins deviate downward from the solid curve, demonstrating that globular proteins take more compact conformations than do the amphiphilic random copolymers bearing dodecyl groups as the hydrophobe, although the snapshot of p(AMPS/C12) with x ) 0.5 shown in Figure 2 looks like a globular protein. The blue squares seem to follow a straight line with a slope of 1/3, which is expected for uniform density spheres. Summary The micellar structure of an amphiphilic random copolymer, sodium (2-acrylamido)-2-methylpropanesulfonate and N-dode-

cylmethacrylamide p(AMPS/C12), in 0.05 M aqueous NaCl was investigated by fully atomistic molecular dynamics (MD) simulation, light scattering, and fluorescence from pyrene added to the copolymer solution. All the results, along with previous light scattering results for similar amphiphilic random and alternating copolymers bearing dodecyl groups, were consistent with the flower micelle model of the minimum loop size, recently proposed by Kawata et al. This model assumes that some hydrophobic (dodecyl) groups exist outside the hydrophobic core, and the density distribution functions determined by the MD simulation (see Figure 4) demonstrated the existence of such hydrophobic groups. A recent work indicated that hydrophobic groups outside the core play an important role in enhancement of the solution viscosity with increasing the copolymer concentration.31 Acknowledgment. This work was partly supported by a Grant-in-Aid for Scientific Research No. 17350058 from the Japan Society for the Promotion of Science as well as by a Grant-in-Aid for Scientific Research on Priority Area “Soft Matter Physics” and a Special Coordination Fund for Promoting Science and Technology (Yuragi Project) of the Ministry of Education, Culture, Sports, Science and Technology, Japan. Supporting Information Available: Details on system preparation and the description of the MD simulation protocol. This material is available free of charge via the Internet at http:// pubs.acs.org. References and Notes (1) For example: (a) McCormick, C. L.; Bock, J.; Schulz, D. N. In Encyclopedia of Polymer Science and Engineering; Kroschwitz, J. I., Ed.; Wiley & Sons: New York, 1989; Vol. 17, pp 730-784. (b) Varadaraj, R.; Branham, K. D.; McCormick, C. L.; Bock, J. In Macromolecular Complexes in Chemistry and Biology; Dubin, P., Bock, J., Davies, R. M., Schulz, D. N., Thies, C., Eds.; Springer-Verlag: Berlin, 1994; pp 15-31. (2) For example: (a) Polymers in Aqueous Media; Glass, J. E., Ed.; Advances in Chemistry Series 223; American Chemical Society: Washington, DC, 1989. (b) Hydrophilic Polymers. Performance with EnVironmental Acceptability; Glass, J. E., Ed.; Advances in Chemistry Series 248; American Chemical Society: Washington, DC, 1996. (c) AssociatiVe Polymers in Aqueous Solutions; Glass, J. E., Ed.; ACS Symposium Series 765; American Chemical Society: Washington, DC, 2000. (3) For example: (a) Qin, A.; Tian, M.; Ramireddy, C.; Webber, S. E.; Munk, P.; Tuzar, Z. Macromolecules 1994, 27, 120–126. (b) Chu, B. Langmuir 1995, 11, 414–421. (c) Fo¨rster, S.; Zisenis, M.; Wenz, E.; Antonietti, M. J. Chem. Phys. 1996, 104, 9956–9970. (d) Moffitt, M.; Khougaz, K.; Eisenberg, A. Acc. Chem. Res. 1996, 29, 95–102. (e) Fo¨rster, S.; Antonietti, M. AdV. Mater. (Weinheim, Ger.) 1998, 10, 195–217. (f) Fo¨rster, S.; Abetz, V.; Mu¨ller, A. H. E. AdV. Polym. Sci. 2004, 166, 173– 210.

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(4) Hashidzume, A.; Morishima, Y.; Szczubialka, K. In Handbook of Polyelectrolytes and Their Applications; Tripathy, S. K., Kumar, J., Nalwa, H. S., Eds.; American Scientific Publishers: Stevenson Ranch, CA, 2002; Vol. 2, pp 1-63. (5) Yamamoto, H.; Mizusaki, M.; Yoda, K.; Morishima, Y. Macromolecules 1998, 31, 3588–3594. (6) Yamamoto, H.; Morishima, Y. Macromolecules 1999, 32, 7469– 7475. (7) Yamamoto, H.; Tomatsu, I.; Hashidzume, A.; Morishima, Y. Macromolecules 2000, 33, 7852–7861. (8) Yamamoto, H.; Hashidzume, A.; Morishima, Y. Polym. J. (Tokyo, Jpn.) 2000, 32, 745–752. (9) Halperin, A. Macromolecules 1991, 24, 1418–1419. (10) Borisov, O. V.; Halperin, A. Langmuir 1995, 11, 2911–2919. (11) Borisov, O. V.; Halperin, A. Macromolecules 1996, 29, 2612–2617. (12) Urakami, N.; Takasu, M. J. Phys. Soc. Jpn. 1996, 65, 2694–2699. (13) Urakami, N.; Takasu, M. Prog. Theor. Phys. Suppl. 1997, 126, 329– 332. (14) Halperin, A. In Supramolecular Polymers; Ciferri, A., Ed.; Marcel Dekker: New York, 2000; pp 93-146. (15) Zhang, G.; Winnik, F. M.; Wu, C. Phys. ReV. Lett. 2003, 90, 035506/035501-035506/035504. (16) Kawata, T.; Hashidzume, A.; Sato, T. Macromolecules 2007, 40, 1174–1180. (17) (a) Yamakawa, H. Modern Theory of Polymer Solutions; Harper & Row: New York, 1971; Chapter II. (b) Yamakawa, H. Helical Wormlike Chains in Polymer Solutions; Springer-Verlag: Berlin, 1997. (18) Yamakawa, H.; Stockmayer, W. H. J. Chem. Phys. 1972, 57, 2843– 2854. (19) Yamakawa, H. Modern Theory of Polymer Solutions: Harper & Row: New York, 1971; Chapter V. (20) Yashiro, J.; Hagino, R.; Sato, S.; Norisuye, T. Polym. J. (Tokyo, Jpn.) 2006, 38, 57–63. (21) According to the Manning theory32 of polyelectrolytes, the ion condensation occurs at l/(1- x) < 0.714 nm (the Bjerrum length) or at x (the content of C12) < 0.65 for AMPS/C12, so that the effective charge density of AMPS/C12 with x ) 0.5 can be identified with that for the AMPS

Tominaga et al. homopolymer with x ) 0. Furthermore, side-chain dodecyl groups affect the excluded volume effect (the long range interaction) but not the local chain stiffness in a first approximation. Therefore, we may approximate q of AMPS/C12 with x ) 0.5 to that of the AMPS homopolymer. (22) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed., Acedemic Press: London, 1992; Chapter 17. (23) Although Kawata et al.16 originally assumed that λ would be proportional to x, we here take this parameter to be constant. (24) (a) Shelley, J.; Watanabe, K.; Klein, M. L. Int. J. Quantum Chem., Quantum Biol. Symp. 1990, 17, 103–117. (b) MacKerell, A. D., Jr. J. Phys. Chem. 1995, 99, 1846–1855. (c) Bruce, D.; Berkowitz, M. L.; Perera, L.; Forbes, M. D. E. J. Phys. Chem. B 2002, 106, 3788–3793. (d) Bruce, D.; Senapati, S.; Berkowitz, M. L.; Perera, L.; Forbes, M. D. E. J. Phys. Chem. B 2002, 106, 10902–10907. (e) Gao, J.; Ge, W.; Hu, G.; Li, J. Langmuir 2005, 21, 5223–5229. (f) Yoshii, N.; Okazaki, S. Chem. Phys. Lett. 2006, 425, 58–61. (25) (a) Khalatur, P. G.; Khokhlov, A. R. AdV. Polym. Sci. 2006, 195, 1–100. (b) Okhapkin, I. M.; Makhaeva, E. E.; Khokhlov, A. R. AdV. Polym. Sci. 2006, 195, 177–210. (c) Khokhlov, A. R.; Khalatur, P. G. Physica A 1998, 249, 253–261. (d) Vasilevskaya, V. V.; Khalatur, P. G.; Khokhlov, A. R. Macromolecules 2003, 36, 10103–10111. (e) Berezkin, A. V.; Khalatur, P. G.; Khokhlov, A. R. J. Chem. Phys. 2003, 118, 8049–8060. (26) Yashiro, J.; Norisuye, T. Polym. Bull. 2006, 56, 467–474. (27) Mizuse, M. Master Thesis, Osaka University, 2010. (28) Ueda, M.; Hashidzume, A.; Sato, T. Polym. Prepr. Jpn. 2009, 58, 3057–3058. (29) The original authors26 chose slightly different wormlike cylinder parameters; here we have chosen q ) 3.0 nm, bead diameter ) 1.5 nm, molar mass per unit contour length ) 90 nm-1, and the excluded volume strength ) 15 nm. (30) Tanford, C. Physical Chemistry of Macromolecules; John Wiley & Sons: New York, 1961; Chapter 6. (31) Sato, T.; Kimura, T.; Hashidzume, A. Prog. Theor. Phys., Suppl. 2008, 175, 54–63. (32) Manning, G. S. J. Chem. Phys. 1969, 51, 924–933.

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