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Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

Kinetics of Morphological Transition between Cylindrical and Spherical Micelles in a Mixture of Anionic−Neutral and Cationic− Neutral Block Copolymers Studied by Time-Resolved SAXS and USAXS Rintaro Takahashi,†,‡,§ Theyencheri Narayanan,† Shin-ichi Yusa,∥ and Takahiro Sato*,‡ †

ESRF−The European Synchrotron, 71 Avenue des Martyrs, F-38043 Grenoble, France Department of Macromolecular Science, Osaka University, 1-1 Machikaneyama-cho, Toyonaka, Osaka 560-0043, Japan § Department of Chemistry and Biochemistry, University of Kitakyushu, 1-1 Hibikino, Wakamatsu-ku, Kitakyushu, Fukuoka 808-0135, Japan ∥ Department of Applied Chemistry, Graduate School of Engineering, University of Hyogo, 2167 Shosha, Himeji, Hyogo 671-2280, Japan ‡

S Supporting Information *

ABSTRACT: This study is concerned with the morphological transition kinetics of polyelectrolyte complex micelles formed from an anionic−neutral block copolymer and a cationic− neutral block copolymer in aqueous NaCl solution. The transformation was induced by changing the mixing ratio of the anionic and cationic monomer units in the copolymers. The kinetics of the morphological transition was directly tracked by time-resolved (ultra) small-angle X-ray scattering coupled with rapid mixing of the block copolymer components by a stopped flow apparatus. The transformation from cylindrical to spherical micelles upon changing the mixing ratio of the copolymers was reversible, and the process occurred via the random scission of the cylindrical micelles along their contours. The reverse transition from spherical to cylindrical micelles was found to be a slow process with high activation energy.

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INTRODUCTION An anionic−neutral block copolymer and a cationic−neutral block copolymer coassemble in aqueous solution via the electrostatic attraction between anionic and cationic blocks or through the entropy gain by the counterion release. The assemblies (called as polyelectrolyte complex micelles or micellar interpolyelectrolyte complexes) are composed of polyelectrolyte complex core and neutral coronal chains, which are good candidates for practical applications such as drug and gene carriers.1−6 In polyelectrolyte complex micellar systems, many factors may influence the morphology, e.g., the salt concentration, the mixing ratio of copolymers, pH, the total copolymer concentration, the ratio of block lengths, etc. Since the first report by Harada and Kataoka,1 various kinds of polyelectrolyte complex micelles have been investigated.2−15 Recently, Takahashi et al.8,9 studied the polyelectrolyte complex micelle formed by mixing an anionic−neutral block copolymer (AP; Chart 1a) and a cationic−neutral block copolymer (MP; Chart 1b). Here, the PMPC in Chart 1 is known to be a biocompatible polymer,16 and its interchain interactions in aqueous solution do not depend on the ionic strength,17 as in the case of neutral chains. When the mixing ratio of AP and MP was changed from stoichiometric to © XXXX American Chemical Society

nonstoichiometric ratio at constant NaCl concentration (CS = 0.1 M), a reversible vesicle-to-sphere morphological transition was observed.9 The nonstoichiometric mixing introduces excess charges on AP or MP component to the micelles, and the resulting electrostatic energy induces the morphological transition.9 Thus, the mixing ratio is an important experimental parameter to control the morphology of the polyelectrolyte complex micelles.10,13−15 In the present work, we have investigated the morphological transition of the polyelectrolyte complex micelles formed by AP and MP by changing the mixing ratio from stoichiometric to a nonstoichiometric value at CS = 0.01 M, instead of CS = 0.1 M in the previous study.9 This enabled us to find the reversible cylinder-to-sphere transition and probe the corresponding kinetic pathway. To monitor this morphological transition kinetics, we have employed time-resolved ultra small-angle and small-angle X-ray scattering (USAXS and SAXS, respectively) combined with rapid stopped-flow mixing. The quantitative Received: January 15, 2018 Revised: March 27, 2018

A

DOI: 10.1021/acs.macromol.8b00101 Macromolecules XXXX, XXX, XXX−XXX

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Chart 1. Chemical Structures of APa (a) and MPb (b)

Article

EXPERIMENTAL SECTION

Materials. AP and MP were synthesized and characterized by a similar manner as in the previous report.14 The molecular characteristics are listed in Table 1. NaCl (>99.5%) purchased from Wako Pure Chemical Industries, Osaka, was used without further purification. Water was deionized by a Synergy system (Merck Millipore, Darmstadt). Preparation of Solutions and Mixing Procedure. AP and MP samples were separately dissolved in aqueous NaCl solutions. The total copolymer mass concentration c and NaCl molar concentration CS were fixed to be 0.005 g/cm3 and 0.01 M, respectively. For static USAXS/SAXS measurements, the AP solution was added dropwise into the MP solution, or the MP solution was added dropwise into the AP solution to adjust the mixing ratio x+, defined by a

Block copolymer consisting of poly[sodium 2-(acrylamido)-2methylpropanesulfonate] (PAMPS) and poly[2-(methacryloyloxy)ethylphosphorylcholine] (PMPC). bBlock copolymer composed of poly{[3-(methacryloylamino)propyl]trimethylammonium chloride} (PMAPTAC) and PMPC.

x+ =

C+ C+ + C −

(1)

where C+ and C− are the molar concentrations of cationic and anionic monomer units, respectively, in the solution after mixing. The former and latter mixing procedures are denoted as “AP → MP” and “MP → AP”, respectively. After mixing, the solution was shaken using a vortex mixer. In the measurements to monitor the morphological transition of micelles from cylinders to spheres by stopped-flow time-resolved USAXS/SAXS, one of the syringes of the SFM-400 stopped-flow device (Bio-Logic Science Instruments, Claix, France)23 was filled with the solution of x+ = 0.55 at CS = 0.01 M. Another syringe was filled with the pure AP solution at CS = 0.01 M, and the mixing ratio was adjusted to get final x+ = 0.35. For the sphere-to-cylinder transition, the syringes were filled with the solution of x+ = 0.35 and the pure MP solution at CS = 0.01 M, and the two solutions were mixed to obtain the final x+ = 0.55. USAXS/SAXS. The synchrotron USAXS/SAXS experiments were performed at the ID02 beamline, ESRF, Grenoble.24 The incident intensity and wavelength λ0 of the X-rays were ca. 1013 photons/s and 0.0995 nm, respectively. The sample-to-detector distance was set to either 30.7 or 3.5 m. Two-dimensional scattering patterns were recorded by the Rayonix MX170 CCD detector. The sample temperature was at ca. 23 °C. For each scattering pattern, the X-ray exposure time was 5 ms in the stopped-flow USAXS/SAXS experiments, and it was 100 ms for the static measurements. The flow rate and dead time of the stopped-flow mixing were 7 mL/s and 0.0025 s, respectively. For these measurement conditions, X-rays did not cause damage of the samples which was verified in a similar manner as in the previous study.25 The measured scattering intensities were normalized to an absolute scale and azimuthally averaged to obtain the one-dimensional scattering profiles. The corresponding normalized background scattering profile was subtracted to obtain the differential scattering cross-section dΣ/dΩ(q) of the sample.26 Here, q denotes the magnitude of the scattering vector defined by q ≡ (4π/λ0) sin (θ/2) with θ the scattering angle. The definition and calculation procedure of the optical constant (denoted as Ke) are given in the Supporting Information. The profiles obtained with the different sample-to-detector distances were merged and rebinned to reduce noise in the high q region.

analysis of the results enabled us to reveal the detailed pathway of the cylinder-to-sphere transition. The kinetics of the micellar morphological transition from the cylinder to sphere was previously investigated for block copolymer micelles in solution by changing the mixed solvent composition19−21 or pH.22 Based on these studies, two kinetic mechanisms have been proposed: (1) the removal of spherical blobs from the cylinder ends18 (Scheme 1A); (2) the Scheme 1. Schematic Illustration of the Two Possible Kinetic Mechanisms for the Cylinder-to-Sphere Transition

fragmentation of cylinder driven by the Rayleigh instability20,21 (Scheme 1B). The latter mechanism may be further classified into the all-or-none fashion and the random scission process. However, these previous studies have led to different conclusions. In this work, we have elucidated the detailed kinetic pathway of the cylinder-to-sphere transition of polyelectrolyte complex micelles triggered by changing the mixing ratio of anionic and cationic block copolymers. Table 1. Molecular Characteristics of AP and MP Samples sample

Mn,1a

Mw,1/Mn,1b

Mw,1c

Mw,1nd

Mw,1−d

PMPC AP MP

6 210 48 500 49 500

1.03 1.07 1.05

6 400 51 900 52 000

6400 6400

45 500

Mw,1+d

N0ne

N0−e

21.7 21.7

198

45 600

N0+e

206

a Number-average molecular weight determined by 1H NMR. bDispersity index determined by SEC. cWeight-average molecular weight calculated from Mw,1/Mn,1 and Mn,1. dWeight-average molecular weight of the neutral block chain Mw,1n, the anionic block chain Mw,1−, or the cationic block chain Mw,1+ calculated from Mw,1 of the PMPC, AP, or MP samples. eWeight-average degree of polymerization of each block chain using the monomer-unit molar mass M0n = 295, M0− = 229, or M0+ = 221.

B

DOI: 10.1021/acs.macromol.8b00101 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules In the low q region, the differential scattering cross-section can be written as the following equation:27

⎡ ⎤1/2 Kec ⎢ ⎥ ⎣ dΣ/dΩ(q) ⎦ =

⎡ 1 1 2 2 4 ⎤ ⎢1 + ⟨S ⟩z ,PCM q + O(q )⎦⎥ 1/2 ⎣ 6 (wPCMM w,PCM)

(2)

Here, Mw,PCM and ⟨S ⟩z,PCM are the weight-average molar mass and the apparent mean-square radius of gyration of the polyelectrolyte complex micelles, and w PCM is the weight fraction of the polyelectrolyte complex micelles in the total copolymers calculated by 2

wPCM

⎧ 1 − (1 − x+)/(1 − x+(c)) ⎪1 − (x+ > x+(d)) ⎪ x+ + (MAP/MMP)(N0 +/N0 −)(1 − x+) =⎨ ⎪ 1 − x+/x+(c) (x+ < x+(d)) ⎪1 − 1 − x+ + (MMP/MAP)(N0 −/N0 +)x+ ⎩

(3)

where MAP and MMP are the molecular weight of AP and MP, and x+(c) and x+(d) are the mole fractions of the cationic monomer unit in the core and in the dilute phase, respectively (in the denominator of the second line of eq 4 in ref 9, x+ and 1 − x+ must be replaced each other). Although the core is negatively charged at x+ = 0.35, the amount of charge is not so large that we approximate x+(c) = 0.5.9 Thus, eq 3 leads to wPCM = 0.70 and 0.90 at x+ = 0.35 and 0.55, respectively. In eq 2, the contribution of the excess chain component to the scattering intensity was ignored because the contribution is much smaller than that of the micellar components in the low q region (see Figure 1a). The radius of gyration ⟨S2⟩z,PCM obtained using eq 2 is affected by the heterogeneity in the contrast factor of the micelle (cf. the Supporting Information).

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RESULTS AND DISCUSSION USAXS/SAXS Profiles in the Equilibrium State. Figure 1a shows the USAXS/SAXS profiles of the solutions for x+ = 0.35 (by MP → AP) and 0.55 (by AP → MP) at CS = 0.01 M in the equilibrium state (about 2 × 105 s elapsed after mixing). At x+ = 0.55, the power law dΣ/dΩ(q) ∝ q−1 was observed in the low q region which is a sign of the cylindrical micelle formation. Similar results were obtained in the previous study.8 On the other hand, the profile at x+ = 0.35 exhibits a weaker slope, and the intensity at low q region resembles the previous result8 at x+ = 0.4 and CS = 0.1 M, suggesting a change of morphology from larger cylinders to smaller spheres. The oscillation at q ∼ 0.2− 0.3 nm−1 comes from the uniform cross-sectional diameter of the micelles. Figure 1a also shows SAXS profiles of the AP (x+ = 0) and MP (x+ = 1) in aqueous NaCl solution at CS = 0.1 M. In the low q region, dΣ/dΩ(q)/(Kec) values of the pure block copolymer components are much lower than those of their complexes. The Gaussian chain model (described in the Supporting Information of ref 8) can fit the data of the pure components represented by the solid curves. Figure 1b demonstrates the reversibility of the USAXS/SAXS profiles for CS = 0.01 M. The red and green symbols are the same data as in Figure 1a, and gray and orange symbols indicate the profiles for the solution of x+ = 0.35 prepared by adding AP solution into the solution of x+ = 0.55 (AP → MP) and for the solution of x+ = 0.55 prepared by adding MP solution into the solution of x+ = 0.35. When x+ changed from 0.55 to 0.35 (or vice versa from 0.35 to 0.55), we obtained almost the same USAXS/SAXS profile as those of the solutions prepared directly. These results indicate that the morphological transitions are reversible.

Figure 1. USAXS/SAXS profiles in the equilibrium state (t ∼ 2 × 105 s) of aqueous NaCl solutions of AP−MP mixtures at different mixing ratios (a) and prepared by different pathways (b). In panel a, red symbols: x+ = 0.35 prepared by MP → AP at CS = 0.01 M; green symbols: x+ = 0.55 prepared by AP → MP at CS = 0.01 M; blue and purple symbols (the data are almost superimposed): x+ = 0 and 1 at CS = 0.1 M, respectively. In panel b, gray symbols: x+ = 0.35 prepared by adding AP solution into the solution of x+ = 0.55 (AP → MP) at CS = 0.01 M; orange symbols: x+ = 0.55 prepared by adding MP solution into the solution of x+ = 0.35 (MP → AP) at CS = 0.01 M. The red and green symbols in panel b are the same data as in panel a. Solid curves represent theoretical curves calculated by eq 4. The measurements were performed at t ∼ 2 × 105 s after mixing. The profiles of x+ = 0.55 in panel b were shifted vertically by multiplying by a factor A for clarity.

Time Evolution of USAXS/SAXS Profiles during the Cylinder-to-Sphere Transition. In order to monitor the morphological transition from the cylindrical to spherical micelles, we performed time-resolved USAXS/SAXS measurements. Figure 2a shows the time evolution of USAXS/SAXS profiles when x+ was changed from 0.55 → 0.35. During t from 0 s (before mixing) to ca. 30 s, the shape of the scattering profiles remained almost constant, indicating the cylinder morphology was kept in this t range. However, the oscillation at q ∼ 0.2−0.3 nm−1 started diminishing at t = 0.0025 s. This C

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Figure 2. Time evolution of USAXS/SAXS profiles during the cylinder-to-sphere transition by changing x+ from 0.55 to 0.35, in double-logarithmic plots (a) and the Berry plots at the early t (b) and at the late t (c). In panel a, successive curves have been shifted by a factor 10 for clarity, and the upper right figure shows the same profiles without vertical shifts. Solid curves in panel a represent the fitted model curves (eq 4). Solid lines in panels b and c represent initial tangents.

molar mass, the particle scattering function, and the second virial coefficient of the component 1, respectively. w′j, wj(M), and Pj(q) (j = cyl, sph) are the weight fraction (in the polyelectrolyte complex micelle), the weight fraction of the component with the molar mass (M), and the particle scattering function28,29 of the micelle j (cf. the Supporting Information), respectively. We assumed a log-normal distribution function for wj(M):

suggests an increase in the dispersity of the cross-sectional diameter of the cylindrical core of the micelles upon mixing at t = 0.0025 s. After t ∼ 150 s, both absolute value and slope of dΣ/dΩ(q) in the low q region decreased with t (see the inset of Figure 2a), which may reflect the decrease of the cylinder length and/or a change of the morphology to spherical micelles. Using the data in the low q region, we constructed the Berry plot,27 as shown in Figure 2b,c. Applying eq 2, we estimated the weight-average molar mass Mw,PCM and the z-average meansquare radius of gyration ⟨S2⟩z,PCM as a function of t. As depicted in Figure 3a, both Mw,PCM and ⟨S2⟩z,PCM started decreasing around t = 10 s and approached the asymptotic values at t ≈ 103 s. Modeling of the USAXS/SAXS Profiles. USAXS/SAXS profiles were analyzed on the basis of the methods reported previously.8,9 Each solution containing the polyelectrolyte complex micelles (PCM) and the excess copolymer component (1) and PCM was modeled as cylindrical (cyl) and/or spherical (sph) micelles. Under the assumption of electroneutrality of the micelles and neglecting the interference effect among PCM particles, we express the scattering function as9

Mwj(M ) =

(5)

where Mw,j and Mn,j are the weight- and number-average molar masses of the micelle j. Although eq 4 contains many parameters, γi and wi can be calculated by eqs S3−S6 (in the Supporting Information) and eq 3, respectively, Mw,1 is given in Table 1 (= 51 900), and P1(q) and A2,1 are approximated by those experimentally obtained at CS = 0.1 M.8 P1(q) and A2,1 are important only at high q where CS dependences of P1(q) and A2,1 are not significant. Furthermore, the scattering functions in equilibrium states at x+ = 0.35 and 0.55 shown in Figure 1 can be fitted by assuming w′cyl (= 1 − w′sph) = 0 and 1, respectively, and Mw,sph and Mw,cyl are given by Mw,PCM at t = 0 and 2 × 105 s, respectively, already obtained experimentally in Figure 3a. The equilibrium scattering function at x+ = 0.35 can be fitted by choosing only two adjustable parameters Mw,sph/Mn,sph and the mass concentration of the core domain ccore for the spherical micelle because ⟨S2⟩corona1/2 has been already determined to be 1.5 nm previously.9 On the other hand, the equilibrium scattering function at x+ = 0.55 is fitted with three adjustable parameters, ccore, the average diameter of the cylinder core ⟨Dcyl,c⟩, and the standard deviation of the diameter σD,

γ12w1M w,1P1(q) γav 2 dΣ (q) = + γPCM 2wPCM 1 + 2A 2,1w1M w,1P1(q) Kec dΩ ⎡ ′ Mwcyl(M )Pcyl(q) dM + w′sph × ⎣⎢wcyl

∫

⎤ dM ⎦⎥

⎡ ln 2(M / M M ) ⎤ w,j n,j ⎥ exp⎢ − ⎢ 2 ln(M w,j /M n,j) ⎥⎦ 2π ln(M w,j /M n,j) ⎣ 1

∫ Mwsph(M)Psph(q) (4)

where γi and wi (i = 1, PCM) are the contrast factor and the weight fraction of the component i in total polymer, respectively. Mw,1, P1(q), and A2,1 are the weight-average D

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where both cylindrical and spherical micelles coexist. However, we can determine ⟨Dcyl,c⟩ and σD almost uniquely from the position and sharpness of the minimum at q ∼ 0.2−0.3 nm−1 and may assume that Mw,cyl/Mn,cyl = 2, Mw,sph/Mn,sph = 1, ⟨S2⟩corona1/2 = 1.5 nm, and ccore = 0.38 g/cm3. Furthermore, Mw,PCM shown in Figure 3a follows the relation ′ M w,cyl + (1 − wcyl ′ )M w,sph M w,PCM = wcyl

(6)

Therefore, the remaining adjustable parameters are only Mw,cyl and w′cyl, and Mw,sph can be calculated using eq 6. The fit results are shown by solid curves in Figure 2a. Figure 4 shows the time evolutions of Mw,cyl, w′cyl, ⟨Dcyl,c⟩, 2Rsph,c, and σD/⟨Dcyl,c⟩ determined from the fits. Values of Mw,cyl, and w′cyl have larger error bars because the scattering function does not change independently with Mw,cyl and w′cyl. However, Mw,cyl definitely decreased for t > 10 s, indicating that the scission of cylindrical micelles is not in an all or none fashion (cf. Scheme 1). The diameter ⟨Dcyl,c⟩ of the cylinder first increased at t ≲ 0.1 s and then decreased over 1 s ≲ t ≲ 100 s. The first increase may be due to the electrostatic repulsion among AP chains absorbed onto the micelle core. The later decrease in ⟨Dcyl,c⟩ may reflect the undulation of the cylindrical micelles driven by the Rayleigh instability.20,21 On the other hand, the diameter 2Rsph,c of the newly formed spherical micelles is larger than ⟨Dcyl,c⟩ for t ≳ 5 s and almost remained constant. As already mentioned in the previous paper,8 the diameter of the core is progressively larger in the order of spherical (2Rsph,c), cylindrical (⟨Dcyl,c⟩), and vesicles due to the interfacial energy. The present results of ⟨Dcyl,c⟩ and 2Rsph,c agree with this rule. Lund et al.20 reported the growth of the spherical micelle via the molecular exchange mechanism in the late stage of the cylinder to sphere transition for poly(ethylene-alt-propylene)b-poly(ethylene oxide) (PEP1−PEO1) in N,N-dimethylformamide-d7 (dDMF)−D2O mixtures by increasing the dDMF content. In our case, there is not such a ripening process after the spherical micelles are formed from the cylindrical micelles. Kinetics of the Cylinder-to-Sphere Transition. In the cylinder-to-sphere transition, both Mw,cyl and w′cyl decreased with time, but the onset of decrease of the former was earlier. If the transition from the cylindrical to spherical micelle would occur in an all or none fashion, Mw,cyl should be constant and only w′cyl should decrease. Thus, the cylinder-to-sphere transition in our system does not take place in the all-ornone fashion but by a process of random scission of the cylindrical micelles. Because the spherical micelles have an optimum aggregation number, the scission of the cylindrical micelles into spherical micelles may be identified with the depolymerization of linear polycondensates, where the spherical micelle is regarded as the monomer, and the scission takes place randomly along the cylindrical micelle contour. Here, it may be inferred that the molecular exchange process does not occur in this time range

Figure 3. Molar mass Mw,PCM and radius of gyration ⟨S2⟩z,PCM1/2 as a function of time t (a) and the plot of ⟨S2⟩z,PCM1/2 vs Mw,PCM (b) during the cylinder-to-sphere transition. The leftmost point in panel a represents the data in the initial state (x+ = 0.55). The solid curves in both panels a and b represent fit results by the random scission model (eqs 7 and 14), and the broken curve in panel b represents the theoretical result for the scission from ends (cf. the Supporting Information).

because we can assume Mw,cyl/Mn,cyl = 2 for the cylindrical micelle (see eq S35). Among these parameters, ⟨Dcyl,c⟩ and σD are almost uniquely determined by the position and sharpness of the minimum at q ∼ 0.2−0.3 nm−1. Thus, the equilibrium scattering functions at x+ = 0.35 and 0.55 can be fitted without significant correlation of the parameters. The fit results are shown by the solid curves for x+ = 0.35 and 0.55 in Figure 1, and parameters obtained are listed in Table 2. The concentrations ccore inside the micelle core at x+ = 0.55 and 0.35 are the same (= 0.38 g/cm3), indicating that negative charges in the core at x+ = 0.35 do not essentially swell the core of spherical micelles. The dispersity Mw,sph/Mn,sph is close to unity, as expected for the spherical micelles. In Figure 1, the solid curve for x+ = 0.55 slightly deviates from the experimental profiles for the cylindrical micelles in the small q region ( 0.55; i.e., some of excess AP chains are absorbed in the spherical core of the spherical polyelectrolyte complex micelle at x+ = 0.35. The electrostatic energy given by the negative charges in the core may contribute to the micellar morphology and structure.9 Electrostatic energies of the cylindrical micelle Ucyl and the spherical micelle Usph are formulated in the Supporting Information and ref 9, respectively. G

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Macromolecules Ucyl =

⎤2 Q cyl 2 ⎧ ⎪⎡ 1 ⎥ ⎨⎢ 4πε0L ⎪⎢⎣ κR cyl,cK1(κR cyl,c) ⎥⎦ ⎩

∞

∫κR

⎫ ε0 ⎪ ⎬ 4εi ⎪ ⎭

K12(y)y dy +

cyl,c

the cylinder to sphere by changing x+ from 0.55 to 0.35 may be explained by this electrostatic energy difference. Figure 7 also indicates that the charged cylindrical micelle can reduce the electrostatic energy by decreasing Rcyl,c. By changing x+ from 0.55 to 0.35, ⟨Dcyl,c⟩ (= 2Rcyl,c) of the cylindrical micelles decreased with time between 1 and 100 s, as shown in Figure 4c. By adding AP chains to the solution of x+ = 0.55, the cylindrical micelle cores absorbed the excess AP chains to be negatively charged and decreased its diameter to reduce the electrostatic energy. After that, the charged cylindrical micelles transformed to energetically more stable spherical micelles.

(17) 1 ⎡ ⎤ Q sph 2 ⎢ 1 + 2 κR sph,c e−κR sph,c ε0 ⎥ = + 8πε0R sph,c ⎢ 5εi ⎥ (1 + κR sph,c)2 ⎣ ⎦

(

Usph

)

(18)

where Qsph and Qcyl are charges of the spherical and cylindrical micelles, respectively, ε0 and εi are the dielectric constant of the solvent and the micellar core, respectively, L is the (effective) core length of the cylindrical micelle (= Lcore,x + (4/3)Rcyl; cf. Figure S1), κ is the reciprocal of the Debye length calculated from the added salt concentration CS, and K1(y) is the firstorder modified Bessel function of the second kind. If a cylindrical micelle is divided into x spherical micelles, we have the following relations: 4π R sph,c3 x = πR cyl,c2L (19) 3 Q cyl

2

2 NQ sph2L 3 ⎛ R cyl,c ⎞ ⎜ ⎟ = (xQ sph) = R sph,c 4 ⎜⎝ R sph,c ⎟⎠

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CONCLUSION We have found a reversible morphological transition between cylindrical and spherical micelles in polyelectrolyte complex micelles induced by changing the mixing ratio. The cylindrical micelles can be charged by changing the mixing ratio from almost stoichiometric to nonstoichiometric, and they become unstable electrostatically. Time-resolved USAXS/SAXS measurements directly revealed the following pathway for the cylinder-to-sphere transition. The cylindrical-to-spherical micelles transition is identified with the depolymerization of linear polycondensates (the spherical micelle is regarded as the monomer), and the random scission along the cylindrical micelle contour takes place. The alternative removal mechanism of spherical blobs from the cylinder ends18 was excluded for the cylinder-to-sphere transition (cf. Scheme 1) in our system. The sphere-to-cylinder transition in the opposite direction is a slow process involving high activation energy.

2

(20)

because the inner concentrations ccore of the spherical and cylindrical micellar cores are identical (cf. Table 2). Combining eqs 17−20, we have Ucyl xUsph

2 3 ⎛⎜ R cyl,c ⎞⎟ = ⎜ 2 ⎝ R sph,c ⎟⎠

[κR cyl,cK1(κR cyl,c)]−2 ∫ ×

∞

κR cyl,c

K12(y)y dy +

⎡ 1 + 1 κR 2 ⎤ −κR sph,c + sph,c /(1 + κR sph,c) ⎦e ⎣ 2

(

)

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ε0 4εi ε0 5εi

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.8b00101. Detailed descriptions of the structure and kinetic models, and electrostatic energy (PDF)

(21)

Figure 7 shows the ratio Ucyl/xUsph plotted against Rcyl,c/Rsph,c, where Rsph,c is chosen to be 18 nm obtained experimentally (cf.

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AUTHOR INFORMATION

Corresponding Author

*(T.S.) E-mail [email protected]. ORCID

Rintaro Takahashi: 0000-0003-2168-9436 Theyencheri Narayanan: 0000-0003-1957-1041 Shin-ichi Yusa: 0000-0002-2838-5200 Takahiro Sato: 0000-0002-8213-7531 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Dr. J. Möller (ESRF) and Dr. R. Dattani (ESRF) for their help during the time-resolved USAXS/SAXS measurements, and we also thank the ESRF for the provision of synchrotron beam time. This work was financially supported by Grant-in-Aid for JSPS Research Fellow (R.T.; Grant No. 16J00359).

Figure 7. Comparison of the electrostatic energy between the cylindrical and spherical micelles in the case of Rsph,c = 20 nm.

Table 2), CS = 0.01 M, and ε0/εi is varied from 1 to 10. The ratio Rcyl,c/Rsph,c was estimated to be ca. 0.9 from the SAXS profiles (cf. Table 2 and Figure 4c). Thus, Ucyl is higher than xUsph, and the spherical micelle is electrostatically more stable than the cylindrical micelle. The morphological transition from

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REFERENCES

(1) Harada, A.; Kataoka, K. Formation of Polyion Complex Micelles in an Aqueous Milieu from a Pair of Oppositely-Charged Block

H

DOI: 10.1021/acs.macromol.8b00101 Macromolecules XXXX, XXX, XXX−XXX

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DOI: 10.1021/acs.macromol.8b00101 Macromolecules XXXX, XXX, XXX−XXX