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Chain Conformation of Poly[2-(methacryloyloxy)ethyltrimethylammonium chloride] in Aqueous Sodium Chloride Solutions Tatsuya Ishikawa,† Moriya Kikuchi,‡ Motoyasu Kobayashi,‡ Noboru Ohta,§ and Atsushi Takahara*,†,‡,∥,⊥ †

Graduate School of Engineering, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan Japan Science and Technology Agency, ERATO, Takahara Soft Interfaces Project, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan § Japan Synchrotron Radiation Research Institute/SPring-8, 1-1-1, Kouto, Sayo-cho, Sayo-gun, Hyogo 679-5198 Japan ∥ Institute for Materials Chemistry and Engineering, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan ⊥ International Institute for Carbon-Neutral Energy Research (WPI-I2CNER), Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan ‡

ABSTRACT: A series of poly[2-(methacryloyloxy)ethyltrimethylammonium chloride] (PMTAC) in aqueous NaCl solutions with a weight-average molecular weight (Mw) of between 6.0 × 103 and 1.5 × 105 were characterized by synchrotron radiation small-angle X-ray scattering (SAXS) and dynamic light scattering (DLS) measurements at 298 K. The hydrodynamic radius (RH) of PMTAC initially decreased with increasing salt concentration (Cs) and then leveled off at Cs = 0.5−1.0 M. The values of z-average root-mean-square radius of gyration (⟨S2⟩z1/2) simply increased with a decrease in Cs for Mw > 2 × 104 g/mol. In contrast, the values of ⟨S2⟩z1/2 for Mw < 2 × 104 g/mol increased with an increase in Cs probably because of a change in local conformation of PMTAC chain. To study local chain conformation of PMTAC, scattering functions and Mw dependences of ⟨S2⟩z1/2 and RH for PMTAC were analyzed in terms of the helical wormlike chain (HW) model which is a simple continuous elastic wire model with bending and torsional energies. The characteristic helix of the HW model in potential energy minimum for PMTAC clearly indicates that the local conformation is strongly influenced by the ionic strength of the solution.

1. INTRODUCTION Polyelectrolytes are polymers containing ionizable groups that can dissociate into polyvalent macroions and counterions of opposite charge in polar solvents such as water.1 Electrostatic interactions between charges lead to complex intra- and intermolecular interactions that have strong consequences for both static and dynamic properties of polyelectrolyte solutions which are qualitatively different from those of neutral counterparts. Polyelectrolytes play an important role in nature because they constitute many of biological macromolecules such as tubulin, actin, and DNA. The molecular conformation of polyelectrolytes has been attracting research interests because of their fundamental importance in biological and biophysical processes. Thus, the response of polyelectrolytes to ionic strength of aqueous solutions has been intensively investigated.2−7 The high charge of the macroion produces a strong electric field which attracts these counterions. Because of the strong electric field of the macroion, a fraction of the counterions will be condensed on the macroion. This is the counterion condensation effect, which is first described by Oosawa1 and formalized by Manning for the case of an infinite line with constant charge density.8 The Manning−Oosawa (MO) model assumes that the polyelectrolyte chain can be © 2013 American Chemical Society

replaced by an infinitely long cylinder because the local chain segments may be considered as a rigid rod due to the strong electrostatic repulsion of the charged groups. The MO model predicts a renormalization of the effective charge of the chain to a constant value, above a condensation threshold at b = lB, where b is the average distance between charged monomers and lB is the Bjerrum length defined by lB = e2/εkBT (ε is the dielectric constant of the solvent, e is the elementary charge, kB is the Boltzmann constant, and T is the absolute temperature). This model is apparently capable of describing the experimental electrical properties of polyelectrolytes.8−10 On the other hand, chain dimensions of polyelectrolytes are very sensitive to changes in charge density and ionic strength of the solution11 because of Coulombic long-range interactions among charged groups. The understanding of dimensional properties of flexible polyelectrolytes is one of the very fundamental and important subjects in polymer physics, but there is currently no consensus.12 As in the case of neutral polymer chain, the conformation of polyelectrolytes is Received: January 30, 2013 Revised: April 17, 2013 Published: April 30, 2013 4081

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dispersity index (Mw/Mn < 1.2), thereby permitting the preparation of samples of PMTAC that are amenable to precise characterization. There are a few reports on the properties of PMTAC with high molecular weights in dilute aqueous solution,25,26 but the molecular weight distributions of the samples of PMTAC used in these studies were rather broad. In the present study, we performed size exclusion chromatography (SEC) measurements with multiangle light scattering (MALS) detector, dynamic light scattering (DLS), and smallangle X-ray scattering (SAXS) measurements on a series of samples of PMTAC with various molecular weights to elucidate the chain conformation of PMTAC in aqueous NaCl solution. The scattering function P(q), the root-mean-square radius of gyration ⟨S2⟩z1/2, and the hydrodynamic radius RH in 0.05, 0.1, and 0.5 M aqueous NaCl solutions were analyzed in terms of the HW chain with an excluded-volume effect.

conceivable to be determined by both short-range and longrange electrostatic interactions. The former interaction affects chain stiffness, i.e., local conformation. The stiffness parameter (λ−1) of polyelectrolyte is considered as a sum of two contributions or λ−1 = λ 0−1 + λel −1

(1)

λ0−1

where is the intrinsic stiffness parameter due to the original stiffness of the backbone and λel−1 is the electrostatic chain stiffness parameter arising from the neighboring charges. According to the so-called Odijk−Skolnic−Fixman (OSF) theory,13,14 λel−1 is given by the equation λel −1 =

lB 2κ 2b2

(2)

where κ−1 is the Debye−Hückel screening parameter. The latter, long-range interaction influences excluded volume effects. The excluded volume strength (B) of polyelectrolyte chains is also considered as a sum of nonelectrostatic (B0) and electrostatic excluded volume strengths (Bel).15 The chain stiffness and the excluded volume strength of polyelectrolytes are a function of ionic strength and in relation of the trade-off. Therefore, it is very difficult but essential to investigate ionic strength dependence of chain stiffness and excluded volume effects of polyelectrolyte chains separately. To determine the factors separately, the wormlike chain model of Kratky and Porod (KP) which is the simplest continuous (elastic wire) model has been used for a wide variety of polyelectrolytes.16−20 The KP chain model theory is applicable not only to stiff polymers but also to ordinary flexible polymers if the characteristic ratio Cn increases monotonically to its coillimiting value C∞ as the number of skeletal bonds n in the chain is increased. However, the breakdown of the KP chain model for some flexible polymers especially for the molecular weight corresponding to oligomer was reported.21 As a more general model, Yamakawa developed the helical wormlike (HW) chain22,23 which is an elastic wire with bending and torsional energies. When its total potential energy is a minimum, the HW chain adopts a regular helical form. This model can be regarded as a hybrid of the three extreme forms of rod, random coil, and regular helix. The HW chain is capable of describing these three forms as well as their intermediate conformations. The KP wormlike chain is considered to be a special case of the HW chain; the former has only the bending energy, and its energy is minimized when its contour becomes a straight line. Thus, the HW chain model enables us to know a local conformation of polymer chains more realistic. In the present study, we investigated the dimensional properties of poly[2-(methacryloyloxy)ethyltrimethylammonium chloride] (PMTAC) as a typical flexible cationic polyelectrolyte chain in aqueous NaCl solutions, especially focused on the change in the local chain conformation with varying ionic strength. On the experimental side, well-characterized samples including oligomers are required for a data analysis based on the HW theory. As far as we know, the experimental data including oligomer samples are very limited especially on polyelectrolytes so far. Recently, our research group reported a direct synthesis of PMTAC by a controlled atom transfer radical polymerization (ATRP) of MTAC monomer in a mixture of 2,2,2-trifluoroethanol and an imidazolium-based ionic liquid as a solvent.24 This ATRP gave PMTAC with predictable molecular weight and low poly-

2. EXPERIMENTAL SECTION 2.1. Materials. Seven samples of PMTAC were prepared by ATRP at 333 K in a mixture of 2,2,2-trifluoroethanol and an imidazolium ionic liquid as a solvent. Details of the reaction conditions have been described elsewhere.24 The chemical structure of the PMTAC is presented in Figure 1. Each of the polymer samples was dissolved in

Figure 1. Chemical structure of PMTAC. deionized water and dialyzed against deionized water to remove unreacted monomer. The solutions were then concentrated in a rotary evaporator and filtered three times through 0.45 μm PTFE membrane filters. The solutions were freeze-dried and then further dried in vacuum at 333 K for 24 h. Deionized water (resistivity = 18.2 MΩ cm) for sample preparation for the DLS and SAXS measurements was purified by using an arium 611UV system (Sartorius Stedim Biotech GmbH, Göttingen, Germany). All polymer solutions were prepared by a gravimetric method, and the polymer mass concentrations Cp (g/ mL) and salt concentrations in the aqueous NaCl solutions were calculated.27 2.2. Molecular Weight Characterization. The values of the weight-average molecular weight (Mw) and the Mw/Mn were determined by SEC equipped with a MALS detector (Dawn Heleos II; Wyatt Technology Corp., Santa Barbara, CA), operating at a wavelength of 658 nm, and a refractive-index (RI) detector (RID-10A; Shimadzu, Kyoto, Japan) calibrated with aqueous NaCl solutions. Aqueous acetic acid (0.5 M) containing sodium nitrate (0.2 M) was used as an eluent at a flow rate of 0.8 mL/min at 313 K. Normalization of the detectors and correction for the delay volume between the MALS and RI detectors were performed by using a sample of poly(ethylene oxide) (PEO) with Mw = 2.22 × 104 g/mol and Mw/Mn = 1.08. The Rayleigh ratio at a scattering angle of 90° was based on that of pure toluene. The polystyrene gel columns TSKgel G5000PWXL-CP × 2 and TSKgel G3000PWXL-CP (Tosoh Corp., Tokyo, Japan) were claimed to separate the polymers with molecular weights in the ranges (PEO) from 5 × 105 to 4 × 102 g/mol and from 5 × 104 to 2 × 102 g/mol, respectively. The angular dependence of the scattering intensity was extrapolated to zero angle by using the linear Berry fitting method. By using a differential refractometer (DRM3000; Otsuka Electronics Co., Ltd., Osaka, Japan) at a wavelength of 633 nm, the specific refractive index increment (dn/dc) of PMTAC in aqueous acetic acid (0.5 M) containing sodium nitrate (0.2 M) at 298 K was determined to be 0.1598 mL/g. This dn/dc value was used for the analysis of the SEC-MALS results, even though the laser 4082

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wavelengths of the refractometer and the MALS detector were different, because the difference between the dn/dc value at a wavelength of 633 nm and that at 658 nm is negligible. 2.3. Dynamic Light Scattering. Dynamic light scattering (DLS) measurements for solutions of PMTAC in aqueous NaCl at 298 K were carried out at scattering angles between 30° and 150° by using a goniometer system (CGS-3-TAC/LSE-5004; ALV-Laser Vertriebsgesellschaft m-b.H., Langen, Germany) with a He−Ne laser (λ = 632.8 nm) to determine the translational diffusion coefficient (D). For each series of measurements, four sample solutions with different polymer concentrations were used. The solutions were filtered through PTFE membranes with suitable pore sizes (usually 0.1 μm) into cylindrical quartz cells with an inner diameter of 8 mm. The apparent diffusion coefficient (Dapp) for each solution with finite polymer concentration (Cp) was determined by applying the cumulant method to the normalized autocorrelation function. At sufficiently small values of Cp, Dapp may be expanded as follows: Dapp = D0(1 + kDC p + ...)

As can be seen from the plots, the data points for each sample follow a straight line, and therefore the values of D0 and kD can be determined from the intercept and the slope, respectively. From the values of the D0 obtained from extrapolation of Dapp to Cp = 0, the hydrodynamic radius (RH) for each sample was evaluated by means of the following equation: RH =

kBT 6πη0D0

(4)

where kB is the Boltzmann constant, T is the absolute temperature, and η0 is the viscosity of the solvent. 2.4. Small-Angle X-ray Scattering. SAXS measurements on solutions of PMTAC in aqueous NaCl at room temperature were carried out at the BL40B2 beamline of the SPring-8 facility (Hyogo, Japan). Incident X-rays with a wavelength of 0.1 nm were used. The scattering vector (q) defined as (4π/λ) sin(θ), where λ is the wavelength and θ is the scattering angle, was calibrated from the Bragg reflection of powdery silver behenate. The scattering intensity I(q) was detected by using an imaging plate with 3000 × 3000 pixels to cover the range of scattering vectors from 0.02 to 7.0 nm−1. Two sample-todetector distances of 1156 and 2164 mm were used. The excess scattering intensity ΔI(q) was obtained as the difference between the value of I(q) for the solution and that of the neat solvent. Correction for the effect of absorption of X-rays by a given solution or solvent was performed by measuring the intensity of the incident X-rays and the scattering intensity of the X-rays. The solution of the sample was contained in a quartz capillary with an external diameter of 2 mm. The excess scattering intensities were analyzed by means of a square-root plot based on the following equation:

(3)

where D0 and kD are the diffusion coefficient at infinite dilution and the concentration coefficient of the diffusion coefficient, respectively. Figure 2 shows plots of values of the first cumulant Γ relative to the square of the scattering vector Γ/k2 against k2 for the sample P2 in 0.5

⎡ ⎤ P (q) [KC p/ΔI(q)]1/2 = [1/M w P(q)]1/2 ⎢1 + M w A 2 2 2 C p + ...⎥ P(q) ⎣ ⎦ (5) where K is the optical constant, A2 is the second virial coefficient, and P2(q) represents the intermolecular interference. P(q) is the particlescattering function and is determined experimentally by the following equation:

Figure 2. Plots of Γ/k2 against k2 for 0.5 M aqueous NaCl solutions of P2 with different polymer concentrations at 298 K.

[C p/ΔI(q)]Cp→ 0 1 = P(q) [C p/ΔI(q)]Cp , q → 0

M aqueous NaCl solution at 298 K, where the magnitude of the scattering vector is defined by k = (4πn0/λ) sin(θ/2), with n0 being the solvent refractive index and θ being the scattering angle. The value of Γ/k2 is almost independent of k2, so that we adopt as the value of Dapp [= (Γ/k2)=0] at each finite Cp the mean value represented by the horizontal line. Figure 3 shows plots of Dapp as a function of Cp for each sample of PMTAC (P1 to P7) in 0.5 M aqueous NaCl solutions.

(6)

P(q) can be expanded as follows: P(q) = 1 −

1 2 2 ⟨S ⟩z q + ... 3

(7)

Figure 4 gives a typical scattering intensity profile obtained from sample P1 in 0.5 M aqueous NaCl solution at various Cp. The plot of

Figure 4. Plots of the reduced scattering intensity at various polymer concentrations for sample P1 in 0.5 M aqueous NaCl solution at 298 K. The open circles were obtained by extrapolation to zero polymer concentration.

Figure 3. Polymer concentration dependence of Dapp for samples of PMTAC in 0.5 M aqueous NaCl solution at 298 K. 4083

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[Cp/ΔI(q)]1/2 against Cp was fitted by a straight line and could be extrapolated to infinite dilution to give the value [Cp/ΔI(q)]Cp=01/2 (the open circle in Figure 4). At values of q above 1.1 nm−1, no significant concentration dependence of [Cp/ΔI(q)]1/2 was observed. We determined the value of ⟨S2⟩z1/2 from the intercept and the initial slope of the Berry plot {[Cp/ΔI(q)]Cp=01/2 against q2} in the Guinier region (a typical example is shown in Figure 5) by applying the following equation, which can be derived from eqs 5 and 7: 1/2 [C p/ΔI(q)]1/2 C p= 0 = [C p/ ΔI (0)]C p= 0 +

+ ...

1 2 2 [C p/ΔI(0)]1/2 C p= 0 ⟨S ⟩z q 6 (8) Figure 6. SEC absolute calibration curve and ⟨S2⟩z1/2 (open squares) for PMTAC samples in 0.5 M aqueous acetic acid solution containing 0.2 M sodium nitrate. Open circles represent molar masses corresponding to maxima in the chromatogram. The SEC chromatograms for each PMTAC sample used in the present study are also shown.

RH value decreased from 9 to 7 nm with increasing Cs and then leveled off at 7 nm for Cs above 0.5 M, as shown in Figure 7. In other words, the chain dimensions of PMTAC were not affected by the presence of hydrated ions above 0.5 M. It is well-known that the chain dimensions of polyelectrolytes decrease with increasing Cs. However, the value of the hydrodynamic radius in 0.5 M aqueous NaCl solution was almost the same as that in a 1.0 M solution. This phenomenon was also observed in a single fully protonated poly(2vinylpyridine) (P2VP) chain at Cs < 1.5 M.29 On the other hand, re-expansion of P2VP coils adsorbed on mica surface from salt solutions with Cs higher than 1.0 M was observed by in-situ atomic force microscopy.5−7 In turn, PMTAC is not salted out even in saturated aqueous NaCl in the present study. Below, we discuss the difference of conformational properties of PMTAC in aqueous NaCl solutions with Cs smaller than 0.5 M at 298 K. 3.3. Molecular Weight Dependence of Chain Dimensions of PMTAC in Aqueous NaCl. The values of ⟨S2⟩z1/2 and RH for PMTAC in aqueous NaCl obtained from SAXS and DLS are listed in Table 1. Figure 8 shows double-logarithmic plots of the ⟨S2⟩z1/2 against Mw for the PMTAC chain in 0.05 M (circles), 0.1 M (squares), and 0.5 M (triangles) aqueous NaCl solutions. The data points are fitted by a straight line with values of exponent 0.85 for Cs = 0.05 M, 0.78 for Cs = 0.1 M, and 0.70 for Cs = 0.5 M. All these values of the exponent are higher than the limiting value of 0.6 for nonionic flexible chains in good solvents. The increase of the slope with decreasing Cs is well-known phenomenon in many polyelectrolyte solutions, resulting from an increase of electrostatic repulsive interaction, which increases the chain stiffness and excluded-volume strength. An interesting and remarkable observation to be noted in Figure 7 is that the values of ⟨S2⟩z1/2 for Mw > 2 × 104 g/mol simply increase with a decrease in Cs, but the values for Mw < 2 × 104 g/mol contrarily decrease with a decrease in Cs. The phenomenon of inversion in Cs dependence of ⟨S2⟩z1/2 appeared more prominently in the Mw dependence of RH for the PMTAC chain as shown in Figure 9. Obviously, the phenomenon of inversion cannot be simply explained by changes in the excluded-volume effect and chain stiffness. One plausible explanation is that a local conformation of PMTAC chain changes in response to Cs of aqueous NaCl solutions.

Figure 5. Berry plot for samples of PMTAC in 0.5 M aqueous NaCl solution at 298 K. The solid line shows the initial slope. Data points for P3 are shifted vertically to different amounts for the viewing clarity.

3. RESULTS AND DISCUSSION 3.1. Molecular Weight Characterization. For reliable SEC measurements of cationic polyelectrolytes, it is necessary to eliminate both ionic and hydrophobic interactions between the cationic polyelectrolytes and the packing material of the column. Stickler and Eisenbeiss controlled non-size-exclusion effects (ion inclusion, ion exclusion, and adsorption) by means of a special surface treatment of the stationary phase and by appropriate selection of the mobile phase.28 In the current study, to reduce hydrophobic and ionic interactions, we used 0.5 M aqueous acetic acid containing 0.2 M sodium nitrate as a mobile phase in conjunction with columns with poly(methacrylate)-bearing cationic groups on their surfaces that are specially designed for the separation of cationic polyelectrolytes. The SEC absolute calibration curve for PMTAC is shown in Figure 6 with chromatograms of each PMTAC sample used in the present study. The Mw and Mw/Mn were calculated from the absolute calibration curve, and the results are summarized in Table 1. The value of ⟨S2⟩z1/2 in the figure for samples of PMTAC with high Mw (> 1.6 × 105 g/ mol) were determined by SEC-MALS. The plot for ⟨S2⟩z1/2 shows a fairly linear relationship between ⟨S2⟩z1/2 and the retention volume, which can be expressed as ⟨S2⟩z1/2 ∝ Mw0.58. These results confirm that reliable Mw and Mw/Mn values for samples of PMTAC are determined in the present SEC condition. 3.2. Salt-Concentration Dependence of the Hydrodynamic Radius. The dependence of RH of PMTAC in aqueous NaCl at 298 K on the concentration of salt (Cs) was investigated by using sample P6 (Mw = 9.01 × 104 g/mol). The 4084

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Table 1. Molecular Characteristics of PMTAC Samples as Determined by SEC-MALS, SAXS and DLS in 0.05 M, 0.1 and 0.5 M Aqueous NaCl Solutions at 298 K ⟨S2⟩z1/2 (nm) for Cs (M) = 3

RH (nm) for Cs (M) =

sample

Mw/10 (g/mol)

Mw/Mn

0.05

0.1

0.5

0.05

0.1

0.5

P1 P2 P3 P4 P5 P6 P7

6.03 12.9 21.0 42.1 67.7 90.1 154

1.08 1.11 1.15 1.15 1.17 1.19 1.22

1.40 2.39 3.83 7.33 10.92 13.0

1.47 2.51 3.80 6.70 9.70 11.5

1.54 2.60 3.73 5.92 8.50 10.1

0.69 1.11 2.35 5.47 8.03 8.96 13.74

0.88 1.66 3.01 5.24 6.76 8.01 11.18

1.48 2.39 3.18 4.62 5.89 7.02 9.44

state. It is known that the molecular weight dependence of ⟨S2⟩z1/2 and the scattering function for PMMA cannot be explained in terms of a Kratky−Porod (KP) wormlike chain;21,30 therefore, the HW model has been applied to PMMA in theta solvents.31,32 The HW chain is a continuous wormlike chain with bending and torsional energies. When its total potential energy is a minimum, the HW chain adopts a regular helical form. The HW chain has four major model parameters: the constant curvature κ0 and the torsion τ0 of the characteristic regular helix at the energy minimum, the stiffness parameter λ−1, and the shift factor ML, which is defined as the molecular weight per unit contour length. The stiffness parameter λ−1 is defined as the bending force constant divided by kBT/2. The λ−1 for the HW chain is related to the Kuhn statistical segment length (AK)

Figure 7. Double-logarithmic plot of RH for P6 against the salt concentration (Cs) in aqueous NaCl solution at 298 K. The solid line is drawn as a guide for the eye.

AK = c∞λ−1

(9)

with c∞ =

4 + τ0 2 4 + κ0 2 + τ0 2

(10)

The theoretical scattering function of the HW model was compared to the experimental data of oligomer PMTAC (P1) to reduce the obscuration resulting from the excluded volume effect. Figure 10 shows plots of the q2[Cp/ΔI(q)]Cp,q=01/2/[Cp/ ΔI(q)]Cp=01/2 against q (in the form of the Kratky plot) for P1 in 0.1 and 0.5 M aqueous NaCl solution at 298 K. As can be seen in the figure, the Kratky plots show a remarkable

Figure 8. Molecular-weight dependence of ⟨S2⟩z1/2 for PMTAC in 0.05, 0.1, and 0.5 M aqueous NaCl solutions at 298 K.

Figure 9. Molecular-weight dependence of RH for PMTAC in 0.05, 0.1, and 0.5 M aqueous NaCl solutions at 298 K.

3.4. Data Analysis and Comparison with Theory. Scattering Function. To determine the local conformation of PMTAC chain in 0.05, 0.1, and 0.5 M aqueous NaCl, we analyzed the scattering profile on the basis of the helical wormlike (HW) chain model proposed by Yamakawa and coworkers, who performed extensive investigations on chain conformation of poly(methyl methacrylate) (PMMA) in a theta

Figure 10. Kratky plots for samples P1 in (a) 0.1 M and (b) 0.5 M aqueous NaCl solutions at 298 K and the theoretical curves for the helical wormlike chain with the parameter sets listed in Table 2. 4085

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Table 2. Helical Wormlike Chain Parameters for PMTAC in Aqueous NaCl Solutions at 298 K Cs (M)

λ−1κ0

λ−1τ0

λ−1 (nm)

AK (nm)

ML (g mol−1 nm−1)

d (nm)

dH (nm)

B (nm)

0.1 0.5

5.2 5.5

3.0 2.4

7.5 ± 0.2 7.0 ± 0.2

2.5 ± 0.06 1.7 ± 0.05

740 ± 40 650 ± 30

0.45 0.50

0.8 1.4

9.26 ± 0.9 3.37 ± 0.5

The values of ⟨S2⟩z1/2 determined by SAXS measurements contain contributions from the finite cross section of the polymer chain. For a continuous chain with a uniform circular cross section of diameter d, ⟨S2⟩ can be expressed as follows:37

oscillation, similar to that observed by Kirste and Wunderlich for a syndiotactic PMMA chain.33 This oscillation was attributed to the local chain conformation, that is, to helical sequences of PMMA. Unfortunately, reliable scattering profile for P1 in 0.05 M aqueous NaCl could not be obtained for q > 1 nm−1 due to the insufficient scattering contrast. To estimate the model parameters for the HW chain from the scattering profiles of PMTAC, we calculated the scattering function F(q;L) for an HW chain of total contour length L by means of empirical equations.34 The effect of the chain thickness (the thickness of the spatial distribution of scatterers around the HW chain contour) was considered by applying the touched spheroid model35 [eq 50 in ref 35]. For sample P1 at Cs = 0.1 and 0.5 M, the best-fit theoretical curve calculated from eq 13 of ref 34 was obtained when we used the parameter values listed in Table 2, where d is the cross-sectional diameter. The HW theory quantitatively reproduced the experimental curve for P1 in 0.1 and 0.5 M aqueous NaCl with a characteristic oscillation in the range q < 6 nm−1. The values of ML increased with decreasing Cs, and the average value of 700 nm−1 yields 0.29 nm for the monomeric contour length l. This l value is close to the value calculated on the assumption that the PMTAC molecule adopts the all-trans conformation. Molecular Weight Dependence of the Radius of Gyration. To demonstrate the validity of the parameters determined from the scattering profiles for P1 at Cs = 0.1 and 0.5 M, we examined whether the HW chain model with these model parameters is capable of describing the molecular-weight dependence of ⟨S2⟩z1/2 for PMTAC at Cs = 0.1 and 0.5 M. The unperturbed mean-square radius of gyration ⟨S2⟩0 of the HW chain with a contour length L is calculated by applying the following expression:23

⟨S2⟩ = ⟨S2⟩0 +

⟨S2⟩ = αs 2⟨S2⟩0

⎡ ⎤2/15 ⎛ 70π 10 ⎞ 2 3/2 3 ⎜ ⎟ z ̃ + 8π z ̃ ⎥ + αs = ⎢1 + 10z ̃ ⎝ 9 ⎣ ⎦ 3⎠ 2

× [0.933 + 0.067 exp(0.85z ̃ − 1.39z 2̃ )]

(13)

ν = (k 0 2 + τ0 2)1/2

(14)

z̃ =

3 K (λ L )z 4

(19)

⎛ 3 ⎞3/2 ⎜ ⎟ (λB)(λL)1/2 ⎝ 2π ⎠

(20)

with z=

and K (λ L ) =

4 7 − 2.711(λL)−1/2 + (λL)−1 3 6

(for λL > 6)

= (λL)−1/2 exp[−6.611(λL)−1 + 0.9198 + 0.03516λL] (for λL ≤ 6) (21)

Here, B is the excluded-volume strength defined by the expression B = β/b2c∞3/2, where β is the binary cluster integral between beads with spacing b. The full curves in Figure 11 represent the theoretical values of ⟨S2⟩1/2 calculated from eq 17 using the model parameters listed in Table 2. The theoretical curves agreed well with the experimental data points with B = 9.26 nm for Cs = 0.1 M and B = 3.37 nm for Cs = 0.5 M. We therefore concluded that the model parameters obtained from the scattering function for P1 were valid for PMTAC chain. The literature values25,26 of ⟨S2⟩z1/2 for PMTAC in a 1.0 M aqueous NaCl solution at 298 K are plotted in Figure 11 as square and triangular symbols. As mentioned above, we have already confirmed that the chain dimensions of PMTAC in both 0.5 and 1.0 M aqueous NaCl solutions are identical. We can therefore compare our present data with the literature data. Note that the PMTAC samples used in the literature showed a high polydispersity; therefore, corrections for this polydispersity

The parameter L is related to the molecular weight Mw by the expression L = Mw/ML. The unperturbed mean-square radius of gyration of the KP chain, ⟨S2⟩0,KP, in eq 11 is calculated from the Benoit−Doty equation36 ⟨S2⟩0,KP =

(18)

where z̃ is a scaled excluded-volume parameter defined as follows:

with

φ = cos−1(2/r )

(17)

and αs can be calculated by using the theoretical scheme of Yamakawa, Shimada, and Stockmayer (YSS)38,39 by applying the modified Domb−Barrett equation:40

κ 2 ⎡ λL 1 cos φ − 2 cos(2φ) ⟨S2⟩0 = 2 2 ⟨S2⟩0,KP + 20 2 ⎢ r λν λ ν ⎣ 3r 2 2 2 cos(4φ) + 4 e − 2λ L + 3 cos(3φ) − 4 r λL r (λL)2 r (λL)2 ⎤ cos(νλL + 4φ)⎥ ⎦ (11)

(12)

(16)

Here, we must consider the excluded-volume effects for the PMTAC chain in aqueous NaCl solutions. The excludedvolume effect on ⟨S2⟩ can be expressed in terms of the radius expansion factor αs as follows:

τ0 2

r = (4 + ν 2)1/2

d2 8

L 1 1 1 − 2 + 3 − 4 2 [1 − exp( −2λL)] 6λ 4λ 4λ L 8λ L (15) 4086

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Figure 12. Molecular weight dependence of RH for PMTAC in aqueous NaCl solutions at 298 K. The solid curves represent the theoretical values for a helical wormlike cylinder with the parameters listed in Table 2.

Figure 11. Molecular weight dependence of ⟨S2⟩z1/2 for PMTAC in 0.1 and 0.5 M aqueous NaCl solutions at 298 K and literature values in 1.0 M aqueous NaCl solution (squares25 and diamonds26). The solid curves represent the theoretical values for a helical wormlike chain with parameters listed in Table 2.

model parameters listed in Table 2. The theoretical curves closely fit the experimental data points except for a lower molecular weight region of Mw < 2 × 104 g/mol. From these results, we confirmed that the HW theory also consistently described the Mw dependence of RH for PMTAC in 0.1 and 0.5 M aqueous NaCl solution as well as the Mw dependence of ⟨S2⟩z1/2. Local Conformation of the PMTAC Chain. For an HW chain, the radius R and the pitch h of the characteristic helix at the minimum potential energy are given by the following expressions: κ R= 2 0 2 κ0 + τ0 (25)

41

were applied by assuming a Schulz−Zimm distribution. The data point for the corrected literature values also fitted the theoretical curve for the HW chain. We found that the model parameters determined from the scattering profile for P1 in 0.1 and 0.5 M aqueous NaCl solution are reasonable. It can be assumed that parameter B for polyelectrolyte chains consist of an intrinsic excluded-volume effect associated with the nonionic polymer backbone and electrostatic contributions as mentioned in the Introduction. Yamakawa and co-workers reported values of B for atactic PMMA in the good solvents acetone, nitroethane, and chloroform.42 The values of B in acetone at 298 K, nitroethane at 303 K, and chloroform at 298 K are 0.22, 0.52, and 1.15, respectively. The values of B = 9.26 nm for Cs = 0.1 M and B = 3.34 nm for Cs = 0.5 M were much larger than those of neutral atactic PMMA, probably as a result of the contribution of electrostatic interactions in PMTAC. Also, the B values in the present study are much larger than those of sodium poly(styrenesulfonate) in aqueous NaCl reported by Norisuye et al.43 Molecular Weight Dependence of the Hydrodynamic Radius. In addition to the analysis of ⟨S2⟩z1/2, we also confirmed that the molecular weight dependence of the RH for PMTAC at Cs = 0.1 and 0.5 M can be explained by means of the HW chain model. The theoretical values of RH as a function of the molecular weight were calculated on the basis of the HW cylinder model.44 The hydrodynamic cylinder model for the HW chain is defined by introducing the diameter of the cylinder, dH. The value of RH,0 for unperturbed HW cylinders may be expressed as follows:44 2 RH,0−1 = fD (λL /c∞ , λd /c∞) (22) L where f D is a known function of λL/c∞ and λd/c∞; f D is derived through the associated KP chain with a Kuhn length identical to that of the HW chain under consideration. The value of RH for a perturbed HW cylinder can be calculated from the expression RH = αHRH,0 (23)

and h=

2πτ0 κ0 2 + τ0 2

(26)

For a solution of PMTAC in 0.1 and 0.5 M aqueous NaCl at 298 K, we have R = 1.08 nm and h = 3.93 nm and R = 1.07 nm and h = 2.93 nm, respectively. From those values, we can draw schematics of the characteristic helices of HW chain model at the minimum potential energy for the PMTAC chain (Figure 13). Note that the actual conformation of the PMTAC chain in dilute solution differs from that of a regular helix because the regular structure is destroyed by thermal fluctuations in dilute solution.

by using the Barrett equation45 for the hydrodynamic radius expansion factor αH in the YSS scheme:38,39 αH = (1 + 6.02z ̃ + 3.59z 2̃ )0.1

(24)

Figure 13. Schematic representation of the conformation of the PMTAC chain in 0.1 and 0.5 M aqueous NaCl solutions at the potential energy minimum.

The solid lines in Figure 12 show the theoretical values of RH calculated from eqs 22, 23, and 24 for an HW cylinder with the 4087

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From Figure 13, it is seen that the characteristic helix for the PMTAC chain is elongated in the direction of helix axis with decreasing Cs. Namely, the PMTAC chain assumes slightly extended conformation with decreasing Cs. This is a result of an increase in short-range interactions caused by decreasing Cs. As mentioned in the Introduction, the chain dimensions of polyelectrolytes are significantly influenced by short- and long-range electrostatic interactions. By applying the HW chain model, these interactions can be analyzed more quantitatively to afford the information on the local chain conformation change of polyelectrolyte with Cs. The values of R and h for the PMTAC chain in 0.5 M aqueous NaCl approach the corresponding values for PMMA, for which R = 1.35 nm and h = 2.33 nm.46 This suggests that the local conformation of PMTAC chain in Cs = 0.5 M at 298 K is similar to that of PMMA chain because the electrostatic repulsive interactions are remarkably suppressed at this Cs.

4. CONCLUSIONS The particle scattering function, the values of ⟨S2⟩z1/2 and RH for a series of samples of PMTAC in Cs = 0.05, 0.1, and 0.5 M were measured at 298 K by means of SAXS and DLS. The scattering function and the molecular weight dependence of ⟨S2⟩z1/2 and RH for the PMTAC chain were quantitatively described in terms of the HW chain model. The RH for PMTAC decreased with increasing salt concentration and then leveled off at salt concentrations above 0.5 M. From the analysis with the HW chain, we revealed that not only the chain dimensions but also the local conformation of PMTAC is strongly influenced by the ionic strength. We conclude that the HW chain is an appropriate model for understanding the solution behavior of the PMTAC chain in aqueous NaCl solutions.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]; Ph +81-92-802-2517; Fax +81-92-802-2518. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS T.I. acknowledges the financial support of a Grant-in-Aid for JSPS Fellows. The authors appreciate Professor Seigou Kawaguchi for helpful comments and suggestions which helped to improve this paper. The synchrotron radiation experiments were performed at beamline BL40B2 at the SPring-8 facility with the approval of the Japan Synchrotron Radiation Research Institute (JASRI) (Proposals 2010B1007 and 2011A1001).

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