Conformations of Ring Polystyrenes in Semidilute Solutions and in

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Conformations of Ring Polystyrenes in Semidilute Solutions and in Linear Polymer Matrices Studied by SANS Takuro Iwamoto,†,& Yuya Doi,‡,†,& Keita Kinoshita,† Atsushi Takano,*,† Yoshiaki Takahashi,§ Eunhye Kim,∥,Δ Tae-Hwan Kim,∥,Δ Shin-ichi Takata,⊥ Michihiro Nagao,#,% and Yushu Matsushita*,† †

Department of Molecular and Macromolecular Chemistry, Nagoya University, Nagoya, Aichi 464-8603, Japan Institute for Chemical Research, Kyoto University, Uji, Kyoto 611-0011, Japan § Institute for Materials Chemistry and Engineering, Kyushu University, Kasuga, Fukuoka 816-8580, Japan ∥ HANARO Research Reactor Utilization Development, Korea Atomic Energy Research Institute (KAERI), Daejeon 305-353, Korea ⊥ J-PARC Center, Japan Atomic Energy Agency (JAEA), Tokai, Ibaraki 319-1195, Japan # NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899-6102, United States % Center for Exploration of Energy and Matter, Indiana University, Bloomington, Indiana 47408, United States

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‡

S Supporting Information *

ABSTRACT: Conformations of highly purified ring polystyrene, R-70, with the molar mass of 70 kg/mol, in a good solvent and in linear polymer homologue matrices were examined by small-angle neutron scattering (SANS) measurements. The radii of gyration Rg of R-70 were estimated by the Guinier’s approximation from the SANS profiles obtained, and the polymer volume fraction Φ dependence of Rg2 was discussed. In deuterated toluene as a good solvent, R-70 exhibits the Rg2 ∼ Φ−0.29±0.01 dependence at high Φ above the overlap volume fraction, Φ0* (i.e., 1 < Φ/Φ0* < 20). This exponent −0.29 shows stronger Φ dependence than that for semidilute solutions of linear polymers, −0.25, predicted from the scaling theory, suggesting that the ring expands more sensitively than linear chains when Φ decreases in semidilute regime. In contrast, the Φ dependence of Rg2 of R-70 is evidently weaker than that of the recent simulation for ring polymer solutions (Rg2 ∼ Φ−0.59) by Reigh et al. This difference is thought to originate from the difference in the ring chain length; i.e., the simulation treated much longer rings than the ring adopted in this study. Therefore, it is expected that the exponent −0.29 for the ring polymer solutions obtained in this study is not a limiting value but is a transit one toward higher Φ/Φ0* region. The size of R-70 is also increased when the ring was diluted with linear polystyrenes. However, the degree of expansion of the rings in linear polymer matrices is considerably lower than that in toluene solutions. Moreover, the molar masses of the linear chains added hardly effect the expansion behavior of the rings. In fact, the dimension of rings gets closer to that of the Gaussian rings as a larger amount of linear chains is added.

1. INTRODUCTION

of ring polymers is one of the fundamental research subjects. To date, conformations of rings in dilute solutions6,17 and in concentrated limit (i.e., bulk)19,23−25 have been experimentally elucidated. In contrast, as for the conformations of rings in semidilute regime, there are no experimental reports as far as we know, and they still have not been understood well. In dilute solutions under good solvent conditions, linear polymer chains reveal expanded conformations compared with the Gaussian chains due to the excluded-volume effects.26−29 In contrast, in bulk, linear polymers behave like ideal Gaussian chains because of the screening of excluded volume

Ring polymers have attracted the attention of researchers, where main concerns are on physical properties because of their intrinsic structural feature: rings have no chain ends. Prior to experiments, some theoretical studies on ring polymers were conducted.1−5 In the 1980s, several pioneering experimental works on the properties of ring polymers were reported.6−11 However, the characteristics, especially the purity, of the ring samples were uncertain in those works. Since the late 1990s, the chromatography techniques called liquid chromatography at the critical condition (LCCC) or interaction chromatography (IC) have been developed, and they made it possible to separate ring samples from linear contaminants.12,13 Since then, studies on the properties of highly purifed ring polymers have been carried out.14−23 Understanding the conformation © XXXX American Chemical Society

Received: May 10, 2018 Revised: July 10, 2018

A

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Macromolecules interactions.30−32 In an intermediate region between dilute solution and bulk, i.e., semidilute solution regime, linear chains gradually vary their conformations from expanded state to ideal one with increasing the concentration c. Theoretically, de Gennes30 predicted that the mean-square radius of gyration ⟨Rg2⟩ of linear chains decreases in proportion to c−0.25 at semidilute regime where c is higher than the overlap concentration c*. Experimentally, Daoud et al.33 performed the small-angle neutron scattering (SANS) measurements of hydrogenous (h-) and deuterated (d-) polystyrene mixture solutions dissolved in carbon disulfide as a good solvent with various concentrations. They confirmed that their experimental results obey the c−0.25 dependence predicted by de Gennes.30 King et al.34 also conducted the SANS measurements for concentrated solutions of h- and d-polystyrene mixtures dissolved in deuterated toluene, and they obtained similar results. Moreover, there are several simulation results for semidilute polymer solutions, and in particular, Pedersen et al.35−37 elaborately analyzed the experimental SANS data by their Monte Carlo simulations. In contrast to linear chains, conformations of ring polymers in semidilute regime are hardly understood. In dilute solutions under good solvent conditions, ring chains are known to swell owing to the similar degree of excluded volume effects to linear ones,4,6,8,17,38−40 while rings in bulk are expected to possess shrunken conformations compared with the Gaussian rings due to the topological constraints on the ring structure.5 These non-Gaussian characteristics of bulk ring chains are one of the distinctive features different from linear ones. Recent simulation and theoretical studies proposed that the Flory’s exponent, ν, which represents the degree of polymerization N dependence of Rg as Rg ∼ Nν, is not constant but varies from 0.5 to 0.33 with increasing N.41−44 Very recently, we experimentally confirmed that ν for bulk rings becomes smaller as N increases by the SANS measurements on highly purified h- and d-ring polystyrenes.23 Conformations of ring chains in semidilute solutions are also expected to be different from those for linear ones since the effect of intermolecular interactions still remains in this concentration regime. To date, there are several simulation results for conformations of ring polymers in semidilute solutions.45−47 Reigh et al.47 examined the concentration c dependence of Rg2 for ring polymers with different chain lengths by the lattice Monte Carlo simulation. It has been clarified that the ring polymer solutions exhibit the relationship of Rg2 ∼ c−0.59 for large rings at high c regime. This c dependence is considerably stronger than that for linear chains (−0.25), and they interpreted this phenomenon by introducing the idea of the chain shrinkage of ring polymers in concentrated regime.47 In relation to the conformations of ring chains in solutions of low molecular weight solvents, those in linear polymer matrices, where the mixtrures give “solid solutions”, are also interesting. As described above, ring chains possess collapsed state in bulk compared to the Gaussian rings due to the topological constraints, i.e., suppression of the intermolecular penetration.5,19,23−25,41−44 When linear chains are surrounding rings in bulk, the linear chains possibly thread through the rings, and consequently the rings expand. In fact, there are several simulations on the conformations of rings in linear polymer matrices,48−54 whereas few systematic experimental studies were conducted. Based on the above background, in this study, conformations of the highly purifed ring polystyrene with the molar mass of

70 kg/mol, R-70, where h- and d-ring homopolymers are blended with the volume fraction of 50/50, dissolved in a low molar mass solvent or in linear polymer matrices are investigated by SANS. Deuterated toluene is used for a solvent, while three h/d-random linear polystyrenes with the molar mass of (20, 80, and 270) kg/mol are prepared for linear polymer matrices. The radii of gyration Rg of R-70 in solutions and in solid solutions are estimated by the Guinier approximation, and the polymer volume fraction Φ dependence of Rg2 is discussed by comparing these results with those from scaling theory and simulation.

2. EXPERIMENTAL SECTION 2.1. Materials. Highly purified hydrogenous (h-) and deuterated (d-) ring polystyrene samples with the molar mass of 70 kg/mol, hRPS-70 and d-RPS-70, respectively, were prepared previously.23 Their molecular characteristics are summarized in Table 1.

Table 1. Molecular Characteristics of h- and d-Ring Polystyrene Samples sample code

Mw/kg mol−1a

Mw/Mnb

purity/%c

h-RPS-70 d-RPS-70

71.4 77.3

1.02 1.02

99.8 99.7

a Estimated by MALS measurements. bEstimated by SEC measurements. cEstimated by interaction chromatography (IC) measurements.

As linear polymer matrices which give no coherent neutron scattering contrasts, three random linear copolymers consisting of hand d-styrene monomers (ran-LPS) with different molar masses of (20, 80, and 270) kg/mol were anionically synthesized. A tetrahydrofuran (THF) solution (20 mL) of h- and d-styrene monomer mixture (∼5 mL) at the volume fraction of 50/50 was very slowly added as droplets into a THF solution (∼100 mL) of secbuthyllithium as an initiator at −78 °C, and after sufficiently stirring the solution (for ca. 10 min), the polymerization reaction was terminated by methanol. Molecular characteristics of three ran-LPS samples obtained are summarized in Table 2. Their mass-average

Table 2. Molecular Characteristics of h/d-Random Linear Polystyrene Samples sample code

Mw/kg mol−1a

Mw/Mnb

ran-LPS-20 ran-LPS-80 ran-LPS-270

22.4 79.5 269

1.02 1.04 1.02

a

Estimated by MALS measurements. bEstimated by SEC measurements. molar mass, Mw, and molar mass distribution, Mw/Mn, were determined by multiangle light scattering (MALS) and size exclusion chromatography (SEC) measurements, respectively. SEC chromatograms of these ran-LPS samples are shown in Figure S1 in the Supporting Information, and they were confirmed to have small Mw/ Mn less than 1.05. 2.2. SANS Measurements. Small-angle neutron scattering (SANS) measurements were performed to evaluate the conformations of ring polymer chains in toluene solutions and in ring/linear polymer blends as solid solutions. For toluene solutions, fully deuterated (d8-) toluene was used as a solvent, which is known as a good solvent for polystyrene, whereas three h/d-random linear polystyrenes shown in Table 2 were used as linear matrices of solid solutions. In principle, h/ d-random copolymers do not show any coherent particle scattering since h- and d-monomers are randomly distributed along the chains with various lengths of h- and d-sequences, and hence they are known B

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Macromolecules to behave like h/d-solvent blends at the composition of the present random copolymers in SANS.55,56,66 Samples for SANS measurements were prepared as follows. First, a mass fraction of 5% benzene solution of h- and d-RPS-70 mixture having the volume of 50/50 was freeze-dried. This h/d-blend ring sample is denoted as R-70, and its chain dimension in bulk was reported previously.23 We set the h/d-blend composition of R-70 as 50/50 in volume to maximize the coherent scattering intensity originating from ring chains in solutions and solid solutions.30 The corresponding h/d-blend linear counterpart, L-70, was also prepared in the same manner. For toluene solution samples, a prescribed amount of R-70 and d8toluene was introduced into a quartz cell with 2 mm thickness. A small iron ball with 1.5 mm diameter was added into a cell, and the mixtures were carefully stirred by hand to accelerate them to be homogeneous solutions. In this study, four R-70/d8-toluene solution samples with different volume fraction of R-70, ΦR, ranging from 0.087 ≤ ΦR ≤ 0.37 were prepared, while four L-70/d8-toluene solutions with the fraction of L-70, ΦL, ranging from 0.075 ≤ ΦL ≤ 0.43 were also prepared as references. For solid solution samples, R-70 and ran-LPS were codissolved in benzene by targetted ratios, and they were freeze-dried. Then the ring/linear mixtures were thermally annealed in a mold with 5 mm diameter at 160 °C under vacuum for 48 h, and disk samples with the thickness between 1 and 2 mm were obtained. Five R70/ran-LPS blends with different ring/linear compositions (ΦR = 0.20, 0.50, 0.70, 0.90, and 0.97) were prepared for each ran-LPS sample. The corresponding L-70/ran-LPS blend samples were not prepared because the dimension of probe linear chains with the degree of polymerization Np in linear matrices with Nm is thought to be unchanged unless Np1/2 > Nm.30,55−60 SANS measurements were performed on the NG7 30m SANS spectrometer61 at the Center for Neutron Research, National Institute of Standards and Technology (NIST) of the U.S. Department of Commerce, and on the 40m-SANS instrument62 at the High-flux Advanced Neutron Application Reactor (HANARO) at Korea Atomic Energy Research Institute (KAERI). For 30m SANS at NIST, the neutron wavelength, λ, was 6 Å, and three sample-to-detector distances (SDD) adopted were (1, 6, and 11) m covering the q-range of 0.003 ≤ q/Å−1 ≤ 0.5. For 40m-SANS at HANARO, λ selected was 7 Å and SDD was 9 m covering the q range of 0.005 ≤ q/Å−1 ≤ 0.07. Here, q is the magnitude of the scattering vector defined as q = (4π/ λ)sin(θ/2), where θ is the scattering angle. SANS measurements for R-70 and L-70 in bulk, R-70 and L-70 solutions in d8-toluene with all of the polymer volume fraction ΦP, and R-70/ran-L blends with ΦR = 0.20 and 0.50 were conducted at NIST, while R-70/ran-L blends with ΦR = 0.70, 0.90, and 0.97 as well as bulk R-70 were measured at HANARO. Data reduction for the NIST data was performed using software developed at NIST.63 The 2D scattering data obtained were azimuthally averaged to convert the data to one-dimensional plots of intensity I(q) versus q. It is noted here that error bars of the data represent a standard deviation throughout the paper. As shown in Figure S2 in the Supporting Information, the raw I(q) data of bulk R70 measured at NIST and HANARO show a good overlap in a whole q range, and hence in this study we can equally treat and compare the data obtained at two instruments. Subtraction of background intensitities is important to correctly extract the information on conformations of ring chains. In this study, we considered two kinds of background corrections for raw data; i.e., one is the incoherent scattering mainly from hydrogenous polymers in R-70, and the other is the scattering from solvents or matrices, i.e., d8toluene or h/d-ran-LPS. The former was estimated from the volume fraction of h-RPS and the sample thickness following the method reported by Shibayama et al.,64 and the latter was determined from direct SANS measurements for d8-toluene and h/d-ran-LPS. We confirmed that both d8-toluene and h/d-ran-LPS exhibit almost qindependent scattering behavior as shown in Figure S3 in the Supporting Information. The detailed treatments of the background corrections are summarized in the Supporting Information (Section S-

3), and in the main text, we only show the SANS profiles where background scatterings were subtracted in advance. Unlike binary blends of h- and d-polymer systems, we need to additionally consider the intermolecular correlations for ternary blends consisting of h-polymer, d-polymer, and solvent. Details of the theoretical interpretations were well described previously in the literature.33,34,65−68 Moreover, we experimentally evaluated the magnitude of the intermolecular correlations for R-70 and L-70 toluene solutions. The measurements were performed at the Material and Life Science Experimental Facility (MLF) in the Japan Protopn Accelerator Research Complex (J-PARC),69 and the details are summarized in the Supporting Information (Section S-4).

3. RESULTS AND DISCUSSION 3.1. Overview of SANS Profiles. Figures 1a and 1b show the double-logarithmic plots of I(q) vs q for series of L-70 and

Figure 1. Coherent SANS profiles for (a) L-70 and (b) R-70 solutions in d8-toluene with different ΦP. The black dashed and blue dotted lines indicate the slope of −2 (I(q) ∼ q−2) and −5/3 (I(q) ∼ q−5/3), respectively, at high q. Error bars of the data are within symbol sizes.

R-70 solutions, respectively, dissolved in d8-toluene with different volume fraction of polymers ΦP. The I(q) profiles shown in Figure 1 are those subtracting the background intensities from raw scattering data in advance. For both linear and ring polymer solutions, it is evident that I(q) decreases regularly with decreasing ΦP covering a whole q range. At low q regime, I(q) gives the information on global structures and dimensions of whole polymer chains, and hence the radius of gyration Rg for each sample was estimated by C

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Macromolecules using the low q data. In contrast, at high q regime, I(q) reveals the information on local structures of polymer chains. For L-70 solutions in Figure 1a, I(q) obeys q−2 dependence at q ≥ 0.05 Å−1 for bulk and concentrated solutions, while I(q) approaches q−5/3 as ΦP decreases. R-70 solutions basically exhibit a similar trend to L-70 solutions at higher q (≥0.07 Å−1), that is, in Figure 1b, concentrated solutions exhibit q−2 dependence, while diluted ones approach q−5/3. I(q) data in the high q regime are known to follow I(q) ∼ q−1/ν, where ν is the Flory’s scaling exponent. The relationship, I(q) ∼ q−2, for the bulk and concentrated solutions corresponds to v = 1/2, while I(q) ∼ q−5/3 for the relatively dilute solutions corresponds to v = 3/5. These experimental facts suggest that excluded volume effects along the chains begin to occur as ΦP decreases, and chain conformations gradually vary from ideal (ν = 1/2) to expanded (ν = 3/5) state for both L-70 and R-70 solutions. Figures 2a and 2b represent the Kratky plots, i.e., q2I(q) vs q, for L-70 and R-70 solutions in d8-toluene, respectively. In

Figure 2. Kratky plots, q2I(q) vs q, for (a) L-70 and (b) R-70 solutions in d8-toluene with different ΦP.

Figure 3. Coherent SANS profiles for three R-70/ran-L blends where the molar masses of linear chains are (20, 80, and 270) kg/mol in panels (a), (b) and (c), respectively, with five different ΦR. The black dashed lines indicate the slope of −2 (I(q) ∼ q−2) at high q. Error bars of the data are within symbol sizes.

Figure 2a, L-70 solutions exhibit a clear plateau at q/Å−1 ≥ 0.03 irrespective of ΦP. In contrast, R-70 solutions exhibit a clear peak at intermediate q range (0.01 ≤ q/Å−1 ≤ 0.07) in Figure 2b. These peaks are known to originate from ringshaped structures.3 Moreover, the peak position for R-70 solutions is slightly shifted to the lower q side as ΦP decreases. This result implies that the size of R-70 in toluene solutions increases as ΦP decreases.19 Figures 3a, b, and c show the double-logarithmic plots of I(q) vs q for the three solid solutions of R-70/ran-L blends, where the molar masses of linear chains are (20, 80, and 270) kg/mol, respectively, with different ΦR (= 0.97, 0.90, 0.70,

0.50, and 0.20). As described in the Experimental Section, h/dran-LPS matrices have basically no coherent scattering contrast. Moreover, the solute R-70 and the matrix h/d-ranLPS are contrast-matched at any ΦP, and hence these systems essentially exhibit the coherent scattering intensities simply originating from the particle scattering of R-70. The details are shown in the Supoorting Information (Section S-4). Similar to the d8-toluene solution samples, I(q) for the ring/linear blends also become lower as ΦR decreases. In some samples, a discrepancy in the order of intensities was observed, but it probably originates from some experimental uncertainties such as sample thickness. Unlike the toluene solution samples, I(q) D

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Macromolecules data at high q for R-70/ran-L samples at 0.70 ≤ ΦP ≤ 0.97 are insufficient, and hence we cannot discuss them further here. 3.2. Estimation of Rg. The radii of gyration Rg of R-70 in toluene solutions and in solid solutions treated in this study are determined by the Guinier’s approximation70 as ln I(q) ∼ −q2R g 2/3

with

q2R g 2 < 1.32

(1)

Figure 4 shows the Guinier’s plots, i.e., ln I(q) vs q2, for the R-70 solutions in d8-toluene. The corresponding plots for the

Figure 4. Guinier plots (ln I(q) vs q2) for R-70 solutions in d8toluene. The black solid lines indicate the best fit lines for the Guinier’s methods (eq 1).

L-70 solutions are shown in Figure S6 in the Supporting Information. The Guinier plots for the ring/linear solid solutions are shown in Figure 5. All of the plots show a good linearity in each adequate q range with reasonably large numbers of the data points, and Rg values were determined from the slope of the plots. Note that upturns observed in the low q region were excluded when deterimining Rg.23 Rg values obtained are summarized in Tables 3 and 4 for the toluene solutions and ring/linear solid solutions, respectively. Errors of Rg values in Table 3 and 4 are originated from uncertainty of the slope of Guinier’s plots. It should be noted that for the toluene solution samples the contribution of intermolecular correlations, which can be represented by using the second virial coefficient A2 or the Flory−Huggins interaction parameter χ, must appear in I(q), especially at low q, in addition to that of the particle scattering. If the former contribution is relatively large, the Rg values obtained from Figure 4 may not be correct. In the Supporting Information (Section S-4), we experimentally confirmed that R-70 and L-70 solutions in d8-toluene with ΦP ∼ 0.1 have little contribution of intermolecular correlations to I(q), especially in the low q region. Moreover, King et al.34 demonstrated that the ratio of intermolecular correlation to intramolecular one (i.e., single-chain particle scattering) becomes smaller as ΦP increases. Therefore, for all solution samples treated in this study Rg values of R-70 (and L-70) dissolved in d8-toluene were assumed to be correctly estimated from Figure 4 without any consideration of the intermolecular correlation. As for the ring/linear solid solutions, the Rg value of R-70 can be regarded as an accurate Rg similar to the toluene solutions since the fraction of R-70 is sufficiently large as ΦR ≥ 0.2. Moreover, in these systems, R-70 and h/d-ran-LPS are

Figure 5. Guinier plots (ln I(q) vs q2) for three R-70/ran-L blends where the molar masses of linear chains are (20, 80, and 270) kg/mol in panels (a), (b), and (c), respectively, with five different ΦR. The black solid lines indicate the best fit lines for eq 1.

contrast-matched at all ΦR adopted, and hence they do not basically show intermolecular correlation term. It has been found from Tables 3 and 4 that Rg values of R-70 in solutions or in solid solutions become larger as a larger amount of d8-toluene or h/d-random linear chains is added. Moreover, the rings are more highly expanded with d8-toluene than with linear chains. It turns out that for the ring/linear solid solutions, the difference in the molar mass of linear chain matrices has little influence on the expansion of R-70. As for E

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Macromolecules Table 3. Rg for R-70 and L-70 in d8-Toluene with Different ΦP ΦR (R-70)

Rg/nm in d8-toluene

ΦL (L-70)

Rg/nm in d8-toluene

1 (bulk) 0.37 0.31 0.17 0.087 0

4.5 ± 0.2 5.2 ± 0.1 5.3 ± 0.1 5.8 ± 0.1 6.3 ± 0.1 7.5a

1 (bulk) 0.43 0.24 0.20 0.075 0

7.1 ± 0.2 7.5 ± 0.2 8.1 ± 0.2 8.2 ± 0.3 8.9 ± 0.2 9.4b

a

Rg values for R-70 in dilution limit in a good solvent were calculated from Rg,R = 0.0090 × Mw0.60 (eq 3).17 bRg values for L-70 in dilution limit in a good solvent were calculated from Rg,L = 0.0121 × Mw0.595 (eq 2).29

Table 4. Rg for R-70 in Three ran-LPS with Different ΦP ΦR (R-70) 1 (bulk) 0.97 0.90 0.70 0.50 0.20

Rg/nm in ran-L-20 4.5 4.4 4.5 4.5 4.7 4.8

± ± ± ± ± ±

0.2 0.2 0.2 0.2 0.2 0.1

Rg/nm in ran-L-80 4.5 4.4 4.4 4.5 4.6 4.7

± ± ± ± ± ±

0.2 0.3 0.2 0.2 0.1 0.2

Rg/nm in ran-L-270 4.5 4.4 4.5 4.6 4.6 4.8

± ± ± ± ± ±

0.2 0.1 0.2 0.3 0.1 0.2

Figure 6. Φ dependence of Rg2 for R-70 and L-70 in d8-toluene and for R-70 in three ran-L matrices where the molar masses of linear chains are (20, 80, and 270) kg/mol. Descriptions of the symbols are shown in the figure panel. The blue and red dashed lines indicate the approximate straight lines of five data points for L-70 and R-70 solutions with slopes of −0.18 and −0.29, respectively. The gray dashed line indicates the approximate line with a slope of −0.10 for three R-70/ran-L blends covering the whole Φ range (0.20 ≤ Φ ≤ 1.0) adopted in this study. The green dotted line indicates the Rg2 value for the Gaussian R-70, Rg,R‑Gauss = √2Rg,L, in bulk as a reference.

dynamic properties of ring polymers, only a few percent of linear contaminants is known to affect drastically,16,19,21 whereas for static properties, it has been clarified that linear chains do not affect so much, at least for R-70 examined in this study. 3.3. Concentration Dependence of Rg. In the previous subsection, it has been confirmed that the size of R-70 increases when the solvent molecule or linear polymer chain is added. Here, the concentration dependence of Rg is discussed. Figure 6 shows the volume fraction Φ dependence of Rg(Φ)2 for R-70 in d8-toluene and in linear polymer matrices, together with that for L-70 in d8-toluene. It turned out in Figure 6 that L-70 and R-70 toluene solutions exhibit the relationships Rg,L2 ∼ ΦL−0.18±0.02 and Rg,R2 ∼ ΦR−0.29±0.01, respectively, when five data points including bulk were connected by an approximate straight line each in double-logarithmic scale. The exponent −0.29 for R-70 shows considerably stronger Φ dependence than that for L-70, −0.18, and hence the ring is found to expand more sensitively than the corresponding linear chain when the low molecular weight solvent was added. Based on the scaling theory,30 linear polymers are expected to show Rg,L2 ∼ Φ−0.25 dependence in a semidilute regime. The details of the theory will be described later. In fact, Daoud et al.33 demonstrated the Φ−0.25 dependence for linear polystyrenes in carbon disulfide as a good solvent, being in a good agreement with the theory. In contrast, the experimental results, Rg,L2 ∼ Φ−0.18, obtained in this study exhibit a slightly weaker dependence than that predicted by the theory. We will discuss this result quantitatively later. The ring polymers diluted with linear chains do not expand much compared with the rings dissolved in d8-toluene. When all data points for R-70/ran-LPS solid solutions including the pure ring bulk in Figure 6 are linearly approximated in doublelogarithmic scale, they exhibit a dependence of Rg,R2 ∼ Φ−0.10±0.03, which is evidently weaker than that in d8-toluene, Rg,R2 ∼ Φ−0.29. Moreover, even for the samples with 80% of linear chains, the dimension of R-70, 4.8 nm as shown in Table 4, is still smaller than that for the Gaussian ring, i.e., Rg,R‑Gauss =

√2 × Rg,L‑Gauss with the same molar mass: 5.0 nm for the Gaussian R-70. Therefore, it has been clarified that R-70 used in this study varies its conformations from shrunken state in bulk (i.e., pure ring) to the Gaussian state as linear chains are added. Goossen et al.19,71 revealed that 1% of hydrogenous ring PS (Mw = 161 kg/mol) diluted with 99% of deuterated linear PS (Mw = 250 kg/mol) behaves as the Gaussian ring from the scattering function analysis of the SANS profile. It is interesting that the expansion of R-70 in linear polymer matrices hardly depends on the molar mass of the linear chains. In the case of linear/linear polymer blends, probe linear chains are expanded only when the degree of polymerization of matrix chains Nm is sufficiently smaller than that of the probe chains Np as Np1/2 > Nm,30,55−60 where short linear matrix chains behave like solvent molecules. In contrast, for the ring/ linear solid solutions treated in this study, the probe rings are expanded even when matrix linear chains have sufficiently large molar masses. Namely, it is considered that the expansion mechanism of probe ring chains in ring/linear blends is essentially different from that in linear/linear blends. In fact, the R-70 sample used in this study is confirmed to possess a shrunken and compact conformation in bulk compared with the Gaussian ring,23 and hence the self-density around its center-of-mass is higher. When linear chains are added to the rings, they are conceived to spontaneously penetrate into the rings so as to lower the density of the rings, and consequently the rings approach the Gaussian chains. Next, our experimental data for the toluene solution samples are compared with the simulation results reported by Reigh et al.47 as shown in Figure 7. Reigh et al. examined the dimensions of four ring polymers with different chain lengths as well as the linear ones in solutions by the lattice Monte Carlo simulations.47 The data for two linear PS (Mw ∼ 100 and 500 kg/mol) in carbon disulfide reported by Daoud et al.33 and F

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Macromolecules Φ0* = (c0*/dP)/(c0*/dP + (1 − c0*)/dS)

(4)

c0* = 3M w /4πR g(0)3 NA

(5)

where c0* is the overlap concentration of polymer solution, dP and dS are the densities of polymer and solvent, respectively, and NA is Avogadro’s number. For our experimental systems, dP = 1.10 g/mL and dS = 0.93 g/mL were used, and Φ0* values for L-70 and R-70 in this study are estimated as 0.030 and 0.061, respectively. In Figure 7, our data for linear polymers are consistent with the previous experimental report by Daoud et al.33 and simulation results by Reigh et al.,47 where both give the relationship Rg(Φ)2/Rg(0)2 ∼ (Φ/Φ0*)−0.25 at high Φ/Φ0* (≫ 1), being proposed from the scaling theory by de Gennes.30 In the scaling argument, the radius of gyration Rg of a polymer chain in semidilute solution is written as Figure 7. Normalized Φ/Φ0* dependence of Rg(Φ)2/Rg(0)2 for R-70 and L-70 solutions in d8-toluene obtained in this study and simulation results for ring and linear polymer solutions reported by Reigh et al.47 The data of linear PS in carbon disulfide reported by Daoud et al.33 and those of small ring and linear PS in bulk reported by our group previously23 are plotted together. Descriptions of the symbols are shown in the figure panel. The blue dashed line indicates the guide line for linear polymer solutions at high Φ/Φ0* regime with a slope of −0.25, which is derived from the scaling argument. The red dashed line indicates the approximate straight line for our R-70 solutions at high Φ/Φ0* (>1) with a slope of −0.29. The black dashed line indicates the simulation results for ring polymers at higher Φ/Φ0* (>10) with a slope of −0.59. The green dotted line indicates Rg(Φ)2/ Rg(0)2 = 1, and its crossing points with the blue and red dashed lines are pointed by the blue and red arrows, respectively.

R g(Φ)2 ∼ R g(0)2 (Φ/Φ0*)m

As described in eqs 4 and 5, Φ0* is represented as Φ0* ∼ N/ Rg(0)3 ∼ N1−3ν0 where N is the degree of polymerization and ν0 is Flory’s exponent in dilution limit (i.e., ν0 = 0.60 in this study as confirmed in eqs 2 and 3). Moreover, Rg in concentrated regime or in bulk is expressed as Rg ∼ Nν. Thus, eq 6 can be rewritten as R g ∼ N ν0(Φ/N1 − 3ν0)m /2 ∼ N ν

ν = ν0 − (1 − 3ν0)m /2

(8)

For linear polymer samples, ν is known as 0.5 and ν0 = 0.6, and hence m is derived as −0.25.30,33 Our experimental data for ring R-70 solutions lie on the simulation results by Reigh et al.47 As described in Figure 6, our experimental five data points exhibit the dependence Rg(Φ)2/Rg(0)2 ∼ (Φ/Φ0*)−0.29 at 1 < Φ/Φ0* < 20. In our previous study,23 R-70 sample was confirmed to possess Flory’s exponent ν = 0.46 in bulk. When this ν value and ν0 = 0.6 are substituted to eq 8, the exponent m of Φ/Φ0* is derived as −0.35, which is slightly stronger but reasonably close to the experimentally obtained exponent m = −0.29 as shown in Figure 7. In contrast, Reigh et al. demonstrated in their simulation that ring solutions exhibit much stronger Φ dependence, Rg(Φ)2/Rg(0)2 ∼ (Φ/Φ0*)−0.59, at higher Φ regime as Φ/Φ0* > 10. They expected that high molar mass ring chains possess ν = 0.36 in bulk, and as a result the sharp exponent, m = −0.59, was derived from eq 8. This result suggests that the moderate exponent m of Φ/Φ0*, −0.29, evaluated in this study is not a limiting value but still in a transition toward higher Φ/Φ0* region. In fact, it is thought that the ν values for ring bulks are not constant unlike linear ones but decrease from 0.5 to 0.33 with increasing the ring chain length.41−44 In experimental aspect, the conformation of high molar mass ring polymers in bulk and solutions is still not fully elucidated. In fact, there is no necessity that experimental data coincide with the simulation results at higher Φ/Φ0*. Thus, it is interesting to compare experimental data for high molar mass rings with the simulation results at concentrated regime of Φ/Φ0* > 10 in the future. Next, we focus on the Φ/Φ0* locations where ring and linear chains start to shrink. They roughly correspond to the

(2)

For ring PS, Takano et al. estimated the R g −M w relationship in d6-benzene by using SANS as 17

R g,R (0) = 0.0090 × M w 0.60 (nm)

(7)

When focusing on the exponent in eq 7, the following relationship stands:

those for small ring and linear PS samples in bulk we previously reported23 are also plotted in Figure 7. In the latter report, the data of R-10, R-30, L-10, and L-30 were used; in the sample codes, R- and L- denote the ring and linear samples, respectively, while the numbers indicate the molar masses in units of kg/mol.23 To compare all of the data equally, both horizontal and vertical axes are normalized; that is, for the horizontal axis Φ is divided by the overlap volume fraction Φ0*, and for the vertical axis Rg(Φ)2 is divided by Rg(0)2, which is the Rg2 value for polymers in a good solvent at dilution limit (i.e., Φ → 0). For linear PS, Miyaki et al.29 precisely examined the molecular weight Mw dependence of Rg in benzene in a wide Mw range by light scattering measurements as R g,L(0) = 0.0121 × M w 0.595 (nm)

(6)

(3)

Although the above measurements were basically carried out in PS/benzene systems, we conceive that the relationships in eqs 2 and 3 can be also applied to the PS/toluene systems adopted in this study. It is because both benzene and toluene are known as athermal solvents for polystyrene.72 By using eqs 2 and 3, Rg(0) values for L-70 and R-70 were calculated, and they are estimated to be Rg,L‑70(0) = 9.4 nm and Rg,R‑70(0) = 7.5 nm, respectively. Moreover, the overlap volume fraction Φ0* was evaluated from the following relationships G

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Macromolecules

ij b b yz Φ Φ I(q) = jjj hP − d P zzz hP 2d P NPΦPvPP(q)B j vhP vd P z{ ΦP (11) k Here, we introduced a constant B as a correction factor for the intensity in eq 11 to correct experimental errors mainly originating from inaccuracy in the concentration of polymer solutions. It is confirmed later that B is close to unity for solution samples, and hence experimental errors are very small. Moreover, the h- and d-PS components of R-70 and L-70 samples used in this study have similar N, i.e., NhP = NdP = NP, and hence eq 11 can hold for our solution samples. For the particle scattering function P(q) for L-70, we used the discrete form of linear chains with introducing the Flory’s exponent ν as67 ÅÄ ÑÉÑ N l ÑÑ o o q2l 2k 2ν | 1 ÅÅÅÅ o o ÑÑ P(q) = 2 ÅÅN + 2 ∑ (N − k)expm − } o o Å o oÑÑÑ 6 N ÅÅÇ (12) n ~ÑÖ k=1 2

red and blue arrows in Figure 7, which indicate the intersection points between Rg(Φ)2/Rg(0)2 ∼ (Φ/Φ0*)m at concentrated limit (m = −0.29 for R-70 and m = −0.25 for L-70) and Rg(Φ)2/Rg(0)2 = 1 at dilution limit. Evidently, the Φ/Φ0* value at the crossing point for ring solutions (∼0.5) is smaller than that for linear ones (∼2). It is interesting that not only experiments but also the simulations give the same tendency. Therefore, the phenomenon that ring polymers in solutions start to shrink at lower concentrations than linear ones is thought to be the intrinsic property of ring polymers. It should be noted that the Φ/Φ0* locations do not depend on the molar mass of ring polymers as shown in Figure 7, where the data are accumulated for the rings with different molar masses in both experiments and simulations. As one of the possibilities, the fact that ring chains have compact conformations in the high-Φ regime might be related to this phenomenon. That is, intermolecular repulsive forces for rings appearing at high Φ may still remain in the lower Φ region. However, the bulk R-10 sample (Mw ∼ 10 kg/mol) also lies on the same line in Figure 7 although this sample was confirmed to behave as a Gaussian ring in bulk.23 As another possibility, the definition of Φ0* (and c0*) shown in eqs 4 and 5 itself might not be appropriate for ring polymers. Thus, we may need to employ a prefactor of Φ0* for ring polymer solutions to consider the topology effect. 3.4. Evaluation of Scattering Profiles. To further analyze the scattering data of the toluene solutions in this study, the fitting of the I(q) profiles by using scattering functions was performed. For the ring/linear solid solutions, I(q) data at high q are insufficient, and hence the analyses were not conducted here. As described above, the toluene solution samples used in this study are prepared by dissolving h- and d-polymer mixtures (volume ratio of 50/50) in d8-toluene at designed ΦP. For these systems, I(q) is theoretically represented as65−68 I(q) = (ρhP − ρd P )2 SS(q) + (ρP − ρS )2 ST(q)

where l is the segment length and k is the segment number (1 ≤ k ≤ N). In the present study, the segment number N is regarded as the degree of polymerization, and hence Rg,L can be simply expressed as Rg,L2 = N2νl2/6. When ν = 0.5 and N ≫ 1, eq 12 is reduced to the Debye function.73 The SANS profiles for L-70 solutions are analyzed by using eqs 11 and 12. The Rg values of L-70 in Table 3 estimated from the Guinier approximation are used as fixed values, while B and ν are two variable parameters to obtain the best fits. In reality, these two parameters are determined to minimize the sum of square errors in log I(q). The fitting results for L-70 solutions are shown in Figure 8, and the parameters used are summarized in

(9)

where ρhP and ρdP are the scattering length densities for h- and d-polymers, and ρP (= ρhPΦhP/ΦP + ρdPΦdP/ΦP) and ρS are those for the h-/d-polymer mixture and solvent. SS(q) represents the single-chain scattering factor, while ST(q) denotes the total chain scattering factor including both intrachain and interchain contributions.66,67 As described in the Experimental Section and in the Supporting Information, SS(q) is confirmed to be much larger than ST(q) for concentrated solutions treated in this study, and hence we can safely ignore the contribution of the ST(q) term. For concentrated solutions, eq 9 can be rewritten as30

Figure 8. Comparison between the experimental SANS profiles for L70/d8-toluene solutions (symbols) and the fitting by using eqs 11 and 12 (solid curves) with B and ν as two variable parameters in doublelogarithmic plots of I(q) vs q.

ij b b yz I(q) = jjj hP − d P zzz j vhP vd P z{ k −1 ij yz 1 1 jj z jj Φ N v P (q) + Φ N v P (q) zzz (10) dP dP dP dP k hP hP hP hP { where bi and vi are the scattering length and volume per monomer, and Φi, Ni, and Pi(q) are the volume fraction, degree of polymerization, and particle scattering function for polymer species i (hP or dP). In this study, 2.33 × 10−12 cm and 1.07 × 10−11 cm are adopted for bhP and bdP, while 0.165 nm3 and 0.163 nm3 are used for vhP and vdP, respectively. When ΦhP = ΦdP = ΦP/2, NhP = NdP = NP, and PhP(q) = PdP(q) = P(q), eq 10 can be simplified as 2

Table 5. The fiiting results represented in the Kratky form are shown in Figure S7 in the Supporting Information. We confirmed that the errors in B are within ca. 10%, which are generated from the errors in the original scattering data, while those of ν are ±0.01 due to accuracy of the fitting. In Figure 8, I(q) data for L-70 solutions are described well by the fitting except for low q range ( 1. This Φ dependence (−0.29) is a little stronger than that predicted for linear polymer solutions Φ−0.25, while it is considerably weaker than that of the recent simulation results by Reigh et al., Rg2 ∼ Φ−0.59, at higher Φ/Φ*. This large gap in the exponent is thought to mainly originate from the difference in the molar mass of solute polymers; i.e., Reigh et al. treated the rings with much longer chain lengths than the present R-70. Therefore, it is expected that the exponent −0.29 obtained in this study is not a limiting value but could be a transit one toward higher Φ/Φ* region. It is interesting to investigate the conformations of higher molar mass rings in solution and to compare them with simulations in the future. Chain dimensions of R-70 in linear polymer metrices are confirmed to approach the Gaussian ring as linear chains are added. It should be stressed that the expanding extent of R-70 does not depend on the molar masses of linear chains, which is essentially different from the results for linear/linear polymer blends. From the above, we confirmed that the ring topology effect (i.e., intermolecular repulsive force) is remarkably generated not only in bulk but also in semidilute solution regime. As far as we know the results obtained in the present study are the first systematical experimental report on the conformations of ring polymers in semidilute solutions and in linear chain matrices. As for the structures and properties of ring polymers, there are still many unknown problems. Further studies especially for high molar mass rings are strongly required. We hope the present results will lead to further understanding of features of ring polymers.

Figure 9. Comparison between the experimental SANS profiles for R70/d8-toluene solutions (symbols) and the fitting by using eqs 11 and 13 (solid curves) with B and ν as two variable parameters in doublelogarithmic plots of I(q) vs q.

In Figure 9, the profiles for R-70 toluene solutions are described well by adequetly choosing the parameters B and ν. Note that an upturn at low q is considered only for the bulk R70 sample as previously reported.23 Similar to L-70 solutions, the correction factors B for R-70 solutions are close to unity within errors of ca. 10%, and the errors of ν are ±0.01. It has been found from the scattering function analysis that ν values for the R-70 solutions are weakly increased from 0.46 in bulk to 0.49 with decreasing ΦP. This trend itself is qualitatively the same as for L-70 solutions. However, the ν value for R-70 at a certain ΦP is considerably lower than that of the corresponding

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.8b00934. I

DOI: 10.1021/acs.macromol.8b00934 Macromolecules XXXX, XXX, XXX−XXX

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SEC profiles of h/d-ran-LPS (Figure S1); SANS profiles of R-70 (Figure S2); SANS profiles of d8-toluene and three ran-LPS (Figure S3); : SANS profiles of R-70 and L-70 solutions (Figures S4 and S5); Guinier plots of L70 solutions (Figure S6); Kratky plots of L-70 and R-70 solutions compared with the fitting (Figures S7 and S8) (PDF)

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. ORCID

Yuya Doi: 0000-0001-8029-7649 Atsushi Takano: 0000-0002-5188-5166 Tae-Hwan Kim: 0000-0001-9113-1281 Present Address Δ

Department of Quantum System Engineering, Chon Buk National University, Deokjin-gu, Jeonju-si, Jeollabuk-do, 561− 756, Korea. Author Contributions &

These authors contributed equally to this work.

Notes

The identification of any commercial product or trade name does not imply endorsement or recommendation by the National Institute of Standards and Technology. The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors are grateful to Dr. S. Y. Reigh at Max-PlanckInstitute for Intelligent Systems and Prof. D. Y. Yoon at Stanford University for providing their data and for their fruitful discussion. We thank Prof. B. Hammouda at NIST for his helpful advices and comments especially for the contrast matching method. We are also grateful to Dr. Sakaue at Kyushu University for his helpful discussion. The authors acknowledge Dr. P. D. Butler at NIST for his help in conducting SANS measurements at NIST. SANS measurements at J-PARC were carried out with experimental assistance of Prof. Y. Nakamura at Kyoto Univeristy (proposal No. 2017B0188; BL15). Travel expenses at 40m SANS at HANARO (Korea) were supported by General User Program for Neutron Scattering Experiments, Institute for Solid State Physics, The University of Tokyo (proposal No. 13064), at JRR-3, Japan Atomic Energy Agency, Tokai, Japan, and the authors are grateful for the support. This work was partly supported by Grant-in-Aid for Scientific Research (No. 16H02292 for Y.M.) from the Japan Society for the Promotion of Science. M.N. acknowledges funding support of cooperative agreement 70NANB15H259 from NIST, U.S. Department of Commerce. Y.D. acknowledges the financial support from the ACCEL program of the Japan Science and Technology Agency (JST).

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DOI: 10.1021/acs.macromol.8b00934 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules (74) Hammouda, B. Form Factors for Branched Polymers with Excluded Volume. J. Res. Natl. Inst. Stand. Technol. 2016, 121, 139− 164.

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DOI: 10.1021/acs.macromol.8b00934 Macromolecules XXXX, XXX, XXX−XXX