Article pubs.acs.org/Macromolecules
SANS Study on Critical Polymer Clusters of Tetra-Functional Polymers Xiang Li,*,† Kazu Hirosawa,† Takamasa Sakai,‡ Elliot P. Gilbert,§ and Mitsuhiro Shibayama*,† †
Institute for Solid State Physics, The University of Tokyo, Kashiwa, Japan Department of Bioengineering, The University of Tokyo, Tokyo, Japan § Australian Centre for Neutron Scattering, Australian Nuclear Science and Technology Organisation, Locked Bag 2001, Kirrawee DC, NSW 2232, Australia ‡
S Supporting Information *
ABSTRACT: A series of critical clusters was prepared by mixing two different kinds of tetra-functional poly(ethylene glycol) (PEG) prepolymers carrying complementary end-groups. The structures of these critical clusters were investigated by small angle neutron scattering (SANS) in different dilution levels. Scaling laws for semidilute polymer solutions were observed in the solution of critical polymer clusters for q < ξ−1, where q is the scattering vector and ξ is the correlation length. The fractal dimension of the critical clusters was estimated to be approximately 2.0, irrespective of the preparation condition of the critical clusters. For q > ξ−1, the size distribution of the critical clusters influenced the scattering intensity. Assuming the validity of the scattering theory for the dilute solution of critical polymer clusters in the q-range q > ξ−1, the Fisher exponent was estimated to be 1.90−2.25, which was found to depend on the preparation condition of the critical clusters.
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INTRODUCTION A polymer gel is a single giant polymer with complex threedimensional network, in which a large amount of solvent can be trapped. Polymer gel can be synthesized by various chemical methods: radical polymerization of monomers and crosslinkers, cross-linking the side groups of linear polymer chains by cross-linkers or by irradiation of γ-rays, coupling end-groups of star polymers, etc.1−5 At the first stage of a gelation reaction, branched polymer clusters are formed and the clusters grow as the reaction proceeds. At a certain point of the reaction, a branched polymer cluster as large as the reaction bath will appear and percolates the system. This critical point of percolation is called gel point and the cluster that percolates the system is an incipient gel. The gelation reaction proceeds even after the gel point, and finally a fully developed gel network is formed. Although there are many different chemical systems for gelation, a set of universal scaling laws is commonly observed for various physical quantities (e.g., size-distribution, weight-averaged molecular weight, correlation length) near the gel point.6,7 Branched critical polymer clusters can be synthesized by mixing the monomers and cross-linkers (or prepolymers and cross-linkers) at the critical ratio, above which the solution eventually becomes a gel but below which the solution remains in sol state.6 In addition to the molar ratio of monomers and cross-linkers, one can achieve the critical condition by changing the total concentration of monomers and cross-linkers; the © 2017 American Chemical Society
sol−gel phase diagram is a two-dimensional (2D) diagram with respect to the molar ratio and the total concentration.8 However, most of the previous studies only dealt with critical clusters prepared at a certain critical point (a critical molar ratio of monomer/cross-liker at a specific concentration).9−13 To further understand the critical behavior of polymer clusters, we need to systematically investigate critical clusters prepared based on the 2D sol−gel phase diagram. Recently, we have synthesized a new type of critical clusters by mixing two different types of tetra-armed poly(ethylene glycol) prepolymers with mutual reactive end-groups in offstoichiometric molar ratios.8 Because each type of the prepolymers reacts exclusively with the other type prepolymers, the prepolymers of excess ratio tend to cover the surface of the cluster and stop the gelation reaction. In a previous study, we determined the 2D sol−gel phase diagram for a series of critical clusters with respect to the molar ratio and the concentration of two prepolymers by rheological measurements.8 According to the critical sol−gel transition curve on the 2D phase diagram, we were able to systematically prepare a series of offstoichiometric critical clusters that is very close to the gel point. In this study, we investigated the static structures of these off-stoichiometric clusters by means of small angle neutron Received: March 12, 2017 Revised: April 14, 2017 Published: April 20, 2017 3655
DOI: 10.1021/acs.macromol.7b00528 Macromolecules 2017, 50, 3655−3661
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the all polymers,14 and thus one can derive the following scaling with eq 7:
scattering (SANS) experiments and compared the obtained critical exponent with the reported experimental values as well as those predicted on the basis of 3D-percolation models.
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ξ* ≅ R z ∼ Mz1/ D ∼ (N *)1/ D ∼ |ε|−1/ Dσ
THEORETICAL BACKGROUND 1. Critical Clusters. There are two main theories for critical clusters near gel point: Flory and Stockmayer model (FS model), and 3D-percolation model.6,7,14 The FS model is based on the assumptions that (1) the reaction of monomers progresses on Bethe lattices, (2) the reaction probability of all the functional groups is the same wherever the functional group is (a Markov process). Although the FS model provides several interesting analytical scaling relationships, the model leads to a fractal dimension of 4 for the branched polymers, which is an unphysical value in our 3-dimensional space. The 3D-percolation model considers the connectivity between the sites or/and the site occupation probability in 3D-lattice. There is no analytical solution for 3D-percolation model, but computer simulations and analytical approximation are available for comparison with experimental data. Here we introduce the common results for both FS and 3Dpercolation models. Let us define the relative distance (ε) from the gel point as ε ≡ |p − pc | /pc
Here, D is the fractal dimension. The all scaling rules shown above are composed of only three independent exponents (τ, σ, and D). The values of τ, σ and D depend on the chemical systems and the models but the resulting relationships are universal irrespective of the systems. The FS model predicts these exponents as τ = 2.50, σ = 0.50, and D = 4.00, while the 3D-percolation model predicts τ = 2.18, σ = 0.46, and D = 2.52. Note that the fractal dimension D shown here is for branched polymers in concentrated state (in a reaction bath) where the excluded volume effect is screened out. In terms of dilute solutions, the 3D-percolation model predicts the fractal dimension of swollen branched chains Ds = 2.0, which is the same as the modified FS theory that takes the excluded volume effect into account by using the Flory approximation.15 The percolation theories work in the critical regime in which system-specific parameters became insignificant while the main domain of application of the FS theory is the pregel or postgel regimes. Thus, in the following discussion for the critical polymer clusters, we mainly compare the experimental result with the percolation model. 2. Scattering from Polymer Solutions. 2.1. Dilute Solution of Polydisperse Polymer Chains. The influence of size distribution of the critical clusters on the scattered intensity has been well discussed for the dilute solution in the previous studies.16−18 At the infinite dilute limit, the scattered intensity from the critical clusters is the averaged sum of the intensities scattered by all the polymers:
(1)
where p is the connectivity or the occupation probability of sites and pc is the p at the gel point. Considering the sites in the FS or percolation models as the multifunctional prepolymers, the number density distribution function of polymer clusters with N prepolymers in both models is generalized as n(p , N ) ∼ N −τf (N /N *), for ε ≪ 1
I(q) ∼ φ
(2)
where τ is the Fisher exponent that is related to the size distribution of the critical polymer clusters, f(N/N*) is a cutoff function that drops sharply to 0 at the characteristic size N*.6 Following the result of the FS model, N* can be regarded as a power law function of the distance from the gel point, which diverges at the gel point.
N * ∼ |ε|−1/ σ
Here, σ is a power law exponent. This scaling law is validated by the computer simulation of 3D-percolation models as well. With eqs 2 and 3, several scaling laws can be derived as following: (4)
M n ∼ (N *)0 ∼ |ε|0
(5)
M w ∼ (N *)3 − τ ∼ |ε|−(3 − τ)/ σ
(6)
Mz ∼ N * ∼ |ε|−1/ σ
(7)
∫0
N*
n(p , N )IN (q)dN
(9)
where φ is polymer volume fraction and IN(q) is the scattered intensity of a polymer chain with the degree of polymerization N. By substituting eq 2 into eq 9, one obtains the following equations:
(3)
Pgel ∼ (N *)2 − τ ∼ |ε|τ − 2/ σ
(8)
I(q) ∼ Mw φ
(10)
I(q) ∼ q−(3 − τ)Ds , for qR z > 1 φ
(11)
where Rz is the z-averaged radius of gyration. 2.2. Semidilute Solution of Monodisperse Polymer Chains. When polymer chains overlap with each other in a solution, the polymer solution is in semidilute state. Scattering profiles of semidilute polymer solutions are often described by an Ornstein−Zernike type function for qξ ≤ 1 and by a powerlaw function with an exponent corresponding to the fractal dimension of a swollen polymer chain for qξ > 1 as follows:6,19,20
Here Pgel is the gel fraction, Mn, Mw, and Mz are the number-, weight- and z-averaged molecular weight, respectively. The last scaling indicates that the characteristic size of the clusters N* is equal to the z-averaged molecular weight of the all polymer clusters. Let us define ξ* as the correlation length or the root-meansquare of the distance between two monomers in the same polymer cluster. Simple calculation shows that ξ* is approximately equal to the z-averaged gyration radius (Rz) of
I(q) =
I(0) , for qξ < 1 1 + (qξ)2
I(q) ∼ φq−Ds , for qξ > 1
(12) (13)
where ξ is the correlation length of concentration that is often simply referred as blob size, and I(0) is the scattering intensity at q = 0. 3656
DOI: 10.1021/acs.macromol.7b00528 Macromolecules 2017, 50, 3655−3661
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Figure 1. Illustration for synthesis of critical clusters by mixing two different types of tetra-armed PEG prepolymers with mutual reactive end groups (maleimide−tPEG and thiol−tPEG). syringe filters. The synthesized critical clusters were not filtered anymore. SANS. SANS experiments were performed on the QUOKKA instrument at the OPAL reactor at Australian Nuclear Science and Technology Organization (ANSTO), Sydney, Australia. Three instrument configurations (distance from samples to detector: 20 m, 8 and 1.3 m with offset) were used to yield a q-range from 0.003 to 0.7 Å−1, where q is absolute value of the scattering vector. The exposure times were 60 min for 20 m, 15 min for 8 m, and 5 min for 1.3 m configuration. The wavelength of neutron beam was 5.0 Å with 10% wavelength resolution. The beam size of the irradiated neutron on the sample was collimated to 12.5 mm in diameter. The polymer solutions of critical clusters were sealed into quartz cells (2 mm in thickness, 20 mm in diameter) for the SANS measurements. All the measurements were carried out at 25.0 ± 0.1 °C. The incoherent scattering from the polymers and solvent were corrected from the scattered intensity of the samples by subtracting the averaged scattered intensity at the highq (q = 0.5−0.7 Å−1), in which q-region the scattered intensity was constant with respect of q, indicating that the scattered intensity is mostly contributed from the incoherent scattering. See the Supporting Information for the scattered intensity before the correction of incoherent scattering.
The blob size is a typical distance by which the excluded volume effect reaches. In the length scale shorter than the blob, the excluded volume effect works; the conformation of polymer chains is that of polymer chains in dilute solution. In contrast, in the length scale longer than the blob, the excluded volume effect is screened out, and the polymer chains are viewed as the ideal chains composed of blobs. As a result, the polymer solution as a whole is viewed as a melt of densely packed blobs. According to the scaling theory of semidilute polymer solutions, the concentration blob scales with the polymer volume fraction as20 ξ ∼ φ−1/(3 −Ds)
(14)
I(0) is proportional to the osmotic compression modulus (Kos) and generally expressed as by using the relation Kos ∼ φ(∂Πmix/ ∂φ) and Πmix ∼ ξ−3:20 I(0) ∼
φ2 φ2 ∼ ∼ φ 2ξ 3 Π mix Kos
(15)
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where Πmix is the osmotic pressure of polymer solutions arising from the free energy of mixing, which is inversely proportional to a cubic of the blob size.
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RESULTS AND DISCUSSION The critical clusters were synthesized by mixing two mutual reactive tetra-armed PEG polymers with complementary endgroups (maleimide−tPEG and thiol−tPEG) (Figure 1). Because maleimide−tPEG reacts exclusively with thiol−tPEG, one can control the reactivity of the tPEGs by mixing maleimide- and thiol−tPEG in off-stoichiometric ratios. Figure 2 shows the illustration of sol−gel phase diagram of maleimideand thiol−tPEG (Mw = 10k) in terms of r and φ. The sol−gel transition curve was determined by a series of dynamic viscoelasticity measurements with the criterion of Winter and Chambon.21,22 When r = 0.5, the synthesized polymer chains percolate and form a gel if φ is higher than 0.0053; one can obtain critical clusters at the point rc = 0.5 and φc = 0.0053. As increasing or decreasing r from the stoichiometric ratio (r = 0.5), more concentrated polymer solutions are needed to synthesize the critical clusters. Note that the sol−gel transition curve is symmetric with respect to the line of r = 0.5. Hence, we only investigated the critical clusters with r ≥ 0.5.
EXPERIMENTS
Samples. Maleimide-terminated tetra poly(ethylene glycol) (maleimide−tPEG) (Mw = 10k, NOF Corporation) and thiolterminated tetra poly(ethylene glycol) (thiol−tPEG) (Mw = 10k, NOF Corporation) were dissolved in D2O buffers (phosphate/citric acid, salt concentration = 0.01M, pH 5.0) at the critical molar ratio (r = rc) and at the corresponding critical polymer volume fraction (φ = φc): (rc, φc) = (0.5, 0.0053), (0.70, 0.0131), (0.80, 0.0259), and (0.85, 0.0504), which were determined by a series of dynamic elastic measurements with the criterion of Winter and Chambon.8,21,22 r is defined as the molar ratio of maleimide−tPEG to the total molar amount of tPEGs (=maleimide−tPEG + thiol−tPEG). The maleimide−tPEG and thiol−tPEG react each other to form critical clusters by the Maleimide−thiol Michael addition reaction. The mixed solutions were placed at room temperature for more than 5 days to ensure all the reaction completed. All the maleimide−tPEG and thiol− tPEG solutions used in this study were filtered with 0.2 μm hydrophilic 3657
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estimated fractal dimension for prepolymers from the Figure 3 was 1.80 ± 0.02 according to the general scattering theory of polymer chains (I/φ ∼ q−Ds).18,20 The estimated fractal dimension is close to the theoretical value for the linear polymer chains in good solvent in dilute state (Ds = 1.67). This similarity of the fractal dimension of star polymer chains and linear polymer chains in good solvents has been reported by Rai et al.25 To investigate the structure and the size distribution of the critical clusters, we diluted the as-prepared critical clusters with the same buffer used to synthesize these clusters, and carried out the same SANS experiments on those dilutions. Figure 4 shows I/φ of various critical clusters (rc = 0.5, 0.7, 0.8, 0.85) and their dilutions. In the prepolymer region (q > 0.1 Å−1), I/φ converged whatever the dilution level is. This is because I/φ in this q-range reflects the structure of prepolymers as mentioned above. The upper-deviations in this q-range for very diluted samples are due to the insufficient statistics for the neutron counting. In the cluster region (q < 0.1 Å−1), by contrast, I/φ increased with decreasing φ and gradually converged on a limiting curve in each subfigure of Figure 4. This kind of convergent on a limiting curve with lowering φ is a common feature of semidilute polymer solutions.27,28 The limiting curve is the scattered intensity profile of very dilute polymer solution, in which I/φ is independent of φ anymore for the full q-range. The scattered intensity of semidilute solution of monodisperse polymers is described with eq 12 and 13. A generalized equation was proposed by Bastide and Candau,26 which smoothly connects eq 12 and 13.
Figure 2. Schematic illustration of the 2D sol−gel phase diagram used in this study. The solid curve represents the sol−gel transition curve. The closed red circles show the critical clusters prepared based on the sol−gel transition curve. The open black circles show diluted polymer clusters that are diluted from the as-prepared polymer clusters. The values on the vertical and horizontal axis denote the critical point where the polymer solution is still a sol but is very close to a gel. The original phase diagram is determined in a previous study.22
The normalized scattered intensities (I/φ) of the as-prepared critical clusters (the red filled circles in Figure 2) were shown as a function of q in Figure 3. In the low q-region (q < 0.1 Å−1), I/
I /φ =
A 1 + g (q)(qξ)2
g (q) =
1 1 + (qξ)2 − Deff
(16)
where A is a fitting parameter equal to I(0)/φ and Deff is an effective fractal dimension representing the exponent of the power law slopes at the cluster region (q < 0.1 Å−1). Although the critical polymer solutions are very polydisperse polymer solutions, we attempted to use eq 16 to fit the data in Figure 4. This trial is supported by the fact that the general physical properties of semidilute polymer solution are independent of the chain length of the polymers.20 The fitting results are shown as the solid curves in Figure 4. All of the fitting curves well reproduced the plots in the cluster region (q < 0.1 Å−1). The estimated parameters (ξ, A, and Deff) are shown in Figure 5, 6 and 7, respectively. Figure 5 shows a double-logarithmic plot of ξ as a function of φ. All the values of ξ fell on a single power law curve of ξ ∼ φ−1.05±0.05 regardless of r. This scaling relationship is consistent with the scaling relationship for semidilute branched polymer chains with monodisperse distribution (ξ ∼ φ−1).29 The effect of polydispersity of the critical polymer clusters has been discussed by Daoud et al, which leads to a scaling relationship of ξ ∼ φ−5/3.30 This prediction is not consistent with our result. Our result suggests that the effect of polydispersity in semidilute region is weaker than the prediction of Daoud et al. A similar scaling relationship with this study was observed in the previous study (ξ ∼ φ−1.0±0.1), in which the critical polymer clusters were synthesized by polycondensation of bifunctional monomers (diisocyanate) and trifunctional monomers (trialcohol).31
Figure 3. Scattered intensity profiles of critical clusters prepared at different critical molar ratios (r = rc) and the corresponding critical concentrations (φ = φc). All the samples shown here were measured in as-prepared condition, corresponding to the red filled plots shown in Figure 2. The error bars show the statistical errors in counting scattered neutrons.
φ decreased with increasing r, while in the high-q region (q > 0.1 Å−1), I/φ converged on the same curve following a scaling law I ∼ q−1.80±0.02. As for q > 0.1 Å−1, the scattered intensity is known to relate to the fractal structure of tetra-armed-PEG polymer (Mw 10k) from previous SANS studies.23,24 Thus, for q < 0.1 Å−1, I/φ relates to the correlations outside the prepolymers, and for q > 0.1 Å−1, I/φ relates to the correlations inside the prepolymers. The convergent of I/φ at the prepolymer region (q > 0.1 Å−1) confirms that all critical clusters are indeed built with the tetra-armed prepolymers. The 3658
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Figure 4. Scattered intensity profiles of the as-prepared critical clusters for each r and their dilutions. The solid curves are the fitting curves of Bastide and Candau model26 for q < 0.1 Å−1. The error bars show the statistical errors in counting scattered neutrons.
Figure 5. Double-logarithm plots of ξ and φ. The solid line is the fitting curve ξ ∼ φ−1.05±0.05. The error bars show the fitting errors. The error bars smaller than the plots are behind the plots. The arrows show the critical clusters in as-prepared condition.
Figure 6. Double-logarithmic plots of A and φ for each r. The solid line is the fitting curve A ∼ φ−1.95±0.10. The error bars show the fitting errors. The error bars smaller than the plots are behind the plots. The arrows show the critical clusters in as-prepared condition.
Figure 6 shows the double-logarithmic plots of A as a function of φ. Similar with the relationship of ξ and φ, A is found to be a power law function of φ and independent of r. A scales with φ as A ∼ φ−1.95±0.10. A similar scaling relationship was observed in the previous study (A ∼ φ−2.0±0.2).31 According to eq 15, the scaling relationship of A and φ obtained in this study gives a relationship Πmix ∼ φ2.95±0.10, which is consistent with the theoretical prediction for the osmotic pressure of semidilute branched polymer chains of monodisperse distribution.29 This result suggests that the polydispersity is not an important factor for osmotic pressure in the semidilute region. By combining the obtained scaling relationships ξ ∼ φ−1.0±0.1 and Πmix ∼ φ2.95±0.10, we can obtain the scaling relationship Πmix ∼ ξ−3 ∼ nξ, where nξ is the total number of blobs in the polymer solution. This scaling law shows that the osmotic pressure is proportional to the number of blobs, which is a general property of semidilute polymer solution.20,32
By substituting eq 14 into eq 15, we obtain the following relationship: I(0) ∼ φ−Ds /(3 − Ds) φ
(17)
By comparing the power law relationship (A vs φ) in Figure 6 with eq 17, we obtain the fractal dimension of the critical clusters as Ds = 1.98 ± 0.10. The fractal dimension can also be estimated by comparing the power law relationship (ξ vs φ) in Figure 5 and eq 14, which gives Ds = 2.05 ± 0.05. The estimated fractal dimensions well agree with each other and both are consistent with the 3D-percolation models for swollen branched polymer chains (Ds = 2.0).14 The other reported values of Ds are in the range of 1.98−2.06,6 which are very close to 2.0. The good consistence in Ds indicates the strong universality of the structure of critical polymer clusters. 3659
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solutions as eq 11 (I ∼ q−(3‑τ)Ds). By comparing the power law exponent Deff and (3-τ)Ds, and substituting the fractal dimension estimated from Figure 5 (or Figure 6), we obtained the Fisher exponent τ for each critical clusters (Table 1). The values of τ were in the range of 1.90 ≤ τ ≤ 2.25, which are close to the theoretical prediction of 3D-percolation model (τ = 2.18) and the reported values for the other polymer clusters (2.18 ≤ τ ≤ 2.33).6 Different from the estimated Ds that is independent of r, the estimated τ increased from 1.90 to 2.25 with increasing r from 0.5 to 0.85. According to eq 3, the bigger τ corresponds to a size-distribution rich with smaller clusters. Thus, the relationship between r and τ indicates that the preparing critical clusters at more off-stoichiometric ratio tend to generate critical clusters rich with small ones. This discussion is reasonable because a mixing of two types of prepolymers at off-stoichiometric ratio tends to generate branched polymer clusters covered with the prepolymers of excess amount, which makes the clusters less-reactive with one another.
Figure 7. Semilogarithm plots of Deff and φ for each r. The solid lines show the averaged values of Deff for each r (r = 0.5, Deff = 2.25 ± 0.04; r = 0.7, Deff = 1.82 ± 0.01; r = 0.8, Deff = 1.64 ± 0.01; r = 0.85, Deff = 1.51 ± 0.01). The error bars show the fitting errors. The error bars smaller than the plots are behind the plots. The dashed line shows the theoretical fractal dimension of swollen branched chains (Ds = 2.0).
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CONCLUSION We synthesized a series of critical polymer clusters by mixing two tetra-armed prepolymers with complementary end-groups based on the 2D sol−gel transition diagram. The structures of these critical polymer clusters were measured by small angle neutron scattering in different dilution levels as functions of the off-stoichiometry parameter, r, and the polymer volume fraction, φ. The normalized scattering intensity (I/φ) converged at q-range of the size of prepolymers because the all the polymer clusters are formed with the same prepolymers. In the q-range of the size larger than a prepolymer, I/φ followed the scaling law for semidilute solutions of monodisperse polymers, reflecting that the physical properties of semidilute polymer solution are dominated by the blobs that are independent of size of polymer chains. The fractal dimension of diluted polymer chains (Ds) was successfully estimated from the scaling law between ξ vs φ, and A vs φ, respectively to be Ds = 2.0 ± 0.1. Ds was an unique value irrespective of preparation condition of the polymer clusters. The Fisher exponents (τ) were estimated from the power law exponents by using the scattering function for diluted critical clusters. τ slightly increased from 1.90 to 2.25 with deviating r from the stoichiometric ratio (r = 0.5), suggesting that smaller polymer clusters tend to be formed in off-stoichiometric ratio. This study clearly shows that the fractal structure of the critical clusters is universal irrespective of the synthesized condition, while the size-distribution of critical clusters is a tunable parameter.
The last fitting parameter Deff was shown as a function of φ in Figure 7. In contrast to ξ and A, Deff depends on r but remains constant while changing φ. The averaged values of Deff for each r were listed in Table 1. According to eq 13, as for a Table 1. Estimated Fractal Dimension and Fischer Exponent r [−] 0.50 0.70 0.80 0.85
Ds [−]
Deff [−]
τ [−]
± ± ± ± ± ± ± ±
2.25 ± 0.04
1.90 ± 0.05
1.82 ± 0.01
2.10 ± 0.02
1.64 ± 0.01
2.20 ± 0.02
1.51 ± 0.01
2.25 ± 0.02
2.05 1.98 2.05 1.98 2.05 1.98 2.05 1.98
0.05a 0.10b 0.05a 0.1b 0.05a 0.1b 0.05a 0.10b
Ds is estimated from the scaling relationship of ξ and φ, bDs is estimated from the scaling relationship of I(0) and φ.
a
semidilute solution of monodisperse polymers, the fitting parameter Deff corresponds to the fractal dimension of dilute polymer solutions. However, the values of Deff change with r and are inconsistent with the fractal dimension estimated above (Ds ∼ 2.0). This discrepancy is explainable by considering that the Deff contains the information on the size distribution of the polymer clusters. The effect of size distribution of polymer clusters in semidilute regime has been considered by Daoud et al.30 According to Daoud, in the semidilute solution of critical polymer clusters, one expects dramatic changes in the configurational properties of the macromolecules. Their basic assumption is that the branched polymer chains shrink because the smaller ones penetrate the larger ones. The branched polymer chains with lower part of the mass distribution penetrate those with the higher part and screen out the excluded-volume interaction as it does in the reaction bath. The scattered intensity for qξ > 1 corresponds to the length scale shorter than the blob size ξ, in which the excluded-volume interaction is present and the polymer chains behave as the swollen chains in dilute state. Thus, the scattering function of critical clusters for qξ > 1 should be similar to that of diluted
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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.7b00528. Scattered intensity of critical clusters and their dilutions (PDF)
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AUTHOR INFORMATION
Corresponding Authors
*(X.L.) E-mail:
[email protected]. *(M.S.) E-mail:
[email protected]. ORCID
Xiang Li: 0000-0001-6194-3676 3660
DOI: 10.1021/acs.macromol.7b00528 Macromolecules 2017, 50, 3655−3661
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(18) Martin, J. E.; Hurd, A. J. IUCr. Scattering From Fractals. J. Appl. Crystallogr. 1987, 20 (2), 61−78. (19) Ornstein, L. S.; Zernike, F. Accidental Deviations of Density and Opalescence at the Critical Point of a Single Substance. Proc. R. Neth. Acad. Arts Sci. 1914, 17, 793−806. (20) de Gennes, P. G. Scaling Concepts in Polymer Physics, 1st ed.; Cornell University Press: Ithaca, NY, 1979. (21) Winter, H. H.; Chambon, F. Analysis of Linear Viscoelasticity of a Crosslinking Polymer at the Gel Point. J. Rheol. 1986, 30 (2), 367− 382. (22) Hayashi, K.; Okamoto, F.; Hoshi, S.; Katashima, T.; Zujur, D. C.; Li, X.; Shibayama, M.; Gilbert, E. P.; Chung, U.; Ohba, S.; Oshika, T.; Sakai, T. Fast-Forming Hydrogel with Ultralow Polymeric Content as an Artificial Vitreous Body. Nature Biomedical Engineering 2017, 1 (3), 0044. (23) Matsunaga, T.; Sakai, T.; Akagi, Y.; Chung, U.; Shibayama, M. Structure Characterization of Tetra-PEG Gel by Small-Angle Neutron Scattering. Macromolecules 2009, 42 (4), 1344−1351. (24) Matsunaga, T.; Sakai, T.; Akagi, Y.; Chung, U.; Shibayama, M. SANS and SLS Studies on Tetra-Arm PEG Gels in as-Prepared and Swollen States. Macromolecules 2009, 42 (16), 6245−6252. (25) Rai, D. K.; Beaucage, G.; Ratkanthwar, K.; Beaucage, P.; Ramachandran, R.; Hadjichristidis, N. Quantification of Interaction and Topological Parameters of Polyisoprene Star Polymers Under Good Solvent Conditions. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2016, 93 (5), 052501. (26) Bastide, J.; Candau, S. J. Structure of Gels as Investigated by Means of Static Scattering Techniques. In Physical Properties of Polymeric Gels; Wiley: 1996; pp 145−295. (27) Brown, W.; Mortensen, K.; Floudas, G. Screening Lengths in Concentrated Polystyrene Solutions in Toluene Determined Using Small-Angle Neutron and Small Angle X-Ray Scattering. Macromolecules 1992, 25 (25), 6904−6908. (28) Pedersen, J. S.; Schurtenberger, P. Scattering Functions of Semidilute Solutions of Polymers in a Good Solvent. J. Polym. Sci., Part B: Polym. Phys. 2004, 42 (17), 3081−3094. (29) Daoud, M.; Joanny, J. F. Conformation of Branched Polymers. J. Phys. (Paris) 1981, 42 (10), 1359−1371. (30) Daoud, M.; Leibler, L. Randomly Branched Polymers: Semidilute Solutions. Macromolecules 1988, 21 (5), 1497−1501. (31) Delsanti, M.; Munch, J. P.; Durand, D.; et al. Conformation of Monodisperse Branched Polymers in Semi-Dilute Solutions. Europhysics Letters (EPL) 1990, 13 (8), 697−702. (32) Teraoka, I. Polymer Solutions; John Wiley & Sons, Inc.: New York, 2002.
Mitsuhiro Shibayama: 0000-0002-8683-5070 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by a Grant-in-Aids for Scientific Research from the Ministry of Education, Culture, Sports, Science, and Technology (No. 16H02277 and 15J10502). The SANS experiment was performed using QUOKKA at the OPAL reactor, Australian Nuclear Science and Technology Organisation, Australia (Proposal No. 4291), which was transferred from SANS-U at JRR-3, Institute for Solid State Physics, Japan (Proposal No. 15568).
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DOI: 10.1021/acs.macromol.7b00528 Macromolecules 2017, 50, 3655−3661
