Heterogeneity and its Influence on the Properties of Difunctional Poly

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Heterogeneity and its Influence on the Properties of Difunctional Poly(ethylene glycol) Hydrogels: Structure and Mechanics Paula Malo de Molina, Sahger Lad, and Matthew E. Helgeson* Department of Chemical Engineering, University of California, Santa Barbara, 3357 Engineering II, Santa Barbara, California 93106, United States S Supporting Information *

ABSTRACT: Difunctional polymer hydrogels, such as those prepared from poly(ethylene glycol) diacrylate (PEGDA) macromers, are widely used for a number of potential applications in biotechnology and advanced materials due to their low cost, mild cross-linking conditions, and biocompatibility. The microstructure of such hydrogels is known to be heterogeneous, yet little is known about the specific structure itself, how it is impacted by the molecular parameters of the macromer, or its impact on macroscopic gel properties. Here, we determine the structure of PEGDA hydrogels using small-angle neutron scattering over a significant range of macromer molecular weights and volume fractions. From this, we propose a structural model for PEGDA hydrogels based on self-excluded, highly branched star polymers arranged into a fractal network. The primary implication of this structure is that heterogeneity arises not from defects in the cross-linking network, as is commonly assumed, but rather from a heterogeneous distribution of polymer concentration. This structural model provides a systematic explanation of the linear elasticity and swelling of PEGDA hydrogels.

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INTRODUCTION Poly(ethylene glycol) (PEG) hydrogels are ubiquitous materials due to their low cost, mild cross-linking conditions, and biocompatibility. PEG is biologically inert and nonimmunogenic, which provides additional advantages over other polymers in drug delivery vehicles and blood-contacting devices.1−3 Moreover, the mechanical properties of PEG hydrogels are similar to those of soft tissues, enabling their use for a variety of tissue engineering and regenerative medicine applications.4−7 In addition, they are permeable to water and other small hydrophilic molecules, for which they have been used in microfluidic devices,8,9 membranes or monoliths for separation processes,10 and structured particles for controlling crystallization processes.11,12 PEG hydrogels are typically formed by chemical reaction of linear or star PEG polymers with reactive end-groups. The type of end-group chemistry determines the functionality of the cross-links (i.e., the number of molecules participating in a cross-link), while the macromer concentration and molecular weight determine the cross-linking density of the hydrogel. There are several types of cross-linking chemistries used to produce PEG hydrogels. Hydrogels formed by PEG star polymers with end-groups that react through a binary © XXXX American Chemical Society

condensation reaction are generally highly homogeneous, which leads to tough hydrogels.13,14 Hydrogels formed by difunctional PEG polymers with polymerizable end-groups, i.e., acrylate or methacrylate groups at both ends (PEGDA and PEGDMA, respectively), react via a radical polymerization with low concentrations of initiator. Diacrylic PEG hydrogels have a highly heterogeneous structure15,16 but are very commonly used because of their low price and availability.17−19 Moreover, the use of photoinitiators allows for fast and simple gelation with UV light, which enables precise control over the size and shape of the hydrogels using lithography methods.20−22 In general, the most important properties of polymer hydrogels for common applications are their mechanical strength, the transport of molecular and biological species through the polymer network, and transparency in the case of optical materials such as contact lenses. These properties all depend on the cross-linked polymer network structure. In the case of a homogeneous structure, the cross-link density and the molecular weight between cross-links can be extracted from Received: May 22, 2015 Revised: July 9, 2015

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DOI: 10.1021/acs.macromol.5b01115 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules macroscopic measurements, such as established models for rubber elasticity23 and equilibrium swelling.24 However, most polymer hydrogel networks are structurally heterogeneous, and this heterogeneity strongly impacts the macroscopic properties of the gel. There are two major types of heterogeneities in polymer networks: (1) topological defects such as dangling chain ends and loops and (2) nanostructured heterogeneities in the form of an inhomogeneous spatial distribution of cross-linking density.25 In general, topological defects have a lesser impact on the network structure and can typically be accounted for by assuming that chains participating in defects do not contribute to the network formation, effectively lowering the cross-linking density. The role of nanostructured heterogeneities on macroscopic gel properties is widely recognized,26 but its molecular origin and impact on macroscopic properties is not yet well understood. Small-angle X-ray (SAXS) and neutron scattering (SANS) are well-suited techniques to characterize structural heterogeneity in polymer gels because they probe length scales relevant to the polymer network structure, and in the case of SANS, scattering contrast can be easily enhanced by using a deuterated solvent. The scattering of homogeneous polymer gels arises primarily from the polymer chain fluctuations and can be analyzed with the Ornstein−Zernike (OZ) model for unstructured polymer fluids.13 By contrast, the presence of long-range heterogeneities results in significant excess scattering intensity at low q-values that has been phenomenologically described by the Debye−Bueche27,28 (DB) or Gaussian29,30 functions for unstructured two-phase materials. Detailed structural characterizations of PEGD(M)A hydrogels of molecular weights up to 8000 g/mol with SANS15 and SAXS16 have revealed two important structural length scales in addition to the polymer chain scattering. Similar to other hydrogels, the first is the length scale associated with the increase in low-q scattering related to the correlation length of large-scale heterogeneities. In addition, PEGDA hydrogels exhibit a correlation peak in the mid-q region that has been attributed to the correlation length between high cross-linking areas.15,16 The model based on a combination of DB and OZ functions does not contemplate a peak due to structural correlations and, thus, is not suitable for such gels. Despite the established presence of these structural features in diacrylic PEG hydrogels, it is unclear whether these features correspond to a specific coherent structure or rather reflect uncorrelated local defects within the network. To our knowledge, all microstructural studies to date on PEGDA hydrogels have therefore been limited to qualitative analysis of the length scales associated with these heterogeneities and lack a detailed structural model. Given the ubiquity of difunctional PEG hydrogels, the aim of this study is therefore to propose a specific model for their heterogeneous structure that describes the observed small-angle scattering data over a wide range of hydrogel compositions, including macromer molecular weight and polymer concentration. With such a model in hand, we hope to assess the importance of heterogeneity in determining the mechanical, swelling, and transport properties of polymer hydrogels that are critical to a majority of their applications.

Table 1. Molecular Parameters of the PEGDAs Used PEG Mw (kDa) Ree (nm) Rg (nm) ϕ*

0.6 2.45 1.01 0.212

6 7.80 3.18 0.067

10 10.1 4.11 0.052

20 14.2 5.81 0.037

35 18.8 7.69 0.028

100 31.8 13.0 0.016

end-to-end distance, Ree, in dilute conditions was estimated by Ree = (C∞nl2)1/2, where n is the number of EG monomers in a macromer, l is the length of the EG unit (l = 0.294 nm), and C∞ = 5.2 is the characteristic ratio for PEG in water.31 The dilute polymer radius of gyration in water, Rg, was estimated by assuming ideal chain statistics, i.e. Rg = Ree/√6. The polymer overlap volume fraction was estimated by ϕ* = (3Mw)/ (ρ4πRg3NA), where Mw is the macromer molecular weight, ρ is mass density, and NA is Avogadro’s number. PEGDA hydrogels were prepared from macromers with varying molecular weight and polymer volume fractions, ϕ0, in water by UV-initiated photopolymerization at 20 °C. Care was taken to ensure that UV exposure was sufficiently long for complete conversion of the gelation reaction. All characterization experiments were performed at the same total polymer volume fraction in the hydrogel as prepared (ϕ0), except for the swelling measurements. Most hydrogels are optically transparent, except for gels prepared from low molecular weight macromers and low polymer volume fractions, which are turbid. This turbidity already indicates the presence of structural heterogeneity through nanoscale spatial concentration fluctuations. Hydrogel Microstructure. To understand the impact of the PEGDA macromer molecular weight and concentration on hydrogel microstructure, we characterized the PEGDA hydrogels using small-angle neutron scattering (SANS). For this, gels were prepared in D2O to enhance contrast between polymer and solvent. Figure 1a shows the scattering curves of hydrogels formed by PEGDA macromers with varying molecular weight at a total polymer volume fraction ϕ0 = 0.2. The scattering curves practically superimpose at high values of the scattering vector q (Figure S4), indicating that the local structure of the polymer chains is very similar for all macromer molecular weights. The SANS profiles for hydrogels with lower molecular weight PEGDAs (up to 10 kDa) exhibit a peak located at qmax, suggesting correlations between regions of distinct network structure separated by a characteristic distance d = 2π/qmax. This peak transforms into a shoulder for higher macromer molecular weights. Such a peak, which is not present for the unpolymerized PEGDA solutions,15 is typical of PEG-acrylate hydrogels15,16,32 and has been attributed to the correlation between regions of relatively high cross-linking density. As the macromer molecular weight increases, the position of qmax is shifted to lower values (larger length scales) that correlate with the theoretical value of the end-to-end distance of the PEG chain (Figure S4). This is not surprising, since if the peak arises from the spatial correlations between cross-links, its characteristic length scale should be proportional to the length of the macromer. In the low-q regime, there is a significant increase in intensity that is similar for all samples. This intensity increase follows a power law of ∼q−df with df ≈ 2.5−3, which indicates that the fractal dimension df of the structure is very similar for all probed molecular weights. A fractal dimension between 2.5 and

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RESULTS AND DISCUSSION Hydrogel Preparation. Table 1 lists the properties of the various PEGDA macromers prepared and used for the hydrogel preparation in this work. The molecular weight of the PEG chain in the macromer ranges from 0.6 to 100 kDa. The chain B

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Figure 2. (a) Scattering intensity of a hydrogel formed from 6 kDa PEGDA at ϕ0 = 0.2 showing all contributions to the final fit to eq 4. Points represent measured SANS data, and lines represent the model fits indicated. (b) Schematic representation of a hydrogel of stars arranged into a fractal structure. In the insets, red lines represent acrylic backbones and blue lines represent PEG macromers.

Figure 1. SANS profiles of hydrogels formed with (a) 0.6, 6, 10, 20, 35, and 100 kDa PEGDA macromers with volume fraction ϕ0 = 0.2 and (b) PEGDA 10 kDa with volume fraction ϕ0 = 0.05, 0.1, 0.2, 0.3, 0.4 and 0.5. Curves have been vertically shifted. Lines: fits with eq 4.

3 is typical for rough surfaces and rules out any significant contribution from free polymer. To further investigate the effect of the polymer concentration on microstructure, we prepared hydrogels from PEGDA macromers with Mw = 10 kDa and studied the scattering for different polymer volume fractions (Figure 1b). All the curves have similar features as the ones described above, but the positions of the various features vary with the total polymer concentration. Here, the scattering intensity at high q-values does not superpose completely, indicating small changes in the polymer chain conformation with the polymer concentration. In addition, the correlation peak moves to higher q-values, which suggests that the separation between the highly crosslinked areas shrinks with increasing polymer volume fraction. This behavior is expected from the concentration dependence of the radius of gyration and blob size of polymer solutions.33 The intensity increase at low q-values follows a similar power law of ∼q−3 in all cases, but the intensity increases significantly with the polymer concentration. Such scattering differs significantly from that typically obtained from homogeneous three-dimensional polymer networks, where the scattering primarily arises from the polymer chains and, more specifically, on the intra- and interchain correlations. In those cases, the scattering curves are well described by the Ornstein−Zernike equation (OZ), which clearly does not describe the data in the mid- and low-q regimes (Figure 2a). Often, the scattering of hydrogels presents an additional upturn in the scattering intensity due to the presence of large heterogeneities that have been phenomenologically described by the Debye−Bueche27 or Gaussian30,34 functions. We find that the experimental scattering intensity of PEGDA hydrogels cannot be represented by an OZ model (dashed red line in Figure 2) or a combination of OZ with Debye−Bueche, squared-Lorentzian, or Gaussian models. In particular, these

models do not predict the significant correlation peak observed for many samples. Assuming that the peak arises from the distance between cross-linking points, the structural unit that produces the scattering is a cross-link, which is a branched or starlike polymer. The idea that the structural unit of a polymer network should be a star polymer rather than the mesh chain was already proposed by Huang35 and Silberberg.36 To model the SANS data, we assume that the PEG chains are connected to an acrylic cross-linking backbone (Figure 2b). There are two limiting structures for such a configuration. If the acrylic backbone is long relative to the PEG chains, the network will resemble a bottle-brush configuration. By contrast, if the acrylic backbone is short relative to the PEG chains, then the PEG chains will asymptotically adopt spherical symmetry, such that the new structural unit will resemble a star polymer. The presence of a correlation peak in concentrated solutions is expected for both star polymers37 and bottle-brush structures.38 The position of the peak is then related to the distance between the star polymer centers or the backbone of the bottle-brush cylindrical structure, respectively. Since here the scattering curves do not show a q−1 power law typical of cylindrical symmetry,39,40 we assume that the cross-links are approximately spherical. Therefore, we choose to model the structure of the PEGDA hydrogels as a network of star polymers. For the form factor, we use an adapted model for the scattering from star polymers:41 nV 2Δρ 2Pstar(q) = IG−P(q) +

−1 4πα sin[μtan (qξ)] Γ(μ) 2 2 qξ [1 + q ξ ]μ/2

(1)

The first term on the right-hand side of eq 1 is a low-q Guinier−Porod contribution to the scattering intensity for aggregates with radius of gyration Rg,42 given by IG−P(q) = I(0)S C

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Table 2. SANS Fit Parameters of Hydrogels Formed from Varying Molecular Weight PEGDAs with Volume Fraction ϕ0 = 0.2 Mw (kDa)

α (cm−1)

ξ (nm)

I(0)S (cm−1)

Rg (nm)

ϕHS

K1′ (cm−1)

DF

Ξ (nm)

0.6 6 10 20 35 100

0.007 0.032 0.031 0.036 0.036 0.040

0.71 0.70 0.76 0.74 0.82 0.78

1.06 1.41 2.18 6.10 163 88.7

1.25 4.2 4.96 12.1 39.5 25.9

0.14 0.15 0.14

4180 3410 1279 1840 2090 21800

2.52 2.78 3.03 2.86 2.89 3.00

50.0 159.2 78.1 193.5 154.7 208.8

exp(−q2Rg2/3) for q ≤ √6/Rg and IG−P(q) = F/q4 for q > √6/ Rg, with F = I(0)Se−236/Rg4. Here I(0)S is the scale factor for the star scattering. The Guinier−Porod model has been used previously to capture the low-q scattering of similar systems, including bottle-brush polymers43 and block copolymer micelles.44 The second term accounts for the blob−blob correlation of star arms according to the Daoud−Cotton model45 with blob size ξ, scaling parameter α, and μ = 1/ν − 1 with excluded volume parameter of PEG in water ν = 0.5831 and Γ(μ) the gamma function with argument μ. The star scattering is then modeled using the deconvolution approximation, such that I = nV 2Δρ 2Pstar(q)S(q)

Equation 4 assumes that the scattering from individual starlike objects and from the mass fractal are decorrelated, which has been successfully used to describe the scattering of fractal structures formed by homogeneous spheres.49 Qualitatively, this model resembles the structure depicted in Figure 2b. The smallest length scales correspond to correlations between blobs of individual polymer chains with size ξ. At intermediate length scales, PEG (blue) is arranged into stars emanating from an acrylic backbone (red) which, absent any loops or dangling ends, are covalently linked to other stars due to the difunctional nature of the precursor polymer. The aggregation of these stars into a network produces the fractal structure, which at the largest scales resembles a porous structure with a characteristic void size set by the fractal cutoff size, Ξ. Such a porous structure with various morphologies has been observed in PEGDA gels by SEM.9,10,50 Fits of the scattering data to our model analysis using eq 4 are in good agreement with the experimental data (Figure 1). The best-fit model parameters for the macromer length variation series are shown in Table 2. We note that both the forward scattering parameter α and the blob size ξ of chains are very similar for all molecular weights, indicating that the polymer chain conformation is locally the same for all PEGDA lengths. However, the star polymer contribution changes significantly with macromer molecular weight. Nevertheless, in all cases, the overall forward scattering of individual stars (without the fractal term), I(0)S = nV2Δρ2Pstar(0)S1(0), increases systematically with increasing macromer molecular weight, as expected. The radius of gyration of the star at fixed polymer volume fraction scales with the macromer molecular weight as Rg = 0.029 nm Mw0.59 (Figure 3a). The scaling factor 0.59 is very similar to the value of 0.58 found for linear PEG in water.31 The fractal dimension ranges from 2.5 to 3, indicating that the structure has rough but persistent interfaces. The cutoff length of the fractal contribution increases moderately with macromer length, indicating an increasing characteristic size of voids within the hydrogel with increasing cross-link density. Table 3 summarizes best-fit parameters for hydrogels from 10 kDa PEGDA hydrogels with varying total polymer volume fraction. The polymer blob size and the star radius of gyration decrease with the total polymer concentration (Figure 3b). The blob size is represented by a power-law function of ϕ with an exponent of −3/4. This exponent is precisely what is predicted from scaling theory for semidilute polymer solutions in a good solvent due to the osmotic compression of chains.33 The prefactor, b = 0.22 nm, which is predicted to be the monomer length, is slightly lower than the length of a PEG monomer (bEG = 0.294 nm).31 The star radius of gyration scales as ϕ−0.12, which compares well with the predicted scaling for linear polymers, Rg ∼ bN1/2ϕ−1/8. The Daoud−Cotton scaling model for star polymers predicts a scaling of Rg ∼ ϕ−3/4 close to the overlap concentration and Rg ∼ ϕ−1/8 for larger concen-

(2)

where n and V are the number density and volume of individual stars, respectively. Δρ is the scattering length density contrast, and Pstar(q) and S(q) are the form and structure factors, respectively. If the star polymers are arranged in several levels of organization due to an heterogeneous distribution of the polymer concentration, the total structure factor S(q) may be approximated by factorization into two terms: S(q) = S1(q) S2(q).46 S1(q) describes the local structure factor of neighboring stars and S2(q) the correlation between star aggregates. The simplest model for the local star structure factor is a hard sphere with excluded volume interactions of objects with size Rg using the Percus−Yevick closure approximation (eq S1),47 S1(q) = SHS(q), with the only adjustable parameter being the local volume fraction of stars, described by the local hard sphere volume fraction ϕ HS . To account for the long-range heterogeneities, we assume that the low-q contribution to the scattering can be described using a mass fractal model, S2(q) = Sfract(q), with characteristic fractal dimension D Sfract(q) = 1 + K1

sin[(D − 1) tan−1(qΞ)] (D − 1)qΞ(1 + q2 Ξ2)(D − 1)/2

(3)

where the functional form corresponds to an exponentially decaying distribution of fractal objects with largest cutoff size Ξ, which has been found to empirically capture the scattering from real mass fractal structures.48 Sfract(q) is approximately 1 at intermediate and high q, and both Pstar(q) and SHS(q) limit to a constant at sufficiently low q. Therefore, the total scattering intensity can be approximated as a sum of the scattering of individual polymer stars and the scattering of a mass fractal I ≈ nV 2Δρ 2Pstar(q)SHS(q) + K1′

sin[(D − 1) tan−1(qΞ)] (D − 1)qΞ(1 + q2 Ξ2)(D − 1)/2 (4)

where K1′ is an amplitude factor that depends on the difference in scattering length density between the elements comprising the fractal and the surrounding medium; since these quantities are not known a priori, it is left as an adjustable parameter. D

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Macromolecules INV =

∫0

∞

I(q)q2 dq

(5)

where the discrete data points were extrapolated to q = 0 and ∞ with standard methods by Guinier and Porod approximations, respectively. The theoretical two-level invariant is modeled as a two-phase medium composed of a high-density network phase (composed of stars) and a low-density void phase (which is rich in solvent) INV = 2π 2ϕvoid(1 − ϕvoid)(ρSL,void − ρSL,network )2 = 2π 2

ϕvoid 1 − ϕvoid

[ϕ0(ρSL,P − ρSL,D O )]2 2

(6)

where ϕvoid is the void volume fraction of voids in the porous structure, and ρSL,void and ρSL,network are the scattering length densities of the voids and the network, respectively. Assuming that the voids are formed entirely by D2O and the network phase is formed by polymer and solvent with a polymer content of ϕ0,n = ϕ0/(1 − ϕvoid), the scattering invariant depends on ϕvoid, the total polymer concentration ϕ0, and the length scattering densities of pure polymer ρSL,P and solvent ρSL,D2O. The void volume fraction as well as its dependence with the macromer concentration and molecular weight are plotted in Figure 4. Overall, we find that ϕvoid decreases with increasing polymer concentration and follows a nearly linear trend that is independent of macromer molecular weight (Figure 4a). Furthermore, using the relationship above to estimate ϕ0,n, we find that the polymer concentration in the dense phase is nearly independent of both macromer molecular weight and concentration. This information helps to elucidate the mechanism of heterogeneity in PEGDA hydrogels. There are several proposed mechanisms for nanoscale heterogeneities in hydrogels. First, polymeric precursors below the overlap concentration form sparse networks when they react because they cannot fill the entire volume.51 Specifically, de Gennes proposed that polymer chains forced to be nearby due to cross-linking have a concentration proportional to the overlap concentration.33 This will necessarily create regions of depleted polymer elsewhere in the material. In this case, the volume fraction of voids should depend on the total polymer concentration relative to the overlap concentration. Specifically, the extent of heterogeneity should decrease with increasing ϕ0 below the overlap concentration and vanish above the overlap concentration. By contrast, our SANS experiments show heterogeneities in gels formed from prepolymer solutions well above the macromer overlap concentration (Table 1). Alternatively, when the polymerization reaction starts and the molecular weight of the newly formed branched polymers increases, there can be an entropically driven phase separation between polymer-rich and polymer-poor phases which leads to

Figure 3. Blob size (circles) and radius of gyration of the star (squares) as a function of (a) the macromer molecular weight with volume fraction ϕ0 = 0.2 and (b) the polymer volume fraction for hydrogels formed by photopolymerization of PEGDA 10 kDa. Lines are best fits to a power law relationship.

trations.45 Here, we only observe the scaling of ∼1/8, such that the polymers behave like linear chains, as has been observed in concentrated solutions of discrete star polymers.37 The same is expected for the blob size at sufficiently high concentrations, so that the blob size is no longer dominated by its distance to the center of the branching, but by the total blob concentration. Here again the fractal dimension is near 3 and is quite insensitive to the total polymer concentration. The fractal cutoff length Ξ varies from 72 to 205 nm with a nonmonotonic trend. However, the forward scattering of the fractal contribution, K1′, increases monotonically with the polymer concentration. Interestingly, the hard sphere volume fraction, or equivalently the local number density of stars, is very similar for all probed macromer molecular weights and concentrations. These results suggest that the gel is formed by a network phase with constant local polymer concentration and voids with essentially no polymer. As the polymer concentration is increased, the amount of network phase increases, leading to a more homogeneous hydrogel. In order to quantify the degree of heterogeneity arising from the fractal-like arrangement of stars, we calculate the scattering invariant, INV. For a two-phase system, assuming the presence of a sharp interface between the two phases, INV is defined as

Table 3. SANS Fit Parameters of Hydrogels Formed with PEGDA 10 kDa ϕ0

α (cm−1)

ξ (nm)

I(0)S (cm−1)

Rg (nm)

ϕHS

K1′ (cm−1)

DF

Ξ (nm)

0.05 0.1 0.2 0.3 0.4 0.5

0.0481 0.0459 0.0279 0.0169 0.0108 0.0076

1.98 1.30 0.77 0.54 0.41 0.34

1.70 1.38 1.58 1.17 0.77 0.35

5.62 5.12 4.96 4.76 4.37 3.94

0.12 0.11 0.14 0.13 0.13 0.14

383 751 1189 3007 5360 20985

3.20 3.20 3.15 3.10 3.08 3.05

100 72.6 78.3 101 125 205

E

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Figure 4. (a) Void volume fraction as a function of the total polymer volume fraction obtained from the scattering invariant using eq 6. (b) Polymer volume fraction in the network phase as a function of the total polymer volume fraction.

networks with heterogeneities of different morphologies depending on the phase behavior and the competition between the reaction rate and the phase separation rate.52 In this case, ϕvoid does not depend strongly on the initial molecular weight but rather on the phase behavior of the branched intermediates, which is very difficult to predict.53 The results shown in Figure 4b therefore tend to support this mechanism. Finally, Panewkov and Rabin proposed a statistical mechanical analysis of a gel model formed by instantaneous cross-linking of semidilute polymer solutions that showed that the thermodynamic conditions (quality of solvent, degree of cross-linking, and monomer concentration) at the preparation time determine the statistical properties of its disordered structure.54 The heterogeneous distribution of cross-links has a characteristic length scale which depends on the conditions of preparation and can vary from microscopic to macroscopic dimensions. The theory’s predictions of the degree of heterogeneity observed from scattering experiments agree qualitatively with the observed trends with concentration and solvent quality. However, quantitative analysis of SANS and light scattering curves was unsuccessful.55,56 From this, we hypothesize that the heterogeneous structure of PEGDA gels arises from polymerization-induced microphase separation. Further testing this hypothesis would require careful measurements of the local polymer concentration during gel formation, which we leave to future studies. Linear Elasticity. The linear elastic modulus G0 of the PEGDA hydrogels was measured using in situ rheological measurements. The modulus of the hydrogels increases with ϕ0 (Figure 5a), as expected due to the increase of cross-linking density in the polymer network. We fit the elastic modulus as a function of the precursor concentration to a power law based on the difference between the polymer volume fraction and the

Figure 5. (a) Concentration-dependent linear elasticity of hydrogels formed with 0.6, 10, and 20 kDa molecular weight PEGDAs. Solid lines are fits to G0 = G0,p(ϕ0 − ϕ0,p)n. Dotted lines are predictions from eq 8. (b) Cross-linking efficiency as a function of the polymer volume fraction. (c) Cross-linking efficiency decreases with void volume fraction obtained from SANS.

percolation threshold G0 = G0,p(ϕ0 −ϕ0,p)n. The best-fit results for G0,p, ϕ0,p, and n are collected in Table 4. The percolation Table 4. Mechanical Parameters of the PEGDA Hydrogels PEG Mw (kDa) ϕP G0,p (MPa) n

0.6 0.055 35.7 2.45

6 0.018 1.16 1.70

10 0.029 0.548 2.82

20 0.038 0.235 1.39

threshold ϕ0,p varies nonmonotonically with the macromer length. To a first approximation, we can rationalize this as follows. As the macromer molecular weight increases, the length of the polymer subchains increases, contributing to a reduction of ϕ0,p. However, the total concentration of acrylic groups that can polymerize decreases, contributing an increase F

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Macromolecules in ϕ0,p. These two dependencies conspire to give the observed nonmonotonic behavior of the percolation threshold. The power-law scaling exponents, n, of G0 are higher than those expected for homogeneous polymer networks. The plateau modulus of an ideal phantom polymer network is related to the cross-linking density, νc, by G0 = AνckBT

efficiency, which is defined as the ratio between the experimental shear modulus and the theoretical one, νeff/ν = G0ex/G0th. Therefore, the cross-linking efficiency is 1 for homogeneous networks and decreases with increasing heterogeneity. Here, the cross-linking efficiencies range from a few percent to 1 as the precursor concentration increases for all macromer molecular weights (Figure 4b), indicating that the polymer networks are more homogeneous as the precursor macromer concentration increases. The molecular weight has a much less pronounced impact than the concentration. This trend has been observed on other heterogenoeus hydrogel networks such as polystyrene, poly(N-isopropilamide), and thiol−ene reaction type PEG hydrogels, where the cross-linking efficiency increases with the total polymer concentration and there is a much lesser effect of the cross-linker concentration or the precursor molecular weight.56,59,60 To more quantitatively compare the deviation of the elastic moduli to the degree of heterogeneity in the PEGDA gels quantified by SANS, we compare the cross-linking efficiency with the void volume fraction obtained from the scattering invariant using eq 6 (Figure 5c). Interestingly, we find that the cross-linking efficiency decreases from unity to almost zero linearly with increasing void volume fraction, regardless of the macromer molecular weight. This result confirms that the heterogeneous structure of PEGDA hydrogels is more dependent on the polymer concentration at the preparation time rather than on the molecular details of the polymer precursor and directly demonstrates the effect of heterogeneous structure on the mechanical properties of PEGDA hydrogels. Equilibrium Swelling. Figure 6a shows the concentration dependence of the equilibrium swelling of the PEGDA hydrogels for several macromer molecular weights. The volumetric swelling ratio Q is much larger for longer precursors

(7)

where kB is the Boltzmann constant and T the absolute temperature. A = 1 for affine network and A = (1 − 2/f) for phantom networks, where f is the cross-link functionality. The validity of these models is currently under debate.57 The phantom model is predicted to be valid for dilute conditions, whereas deviations toward the affine network are expected at higher concentrations.58 The elastic modulus normalized by the number density of PEG chains, G0/νPEGkBT, was determined for all PEGDA hydrogels (Figure S5). We find that for most conditions G0/ νPEGkBT increases from the phantom network limit toward the affine limit with increasing polymer concentration. However, we note that for sufficiently low macromer molecular weights (e.g., 0.7 and 6 kDa), G0/νPEGkBT > 1, which is above the affine limit. Therefore, there must be an additional contribution to the linear elasticity. The polymer chains are too short for trapped entanglements to occur, so the only remaining possibility is that the acrylate backbones contribute to an increase in G0. Assuming that the PEGDA hydrogels are formed by two kinds of chains, the acrylic backbones and the PEG chains, the elasticity of a perfectly homogeneous PEGDA network in the affine limit (A = 1) for both kinds of chains can thus be approximated as the sum of the two contributions ⎛ϕ ρ ϕ ρ ⎞ G0 ≅ G PEG + Gacr = RT ⎜ PEG PEG + acr acr ⎟ Macr ⎠ ⎝ MPEG

(8)

where R is the gas constant, and ϕi, ρi, and Mi are respectively the volume fraction, mass density, and the molecular weight of the acrylate backbone (i = acr) and PEG chains (i = PEG). Since eq 8 assumes affine contributions of the PEG and acrylic strands to the network elasticity, it provides an upper bound on the modulus of the gels. Indeed, we find that all macromer molecular weights studied asymptotically approach this limit at high polymer concentration. Furthermore, the theoretical values of G0 according to eq 8 (dotted lines in Figure 5) provide a nearly quantitative prediction of the experimental data for low molecular weight macromers and high polymer volume fractions. We note that for the shortest polymer, 0.6 kDa PEG, the acrylic backbones contribute a significant fraction (51%) of the total elasticity. As the macromer molecular weight increases, the backbone contribution becomes less important. However, eq 8 fails to predict the experimental values of G0 at intermediate and low polymer volume fractions. We attribute the disagreement between eq 8 and the data at low PEGDA concentrations and macromer molecular weights to the heterogeneous structure of the polymer network found SANS experiments. It has been shown that the elastic shear modulus of a hydrogel decreases with increasing degree of heterogeneity.56,59,60 The elasticity of gels with heterogeneous nanostructure depends on both the type and on the size of network heterogeneities compared to the length scale of the gel region probed.61 In general, the effect of heterogeneity on the elastic moduli of gels is quantified by the cross-linking

Figure 6. (a) Equilibrium swelling ratio as a function of the PEGDA volume fraction. (b) Equilibrium swelling ratio of the hydrogels as prepared as a function of their elastic modulus. Macromer molecular weights are indicated in the legend. G

DOI: 10.1021/acs.macromol.5b01115 Macromolecules XXXX, XXX, XXX−XXX

Macromolecules

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CONCLUSION We have proposed a new model to describe the heterogeneous structure of poly(ethylene glycol) diacrylate hydrogels in water based on starlike polymers arranged in a fractal network. The source of this specific structure is due to the radical polymerization process that produces these hydrogels, which results in highly functional acrylic cross-linking backbones with side PEG chains, which are connected primarily to different backbones on each end. Therefore, the structural unit of these hydrogels is a starlike polymer rather than a mesh chain. Interconnection of stars into a fractal network leads to polymerlean voids with characteristic length scales of tens to hundreds of nanometers, much larger than the average distance between cross-links (i.e., the radius of the stars), which we hypothesize form due to a process of polymerization-induced phase separation. This structural picture has two main consequences for the properties of these ubiquitous hydrogels. First, the polymer composition and chain length are intrinsically heterogeneous. As such, both the PEG chains and the acrylic backbones contribute significantly to the elastic modulus as well as the equilibrium swelling. Second, nanoscale heterogeneities result in a loss of elasticity at low to moderate polymer concentration. Overall, we anticipate that understanding the starlike, heterogeneous structure of PEGDA hydrogels will be useful in controlling the morphology and mesoscopic properties of PEGDA hydrogels for a number of applications that rely on their mechanical, swelling, and transport properties.

and decreases with the polymer concentration in the asprepared state ϕ0. The equilibrium swelling of a hydrogel occurs when the total free energy per volume is minimized. There are two contributions to the free energy of uncharged polymer networks: an osmotic part Fos acting to swell the gel and an elastic part Fel that restricts swelling. Thus, equilibrium swelling will take place when the osmotic pressure Π equals the elastic modulus of the swollen hydrogel, Gswell: Q≡

Veq Vdry

1 ϕ

=

when Gswell ≈ Π

(9)

where Veq is the equilibrated swollen hydrogel volume, Vdry is the dry volume, and ϕ is the polymer volume fraction in the swollen state. The osmotic pressure of a semidilute polymer solution with Flory exponent ν is33

Π≈

kBT b3

ϕ3ν /(3ν− 1)

(10)

According to eq 9, the elastic modulus in the swollen state is therefore Gswell ≈

kBT b3

Q−3ν /(3ν − 1)

(11)

The elastic modulus after photopolymerization therefore ideally depends on the equilibrium swelling ratio as G0 ≈ ϕ01/3

kBT b3

Q(1/3) − (3ν /(3ν − 1))

Article

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EXPERIMENTAL SECTION

Materials. Poly(ethylene glycol) diacrylate (PEGDA) (Mw ∼ 700 g/mol), poly(ethylene glycol) (PEG) (Mn = 6, 10, 20, 35, and 100 kDa), 2-hydroxy-4-(2-hydroxyethoxy)-2-methylpropiophenone (PI), acryloyl chloride, triethylamine, dichloromethane, and ethyl ether were purchased from Sigma-Aldrich (USA) and used without further purification. Deuterated water (D > 99%) was obtained from Cambridge Isotopes. Synthesis of PEGDAs and Their Hydrogels. PEGDAs (6K, 10K, 20K, 35K, and 100K) were prepared by reacting PEGs with acryloyl chloride. Briefly, PEG, 2.2 equiv of acryloyl chloride and triethylamine was reacted in dichloromethane for 8 h in a dark environment at room temperature. The solution was filtered and precipitated into ethyl ether. The product was collected by filtration and then dried in a vacuum oven prior to use. The molecular mass and molecular mass distribution were determined using a combination of 1H NMR and GPC. Hydrogels were made by UV irradiating prepolymer solutions prepared with polymer volume fraction ϕ0 and 1 vol% photoinitiator. The time required to achieve complete the gelation reaction, determined using in situ rheology and irradiation energy or 150 W/ cm2, varies with the PEGDA molecular weight and concentration and ranges from 5 to 50 s (Figure S1). Small-Angle Neutron Scattering (SANS). Experiments were performed with the NG3 (Figure 1a) and NG7 (Figure 1b) 30 m SANS instruments at the NIST Center for Neutron Research.64 SANS data were collected using neutrons with a wavelength of λ = 6 Å and Δλ/λ = 0.11, a detector distance of 1, 4, and 13.5 m, and a wavelength of λ = 8.4 Å and Δλ/λ 0.11 at a detector distance of 15.3 m in NG7 and 13.5 m in NG3. Samples were loaded into 1 mm thick titanium scattering cells with transparent quartz windows and placed in the 10CB sample environment, whose temperature was controlled using a Julabo temperature-controlled bath. The 2D detector image data were reduced and radially averaged using the National Institute of Standards and Technology IGOR software package.65 In the SANS experiments, q was varied from 0.03 to 4 nm−1, corresponding to length scales from 1.5 to 200 nm. The incoherent background was determined at high scattering angle, set as a constant and subtracted from the data.

(12)

where ϕ0 is the polymer volume fraction before cross-linking. We find experimentally that Q ∼ G0−0.4 for all macromer molecular weights (Figure 6b), which would correspond to a Flory exponent of ν ≈ 0.52. This value is in good agreement with the value 0.59 obtained from the molecular scaling of Rg from the SANS analysis (Figure 3a). Therefore, the equilibrium swelling of PEGDA hydrogels is determined entirely by the balance between the osmotic pressure and the elasticity of the network as it is predicted for nonionic polymer networks. However, equilibrium swelling theories for homogeneous networks, such as the Flory−Rehner theory24 typically used to estimate the local microstructure of the gel (e.g., the “mesh size”), clearly do not apply for PEGDA gels. It is well-known that significant density fluctuations lead to failure of the central assumption of the theory: that the elastic and mixing contributions to the free energy can be separated by assuming that the energy of mixing of a cross-linked network is identical to an equivalent un-cross-linked solution.62 Specifically, it has been shown that such a failure is expected for polymer networks with a fractal-like microstructure at length scales larger than the solution correlation length between crosslinks.63 Nevertheless, it is important to note that Figure 6b clearly indicates that the contribution of heterogeneity to the swelling arises entirely from its influence on the elastic modulus, provided that the swelling ratio is appropriately normalized as in eq 12. This suggests that a theory of network elasticity appropriate for systems with large-scale density fluctuations (at length scales larger than the correlation length between cross-links) that resemble a fractal structure could be used to extend the Flory−Rehner theory to PEGDA gels. H

DOI: 10.1021/acs.macromol.5b01115 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules Rheology. Measurements were carried out using a TA Instruments ARG2 stress-controlled rheometer with a 20 mm quartz lower-plate geometry and a temperature-controlled electrical heated upper-plate geometry. Samples were loaded and equilibrated at the desired temperature and then exposed to ultraviolet light with wavelength λ = 254 nm and irradiance 150 mW cm−2 for 5 s until the measured value of G0 showed no significant change between pulses to ensure complete polymerization of the sample. Subsequently, frequency sweeps were carried out over increasing frequencies from ω = 0.1−20 rad s−1 at a fixed strain amplitude of γ0 = 0.1%. Equilibrium Swelling Experiments. Hydrogel disks of 20 mm diameter and 0.8 μm thickness were synthesized using the UV rheometer setup. The mass of the disks was recorded after photopolymerization and after swelling in DI water for 24 h.

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(12) Diao, Y.; Whaley, K. E.; Helgeson, M. E.; Woldeyes, M. A.; Doyle, P. S.; Myerson, A. S.; Hatton, T. A.; Trout, B. L. J. Am. Chem. Soc. 2012, 134, 673−684. (13) Matsunaga, T.; Sakai, T.; Akagi, Y.; Chung, U.; Shibayama, M. Macromolecules 2009, 42, 6245−6252. (14) Akagi, Y.; Matsunaga, T.; Shibayama, M.; Chung, U.; Sakai, T. Macromolecules 2010, 43, 488−493. (15) Lin-Gibson, S.; Jones, R. Macromolecules 2005, 38, 2897−2902. (16) Waters, D. J.; Engberg, K.; Parke-Houben, R.; Hartmann, L.; Ta, C. N.; Toney, M. F.; Frank, C. W. Macromolecules 2010, 43, 6861− 6870. (17) Cushing, M. C.; Anseth, K. S. Science 2007, 316, 1133−1134. (18) Ward, J. H.; Peppas, N. a. J. Controlled Release 2001, 71, 183− 192. (19) Vogus, D. R.; Mansard, V.; Rapp, M. V.; Squires, T. M. Lab Chip 2015, 15, 1689−1696. (20) Lin-Gibson, S.; Bencherif, S.; Cooper, J. A.; Wetzel, S. J.; Antonucci, J. M.; Vogel, B. M.; Horkay, F.; Washburn, N. R. Biomacromolecules 2004, 5, 1280−1287. (21) Zhang, H. B.; Wang, L.; Song, L.; Niu, G. G.; Cao, H.; Wang, G. J.; Yang, H. A.; Zhu, S. Q. J. Appl. Polym. Sci. 2011, 121, 531−540. (22) Helgeson, M. E.; Chapin, S. C.; Doyle, P. S. Curr. Opin. Colloid Interface Sci. 2011, 16, 106−117. (23) Shibayama, M. Soft Matter 2012, 8, 8030. (24) Peppas, N. A.; Bures, P.; Leobandung, W.; Ichikawa, H. Eur. J. Pharm. Biopharm. 2000, 50, 27−46. (25) Shibayama, M. Polym. J. 2011, 43, 18−34. (26) Di Lorenzo, F.; Seiffert, S. Polym. Chem. 2015, in press. (27) Shibayama, M. Macromol. Chem. Phys. 1998, 199, 1−30. (28) Diao, Y.; Helgeson, M. E.; Siam, Z. A.; Doyle, P. S.; Myerson, A. S.; Hatton, T. A.; Trout, B. L. Cryst. Growth Des. 2012, 12, 508−517. (29) Hecht, A.; Duplessix, R.; Geissler, E. Macromolecules 1985, 18, 2167−2173. (30) Mallam, S.; Horkay, F.; Hecht, A.-M.; Rennie, A. R.; Geissler, E. Macromolecules 1991, 24, 543−548. (31) Kawaguchi, S.; Imai, G.; Suzuki, J.; Miyahara, A.; Kitano, T.; Ito, K. Polymer 1997, 38, 2885−2891. (32) Saffer, E. M.; Lackey, M. A.; Griffin, D. M.; Kishore, S.; Tew, G. N.; Bhatia, S. R. Soft Matter 2014, 10, 1905−1916. (33) De Gennes, P.-G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979. (34) Horkay, F.; Basser, P. J.; Hecht, A. M.; Geissler, E. Polymer 2005, 46, 4242−4247. (35) Huang, Y.; Szleifer, I.; Peppas, N. A. Macromolecules 2002, 35, 1373−1380. (36) Silberberg, A. In Polyelectrolyte Gels; ACS Symposium Series; American Chemical Society: Washington, DC, 1992; Vol. 480, pp 146−158 SE-9. (37) Richter, D.; Willner, L.; Fetiers, L. J.; Lin, M.; Huang, J. S. J. Phys. IV 1993, 3, 3−12. (38) Rathgeber, S.; Pakula, T.; Wilk, A.; Matyjaszewski, K.; Lee, H.; Il; Beers, K. L. Polymer 2006, 47, 7318−7327. (39) Rathgeber, S.; Pakula, T.; Wilk, A.; Matyjaszewski, K.; Beers, K. L. J. Chem. Phys. 2005, 122, 124904 . (40) Zhang, B.; Gröhn, F.; Pedersen, J. S.; Fischer, K.; Schmidt, M. Macromolecules 2006, 39, 8440−8450. (41) Dozier, W. D.; Huang, J. S.; Fetters, L. J. Macromolecules 1991, 24, 2810−2814. (42) Hammouda, B. J. Appl. Crystallogr. 2010, 43, 716−719. (43) Pesek, S. L.; Li, X.; Hammouda, B.; Hong, K.; Verduzco, R. Macromolecules 2013, 46, 6998−7005. (44) Hammouda, B. Eur. Polym. J. 2010, 46, 2275−2281. (45) Daoud, M.; Cotton, J. P. J. Phys. (Paris) 1982, 43, 531−538. (46) Oberdisse, J.; Rharbi, Y.; Boué, F. Comput. Theor. Polym. Sci. 2000, 10, 207−217. (47) Wertheim, M. Phys. Rev. Lett. 1963, 10, 321−323. (48) Helgeson, M. E.; Gao, Y.; Moran, S. E.; Lee, J.; Godfrin, M.; Tripathi, A.; Bose, A.; Doyle, P. S. Soft Matter 2014, 10, 3122−3133.

ASSOCIATED CONTENT

S Supporting Information *

Polymer synthesis and characterization; hydrogel formation and linear rheology; additional SANS plots. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.5b01115.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (M.E.H.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are grateful to Dr. Johannes Sprafke and Dr. Will Gutekunst from Craig Hawker’s group for assistance during the polymer synthesis. This work was funded in part by the Defense Threat Reduction Agency under the Natick Soldier Research, Development and Engineering Center Agreement No. W911QY-13-2-0001. We acknowledge the support of the National Institute of Standards and Technology, U.S. Department of Commerce, in providing the neutron research facilities used in this work. We thank Dr. Mary Raven for help with confocal microscope facility, which was supported by National Institutes of Health Grant 1 S10 OD010610-01A1. The MRL Shared Experimental Facilities are supported by the MRSEC Program of the NSF under Award DMR 1121053, a member of the NSF-funded Materials Research Facilities Network.

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REFERENCES

(1) Griffith, L. G. Acta Mater. 2000, 48, 263−277. (2) Vermonden, T.; Censi, R.; Hennink, W. E. Chem. Rev. 2012, 112, 2853−2888. (3) Gombotz, W. R.; Wang, G. H.; Horbett, T. A.; Hoffman, A. S. J. Biomed. Mater. Res. 1991, 25, 1547−1562. (4) Nguyen, K. T.; West, J. L. Biomaterials 2002, 23, 4307−4314. (5) Bryant, S. J.; Anseth, K. S. J. Biomed. Mater. Res. 2003, 64, 70−79. (6) Burdick, J. A.; Anseth, K. S. Biomaterials 2002, 23, 4315−4323. (7) Slaughter, B. V.; Khurshid, S. S.; Fisher, O. Z.; Khademhosseini, A.; Peppas, N. A. Adv. Mater. 2009, 21, 3307−3329. (8) Paustian, J. S.; Azevedo, R. N.; Lundin, S. T. B.; Gilkey, M. J.; Squires, T. M. Phys. Rev. X 2013, 3, 1−13. (9) Traore, M. A.; Behkam, B. J. Micromech. Microeng. 2013, 23, 085014. (10) Wu, Y. H.; Park, H. B.; Kai, T.; Freeman, B. D.; Kalika, D. S. J. Membr. Sci. 2010, 347, 197−208. (11) Diao, Y.; Helgeson, M. E.; Myerson, A. S.; Hatton, T. A.; Doyle, P. S.; Trout, B. L. J. Am. Chem. Soc. 2011, 133, 3756−3759. I

DOI: 10.1021/acs.macromol.5b01115 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules (49) Pontoni, D.; Narayanan, T. J. Appl. Crystallogr. 2003, 36, 787− 790. (50) Lin, S.; Sangaj, N.; Razafiarison, T.; Zhang, C.; Varghese, S. Pharm. Res. 2011, 28, 1422−1430. (51) Nishi, K.; Asai, H.; Fujii, K.; Han, Y.-S.; Kim, T.-H.; Sakai, T.; Shibayama, M. Macromolecules 2014, 47, 1801−1809. (52) Boots, H.; Kloosterboer, J.; Serbutoviez, C.; Touwslager, F. J. Macromolecules 1996, 29, 7683−7689. (53) Irzhak, V. I.; Mezhikovskii, S. M. Russ. Chem. Rev. 2009, 78, 165−194. (54) Panyukov, S.; Rabin, Y. Macromolecules 1996, 29, 7960−7975. (55) Norisuye, T.; Kida, Y.; Masui, N.; Tran-Cong-Miyata, Q.; Maekawa, Y.; Yoshida, M.; Shibayama, M. Macromolecules 2003, 36, 6202−6212. (56) Nie, J.; Du, B.; Oppermann, W. Macromolecules 2004, 37, 6558− 6564. (57) Akagi, Y.; Gong, J. P.; Chung, U.; Il; Sakai, T. Macromolecules 2013, 46, 1035−1040. (58) Flory, P. J. Molecular theory of rubber elasticity. Polym. J. 1985, 17, 1−12. (59) Kizilay, M. Y.; Okay, O. Polymer 2003, 44, 5239−5250. (60) Lindemann, B.; Schröder, U. P.; Oppermann, W. Macromolecules 1997, 30, 4073−4077. (61) Di Lorenzo, F.; Hellwig, J.; von Klitzing, R.; Seiffert, S. ACS Macro Lett. 2015, 4, 698−703. (62) Ball, R. C.; Edwards, S. F. Macromolecules 1980, 13, 748−761. (63) Sommer, J.-U.; Vilgis, T. A.; Heinrich, G. J. Chem. Phys. 1994, 100, 9181−9191. (64) Glinka, C. J.; Barker, J. G.; Hammouda, B.; Krueger, S.; Moyer, J. J.; Orts, W. J. J. Appl. Crystallogr. 1998, 31, 430−445. (65) Kline, S. R. J. Appl. Crystallogr. 2006, 39, 895−900.

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