Mechanical Properties of Polymer Gels with Bimodal Distribution in

Aug 20, 2013 - has been controversial because hydrogels inherently have a substantial amount of various heterogeneities in their structures. In this s...
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Mechanical Properties of Polymer Gels with Bimodal Distribution in Strand Length Shinji Kondo, Hayato Sakurai, Ung-il Chung, and Takamasa Sakai* Department of Bioengineering, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan S Supporting Information *

ABSTRACT: The understanding of the physical properties of conventional hydrogels has been controversial because hydrogels inherently have a substantial amount of various heterogeneities in their structures. In this study, we focused on one of the simplest heterogeneities, heterogeneous distribution of strand length, and investigated its influence on physical properties. We prepared tetra-PEG gels with bimodal distribution in strand length (tetra-PEG bimodal gels) by combining tetra-PEG prepolymers with different molecular weights and measured the physical properties. The physical properties of tetra-PEG bimodal gels formed above the overlapping concentration of prepolymers were well described by the models for conventional tetra-PEG gels with the average polymerization degrees between cross-links. We conclude that the mechanical properties of hydrogels that have heterogeneous distribution in strand length can be predicted from those of hydrogels with the average strand length in the range tested in this study.

I. INTRODUCTION Polymer gels are three-dimensional polymer networks swollen in solvent and are applied to a variety of commercial products.1−3 In order to tailor the physical properties for each application, a precise control of the physical properties is required. However, polymer gels inherently have heterogeneous structures, prohibiting the precise control of their physical properties. The heterogeneities are categorized into three types: spatial, connective, and topological ones;4,5 they are detected as discrepancies of their physical properties from the theoretical predictions or are directly observed by microscopes and scattering measurements.6 The heterogeneities cause another serious problem; it is impossible to examine the validity or the requirement condition of theoretical predictions, because we have no polymer gels obeying these theories. The ambiguities in theories, on the other hand, prohibit the confirmation of homogeneity of polymer gels. Thus, it has been desired to develop a versatile methodology to suppress the heterogeneity of polymer gels and create ideal polymer gels free from any heterogeneity. Recently, we developed a novel network forming method named the AB-type cross-link coupling.7−9 This reaction is different from conventional AB-type end-cross-linking,10 which forms a polymer network from mutually reactive telechelic polymer and multifunctional cross-linkers; the reaction occurs between chain ends in AB-type cross-link coupling, while it occurs at the cross-linking point in the AB-type end-crosslinking. Using the AB-type cross-link coupling, we have fabricated tetra-PEG gels from tetra-PEG prepolymers with same molecular weight (Figure 1a)7 or with different molecular weights (Figure 1b),11 which have unimodal strand length. Our previous analysis of reaction kinetics showed that the reaction rate constant is invariant from the initiation through the gelation threshold to almost completion,12,13 and final reaction © XXXX American Chemical Society

Figure 1. Conventional tetra-PEG gels with unimodal network formed from tetra-PEG prepolymers (a) with same molecular weight and (b) with different molecular weights and (c) tetra-PEG gels with bimodal network formed from different sized tetra-PEG prepolymers with the molar ratio being tuned.

conversion is up to 95%.14 Small angle neutron scattering and NMR studies detected extremely small amounts of spatial and connective heterogeneity, compared with that of conventional gels.15−18 We also investigated the mechanical properties including elastic modulus, fracture energy, and ultimate elongation ratio of tetra-PEG gels with a series of feed conditions.14,19 Through these studies, we confirmed the validity and the requirement condition of the models, and the homogeneity of the tetra-PEG gel. These data strongly suggest that tetra-PEG gels are near-ideal polymer networks, and the theory predicting the mechanical properties of an ideal polymer Received: July 22, 2013 Revised: August 1, 2013

A

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and the fracture energy (T0) by tearing measurements. By comparing these mechanical properties with those of conventional tetra-PEG “unimodal” gels and with the predictions from models, we investigated the effect of the bimodal distribution in strand length on the mechanical properties.

network was identified. We believe these results on ideal polymer networks are a first step toward full understanding of the mechanical properties of conventional polymer gels, which have a variety of heterogeneities. In this study, we try to introduce “controlled heterogeneity” to tetra-PEG gels, in order to understand the effect of the heterogeneity on mechanical properties. We focus on the heterogeneous distribution of strand length, which is one of the common heterogeneities in polymer gels. The heterogeneous distribution of strand length may influence especially the ultimate mechanical properties, such as ultimate elongation ratio and fracture energy, because scissions of the shorter chains may occur preferentially before those of longer chains. Alternatively, when the shorter and longer chains distribute homogeneously, the stress may distribute homogeneously. In this case, the mechanical properties will be predicted from the average of strand length (Navg). Mark et al.20−23 first started a series of studies on this issue; they fabricated polymer networks with bimodal strand length (bimodal networks) from AB-type end-cross-linking of telechelic prepolymers with bimodal chain lengths. The bimodal networks with increased amount of short strands showed drastically decreased ultimate elongation ratio (λmax), compared with that predicted from Navg.20−22 They also observed the aggregation of the short chains by small-angle X-ray scattering (SAXS)23 and concluded that the abnormally decreased λmax is caused by the limited extensibility of short network chains. Urayama et al.24,25 investigated the structure and mechanical properties of bimodal networks swollen in solvent. Their SAXS measurement24 showed that the correlation length drastically decreased with an addition of 10 mol % of short chain to 90 mol % of long chain. Further addition of the short chain did not decrease the correlation length. As for mechanical properties,25 the values of λmax decreased with an increase in the amount of short chains, which is qualitatively expected from the SAXS data. However, a drastic decrease in λmax occurred when 5 mol % of long chain was added to 95 mol % of short chain. This discrepancy was explained by the difference in the nature of heterogeneity detected by each measurement. The studies of Mark and Urayama conclude that the physical properties of bimodal networks are not predicted by the value of Navg. There are two essential problems in these two studies. First, they made the polymer networks without any diluent. This fabrication method induces a large amount of trapped entanglements (topological heterogeneity).26,27 Second, the bimodal chains were not distributed homogeneously (spatial heterogeneity). These heterogeneities severely affect both the mechanical properties and the resultant network structures, inhibiting the investigation of pure effect of bimodality. In order to examine the pure effect of multimodality, the mechanical properties of bimodal networks free from any other heterogeneities should be investigated. In order to fabricate the bimodal network free from the other heterogeneities, we utilized AB-type cross-link coupling of different sized tetra-PEG prepolymers with the molar ratios being tuned, while maintaining the equimolar condition of amine and activated ester (tetra-PEG bimodal gels, Figure 1c). Because the mixing of prepolymers with different molecular weight does not introduce heterogeneity,11 we can focus on the effect of bimodal distribution on the mechanical properties. We investigated the reaction conversion (p) of these networks by infrared (IR) measurements, the elastic modulus (G) and ultimate elongation ratio (λmax) by stretching measurements,

II. EXPERIMENTAL PROCEDURES A. Characterization of Tetra-PEG Modules. Tetra-amineterminated PEG (tetra-PEG-NH2) and tetra-NHS-terminated PEG (tetra-PEG-OSu) were purchased from NICHIYU (Tokyo, Japan). Here, NHS is N-hydroxysuccinimide. The details of the preparation of tetra-PEG-NH2 and tetra-PEG-OSu were reported previously.7 The molecular weight, the polydispersity, the end-group functionality (X0), and the overlapping polymer volume fraction (ϕ*) of the tetra-PEGs are shown in the Supporting Information (Table S1). B. Fabrication of Tetra-PEG Bimodal Gels. We combined two different molecular weights (Mw) of tetra-PEG-NH2 (Mw = 5 or 20 kg/ mol) and tetra-PEG-OSu (Mw = 5 or 20 kg/mol) to fabricate tetraPEG gels with bimodal strand length (tetra-PEG bimodal gels). Equimolar amounts of tetra-PEG-NH2 and tetra-PEG-OSu were dissolved in 3 mL of phosphate buffer (pH 7.4) and 3 mL of phosphate−citric acid buffer (pH 5.8), respectively (Table 1). The

Table 1. The Molar Concentrations of Each Tetra-PEG Prepolymer in Hydrogel, The Molar Fraction of Prepolymer with Mw = 20 kg/mol in All Prepolymers (r), the Initial Polymer Volume Fractions (ϕ0), and the Overlapping Polymer Volume Fractions (ϕ*) of Tetra-PEG Bimodal Gels tetra-PEGNH2 [mM]

tetra-PEG-Osu [mM]

5 kg/ mol

20 kg/ mol

5 kg/ mol

20 kg/ mol

20 kg/mol tetra-PEG mol fraction (r)

ϕ0

ϕ*

4.0 4.0 4.0 4.0 4.0 3.0 2.0 1.0 0

0 0 0 0 0 1.0 2.0 3.0 4.0

4.0 3.0 2.0 1.0 0 0 0 0 0

0 1.0 2.0 3.0 4.0 4.0 4.0 4.0 4.0

0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1

0.034 0.046 0.058 0.070 0.081 0.092 0.103 0.113 0.124

0.090 0.083 0.076 0.069 0.062 0.055 0.048 0.041 0.033

corresponding initial polymer volume fractions, ϕ0, were in the range between 0.034 and 0.124 (mass density = 1.129 g/cm3). In order to control the reaction rate, the ionic strengths of the buffers were chosen to be 25 mM for lower polymer concentrations (ϕ0 = 0.034−0.070), 50 mM for middle polymer concentrations (ϕ0 = 0.081−0.092), and 100 mM for higher polymer concentrations (ϕ0 = 0.103−0.124).12,13 The two polymer solutions were mixed, and the resulting solution was poured into the mold. We waited at least 12 h for the completion of the reaction before the subsequent experiment was performed. C. Infrared (IR) Measurements. The gels were prepared as rectangular films (height 20 mm, thickness 10 mm). Prepared gel samples were soaked in H2O for 2 days at room temperature and then dried. The dried samples were cut into thick films (thickness 40 μm) using a microtome (SM2000R, Leica, Wetzlar, Germany). These samples were swollen in D2O until equilibrium was reached. The IR spectra of these samples were obtained at 25 °C using a JASCO FTIR-6300 equipped with a deuterated triglycine sulfate (DTGS) detector, in which 128 scans were coadded at a resolution of 4 cm−1 for samples. More than two samples were tested for each network concentration. D. Stretching Measurements. The stretching measurements were performed on a dumbbell shape (2 mm thick) at room temperature with a mechanical testing apparatus (Autograph AG-X B

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Figure 2. (a) The concept for fabrication of tetra-PEG bimodal gels. (b) The reaction conversion (p) as a function of 20 kg/mol tetra-PEG molar ratio (r) in tetra-PEG bimodal gel. plus; SHIMADZU, Kyoto, Japan) at a crosshead speed of 60 mm/min. During the stretching, the distance between the gauge points was measured with a CCD camera (XCL-5005CR; SONY, Tokyo, Japan). Because the gel samples were used in the as-prepared state, the stretching was performed in air. The stress−strain relations obtained here corresponded to the equilibrium relations without time effects because no appreciable relaxation was observed for the stress after the imposition of a constant large strain at this crosshead speed. We stretched a sample repeatedly and confirmed that the evaporation of water did not affect the experimental results. At least four samples were tested for each network concentration, and the observed moduli were arithmetically averaged. E. Tearing Measurements. The tearing measurements were performed at room temperature with a mechanical testing apparatus (Autograph AG-X plus; SHIMADZU, Kyoto, Japan). Because the gel samples were used in the as-prepared state, the tearing was performed in air. The gels were cut into the shape specified by JIS K 6252 at 1/2 size (50 mm × 7.5 mm × 1 mm, with an initial notch of 20 mm). The two arms of the test samples were clamped, and one arm was pulled downward at a constant velocity of 40 mm/min while the other arm remained stationary. The tearing force F was recorded.

ester. We started from the mixing of 5 kg/mol tetra-PEG-NH2 and 5 kg/mol tetra-PEG-OSu, which forms a tetra-PEG unimodal gel with molecular weight of network strand of 2.5 kg/mol (r = 0), then we gradually exchanged 5 kg/mol tetraPEG-OSu with 20 kg/mol tetra-PEG-OSu to achieve the bimodal network (Figure 2a, left). It should be noted that tetraPEG gel formed from 5 kg/mol tetra-PEG-NH2 and 20 kg/mol tetra-PEG-OSu (r = 0.5) corresponds to a tetra-PEG unimodal gel with molecular weight of network strand of 6.25 kg/mol. Then, we gradually exchanged 5 kg/mol tetra-PEG-NH2 with 20 kg/mol tetra-PEG-NH2, finally resulting in tetra-PEG unimodal gel with molecular weight of network strand of 10 kg/mol (r = 1.0) (Figure 2a, right). First, we estimated the reaction conversion (p) of tetra-PEG bimodal gels with r = 0, 0.125, 0.25, 0.375, 0.5, 0.625, 0.75, and 1.0 (Figure 2b). The value of p was almost constant as a function of r and was 0.9−0.95, which is as high as that of conventional tetra-PEG gels (0.82−0.95).14 This result suggests that the mixing of prepolymers with different sizes does not affect the reaction conversion, and the reaction conversion only depends on the reaction group, at least in the range examined. These data well correspond to our previous measurement on tetra-PEG gels.11 B. Elastic Modulus. Then we performed a stretching test for tetra-PEG bimodal gels. According to the linear elasticity theory,28 we estimated the elastic moduli (G) from the initial

III. RESULTS AND DISCUSSION A. Reaction Conversion (p). In order to achieve tetra-PEG gels with bimodal strand length, we mixed 5 and 20 kg/mol tetra-PEG prepolymers with the molar ratio being tuned (r = (20 kg/mol tetra-PEG prepolymer)/(total prepolymer)), while maintaining the equimolar condition of amine and activated C

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prepolymers decreases without changing ρ. The value of ϕ* for the mixed solutions of the 5 and 20 kg/mol tetra-PEG prepolymers were calculated as11

slope of the stress (σ)−elongation (λ) curves. Our previous studies showed that the models predicting G of tetra-PEG unimodal gels shift from the phantom to affine network models with an increase in the concentration of prepolymers.14 In this study, we set the experimental conditions in the range where the phantom network model is valid. In the phantom network model, G is predicted to be Gcal = (ν − μ)kBT

* + rϕ * ϕ* = (1 − r )ϕ5k 20k

where ϕ*5k and ϕ*20k are the overlapping concentrations of 5 and 20 kg/mol tetra-PEG prepolymers, respectively (Table 1). From the calculation, it is revealed that the samples with r ≥ 0.375 are above ϕ*, but those with r ≤ 0.375 are below ϕ*. Because the point where the downward deviation starts well corresponds to ϕ*, it is strongly suggested that this deviation is not caused by the bimodal structure but by the formation of elastically ineffective loops. This tendency was also observed in our previous measurements for tetra-PEG unimodal gels,11,14 suggesting that this is a normal behavior for our system. These data indicate that the elasticity of tetra-PEG bimodal gel can be predicted by the phantom network model in the region ϕ0 > ϕ*, regardless of the bimodal distribution of the strand lengths, at least in the range tested. As for G, we can treat a tetra-PEG bimodal gel as the tetra-PEG unimodal gel with the numberaverage degree of polymerization of network strands (Navg), which is given by

(1)

where kB is Boltzmann’s constant and T is the absolute temperature. The number densities of the elastically effective chains (ν) and the active cross-links (μ) for the tetra-functional network are predicted based on a tree-like approximation. Although the tetra-PEG gel is formed by the AB-type coupling of tetra-arm polymers, we can treat this coupling as AA-type coupling of tetra-arm polymers when we consider the stoichiometric conditions.29 As for AA-type coupling of tetraarm polymers, the probability that an arm does not lead to an infinite network (P∞) is correlated with p by30 P∞ = pP∞ + (1 − p)

(2)

Using P∞, μ and ν are predicted as follows: ⎧⎛ 4 ⎞ ⎫ ⎛4⎞ μ = ρ⎨⎜ ⎟(1 − P∞)3 P∞ + ⎜ ⎟(1 − P∞)4 ⎬ ⎝4⎠ ⎩⎝ 3 ⎠ ⎭

⎧ 3 ⎛4⎞ ⎫ 4 ⎛4⎞ ν = ρ⎨ ·⎜ ⎟(1 − P∞)3 P∞ + ·⎜ ⎟(1 − P∞)4 ⎬ 2 ⎝4⎠ ⎩ 2 ⎝3 ⎠ ⎭

(5)

Navg = (3)

1 Mavg 2 Mm

(6)

where Mavg is the number-average molecular weight of mixed prepolymers (5000(1 − r) + 20000r) and Mm is the molecular weight of the monomer unit. Hereafter, we use Navg as the representative values of N for tetra-PEG bimodal gels. C. Fracture Energy. We performed the tearing measurement for the tetra-PEG bimodal gels in order to investigate the fracture energy (T0), which is defined as the energy required for the development of a crack with unit length.31,32 The Lake− Thomas model is a popular model for the prediction of the fracture energy of elastomers and well predicts the fracture energy of conventional tetra-PEG gels.11,14 According to the Lake−Thomas model, T0 is estimated as the energy needed to break the chemical bonds on the fracture surface and is given by33

(4)

where ρ is the number density of the tetra-PEG prepolymers, ⎛x⎞ and ⎜ y ⎟ is the usual notation for the number of combinations ⎝ ⎠ of x items taken y at a time: x!/y!(x − y)!. In this study, we made tetra-PEG bimodal gel samples with the same ρ (8.0 mM), which showed almost the same p according to the IR measurement. If the bimodal distribution in strand length does not affect G, G will be constant regardless of r and be predicted by the phantom network model. The variations of G and Gcal as a function of r are shown in Figure 3. G was almost constant and corresponded well with Gcal in the region r ≥ 0.375, which agreed well with the phantom prediction. On the other hand, in the region r ≤ 0.375, the downward deviation of G from Gcal was increasingly pronounced as r decreased. Here, it is important to know that the initial polymer volume fraction (ϕ0) also decreases with a decrease of r, because the average molecular weight of

T0 = NUdν

(7)

where U is the energy required to rupture a monomer unit and d is the displacement length. The following scaling relation is valid for an ideal polymer network: ν≈

ϕ0 (8)

N

d ≈ R g ≈ aN1/2

(9)

where a is the monomer length. According to eqs 7, 8, and 9, T0 scales with the ϕ0 and N by T0 ≈ ϕ0N1/2

(10)

We have confirmed the validity of eq 10 for tetra-PEG unimodal gels with the value of N computed from the molecular weight of prepolymers (N = Mw/2Mm). Thus, by examining eq 10 for tetra-PEG bimodal gels with Navg, we can assess the effect of bimodal network structure on fracture process. We plotted the T0 of tetra-PEG bimodal gels against ϕ0Navg1/2 in Figure 4. The data of tetra-PEG unimodal gels are also shown in Figure 4. Almost all the data fell onto the master

Figure 3. The value of G estimated from the stretching measurements and Gcal estimated from the reaction conversion (p) as a function of r in the tetra-PEG bimodal gels. D

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σ=

α G (λ − λ − 2 ) + (1 − α)G(1 − λ−3) 1 − (λ 2 − 2λ−1 − 3)/lm (11)

lm = (λmax 2 + 2λmax −1) − 3

(12)

where the parameter α was estimated from the fit of stress− elongation curves for tetra-PEG gels (20 kg/mol) under biaxial stretching. Because G is fixed as the value obtained by the linear fit, the fitting parameters are only λmax. The fit results of tetra-PEG bimodal gels are shown in the Supporting Information (Figure S1). The stress−elongation relationships were well predicted by the extended Gent model, regardless of r, suggesting that the uniaxial stretching behavior of tetra-PEG bimodal gels is similar to that of tetra-PEG unimodal gel. Although the uniaxial measurement gives us limited information, the whole range of stress−elongation relationship of tetra-PEG bimodal gel was well reproduced by the extended Gent model with intrinsic parameters including the elastic modulus (G) and the ultimate elongation (λmax). Thus, we try to investigate the effect of r on λmax. The values of λmax for tetra-PEG bimodal gels are shown against Navg in Figure 6. In the Kuhn model, λmax is estimated as

Figure 4. Fracture energy as a function of ϕ0Navg1/2 in the tetra-PEG bimodal gels. The dashed line is the master curve. The data for the tetra-PEG unimodal gels (N = 57, 114, 228) were reproduced from Akagi et al.14 with permission from the American Chemical Society.

curve of tetra-PEG unimodal gels showing T0 ≈ ϕ0Navg1/2, suggesting that the stress distributes homogeneously and shorter chains do not break before the breakage of longer chains in tearing process. D. Ultimate Elongation Ratio. Finally, we investigated the ultimate elongation ratio (λmax) of tetra-PEG bimodal gels. In Figure 5, the stress (σ) normalized by G is plotted against the

Figure 6. Ultimate elongation ratio as a function of the average polymerization degree between the cross-links (Navg) in the tetra-PEG bimodal gel. The dash line is the guide showing the relationship, λmax ≈ Navg1/2.

Figure 5. Stress−elongation curves of tetra-PEG bimodal gels with (a) r = 0, (b) r = 0.25, (c) r = 0.5, (d) r = 0.75, or (e) r = 1. Stress is normalized by G. The dashed line is the prediction of the NeoHookean model (σ = (λ − λ−0.5)).34

the ratio of the contour length (L) to the root-mean-square end-to-end distance (Rg) of the network strands:37 λmax =

elongation ratio (λ). The prediction of the Neo-Hookean (NH) model,34 which predicts the stress−elongation relation of polymer networks with infinite extensibility and without structural defects, with G = 1 is also shown in Figure 5. The stress−elongation curve of tetra-PEG bimodal gels obeyed the NH model in the low λ region, and then deviated upward from the prediction of the NH model, and λ where the deviation occurs shifted to higher λ with an increase in r. Because the upward deviation from the NH prediction is caused by the effect of finite extensibility,35 this result suggests that the introduction of longer strand increases λmax. In order to analyze λmax qualitatively, we utilized the extended Gent model.36 Our previous study confirmed that the extended Gent model successfully predicted the biaxial stress−elongation relationship of tetra-PEG gel. 19 The extended Gent model predicts the uniaxial stretching behavior as follows:36

L aN = ≈ N1/2 Rg aN1/2

(13)

The Kuhn prediction is also displayed as dotted line in Figure 6; the Kuhn model failed to predict the experimental result. These results indicate that the Kuhn model cannot predict the λmax of tetra-PEG bimodal gels, in a similar fashion to tetra-PEG unimodal gels.11,19 In our previous study, we proposed a novel semiempirical model that successfully predicted the λmax of tetra-PEG unimodal gels. We changed only the denominator of eq 13 to the geometrical distance between neighboring prepolymers (d) as19 λmax ≈

L aN ≈ −1/3 ≈ ϕ01/3N 2/3 d ν

(14)

ϕ01/3Navg2/3,

We replotted all the data of λmax against and also show the guides of eq 13 (Figure 7). The data of tetra-PEG unimodal gels are also shown in Figure 7. As expected, all the E

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than that of previous studies (∼100 kg/mol). The existence of longer strands can influence the mechanical properties in a different manner. The pure effect of such a long strand may be difficult to study, because of the introduction of trapped entanglements. In the case of the bimodal structure up to 100 nm, which may be categorized as spatial heterogeneity, the mechanical properties are strongly affected by the softer network.38 In the case of bimodal network on smaller scale as seen in our case, however, the mechanical properties correspond to those predicted by the number-average molecular weight of strands.

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ASSOCIATED CONTENT

S Supporting Information *

Properties of the tetra-PEG polymers and fit results of stress− elongation curves of the bimodal gels. This material is available free of charge via the Internet at http://pubs.acs.org.

Figure 7. Ultimate elongation ratio as a function of ϕ01/3Navg2/3 in the tetra-PEG bimodal gels. The dashed line is the guide showing the relation λmax ≈ ϕ01/3Navg2/3. The data for the tetra-PEG unimodal gels (N = 57, 114, 228) were reproduced from Akagi et al.19 with permission from the Royal Society of Chemistry.

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AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

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data fell onto a master curve of tetra-PEG unimodal gels showing the relationship, λmax ≈ ϕ01/3Navg2/3, suggesting that the ultimate elongation ratio of tetra-PEG bimodal gels can be predicted from Navg and ϕ0 using the same model as that of tetra-PEG gel, regardless of the bimodal distribution of strand length. These data also indicate that the stress distributes homogeneously in the network and the shorter chains are not ruptured prior to the rupture of the longer chains.

ACKNOWLEDGMENTS This work was supported by the Japan Society for the Promotion of Science (JSPS) through Grants-in-Aid for Scientific Research, the Center for Medical System Innovation (CMSI), the Graduate Program for Leaders in Life Innovation (GPLLI), the International Core Research Center for Nanobio, and the Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST program); the Ministry of Education, Culture, Sports, Science, and Technology in Japan (MEXT) through the Center for NanoBio Integration (CNBI); the Japan Science and Technology Agency (JST) through the S-innovation program; and Grants-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology (No. 23700555 to T.S. and No. 24240069 to U.C.).

IV. CONCLUSION The elastic modulus, the fracture energy, and the ultimate elongation ratio of the tetra-PEG bimodal gels corresponded well to those predicted by the models for conventional tetraPEG unimodal gels with Navg and ϕ0. From these results, we conclude that the mechanical properties of a hydrogel with bimodal strand length correspond to those of a homogeneous hydrogel with unimodal strand length, Navg. This conclusion seems to include two important points. First, the heterogeneous distribution in the precursor length does not strongly affect the mechanical properties of polymer gels. These results are quite different from those by Mark20−23 and Urayama;24,25 in their studies, the smaller chains strongly affected the mechanical properties of bimodal network. Our result suggests that the mechanical properties of their bimodal networks are affected by other heterogeneities collaterally introduced. When there is only a bimodal distribution on strand length, the mechanical properties are predicted by number-average of strand length. Second, we cannot obtain the information about the strand length heterogeneity from the mechanical measurement performed in this experiment. The molecular weight of the network strand predicted by these mechanical measurements is the number-average value. In order to obtain the information about the strand length heterogeneity, we should develop a new methodology that is sensitive to the distribution of strand length. The 1H multiple-quantum NMR microscopy may be the candidate method to detect such heterogeneous distribution of strand length. Finally, we would like to address the limitation of this study. Although the shorter strand we used is 2.5 kg/mol, which is short enough comparable to the previous studies, the longer strand is up to 10 kg/mol. The longer strand is much shorter

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REFERENCES

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dx.doi.org/10.1021/ma401533z | Macromolecules XXXX, XXX, XXX−XXX